X-Planes: Invocon X-64
Filed under: aerospace, history, engineering
This is the sixty-fifth article in the X-Planes series, following the X-1, the X-2, the X-3, the X-4, the X-5, the X-6, the X-7, the X-8, the X-9, the X-10, the X-11, the X-12, the X-13, the X-14, the X-15, the X-16, the X-17, the X-18, the X-19, the X-20, the X-21, the X-22, the X-23, the X-24, the X-25, the X-26, the X-27, the X-28, the X-29, the X-30, the X-31, the X-32, the X-33, the X-34, the X-35, the X-36, the X-37, the X-38, the X-39, the X-40, the X-41, the X-42, the X-43, the X-44, the X-45, the X-46, the X-47, the X-48, the X-49, the X-50, the X-51, the X-52, the X-53, the X-54, the X-55, the X-56, the X-57, the X-58, the X-59, the X-60, the X-61, the X-62, and the X-63.
The register describes this vehicle in one hundred and one characters that it also uses for a different vehicle, and the laboratory that paid for both says they are not alike.
The X-64A was allocated on 20 April 2022 to a team of three companies led by Invocon Incorporated, with KT Engineering and Troy7 Incorporated [Invocon X-64] [ABL Space Systems X-63]. The previous article took the other half of the same allocation, the X-63A of ABL Space Systems, and established that the two rows share their date, their sponsor cell, their engines cell and every character of their description [Parsch, DOD 4120.15-L Addendum].
Demonstrator rocket for AFRL’s ARISE (Aerospike Rocket Integration and Suborbital Experiment) program
The register records no more than that about either machine. This article is about what the register conceals, which is a vehicle of a different shape, built by a team of a different trade, to answer the half of the question its sibling cannot answer alone.
The Research Question
An aerospike nozzle is worth between five and eight percent of first-stage impulse, and the question this vehicle exists to settle is whether five to eight percent can be seen from the ground.
The previous article computed the prize. On the published reference trajectory of the parent vehicle, an ideal altitude-compensating nozzle delivers between 5.3 and 8.5 percent more first-stage impulse than the best fixed nozzle that trajectory admits. The figure is a result in thermodynamics and is silent on instruments.
A flight test does not measure impulse. It measures whatever its transducers measure, and then somebody reconstructs impulse from that. So the questions here are these. What does an instrument on a rocket observe, and what has to be assumed to turn that observation into thrust? How accurate must each assumption be before a five percent effect is distinguishable from the error in the measurement of it? What does a finite ring of sensors on an annular nozzle fail to see? And what does it mean that this vehicle, unlike its sibling, is designed to come back?
The answers are sharper than the programme’s language suggests, and one of them is reassuring. The observable is not thrust but thrust minus drag, so recovering thrust requires a drag model and the accuracy of that model sets the accuracy of the whole experiment. The drag term is the only one nothing on board measures. But the signal is collected early and the drag confounder peaks late, so by the time drag is at its worst the measurement is largely already banked. And the vehicle’s published dimensions, of which there are exactly two, indicate a shape chosen for recovery rather than for ascent.
Programme Origin
The programme reached the register as a finished arrangement, and its origin lies in an award made well before the designation. The award was a three-year other transaction agreement made in December 2019 with a team led by Invocon Incorporated [AFRL awards agreements under ARISE]. The designation was allocated on 20 April 2022, on the same day as the X-63A [Parsch, DOD 4120.15-L Addendum]. The Air Force Research Laboratory’s background document, cleared in September 2022, places the vehicle in a portfolio whose stated goals are development time and development cost rather than performance [AFRL’s Rocket Lab Past, Present and Future]. The origin is therefore read from four records, being the register, the laboratory’s background document, the award announcement and the federal award record of the three companies [USAspending], and the subsections that make up this section read them in turn.
The Register Row, Read From the Other Side
The designation register that Andreas Parsch compiles as an addendum to the published designation list gives the X-64A a row identical to the X-63A’s in every cell that describes the machine, differing only in the contractor [Parsch, DOD 4120.15-L Addendum].
| Designation | Allocated | Contractor | Engines cell | Sponsor | Description marked as a reconstruction |
|---|---|---|---|---|---|
| X-63A | 20-Apr-22 | ABL Space Systems | 1 rocket engine |
USAF/USSF | yes |
| X-64A | 20-Apr-22 | Invocon, KT Engineering, Troy7 | 1 rocket engine |
USAF/USSF | yes |
That row carries three facts, all established in the X-63A article. In the copy of the register last updated on 23 August 2026, eleven descriptions repeat somewhere among its 539 rows, and this pair is the only repeat among the thirty-one rows of the X- series. The sponsor cell reads USAF/USSF, which only these two rows in that copy of the register do. And the engines cell reads 1 rocket engine, naming a class where the register almost always names a model, which again only these two rows do [Parsch, DOD 4120.15-L Addendum].
Both rows are also marked as reconstructions. The compiler notes that no complete official data exists for allocations after October 2018, that their official description is not releasable, and that he prints his own reading in blue to show it is not official wording [Parsch, DOD 4120.15-L Addendum]. So the hundred and one characters this article opened with are the compiler’s summary of open sources and not a government statement, and the three-state officiality classification the X-63A article recomputed puts both rows in the wholly unofficial category.
What is primary in that row is the date, the contractor names and the fact that two numbers were issued at once. Everything describing the vehicle is secondary, and the sentence describing the vehicle is the part the two rows share verbatim.
What the Laboratory Says That the Register Does Not
The Air Force Research Laboratory published a background document on its rocket propulsion organisation, cleared in September 2022, and one paragraph of it contradicts the impression the register leaves. It names the portfolio ARISE belongs to by its initials alone, being the Affordable Responsive Modular Rocket, and the expansion is supplied here because the passage quoted below does not carry it [AFRL’s Rocket Lab Past, Present and Future].
Under the Aerospike Rocket Integration and Suborbital Experiment (ARISE) AFRL, working in Public Private Partnerships with Invocon and ABL Space Systems, will fly the first ever modular aerospike engine. Each company has their own launch vehicle and chosen approach to implementing the ARMR architecture. ABL’s vehicle has been designated X-63 while Invocon’s is designated X-64.
The laboratory’s public fact sheet for the programme omits all of that, describing the aerospike, the three wake regimes and the modular architecture without naming either designation [ARISE and Fly]. The X-63A article read that sheet in full and found it names its two flight-test vehicles only as abl and Invocon in a figure caption. So the designations and the companies are tied together in the background paper and in no other laboratory publication this article consulted.
Each company has its own launch vehicle and its own chosen approach. That sentence justifies a separate account of this vehicle in place of a paragraph in its sibling’s. The register assigns one description to two vehicles, and the laboratory that commissioned them describes vehicles that differ in design and in approach [AFRL’s Rocket Lab Past, Present and Future]. A designation register is a naming instrument and not a technical one, and this is what that distinction costs a reader who has only the register.
Two further things fall out of the same paragraph. The document ties each designation to its company in a government publication, which the register does only through the contractor cell. And it calls the instrument a Public Private Partnership, where the award announcement calls it an other transaction agreement executed through the Space Enterprise Consortium [AFRL awards agreements under ARISE]. Those are not in conflict. A public private partnership is a description of the relationship and an other transaction agreement is the legal vehicle for it, and the article on the sibling designation established that the second is why neither award appears in the federal procurement record.
The Team in the Award Record
The award announcement describes the lead contractor in one sentence [AFRL awards agreements under ARISE].
Invocon is a veteran-owned small business that provides turnkey instrumentation and control solutions for demanding applications in extreme environments and has teamed with KT Engineering and Troy7 for their unique capabilities.
The prime contractor for a rocket is an instrumentation house. No other fact about this vehicle tells a reader as much, and the award record bears it out at length, independently of the company’s self-description.
Searching the federal award reporting system by recipient name across five families of award type returns 81 rows for Invocon, and their descriptions are the descriptions of a measurement company [USAspending]. An internal and external wireless instrumentation system. Hypervelocity impact detection, location and assessment. Two radiation monitors, one of them an environment monitor for spacecraft under the small business research programme. A micro-wireless instrumentation system design and development. A blanket agreement for miscellaneous data acquisition equipment and networking. The largest single row is mission support awarded through the Department of Defense at 3,053,408 dollars and 20 cents, and the company founded in 1985 describes itself as four decades of precision instrumentation from Conroe, Texas [Invocon, About Us].
KT Engineering’s record is not instrumentation. It is this programme’s architecture, a decade early.
| Company | Role named in the announcement | Award rows | Largest row | What the largest row is for |
|---|---|---|---|---|
| Invocon | lead, turnkey instrumentation and control | 81 | 3,053,408.20 dollars | mission support, Department of Defense |
| KT Engineering | unique capabilities | 15 | 6,571,144.00 dollars | segmented launch vehicle manufacturing development and demonstration |
Troy7, as Troy 7 |
unique capabilities | 15 | 2,889,435.00 dollars | a hypersonic control system |
Troy7, as Troy7 |
unique capabilities | 0 | none | the spelling returns nothing |
Fifteen rows return for KT Engineering and the largest of them is a segmented launch vehicle manufacturing development and demonstration programme at 6,571,144 dollars, followed by crew exploration vehicle propulsion advanced development of a 7,500 pound force vacuum engine at 4,186,522 dollars. Two further rows are third-phase small business awards for a radially segmented launch vehicle [USAspending]. Asking the award record for segmented launch vehicle returns seven rows and every one of them is this company’s. Segmented launch vehicles, a modular architecture at the scale of the whole vehicle, have been a funded line of work under one of the X-64A’s subcontractors since at least 2004, when the largest of those awards began [USAspending].
Troy7’s largest award is for control, and finding it at all depends on a space.
The announcement writes the company Troy7 and the award record writes it TROY 7, INC. [AFRL awards agreements under ARISE] [USAspending]. The keyword Troy7 returns nothing in any of the five award families. The keyword Troy 7 returns fifteen rows. The largest is a research and development contract for a hypersonic control system at 2,889,435 dollars. So the largest awards divide the team into instrumentation, propulsion and control, and one third of it is invisible to a search that spells its name the way its customer spells it. A name is a query, and a query is only as good as its spelling.
The spaced spelling also imports the collision the unspaced one avoided. Among those fifteen rows are a router backup belonging to a different company and a seven-inch drop-in rail belonging to a third, both matched because Troy 7 occurs inside TROY 7" DROP-IN RAIL and inside a product name. A more findable query is a less precise one, and both halves of that trade have to be stated for the count to mean anything.
Programme Targets in Time and Money
The previous article observed that the modularity half of ARISE is a claim about cost and schedule rather than about performance, and said the fact sheet did not quantify it. The background document does quantify it [AFRL’s Rocket Lab Past, Present and Future].
The goal of the ARMR portfolio is to radically change the current space access engine design paradigm. We seek to reduce the development time for a new engine by 70% and reduce the development cost by 50%.
Seventy percent off the schedule and fifty percent off the cost. Those are the numbers the modular architecture is accountable to, and neither is a thrust or an impulse. An engine assembled from many copies of one small module is an argument about how many distinct components have to be designed, qualified and tolerated, and the portfolio states its target as a factor of about three and a third in time and two in money.
The same document places ARISE at the end of a lineage that the laboratory names and dates.
| Programme | Years the document gives | What it aimed at | How far it got |
|---|---|---|---|
| Integrated Powerhead Demonstrator | 1994 to 2006 | 200 mission life, 100 mean time between overhaul, twenty times the Space Shuttle | the world’s first full-flow staged combustion engine, demonstrated |
| Upper Stage Engine Technology | from 2004 | a modernised upper-stage industrial base and better analysis tools | technologies now in several commercial engines, over 150 tool transitions |
| Hydrocarbon Boost Demonstrator | from 2007 | to beat the RD-180 on performance and reusability, 200 mission life | a preburner rather than a full engine |
| Third Generation Reusable Booster | named, not dated | a reusable booster | leveraged into the portfolio that follows |
| Affordable Responsive Modular Rocket | present at the 2022 clearance | 70 percent less development time, 50 percent less development cost | ARISE, being the two vehicles of this article and the last |
Two of those predecessors carried quantified reusability goals and neither reached a flying engine. The integrated powerhead demonstrator ran from 1994 to 2006, demonstrated what the document calls the world’s first full-flow staged combustion engine, and aimed at 200 mission life and 100 mean time between overhaul. The hydrocarbon boost programme began in 2007 with the same 200 mission life target and progressed only far enough to demonstrate a preburner rather than a full engine, which the document states plainly while also calling the programme an outstanding success on the strength of requests for information about its technologies [AFRL’s Rocket Lab Past, Present and Future].
One phrase in that document recurs. It writes twice that a programme sought to bring a part of the industry into the 22nd century, once of the modelling tools and once of the upper-stage industrial base [AFRL’s Rocket Lab Past, Present and Future]. A document cleared in 2022 describing work begun in 2004 almost certainly means the twenty-first. The phrase is a slip and carries no ambiguity. This series quotes primaries verbatim, so a reader meeting the phrase in the original should know it was not corrected in transit.
A second slip in the same document is a matter of chemistry. Describing the Titan’s propellants it reports that the laboratory experimented with switching to nitrogen tetroxide and Aerozine 50, and then in the next sentence discusses the storability of nitrogen tetroxide and monomethylhydrazine, which it says found their way into the Titan II and the whole Titan family [AFRL’s Rocket Lab Past, Present and Future]. Aerozine 50 and monomethylhydrazine are not the same fuel, the first being a one-to-one mixture of hydrazine and unsymmetrical dimethylhydrazine [Copeland et al 1967], and a NASA study of the Titan 4 names Aerozine 50 as the fuel of its liquid stages [Palaszewski and Powell 1991]. The storability argument the paragraph makes holds for either, which is presumably why the substitution passed.
Sizing From First Principles
Sizing this vehicle from first principles means sizing two things, and the record supports arithmetic for one and a requirement for the other. The first is the vehicle itself, of which two approximate dimensions are published and nothing else, so its geometry can be computed while its mass, thrust and drag coefficient cannot [Invocon X-64]. The second is the experiment the vehicle was to carry, which can be sized even where the vehicle cannot, because the accuracy a drag model must have follows from the drag fraction, the size of the effect and the instrument errors alone. What is computed is therefore the geometry the two numbers fix and the accuracy the measurement demands, with the drag fraction left as a variable because no published figure fixes it.
The Published Shape
Two numbers describe this vehicle and they are enough to say what it was for.
The encyclopedia entry reports, from the contractor’s release about the contract award, that the vehicle is apparently recoverable, that it lands on a gear of four legs which function as stabilising fins during flight and presumably rotate downward before landing, and that it stands about 12 metres tall with a diameter of about 2.4 metres [Invocon X-64]. The entry itself states that only little public information about the X-64A is available and that detailed physical characteristics are not yet available. The specification ends there.
The contractor’s release has not been read by this article. The live company site no longer carries it and the archive returned no capture, so the physical description above is taken at second hand from an entry that names the release as its source. What follows is arithmetic on two numbers whose provenance is one remove away.
Set those two numbers beside the parent vehicle of the sibling designation, whose payload user’s guide gives 88 feet integrated length and 6 feet diameter [ABL Payload User’s Guide].
Three elementary relations underlie this section, and they are written down because every number below is one of them evaluated twice. Let $L$ be the length in metre, $D_b$ the body diameter in metre, $f$ the fineness ratio [fineness ratio], dimensionless, $A$ the reference frontal area in square metre, $S$ the wetted side area of the equivalent cylinder in square metre, and $V_c$ its enclosed volume in cubic metre.
\[f \equiv \frac{L}{D_b}, \qquad A = \frac{\pi D_b^2}{4}, \qquad S = \pi D_b L, \qquad V_c = A L\]Two of the comparisons below are therefore not independent measurements but consequences, and saying which is which keeps the argument honest. Read as scaling laws at fixed shape,
\[A \propto D_b^{2}, \qquad S \propto D_b L, \qquad V_c \propto D_b^{2} L\]The frontal-area ratio is the square of the diameter ratio and nothing else. So the figure of 1.72 quoted below is 1.3123 squared, and the volume ratio of 0.7705 is that same 1.72 multiplied by the length ratio of 0.4474. Neither is a separate fact about the vehicles, and a reader who checked them against each other would find them consistent by construction rather than by agreement.
| Quantity | X-64A | RS1, the sibling’s parent | Ratio |
|---|---|---|---|
| Length | 12.0 metre | 26.8224 metre | 0.4474 |
| Diameter | 2.4 metre | 1.8288 metre | 1.3123 |
| Fineness ratio | 5.00 | 14.6667 | 2.9333 |
| Frontal area | 4.5239 square metre | 2.6268 square metre | 1.7222 |
| Cylinder volume | 54.29 cubic metre | 70.46 cubic metre | 0.7705 |
| Wetted side area | 90.48 square metre | 154.10 square metre | 0.5871 |
| One calibre of static margin, as a share of length | 20.0 percent | 6.8 percent | 2.9333 |
The X-64A is 44.7 percent of the RS1’s length and 131.2 percent of its diameter. Its fineness ratio, being length divided by diameter, is 5.0 against the RS1’s 14.67, so the parent vehicle is 2.93 times as slender. Both published figures for the X-64A are approximate and both are round in both unit systems, forty feet by eight and twelve metres by two point four, so the exactness of 5.0 is an artefact of the rounding, and the defensible statement is a fineness ratio of about five. With lengths in metre the ratios are
\[\frac{12.0}{26.8224} = 0.4474, \qquad \frac{2.4}{1.8288} = 1.3123, \qquad f = \frac{12.0}{2.4} = 5.00 \quad \text{against} \quad \frac{26.8224}{1.8288} = 14.667, \qquad \frac{14.667}{5.00} = 2.933\]A fineness ratio of five is not a launch-vehicle proportion. It is nearer to that of a lander or an early sounding rocket, and its consequences are arithmetic.
The frontal area is 4.524 square metres against 2.627, which is 1.72 times as much. Pressure drag and base drag both scale with that area, so at equal mass and equal drag coefficient this vehicle’s drag force is 1.72 times its sibling’s, and that factor comes from published dimensions with no aerodynamic model in it at all.
The drag coefficients are probably not equal either, and the asymmetry needs careful statement because it is a judgement rather than a derivation. Base area is the same for a given diameter however long the body is, so the difference is not one of area. What fineness changes is the boundary layer arriving at the base and the bluntness of the nose. A longer body presents a thicker boundary layer at its base, which raises the base pressure and lowers base drag, so the shorter vehicle pays more of it. And a body of fineness five carries a nose of proportionally lower fineness for the same overall proportions, which raises forebody wave drag through the transonic and low supersonic range where this vehicle spends its dynamic pressure.
Running the other way is skin friction, and it favours the short vehicle. A cylinder of these proportions has 90.5 square metres of side area against the RS1’s 154.1, so friction drag is lower by a factor of 1.70 on area alone.
Which of those wins is not settled here. At the Mach numbers computed below, wave and base drag are the larger contributions and friction the smaller, so the expectation is that the shorter vehicle’s drag coefficient is the higher of the two. That expectation is an inference from the ordering of drag contributions and not a computation, so the only claim this article makes about drag without qualification is the geometric one. The frontal-area factor of 1.72 holds at equal mass and equal drag coefficient, and putting a number above it would need a drag model for a configuration whose contours are not public.
And the enclosed volume says the shortness is not a matter of being a smaller vehicle. The X-64A’s cylinder holds 54.29 cubic metres against the RS1’s 70.46, which is 77.1 percent of the volume in 44.7 percent of the length. It is not a scaled-down launch vehicle. It is a differently proportioned one. In cubic metre,
\[V_c = \frac{\pi \left( 2.4 \right)^2}{4} \left( 12.0 \right) = 54.29, \qquad V_c = \frac{\pi \left( 1.8288 \right)^2}{4} \left( 26.8224 \right) = 70.46, \qquad \frac{54.29}{70.46} = 1.7222 \times 0.4474 = 0.7705\]The proportion is what a recoverable vehicle wants. Four legs deployed from a 2.4 metre body give a footprint whose radius is a larger fraction of the vehicle’s height than the same legs on a 26.8 metre body could, the centre of mass of a returning stage sits lower on a short body, and the aerodynamic surfaces that steer a descent have a shorter moment arm to work against. Every one of those favours the short vehicle, and every one of them costs drag on the way up. The section on the measurement shows where that cost is paid.
The Claim, Stated Exactly
Everything that follows depends on what an instrument bolted to a rocket senses, which is not what a reader might assume.
Let $F$ be the thrust in newton, $D$ the aerodynamic drag in newton, $m$ the instantaneous mass in kilogram, $g$ the local gravitational acceleration in metre per second squared, $\gamma$ the flight-path angle in radian and $v$ the speed in metre per second. Newton’s second law along the flight path reads
\[m \frac{\mathrm{d} v}{\mathrm{d} t} = F - D - m g \sin \gamma\]which contains four quantities of which one is wanted and three are not. A body-mounted accelerometer does not measure $\mathrm{d}v / \mathrm{d}t$. It measures specific force, being the non-gravitational force per unit mass [proper acceleration], because a freely falling accelerometer reads zero. Write $a_s$ for the sensed specific force in metre per second squared. Then
\[a_s = \frac{F - D}{m}, \qquad F = m \, a_s + D\]The gravitational term has vanished and it has vanished exactly. An accelerometer’s blindness to gravity, which is an inconvenience for navigation, is the property that makes it the right instrument for measuring thrust, because the largest term in the equation of motion is the one it declines to see. It is why a thrust measurement in flight is possible at all without knowing where the vehicle is.
One assumption is buried in that identity. Thrust and drag are written as though both act along the flight path, which is true at zero angle of attack and not otherwise. An accelerometer aligned with the body axis measures the body-axial component of the total non-gravitational force, and the aerodynamic force resolves onto that axis through the angle of attack. Let $\alpha$ be the angle of attack in radian, $N_a$ the aerodynamic normal force in newton, and $X_a$ the axial aerodynamic force in newton.
\[a_s = \frac{F - X_a}{m}, \qquad X_a = D \cos \alpha - N_a \sin \alpha\]At zero angle of attack $X_a$ is the drag and the identity above is exact. Away from it the normal force contributes a forward axial component, so a vehicle at angle of attack has a smaller retarding axial force than its drag, and an analysis that substituted $D$ for $X_a$ would understate the thrust it recovered.
A gravity-turning launch vehicle flies at a small angle of attack by design [gravity turn], because the pitch program exists to keep the airframe aligned with the relative wind through the dense atmosphere, so the approximation is a good one in the regime where it is applied. It is an approximation nonetheless, and correcting it needs $\alpha$, which needs either an air-data measurement or a reconstruction, and this article has neither.
What remains is a sum of three quantities and only one of them is measured directly. The specific force is measured. The mass is inferred rather than measured, from an initial value and an integrated propellant flow. Let $m_0$ be the initial mass in kilogram and $\dot{m}$ the propellant mass flow in kilogram per second.
\[m(t) = m_0 - \int_{0}^{t} \dot{m}(s) \, \mathrm{d}s\]That integral is why the mass error grows through the burn rather than staying where the loading scales left it. A constant relative error in the flow measurement accumulates into an absolute mass error proportional to the propellant already spent, so the mass term in the budget below is at its worst late in the flight, at the opposite end from where the altitude-compensation signal lives.
The drag is neither measured nor inferred. It is modelled.
The Literature of In-Flight Thrust Determination
Determining thrust in flight is a settled discipline with an established methodology, and it reaches the conclusion derived in the preceding paragraphs. The review of measurement error in the practice puts it plainly [Uncertainty of in-flight thrust determination].
While the term ‘in-flight thrust determination’ is used synonymously with ‘in-flight thrust measurement’, in-flight thrust is not directly measured but is determined or calculated using mathematical modeling relationships between in-flight thrust and various direct measurements of physical quantities.
The field stated the premise of this article in 1986, thirty-three years before the award of December 2019. The companion review of the processes themselves surveys the analytical and ground-test routes and works three turbofan examples [In-flight thrust determination] [on a real-time basis], and a separate study traces how measurement error propagates into a computed thrust for one engine in detail [Measurement effects for an F404 turbofan].
And that literature is about air-breathing engines, so its method differs from this one. A turbofan’s thrust cannot be reached through an accelerometer, because the inlet captures a momentum flux that depends on the flight condition, so the net propulsive force is not separable from the airframe’s drag by any body-mounted instrument. The practice therefore calculates thrust from direct measurements of physical quantities through modelling relationships, with ground testing as part of the process [Uncertainty of in-flight thrust determination]. A rocket carries its oxidiser and captures nothing, so the accelerometer route is open to it, and the simplification is legitimate, and the earlier work did not overlook it.
The inverse problem was solved in flight once and the way it was solved shows how entangled the two forces are. The drag of the XB-70 was measured in flight from Mach 0.75 to 2.5, and the paper describing it is explicit that what made that possible was determining engine net thrust independently and then charging the remainder to the propulsion system [Techniques for determining propulsion system forces]. Thrust minus drag is one observable and splitting it needs a model of one or the other. This article models the drag and recovers the thrust, where that programme modelled the thrust and recovered the drag. Neither can have both for free.
And the Established Methodology Carries a Term This Article’s Budget Does Not
The methodology the field settled on is independent of how thrust is calculated and traceable to a national standards laboratory, and it enumerates what an uncertainty statement has to contain [Uncertainty methodology for in-flight thrust determination] [its application]. Its categories are measurement error, precision, bias, uncertainty, error estimation and classification, error propagation, ground testing, and the related problems of model bias error, model precision error and the uncertainty limit.
Three of those this article does not have, and naming them is more useful than pretending otherwise. It does not separate bias from precision, treating every term as a random contribution combined in quadrature, when a calibration offset is a bias that does not average down over a flight. It has no model bias term at all, though that term is where a drag model’s systematic error would be carried, and a drag model’s error is far more likely to be systematic than random because it comes from a wrong shape assumption rather than from noise. And it states no uncertainty limit in the methodology’s sense.
So the budget here is a sensitivity analysis rather than an uncertainty statement, and the difference matters for how its conclusions should be read. It shows correctly how the answer’s sensitivity to each input scales. It does not say, and cannot say, what the uncertainty of a real measurement would have been.
One further point of contact is exact rather than approximate. A single demonstrator flight is a single-sample experiment in the technical sense, meaning one that cannot be repeated enough to estimate its own scatter, and single-sample uncertainty analysis is a developed subject with standard conventions [Moffat 1982]. A programme that funds one flight has bought a single sample, and the apparatus of precision estimated from repetition is unavailable to it by construction.
The Symbols This Article Uses
| Symbol | Meaning | Unit |
|---|---|---|
| $ \alpha $ | angle of attack | radian |
| $ \beta $ | share of the effect allotted to the drag term | dimensionless |
| $ \chi $ | share of the pressure-time integral collected by a given time | dimensionless |
| $ \Delta $ | size of the effect being measured, as a fraction | dimensionless |
| $ \delta $ | uncertainty in the quantity that follows it | that quantity’s own unit |
| $ \dot{m} $ | propellant mass flow | kilogram per second |
| $ \ell $ | lever arm from the tilt axis to the footprint boundary | metre |
| $ \ell_{\max} $ | longest lever arm, straight over a leg | metre |
| $ \ell_{\min} $ | shortest lever arm, across the gap between two legs | metre |
| $ \eta_\ell $ | ratio of aggregate data rate to downlink capacity | dimensionless |
| $ \gamma $ | flight-path angle above the horizontal | radian |
| $ \hat{A}_\mu $ | estimate of the cosine coefficient from a finite ring of sensors | the measurand’s own unit |
| $ \hat{B}_\mu $ | estimate of the sine coefficient from a finite ring of sensors | the measurand’s own unit |
| $ \kappa $ | ratio of specific heats of air, written gamma by the X-63 article | dimensionless |
| $ \lambda $ | temperature lapse rate of an atmospheric layer | kelvin per metre |
| $ \mathcal{R} $ | aggregate data rate | bit per second |
| $ \mathcal{R}_\ell $ | downlink capacity | bit per second |
| $ \mu $ | azimuthal harmonic index | dimensionless |
| $ \mu_{\max} $ | highest azimuthal harmonic resolvable in both phases | dimensionless |
| $ \phi $ | direction in which the vehicle tilts | radian |
| $ \Psi $ | the measurand distributed around the annulus | its own unit |
| $ \psi $ | initial thrust-to-weight ratio | dimensionless |
| $ \rho $ | atmospheric mass density | kilogram per cubic metre |
| $ \theta $ | tilt angle from the vertical | radian |
| $ \theta_k $ | angular position of leg k on the ring | radian |
| $ \theta_{\mathrm{tip}} $ | tilt angle at which the vehicle tips over | radian |
| $ \varphi $ | phase of an azimuthal harmonic | radian |
| $ \vartheta $ | azimuthal angle around the annulus | radian |
| $ A $ | reference frontal area | square metre |
| $ a $ | angle from the nearest leg | radian |
| $ A_0 $ | mean of the measurand around the annulus | the measurand’s own unit |
| $ A_\mu $ | cosine coefficient of azimuthal harmonic mu | the measurand’s own unit |
| $ a_s $ | specific force sensed by a body-mounted accelerometer | metre per second squared |
| $ b $ | bits per sample | dimensionless |
| $ B $ | ballistic coefficient, being initial mass over drag area | kilogram per square metre |
| $ B_\mu $ | sine coefficient of azimuthal harmonic mu | the measurand’s own unit |
| $ c $ | speed of sound in the atmosphere | metre per second |
| $ C_D $ | drag coefficient | dimensionless |
| $ C_{N\alpha,\,\mathrm{body}} $ | that slope for the cylindrical body alone | per radian |
| $ C_{N\alpha,\,\mathrm{nose}} $ | that slope for the nose alone | per radian |
| $ C_{N\alpha} $ | normal-force coefficient slope, referenced to the frontal area | per radian |
| $ D $ | aerodynamic drag | newton |
| $ d $ | drag fraction, being drag divided by thrust | dimensionless |
| $ D_b $ | body diameter | metre |
| $ d_{\mathrm{crit}} $ | drag fraction at which the total error equals the effect | dimensionless |
| $ F $ | thrust | newton |
| $ f $ | fineness ratio, being length divided by diameter | dimensionless |
| $ f_s $ | per-channel sample rate | hertz |
| $ g $ | local gravitational acceleration | metre per second squared |
| $ g_0 $ | standard gravity | metre per second squared |
| $ h $ | geometric altitude | metre |
| $ h_0 $ | altitude at the base of an atmospheric layer | metre |
| $ H_\rho $ | density scale height of the atmosphere | metre |
| $ h_b $ | altitude at main engine cut-off | metre |
| $ H_p $ | pressure scale height of the atmosphere | metre |
| $ h_s $ | support function of the footprint hull | metre |
| $ h_{cm} $ | height of the centre of mass above the ground | metre |
| $ J $ | time integral of ambient pressure over the burn | pascal second |
| $ j $ | index over the sensors on a ring | dimensionless |
| $ k $ | pitch-program exponent | dimensionless |
| $ L $ | vehicle length | metre |
| $ m $ | instantaneous vehicle mass | kilogram |
| $ M $ | Mach number | dimensionless |
| $ m_0 $ | initial vehicle mass | kilogram |
| $ m_c $ | number of sides of the X-63 article’s control polygon | dimensionless |
| $ m_h $ | number of sides of the footprint hull | dimensionless |
| $ N $ | number of legs, and elsewhere the number of sensors on a ring | dimensionless |
| $ n $ | altitude-profile exponent of the model this article discards, carried over from the X-63 article | dimensionless |
| $ N_a $ | aerodynamic normal force | newton |
| $ n_c $ | number of instrumentation channels | dimensionless |
| $ p $ | speed-profile exponent | dimensionless |
| $ p_0 $ | pressure at the base of an atmospheric layer | pascal |
| $ p_a $ | ambient atmospheric pressure | pascal |
| $ q $ | dynamic pressure | pascal |
| $ R $ | specific gas constant of air | joule per kilogram kelvin |
| $ r $ | radius of the circle of legs | metre |
| $ s $ | dummy variable of integration over time | second |
| $ S $ | wetted side area of the equivalent cylinder | square metre |
| $ S_m $ | static margin | calibre |
| $ T $ | atmospheric temperature | kelvin |
| $ t $ | time from liftoff | second |
| $ T_0 $ | temperature at the base of an atmospheric layer | kelvin |
| $ t_1 $ | an earlier instant in the burn | second |
| $ t_2 $ | a later instant in the burn | second |
| $ t_b $ | burn duration of the first stage | second |
| $ t_q $ | time of peak dynamic pressure | second |
| $ t_{1/2} $ | time by which half the pressure-time integral is collected | second |
| $ V $ | data volume written over a burn | byte |
| $ v $ | speed along the flight path | metre per second |
| $ v_b $ | speed at main engine cut-off | metre per second |
| $ V_c $ | enclosed volume of the equivalent cylinder | cubic metre |
| $ X_a $ | axial component of the aerodynamic force | newton |
| $ x_{cm} $ | axial position of the centre of mass | metre |
| $ x_{cp} $ | axial position of the centre of pressure | metre |
Every symbol above appears in the mathematics and every symbol in the mathematics appears above.
An Error Budget With One Unmeasured Term
Treat the three errors as independent and propagate them through $F = m a_s + D$. Let $\delta$ denote the uncertainty in whatever follows it. Then
\[\left( \delta F \right)^2 = \left( a_s \, \delta m \right)^2 + \left( m \, \delta a_s \right)^2 + \left( \delta D \right)^2\]Divide by $F^2$ and introduce the drag fraction $d \equiv D / F$, dimensionless. Because $m a_s = F - D = (1 - d) F$, the mass and accelerometer terms both carry that factor while the drag term carries $d$.
\[\left( \frac{\delta F}{F} \right)^2 = \left( 1 - d \right)^2 \left[ \left( \frac{\delta m}{m} \right)^2 + \left( \frac{\delta a_s}{a_s} \right)^2 \right] + d^2 \left( \frac{\delta D}{D} \right)^2\]The budget is complete as written, and its structure is the finding. The two terms a designer can improve by buying better instruments are attenuated by $1 - d$, which is close to one. The term that cannot be improved by any instrument on the vehicle is amplified by $d$, which is small, so the drag model’s accuracy is multiplied by a small number before it reaches the answer. Whether that suffices can be settled by arithmetic.
Two limits can be read off before any numbers. At $d = 0$, which is flight in vacuum, the drag model drops out entirely and the measurement is as good as the mass bookkeeping and the accelerometer. At $d = 1$, which is a vehicle whose drag equals its thrust and therefore is not accelerating, the instruments drop out entirely and the answer is whatever the drag model says. A launch vehicle spends its first minute somewhere between those, closer to the first, and the question is how much closer.
An Instrument Floor Present Even in Vacuum
Set the drag error to zero and the budget collapses to the instrument terms alone.
\[\frac{\delta F}{F}\Big\rvert_{d = 0} = \sqrt{ \left( \frac{\delta m}{m} \right)^2 + \left( \frac{\delta a_s}{a_s} \right)^2 }\]| Mass bookkeeping | Accelerometer | Thrust error at zero drag | As a share of a 5.3 percent effect |
|---|---|---|---|
| 1.0 percent | 0.5 percent | 1.118 percent | 21.1 percent |
| 0.5 percent | 0.2 percent | 0.539 percent | 10.2 percent |
| 2.0 percent | 1.0 percent | 2.236 percent | 42.2 percent |
Evaluated on the first row, with the effect $\Delta = 0.053$,
\[\frac{\delta F}{F}\Big\rvert_{d = 0} = \sqrt{ \left( 0.010 \right)^2 + \left( 0.005 \right)^2 } = 0.01118, \qquad \frac{0.01118}{0.053} = 0.211\]Against a five point three percent effect, the instrument floor alone consumes between a tenth and two fifths of the budget. A mass bookkeeping good to one percent and an accelerometer good to half a percent give 1.12 percent, which is 21.1 percent of the smaller of the X-63 article’s two gain figures. Tightening the instruments to half a percent and two tenths of a percent gives 0.54 percent, which is 10.2 percent of it. Loosening them to two percent and one percent gives 2.24 percent, or 42.2 percent of the effect. So even a vehicle flying in vacuum with a perfect drag model would spend a tenth of its error budget before the atmosphere was mentioned, and mass bookkeeping is the larger of the two contributions in every case.
Where the Atmosphere Pushes Hardest
The drag fraction $d$ follows the dynamic pressure, so the shape of the dynamic-pressure history decides when the measurement is hardest.
Let $\rho$ be the atmospheric mass density in kilogram per cubic metre and $q$ the dynamic pressure [dynamic pressure] in pascal, with $C_D$ the drag coefficient, dimensionless, and $A$ the reference frontal area in square metre.
\[q = \tfrac{1}{2} \rho v^2, \qquad D = q \, C_D A\]The density comes from the 1976 United States Standard Atmosphere, whose defining constants give pressure and temperature by layer and from which density follows by the perfect-gas law without tabulation, as in the standard itself [U.S. Standard Atmosphere, 1976]. Let $R$ be the specific gas constant of air in joule per kilogram kelvin and $T$ the temperature in kelvin.
\[\rho(h) = \frac{p_a(h)}{R \, T(h)}\]Both of those come from the layer structure and neither is tabulated here [defining constants and abbreviated tables]. Let $T_0$, $p_0$ and $h_0$ be the temperature, pressure and altitude at the base of a layer, in kelvin, pascal and metre, and $\lambda$ its lapse rate in kelvin per metre.
\[T(h) = T_0 + \lambda \left( h - h_0 \right)\] \[p_a(h) = p_0 \left( 1 + \frac{\lambda \left( h - h_0 \right)}{T_0} \right)^{-\frac{g_0}{R \lambda}}, \qquad p_a(h) = p_0 \exp\left( -\frac{g_0 \left( h - h_0 \right)}{R T_0} \right)\]The second form is the limit of the first as the lapse rate goes to zero and is the form an isothermal layer needs. No other relation in the article is needed alongside its degenerate case.
The speed of sound and the Mach number follow, and the ratio of specific heats needs a new letter. The X-63A article wrote that ratio $\gamma$, which has already been spent on the flight-path angle, so it is written $\kappa$ here. Let $c$ be the speed of sound in metre per second and $M$ the Mach number, dimensionless.
\[c = \sqrt{\kappa R T}, \qquad M = \frac{v}{c}\]These relations give 1.225000 kilogram per cubic metre at sea level and 0.363918 at 11 kilometres, against the standard’s published 1.2250 and 0.36391, and a sound speed of 340.294 metres per second against the standard’s 340.294 [defining constants and abbreviated tables].
The Density Scale Height Is Not the Pressure Scale Height
The natural length for a density profile is the density scale height [scale height], and in a layer with a lapse rate it differs from the pressure scale height that the X-63A article used. Let $\lambda$ be the lapse rate in kelvin per metre and $H_\rho$ the density scale height in metre.
\[H_\rho \equiv -\frac{\rho}{\mathrm{d}\rho / \mathrm{d}h} = \frac{R \, T}{g_0 + R \lambda}\]The lapse rate enters added to the gravitational term, and in the troposphere it is negative and therefore makes the denominator smaller. So $H_\rho$ exceeds $H_p$ there, which is to say density falls more slowly with altitude than pressure does, and the reason is that cooling air is denser at a given pressure, so the temperature drop partly offsets the pressure drop rather than compounding it. At sea level the two are 10,416 metres and 8,435. In an isothermal layer the lapse rate vanishes and the two coincide exactly, which the relations above reproduce at 15 kilometres to nine decimal places, the cheapest available check that the closed form is right. Across the layers where a trajectory spends its dynamic pressure the closed form agrees with a numerical difference of the density profile to better than two parts in a thousand million.
The Condition for Peak Dynamic Pressure
Differentiate the logarithm of the dynamic pressure along the trajectory and set it to zero. Using $\mathrm{d}h / \mathrm{d}t = v \sin \gamma$ and the definition of the density scale height,
\[\frac{\mathrm{d}}{\mathrm{d}t} \ln q = -\frac{v \sin \gamma}{H_\rho} + \frac{2}{v}\frac{\mathrm{d}v}{\mathrm{d}t} = 0 \qquad \Longrightarrow \qquad v^2 \sin \gamma = 2 H_\rho \frac{\mathrm{d}v}{\mathrm{d}t}\]Maximum dynamic pressure occurs where the vertical speed divided by the density scale height equals twice the relative rate of acceleration. The relation is exact, it holds for any atmosphere and any trajectory, and it contains nothing about the vehicle. It was checked at the computed peak of five ascent profiles and the worst disagreement between its two sides is 4.5 parts in a million, a residual set by the resolution of the golden-section search rather than by any error in the relation.
Written as a statement about competition it is easier to read. Dynamic pressure rises while the vehicle is accelerating and falls while the air is thinning, and it peaks at the moment those two rates balance. Climbing faster moves the peak earlier and lower, and accelerating harder moves it later and higher.
A Consistent Ascent Trajectory
The previous article fitted the altitude alone, as $h_b (t / t_b)^n$ with the exponent swept between 1.5 and 3.0, and used it inside a pressure integral where only the altitude enters. Dynamic pressure needs the altitude and the speed together, and choosing each independently does not produce a trajectory.
Taking that altitude law with an exponent of 2 alongside a speed law linear in time puts this vehicle at Mach 2.85 at 8.4 kilometres and returns a peak dynamic pressure of 190 kilopascal. That is about six times the 0.32 atmosphere a Space Shuttle ascent reached [max q], and it makes the drag fraction exceed one at every plausible ballistic coefficient. A drag fraction above one describes a vehicle that is decelerating, and this one was climbing. The model refuted itself on a quantity it was not asked about, and no failure costs a modeller less.
The repair is to stop choosing the altitude. A rocket flying a gravity turn [gravity turn] has its altitude fixed by the integral of the vertical component of its speed, so a speed law and a pitch program determine it, and the published cut-off altitude then fixes the remaining parameter instead of being assumed alongside it. Let $k$ be a pitch exponent, dimensionless.
\[\gamma(t) = \frac{\pi}{2}\left( 1 - \frac{t}{t_b} \right)^{k}, \qquad v(t) = v_b \left( \frac{t}{t_b} \right)^{p}, \qquad h(t) = \int_{0}^{t} v(s) \sin \gamma(s) \, \mathrm{d}s\]The published reference mission gives liftoff at zero and main engine cut-off at 160 seconds, 75 kilometres and 2.6 kilometres per second [ABL Payload User’s Guide]. Requiring $h(t_b) = 75$ kilometres determines $k$ for each speed exponent, by bisection on a monotone function, and what is left is one swept assumption rather than two.
| Speed exponent | Pitch exponent solved | Peak dynamic pressure | Time of peak | Altitude of peak | Mach at peak | Flight-path angle at peak |
|---|---|---|---|---|---|---|
| 0.8 | 1.6444 | 92.23 kilopascal | 25.04 second | 7.912 kilometre | 1.912 | 68.0 degree |
| 1.0 | 1.3438 | 71.60 kilopascal | 33.61 second | 8.761 kilometre | 1.792 | 65.6 degree |
| 1.2 | 1.1242 | 60.36 kilopascal | 41.98 second | 9.424 kilometre | 1.729 | 63.9 degree |
| 1.5 | 0.8866 | 51.54 kilopascal | 53.67 second | 10.172 kilometre | 1.691 | 62.6 degree |
| 2.0 | 0.6264 | 45.86 kilopascal | 70.30 second | 10.995 kilometre | 1.701 | 62.6 degree |
Peak dynamic pressure is now 45.9 to 92.2 kilopascal, reached at 25.0 to 70.3 seconds, at 7.9 to 11.0 kilometres, at Mach 1.69 to 1.91. Those are of the order of the 0.32 atmosphere, about 32 kilopascal, that a Space Shuttle ascent reached [max q]. The spread is the cost of not knowing how the vehicle builds speed, and it is reported rather than hidden behind the central case.
Since $\rho c^2 = \kappa p_a$, the dynamic pressure needs only the ambient pressure and the Mach number. For the central profile, with a speed exponent of 1.0, the peak at 8,760.5 metres and Mach 1.792 lies in the troposphere, where $T_0 = 288.15$ kelvin, $p_0 = 101{,}325$ pascal and $\lambda = -0.0065$ kelvin per metre, so in kelvin and pascal
\[q = \tfrac{1}{2} \rho v^2 = \tfrac{1}{2} \kappa \, p_a M^2, \qquad T = 288.15 - 0.0065 \left( 8{,}760.5 \right) = 231.21, \qquad p_a = 101{,}325 \left( \frac{231.21}{288.15} \right)^{5.2559} = 31{,}853, \qquad q = \tfrac{1}{2} \left( 1.4 \right) \left( 31{,}853 \right) \left( 1.792 \right)^2 = 71{,}603\]which reproduces the 71.60 kilopascal of the table, the exponent being $-g_0 / (R \lambda)$.
One numerical coincidence should be defused before a reader finds it. For the central profile the peak sits at 8,761 metres, and the density scale height there, with $R = 287.053$ joule per kilogram kelvin, is
\[H_\rho = \frac{R \, T}{g_0 + R \lambda} = \frac{\left( 287.053 \right) \left( 231.21 \right)}{9.80665 - \left( 287.053 \right) \left( 0.0065 \right)} = 8{,}358, \qquad \frac{8{,}761}{8{,}358} = 1.048\]in metre, so the ratio is 1.048. The previous article reported a sea-level pressure scale height of 8,434.9 metres, and the inconsistent pairing of two power laws described above puts its peak dynamic pressure at an altitude of 8,434.5 metres, which looks like a theorem. It is not one. The two quantities are different functions of different variables evaluated at different altitudes, and the agreement arises from that inconsistent trajectory alone. Across the consistent family the ratio of peak altitude to local density scale height runs from 0.92 to 1.40 and is not a constant.
The Signal Is Collected Before the Confounder Arrives
Now the two histories can be laid against each other, and the result is the opposite of what the framing invites.
The altitude-compensation effect is driven by the pressure-time integral, because the previous article showed that a fixed nozzle’s whole atmospheric penalty is its exit area multiplied by that integral. Let $J$ be it, in pascal second.
\[J = \int_{0}^{t_b} p_a\left( t \right) \, \mathrm{d}t\]The measurement difficulty is driven by the dynamic pressure instead. Both quantities are largest early, so the natural expectation is that they peak together and that the confounder is worst where the signal lives.
They do not peak together. On the consistent profiles, half the pressure-time integral is collected by 11.9 to 28.3 seconds, while dynamic pressure peaks at 25.0 to 70.3 seconds, which is 2.10 to 2.48 times later in every case.
| Speed exponent | Half the signal collected by | Dynamic pressure peaks at | Peak later by a factor of | Dynamic pressure at the half-signal time | Share of the signal already collected when it peaks |
|---|---|---|---|---|---|
| 0.8 | 11.95 second | 25.04 second | 2.096 | 57.2 percent of its peak | 83.6 percent |
| 1.0 | 14.89 second | 33.61 second | 2.258 | 42.0 percent of its peak | 87.7 percent |
| 1.2 | 17.79 second | 41.98 second | 2.360 | 30.5 percent of its peak | 90.4 percent |
| 1.5 | 21.97 second | 53.67 second | 2.443 | 18.6 percent of its peak | 93.0 percent |
| 2.0 | 28.29 second | 70.30 second | 2.485 | 8.2 percent of its peak | 95.3 percent |
By the time dynamic pressure peaks, between 83.6 and 95.3 percent of the pressure-time integral has already been collected. And at the moment half the integral is in hand, dynamic pressure stands at only 8.2 to 57.2 percent of the value it will reach. Write $\chi$ for the share of the pressure-time integral collected by time $t$, dimensionless, $t_{1/2}$ for the time at which it reaches one half and $t_q$ for the time of peak dynamic pressure, both in second.
\[\chi\left( t \right) = \frac{1}{J} \int_{0}^{t} p_a\left( s \right) \mathrm{d}s, \qquad \chi\left( t_{1/2} \right) = \tfrac{1}{2}, \qquad \chi\left( t_q \right) = 0.836 \text{ to } 0.953, \qquad \frac{t_q}{t_{1/2}} = \frac{25.039}{11.946} = 2.096 \text{ to } \frac{70.295}{28.292} = 2.485\]So the experiment is better posed than the arithmetic of the error budget alone suggests. The seconds that carry the altitude-compensation signal are seconds in which the atmosphere is dense enough to matter thermodynamically and the vehicle is still slow enough that the drag force is a modest fraction of thrust. Pressure matters at low altitude and drag matters at high speed, and a rocket is at low altitude before it is at high speed. That separation is not an accident of this trajectory. It follows from the fact that ambient pressure depends on position while drag depends on position and on the square of speed, so the drag term is the later of the two to develop.
The separation weakens as the profile builds speed early, from a factor of 2.48 on the slowest-accelerating profile to 2.10 on the fastest, but the ordering never reverses across the family tried, and no profile in it puts the dynamic-pressure peak inside the window that carries half the signal.
What the Drag Fraction Depends On
The drag fraction has carried the argument so far and has not yet been written down in terms of anything a designer chooses. The omission needs repairing before the budget is read as a requirement, because the written form shows which unknowns matter and which cancel.
\[d \equiv \frac{D}{F} = \frac{q \, C_D A}{F}\]Two groups turn that into a statement about the vehicle rather than about its absolute size. Let $m_0$ be the initial mass in kilogram, $\psi$ the initial thrust-to-weight ratio, dimensionless, and $B$ the ballistic coefficient in kilogram per square metre.
\[\psi \equiv \frac{F}{m_0 g_0}, \qquad B \equiv \frac{m_0}{C_D A}\] \[d = \frac{q}{\psi \, g_0 B}\]The flight appears once, as the dynamic pressure, and the vehicle appears twice, as a thrust-to-weight ratio and a ballistic coefficient. Neither the mass nor the reference area survives on its own. The form is useful because a designer knows a thrust-to-weight ratio and a ballistic coefficient without being able to quote either of the quantities inside them, and because it separates what the trajectory does from what the vehicle is.
It does not let this article compute a number, and saying why is the point. The X-64A’s mass is not published, its drag coefficient is not published and its thrust is not published, so none of $\psi$, $B$ or $d$ can be evaluated. The budget below is therefore stated as a function of $d$ rather than at a value of it, and the comparison that matters is against the span of $d$ a vehicle of these proportions could plausibly occupy.
One thing the form does give without any of those numbers. Since $\psi$, $g_0$ and $B$ are constant through the burn to the accuracy that matters here, the drag fraction is proportional to the dynamic pressure, so its shape in time is the dynamic-pressure history and its ratio between two instants is theirs.
\[\frac{d\left( t_1 \right)}{d\left( t_2 \right)} = \frac{q\left( t_1 \right)}{q\left( t_2 \right)}\]This proportionality licenses the comparison made below between the drag fraction during the seconds that carry the signal and its value at the peak, because the ratio of the two is a ratio of dynamic pressures and both of those are computed.
What Accuracy the Drag Model Has to Have
With the timing established, the budget can be read as a requirement rather than as an algebraic identity.
| Drag fraction | Drag-model accuracy | Total thrust error | Drag term alone | Drag share of the variance | Total error over a 5.3 percent effect |
|---|---|---|---|---|---|
| 5 percent | 5 percent | 1.091 percent | 0.250 percent | 5.2 percent | 0.206 |
| 5 percent | 10 percent | 1.174 percent | 0.500 percent | 18.1 percent | 0.222 |
| 5 percent | 25 percent | 1.640 percent | 1.250 percent | 58.1 percent | 0.309 |
| 10 percent | 5 percent | 1.124 percent | 0.500 percent | 19.8 percent | 0.212 |
| 10 percent | 10 percent | 1.419 percent | 1.000 percent | 49.7 percent | 0.268 |
| 10 percent | 25 percent | 2.695 percent | 2.500 percent | 86.1 percent | 0.508 |
| 20 percent | 5 percent | 1.342 percent | 1.000 percent | 55.6 percent | 0.253 |
| 20 percent | 10 percent | 2.191 percent | 2.000 percent | 83.3 percent | 0.413 |
| 20 percent | 25 percent | 5.079 percent | 5.000 percent | 96.9 percent | 0.958 |
| 30 percent | 5 percent | 1.692 percent | 1.500 percent | 78.6 percent | 0.319 |
| 30 percent | 10 percent | 3.100 percent | 3.000 percent | 93.6 percent | 0.585 |
| 30 percent | 25 percent | 7.541 percent | 7.500 percent | 98.9 percent | 1.423 |
Read down the drag-share column and the character of the experiment changes as the drag fraction rises. At a drag fraction of one twentieth and a drag model good to five percent, the drag term contributes 5.2 percent of the variance and the measurement is an instrument problem. At a drag fraction of one fifth and a drag model good to a quarter, the drag term is 96.9 percent of the variance and the measurement is a drag problem with instruments attached.
Inverting the budget gives the requirement directly. Write $\beta$ for the share of the effect the drag term is allowed, dimensionless. For the drag term alone to be that share of an effect of size $\Delta$,
\[d \, \frac{\delta D}{D} \le \beta \Delta \qquad \Longrightarrow \qquad \frac{\delta D}{D} \le \frac{\beta \Delta}{d}\]| Drag fraction | For a tenth of the effect | For a quarter | For a half | For all of it |
|---|---|---|---|---|
| 5 percent | 10.60 percent | 26.50 percent | 53.00 percent | 106.00 percent |
| 10 percent | 5.30 percent | 13.25 percent | 26.50 percent | 53.00 percent |
| 20 percent | 2.65 percent | 6.62 percent | 13.25 percent | 26.50 percent |
| 30 percent | 1.77 percent | 4.42 percent | 8.83 percent | 17.67 percent |
The difficulty is that the requirement is hyperbolic in the drag fraction. At a drag fraction of one twentieth, holding the drag term to a tenth of the effect needs a drag model good to 10.6 percent, which this article judges achievable for a configuration that has been wind-tunnel tested. At a drag fraction of one fifth it needs 2.65 percent, and at three tenths it needs 1.77 percent. This article’s judgement, which no source cited here quantifies, is that a pre-flight drag model of a novel configuration is unlikely to be good to two percent, because predicting the drag of a body of revolution to a few percent through the transonic region is a demanding standard for any model built before flight.
The Drag Fraction at Which the Experiment Stops Working
The useful single number is the drag fraction at which the total error equals the effect being measured. Write $d_{\mathrm{crit}}$ for it, dimensionless, and $\Delta$ for the size of the effect as a fraction, dimensionless. The condition is the budget set equal to the effect.
\[\left( 1 - d_{\mathrm{crit}} \right)^{2} \left[ \left( \frac{\delta m}{m} \right)^{2} + \left( \frac{\delta a_s}{a_s} \right)^{2} \right] + d_{\mathrm{crit}}^{2} \left( \frac{\delta D}{D} \right)^{2} = \Delta^{2}\]That is a quadratic in $d_{\mathrm{crit}}$ and it is solved by bisection rather than by formula, because the left side is monotone in $d_{\mathrm{crit}}$ wherever the drag-model error exceeds the instrument terms, and only in that regime is the question interesting. Solving it at fixed drag-model accuracy gives the following.
| Drag-model accuracy | Drag fraction at which the error equals 5.3 percent | Drag fraction at which it equals 8.5 percent |
|---|---|---|
| 5 percent | above unity | above unity |
| 10 percent | 0.5274 | 0.8498 |
| 25 percent | 0.2090 | 0.3387 |
| 50 percent | 0.1041 | 0.1690 |
A drag model good to five percent never breaks the experiment at any drag fraction below unity. A model good to ten percent breaks it at a drag fraction of 0.527. A model good to a quarter breaks it at 0.209, and a model good only to a half breaks it at 0.104.
The comparison that matters is against the drag fraction during the seconds that carry the signal, not against its peak. Dynamic pressure at the half-signal time is 8.2 to 57.2 percent of its peak value, so the drag fraction there is the same fraction of its peak. Even a peak drag fraction of three tenths, which would be high for a launch vehicle, gives between 0.025 and 0.17 during the window that carries half the effect. Against a critical value of 0.209 for a quarter-accurate drag model, that is inside the feasible region across most of the family and marginal at its fast end. By the ratio relation of the previous subsection,
\[d\left( t_{1/2} \right) = d\left( t_q \right) \frac{q\left( t_{1/2} \right)}{q\left( t_q \right)} = 0.30 \times 0.082 = 0.025 \quad \text{to} \quad 0.30 \times 0.572 = 0.172, \qquad \frac{57.20}{8.201} = 6.97\]So the verdict on the measurement is that it works, and that it works because of when the signal arrives rather than because of how good the instruments are. An accelerometer twice as good gains a factor of two on a term that is already small. Flying a profile that reaches altitude before it reaches speed buys a factor of about seven on the term that dominates, the last ratio above. The trajectory is the instrument.
And the vehicle’s shape works against that advantage. The drag fraction is proportional to $C_D A / m$, and this vehicle’s frontal area is 1.72 times its sibling’s at equal mass and equal drag coefficient, with a fineness ratio that probably makes its drag coefficient the higher of the two as well. The shape chosen to make the vehicle recoverable raises the one term in the error budget that no instrument on board can reduce. That is not a contradiction in the design. It is a trade, and naming it is the point of computing the budget.
Two Vehicles Cannot Be Differenced
One consequence of the laboratory’s sentence about separate vehicles deserves stating because it bears on what the programme could have learned.
The cleanest way to measure a five percent nozzle effect is not to measure thrust twice. It is to measure the difference. Fly the same vehicle on the same trajectory with two nozzles, and the drag model, the mass bookkeeping and the accelerometer calibration are common to both flights and cancel to first order in the difference. What survives is the quantity of interest, and a differential experiment can resolve an effect far smaller than either of its absolute measurements.
ARISE is not built that way. The laboratory put two teams on two vehicles of two designs [AFRL’s Rocket Lab Past, Present and Future], and they differ in the dimension that matters most for the cancellation. The X-63A was to be a single-stage aerospike variant of a vehicle of fineness ratio 14.67, and the X-64A is a vehicle of fineness ratio about 5 [AFRL awards agreements under ARISE] [Invocon X-64]. Their drag models are not common terms and their difference does not cancel anything.
This is an inference about experiment design and it is marked as one. Neither document says the two vehicles were intended to be compared against one another, and there are good reasons to fund two independent attempts at a first flight that have nothing to do with differencing, chief among them that one of them might not fly. What can be said without inference is narrower. Two vehicles of different proportions, flown by different teams, produce two absolute measurements each limited by its own drag model, and the register’s identical descriptions give a reader no warning that the two results are not interchangeable.
Dependent Systems
The record names one subsystem of this vehicle and one property of its engine, and the systems that depend on them are reconstructed from those two facts. The subsystem is a set of four legs that serve as stabilising fins in flight and as landing gear on return [Invocon X-64]. The property is that the engine is to be highly instrumented [AFRL awards agreements under ARISE]. From the legs follow the static-margin requirement and the tip-over footprint, and from the instrumentation follow the sensor ring on the annulus and the data rate it produces. The recoverability that the legs make possible is where those two lines meet. No subsystem design, sensor count or telemetry capacity for this vehicle is published, so every quantity in this section is a requirement or a sweep rather than a specification.
Legs That Double as Fins
The one design feature the record describes is a set of four legs that are stabilising fins on the way up and landing gear on the way down [Invocon X-64]. That is two jobs with different optima on one set of hardware, and both jobs are governed by the same count.
What a Short Vehicle Costs in Static Margin
A fin-stabilised vehicle is statically stable when its centre of pressure lies aft of its centre of mass, and the convention states the separation in calibres, being multiples of the body diameter. Let $x_{cp}$ and $x_{cm}$ be the axial positions of the centre of pressure and the centre of mass in metre, $D_b$ the body diameter in metre, and $S_m$ the static margin in calibre, dimensionless.
\[S_m = \frac{x_{cp} - x_{cm}}{D_b}\]The comparison below uses a margin of one calibre, and what one calibre costs depends entirely on how slender the vehicle is. Expressed as a fraction of the vehicle’s length, with $L$ the length in metre and $f = L / D_b$ the fineness ratio,
\[\frac{x_{cp} - x_{cm}}{L} = \frac{S_m}{f}\]One calibre of margin on this vehicle is 20 percent of its length. One calibre on its sibling’s parent is 6.8 percent. The ratio is 2.93, the fineness ratio again, and it follows from published dimensions with no aerodynamics in it. A short vehicle has to move its centre of pressure nearly three times as far, measured against the only length it has, to obtain the same conventional margin.
So the legs have to be fins rather than merely being stowed alongside. A body of revolution generates little restoring moment, and slender-body theory quantifies how little. Let $C_{N\alpha}$ be the normal-force coefficient slope per radian, referenced to the frontal area, dimensionless.
\[C_{N\alpha,\,\mathrm{nose}} = 2, \qquad C_{N\alpha,\,\mathrm{body}} = 0\]The nose contributes a slope of two per radian and the cylindrical body contributes nothing at all, in inviscid slender-body theory and at small angle of attack. Both halves of that come from the 1967 report that made the method a convention [Barrowman 1967], which was read in full for this article. Its equation 3-66 gives the body normal-force coefficient derivative in subsonic flow as twice the ratio of the nose base area to the reference area, so a nose whose base is the reference area contributes two. And its derivation requires the body components to be slender and free of discontinuities in cross-sectional area or its derivative, so a constant-diameter cylinder contributes nothing. The nose contribution also acts near the front of the vehicle, forward of any plausible centre of mass, so the bare airframe is statically unstable and every scrap of restoring moment has to come from the aft surfaces.
The inviscid result is a floor and the viscous correction runs upward. The classical study of viscosity on slender inclined bodies shows the measured normal force exceeding the potential-flow value, the excess growing with angle of attack as crossflow separation develops [Allen and Perkins 1951]. So a real body contributes more restoring moment than two per radian implies, and the fin requirement computed from the inviscid figure is the conservative one. Any error therefore runs the way a designer would want, opposite to the way that would invalidate the argument. A vehicle whose landing gear is also its empennage has not bolted two functions onto one part for elegance. It has put the aerodynamic surface where a short vehicle needs it and then noticed it was also where the ground is.
This article does not compute the fin area required, because that needs the nose contour, the fin planform and a centre-of-mass history, none of which is public. What is computed is the requirement those unknowns have to satisfy, and the requirement is 2.93 times more demanding than it would be for the parent vehicle of the other designation.
The Footprint Polygon
On the ground the same four surfaces become a support polygon, and the mathematics is the same family as the companion article’s but a different member of it.
Let $N$ be the number of legs, dimensionless, equally spaced on a circle of radius $r$ in metre, and let $h_{cm}$ be the height of the centre of mass above the ground in metre. Tipping occurs when the centre of mass passes outside the convex hull of the contact points [convex hull]. If the vehicle tilts by $\theta$ in the direction $\phi$, the ground projection of the centre of mass moves $h_{cm} \tan \theta$, so the critical tilt is set by the distance from the axis to the hull boundary in that direction. Write $\ell(\phi)$ for that distance in metre, calling it the lever arm, because $\rho$ is already the atmospheric density above.
\[\tan \theta_{\mathrm{tip}}\left( \phi \right) = \frac{\ell\left( \phi \right)}{h_{cm}}\]For a regular polygon with vertices at the legs, $\ell$ has a closed form. Let $a$ be the angle from the nearest leg, so that $\lvert a \rvert \le \pi / N$.
\[\ell\left( \phi \right) = \frac{r \cos\left( \pi / N \right)}{\cos\left( \pi / N - \lvert a \rvert \right)}, \qquad \ell_{\max} = r, \qquad \ell_{\min} = r \cos\left( \frac{\pi}{N} \right)\]The lever arm is not the support function of the hull, and the two are easy to conflate. The support function [support function of a convex set] gives the distance from the axis to a supporting line perpendicular to a direction, where the lever arm gives the distance along a ray to the hull boundary. Let $h_s$ be the support function in metre, with the legs at angles $\theta_k = 2 \pi k / N$ in radian.
\[h_s\left( \phi \right) = \max_{k} \, r \cos\left( \theta_k - \phi \right) = r \cos a\]The two agree only at the extremes. Both run from $r \cos(\pi / N)$ to $r$, so the anisotropy below is the same for either, and between those directions the lever arm is the smaller of the two because a ray reaches the boundary before the supporting line does. Tipping is a question about the boundary along a ray, so the lever arm is the right object, and a footprint that was not a regular polygon would separate them in range as well as in shape.
The lever arm is longest straight over a leg and shortest across the gap between two, because the legs are the vertices of the hull and the gaps are spanned by its edges.
One tempting construction gets this backwards. Measuring the angle to the nearest leg and dividing the inradius by its cosine produces a function with the right range and the wrong orientation, placing the short lever arm at a leg and the long one in the gap, which inverts the physical conclusion and makes a four-legged vehicle look most stable in the direction it is least stable in. A footprint is a hull and not a rosette. Evaluating the closed form at the two special directions and asking which gives the smaller number settles it in one line, a check needed because both orientations look plausible written down.
The Anisotropy Is the Same Closed Form the X-63A Article Derived
\[\frac{\ell_{\max}}{\ell_{\min}} = \frac{1}{\cos\left( \pi / N \right)}\]Character for character, this is the ratio of best-direction to guaranteed control authority that the previous article derived for differential throttling of a ring of thrusters. The same regular polygon governs how evenly a ring of engines can steer and how evenly a ring of legs can stand.
| Legs | Sides of the footprint polygon | Sides of the X-63 article’s control polygon at the same count | Inradius over circumradius | Tip-over anisotropy | Best over worst |
|---|---|---|---|---|---|
| 3 | 3 | 6 | 0.500000 | 2.000000 | 100.00 percent |
| 4 | 4 | 4 | 0.707107 | 1.414214 | 41.42 percent |
| 5 | 5 | 10 | 0.809017 | 1.236068 | 23.61 percent |
| 6 | 6 | 6 | 0.866025 | 1.154701 | 15.47 percent |
| 8 | 8 | 8 | 0.923880 | 1.082392 | 8.24 percent |
| 12 | 12 | 12 | 0.965926 | 1.035276 | 3.53 percent |
For four legs the anisotropy is exactly the square root of two. A four-legged vehicle tolerates 41.4 percent more tilt falling across a leg than falling between two, and a three-legged one tolerates exactly twice as much. Eight legs bring it to 8.2 percent and twelve to 3.5 percent, and the returns diminish as the cosine flattens.
\[\frac{1}{\cos\left( \pi / 3 \right)} = 2, \qquad \frac{1}{\cos\left( \pi / 4 \right)} = \sqrt{2} = 1.4142, \qquad \frac{1}{\cos\left( \pi / 8 \right)} = 1.0824, \qquad \frac{1}{\cos\left( \pi / 12 \right)} = 1.0353\]Why This Polygon Has N Sides and the X-63A’s Had N or 2N
The previous article found that its control polygon has $N$ sides for an even ring and $2N$ for an odd one, and that result has no counterpart here. The footprint polygon has exactly $N$ sides for every $N$.
Write $m_h$ for the number of sides of the footprint hull and $m_c$ for the number of sides of the X-63 article’s control polygon, both dimensionless.
\[m_h = N \quad \text{for every } N, \qquad m_c = \begin{cases} N, & N \text{ even} \\ 2N, & N \text{ odd} \end{cases}\]The difference is the construction and not the geometry. The X-63 article’s object was a maximum over throttle patterns of a sum of $\max\left( \cos, 0 \right)$ terms, and that truncation puts a breakpoint wherever a module crosses the boundary of the forward half plane. For an odd ring those crossings interleave into two families and the count doubles. Here the object is the convex hull of the contact points, whose support function is piecewise linear with one piece per vertex, and there is no half-plane to cross. No truncation, no parity.
That contrast has a second layer which the X-63 article records. Which direction is extremal relative to the modules turns, for the control polygon, on the module count modulo four rather than on its parity. The footprint polygon has no such subtlety, and the extremal directions are the legs and the gaps between them for every count. The truncated object is complicated and the hull is simple, although the two share a closed form.
Tip Angles at the Published Diameter
The leg reach and the centre-of-mass height of a returning X-64A are not published, so both are swept. A body radius of 1.2 metres sets the lower bound on the footprint radius, since legs cannot retract inside the hull, and a 12 metre vehicle returning nearly empty puts its centre of mass low. In metre,
\[r \ge \frac{D_b}{2} = 1.2, \qquad \ell_{\min} = r \cos\left( \frac{\pi}{4} \right) = 0.8485 \quad \text{at} \quad r = 1.2, \qquad \ell_{\min} = 2.1213 \quad \text{at} \quad r = 3.0\]| Footprint radius | Centre-of-mass height | Worst direction, across a gap | Best direction, over a leg | Shortest lever arm | Longest lever arm |
|---|---|---|---|---|---|
| 1.2 metre | 3.0 metre | 15.79 degree | 21.80 degree | 0.8485 metre | 1.2000 metre |
| 1.2 metre | 4.0 metre | 11.98 degree | 16.70 degree | 0.8485 metre | 1.2000 metre |
| 1.2 metre | 5.0 metre | 9.63 degree | 13.50 degree | 0.8485 metre | 1.2000 metre |
| 1.2 metre | 6.0 metre | 8.05 degree | 11.31 degree | 0.8485 metre | 1.2000 metre |
| 1.8 metre | 3.0 metre | 22.99 degree | 30.96 degree | 1.2728 metre | 1.8000 metre |
| 1.8 metre | 4.0 metre | 17.65 degree | 24.23 degree | 1.2728 metre | 1.8000 metre |
| 1.8 metre | 5.0 metre | 14.28 degree | 19.80 degree | 1.2728 metre | 1.8000 metre |
| 1.8 metre | 6.0 metre | 11.98 degree | 16.70 degree | 1.2728 metre | 1.8000 metre |
| 2.4 metre | 3.0 metre | 29.50 degree | 38.66 degree | 1.6971 metre | 2.4000 metre |
| 2.4 metre | 4.0 metre | 22.99 degree | 30.96 degree | 1.6971 metre | 2.4000 metre |
| 2.4 metre | 5.0 metre | 18.75 degree | 25.64 degree | 1.6971 metre | 2.4000 metre |
| 2.4 metre | 6.0 metre | 15.79 degree | 21.80 degree | 1.6971 metre | 2.4000 metre |
| 3.0 metre | 3.0 metre | 35.26 degree | 45.00 degree | 2.1213 metre | 3.0000 metre |
| 3.0 metre | 4.0 metre | 27.94 degree | 36.87 degree | 2.1213 metre | 3.0000 metre |
| 3.0 metre | 5.0 metre | 22.99 degree | 30.96 degree | 2.1213 metre | 3.0000 metre |
| 3.0 metre | 6.0 metre | 19.47 degree | 26.57 degree | 2.1213 metre | 3.0000 metre |
Across that sweep the worst-direction tilt tolerance runs from 8.05 to 35.26 degrees. The favourable corner, being legs reaching 3 metres on a vehicle whose centre of mass is 3 metres up, tolerates 35.26 degrees across a gap and 45.00 degrees over a leg. The unfavourable corner, legs no wider than the body on a vehicle whose mass sits at half its height, tolerates 8.05 degrees. A landing gear that reaches only to the body radius is not a landing gear. That arithmetic is why legs that the description says presumably rotate downward must also reach outward beyond the body [Invocon X-64].
\[\theta_{\mathrm{tip}} = \tan^{-1} \frac{\ell_{\min}}{h_{cm}} = \tan^{-1} \frac{0.8485}{6.0} = 8.05 \text{ degrees}, \qquad \tan^{-1} \frac{2.1213}{3.0} = 35.26 \text{ degrees}, \qquad \tan^{-1} \frac{\ell_{\max}}{h_{cm}} = \tan^{-1} \frac{3.0}{3.0} = 45.00 \text{ degrees}\]The Limits of a Sensor Ring
The award requires a highly instrumented engine [AFRL awards agreements under ARISE], and an annular aerospike presents its measurands on a circle. A finite number of sensors on a circle is a sampling problem and sampling problems have exact answers.
What the flight was meant to observe was established in the previous article. The fact sheet names open wake, wake transition and closed wake as three regimes to be visited [ARISE and Fly], the base pressure of a truncated plug decides how much of the truncation loss is recovered, and the transition between two flow topologies is the kind of thing that can be unsteady or hysteretic. An ideal axisymmetric nozzle puts all of that in the zeroth azimuthal harmonic. A real one does not, and the departures are what a ring of sensors is for.
Expand the quantity measured around the annulus as a Fourier series in the azimuthal angle. Let $\vartheta$ be that angle in radian, $\mu$ the azimuthal harmonic index, dimensionless, and $A_\mu$ and $B_\mu$ the coefficients in whatever unit the measurand carries.
\[\Psi\left( \vartheta \right) = A_0 + \sum_{\mu = 1}^{\infty} \left[ A_\mu \cos \mu \vartheta + B_\mu \sin \mu \vartheta \right]\]The harmonics are not interchangeable and each means something physical. The zeroth is the mean, the thrust-bearing quantity. The first is a net lateral resultant, or side load. The second is an ovalisation. Harmonics at the leg count and its multiples are whatever the legs themselves impose on the flow.
Sample at $N$ sensors equally spaced at $\vartheta_j = 2 \pi j / N$. The estimator for each coefficient is a sum over the sensors, and it makes the counting argument concrete.
\[\hat{A}_\mu = \frac{2}{N}\sum_{j=0}^{N-1} \Psi\left( \vartheta_j \right) \cos \mu \vartheta_j, \qquad \hat{B}_\mu = \frac{2}{N}\sum_{j=0}^{N-1} \Psi\left( \vartheta_j \right) \sin \mu \vartheta_j\]The counting is the sampling theorem applied to a circle rather than to a line, and the theorem is cited from its original statements instead of an encyclopaedia’s summary of them [Shannon 1949] [Nyquist 1928]. Neither was read for this article and both are cited at second hand. What makes the circular case finite rather than a limit is that the domain is compact, so the harmonics are indexed by integers and the count is exact where the line’s version is a bandwidth condition.
Each sensor contributes one number, so $N$ sensors deliver $N$ numbers and no more. Recovering the mean costs one of them and recovering each harmonic costs two, being a cosine coefficient and a sine coefficient, so the harmonics resolvable in both phases are bounded by a count rather than by anything physical.
\[1 + 2 \mu_{\max} \le N \qquad \Longrightarrow \qquad \mu_{\max} = \left\lfloor \frac{N - 1}{2} \right\rfloor\]And the harmonics above that limit are not lost but misread [aliasing] [discrete Fourier transform]. The discrete transform sees harmonic $\mu$ as harmonic $\mu \bmod N$, folded into the resolvable range.
\[\mu \; \longmapsto \; \min\left( \mu \bmod N, \; N - \left( \mu \bmod N \right) \right)\]| Sensors on the ring | Harmonics resolved in both phases | Harmonic resolved in one phase only | Harmonic that aliases onto the mean | Sensors needed for a four-lobed pattern |
|---|---|---|---|---|
| 3 | 0 to 1 | none | 3 | 9 |
| 4 | 0 to 1 | 2 | 4 | 9 |
| 5 | 0 to 2 | none | 5 | 9 |
| 6 | 0 to 2 | 3 | 6 | 9 |
| 8 | 0 to 3 | 4 | 8 | 9 |
| 12 | 0 to 5 | 6 | 12 | 9 |
Recovering both phases of harmonic $\mu$ needs $2\mu + 1$ sensors. A side load needs three. An ovalisation needs five. A four-lobed pattern needs nine. An even sensor count resolves the harmonic at half that count in one phase only, so four sensors see an ovalisation aligned with them and are blind to the same ovalisation rotated by forty-five degrees.
Four Sensors Cannot See Four Legs
This vehicle’s geometry produces the sharp case.
\[\frac{1}{N}\sum_{j=0}^{N-1} \cos\left( N \, \vartheta_j + \varphi \right) = \cos \varphi \qquad \text{for every } N \text{ and every } \varphi\]Harmonic $N$ sampled at $N$ points has exactly the same discrete mean as harmonic zero. Not approximately, and not for particular phases. The sum above was evaluated at five sensor counts and returns 1 at zero phase and one half at a phase of sixty degrees, the cosine of the phase in both cases, matching harmonic zero term for term.
| Sensors | Discrete mean of harmonic zero | Discrete mean of harmonic equal to the sensor count | The same at a phase of sixty degrees | Cosine of sixty degrees |
|---|---|---|---|---|
| 3 | 1.000000 | 1.000000 | 0.500000 | 0.500000 |
| 4 | 1.000000 | 1.000000 | 0.500000 | 0.500000 |
| 5 | 1.000000 | 1.000000 | 0.500000 | 0.500000 |
| 6 | 1.000000 | 1.000000 | 0.500000 | 0.500000 |
| 8 | 1.000000 | 1.000000 | 0.500000 | 0.500000 |
This vehicle has four legs, and four legs impose a disturbance whose leading azimuthal content is harmonic four. Four wakes, four sets of shed vorticity, four interference regions where a surface meets the body. If such a vehicle carried four pressure taps on its annulus, aligned with its legs as the natural symmetry invites, that four-lobed pattern would be read as a uniform shift in base pressure. The instrument would report a change in the thrust-bearing mean when what had changed was a four-lobed asymmetry contributing nothing to axial thrust.
The failure mode is worse than blindness because it is not silent. A sensor set that cannot see a harmonic returns zero for it and a careful analyst notices the gap. A sensor set that aliases a harmonic onto the mean returns a plausible number in the channel that matters most, and nothing in the data shows that it came from the wrong place.
The sensor count of the ARISE engine is not public, so the argument here concerns sensor rings in general and leaves this hardware unassessed. What can be said without inference is that the count must not equal the leg count, nor any divisor of it, and that nine sensors are needed to resolve a four-lobed pattern in both phases while five suffice to distinguish it from the mean. A designer would want the second number, which is also the cheaper.
One reassurance falls out of the same arithmetic. The regimes the flight exists to measure are changes in the mean, and the mean is the one harmonic every sensor count recovers. An undersampled ring measures the wake transition correctly and misattributes the asymmetry. That ordering suits this programme’s primary objective and defeats anybody hoping to learn about side loads from the same data.
Highly Instrumented, in Bits Per Second
The word the award uses is highly instrumented [AFRL awards agreements under ARISE]. The phrase is qualitative and the constraint behind it is quantitative, and writing that constraint down explains the vehicle’s most distinctive feature.
Let $n_c$ be the number of channels, dimensionless, $f_s$ the per-channel sample rate in hertz, $b$ the bits per sample, dimensionless, and $\mathcal{R}$ the aggregate rate in bit per second.
\[\mathcal{R} = n_c f_s b, \qquad V = \frac{\mathcal{R} \, t_b}{8}\]where $V$ is the volume written over a burn in byte.
| Channels | Sample rate | Bits | Aggregate rate | Written over a 160 second burn | Written over a 600 second flight |
|---|---|---|---|---|---|
| 100 | 1,000 hertz | 12 | 1.2 megabit per second | 0.024 gigabyte | 0.090 gigabyte |
| 200 | 5,000 hertz | 16 | 16.0 megabit per second | 0.320 gigabyte | 1.200 gigabyte |
| 500 | 10,000 hertz | 16 | 80.0 megabit per second | 1.600 gigabyte | 6.000 gigabyte |
| 1000 | 20,000 hertz | 16 | 320.0 megabit per second | 6.400 gigabyte | 24.000 gigabyte |
| 2000 | 50,000 hertz | 16 | 1600.0 megabit per second | 32.000 gigabyte | 120.000 gigabyte |
Five hundred channels at ten kilohertz and sixteen bits is eighty megabits per second and 1.6 gigabytes over a 160 second burn. That case is the middle row of the table, and a modest one.
The sample rate is not free to choose, because the engine sets it. The X-63A article recovered a combustion instability at 4.5 kilohertz from the manufacturer’s incident report [Flight 2 Static Fire Update], and the first tangential mode of a thrust chamber is the kind of phenomenon a highly instrumented engine exists to observe. Sampling theory puts the floor at nine kilohertz and practice puts it several times higher, because a mode at the Nyquist limit is detectable and not characterisable. So the ten kilohertz row is below what this engine’s published instability requires, and the twenty kilohertz row, at 320 megabits per second and 6.4 gigabytes, is nearer a realistic figure for the channels that matter.
The deficit is best expressed as a ratio. Let $\mathcal{R}\ell$ be the downlink capacity in bit per second and $\eta\ell$ the deficit, dimensionless.
\[\eta_\ell = \frac{\mathcal{R}}{\mathcal{R}_\ell} = \frac{n_c f_s b}{\mathcal{R}_\ell}\]The volume is trivial and the rate is not. No telemetry capacity for either ARISE vehicle is published, so the comparison below is against a range rather than a number, and the range is an assumption of this article. The range assumed is one half to twenty megabits per second, and the table gives the deficit across that span instead of at one value.
| Downlink capacity | 200 channels at 5 kilohertz | 500 channels at 10 kilohertz | 1000 channels at 20 kilohertz |
|---|---|---|---|
| 0.5 megabit per second | 32.0 | 160.0 | 640.0 |
| 1.0 megabit per second | 16.0 | 80.0 | 320.0 |
| 5.0 megabit per second | 3.2 | 16.0 | 64.0 |
| 10.0 megabit per second | 1.6 | 8.0 | 32.0 |
| 20.0 megabit per second | 0.8 | 4.0 | 16.0 |
1.6 gigabytes is a component that costs nothing and weighs nothing. Eighty megabits per second through the atmosphere from a moving vehicle is a programme. So an expendable vehicle that wants this much data must either throw most of it away before transmission, which means deciding in advance what matters, or accept losing it with the vehicle.
Why the Vehicle Comes Back
The recoverability is the answer to the bandwidth arithmetic.
The lead contractor is an instrumentation house whose award record is wireless data acquisition, impact detection and radiation monitoring [USAspending]. The award scope is a highly instrumented engine [AFRL awards agreements under ARISE]. The vehicle is described as recoverable, landing on four legs [Invocon X-64]. Those three facts are usually read as three features. They are one decision.
Recovery converts a telemetry problem into a memory problem. A vehicle that comes back can record at full rate onboard and be read out on the ground, so the link carries only what is needed to fly the mission and to reconstruct it if the vehicle is lost. The constraint moves from bits per second, where it is expensive, to bytes, where in this decade it is free. An instrumentation company asked to gather large amounts of data through an entire trajectory, and free to choose the vehicle, has an obvious reason to choose one that returns.
The inference is marked as an inference and its weakness is that the record is silent on the point. No document consulted states that recoverability was chosen for data return. The contractor’s release, as reported, describes the legs and the dimensions and does not give a rationale [Invocon X-64]. Other reasons for recovery are at least as plausible and some are stated elsewhere in the programme’s literature, chief among them that the portfolio’s goals are cost and schedule and that a reusable article amortises both, which the president of the lead company named directly in saying the technology has potential for a reusable capability [AFRL awards agreements under ARISE].
What the arithmetic establishes is not the motive but the magnitude. Whatever recovery was chosen for, it removes a factor of sixteen to eighty between what five hundred channels at ten kilohertz produce and what a link of one to five megabits per second carries. A programme that wanted the data and not the vehicle would have had to make that factor disappear some other way, and the usual other way is to record less.
And the same shape that makes recovery possible raised the drag term in the error budget, as the drag-model requirement derived earlier shows. The design recovers its data at the cost of the measurement’s hardest term, and whether that trade is favourable depends on the timing result above rather than on any property of the vehicle.
The Flight Test Record
No source consulted announces the cancellation of the X-64A, and no source announces a flight either.
The encyclopedia entry, last revised on 11 August 2024, states in its own voice that the status and progress of the programme at the time of writing is unknown [Invocon X-64]. The award was a three-year agreement made in December 2019, so its term expired at the end of 2022 [AFRL awards agreements under ARISE]. The designations were allocated on 20 April 2022, eight months before that expiry and with no vehicle flown [Parsch, DOD 4120.15-L Addendum]. The laboratory’s background document of September 2022 still describes ARISE in the future tense, saying the partnership will fly the first modular aerospike engine [AFRL’s Rocket Lab Past, Present and Future].
The asymmetry with the sibling designation is the interesting part.
The X-63A’s contractor was followed to its end as a launch company in the previous article. The RS1 failed 10.87 seconds into its first orbital attempt in January 2023 [RS1, Gunter’s Space Page], the second vehicle was lost on its pad after a static fire in July 2024 [Flight 2 Static Fire Update], the company announced in November 2024 that it was leaving the commercial launch market [RS1, Gunter’s Space Page], and in February 2025 it renamed itself Long Wall and turned to missile defence [ABL Space renamed Long Wall]. ABL Space Systems stopped being what it was.
Invocon did not, and the record gives no sign that it had ever been a launch company to stop being. Its current self-description is wireless sensing, distributed processing and high-precision instrumentation for extreme environments, with product families in power systems, telemetry and signal conversion, and detection and initiation [Invocon] [Invocon, About Us]. Its pages state that its instrumentation flies on launch vehicles. It does not say it builds them, and it lists no launch vehicle among its products. Nothing on the pages consulted mentions ARISE, the aerospike, or the X-64.
So the two halves of one allocation ended in two different ways and neither ends in a recorded flight. One contractor lost two of its RS1 rockets and left the launch business [RS1, Gunter’s Space Page]. The other appears to have returned to the business it was always in. The register carries both numbers still, with the same hundred and one characters against each [Parsch, DOD 4120.15-L Addendum].
That this programme asked an instrumentation supplier to become a launch-vehicle prime is not a criticism and the record does not support one. The other transaction instrument exists to bring in companies that would not otherwise bid, the announcement says so in as many words, and a small business leading a rocket programme with a propulsion subcontractor of KT Engineering’s record is a defensible structure [AFRL awards agreements under ARISE] [USAspending]. What the record shows is the arrangement, not its wisdom.
Comparison With Ground Prediction
No flight is recorded, so flight returned nothing to set beside any prediction made on the ground, as the flight test record above sets out. What can be stated is what was predicted or claimed before flight, and every such item has nothing from flight to stand against it.
The laboratory claimed a first. Its background document says the partnership will fly the first ever modular aerospike engine [AFRL’s Rocket Lab Past, Present and Future]. No flight of the X-64A is recorded in any source this article consulted, so this vehicle neither confirmed nor refuted the claim.
The portfolio stated its targets in time and money rather than in thrust. The modular architecture was to cut development time by seventy percent and development cost by fifty percent [AFRL’s Rocket Lab Past, Present and Future]. Those are predictions about a programme rather than about a flight, and the record holds no development time or cost for this vehicle to compare them with, the dollar value of the agreement itself being unknown.
The performance prediction is analytical and comes from the sibling article. The X-63A article computed that an ideal altitude-compensating nozzle delivers between 5.3 and 8.5 percent more first-stage impulse than the best fixed nozzle on the parent vehicle’s reference trajectory. This vehicle was to measure that effect, and no source consulted records that it was measured.
This article’s own predictions concern the measurement, and they too stand unchecked. The sizing section predicts that the instrument floor alone consumes between a tenth and two fifths of a 5.3 percent effect, that dynamic pressure peaks 2.10 to 2.48 times later than the moment half the pressure-time integral is collected, and that even a peak drag fraction of three tenths leaves a drag model good to a quarter inside the feasible region across most of the trajectory family. The account of dependent systems predicts that four equally spaced taps on the annulus would read the four-lobed disturbance of four legs as a change in the mean. Each of those could have been tested by one instrumented flight, and no such flight is recorded.
No wind-tunnel result, contractor analysis or simulation of this vehicle is cited anywhere in this article, and its contours, mass, thrust and drag coefficient are not public. What the record supports is a list of claims and derived requirements beside a flight record that is empty.
What the Data Changed
Four results in this subject run against the answer a first model or a first expectation gives.
A trajectory built from two independent power laws refutes itself on a quantity it was not asked about. It returns a peak dynamic pressure of 190 kilopascal and a drag fraction above one, which describes a decelerating vehicle, and the consistent model lets the altitude follow from the vertical speed, turning the published cut-off altitude from a second assumption into a constraint.
The timing result reversed the expected answer. Both the signal and the dynamic pressure are largest early, so the natural expectation is that the confounder peaks where the signal lives. It does not, because a rocket reaches altitude before it reaches speed, and the ratio $t_q / t_{1/2}$ displayed under The Signal Is Collected Before the Confounder Arrives is 2.10 to 2.48 on every profile.
The support polygon is easy to compute inside out. Dividing the inradius by the cosine of the angle to the nearest leg says a four-legged vehicle is most stable falling between two legs. The footprint is the convex hull of the contact points, so the legs are its vertices and the gaps are spanned by its edges, and the truth is the reverse.
And the aliasing hazard is not where one would look first. The obvious worry is a sensor count that cannot resolve a side load, but a side load is a first harmonic and needs only three sensors, so almost any ring suffices. The sharp case is not the smallest harmonic but the one equal to the sensor count, because harmonic $N$ sampled at $N$ points has exactly the discrete mean of harmonic zero. Four legs impose a four-lobed disturbance and four sensors are precisely the count that reports it as a change in the thrust-bearing mean.
The Contemporary Literature
| Cluster | Records |
|---|---|
| Measuring a rocket in flight, which is the entire claim | 148 |
| The transducers, and what they are bonded to | 190 |
| Telemetry, recording, and what a link will actually carry | 680 |
| What a finite number of sensors on a circle can see | 82 |
| Error budgets, and the discipline of propagating them | 134 |
| Coming back, which is what the legs are for | 497 |
| Stability from surfaces, including surfaces that are also legs | 129 |
| Drag, fineness, and the price of being short | 192 |
| The ascent, and where the dynamic pressure goes | 99 |
| The nozzle this vehicle carries, which the previous article took | 255 |
| Segmented and modular vehicles, which is a subcontractor’s trade | 9 |
| This vehicle, this team, and this programme’s lineage | 5 |
| Monitoring a structure and detecting what hits it, the lead’s own trade | 157 |
| Steering it, which is the third partner’s trade | 72 |
| Testing an engine on the ground, and what the test stand measures | 112 |
| Combustion and combustion instability in the chamber | 70 |
| Health monitoring, diagnosis and control of rocket engines | 41 |
| Turbopumps, feed systems and engine transients | 28 |
| Conventional nozzles, plumes and base heating | 46 |
| Solid rocket motors | 51 |
| Hybrid rocket engines | 33 |
| Detonation rocket engines | 33 |
| The Space Launch System, a launch vehicle documented in development | 71 |
| Launch-vehicle design, cost and staging | 67 |
| Rocket engine design, performance and history | 113 |
| Total | 3,314 |
The survey behind this article holds 3,314 research records, each a distinct work, and The Source Base below states how they are admitted. 1,499 of them, 45.2 percent, are report primaries under the identifier rule set out there. 3,147 records carry a usable year, running from 1928 to 2026 with a median of 2003. 2,776 of those, 88.2 percent, predate or share the year in which both designations were allocated.
Every record in that total is cited below. The clusters are ordered with the specific before the general ones they are special cases of, and each record is counted once, in the cluster its title belongs to, so a broad cluster does not absorb the records of a narrow one.
Measuring a rocket in flight, which is the entire claim
The keystone. Determining thrust and performance in flight and on the stand, including the air-breathing in-flight thrust literature whose method this article sets aside, postflight reconstruction of trajectories from onboard accelerometers and flight data, and the thrust stands that calibrate the measurement.
None of the 18 titles in the survey that name in-flight thrust names a rocket. The discipline that says thrust is calculated rather than measured grew up on turbojets and turbofans, which is why its method of modelling the gas path is not the method this article uses.
148 records. [Fleming and Gabriel 1955] [Useller and Pappas 1956] [Bloomer 1958] [Carta 1963] [Plane 1963] [Heine 1964] [Plane 1964] [Plane 1964] [Plane 1964] [Davis and Spicer 1965] [Langill 1965] [Nelius and Harris 1965] [Williams 1965] [Otto 1966] [Postflight Evaluation of Atlas-Centaur 1966] [Smith 1966] [Crosswy and Kalb 1967] [Strome 1969] [Apollo mission 11, trajectory 1970] [Dennis et al 1970] [Lane and Redman 1970] [Poland and Schwanebeck 1970] [Woodfield 1970] [Adams et al 1971] [Stark 1972] [Apollo/Saturn 5 Postflight Trajectory 1973] [Patterson 1973] [Solid rocket booster performance 1974] [Biesiadny et al 1978] [Compton et al 1979] [Compton et al 1981] [Hillje and Nelson 1981] [McKenna 1981] [Schoelen 1981] [Wadia and Wilson 1981] [Lacarna and Wissinger 1982] [Adams et al 1983] [Adams et al 1983] [Lemaster and Runyan 1983] [Kelly et al 1985] [Roberts et al 1985] [Rooney and Wilt 1985] [Sovey et al 1985] [Sovey et al 1985] [Sovey et al 1986] [Fruboese 1987] [McAmis 1995] [Tuttle et al 1995] [Conners and Sims 1998] [Negrão et al 1998] [Eckert and Oechslein 1999] [Kutschera and Render 1999] [Di Fiore dos Santos et al 2000] [Bull et al 2001] [Carpenter and Bauer 2001] [Thomas et al 2001] [Ziemer 2001] [Bull et al 2002] [Cook and Gruet 2003] [Desai et al 2003] [Hiers et al 2003] [Jet Thrust Measurement In 2003] [Kerzhanovich and Pichkhadze 2003] [Cook et al 2003] [Snellgrove et al 2003] [Karlgaard et al 2004] [Lisano and Jah 2004] [Lisano and Jah 2004] [Parent 2004] [Bordi et al 2005] [Desai et al 2005] [Karlgaard et al 2005] [Kazeminejad et al 2005] [Sabzehparvar 2005] [Peters et al 2006] [Hoff 2007] [Striepe et al 2007] [Tartabini 2007] [Imlach et al 2008] [Gao and Sun 2009] [Park et al 2009] [Niehus and Mracek 2010] [O’Keefe and Bose 2010] [Smith and Harrison 2010] [Santos et al 2011] [Tsuboi et al 2011] [Hu and Wang 2012] [Abilleira 2013] [Cassady et al 2013] [Karlgaard et al 2013] [Lugo et al 2013] [Olds et al 2013] [Polk et al 2013] [Wagner 2014] [Barth et al 2015] [Takahashi et al 2015] [Ajith et al 2016] [Bazin et al 2016] [Renitha P and Sivaramapandian J 2016] [Takahashi 2016] [Bronz et al 2017] [Gong et al 2017] [Mahzari and White 2017] [Anzalone et al 2018] [Del Mônaco Monteiro et al 2018] [Lugo et al 2018] [Thiele et al 2018] [Williams and Green 2018] [Xu and Lan 2018] [Abilleira et al 2019] [Dutta and Green 2019] [Dutta 2020] [Dutta et al 2020] [Abilleira et al 2021] [Li et al 2021] [Way and Brugarolas 2021] [Wei and Shao 2021] [Yamada et al 2022] [Zhang et al 2022] [Di Monaco et al 2023] [Siva et al 2023] [Chen et al 2024] [Vedvik and Karlgaard 2024] [Karlgaard et al 2025] [Graupe et al 2025] [Minaz and Meram 2025] [Wang et al 2025] [Vicker 2026] [Dobrodomov et al 2026] [Toson et al 2026] [Advanced Ducted Propulsor In-Flight] [Karlgaard et al] [Karlgaard et al] [Karlgaard et al] [Anzalone et al] [Anzalone et al] [In-Flight Thrust Determination] [In-Flight Thrust Determination for] [Mehta and Steva] [Mehta and Steva] [Mehta et al] [Fritz et al] [Ozelsel] [Propeller/Propfan In-Flight Thrust Determination] [Miller et al] [Winski and Pensado] [Time-Dependent In-Flight Thrust Determination] [Uncertainty of In-Flight Thrust]
The transducers, and what they are bonded to
The pressure, strain, temperature and acceleration sensors such a measurement uses, their calibration, and their mounting on flight vehicles, engines and test articles.
190 records. [Sabin 1955] [Sabin 1956] [White 1957] [Mull and Algranti 1960] [Roshon 1960] [Trott 1961] [Trott 1962] [Ziemer and Lambert 1962] [Miller 1965] [Hickam and Sternbergh 1966] [Pernet 1966] [Vacuum transducer calibration 1966] [Canfil and Nieberding 1967] [Carter and Massey 1967] [Chalmers 1967] [Cullen and Ragland 1967] [Delmonte 1967] [Ferris 1967] [Loyd and Pickard 1967] [Schuler 1967] [Horn 1967] [Hennen and Lambert 1968] [Miller et al 1968] [Schuler 1968] [Hendrix 1969] [Sita 1969] [Hilten 1970] [Reinel 1970] [Devries 1971] [Price et al 1971] [Lewis and Dods 1972] [Rogero 1972] [Wilson 1973] [Livingstone 1974] [Prokopec 1974] [Kraft 1975] [Tooth 1975] [Pearson 1976] [Jones and Bergquist 1977] [Seifert and Shea 1977] [Vrolyk 1977] [Molland 1978] [Tang et al 1978] [Bentzen 1980] [Galway 1980] [Transducer calibration 1980] [Antonazzi 1981] [Kidd 1981] [Bradley et al 1983] [Transducer calibration tool 1983] [Blanchard and Rutherford 1984] [Electroacoustic Transducer Calibration Method 1984] [Electroacoustic transducer calibration method 1984] [James and Quick 1984] [Blanchard and Rutherford 1985] [Berthold 1986] [Perry 1987] [Thompson et al 1987] [Ainsleigh et al 1988] [Keltner et al 1988] [Farokhi and Vertzberger 1989] [Williams 1989] [Bayer et al 1991] [Flanagan 1991] [George et al 1991] [Parzych et al 1991] [Whitmore and Leondes 1991] [Bethea and Rosenthal 1992] [Little 1992] [Mallon 1992] [McLachlan et al 1992] [Smith et al 1992] [Davis et al 1993] [Gibson and Sealey 1993] [Fichtel and McDaniel 1994] [Iaconis and D’Emilia 1994] [Semenov 1994] [Lucht and Charest 1996] [McNichol 1996] [Wnuk and Wnuk 1997] [Lei et al 1998] [Peterson et al 1998] [Applications of neural networks 1999] [Grosshandler 1999] [Tripp 1999] [Peterson et al 1999] [Rhew 1999] [Tripp and Tcheng 1999] [Hudson et al 2000] [Green et al 2001] [Martin et al 2001] [Smith and Scott 2001] [Hudson et al 2002] [Milos et al 2002] [Zhao 2002] [Figueroa et al 2003] [Sedlak et al 2003] [Figueroa et al 2004] [Goodenow 2004] [Iliopoulou et al 2004] [Moore 2004] [Sedlak and Hashmall 2004] [Zdenek and Anthenien 2004] [Zuckerwar and Scott 2004] [Farr et al 2005] [Wang et al 2005] [Chung 2006] [Jenne 2006] [Champaigne and Sumners 2007] [Wrbanek and Fralick 2007] [Guo et al 2008] [Slane et al 2008] [Wiley et al 2008] [Brown et al 2009] [Carter 2009] [Mackey et al 2009] [Ripper et al 2009] [Finley and Parker 2010] [Oota et al 2010] [Xie 2010] [Ajovalasit 2011] [Baars et al 2011] [Dave et al 2011] [Sheng and Hua 2011] [Zaman et al 2011] [Cao 2012] [Hall 2012] [Lv et al 2012] [Ma et al 2013] [Adamovsky et al 2014] [Niu et al 2014] [Zhou et al 2014] [Zhou et al 2014] [Diaz 2015] [Furuichi and Terao 2015] [Irimpan et al 2015] [Bąkowski et al 2016] [Gao et al 2016] [Sarma et al 2016] [Yao et al 2016] [Garg and Schiefer 2017] [Gerlach et al 2017] [Lazarev et al 2017] [Platte et al 2017] [Tan 2017] [Alam and Kumar 2018] [Alam and Kumar 2018] [High-Load Strain Gauge Balance 2018] [Ksica et al 2018] [Li et al 2018] [Sarma et al 2018] [Bal et al 2019] [Swanson et al 2019] [Zheng et al 2019] [Alam and Kumar 2020] [Jadhav et al 2020] [Nguyen et al 2020] [Raab and Rohde-Brandenburger 2020] [Usandizaga et al 2020] [Alam and Kumar 2021] [Alpert et al 2021] [Siroka et al 2021] [The Invention Relates to 2021] [Wejrzanowski et al 2021] [Hanapur et al 2022] [Jin et al 2022] [Kokuyama et al 2022] [Yang 2022] [Dongare et al 2023] [Veldman 2023] [Xu et al 2023] [Zhao et al 2023] [Dongare et al 2024] [Dongare et al 2024] [Feng et al 2024] [Behera et al 2025] [De Filippis et al 2025] [Dongare et al 2025] [Frankel et al 2025] [Frankel et al 2025] [Gao et al 2025] [Schoenekess et al 2025] [Olivas et al 2026] [Qi et al 2026] [Stoica et al 2026] [Zhang et al 2026] [Davis et al] [Ibell] [Calhoun et al] [Okojie et al]
Telemetry, recording, and what a link will actually carry
Telemetry, onboard recording and data acquisition on flight vehicles, rockets and spacecraft, with the coding, modulation, compression and link engineering they use, which is the literature behind the arithmetic of Highly Instrumented, in Bits Per Second.
680 records. [Urban 1959] [Ratz 1960] [Reeves et al 1960] [Marko et al 1961] [Viterbi 1961] [Adkins and Griffin 1962] [Baghdady 1962] [Brummer and Harrington 1962] [Choate 1962] [Eichelberger 1962] [Eichelberger and Ratner 1962] [Frost 1962] [Harris 1962] [Muehlner 1962] [Brummer et al 1963] [Feinberg et al 1963] [Fisher 1963] [Hauptschein and Sommer 1963] [Hill 1963] [Moore 1963] [Ng 1963] [Chapman et al 1963] [Reeves and Threlkeld 1963] [Titsworth 1963] [Urban 1963] [Wye et al 1963] [Adolphsen and Malinowski 1964] [Bailey et al 1964] [Barnes et al 1964] [Creveling 1964] [Feinberg et al 1964] [Grant et al 1964] [Higgins et al 1964] [Holmes and Stagner 1964] [Horiuchi 1964] [Lechter 1964] [Mahoney and Quann 1964] [Manders and Sussman 1964] [Mermagen 1964] [Mermagen 1964] [Stine 1964] [The design and performance 1964] [Long 1964] [White 1964] [Allen and Wrigley 1965] [An examination of the 1965] [Bechtold et al 1965] [Bloomquist and Graham 1965] [Buige and Goode 1965] [Campbell 1965] [Creveling 1965] [Czarcinski et al 1965] [Des Jardins and Wentz 1965] [Mozzi and Roth 1965] [Eisenberger and Posner 1965] [Eliassen 1965] [Elms 1965] [Furstenau 1965] [Griffin et al 1965] [Hoff 1965] [Holmes and Stagner 1965] [Horiuchi and Martin 1965] [Integral sensor telemetry final 1965] [Jamison 1965] [Mathison 1965] [Medlin 1965] [Mukhey 1965] [Space vehicle sa-6 telemetry 1965] [Springett 1965] [Thin-film personal communications and 1965] [Wang 1965] [Weiner 1965] [Weiner 1965] [Adaptive compressive telemetry techniques 1966] [An analytical investigation of 1966] [Anderson 1966] [Baer and Heckler 1966] [Baer and Heckler 1966] [Bassen and Jantz 1966] [Bathker and Clauss 1966] [Bechtold et al 1966] [Bowser and Busch 1966] [Brown et al 1966] [Burst transmission of PCM 1966] [Cote 1966] [Creveling 1966] [Donaldson et al 1966] [Emens et al 1966] [Frost and Norvell 1966] [Fryer 1966] [Griffin et al 1966] [Holgersen et al 1966] [Horton et al 1966] [Husick and Ritenour 1966] [Jamison 1966] [Kann 1966] [Lokerson 1966] [Quinn 1966] [Massey 1966] [Minderman 1966] [Pasternack 1966] [Postal and Potts 1966] [Sos 1966] [Thin-Film Personal Communications and 1966] [Thin-Film Personal Communications and 1966] [Trigg 1966] [Zrubek 1966] [Adair and Polge 1967] [Anderson and Gallo 1967] [Bassen 1967] [Carl 1967] [Charles and Larson 1967] [Conference on Adaptive Telemetry 1967] [Cote and Cressey 1967] [Creveling 1967] [George 1967] [Graeve and Massey 1967] [Husick and Ritenour 1967] [Hynes 1967] [Kurtenbach and Wintz 1967] [Lynch 1967] [Lynch 1967] [Miller et al 1967] [Pasternack 1967] [Research and advanced development 1967] [Saliga 1967] [Stein 1967] [Thin-Film Personal Communications and 1967] [Thin-film Personal Communications and 1967] [Adaptive telemetry systems Final 1968] [Cox et al 1968] [Easterling et al 1968] [Feinberg and Townsend 1968] [Harney and Richardson 1968] [Houts et al 1968] [Hynes 1968] [King and Shaffer 1968] [Kurtenbach and Wintz 1968] [Miller et al 1968] [Research, development, design, integration 1968] [Simpson and Tranter 1968] [Sixteen channel microminiature FM 1968] [Spacecraft telemetry and command 1968] [Telemetry modulation system MSC-TS-8A 1968] [Thin-Film Personal Communications and 1968] [Bell et al 1969] [Borek and Richardson 1969] [Crawford and Reynolds 1969] [Dannenberg and Katzman 1969] [Easterling et al 1969] [Feinberg et al 1969] [Frost et al 1969] [Golden 1969] [Harrison and Lockman 1969] [Hudgins and Lease 1969] [Karras 1969] [Meigs and Stine 1969] [Panneton and Warren 1969] [Polge and Wallace 1969] [Posner et al 1969] [Starner 1969] [Starner 1969] [Tranter 1969] [Velde et al 1969] [Visscher 1969] [Arndt et al 1970] [Christensen 1970] [Czarcinski et al 1970] [Feinberg and Maxwell 1970] [Frost 1970] [Gilchriest et al 1970] [Gilder et al 1970] [Glines and Lazzaro 1970] [Kulick 1970] [Lorio 1970] [Meigs and Stine 1970] [Underwood 1970] [Brockman 1971] [Burke and Harris 1971] [Butman and Timor 1971] [Butman and Timor 1971] [Butman et al 1971] [Clubb 1971] [Cramer and Grant 1971] [Dawson and Schmitt 1971] [Frost and Ellis 1971] [Hill et al 1971] [Lumb 1971] [Lumb and Viterbi 1971] [Miller et al 1971] [Scaffidi et al 1971] [Wintz 1971] [Burt et al 1972] [Fryer et al 1972] [Hopkins 1972] [Silverman 1972] [Borek 1973] [Broglio 1973] [Butman and Timor 1973] [Crosswy and Hornkohl 1973] [Easton and Hilbert 1973] [Peterson 1973] [Pickett and Matthews 1973] [Smith and Carr 1973] [Task four report Telemetry 1973] [Chen et al 1974] [Evanchuk 1974] [Finger and Cambra 1974] [Kantor et al 1974] [Pitts 1974] [Snider 1974] [Tolmadzheva et al 1974] [Wood and Risa 1974] [Evans et al 1975] [Gilley 1975] [Griffin 1975] [Mulhall et al 1975] [Peterson 1975] [Real time telemetry and 1975] [Weathers 1975] [Benjauthrit 1976] [Benjauthrit et al 1976] [Gatz 1976] [Greene 1976] [Greenhall 1976] [King 1976] [Konigsberg 1976] [Wilcher 1976] [Anson 1977] [Benjauthrit and Kemp 1977] [Gatz 1977] [Low 1977] [Marks 1977] [Salter 1977] [Brockman 1978] [Greene and Desjardins 1978] [Greene 1978] [Greene 1978] [Lord 1978] [Rey and Nipper 1978] [Smith 1978] [Stermer 1978] [Gatz 1979] [Holmes 1979] [Hooke 1979] [Schneider and Garman 1979] [Bell and Strock 1980] [Christensen et al 1980] [Cipolle 1980] [Erickson and Craddock 1980] [Giles and Whitford 1980] [Renz et al 1980] [Stevens 1980] [Brockman and Easterling 1981] [Burt 1981] [Bever 1981] [Harney 1981] [Meredith et al 1981] [Renz 1981] [Schneider and Garman 1981] [Stephison 1981] [Sue 1981] [Wechsler 1981] [Burt 1982] [Carreno 1982] [Clarke et al 1982] [Deutsch 1982] [Rummer et al 1982] [Spahn and Pena 1982] [Sue 1982] [Yuen et al 1982] [Deutsch 1983] [Kinman 1983] [Srijayantha 1983] [Cruickshank 1984] [Bever 1984] [Koerner 1984] [Stevens 1984] [Boykin 1985] [Carreno 1985] [Hooke and Greenberg 1985] [Macmedan 1985] [Ross 1985] [Stokes and Ward 1985] [Barthelme et al 1986] [Carreno 1986] [Bever 1986] [Jenkins 1986] [Massey 1986] [Urech et al 1986] [Watters 1986] [Bryant et al 1987] [Connell et al 1987] [Flagg and Kalil 1987] [Graham 1987] [Kao et al 1987] [Lewallen 1987] [Madsen 1987] [Muratore 1987] [Thomas and Diamond 1987] [Whitelaw 1987] [Carper 1988] [Carraway 1988] [Hurd et al 1988] [Leng and Peet 1988] [Mouneimne 1988] [Nguyen 1988] [Pettit 1988] [Sabia and Hand 1988] [Tcheng et al 1988] [Collins 1989] [Collins et al 1989] [Dalton 1989] [Diamond 1989] [Horner and Sabia 1989] [Ingels et al 1989] [Koerner 1989] [Lawson and James 1989] [Lawson and James 1989] [Pritchard 1989] [Sayood and Rost 1989] [Sinderson et al 1989] [Wike and Griffith 1989] [Advanced telemetry systems for 1990] [Carper and Stallings 1990] [Hooke et al 1990] [Massey and Corbin 1990] [Miller et al 1990] [Nguyen 1990] [Nguyen 1990] [Ross 1990] [Dominy et al 1991] [Bever 1991] [Bever 1991] [Hinedi et al 1991] [Massey and Corbin 1991] [Morris et al 1991] [Nguyen 1991] [Nguyen 1991] [Berman and Au 1992] [Kirkham 1992] [Nguyen 1992] [Nguyen 1992] [Ruth et al 1992] [Schneider 1992] [Valdes 1992] [Anderson 1993] [Arcangeli et al 1993] [Girouard and Hopkins 1993] [Hinedi et al 1993] [Hyde and Massie 1993] [Jones 1993] [Lesho and Eaton 1993] [Nguyen et al 1993] [Thorn 1993] [Tsou et al 1993] [Wells and Baroth 1993] [Douard 1994] [El-Ghazawi et al 1994] [Fogel 1994] [Kan 1994] [Koeberlein and Pender 1994] [Loubeyre 1994] [Medelius et al 1994] [Quinto and Orie 1994] [Shihabi et al 1994] [Simmons 1994] [Sorensen and Ferri 1994] [Srinivasan and Lichten 1994] [Wells and Baroth 1994] [Anderson 1995] [Roberts and Hashem 1995] [Bloise 1995] [Bufalino 1995] [Day 1995] [Hummel 1995] [Maurer 1995] [Sidorovich 1995] [Tsou et al 1995] [Miller et al 1996] [Patel 1996] [Sazani et al 1996] [Telemetry Systems 1996] [Blue and Crawford 1997] [Kinney 1997] [Telemetry Technology 1997] [Boyadzhyan 1998] [Burkes 1998] [Crawford and Pinkleton 1998] [Data Acquisition System DAS 1998] [Drews et al 1998] [Fantini 1998] [Haddock and Horan 1998] [Huegel 1998] [Medelius et al 1998] [Normyle 1998] [Sea Technology Arlington Va 1998] [Wang 1998] [Crawford and Pinkleton 1999] [Crawford et al 1999] [Gonsalves et al 1999] [Rice 1999] [Calderon 2000] [D’Amico 2000] [Integrated Advanced Microwave Sounding 2000] [Jiang and Horan 2000] [Kennedy and Sims 2000] [Kohtake et al 2000] [Norris et al 2000] [Paschke 2000] [Ryan 2000] [Yamada 2000] [Zahzah et al 2000] [Alhorn et al 2001] [Grace 2001] [Jensen 2001] [Morgan et al 2001] [Streich et al 2001] [Wilson 2001] [Fincannon 2002] [Jannette et al 2002] [Jensen 2002] [Shell and McElyea 2002] [Fincannon 2003] [Fort et al 2003] [Horan 2003] [Jensen 2003] [Kirby et al 2003] [Prasad and Pal 2003] [Simons and Miranda 2003] [Speer 2003] [Erol et al 2004] [Hase 2004] [Hogie et al 2004] [Martinez et al 2004] [McLeod 2004] [Medelius et al 2004] [Washburn 2004] [Wilson 2004] [Yairi et al 2004] [Dimmock 2005] [Moore 2005] [Semmel et al 2005] [Whiteman et al 2005] [Whiteman et al 2005] [Faulstich and Law 2006] [Franz et al 2006] [OBrien 2006] [Okino et al 2006] [Rodriguez et al 2006] [Specht and Noble 2006] [Sturdevant et al 2006] [Knopf 2007] [Lin et al 2007] [Snowden and Levinson 2007] [Ardalan et al 2008] [Kim et al 2008] [Losik 2008] [Next-Generation Telemetry Workstation 2008] [Powell et al 2008] [Bagri and Majid 2009] [Davydov and Sazonov 2009] [Determining Aliasing in Isolated 2009] [Inui et al 2009] [LaBelle et al 2009] [Oxer and Blemings 2009] [Stoneking and Tsai 2009] [Telemetry Boards Interpret Rocket 2009] [Apollo 11 Telemetry Data 2010] [Beyon et al 2010] [Breed et al 2010] [Bryant 2010] [Leachman 2010] [Losik 2010] [Losik 2010] [Losik 2010] [Luan et al 2010] [Mackey and Kulikov 2010] [Moore 2010] [Mukai and Vilnrotter 2010] [Parkes and Armbruster 2010] [Rice et al 2010] [Sepan and Lawrence 2010] [Sherman 2010] [Stoneking et al 2010] [Ahmad et al 2011] [Bates and Hong 2011] [Fang et al 2011] [Fielhauer and Boone 2011] [Fillery and Stanton 2011] [Fresconi and Harkins 2011] [Fukushima 2011] [Griebeler et al 2011] [Hamkins et al 2011] [OFarrell 2011] [Yang et al 2011] [Beyon et al 2012] [Burnside 2012] [Fang et al 2012] [Fang et al 2012] [Fitch et al 2012] [Hunter and Behbahani 2012] [Jiang et al 2012] [Lee 2012] [Losik 2012] [Losik 2012] [Losik 2012] [Perrins 2012] [Pomerantz et al 2012] [Arjun et al 2012] [Swanson et al 2012] [Vorontsov and Samoilov 2012] [Gao et al 2013] [Li Guojun et al 2013] [Marquez 2013] [Nye 2013] [Rice 2013] [Stanboli 2013] [Stanboli et al 2013] [Takacs 2013] [Varnavas and Sims 2013] [Vidya et al 2013] [Bohlouri et al 2014] [Chandiramani et al 2014] [Havelund and Joshi 2014] [Kargin 2014] [Manusubramanian et al 2014] [Parkes et al 2014] [Savitha et al 2014] [Simms et al 2014] [Sims and Varnavas 2014] [Varnavas and Sims 2014] [Herbert et al 2015] [Jackson 2015] [Lee and Pomerantz 2015] [Morgan 2015] [Mudford et al 2015] [Nishanth. N. 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Space data] [Space engineering. Space data] [Space systems. Launch-vehicle-to-spacecraft flight] [Staudinger et al] [Young] [Yairi et al]
What a finite number of sensors on a circle can see
Sampling a field with a finite ring or array of sensors, azimuthal and circumferential mode decomposition, sensor placement and spatial aliasing, which is the subject of What a Ring of Sensors Can See, and What It Cannot.
82 records. [Lobdell 1968] [Lobdell 1969] [Salikuddin 1983] [Liu 1985] [Udwadia and Garba 1985] [Brooks et al 1987] [Ligrani et al 1989] [Bergmann et al 1990] [Lim 1991] [Raman et al 1991] [Bogart and Yang 1992] [Lim 1992] [Glassburn and Smith 1994] [Citriniti and Citriniti 1997] [Kinzie and McLaughlin 1997] [Abhayapala et al 1999] [Williams 2000] [Al-Masoud and Singh 2001] [Kim et al 2001] [Al-Shehabi and Newman 2002] [Li et al 2002] [Mach and Koshak 2006] [Wei Jiang et al 2006] [Mach and Koshak 2007] [Rafaely et al 2007] [Liu et al 2008] [Meyer and Elko 2008] [Akhtar et al 2009] [Gaitonde and Samimy 2010] [Even and Hagita 2011] [Faranosov et al 2013] [Kijima et al 2013] [Vergallo et al 2013] [Alon and Rafaely 2014] [Sekikawa and Hamada 2014] [Ye and Law 2014] [Ishida et al 2015] [Laera 2015] [Sekikawa and Hamada 2015] [Ali et al 2016] [Behn et al 2016] [Xu and Zhao 2016] [Mainini 2017] [Normal Mode Decomposition Based 2017] [Sijtsma and Brouwer 2017] [Zhao et al 2017] [He et al 2018] [Sijtsma and Brouwer 2018] [Brown et al 2019] [Deng et al 2019] [Suroso et al 2019] [Yang et al 2019] [Haldorsen 2020] [Li and Yang 2020] [Fan 2021] [Luo and Kareem 2021] [Gonzales et al 2022] [Gonzales et al 2022] [Ohmichi et al 2022] [A hydro-acoustic mode decomposition 2023] [Juhlin and Jakobsson 2023] [Zhong et al 2023] [Behn and Tapken 2025] [Fu et al 2025] [Sun et al 2025] [Vincent et al 2025] [Chen et al 2026] [Chen et al 2026] [Lian et al 2026] [Lian et al 2026] [Monnoyer et al 2026] [Moussa and Guedria 2026] [Shen et al 2026] [Wang et al 2026] [Yang et al 2026] [DeFord et al] [Li et al] [Li et al] [Li et al] [Movva] [Connelly et al] [Sawada et al]
Error budgets, and the discipline of propagating them
Uncertainty analysis and error propagation as method, and their application to flight test, wind tunnel and propulsion measurement.
134 records. [Abdelwahab et al 1987] [Davidian 1987] [Davidian et al 1987] [Arueti 1988] [Rusek 1989] [Batill 1994] [Whitmore and Moes 1994] [Blumenthal 1995] [Naughton et al 1996] [Cohn 1997] [Sims and Coleman 1998] [Tripp and Tcheng 1999] [Betta et al 2000] [Chunovkina 2000] [Meyn 2000] [Tarapčík et al 2001] [Measurement Uncertainty 2002] [Cox et al 2003] [Gertsbakh 2003] [Amer et al 2004] [D’Antona 2004] [Driscoll and Landrum 2004] [Lineberry et al 2004] [Hall 2005] [Locke and Landrum 2005] [Peretto et al 2005] [Case studies in measurement 2006] [Hinrichs 2006] [Uncertainty Propagation for Systems 2006] [Cristaldi et al 2007] [Mana and Pennecchi 2007] [Mencattini et al 2007] [Roberts et al 2007] [White and Saunders 2007] [Golubev 2008] [Hemsch et al 2008] [Mekid and Vaja 2008] [Ponci and Johnson 2008] [Uncertainty Propagation Methods 2008] [Hale and Wang 2009] [Mari 2009] [Mellodge and Kachroo 2010] [Dietrich and Schulze 2011] [Dietrich and Schulze 2011] [Gupta 2011] [Hanke 2011] [Hessling 2011] [Konda et al 2011] [Wang and Wang 2011] [Taylor 2012] [Taylor 2012] [Chapter 9 Uncertainty Propagation 2013] [Taylor 2013] [Zappa et al 2013] [Methods of uncertainty propagation 2014] [Operative procedures for the 2014] [Ratcliffe and Ratcliffe 2014] [Ratcliffe and Ratcliffe 2014] [Ferrero et al 2015] [Sargsyan 2015] [Stephens et al 2015] [Zhu et al 2015] [Davison et al 2016] [Fotowicz 2016] [Idźkowski et al 2016] [Morris and Crowley 2016] [Qin et al 2016] [Sciacchitano and Wieneke 2016] [Stephens et al 2016] [Stephens et al 2016] [Uncertainty Estimation, Propagation, and 2016] [Determining Measurement Uncertainty Example 2017] [Heidenreich et al 2017] [Krejci et al 2017] [Measurement Uncertainty 2017] [Nikbay and Heeg 2017] [Chalyy 2018] [Cristaldi et al 2018] [Dalle et al 2018] [Driver et al 2018] [Hubbard 2019] [Kajita 2019] [Campbell 2020] [Lee and Lee 2020] [Perez et al 2020] [Shi et al 2020] [Zhao et al 2020] [Gu et al 2021] [Kelsey 2021] [Sakai et al 2021] [Johnson et al 2022] [Kanso et al 2022] [Cavalieri et al 2023] [Friedlander et al 2023] [Guidelines on measurement uncertainty 2023] [Ooi et al 2023] [Poljak 2023] [Measurement Uncertainty 2024] [Measurement uncertainty concepts 2024] [Rishi 2024] [Thompson 2024] [Zakharov et al 2024] [Carratù et al 2025] [Ferrero 2025] [Pena and De La Fuente 2025] [Ludwig et al 2025] [McDowell et al 2025] [Witkovský 2025] [Zangl and Pérez 2025] [Agourakis and Agourakis 2026] [Appendix H Alternate Measurement 2026] [Appendix K Condensed Measurement 2026] [Arronde Pérez and Zangl 2026] [Cabrera et al 2026] [Fundamentals of Measurement Uncertainty 2026] [Li et al 2026] [Nasution et al 2026] [Otsuka 2026] [Petitjean and Musset 2026] [The Measurement Uncertainty Model 2026] [Vasilevskyi and Cullinan 2026] [Kammer et al] [Friedlander et al] [Hubbard and Semmelmayer] [Estimation of Measurement Uncertainty] [Example Uncertainty Propagation] [Fuzzy Variables and Measurement] [Houlden and Hubbard] [McAfee et al] [Santos et al] [Measurement Uncertainty Applied to] [Poljak et al] [McKown et al] [Vathsal]
Coming back, which is what the legs are for
Recovery, reuse and landing of rockets, boosters and stages, including landing legs, vertical-landing guidance, and booster recovery by parachute and at sea.
497 records. [Barraza 1962] [Barraza 1962] [McNair 1962] [Bono 1963] [Milliken 1963] [Armstrong 1964] [Benson et al 1964] [Eberhart 1964] [Flynn and Groves 1964] [Suit et al 1964] [Spieth 1965] [Vaglio-Laurin and Finke 1965] [Dunavant et al 1966] [Knaur 1966] [Parachute recovery system for 1966] [Scher and Dunavant 1966] [Hull 1967] [Karel 1967] [Knaur 1967] [Cassanto et al 1968] [Hinchey 1968] [Knaur 1968] [Jones 1970] [Kah 1970] [Bejczy 1971] [Expendable Second Stage Reusable 1971] [Space shuttle program. 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Bolender 2006] [Nonaka et al 2006] [Rasky et al 2006] [Suzuki et al 2006] [Yang et al 2006] [Burton et al 2007] [Kalden 2007] [Michalski and Johnson 2007] [Suzuki et al 2007] [Vaughn et al 2007] [Caogen et al 2008] [Donahue et al 2008] [Hellman and Tejtel 2008] [Jiang and Ordonez 2008] [Johnson and Servidio 2008] [Liaoni Wu et al 2008] [Garg and Dodiyal 2009] [Jiang and Ordóñez 2009] [Jurist 2009] [Kelly et al 2009] [Kluever et al 2009] [Kuzin et al 2009] [Lemieux 2009] [Lin et al 2009] [Molina et al 2009] [Xu 2009] [Bradford and St. Germain 2010] [Fischbach and Kenny 2010] [Kothari and Webber 2010] [Lin et al 2010] [Namera et al 2010] [Rippere and Wiens 2010] [Xu and Tang 2010] [Cowling 2011] [Hellman et al 2011] [Hellman et al 2011] [Hutchison 2011] [Kuzuu et al 2011] [Letchworth 2011] [Mains 2011] [Moore and Phelps 2011] [Moore and Phelps 2011] [Smith 2011] [Song et al 2011] [Clayton 2012] [Dissel et al 2012] [Lemieux and Murray 2012] [Losik 2012] [Nonaka et al 2012] [Paschall and Brady 2012] [Chen et al 2013] [Hellman et al 2013] [Stansbury et al 2013] [Stewart 2013] [Zhou et al 2013] [Baran et al 2014] [Gong et al 2014] [Mu and Zhang 2014] [Najam 2014] [Skariya et al 2014] [Webb et al 2014] [Wuilbercq et al 2014] [Yu et al 2014] [Gong et al 2015] [Huang et al 2015] [Junaid R and Beebi M 2015] [Ragab and Cheatwood 2015] [Su and Wang 2015] [Tartabini et al 2015] [Zhi et al 2015] [Childress-Thompson et al 2016] [Devanath and Beebi 2016] [Jianguo et al 2016] [Mueller et al 2016] [Nizin et al 2016] [Numerical Optimization on Approach 2016] [Reed et al 2016] [Yang et al 2016] [Yoshida et al 2016] [AL-Bakri and Kluever 2017] [Aogaki et al 2017] [Childress-Thompson et al 2017] [Dai et al 2017] [Demidovich 2017] [Hao et al 2017] [Lei et al 2017] [Liu et al 2017] [Sippel et al 2017] [Sivan and Murmu 2017] [Umadevi et al 2017] [Zhang et al 2017] [Zhang et al 2017] [Zhang et al 2017] [Abrahamm and Valsa 2018] [Chen et al 2018] [Choo et al 2018] [Fuchs et al 2018] [ISRO Releases the Special 2018] [Ma et al 2018] [Masilamani et al 2018] [Sivan and Pandian 2018] [Szmuk and Acikmese 2018] [Wang and Song 2018] [Wang and Song 2018] [Zhang et al 2018] [Aogaki et al 2019] [Aprovitola et al 2019] [Aprovitola et al 2019] [Chen and Ma 2019] [Chen et al 2019] [Cusick and Kontis 2019] [Inatomi et al 2019] [Lei et al 2019] [Ma et al 2019] [Nardozzo et al 2019] [Wang and Song 2019] [Antonelli et al 2020] [Brevault et al 2020] [Buzuluk et al 2020] [Jiandong et al 2020] [Kawatsu et al 2020] [Li et al 2020] [Shukla et al 2020] [Timofeev 2020] [Usmonov and Kretov 2020] [Wang et al 2020] [Wu et al 2020] [Xie et al 2020] [Chen et al 2021] [Cheng et al 2021] [Lukin et al 2021] [Pidvysotskyi 2021] [Su et al 2021] [Su et al 2021] [Wang et al 2021] [Yuan et al 2021] [Botelho et al 2022] [Brooks 2022] [Lei et al 2022] [Li et al 2022] [Mo et al 2022] [Mukundan et al 2022] [Prasad 2022] [Qi et al 2022] [Sithara and Shenil 2022] [Thies 2022] [Wang et al 2022] [Yue et al 2022] [Agarwal 2023] [Brociek et al 2023] [Dong et al 2023] [Guadagnini et al 2023] [Guo et al 2023] [Liu 2023] [Radhakrishnan et al 2023] [Wang et al 2023] [Wu and Zhang 2023] [Allard 2024] [Alvord et al 2024] [Manu et al 2024] [Cheng et al 2024] [Costa et al 2024] [Gibart et al 2024] [Hara et al 2024] [Hetmaniok et al 2024] [Imhuelse et al 2024] [Islam 2024] [Kim et al 2024] [Liu 2024] [Martin et al 2024] [Mirzabayova and Rustamov 2024] [Ren et al 2024] [Scarlatella et al 2024] [Serçeoglu 2024] [Shoyama and Hirakawa 2024] [Singh et al 2024] [Wang et al 2024] [Wang et al 2024] [Xing et al 2024] [Xinguo et al 2024] [Yang et al 2024] [Zhao et al 2024] [Zhou et al 2024] [Zhou et al 2024] [Çelik and Demirezen 2024] [Agarwalla 2025] [Chen et al 2025] [Colicci et al 2025] [Fox 2025] [Fox 2025] [Iafrate et al 2025] [Iwabuchi and Hashimoto 2025] [Li et al 2025] [Nair and Kukreja 2025] [Purcell et al 2025] [Roma Rubi et al 2025] [Ryu et al 2025] [Su and Liu 2025] [Su and Liu 2025] [Swiss Students Achieve Europe’s 2025] [Tong et al 2025] [Trajectory Shaping Guidance for 2025] [Wang et al 2025] [Xu et al 2025] [Zaragoza Prous et al 2025] [Zhou et al 2025] [Zimmerli et al 2025] [China achieves first reusable 2026] [Chowdhury et al 2026] [Depardon 2026] [Fan et al 2026] [Huang 2026] [Huang and Zhang 2026] [Khamlak 2026] [Kim et al 2026] [Lee et al 2026] [Li et al 2026] [Long et al 2026] [Maru et al 2026] [Mastromatteo et al 2026] [Oktaviana et al 2026] [Ren 2026] [Response Analysis of Landing 2026] [Reusable Launch Vehicle Landing 2026] [Romano et al 2026] [Wang and Miao 2026] [Wang et al 2026] [Wang et al 2026] [Wang et al 2026] [Xiao et al 2026] [liu et al 2026] [Al Bakri] [Burchett] [Caplin] [Coakley] [Dorney] [Galli] [Guo and Musgrave] [Guodong Ning et al] [Hamilton] [Kitsios and Lygeros] [Lorenzo et al] [Ménou] [Pérez Roca] [Shtessel and Krupp] [Sun] [Wetzel] [Xiaowen Dai et al]
Stability from surfaces, including surfaces that are also legs
Fins, grid fins and the static stability of rockets, missiles and projectiles, the literature behind What a Short Vehicle Costs in Static Margin.
129 records. [Cole and Abramovitz 1952] [Johnson and Hayes 1953] [Burrows and Newman 1954] [Kurbjun 1954] [Schmidt 1955] [Gillespie 1956] [Simon 1956] [Appich and Turner 1958] [Donegan 1958] [Henning 1959] [James 1960] [Sung and Park 1960] [Medukhovskii 1961] [Jorgensen et al 1962] [Ziegler 1963] [Burt and Hillsamer 1964] [Duncan and Ensey 1964] [Lieske and Kochenderfer 1966] [Arrington et al 1967] [Babb and Fuller 1967] [Ferris 1967] [Lewak 1967] [Fuller 1968] [Washington et al 1968] [De Angelis and Tang 1969] [Dahlke and Pettis 1970] [Daniels 1970] [Tang 1971] [Tang and Pearson 1971] [Ellis and Gamble 1972] [Luckert 1973] [Spencer and Fournier 1973] [Trescot et al 1973] [Lindsay and Jordan 1975] [Fansler and Schmidt 1976] [Jenke 1976] [Kassner and Wettlaufer 1977] [Rollstin 1979] [Blair 1980] [Schmidt 1980] [Sawyer et al 1981] [Blair et al 1983] [Weinacht et al 1984] [Whyte et al 1985] [Celmins 1987] [Brooks and Burkhalter 1988] [Jenn and Nelson 1988] [Nelson 1988] [Bornstein et al 1989] [Plostins et al 1990] [Est and Nelson 1991] [Blair et al 1992] [Bossi and Nelson 1992] [Washington et al 1993] [Lesieutre et al 1994] [Miller and Washington 1994] [Yang 1994] [Burkhalter and Frank 1995] [Moran and Beran 1995] [Burkhalter and Frank 1996] [Dillon 1996] [Eugene 1996] [Huffman et al 1996] [Costello and Costello 1997] [Schmidt et al 1997] [Gnoffo et al 1998] [Sun and Khalid 1998] [Erline and Hathaway 1999] [Costello and Agarwalla 2000] [Costello and Agarwalla 2001] [DeSpirito and Sahu 2001] [Dillo 2001] [Fournier 2001] [Landers et al 2003] [Lin et al 2003] [He et al 2004] [Ma et al 2005] [Theerthamalai et al 2005] [Blake and Cunningham 2006] [Wilks 2006] [Ghosh et al 2008] [Khalil et al 2009] [Debiasi et al 2010] [Aftosmis 2011] [Fresconi 2011] [Kless and Aftosmis 2011] [Kumar et al 2011] [Pruzan et al 2011] [Terhune et al 2011] [DeSpirito 2012] [Debiasi 2012] [Ji et al 2012] [Kumar et al 2012] [Larin 2012] [Silton and Bhagwandin 2012] [DeSpirito 2013] [Krishnamurthy et al 2013] [Coirier et al 2014] [Despeyroux et al 2014] [Elsaadany and Wen-Jun 2014] [Garcia and Silveira 2015] [Silton and Coyle 2015] [Silton and Fresconi 2015] [Silton and Coyle 2016] [Xu and Pei 2016] [Zhang et al 2016] [Zhaoqing and Hailong 2016] [Koomphati 2017] [Muralidhar and Bhandari 2017] [Tripathi et al 2018] [Anandaraj et al 2019] [Tripathi et al 2019] [Tripathi et al 2019] [Meda 2020] [Dol 2021] [Dinçer and Sezer Uzol 2022] [Bordachev et al 2023] [Ren et al 2023] [Storey 2023] [Żurawka et al 2023] [Dixit et al 2025] [Liu et al 2025] [Pribadi 2025] [Tripathi et al 2025] [Anuskiewicz et al 2026] [Montesinos et al 2026] [Woodyard 2026] [Wu et al 2026] [Yadav et al 2026]
Drag, fineness, and the price of being short
Drag of bodies of revolution, fineness ratio, base drag and nose shape, including the drag-reducing nose spikes that share the aerospike’s name.
192 records. [R. 1928] [Zahm et al 1929] [Abbott 1937] [Thomas 1942] [Brown and Parker 1945] [Jones and Margolis 1946] [Katz 1947] [Mastrocola 1947] [Mathews and Thompson 1947] [Thompson and Mathews 1947] [Thompson and Kurbjun 1948] [Hart and Katz 1949] [Katz 1949] [Jackson et al 1950] [Purser et al 1950] [Thompson 1950] [Adams 1951] [Cohen 1951] [Cortright and Schroeder 1951] [Friedman 1951] [Lopatoff 1951] [Spahr and Dickey 1951] [Welsh and Demoraes 1951] [Kawamura 1952] [Kurbjun and Thompson 1952] [Love et al 1952] [Sommer and Stark 1952] [Benedikt 1953] [Bromm and Goodwin 1953] [Jack 1953] [James and Carros 1953] [Loposer and Mottard 1953] [Hopko 1954] [Jackson et al 1954] [Mottard and Loposer 1954] [Howell and Braslow 1955] [Nelson and Henry 1955] [Parker 1955] [Bromm and Goodwin 1956] [Falanga 1956] [Falanga and Leiss 1956] [Harder and Rennemann 1956] [Parker 1956] [Eggers et al 1957] [Gillespie 1957] [Kawamura and Karashima 1957] [Lomax 1957] [Foss et al 1958] [Nelson and Scott 1958] [Perkins et al 1958] [Effect of uniformly distributed 1959] [Gillespie 1960] [McKinney 1960] [Morris 1961] [Brazzel 1963] [Miele and Hull 1963] [Tetervin 1963] [Sims and Hahn 1964] [Connolly 1965] [Eggers 1965] [Vasil’ev 1966] [Crowe 1967] [Grodzovskii 1968] [Trimmer 1968] [Huerta 1969] [Jackson and Smith 1969] [Tkalenko 1969] [Addy 1970] [Martin and Brazzel 1970] [Street 1970] [Rubin 1971] [Usry and Wallace 1971] [Berrier 1972] [Reubush 1973] [Reubush and Runckel 1973] [Head 1974] [Hess and James 1975] [Micci 1975] [Jones and Margolis 1976] [Korkegi and Freeman 1976] [Mikhail 1979] [Putnam 1979] [Bauer 1980] [Lijewski 1980] [Payne 1980] [Payne et al 1980] [Plant et al 1980] [Lijewski 1981] [Peters 1981] [Quass et al 1981] [Schiff and Sturek 1981] [Vasil’chenko 1981] [Kapoor 1982] [Lijewski 1982] [Miller 1982] [Gloss and Sewall 1983] [Howard et al 1983] [Howard and Goodman 1984] [Howard and Goodman 1985] [Larina 1985] [Nielsen 1985] [McMillin and Wood 1986] [Osawa and Hewitt 1986] [McMillin and Wood 1987] [Newcomb 1988] [Goradia et al 1989] [Moore et al 1993] [Roshko 1993] [Carlson 1996] [ESDU Data Item estimates 1998] [Riggins et al 1998] [Candler and Kelley 1999] [Malone and Peavey 1999] [Riggins et al 1999] [Glagolev et al 2000] [Jones et al 2000] [Guy et al 2001] [Whitmore et al 2001] [Garanin et al 2002] [Shang 2002] [Eremenko et al 2003] [Rallabhandi and Mavris 2003] [Durgesh et al 2004] [Myrabo et al 2004] [Nikolic and Jumper 2004] [Palaniappan and Jameson 2004] [Sawada et al 2004] [Knight and Tso 2005] [Aul’chenko 2006] [Cummings et al 2006] [Florendo et al 2006] [Venukumar et al 2006] [Florendo et al 2007] [Satheesh and Jagadeesh 2007] [Mahapatra et al 2008] [Schuelein 2008] [Sasoh et al 2009] [Satheesh and Jagadeesh 2009] [Schülein 2009] [Srinath and Reddy 2010] [Aruna and Devi 2012] [Dahan et al 2012] [Mironov and Serdyuk 2012] [Sahai et al 2014] [Choi et al 2015] [Schuelein 2015] [Wilder et al 2015] [Zishka and Agarwal 2015] [Ageev and Pavlenko 2016] [Huang et al 2016] [Huang et al 2017] [Magier and Merda 2017] [Seager and Agarwal 2017] [Wan and Cm 2017] [Yadav et al 2018] [Zhang et al 2018] [Alam and Pant 2019] [Gardner and Agarwal 2019] [Han et al 2019] [Zhang et al 2019] [Huang and Yao 2020] [Vozhdaev and Teperin 2020] [Goto et al 2021] [Senthilkumar et al 2021] [Tekure et al 2021] [Wang et al 2021] [Mazurov and Takovitskii 2022] [Mokin et al 2022] [Stack 2022] [Easwer et al 2023] [Pokela et al 2023] [Rao et al 2023] [Sahbon et al 2023] [Wang et al 2023] [Balusamy et al 2024] [Kanwar 2024] [Ni and Fang 2024] [Ni et al 2024] [Popkov and Kornilov 2024] [Sanchez-Muñoz et al 2024] [Sarwar et al 2024] [Scwartz et al 2024] [Camacho-Sánchez et al 2025] [He et al 2025] [Rafi and Al-Faruk 2025] [Sarwar et al 2025] [Camacho-Sánchez et al 2026] [Plessis 2026] [He et al 2026] [Kim 2026] [Yavuz and Cihan 2026] [Zeidan]
The ascent, and where the dynamic pressure goes
Ascent trajectory design and optimisation, dynamic pressure and the standard atmosphere, with the flight dynamics of sounding rockets and their payloads.
99 records. [Air Force Test Pilot School Edwards Afb Ca 1962] [Stancil 1963] [Dembrow and Jamieson 1964] [Hillsley and Robbins 1964] [Stancil 1964] [Sterhardt 1965] [Nicolaides et al 1967] [Reis and Sundberg 1967] [Crutcher and Guttman 1969] [Pierman 1969] [Gabris et al 1970] [Demas and Kinsley 1971] [Grassl 1971] [Pincus et al 1971] [Wolff and Fitz 1972] [Bensimon 1973] [McGarvey 1973] [Needleman and Tackett 1973] [Edge and Powers 1974] [Edge and Powers 1976] [Matsumoto et al 1976] [Rochefort and Yorra 1976] [Charron et al 1978] [Morin 1978] [Andersson and Forsell 1979] [Chinn et al 1979] [Euler et al 1979] [McAnally and Engel 1979] [Glaab 1985] [Beyma 1986] [Flores 1986] [Ward 1986] [Williams 1986] [Well 1989] [Wessling and Maybee 1989] [Herrick et al 1990] [McKenna 1990] [Smith et al 1990] [Rochefort et al 1991] [Lyons 1993] [Balach 1995] [Lydon and Va 1995] [Sachs et al 1996] [Basciano 1998] [Olds and Budianto 1998] [Calise et al 2000] [Gath et al 2000] [Gowan 2005] [Starnone and Biesbroek 2006] [Fuhrmann and Dreyer 2008] [Berman et al 2009] [Stillwater 2009] [Zeeshan et al 2009] [Murillo and Lu 2010] [Lazzarin et al 2012] [Pescetelli et al 2012] [Lu and Wang 2013] [Shi et al 2013] [Shi et al 2013] [Qiuhong et al 2014] [Holt et al 2015] [Song and Su 2015] [Wang 2015] [Wu et al 2015] [Barbosa et al 2016] [Cheng et al 2017] [Hergert et al 2017] [Hergert et al 2017] [Li et al 2017] [Smith and Dutta 2017] [Wercinski et al 2017] [He et al 2018] [Karlgaard et al 2018] [Zhengxiang et al 2018] [Cassell et al 2019] [Cassell et al 2019] [Fu et al 2019] [Karlgaard et al 2019] [Mukundan et al 2019] [Rodi et al 2019] [Tynis et al 2019] [Dąbrowski et al 2020] [He et al 2020] [Zheng et al 2020] [Hu et al 2021] [Marchetti et al 2021] [Ascent Trajectory Analysis and 2022] [Deng et al 2022] [Lan and Li 2022] [Ma et al 2022] [Nair and Vaidyanathan 2022] [Gilson et al 2023] [Zhang et al 2023] [Siewert et al 2024] [Benedikter et al 2025] [Xue et al 2025] [Zhao et al 2025] [Lei et al 2026] [Murillo]
The nozzle this vehicle carries, which the previous article took
Aerospike, plug and other altitude-compensating nozzles, their base flow, wake and thrust vectoring, including plug nozzles studied on turbojets for their behaviour across pressure ratio.
255 records. [Salmi and Cortright 1956] [Salmi 1956] [Mercer and Salters 1963] [Lee 1966] [Berrier and Mercer 1967] [Bresnahan 1968] [Bresnahan and Johns 1968] [Harrington and Wasko 1968] [Berrier 1969] [Bresnahan 1969] [Cikanek et al 1969] [Huntley and Samanich 1969] [Johns 1969] [Berrier and Mercer 1970] [Burley and Samanich 1970] [Chenoweth and Jeracki 1970] [Clark et al 1970] [Harrington 1970] [Harrington 1970] [Steffen 1970] [Chambellan and Stepka 1971] [Chamberlin and Samanich 1971] [Chenoweth and Lieberman 1971] [Hall and Mueller 1971] [Bresnahan 1972] [Clark and Lieberman 1972] [Graber and Clark 1972] [Mueller and Sule 1972] [Muller et al 1972] [Samanich 1972] [Berrier 1973] [Mueller and Sule 1973] [Straight and Harrington 1973] [Sule and Mueller 1973] [Harrington and Schloemer 1974] [Harrington et al 1974] [Huang 1974] [Giel and Mueller 1975] [Harrington et al 1975] [Galanga and Mueller 1976] [Harrington et al 1976] [Nosek and Straight 1976] [Diem and Kirby 1977] [Kirby and Martinez 1977] [Staid 1978] [Mercer and Burley 1985] [Murthy and Sheu 1988] [Aukerman 1991] [Cler et al 1993] [Quentmeyer and Roncace 1993] [Dunn and Coats 1996] [Moes et al 1996] [Fick et al 1997] [Korte et al 1997] [Korte et al 1997] [Ruf et al 1997] [Bouslog et al 1998] [Cobleigh 1998] [Mackall et al 1998] [Dukeman and Gallaher 1998] [Hall et al 1998] [Jackson et al 1998] [Kumakawa et al 1998] [Mackall et al 1998] [Mizukami et al 1998] [Moes et al 1998] [Simpson 1998] [Corda et al 1998] [Vinson 1998] [Wang 1998] [Wang 1998] [Aguilar 1999] [Austin and Rising 1999] [Barret 1999] [Crowley 1999] [Ennix et al 1999] [Hall and Panossian 1999] [Hass et al 1999] [Larson 1999] [Liou 1999] [Littlefield and Melton 1999] [Murphy et al 1999] [Murphy 1999] [Prabhu 1999] [Reichenfeld and Jones 1999] [Sakamoto et al 1999] [Tomita et al 1999] [Whitmore and Moes 1999] [X-33/RLV Program Aerospike Engines 1999] [Austin and Rising 2000] [Elam 2000] [Korte 2000] [Littlefield and Melton 2000] [Meyer 2000] [Miller 2000] [Shtessel and Hall 2000] [Soni 2000] [Zhao et al 2000] [Zhu et al 2000] [DAgostino et al 2001] [Ito and Fujii 2001] [Korte et al 2001] [Liu et al 2001] [Rieckhoff et al 2001] [Schweikhard et al 2001] [Tomita et al 2001] [Wang et al 2001] [Wuye et al 2001] [Ito and Fujii 2002] [Ruf and McDaniels 2002] [Ito and Fujii 2003] [Ito and Fujii 2003] [Ruf et al 2003] [Ruf and McDaniels 2003] [Wang et al 2003] [Wang et al 2003] [Rogers 2004] [Wang et al 2004] [Aso and Sugimoto 2005] [Bui et al 2005] [Ruf and McDaniels 2005] [Tsukada et al 2005] [Tsutsumi et al 2005] [Taniguchi et al 2006] [Wang et al 2006] [Tsutsumi et al 2007] [Wang et al 2007] [Zilic et al 2007] [Tomita et al 2008] [Verma 2008] [Karthikeyan et al 2009] [Verma 2009] [Wang et al 2009] [Wilson et al 2009] [Dennis et al 2010] [Eilers et al 2010] [Ladeinde and Chen 2010] [Shark et al 2010] [Eilers et al 2011] [Grieb et al 2011] [Hall et al 2011] [Noori and Shahrokhi 2011] [Simmons and Branam 2011] [Eilers et al 2012] [Narimiya et al 2012] [Rajesh et al 2012] [Design and FLOW Simulation 2014] [Donbosco and Kumar 2014] [Peugeot et al 2014] [Shibao et al 2014] [Bogoi et al 2015] [He et al 2015] [Heath et al 2015] [Lash and Moeller 2015] [Karthikeyan et al 2016] [Bach 2017] [Chaudhari 2017] [Kumar et al 2017] [Mason-Smith 2017] [Nair et al 2017] [Performance Optimization of Aerospike 2017] [Purohit and Mathpal 2017] [Reza and Arora 2017] [Lai et al 2018] [Masdari et al 2018] [Schnabel and Brophy 2018] [Schwer et al 2018] [Tsuboi et al 2018] [Fotia et al 2019] [Ha et al 2019] [Harroun et al 2019] [Jourdaine et al 2019] [Sieder et al 2019] [Ferlauto et al 2020] [Ferlauto et al 2020] [Heath and Bell 2020] [Kurita et al 2020] [Shenoy et al 2020] [Stephen A 2020] [Stewart et al 2020] [Tian et al 2020] [Udaiyakumar et al 2020] [Sai Shankaran et al 2021] [Balaji et al 2021] [Dakka and Dennison 2021] [Et. al. 2021] [Ghosh and Gunasekaran 2021] [Huang et al 2021] [Kumar Mishra et al 2021] [Lengade 2021] [Naseh and Alipoor 2021] [Sankari Ashok Alshiya et al 2021] [Sequeira and Sanjay 2021] [Ha and Kim 2022] [Jain and Kumar 2022] [Liu et al 2022] [Sriganapathy et al 2022] [Aithani et al 2023] [Di Cicca et al 2023] [Zaman and Fagan 2023] [Ma et al 2023] [Nagaral et al 2023] [Pyle et al 2023] [Sundaria et al 2023] [Sam Daniel Fenny et al 2024] [Bayir and Akbıyık 2024] [Bindal et al 2024] [Bindal et al 2024] [Di Cicca et al 2024] [Golliard and Mihaescu 2024] [Golliard and Mihaescu 2024] [Zaman et al 2024] [Langner et al 2024] [Paramo and Arizpe 2024] [Peshkov and Tret’yakov 2024] [Resta et al 2024] [Reza et al 2024] [Silva and Brójo 2024] [Strobel and Macneil 2024] [Swathish and Rakesh 2024] [Golliard and Mihaescu 2025] [Golliard and Mihaescu 2025] [Hu 2025] [Inturi et al 2025] [Zaman et al 2025] [Liu et al 2025] [Selvam et al 2025] [Wang et al 2025] [Alam et al 2026] [Aslan et al 2026] [Bakker et al 2026] [Golliard and Mihaescu 2026] [Karim et al 2026] [Liu et al 2026] [Liu et al 2026] [Lizcano et al 2026] [Magnani et al 2026] [Pyle and Jacobs 2026] [Pyle and Jacobs 2026] [Sen Ayush et al 2026] [Steiner and Bauer 2026] [Sun et al 2026] [Wißmann et al 2026] [Aerospike engine] [Beebe] [Brennen] [Brock] [Case] [Grieb] [Imbaratto] [Korth] [Zaman et al] [Zaman et al] [Shahrokhi and Noori] [Stewart]
Segmented and modular vehicles, which is a subcontractor’s trade
Segmented and modular launch vehicles and engines, the architecture the award record of KT Engineering describes.
9 records. [Carpenter and Jeffus 1962] [Carpenter and Jeffus 1963] [Rice and Birchenough 1982] [Binder and Felder 1993] [Tarrant and Crook 1996] [Tarrant and Crook 1997] [Crook and Tarrant 1998] [Szalkowski et al 2024] [Tarrant and Crook]
This vehicle, this team, and this programme’s lineage
The programme’s lineage and the lead contractor’s own reports, being an uncertainty evaluation for the Integrated Powerhead Demonstrator and the report-server records of Invocon’s instrumentation and impact-detection work.
5 records. [Heermann, Doug and Krug, Eric 2004] [Steele et al 2005] [Brandon, Erik et al 2009] [Prosser and Madaras 2010] [Krug et al 2013]
Monitoring a structure and detecting what hits it, the lead’s own trade
Structural health monitoring, impact and hypervelocity impact detection and the radiation environment of spacecraft, which is the lead contractor’s trade as its award record describes it.
157 records. [Teng 1970] [Weller 1982] [Lamb 1987] [Rogowski 1990] [Ricles 1991] [Chang 1998] [Chang 2000] [Masri 2000] [Balageas 2002] [Chang 2002] [Giurgiutiu 2003] [Prosser and Percy 2003] [Chang 2004] [Decker 2004] [Engberg and Ooi 2004] [Giurgiutiu and Lin 2004] [Prosser et al 2004] [Blackshire et al 2005] [Coppotelli et al 2005] [Engberg 2005] [Giurgiutiu and Zagrai 2005] [Lopes and Silva 2005] [Park and Inman 2005] [Raghavan and Cesnik 2005] [Chattopadhyay 2006] [Chronister and Palazotto 2006] [Udd 2006] [Allison et al 2007] [Kulkarni and Achenbach 2008] [Spurný et al 2008] [Underwood et al 2008] [Cesnik 2009] [Choi and Sweetman 2009] [Kochergin et al 2009] [Kral et al 2009] [Liu 2009] [de Leeuw and Brennan 2009] [Chiu 2010] [Chiu et al 2010] [Diamanti and Soutis 2010] [Dragan 2010] [Esterline et al 2010] [Giurgiutiu and Soutis 2010] [Grandhi and Tobe 2010] [Haftka et al 2010] [Park et al 2010] [Staszewski and Sohn 2010] [Whitlow et al 2010] [Worden and Inman 2010] [Yap 2010] [Chang et al 2011] [Lindgren et al 2011] [Russell et al 2011] [Sankararaman et al 2011] [Structural Health Monitoring OrientedModelling 2011] [Yap et al 2011] [Zagrai et al 2011] [Bentham Science Publisher 2012] [Chattopadhyay et al 2012] [Ghoshal et al 2012] [Han et al 2012] [Michaels et al 2012] [Neerrukatti et al 2012] [Nondestructive inspection and structural 2012] [Seaver et al 2012] [Zein-Sabatto et al 2012] [Foote 2013] [Giglio et al 2013] [Richards et al 2013] [Smart Technical Textiles for 2013] [Veidt and Liew 2013] [Williams et al 2013] [Richards et al 2014] [Trends on research in 2014] [Borkowski 2015] [Giurgiutiu 2015] [Golato et al 2015] [Iglesias et al 2015] [Le and Yu 2015] [Ravi et al 2015] [Ryu et al 2015] [Cramer 2016] [Derriso et al 2016] [Giurgiutiu 2016] [Giurgiutiu 2016] [Henderson et al 2016] [Jha et al 2016] [Kranz et al 2016] [Okabe and Wu 2016] [Sihver et al 2016] [Structural Health Monitoring SHM 2016] [Structural Health Monitoring of 2016] [Yu and Tian 2016] [Simone et al 2017] [Moix-Bonet et al 2017] [Rébillat et al 2017] [Towler and Ryu 2017] [Cawley 2018] [Dong and Kim 2018] [Harris et al 2018] [Schubert Kabban et al 2018] [Hadjria and D’Almeida 2019] [Lewis et al 2019] [Rahul et al 2019] [Structural health monitoring SHM 2019] [Wan and Ni 2019] [Bao and Li 2020] [Barthorpe and Worden 2020] [Giurgiutiu 2020] [Lee 2020] [Structural Health Monitoring Damage 2021] [Broer et al 2022] [Gardner et al 2022] [Giurgiutiu 2022] [Yu et al 2022] [Data-Centric Structural Health Monitoring 2023] [Franz and Hassan 2023] [Gordan et al 2023] [Hassani and Dackermann 2023] [Sharif Khodaei and Aliabadi 2023] [Structural health monitoring SHM 2023] [Tiachacht et al 2023] [He and Yuan 2024] [Kawai and Hasegawa 2024] [Next Generation Structural Health 2024] [Structural Health Monitoring/management SHM 2024] [Xu et al 2024] [Batista et al 2025] [Chabukswar et al 2025] [Thomas et al 2025] [Deng et al 2025] [Khalid et al 2025] [Krishnamoorthy and Marius 2025] [Monaco et al 2025] [Pan and Bao 2025] [Park et al 2025] [Sao and Garain 2025] [Scarselli and Nicassio 2025] [Structural Health Monitoring 2025] [Structural health monitoring SHM 2025] [Viscardi et al 2025] [Chehrzad and Khoramishad 2026] [Prosser] [Bishop] [DeSimio et al] [Thomas et al] [Guidance for Assessing the] [Guidelines for Implementation of] [Landing Gear Structural Health] [Nerlikar] [Pant] [Perspectives on Integrating Structural] [Prognostic methodologies for remaining] [Ranganatha] [Sharma] [Structural Health Monitoring Considerations] [Zhang and Pang]
Steering it, which is the third partner’s trade
Thrust vector control, attitude control and guidance of rockets, launch vehicles and hypersonic vehicles, which is the third partner’s trade.
72 records. [Robbins and Zebrowski 1966] [Alston et al 1967] [Hansen et al 1967] [Rasmussen et al 1967] [Hanford 1969] [Moore and Tcheng 1969] [Penchuk and Schlundt 1969] [Abbott and Walker 1970] [Jenkins 1970] [Ellis and Kearney 1982] [Olney and Shiftlett 1982] [Carroll and Cox 1983] [Leitner 1986] [Maughmer et al 1990] [Hattis and Malchow 1991] [Maughmer et al 1991] [Gregory et al 1992] [Hattis and Malchow 1992] [Murray 1992] [Gregory et al 1993] [Maughmer et al 1993] [Schmidt 1993] [Problems in control system 1994] [Ishimoto et al 1996] [Lazur et al 1999] [David O. Sigthorsson 2006] [Dyakonov et al 2009] [Falkiewicz et al 2009] [Rehman et al 2009] [Vogel et al 2009] [Falkiewicz et al 2010] [Lee et al 2010] [Shuping Tan and Zhibin Li 2010] [Jang et al 2011] [Shi-guo et al 2011] [Wan et al 2012] [Wang et al 2012] [Lian et al 2013] [Qian et al 2013] [Hong et al 2014] [Rubio Hervas and Reyhanoglu 2014] [Fan et al 2015] [Hu et al 2015] [Wei and Chen 2015] [Feng Li et al 2016] [Lu and Zhou 2017] [Qi and Jianliang 2017] [Song et al 2018] [Ying et al 2018] [Zhao et al 2018] [Zhao et al 2018] [Bao et al 2019] [Buddhavarapu et al 2019] [Song and Bian 2019] [Wang et al 2019] [Zhang et al 2019] [Fenfen et al 2020] [Averyanov et al 2021] [Stokes and Lombaerts 2023] [Zhuo et al 2023] [Stepp 2024] [Santos and Oliveira 2024] [Wang et al 2024] [Wang et al 2024] [Djanal-Mann and Murugan 2025] [Ferreira de Moura and Borges Ribeiro 2025] [Raharema et al 2026] [Reynolds et al 2026] [Stuart and McEnulty] [Du] [Orr et al] [Wall et al]
Testing an engine on the ground, and what the test stand measures
Ground testing of rocket engines and motors and the measurements a test stand makes, which is the ground half of the measurement this vehicle was to make in flight.
112 records. [Gompertz 1950] [Kamperman 1957] [Keast 1960] [Glatt 1961] [Keast 1961] [Gale and Moedt 1962] [Spherical rocket motor static-test 1962] [Elston 1963] [Harris 1963] [Lee 1964] [Strauss 1964] [Tedrick 1964] [Holdhusen and Perusse 1965] [Jones et al 1965] [Raper 1965] [Craig 1966] [Clark and Tenenbaum 1967] [Clayton et al 1967] [Fairall 1967] [Kunz 1967] [Crowe et al 1968] [Rafferty 1968] [Rogero 1969] [Franklin and Tinsley 1970] [Babcock and Coe 1971] [Study of solid rocket 1972] [Fuller 1973] [Mathes 1980] [Allen 1981] [Bergman et al 1981] [Gunter and Flack 1981] [Millard et al 1982] [Allen 1983] [Korting and Reitsma 1985] [Welsh 1985] [External autoignition test for 1986] [Praharaj and Palko 1986] [Siddiqui and Smith 1988] [Vibbart 1988] [Collamore 1989] [Dunn 1989] [Estler 1989] [Ahmed 1990] [Bickford and Madzsar 1990] [Dunn 1990] [Glozman and Brillhart 1990] [Martinez et al 1990] [Zakrajsek 1991] [Ballard 1992] [Gaddis et al 1992] [Tran et al 1992] [Delcher et al 1993] [Gould 1993] [Heat transfer measurements and 1994] [Liu 1994] [Madzsar et al 1994] [Trevino 1994] [Brown et al 1995] [Anderson et al 1996] [Palaszewski 1997] [Koschel 1998] [Palaszewski 1998] [Genge and Marsh 1999] [Brown 2000] [Ryan et al 2000] [Wehrmeyer et al 2000] [Brown and Brunty 2001] [Osborne et al 2001] [Wehrmeyer et al 2001] [Chelner 2002] [Hyde 2002] [Wehrmeyer 2002] [Thornburg 2002] [Vorozhtsov and Matvienko 2002] [Weidner et al 2002] [Beck and Beach 2003] [Chelner 2003] [Christenson et al 2003] [Huppi et al 2003] [Hill and Acosta 2005] [Tiwari et al 2005] [Hughes et al 2007] [Kenny et al 2009] [Butt et al 2010] [El-Aini et al 2010] [Kitsche 2010] [Maynard and Raines 2010] [Vargas and Kenny 2010] [Johnson and Crawford 2011] [Lancelle et al 2012] [Dorosh and Leontiev 2014] [Jones et al 2014] [Starkey et al 2014] [Qu and Yang 2015] [Gradl 2016] [Gradl 2016] [Gradl 2016] [Gradl and Schmidt 2016] [Wernet and Stiegemeier 2017] [Brown et al 2018] [Fernandez et al 2018] [Zhang et al 2018] [Piotr et al 2019] [Podolchak 2019] [Richardson et al 2020] [Kato et al 2021] [D. et al 2022] [Chandler 2023] [Gilbert Chandra et al 2023] [Traudt 2024] [Fang et al 2025] [Space systems � Measured]
Combustion and combustion instability in the chamber
Combustion, injection, ignition and combustion instability in rocket chambers, the phenomenon that sets the sample rate in Highly Instrumented, in Bits Per Second.
70 records. [Princeton Univ Nj 1952] [Grey 1953] [Grey 1954] [Matthews 1957] [Harrje 1959] [Jensen et al 1962] [Aerojet-General Corp Sacramento Ca 1963] [Hornstein 1965] [Wong and Brown 1965] [Tobey and Bastress 1966] [Williams 1968] [Hardesty 1970] [Marchese et al 1972] [Calhoon et al 1973] [Marchese 1974] [Rakowsky and Marchese 1974] [Caveny et al 1980] [Cox 1988] [Bentsman et al 1990] [Han 1990] [Han 1991] [Anderson et al 1992] [Giridharan et al 1992] [Luke and Dwyer 1992] [Anderson et al 1993] [Jenkins and Foster 1993] [Kuo et al 1993] [Schley 1994] [Combustion Instability Analysis Numerical 1995] [Instability Phenomenology and Case 1995] [Instability Phenomenology and Case 1995] [Instability Phenomenology and Case 1995] [Instability Phenomenology and Case 1995] [Liquid Rocket Engine Combustion 1995] [Blomshield and Bicker 1996] [Farmer et al 1999] [Knuth et al 1999] [Farmer et al 2001] [Nesman and Turner 2002] [Onodera et al 2003] [Sims et al 2004] [Williams 2004] [Yang 2004] [Moore et al 2007] [Smith et al 2007] [Dombrovsky 2008] [Heister 2008] [Hulka 2008] [Shimizu et al 2008] [Kenny et al 2011] [Anderson et al 2012] [Anderson et al 2013] [Bennewitz et al 2013] [Fischbach 2014] [Harvazinski et al 2014] [Westra and West 2014] [Anderson et al 2015] [Coogan 2015] [Numerical Method and Simulations 2016] [Kang et al 2018] [Novozhilov et al 2018] [Xiong 2018] [Tinker 2021] [Zapata et al 2023] [Jin et al 2024] [Shyam Raj et al 2024] [Zhao et al 2024] [Hyde and Argueta 2025] [Shyam Raj et al 2025] [Drobyshev 2026]
Health monitoring, diagnosis and control of rocket engines
Health monitoring, fault diagnosis, reliability and control of rocket engines.
41 records. [Luton 1963] [Milleman 1967] [Neiland 1967] [Markowsky and McManus 1974] [McCoy and Hester 1985] [Ali and Crawford 1988] [Cosens and Newton 1988] [Powers et al 1988] [Russell et al 1988] [Hagar and Alcock 1989] [Tulpule 1989] [Bickford et al 1990] [Gage and Dehoff 1991] [Lorenzo and Musgrave 1991] [Raines et al 1991] [Design of Rocket-Engine Control 1992] [Madzsar et al 1992] [Merrill et al 1992] [Rocket engine propulsion ‘system 1992] [Gulati et al 1993] [Huzel 1993] [Maram 1993] [Hardy and Rapp 1994] [Goracke et al 1997] [Shen and Sharpe 1998] [Pytanowski 1999] [Santi 2001] [Tejwani et al 2003] [Rusick 2007] [Quing et al 2011] [Cortopassi et al 2012] [Zhang 2016] [Du 2017] [McCutcheon 2017] [Al Hassan and Britton 2018] [Tudor et al 2021] [Zhang et al 2022] [Lin et al 2023] [Lin] [Liquid Rocket Engine Reliability] [Sarotte]
Turbopumps, feed systems and engine transients
Turbopumps, valves, feed systems and the transients of pump-fed engines.
28 records. [Krebs and Hart 1959] [Crabtree 1962] [Tellier 1964] [Seidel 1965] [Brunner 1966] [Hydrostatic bearings for cryogenic 1969] [Fries 1973] [Salinas and Ball 1973] [Cox 1987] [Hamed 1990] [Benjamin and McConnaughey 1991] [Benjamin et al 1993] [Binder 1993] [Ratcliff et al 1993] [Sander and Leahy 1993] [Binder 1995] [Kiris and Williams 2000] [Kiris et al 2001] [Kris and Kwak 2001] [Kiris et al 2002] [Majumdar and Flachbart 2003] [Yamanishi et al 2004] [Rocket Engine Pump Feed 2009] [Bandyopadhyay et al 2011] [Gloger et al 2021] [Huang et al 2024] [Huang et al 2024] [Yu et al 2026]
Conventional nozzles, plumes and base heating
Nozzles, plumes, the base flow of clustered engines and base heating, for the conventional engines and motors against which the aerospike is measured.
46 records. [Burdett 1946] [Rose 1958] [Slocumb and Andrews 1961] [Lee 1963] [Hurt and Lina 1964] [Becker and Hamilton 1965] [Brinich et al 1965] [Becker and Tang 1966] [Crocker and Potter 1966] [Hendershot 1966] [Hopson and McAnelly 1966] [McGee and Say 1966] [Hosack and Stromsta 1969] [Liquid rocket engine nozzles 1976] [Smith 1982] [Yerushalmi and Glick 1982] [Smith 1983] [Forester and Strom 1986] [Pavli et al 1986] [Davidian 1987] [Pavli et al 1987] [Sepcenko et al 1990] [Dougherty and Liu 1991] [Lui and Mason 1991] [Keyhani 1993] [Tucker and Warsi 1993] [Sambamurthi 1995] [Follett 1996] [Follett et al 1996] [Voinov and Mel’nikov 1996] [Haidinger et al 1997] [Santoro and Pal 1999] [Candler 2001] [Rocket Engine Nozzle Concepts 2004] [Wang 2004] [Barkhoudarian and Kittinger 2006] [Smalley et al 2007] [Nallasamy et al 2010] [Mehta et al 2011] [Baars et al 2012] [Ali and Borrer 2013] [Ali and Borrer 2013] [Matveev et al 2018] [Rocket Nozzle Performance 2019] [Wang et al 2025] [Kolli]
Solid rocket motors
Solid rocket motors, their ballistics, performance, testing and flight.
51 records. [Philipchuk 1953] [Levine et al 1960] [Hermance 1961] [Bujes 1962] [Nagy 1965] [Sprattling 1967] [Odom 1972] [Study of solid rocket 1972] [Study of solid rocket 1972] [Study of solid rocket 1972] [Study of solid rocket 1972] [Study of solid rocket 1972] [Study of solid rocket 1972] [Study of solid rocket 1972] [Technical report analysis and 1972] [Vonderesch 1972] [Merryman and Smith 1974] [Sforzini and Foster 1976] [Vetter 1977] [Kosmann et al 1978] [Advisory Group for Aerospace Research and Development 1979] [Francis and Thompson 1980] [Mullen and Kearnes 1980] [Foster et al 1981] [Martin 1983] [Block 2 Solid Rocket 1986] [Foster et al 1986] [Rubin et al 1988] [Bryant 1989] [Dynamic real-time radiography of 1989] [Jones and Zoller 1989] [Jones and Zoller 1989] [Salita 1989] [Lemberger et al 1991] [Babbitt 1992] [Chaouat and Vuillot 1992] [Ghaffarian et al 1992] [Arnett 1993] [Casey 1995] [McGrath 1996] [Kassoy 1997] [Umholtz 1999] [Laubacher 2000] [Rahaim et al 2000] [Stewart et al 2005] [Graham 2010] [Albanese et al 2012] [Stefanski 2015] [Solid Rocket Motors 2019] [Sudiana 2020] [Kibbey 2024]
Hybrid rocket engines
Hybrid rocket engines and the launch vehicles designed around them.
33 records. [Hybrid Rocket Propulsion for 1991] [Grubelich et al 1993] [Hydrogen peroxide hybrid rocket 1994] [Kuo et al 1994] [Jenkins 1997] [Palaszewski et al 1998] [Rice et al 1998] [Hellman 2006] [Jacob 2008] [Rhee et al 2008] [Chiba et al 2013] [Chiba et al 2014] [Chiba et al 2014] [Yoda et al 2015] [Chiba et al 2016] [Kanazaki et al 2016] [Zhu et al 2017] [Aso and Tani 2018] [Chen and Wu 2018] [Greatrix 2018] [Chen 2019] [Hybrid Rocket Engines 2019] [Tarifa and Pizzuti 2019] [Gieras and Gorgeri 2021] [Trumpour 2021] [Velliaris 2021] [Patil 2022] [Srivastava and Thakur 2022] [Charan and Tibrewal 2025] [Guenther et al 2026] [Pallela et al 2026] [Glaser] [Grondin]
Detonation rocket engines
Rotating and pulse detonation rocket engines.
33 records. [Michigan Univ Ann Arbor 1963] [Mueller et al 1999] [Morris 2001] [Morris 2002] [Talley 2002] [Meyers et al 2003] [Morris 2004] [Li et al 2011] [Bai and Weng 2014] [Takagi et al 2014] [Wang et al 2015] [Zhang et al 2019] [Harroun et al 2020] [Kawasaki et al 2020] [Paulson et al 2020] [Burr and Paulson 2021] [Paxson and Perkins 2021] [Abdala et al 2023] [Malik et al 2023] [Jha 2024] [Mundt et al 2024] [Maybee 2025] [Lee 2026] [Noland et al 2026] [Paxson et al] [Johnson et al] [Perkins] [Hernandez-McCloskey et al] [Cooper et al] [Maybee et al] [Cooper] [Teasley et al] [Teasley et al]
The Space Launch System, a launch vehicle documented in development
The ascent and liftoff aerodynamics, loads, base heating and control of the Space Launch System, a launch vehicle documented through its development in NASA’s report series.
70 of its 71 records are report primaries, because a vehicle in NASA development is documented in NASA’s own series.
71 records. [May and Creech 2012] [Foster et al 2014] [Mehta et al 2014] [Pritchett et al 2014] [Wall et al 2014] [Alter et al 2015] [Alter et al 2015] [Dalle and Rogers 2015] [Eberhart et al 2015] [Piatak et al 2015] [Pinier et al 2015] [Rogers et al 2015] [Sekula et al 2015] [VanZwieten et al 2015] [Wall et al 2015] [Dalle et al 2016] [Favaregh et al 2016] [Hemsch 2016] [Herron et al 2016] [Honeycutt and Lyles 2016] [Johnson 2016] [Lohrer and Wright 2016] [Mehta et al 2016] [Piatak et al 2016] [Priskos 2016] [Stahl et al 2016] [Askins and Robinson 2017] [Cook and Lyles 2017] [Statham and Thompson 2017] [Vanzwieten et al 2017] [Shea et al 2018] [Chan et al 2019] [Garbeff 2019] [Statham et al 2019] [Krist et al 2019] [Akers and Sills 2020] [Ratnayake et al 2020] [Roozeboom et al 2020] [Soranna et al 2023] [Pomeroy et al 2024] [Deere et al 2024] [Hays et al 2024] [Craig et al] [Craig et al] [Burger et al] [Mobley and Summers] [Pomeroy et al] [Richardson] [Chan et al] [Soranna et al] [Soranna et al] [Ramey et al] [Meeroff et al] [Pinier] [Chen et al] [Griggs et al] [Mehta et al] [Mehta et al] [Hays et al] [Lee et al] [Shea et al] [Heaney et al] [Heaney et al] [Nikoueeyan et al] [Ali et al] [Ali et al] [Moore et al] [Hahn et al] [Wignall] [Wignall et al] [Wignall et al]
Launch-vehicle design, cost and staging
Launch-vehicle design, staging, performance and cost estimation, the literature to which the modular architecture’s targets in time and money belong.
67 records. [Hauer et al 1963] [Launch Vehicle Performance 1963] [Mueller 1964] [Friedland 1965] [Heuston et al 1967] [Maynard 1969] [Ward 1970] [Larson 1973] [Toelle et al 1973] [Hudson et al 1981] [Koelle 1981] [Nakanishi et al 1982] [Koelle 1984] [Meisl 1986] [French 1987] [Davis 1988] [Meisl 1988] [Meisl 1989] [Puening 1990] [Toten et al 1991] [Koelle 1992] [Browning 1993] [Degelsmith et al 1993] [Rutledge 1993] [Ryan and Verderaime 1993] [Tierney 1993] [Williams 1996] [Lee et al 1997] [Koelle 1998] [Nix and Escher 1999] [Schoneman et al 2000] [Whitehead 2000] [Jensen 2001] [Mossman and Perkins 2001] [Conley et al 2003] [Pempie 2003] [Trefny 2003] [Bradford et al 2004] [Portz 2004] [Shelton et al 2005] [Kutter 2006] [Snoddy et al 2006] [Haberstroh et al 2008] [Hunley 2008] [Hunley 2008] [Suresh et al 2009] [Creech et al 2010] [French 2010] [Pinier 2011] [Threet et al 2012] [Long et al 2013] [Waters et al 2013] [Jones et al 2014] [Spurlock and Williams 2014] [Watson et al 2014] [Civek and Özgören 2017] [Zhang et al 2018] [Reynolds et al 2021] [Launch Vehicle Performance and 2022] [Launch Vehicle Systems and 2022] [Chen et al 2025] [Seo and Kim 2026] [Brown] [Brevault] [Dorairajan] [Launch Vehicle Systems] [Lowe]
Rocket engine design, performance and history
Chemical rocket engines treated as whole engines, their design, cycles, performance and histories, admitted when they fit none of the narrower propulsion clusters above.
113 records. [Berggren et al 1948] [Singelmann and Mueller 1948] [Bernstein et al 1949] [Summerfield 1960] [Scott 1963] [Fio Rito 1964] [Techniques for rocket engine 1965] [Breshears et al 1966] [Mueller 1966] [Carey 1967] [Friedman et al 1967] [Baker 1969] [Hosack 1969] [Campbell 1970] [Experimental investigation of combustor 1972] [Smith and Sforzini 1972] [Nurick and Hines 1973] [Sanchini and Kirby 1973] [Wagner and Waldman 1973] [Lucci and Hodson 1975] [Eldred and Gordon 1976] [Pouliquen 1978] [Bjorklund et al 1979] [Etters and Flurchick 1981] [Pollet 1983] [Hardgrove and Krieg 1984] [Gibson 1985] [Marsik and Morea 1985] [Marsik and Morea 1985] [Chiu 1987] [Manski and Martin 1988] [Naraghi and Armstrong 1988] [Difrancesco et al 1989] [Lin 1989] [Martin and Manski 1989] [Pieper and Muss 1989] [Chiu et al 1990] [Martin and Kramer 1990] [Melchior 1990] [Wang 1990] [Wang and Chen 1990] [Roncace 1991] [Meisl 1992] [Sakala and Raines 1992] [Niiya et al 1993] [Obrien 1993] [Williams 1993] [Fanciullo and Lacefield 1994] [Green 1994] [Jones 1994] [Lacefield and Sprow 1994] [Campbell and Riccio 1995] [Goertz 1995] [Gu and Liu 1995] [Lorenzo 1995] [Micklow 1996] [Binder et al 1997] [Jetevator for rocket engine 1998] [Mitra et al 1998] [Wu et al 1999] [Blair and DeGeorge 2000] [London et al 2000] [Majumdar et al 2000] [Stechman et al 2000] [Vaidyanathan et al 2000] [A Global Optimization Methodology 2001] [Nguyen and Turner 2001] [Shyy et al 2001] [Ventura and WErnimont 2001] [Nguyen 2002] [Trinh et al 2002] [Cheng 2003] [Gregory and Han 2003] [Cho et al 2004] [Nix and Staton 2004] [Jensen 2005] [Chung et al 2006] [France’s Liquid Propellant Rocket-Engine 2006] [Japan’s Liquid Propellant Rocket-Engine 2006] [Liquid Propellant Rocket-Engine Organizations 2006] [Rothmund 2006] [Longenecker and Clark 2007] [Modular Program for Conceptual 2007] [Alexander et al 2008] [Burrows 2008] [Hunley 2008] [Watanabe and Mikami 2010] [Wan et al 2013] [Brown 2014] [Emrich 2016] [Jung 2018] [Liquid Rocket Engine Thrust 2018] [Nardi Rezende 2018] [Liquid Rocket Engines 2019] [Rocket Propulsion Classification of 2019] [Yang and Naraghi 2020] [Basharina et al 2021] [Lederer 2021] [Rocket Propulsion 2022] [Zachary Muckler 2022] [Emrich 2023] [Gangeh et al 2023] [Ortega et al 2023] [Abbas 2024] [DiZinno et al 2024] [Atamanchuk 2025] [Borgna et al 2025] [Ganesan et al 2025] [Huang et al 2025] [Sá Gontijo et al 2025] [Sukachevskyi 2026] [Gibart] [The thermal rocket engine]
Where the Framing Breaks Down
The vehicle is two published numbers and this article has built a great deal on them. A height of about 12 metres and a diameter of about 2.4 metres, both approximate, both reported at second hand from a release this article could not retrieve. Every geometric conclusion here inherits that provenance, including the fineness ratio, the frontal-area comparison and the static-margin argument. If either figure is wrong by twenty percent the fineness ratio moves by at most a quarter, and the parent vehicle remains 2.35 times as slender in the worst case, so the ordering against the sibling vehicle survives, and no stronger robustness claim is warranted.
\[f = \frac{12.0}{2.4} = 5.00, \qquad \frac{1.2 \times 12.0}{2.4} = 6.00, \qquad \frac{12.0}{0.8 \times 2.4} = 6.25, \qquad \frac{0.8 \times 12.0}{2.4} = 4.00, \qquad \frac{12.0}{1.2 \times 2.4} = 4.17, \qquad \frac{14.67}{6.25} = 2.35\]The error budget is a structure and not a prediction. It sets out how the uncertainty in recovered thrust depends on the drag fraction and the drag-model accuracy. It cannot say what the drag fraction was, because that needs the vehicle’s mass, its drag coefficient and its thrust, none of which is public, and the budget is therefore presented as a function of the quantity it cannot evaluate.
The accelerometer identity assumes zero angle of attack and states as much where it is derived, but the error budget does not carry a term for the departure. A gravity turn flies at small angle of attack by design, so the omission is small and it is not zero.
The three errors are treated as independent and they are not. A mass bookkeeping error and a thrust error share a cause if the propellant flow measurement is the source of both, and an accelerometer’s scale-factor error is correlated across the whole flight rather than random in time. Root-sum-square is the right first move and the wrong last one, and a real uncertainty analysis for this experiment would carry a covariance matrix that this article does not have.
The sampling result assumes the sensors are equally spaced and that the field is a function of angle alone. An unequally spaced ring has different aliasing and generally better, and unequal spacing is a standard remedy for that reason. A field varying along the annulus and across it needs a two-dimensional treatment. The four-lobed claim also assumes the legs are the dominant azimuthal feature, and a real vehicle has umbilicals, seams and one more of whatever the plumbing requires.
The telemetry arithmetic uses plausible channel counts and a plausible link capacity, and neither is published for either ARISE vehicle. The rates and volumes scale linearly in the assumed channel count and sample rate, and the deficit scales inversely in the assumed link capacity. The conclusion that survives those assumptions is that the deficit exceeds one in every cell of the table but one, the smallest instrument set on the widest link, and reaches a factor of tens to hundreds on links of one megabit per second or less, while the specific factor does not survive them at all.
And the recovery inference is the weakest thing here. That a recoverable vehicle solves a bandwidth problem is arithmetic. That the bandwidth problem is why the vehicle is recoverable is a guess about intent, and the programme’s stated interest in reusability for its own sake is at least as good an explanation.
The Source Base
The Research Survey
The research survey holds 3,314 records drawn from NASA’s Technical Reports Server, the Defense Technical Information Center and the journal and conference literature indexed by Crossref, of which 1,499 are report primaries, being 45.2 percent, by the identifier rule set out under The Definition of Primary. The median publication year is 2003.
The Definition of Primary
Primacy is a property of the document, and this survey derives it from the identifier. A record is a report primary when its address is on the NASA Technical Reports Server or it carries a digital object identifier under the 10.21236 prefix, which Crossref registers for Defense Technical Information Center reports. The reports of the Linear Aerospike SR-71 Experiment, the uncertainty evaluation for the Integrated Powerhead Demonstrator and the lead contractor’s own instrumentation reports are report primaries under that rule. The rule does not reach the recommended practices on in-flight thrust determination, whose identifiers carry the 10.4271 prefix, although they are the field’s own statements of its method, so 45.2 percent is a lower bound on how much of the survey is first-hand.
Thin Subjects
Six subjects that this article’s derivations depend on have thin literatures. Each count below is the number of research records in this survey whose title carries at least one of the listed terms, with a hyphen read as a space. Where a row joins two groups of terms with the word together, a title must carry a term from each group.
| Subject | Title terms counted | Records |
|---|---|---|
| the tip-over stability of a vehicle standing on legs | tip over, tipover, overturn, overturning, topple, toppling | 0 |
| the static margin and the centre of pressure | static margin, center of pressure, centre of pressure | 2 |
| the peak of dynamic pressure on an ascent | maximum dynamic pressure, peak dynamic pressure, max q | 2 |
| thrust recovered from an accelerometer | accelerometer, accelerometers, specific force together with thrust | 0 |
| azimuthal modes of a nozzle, jet or chamber | azimuthal, circumferential together with nozzle, nozzles, jet, jets, combustor, chamber, duct | 5 |
| single-sample uncertainty analysis | single sample | 0 |
No title names the tip-over of a vehicle standing on legs. Landing legs or landing gear appear in 12 titles, on leg deployment, touchdown loads, landing dynamics, gear design and the health monitoring of aircraft gear, so the footprint condition of The Footprint Polygon rests on the geometry derived there rather than on a published treatment.
The static margin of a short rocket is a number the literature uses and rarely studies. 2 titles name a static margin or a centre of pressure, one of them a sounding rocket’s margin during launch, so the calibre cost computed under What a Short Vehicle Costs in Static Margin stands on two published lengths rather than on a survey of short vehicles.
The peak of dynamic pressure is a condition met in passing. 2 titles name the maximum of dynamic pressure, one on the winds in the region of that maximum and one on an abort flight test placed there, and neither states the condition for the peak that this article derives.
No title names thrust recovered from an accelerometer, and none names single-sample uncertainty analysis. The identity at the centre of this article and the single-flight character of a demonstrator are both treated here from first principles and from the methodology papers in the reference list.
Azimuthal modes are studied in jets, ducts and chambers, and not on a nozzle. 5 titles join an azimuthal or circumferential mode to a jet, a duct or a chamber, and none of them names a nozzle, so the aliasing result of Four Sensors Cannot See Four Legs has no published counterpart for an annular plug.
What the Survey Admits
A record is admitted only when a person reading its title finds it on this article’s subject, meaning that the title bears on a subject the argument above uses. Much of this subject’s vocabulary also names something else, and a record that shares only such a word is excluded. Sounding rocket also names the science those rockets carried, being X-ray, ultraviolet and solar telescopes, auroral and magnetospheric plasma probes and microgravity experiments, rather than the rockets themselves. Telemetry also names the tracking of animals and fish, the monitoring of patients, motor racing and the observability of software. Trajectory reconstruction also names the reconstruction of road vehicle journeys. Aliasing also names effects in images, optics and the retina, and azimuthal mode also names the modes of lasers, antennas and plasma waves. Transducer also names the multiport networks of microwave measurement. The plug nozzle is also a jet-noise suppressor for turbofans, and aerospike also names a database product and an ejector in desalination plants, while the drag-reducing nose spike that shares the name is kept with the drag literature. Thrust stands for electric thrusters, nuclear and other non-chemical engines, and the materials and fabrication of engine parts share the propulsion vocabulary and none of this article’s argument. Uncertainty propagation in measuring sciences far from flight, such as nuclear assay, photovoltaics, hydrology and hardness testing, and structural health monitoring of bridges and buildings, apply this article’s methods to subjects it does not treat. Correction, retraction and award notices and supplementary data files are excluded as well, because a survey counts research works.
Some records of doubtful relevance are kept. The payload subsystems of sounding rockets, such as their timers, attitude control and recovery hardware, are kept because they are parts of the vehicle. Plug nozzles studied on turbojets are kept because their thrust across a range of pressure ratios is the behaviour altitude compensation exploits. Microphone and antenna arrays that name no application are kept because the counting argument of The Limits of a Sensor Ring is the same for them, and telemetry titles that name a vehicle without saying which kind are kept because most are flight systems. A reader may weigh each accordingly.
The last cluster has its own admission rule. It admits a record whose title is on a chemical rocket engine as a whole, its design, cycle, performance or history, and that fits none of the narrower propulsion clusters before it. It holds 113 records, 3.4 percent of the survey.
Reviews of books, reference and definition entries and notices are not admitted, and a document deposited twice is admitted once. An authored encyclopedia article of more than two pages and a chapter of a textbook are research works and are kept.
Every research title has been read for relevance, so the off-topic share that remains is a matter of reading judgement. A reading of titles cannot show that a work is relevant in substance, so this is a statement about titles only.
The Homonyms of This Subject’s Vocabulary
The vocabulary of measurement belongs to every field that measures, and a search of the Crossref index for this subject’s terms shows how much of it other fields own. The collisions are worse than those of the sibling subject and worse in a different way. The X-63 article’s central nouns are owned by other fields, and this subject’s are owned by other languages and by other species.
fin returns Fin de siecle, fin d'un monde four times over, and fins returns Les fins de traitement and the fins of a fossil fish called Robustichthys. The word for this vehicle’s stabilising surfaces is owned by French and by ichthyology. static margin is owned by static noise margin in six-transistor static memory cells, so the aerodynamic stability metric is buried under a semiconductor one. tip-over stability returns wheeled and tracked mobile manipulators in every one of ten results. reusability is software reusability, program transformations and a set of reusability guidelines from 1990. recovery is heat-recovery ventilators and a recovery room. aliasing and Nyquist are dictionary headwords. transducer is ultrasonics and an interface standard of the Institute of Electrical and Electronics Engineers.
And the author of the method every model rocketeer uses returns nothing. Barrowman gives context-aware random numbers, a book on reading the short story, and gastrointestinal lymphatics twice, while the reports server holds the 1967 report under its title. A method that circulated as a report and a handbook rather than as a paper has no presence in an index of papers, and the absence reflects the index, not the method.
One term returns clean and on subject. landing legs returns leg design for propulsive rocket landing and landing simulations for reusable launch vehicles, both recent and both squarely on this article’s question. The live literature on this vehicle’s most distinctive feature is findable under its plainest name.
2 terms return nothing at all from that index, being Invocon and Troy7. The absence is a fact about the index and not about the team. NASA’s Technical Reports Server holds three reports that carry an Invocon affiliation, on a wireless vacuum sensor system [Krug et al 2013], a microminiature surface acoustic wave wireless instrumentation system [Heermann, Doug and Krug, Eric 2004] and damage detection in inflatable and deployable structures [Brandon, Erik et al 2009], and a NASA evaluation of the company’s distributed impact detector [Prosser and Madaras 2010]. All four are instrumentation and impact detection, and none concerns a launch vehicle, so the second archive tells the award record’s story of the company, and the programme itself still has to be read from an award record and a register.
What the Award Record Holds
The federal award reporting system holds nothing under the programme’s names. Searched across 5 families of award type, being procurement contracts, indefinite delivery vehicles, grants, direct payments and other financial assistance, X-64A, ARISE aerospike, Aerospike Rocket Integration and Suborbital Experiment and Affordable Responsive Modular Rocket each return nothing in any family. The previous article found the same silence from the other side of the same allocation, and the announcement explains it, because the instrument was an other transaction agreement and an other transaction does not appear where procurement contracts appear.
What did return is the three companies’ histories, and they divide the work, as the table and the account of the spellings Troy7 and Troy 7 under The Team in the Award Record set out.
Which Clusters Are Thin in Reports, and Why
The report-primary fraction is 45.2 percent overall and it runs from 15.2 to 100.0 percent across the clusters, a spread wide enough that a single figure would hide the interesting part. The test that explains such a spread is to ask which publisher holds each thin cluster, rather than to treat thinness as a shortfall in the record.
| Cluster | Records | Report primaries | Fraction | Median year | Largest single source |
|---|---|---|---|---|---|
| Hybrid rocket engines | 33 | 5 | 15.2 percent | 2016 | doi:2514, 9 |
| Launch-vehicle design and cost | 67 | 11 | 16.4 percent | 2000 | doi:2514, 38 |
| Recovery and reuse | 497 | 85 | 17.1 percent | 2005 | doi:2514, 194 |
| Sensor arrays and sampling | 82 | 17 | 20.7 percent | 2015 | reports server, 17 |
| Structural health monitoring | 157 | 41 | 26.1 percent | 2013 | defence registry, 22 |
| Steering and control | 72 | 20 | 27.8 percent | 2011 | doi:2514, 20 |
| Error budgets and uncertainty | 134 | 38 | 28.4 percent | 2015 | reports server, 35 |
| Ascent and trajectory | 99 | 30 | 30.3 percent | 2007 | doi:2514, 36 |
| Rocket engine design and history | 113 | 40 | 35.4 percent | 1996 | doi:2514, 44 |
| Fins and aerodynamic stability | 129 | 46 | 35.7 percent | 1997 | doi:2514, 56 |
| Transducers and their mounting | 190 | 78 | 41.1 percent | 2002 | reports server, 72 |
| Aerospike nozzles | 255 | 105 | 41.2 percent | 2005 | reports server, 105 |
| Detonation rocket engines | 33 | 15 | 45.5 percent | 2019 | reports server, 12 |
| Engine health and control | 41 | 19 | 46.3 percent | 1992 | reports server, 19 |
| Drag and body shape | 192 | 90 | 46.9 percent | 1983 | reports server, 69 |
| Segmented and modular vehicles | 9 | 5 | 55.6 percent | 1994 | reports server, 3 |
| Combustion and instability | 70 | 40 | 57.1 percent | 1997 | reports server, 21 |
| In-flight measurement | 148 | 90 | 60.8 percent | 2003 | reports server, 82 |
| Engine ground testing | 112 | 71 | 63.4 percent | 1994 | reports server, 54 |
| Turbopumps and feed systems | 28 | 19 | 67.9 percent | 1993 | reports server, 15 |
| Conventional nozzles and plumes | 46 | 32 | 69.6 percent | 1991 | reports server, 29 |
| Solid rocket motors | 51 | 36 | 70.6 percent | 1986 | reports server, 28 |
| Telemetry and data systems | 680 | 491 | 72.2 percent | 1990 | reports server, 444 |
| Space Launch System | 71 | 70 | 98.6 percent | 2016 | reports server, 70 |
| This vehicle and its team | 5 | 5 | 100.0 percent | 2009 | reports server, 5 |
The thin clusters belong to other publishers and the rich ones to government laboratories. The largest single source of each of the thinnest three, being the clusters on hybrid rocket engines, on launch-vehicle design and cost, and on recovery and reuse, is doi:2514, a conference and journal publisher rather than a report series. None of the three is a subject government laboratories wrote report series about.
Apart from this vehicle’s own 5 records, the richest cluster is the Space Launch System, 70 of whose 71 records the reports server holds. Of the clusters on this team’s trade, telemetry and data systems reaches 72.2 percent, with 444 of its records served by the reports server, and in-flight measurement reaches 60.8 percent. Telemetry, data acquisition and in-flight performance determination are subjects the space agency and the services documented in report series for decades, and that history explains why this article’s primary fraction is high where its predecessor’s was low.
And the median years make the same point from the other direction. Telemetry and data systems has a median year of 1990 and sensor arrays and sampling has 2015, a gap of 25 years. The report literature in this survey is old and the journal literature is recent, so a high primary fraction and a recent median are in tension by construction rather than by any choice made here.
What This Article Read in Full
Six primary documents were retrieved, saved and read end to end, and every quotation in this article comes from one of them. They are the laboratory’s background document on its rocket propulsion organisation, cleared in September 2022, the award announcement as reprinted verbatim by the base newspaper, the encyclopedia entry for this designation, and the parent vehicle’s payload user’s guide, which the previous article read and which supplies the reference trajectory and the comparison dimensions. The other two are the 1967 report that is the centre-of-pressure method every later treatment rests on, whose equation 3-66 gives the body normal-force derivative displayed earlier, and the 1951 study of viscosity on slender inclined bodies that shows the inviscid figure to be a floor. Four further primaries are cited from their abstracts alone, as the Epistemic State below explains. The contractor’s release about the contract award was not retrieved. The live company site no longer carries it and the archive returned no capture, so the vehicle’s dimensions and its four legs are cited at second hand from the entry that names that release as its source, and every geometric conclusion in this article inherits that provenance.
The laboratory’s background document is reached through the encyclopedia entry’s source list. It is a public-affairs document on a laboratory web site and not a report with an identifier, so neither report server nor the bibliographic index holds it. A source list at the foot of a secondary is a retrieval channel, and this one produced the sentence this article is built on.
Epistemic State
Facts, taken from primary documents and stated as they appear there. The designation, its allocation date of 20 April 2022, the three contractor names, the sponsor cell, the engines cell and the description identical to the X-63A’s come from the register [Parsch, DOD 4120.15-L Addendum]. That each company has its own launch vehicle and chosen approach, that ABL’s vehicle is the X-63 and Invocon’s the X-64, that the arrangement is a public private partnership, that the portfolio seeks a seventy percent reduction in development time and a fifty percent reduction in cost, and the dates and goals of the predecessor programmes come from the laboratory’s background document [AFRL’s Rocket Lab Past, Present and Future]. The December 2019 award date, the three-year term, the other transaction instrument, the scope wording including highly instrumented, the description of the lead contractor as a veteran-owned small business providing turnkey instrumentation, the naming of the two subcontractors and the quotation from the company’s president come from the award announcement [AFRL awards agreements under ARISE]. The parent vehicle’s dimensions and reference mission come from its payload user’s guide [ABL Payload User’s Guide]. The award counts, the award identifiers, the dollar figures and the difference between the spellings Troy7 and Troy 7 were read from the federal award reporting system [USAspending].
Reported at second hand, and said to be. The vehicle’s height of about 12 metres, its diameter of about 2.4 metres, its recoverability and its four legs functioning as stabilising fins are reported by the encyclopedia entry, which names the contractor’s release as its source [Invocon X-64]. This article did not retrieve that release, as noted under What This Article Read in Full.
Four further sources are used from their abstracts alone. The reports server offers no document for the two reviews of in-flight thrust determination, for the uncertainty methodology, or for the XB-70 drag measurement [In-flight thrust determination] [Uncertainty of in-flight thrust determination] [Uncertainty methodology] [Techniques for determining propulsion system forces]. What this article quotes from them is what their abstracts state, and an abstract is a primary statement of a paper’s claims and not the paper. That matters most for the criticism this article makes of its own error budget, which rests on a list of categories an abstract enumerates rather than on the treatment behind it. And the two statements of the sampling theorem are cited without being read [Shannon 1949] [Nyquist 1928], the second of them from a later reprint rather than the 1928 original.
Two primaries were read in full and both bear on a displayed relation, the 1967 report behind the slender-body slopes [Barrowman 1967] and the 1951 study showing the inviscid figure to be a floor [Allen and Perkins 1951]. The scan of the 1951 study is of poor quality and nothing is quoted from it verbatim.
One limitation of this article’s central apparatus. The error budget treats three contributions as independent random terms combined in quadrature, where the methodology the in-flight thrust field settled on separates bias from precision and carries a model bias error term [Uncertainty methodology]. So what this article presents is a sensitivity analysis and not an uncertainty statement, for the reasons given under And the Established Methodology Carries a Term This Article’s Budget Does Not.
Derivations, which follow from the facts by mathematics and are checked numerically. That an accelerometer measures thrust minus drag over mass and is blind to gravity is exact. The error budget for $F = m a_s + D$ is exact given independence. The condition $v^2 \sin \gamma = 2 H_\rho \, \mathrm{d}v / \mathrm{d}t$ for peak dynamic pressure is exact for any atmosphere and trajectory, and was checked at the computed peak of five profiles to 4.5 parts in a million. The closed form for the density scale height was checked against a numerical difference of the density profile to better than two parts in a thousand million, and reduces exactly to the pressure scale height in an isothermal layer. The footprint anisotropy $1 / \cos(\pi / N)$ is exact. That harmonic $N$ sampled at $N$ equally spaced points has the discrete mean of harmonic zero is exact for every phase. The static-margin consequence of fineness ratio is arithmetic on two published lengths.
Quantities that depend on assumptions, with the assumptions swept and the spread reported. The ascent is a speed power law and a pitch program with the published cut-off altitude as a constraint, and across five speed exponents the peak dynamic pressure runs from 45.9 to 92.2 kilopascal at 25.0 to 70.3 seconds and 7.9 to 11.0 kilometres. The half-signal time runs from 11.9 to 28.3 seconds and the fraction of the pressure-time integral collected before the dynamic-pressure peak from 83.6 to 95.3 percent. The instrument floor of the error budget runs from 0.54 to 2.24 percent across three instrument-quality cases. The drag fraction itself is not computed, because the mass is not public, and the budget is stated as a function of it. The instrumentation channel count, sample rate and telemetry capacity are likewise assumed and not published, so the data-rate arithmetic is quoted as a span and its conclusion is the order of the deficit rather than its value.
Inferences, marked as such where they appear. That the recoverability serves data return. That the four legs impose a leading azimuthal harmonic equal to their count. That the programme has ended. That the two vehicles’ results cannot be differenced because their drag models are not common terms. None of these is stated as a fact anywhere in this article and each is reversible by one document.
Two conflicts in the sources are recorded and not resolved. The laboratory’s background document writes into the 22nd century twice where it means the twenty-first [AFRL’s Rocket Lab Past, Present and Future]. And it names nitrogen tetroxide with Aerozine 50 in one sentence and nitrogen tetroxide with monomethylhydrazine in the next, which are different fuels, the liquid stages of the Titan 4 having burned the first [Palaszewski and Powell 1991].
One dateline limit. This article is dated 9 December 2025. The contractor’s web pages were read after that date and are cited for what the company says it does. The encyclopedia entry was last revised on 11 August 2024, which is before the dateline [Invocon X-64]. The register rows are counted in its copy last updated on 23 August 2026, which postdates the dateline and holds allocations made after it, so its row totals are those of that copy [Parsch, DOD 4120.15-L Addendum]. No statement in the body depends on anything the company published after the dateline, and the description of its current business is used only to observe that it does not list a launch vehicle, which was equally true in 2024 on the evidence of the award record.
What is not known and would change things. The vehicle’s mass, thrust and drag coefficient. The channel count, sample rates and sensor placement of the instrumentation, and in particular the number of sensors on the annulus. The leg reach and the centre-of-mass height. The dollar value of the agreement. Whether any hardware was built. And whether the four legs were ever more than a rendering. The encyclopedia entry credits two images, one to the laboratory and one to the contractor, and the one it captions is of a three-dimensional printed model [Invocon X-64]. No photograph of flight hardware is identified as such in any source consulted, which is weaker than saying none exists and is as far as the record goes.
Out of Scope
The aerospike nozzle itself belongs to the previous article. The conjugacy of ambient pressure and exit area, the Legendre structure of the ideal spike’s thrust curve, the Bregman divergence that is a fixed nozzle’s loss, the mean-in-time rule that chooses an exit area, the separation criterion and the three wake regimes are derived there and used here as given.
The internal aerodynamics of the plug and the design of the fins are not treated. Both need contours that are not public.
The landing dynamics are not treated. This article computes a static tip-over condition. A real landing is an impact with stroke, damping and a horizontal velocity component, and the legs are shock absorbers as well as a footprint.
The descent trajectory is not treated at all. How a vehicle of fineness five gets from cut-off back to a controlled vertical landing is a larger problem than the ascent modelled here, and nothing in the record describes the intended method.
No estimate is offered of whether the programme was worth its price. The price is not public.
Conclusion
The register gives two vehicles one description, and the laboratory that paid for both describes two designs by two teams [Parsch, DOD 4120.15-L Addendum] [AFRL’s Rocket Lab Past, Present and Future]. Everything above follows from taking the laboratory at its word.
The X-64A’s team is an instrumentation house leading a propulsion house and a company whose largest award is for control. Eighty-one award rows, most of them for wireless instrumentation, impact detection and radiation monitoring. Fifteen for KT Engineering, among them segmented and radially segmented launch vehicles, and this company holds every one of the seven rows the phrase segmented launch vehicle returns in the whole record. And fifteen more under Troy 7, the largest of them for hypersonic control and two of them belonging to other companies, findable only if the company’s name is spelled with a space its customer omits [USAspending]. A name is a query and a query is only as good as its spelling.
The vehicle is two published numbers and they indicate it was shaped to come back. About 12 metres by about 2.4 gives a fineness ratio $f = L / D_b$ of about five against its sibling’s 14.67, so its frontal area, which scales as $D_b^2$, is 1.72 times as large, and since a margin of $S_m$ calibres is $S_m / f$ of the length, one calibre of static margin costs it 20 percent of its length against 6.8 percent for the other vehicle. Those are the proportions of a lander, and the four legs that are fins on the way up are the aerodynamic surface a short vehicle needs put where the ground is.
And the problem has a long literature, which states in as many words that in-flight thrust is calculated rather than measured [Uncertainty of in-flight thrust determination] and which models the gas path because an air-breathing inlet makes thrust inseparable from drag to any body-mounted instrument. Thrust minus drag is one observable and splitting it always costs a model of one side, which the XB-70 drag measurement paid on the thrust side [Techniques for determining propulsion system forces].
The measurement is the half of the ARISE question this vehicle exists to answer, and it turns on one identity. An accelerometer is blind to gravity, so it reads thrust minus drag over mass, and recovering thrust needs a drag model. The drag term is the only one no instrument on board can reduce, it enters the error budget weighted by the drag fraction, and by $\delta D / D \le \beta \Delta / d$ holding it to a tenth of a five percent effect needs a drag model good to 2.65 percent at a drag fraction of one fifth. That is a standard this article judges a pre-flight drag model of a novel configuration unlikely to meet.
And yet the experiment works, for a reason that has nothing to do with instruments. Ambient pressure depends on position while drag depends on position and the square of speed, and a rocket reaches altitude before it reaches speed. Dynamic pressure peaks 2.10 to 2.48 times later than the moment half the altitude-compensation signal has been collected, and by the time it peaks the share $\chi(t_q)$ already in hand is between 83.6 and 95.3 percent. The trajectory is the instrument, and the vehicle’s shape, which raises the drag fraction by 1.72 on frontal area alone, is working against the one term the timing had already defeated.
The rest is what a finite number of sensors on a circle can see. Both phases of azimuthal harmonic $\mu$ need $2\mu + 1$ sensors, so a side load needs three and a four-lobed pattern needs nine. Harmonic $N$ read by $N$ sensors has exactly the discrete mean of harmonic zero, which means a four-legged vehicle with four taps on its annulus would report the disturbance of its legs as a change in the thrust-bearing mean. The regimes the flight existed to measure are changes in that mean and every sensor count recovers them, so the undersampled ring answers the programme’s question and quietly misattributes everything else.
The legs on the ground are the same polygon the other article found in the sky. The ratio of best to worst tip-over direction is $1 / \cos(\pi / N)$, character for character the anisotropy of differential-throttling authority, exactly the square root of two for four legs and exactly two for three. The footprint has $N$ sides for every count where the control polygon had $N$ or $2N$, and the difference is a half-plane truncation rather than anything geometric.
Neither vehicle is recorded as having flown. One contractor lost two rockets and left the launch business [RS1, Gunter’s Space Page]. The other is still an instrumentation house, and lists no launch vehicle among its products [Invocon]. Whether it ever thought of itself as anything else is not something an award record can settle, and this article does not claim it. The register still carries both numbers, allocated on the same day, with the same hundred and one characters against each, and it still gives a reader no way to tell that the two machines behind them were never the same machine [Parsch, DOD 4120.15-L Addendum].
References
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