**Update 4-8-2024:**
Should any readers want to learn how to do what I do (estimating
performance of launch rockets or other space vehicles), be aware that I have created a series of
short courses in how to go about these analyses, complete with effective tools for actually
carrying it out. These course materials are
available for free from a drop box that can be accessed from the Mars Society’s
“New Mars” forums, located at http://newmars.com/forums/, in the “Acheron labs” section, “interplanetary transportation” topic, and conversation thread titled “orbital
mechanics class traditional”. You may
have scroll down past all the “sticky notes”.

The first posting in that thread has a list of the classes
available, and these go far beyond just the
two-body elementary orbital mechanics of ellipses. There are the empirical corrections for
losses to be covered, approaches to use
for estimating entry descent and landing on bodies with atmospheres, and spreadsheet-based tools for estimating
the performance of rocket engines and rocket vehicles. The same thread has links to all the materials
in the drop box.

The New Mars forums would also welcome your
participation. Send an email to newmarsmember@gmail.com to find out
how to join up.

A lot of the same information from those short courses is
available scattered among the postings here.
There is a sort of “technical catalog” article that I try to main
current. It is titled “Lists of Some
Articles by Topic Area”, posted 21
October 2021. There are categories for
ramjet and closely-related,
aerothermodynamics and heat transfer,
rocket ballistics and rocket vehicle performance articles (__of
specific interest here__), asteroid
defense articles, space suits and
atmospheres articles, radiation hazard
articles, pulsejet articles, articles about ethanol and ethanol blends in
vehicles, automotive care articles, articles related to cactus eradication, and articles related to towed decoys. All of these are things that I really
did.

To access quickly any article on this site, use the blog archive tool on the left. All you need is the posting date and the
title. Click on the year, then click on the month, then click on the title if need be (such as if
multiple articles were posted that month).
Visit the catalog article and just jot down those you want to go see.

Within any article,
you can see the figures enlarged, by the expedient of just clicking on a
figure. You can scroll through all the
figures at greatest resolution in an article that way, although the figure numbers and titles are
lacking. There is an “X-out” top right
that takes you right back to the article itself.

----------

** Update
23 March 2024:**
For the readers of this and other similar articles about ramjet
propulsion, be aware that GW’s ramjet
book is finally available as a self-published item. Its title is “A Practical Guide to Ramjet
Propulsion”. Right now, contact GW at gwj5886@gmail.com to buy your copy.

He will, upon receipt of payment by surface mail or
Western Union (or similar), manually email
the book to you as pdf files. This will
take place as 9 emails, each with 3
files attached, for a total of 27 files
(1 for the up-front stuff, 1 each for 22
chapters, and 1 each for 4
appendices). The base price is
$100, to which $6.25 of Texas sales tax
must be added, for an invoice total of
$106.25.

This
procedure will get replaced with a secure automated web site, that can take credit cards, and automatically send the book as
files. However, that option is not yet available. Watch this space for the announcement when it
is.

GW is working
on a second edition. No projections yet
for when that will become available.

--------------

**It is intended to give readers a means to compute accurate and realistic thrusts. This plus a knowledge of chamber characteristic velocity c* is sufficient to do very elementary rocket ballistics.**

*This article applies to anything that thrusts from a subsonic chamber through (at least) a sonic throat.***see stuff added at the very end, below, past the original figures.**

__Update 11-16-18__:**see stuff added at the very end below, past the first update.**

__Update 1-26-19__:**see stuff added at the very, very end below, past the second update.**

__Update 1-31-19__:**see plume spread stuff added at the very end**

__Update 10-1-19__:**this article uses expansion area ratio and assumed geometry to estimate aerospike and other free expansion designs. I have since posted a new article "Rocket Nozzle Types", posted 4 Feb 23, that revisits the aerospikes using actual Prandtl-Meyer expansion-corner analysis. It gets about the same answers ultimately, but allows determination that the gas generators benefit from some bell-confined expansion followed by the free expansion on the spike.**

__Update 2-9-23__:**Fundamentals**

**control volume**drawn about the rocket engine is

**pierced by the exit stream exactly at its exit area**, and the momentum of the propellant feeds are either inconsequential, or perpendicular to the thrust axis, or they come from tanks inside the control volume. This could be any combination of those situations, or even all three. Balancing stream momentum and the pressure forces against a restraining force, leads to evaluating the thrust.

__All figures are at the end.__

_{p}. (Internal energy change uses the specific heat at constant volume c

_{v}.)

**. That last means we may use as our equation of state P = ρ R T, and we may use as the change in enthalpy Δh = c**

*as long as the ideal gas assumption applies*_{p}ΔT.

_{t}and temperatures T

_{t}. Assuming no appreciable friction losses, flow is “isentropic”,

__meaning total pressure and total temperature are constant through the nozzle__, a very good assumption in almost every conceivable case.

_{p}/c

_{v}becomes a very useful value to relate totals to statics. At a location where the Mach number is M, the total/static temperature ratio TR = 1 + 0.5*(γ – 1) M

^{2}, and the total/static pressure ratio is PR = TR

^{exp}, where exp = γ / (γ – 1).

_{t}. If you know the Mach number M at another station where the area is A, you can find that area ratio AR = A/A

_{t}as easily as the total/static ratios TR and PR. If you know instead the area ratio A/A

_{t}, finding the Mach number M is inherently a transcendental (iterative) solution:

_{t}= (1 / M) [TR / const1]

^{const2}

**We**

*this is driven not by static temperature but by recovery temperature!*__must__do this because the supersonic flow in the nozzle is both highly compressible, and highly dissipative. At any given Mach number M, recovery temperature T

_{r}is very nearly the same as total temperature T

_{t}. How it varies depends upon laminar versus turbulent flow, and the gas property Prandtl number P

_{r}:

_{r}= T + r (T

_{t}– T) where the recovery factor r = P

_{r}

^{0.5}laminar, P

_{r}

^{0.33}turbulent

_{r}– T

_{s}) where T

_{s}is the material surface temperature

_{r}= 4 γ /(9γ – 5).

**Conventional Nozzle Thrust Coefficient C**

_{F}

_{}__ideal gas model__of the gas flowing through the nozzle comprises its constant specific heat ratio γ, and its constant molecular weight MW. These can come from thermochemistry calculations, and need to reflect the high temperatures involved.

_{e}times the difference in pressure between the exiting stream static pressure P

_{e}and the ambient backpressure of the surroundings P

_{b}. Ideally, all the streamlines are parallel to the thrust axis.

_{ke}. It applies to the momentum term in thrust,

__but not the pressure term__, as long as the exit plane is perpendicular to the axis.

_{ke}m V

_{e}+ (P

_{e}– P

_{b}) A

_{e}where m = mass flow rate and V

_{e}= exit velocity

_{e}A

_{e}V

_{e}massflow definition, the ideal gas equation of state P

_{e}= ρ

_{e}R T

_{e}with R = R

_{univ}/MW, and the exit plane speed of sound for an ideal gas c

_{e}= (γ g

_{c}R T

_{e})

^{0.5}. The variable g

_{c}is the “gravity constant” to make the equation consistent with inconsistent mass and force units. If those units are consistent, g

_{c}will be 1.

_{ke}(P

_{e}/ R T

_{e}) Ve

^{2}Ae + (P

_{e}– P

_{b})A

_{e}using massflow, then equation of state

_{ke}P

_{e}A

_{e}M

_{e}

^{2}+ (P

_{e}– P

_{b})A

_{e}using speed of sound

**Distribute the A**

*Note that the first term in the equation just above is the momentum term, and the second term is the static pressure difference term.*_{e}so that there are 3 separate terms, and regroup.

_{e}A

_{e}[1 + γ η

_{ke}M

_{e}

^{2}] – P

_{b}A

_{e}recombining terms such that P

_{e}A

_{e}factors out

_{e}A

_{e}factored out, and the γ η

_{ke}M

_{e}

^{2}corresponds to the momentum term with P

_{e}A

_{e}factored out. The backpressure effect is still a separate force term,

*with the recombined bracket-containing force term really just being thrust into vacuum.*

_{F}= F / P

_{c}A

_{t}with the understanding that the P

_{c}is the total (stagnation) pressure feeding the nozzle. If the contraction from chamber to throat area is large enough, there is no measurable difference between total and static pressure at the nozzle entrance.

_{F}= F / P

_{c}A

_{t}= (P

_{e}A

_{e}/ P

_{c}A

_{t})[1 + γ η

_{ke}M

_{e}

^{2}] – (P

_{b}A

_{e}/ P

_{c}A

_{t})

_{e}/P

_{c})(A

_{e}/A

_{t})[1 + γ η

_{ke}M

_{e}

^{2}] – (P

_{b}/ P

_{c})(A

_{e}/ A

_{t}) regrouping P’s and A’s together

**CF = (AR / PR) [1 +**

**γ η**

_{ke}M_{e}^{2}] – AR / PR_{op}**(**

__the__thrust coefficient equation)_{op }= P

_{c}/P

_{b}using the actual design P

_{c}and P

_{b}

_{c}/P

_{e}= (1 + 0.5 (γ – 1) M

_{e}

^{2})

^{exp}= TR

^{exp}where exp = γ / (γ – 1)

_{e}/A

_{t}= (1/M

_{e})[ TR/const1]

^{const2}

_{e}, because AR = A

_{e}/A

_{t}and PR = P

_{c}/P

_{e}are easily calculated from M

_{e}using the ideal gas γ. Otherwise, if conditions at a certain AR are desired, one iteratively tries M

_{e}values until the desired AR obtains, then computes PR. Essentially, M

_{e}and P

_{c}/P

_{e}are “locked in” by the AR value regardless of the value of P

_{c}, although they are not most conveniently figured in that order.

_{op}= P

_{c}/P

_{b}depends directly upon your design choices for P

_{c }and P

_{b}. One had to choose a P

_{c}to do the thermochemistry, and P

_{b}is set by the altitude, or else 0 if vacuum.

_{e}, PR, AR, and PR

_{op}are all known, evaluating C

_{F}is easy, per the above equation. If you have used a value of c* to size a throat A

_{t}elsewhere in your fundamental ballistics, then the nozzle thrust is easily obtained as F = C

_{F}P

_{c}A

_{t}. From ballistics, choked nozzle massflow w = P

_{c}C

_{D}A

_{t}g

_{c}/ c*, see Figure 2 below. C

_{D}is the nozzle throat’s discharge coefficient (or efficiency).

_{sp}= C

_{F}c* / g

_{c}.

*If not, you must ratio down your calculated I*_{sp}, F, and A_{t}by 1 + f, where f is the fraction of generated hot gas massflow that does__not__go through the nozzle.

**Example Problem: Conventional Nozzle, Sea Level and 20 kft Designs**

_{t}= 1.0 square inch, with a nozzle C

_{D}= 0.99 to size flow rate. The resulting design is a nominal 3000 lb thrust design, completely immune to backpressure-induced separation, since it is never over-expanded. How the nozzle kinetic energy efficiency is calculated from half angle is discussed below.

**Flow Separation Risks**

_{c}/P

_{e}. P

_{sep}is the estimated backpressure, at and above which nozzle flow separation is to be expected. It is empirical, and it is a rough estimate.

__The designer should allow significant margin__.

_{sep}/ P

_{c}= (1.5 P

_{e}/P

_{c})

^{0.8333}

_{c}/P

_{e}= 266.3, so that P

_{e}/P

_{c}= 0.003755. Thus P

_{sep}/P

_{c}= 0.013355, and for P

_{c}= 1800 psia, the expected P

_{sep}= 24.04 psia, quite a margin above sea level pressure. We can conclude that there is no risk of separation in the example nozzle, all the way down to sea level, where P

_{b}is only 14.7 psia. The risky backpressure is even higher at about 45 psia for the sea level design.

**KE-Efficiency Correlations**

^{o}) = 1.00. Those near the nozzle wall are aligned

__at the angle of that wall off the axis__. For a conical nozzle, this is the half angle of the cone. The cosine factor for off-angle alignment is cos(a) where “a” is the half angle of the cone. See Figure 1 again.

__effective cosine factor for the distribution__would be to integrate them for an average,

**. Simply compute the arithmetic average of the centerline cosine factor value (1.00) and the wall cosine factor value cos(a), and call that the nozzle kinetic efficiency factor:**

*there is an easier model that is just as good***η**

_{ke}= 0.5 [1 + cos(a)] where “a” is the effective average half-angle of the nozzle wall

_{ke}= 0.983. For a curved bell, there is a local “a” near the throat, and a smaller local “a” at the exit lip. One simply averages the two local a’s, and uses that average as “a” in the kinetic energy efficiency formula. For most practical curved bell designs, that average “a” won’t be far from 15 degrees. See again Figure 1.

**Free-Expansion Designs By “Last Point of Contact = Perpendicular Exit Plane Model”**

*We want the components of the actual distribution of exhaust velocities, that are aligned with the engine axis.***Note that the very same gas-generating chamber and throat area serves as the feed to the free-expansion “nozzle” at all values of P**

*It is just that the order in which things need to be done is revised.*_{b}.

_{e}= P

_{b},

__as long as P__! Once a P

_{b}is not exactly zero_{b}is known, then PR = P

_{c}/P

_{e}= P

_{c}/P

_{b}is known. One solves the PR equation for M

_{e}at this value of PR, which is

__not__a transcendental iteration, just a simple direct solution:

_{e}= { 2/(γ – 1) [PR

^{(}

^{γ-1)/}

^{γ}– 1]}

^{0.5}

_{e}now known, find the area ratio from the streamtube relation, and use it with the throat area to find the effective value of the exit area A

_{e}:

_{e}

^{2}

_{e}) [TR / const1]

^{const2}

_{e}= AR A

_{t}

**.**

*which is different for each backpressure*__Each geometry is different__, but the idea is to find the largest half-angle off of axial and use it as “a”. This goes into the correlation for kinetic energy efficiency. That correlation is generally for “a” < 30 degrees,

__so we are misusing this here!__But, it is the best I have at this time to offer.

*Any such “a”-dependent model, even if flawed, is better than no model at all!*

_{e}= P

_{b}, and from that thrust, the thrust coefficient (to use with c* for I

_{sp}):

_{ke}γ P

_{e}A

_{e}M

_{e}

^{2}

_{F}= F / P

_{c}A

_{t}

_{e}nor η

_{ke}are constants here, as P

_{b}changes. At high backpressures (low altitudes), “a” is small, η

_{ke}is high, and A

_{e}and M

_{e}are lower. At low backpressures (high altitudes), “a” is quite large, η

_{ke}is lower, and A

_{e}and M

_{e}are high.

*Exactly how “a” varies is quite geometry-dependent.*_{b}= 0 (vacuum of space), PR = infinite, leading to infinite M

_{e}and A

_{e}. There can be no planar exit plane, and Prandtl-Meyer expansion says “a” > 90 degrees by a small amount.

**, quite unlike a conventional nozzle! (Which means this free-expansion design approach is inappropriate in vacuum!)**

*There is no point trying to use this compressible flow analysis technique on a free-expansion nozzle in vacuum*^{-1}[(R

_{e}-R

_{t})/L

_{spike}]. For this, R

_{e}= (A

_{e}/pi)

^{0.5}, and R

_{t}= approximately (A

_{t}/pi)

^{0.5}. L

_{spike}is the distance from throat plane to exit plane. Longer is lower effective “a”, but higher weight, and a tougher cooling design.

**Example Axisymmetric Aerospike Problem**

_{c}, A

_{t}, and γ. If thrust is the issue, and it usually is for launch vehicles, then the preferred performance variable to examine is thrust.

_{c}= 1800 psia, A

_{t}= 1.0 in

^{2}, and γ = 1.20, same as the conventional nozzle examples earlier. The same c* and nozzle throat C

_{D}are used.

__In effect, this engine shares the very same gas generator as the two conventional examples__. The same altitude backpressures are also used, so that this design can be compared directly to the earlier examples,

**.**

*except that vacuum performance cannot be included***Comparisons Among the Example Nozzle Designs**

_{c}= 1800 psia, A

_{t}= 1 square inch, γ = 1.20, chamber c* = 5900 ft/sec, and nozzle throat discharge coefficient C

_{D}= 0.99. All three are roughly the same 3000 lb thrust

__at their design points__, within a percentage point or three.

__by far__over the pressure difference term, in thrust.

**, something that in turn depends upon the effective average half-angle of the propulsion stream bondary. This is really nothing but the cosine factors of streamlines that are aligned off-axis. Kinetic energy efficiency trends are given in Figure 11.**

*traces directly to the trends of nozzle kinetic energy efficiency*__locked-in by the physical bell__, right up to the exit plane. Downstream of the exit lip, gas expands laterally into the vacuum, but this happens

__downstream__of the “last point of contact”, where thrust is actually calculated. This is implied by how we draw the control volume about the engine and nozzle, something shown in the lower right corner of Figure 1,

__touching at that last point of contact__.

__inside__the control volume, as is the bell of the conventional nozzle. At high altitudes where the air pressure is low, the plume boundary must expand quite far laterally, between the throat, and the “exit plane” at the last point of contact. This is precisely how large AR and M

_{e}are achieved, in order to match P

_{e}= P

_{b}. Since the length of the free-expansion zone is fixed, the boundary half-angle must be quite large at high AR. That reduces kinetic energy efficiency.

_{spike}used), but drops below conventional at about 20 kft, and falls ever more rapidly to only about 77.7% at 100 kft.

__This traces directly to the effective half-angle__of the plume boundary between the throat, and the exit plane at last point of contact.

^{o}) = 0.5543. Averaging that with 1 inherently produces η

_{ke}= 77.7%.

**Conclusion**

__entirely inappropriate for use in vacuum__! The complications with cooling the spike outweigh any tangible performance benefits, which are low (unless you cheat by not accounting for the streamline divergence effects).

**This conclusion is correct as far as it goes, but it is also incomplete. See the update just below the original 12 figures. I explored the free-expansion nozzle approach a lot further, in two different forms. It can be made to work better at high altitudes, but NOT in vacuum, and not really significantly better than a conventional nozzle sized at the highest altitude consistent with not flow-separating at sea level.**

__Update 11-16-18__:Figure 2 – Modeling Nozzles with Compressible Flow

Figure 3 – Spreadsheet Image for 15 Degree Conical Nozzle As Sea Level Design

Figure 4 – Spreadsheet Image for 15 Degree Conical Nozzle As 20 Kft Design

Figure 5 – Analogous Procedure for Free-Expansion Designs

Figure 6 – Example Axisymmetric Aerospike Nozzle Results, Part A

__Update 11-16-18__:

**).**

*all figures at the end of this update***: I call it the “twin aerospike”. Instead of being axisymmetric with the spike immersed along the plume centerline, in the twin aerospike, the nozzle bell is conceptually cut away, from throat to exit, top and bottom, leaving two symmetrically-placed spikes along each side, at the physical bell half angle. This is shown in Figure 14.**

*that limits expanded half-angle better*

*I have never before seen a proposal like this; therefore,*__it is my idea__. Please give me credit for it, if you pursue it.**, in order to understand what happens to nozzle kinetic energy efficiency. That in turn governs the thrust and impulse performances that can be achieved.**

*It is very important to understand what happens to effective average boundary half-angle for these various designs***That item is plotted vs altitude in the second comparison plot (Figure 16) for all 7 designs. The other performance measures are thrust, specific impulse, and thrust coefficient, as given in Figures 17, 18, and 19.**

*Its cosine averaged with 1 is the nozzle kinetic energy efficiency.***The nozzle efficiencies in Figure 16 show both those designs with similar trends to 200 kft altitude. An “eyeball guess” says the twin 100k design might “bottom out” somewhere near 70% efficiency at 300 kft, while the “axi a 100k” design’s efficiency will probably just continue to fall precipitously.**

*Designing the free expansion nozzles at higher altitude (getting far longer spikes) is quite evidently a better deal*__from a fluid mechanics standpoint__.

*These show quite clearly that we do*__not__want the lower-altitude free-expansion designs, since by about 40 kft,**. The 30k twin aerospike design does better than the 20k axisymmetric aerospike design, mainly because of the half-angle averaging, but neither maintains equal or better performance than conventional, past 50 kft “for sure”.**

*the conventional nozzles outperform them, just as I said in the original article*

*There is no fluid mechanical optimum here!***from both the constructional, and the thermo-structural, viewpoints. It is**

*increasingly-long spikes are increasingly infeasible*__that__trade-off,

__not fluid mechanics__, which determines whether either of the free-expansion design approaches is “better” than a conventional nozzle design. And, as shown in Figures 17-19, the performance advantage of the free-expansion design over a 30 kft conventional design is never very large at all!

__Just as the original article conclusion indicated__.

**Updated Conclusions**

**Free-expansion designs can NEVER outperform conventional designs in vacuum, just as originally concluded. The trends are just wrong to support such a conclusion.**

*The conclusions in the original article are correct, but a bit incomplete.*__does not__limit this!

__up to this time__. They thus cannot be considered a well-established technology, the failed X-33 program notwithstanding.

Figure 14 – Pertinent Dimensions and Conditions for Twin Aerospike Designs

Figure 15 -- Comparison of Effective Half-Angle vs Altitude for the 7 Designs

Figure 16 -- Comparison of Nozzle Kinetic Energy Efficiency vs Altitude for the 7 Designs

Figure 18 -- Comparison of Specific Impulse vs Altitude for the 7 Designs

__Update 1-26-19__:_{t}= 1 square inch, at a fixed throat discharge coefficient C

_{D}= 0.99. Whether conventional bell or free-expansion, the streamtube area ratio at design is such that expanded pressure equals ambient atmospheric pressure at the design altitude.

*Bear in mind that if the very long spikes are not practically achievable, then this advantage cannot be realized!*

*My overall conclusions remain the same as they have been, in both the original article and the first update.*__required very long spikes__are really practically-achievable designs.

**. That is because its exit streamline directions are well-collimated, while those of the free-expansion designs inherently cannot be.**

*out in hard vacuum, the conventional bell is just inherently superior*Figure C – Effective Nozzle Kinetic Energy Efficiency Trends vs Altitude

__Update 1-31-19__:

**Yet Another Study: “Best Long” Designs**

**All three had their throat areas At resized slightly, so as to provide exactly 3000 lb thrust at sea level.**

*These produce quite long expansion spikes compared to the conventional bell length, hence the terminology “long”.*

*See Figure I below.*

**. Of course, that trend is constant with the conventional bell. With the two aerospikes, it varies quite strongly, reflecting the perfectly-expanded condition, although with slightly-different curve shapes. These differences reflect the circular plume shape for the axisymmetric design, versus the elliptical plume shape for the twin design. Those half-angle trends produce the nozzle kinetic energy efficiency trends**

*shown in Figure II***. Again, the aerospike designs show varying efficiencies, versus the constant efficiency for the conventional bell.**

*shown in Figure III***. They all tell exactly the same story. Not shown is expanded Mach number. This is fixed for the conventional bell, and very variable for both aerospike designs.**

*given in Figures IV, V, and VI***Conclusions:**

**Still Another Study: “Short” Designs**

*See Figure VII below.*

**. Of course, that trend is constant with the conventional bell, as before. With the two aerospikes, it varies quite strongly, reflecting the perfectly-expanded condition, although with slightly-different curve shapes. These differences reflect the circular plume shape for the axisymmetric design, versus the elliptical plume shape for the twin design. They are larger, because the spikes are very much shorter, for the same expanded exit area.**

*shown in Figure VIII***. Again, the aerospike designs show varying efficiencies, versus the constant efficiency for the conventional bell, just not quite as good as the “long” designs, especially at very high altitudes.**

*shown in Figure IX***. They all tell exactly the same story. Not shown is expanded exit-plane Mach number. This is fixed for the conventional bell, and very variable for both aerospike designs.**

*given in Figures X, XI, and XII***Conclusions:**

**Final Observations:**

Figure IX – Nozzle Kinetic Energy Efficiency Trends With Altitude, “Short”

**Jet Plume Spreading Stuff**

__Update 10-1-19__:This topic is important for the plume impact forces that can blow around surface materials and excavate blast craters during retropropulsive landings.

The main point is the our intuitive feel for the effects of jet plume impact here on Earth

__is entirely inappropriate__for jet plumes in vacuum, or near-vacuum, conditions.

As the Figure U-1 indicates, the difference in fundamental behavior is quite substantial. In evaluating this figure bear in mind that exit Mach number Me varies from around 4.2 at area ratio 40 to around 5.4 at area ratio 200, which pretty well covers the range of practical rocket engines.

Figure U-1 -- Jet Plume Behaviors in Dense Atmospheres and in Vacuum

Accordingly, the jet blast wind pressure is essentially axially directed, and equal to the nozzle thrust force, at least up close where no significant mixing has yet occurred. The mixing and dilution effect starts getting to be barely significant about 10 plume diameters downstream of the exit, and has become very significant somewhere around 100 (or more) diameters downstream.

Jet plumes in vacuum (or near-vacuum) behave very differently! There is a short core that remains collimated axially, but its extent is of very limited proportion, being

__at most__about 10 exit diameters long. This region is bounded roughly by the similar right triangle of exit velocity axial, and speed-of-sound radial. In other words, the core cone length is about twice the exit Mach number times the exit diameter.

The rest of the plume outside that core cone immediately and drastically "fans out" radially. This is the effect of a compressible flow physics phenomenon called "Prandtl-Meyer expansion". This happens so as to take the finite exit-plane expanded pressure, and immediately reduce it to the surrounding vacuum or near-vacuum pressure by sharply turning. In vacuum, this sharp turn is slightly more than 90 degrees.

The resulting jet blast effect is reduced to low force values very quickly by the pressure drop (wind pressure is proportional to gas pressure times gas velocity squared), and rapidly becomes mostly radial, not at all axial. Only the very-limited-extent core is strong enough to disturb anything!

The net result of this is that a rocket jet blast which on Earth would dig a big blast crater and fling copious amounts of rocks and dirt, will have very little effect upon a plume-impacted surface out in vacuum. This is exactly why there was only a little blowing dust and a couple of pebbles seen during the Apollo moon landings, and no big excavated craters were seen under the lunar modules after landing.

There are conspiracy theorists out there who claim we never actually landed on the moon, and one of the things they point to is the lack of a blast crater under the lunar modules in the photos. I just told you why they are wrong to make that claim!