**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.

----------

I want to keep this short and sweet. It makes use of things computed in other
posted articles, as well as what I did
for this one with a simple spreadsheet.
I often run into correspondents who advocate single-stage-to-orbit (SSTO)
as the better means to send payload to low Earth orbit. I disagree that it is “better” than
two-stage-to-orbit (TSTO), but it is actually
possible to do SSTO with chemical propulsion!
However, it is possible
technologically __only if it is done as
an expendable__, which simply cannot
be inexpensive! Equal payload but more
expensive is __not__ better! See **Figure
1**.

Figure 1 – Overall Results for SSTO Bounding Calculations

There is a lot contained in that figure. Upper left is a plot of SSTO vehicle inert
mass fraction versus its propulsion’s __ascent-averaged__ specific impulse
(Isp). This is for a constant 1% payload
mass fraction, which is quite low. I indicated which ranges of Isp could be
attained with chemical propulsion,
versus nuclear. This is very
simple rocket equation-based stuff.

The equations used to compute these things are given
underneath the sketch of the vehicle concept lower left. The flat lines in the plot are the levels of
vehicle inert mass fraction that might be “credible” under different
circumstances. The arrows indicate where
there might be payload fraction added to the baseline 1%. __5% inert is only credible for expendable
stages__!

10% inert is __almost not credible at all__ for a
reusable vehicle, due to the airframe
items required as a heat shield, and retro-propulsion
to effect some sort of landing, which
also increases the dV requirement further!
__15-20% inert are more credible__ as the inert mass fractions of
some sort of lifting body vehicle with internal tanks, heat protection, and landing gear for a glide landing.

Most bomber and transport aircraft are nearer 40%
inert, and so was the X-15 rocket
plane! None of those has (or had) the
heat protection for full orbital entry! Naval
carrier aircraft are closer to 50% inert,
because of the hard knocks they must take.

Upper right is a depiction of the mission and the resulting
delta-vee (dV) requirements. The mission
is eastward low Earth orbit at 300 km altitude,
at low inclination. A surrogate
for reaching this energy is the surface circular orbit speed of 7.913
km/s. To this must be added losses-to-cover. These losses I presumed as 5.6% of surface
circular each, for the gravity and drag
losses. That gets us to just about 8.8
km/s to just barely reach orbit, to
include a circularization burn.

Once in orbit, there
needs to be a dV budget for rendezvous with “something” in order to be
useful, plus a deorbit burn budget. I used 0.6 (arbitrary but ballpark) and 0.1
km/s (fairly precise) for these,
respectively. __Excluding
any sort of landing burns__, the
mission mass ratio-effective dV is thus just about 9.5 km/s.__That’s close enough for a realistic
bounding calculation__.

The other point to be made is the variation of propulsion
Isp with increasing altitude during the ascent.
This is sketched in a sort-of-sketch plot in the lower right of the
figure. Behavior is quite different for
engines with conventional bells versus engines with free-expansion nozzle
designs. *The point is this: you need an ascent-averaged Isp to use
in the rocket equation SSTO model. *

**However,
the figure makes clear that regardless of what inert mass fraction you might
presume to be credible, only nuclear
propulsion could supply enough Isp to make a reusable lifting body SSTO
feasible at 15-20% inert. Chemical
propulsion is probably only feasible for expendable SSTO designs, with the rather low stage inert mass fraction
that is typical of such things. **

**Alternatively, you might increase the payload fraction a
little bit at 5% inert, in an expendable
LOX-LH2 SSTO, possibly to as high as
5-6%. That would be an equal payload
fraction to an expendable TSTO. But, expendable is just not inexpensive to
use, because you are losing all the
hardware! The TSTO would be cheaper use
even if only its first stage were reusable,
and that has already been demonstrated feasible at low inert fractions
by SpaceX with its Falcons!**

**Details (read on only if you are interested in the
details)**

Isp is __not__ solely a property of the propellant
combination, but characteristic velocity
c* is, contrary to prevailing
opinions. This and the mixture ratio are
weak functions of the downstream chamber pressure Pc that feeds the nozzle
(whatever it is). Isp would be
proportional to c* if all else were equal,
but it is not! The thrust
coefficient that is achievable with the expansion is just as important to Isp
as is c*.

Once you determine an expansion thrust coefficient, then Isp actually is proportional to c*. See **Figure 2 below** for typical c* data
(blue) versus propellant combinations.
At any design point Pc, c* = (c*
at 1000 psia)*(Pc/1000 psia)^{m}.
Thrust coefficient is C_{F} = Fth/Pc At = (Ae/At)*(Pe/Pc)[1 + γ
η_{KE} Me^{2}] – (Pa/Pc)*(Ae/At), where the expansion details set your values
of Me, Ae/At, and Pe/Pc,
plus your nozzle kinetic energy efficiency η_{KE}.

The basic message here is that once the expansion and thrust
coefficient are defined (along with the dumped bleed fraction BF), you can rough-estimate Isp for a different
propellant combination as being proportional to c*. That is because Isp = C_{F} c* (1 –
BF)/g_{c} C_{D}. _{D} ~ .99.

Designing fixed-bell engines can be to a fixed area
expansion ratio Ae/At, or to a fixed
expanded pressure ratio Pe/Pc (both depend explicitly on Me). The ambient atmospheric pressure Pa also
affects the value of C_{F}, as
indicated above. A __traditional sea
level-optimized design__ sets its expansion such that Pe = Pa at sea
level, figured at a suitable design
value of Pc. What I call a “compromise
bell” does that same thing, __just at
an altitude above sea level__, such as
10,000 feet, 20,000 feet, or even 30,000 feet. The “compromise bells” get higher values of
Ae/At (and higher resulting vacuum Isp performance but lower near sea level) at
the cost of part-throttle flow separation at sea level while testing in
“open-air nozzle” mode.

Figure 2 – Models for c* and r Versus Some Propellant
Combinations (1969 P&W Handbook)

There are multiple kinds of free-expansion nozzles. These always reach Pe = Pa at any
altitude, thus varying the effective
Ae/At. This has to be measured at the
last point of physical contact with the expansion spike or surface. The shape
of that expanded Ae varies, as does the
divergence angle of the outer streamline with respect to the axial thrust
direction.

The nozzle kinetic energy efficiency η_{KE} is the
integrated average of all the cosine factors-to-axial of all the streamlines in
the exiting stream. This is fixed for a
fixed bell, and __highly-variable with
altitude__ (becoming more extreme at very high altitudes) for a free
expansion nozzle of any kind. The effect
is quite serious in a simple coaxial spike design with a sonic-only gas
generator, as shown in **Figure 3 below**. Such are really-lousy vacuum engines.

I explored multiple design variations of free-expansion
nozzles, until settling on an aerospike
design fed with multiple gas generators** .
The streamline divergence was extreme until I added some fixed-bell
supersonic expansion, before further
expanding that supersonic exit plume against the aerospike.** That was the key to improved performance
flying out into vacuum, because there
was reduced potential for Prandtl-Meyer expansion effects. See

**Figure 4 below**. These are actually “good” vacuum engines, but they still fall slightly short of fixed-bell vacuum Isp.

These partial-supersonic free-expansion estimates are
somewhat over-estimates, since they do
not account for the oblique shocks incurred turning a supersonic stream. But this design approach gets an Isp trend vs
altitude that ascent-averages pretty much the same as that of a good compromise
bell. ** I do not show a distinct
advantage of either nozzle type, either
way, for average performance during
ascent!** However, the “trap” is not properly optimizing the
free-expansion design. It “falls off the
cliff” easier than the fixed bell, if
you fail to optimize it “just so”.

I sized most of these nozzles at modest Pc values, using LOX-RP1, at a nominal c* = 5900 sec. To __approximately__ convert that trend to
LOX-LH2, just multiply the Isp values by
7950/5900 = 1.347. For LOX-LCH4, multiply by 6120/5900 = 1.037. *It is actually better to just do the
ballistics.*

For the compromise bell at 100,000 feet in **Figure 4**, 325 s max Isp on LOX-RP1 becomes about 438
sec with LOX-LH2. For the
partial-supersonic aerospike, 350 sec
max Isp on LOX-RP1 becomes about 472 sec with LOX-LH2. Those numbers might get you to 10% inert (__not
very credible for reusability__) in the **Figure 1** plot of inert
fraction vs ascent-average Isp. They do __not__
get you even close to the more-credible 15-20% inert range! That requires 530+ sec of Isp, which is nuclear.

Figure 3 – Free Expansion Nozzle Performance Falls Quickly
With Sonic-Only Gas Generators

Figure 4 – No Advantage Either Way With Bells Vs
Partial-Supersonic Aerospike Free Expansion

**References:**

All of these listed below are earlier articles posted on
this site that relate to this topic in some way. Use the blog archive on the left side of this
page as a fast navigation tool to find them quickly. All you need is the article title and its
posting date. Click on the year, then on the month, then on the title if more than one article
was posted that month. The 2-4-23
article (bold italic) has a lot of the earlier work used here. The other 11-12-18 article (also bold italic)
was the first to explore free expansion.

3-3-24 Launch
to Low Earth Orbit: 1 or 2 Stages?

3-4-24 Launch
to Low Earth Orbit: Fixed Geometry Options

6-20-23 TSTO
Launch Fundamentals

*2-4-23 Rocket
Nozzle Types (bells and aerospikes)*

10-1-22 Rocket
Engine Calculations

2-9-21 Rocket
Vehicle Performance Spreadsheet

2-16-20 Solid
Rocket Analysis (solid ballistics and more)

*11-12-18 How
Propulsion Nozzles Work*

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