This author has tried several times to automate
pencil-and-paper design analyses with spreadsheet software, for the purpose of rough-sizing
Earth-to-orbit launch vehicles, be they
1 or 2 stage. None of those were as
successful and suitable as he would like,
until now. Bear in
mind that these spreadsheet results are not real performance estimates, only bounding calculations! The choice of stage inert mass fractions is
entirely arbitrary at this level of analysis,
totally unrefined by any sort of inerts-buildup design activity.
In this particular case,
the spreadsheet file is an Excel spreadsheet file named “stage
studies.xlsx”. It has 3 worksheets
within, 1 being for
single-stage-to-orbit (SSTO) scenarios,
the other 2 worksheets for two-stage-to-orbit (TSTO) scenarios.
The worksheet named “SSTO” creates plotted trends of payload mass fraction vs ascent-averaged specific impulse (Isp), parametric upon values of vehicle inert mass fraction Winert/Wignition. Metric units are presumed, being metric tons (m.ton) for masses, and speeds in kilometers per second (km/s). Isp, measured in seconds (s), is not defined in terms of consistent units! The corresponding effective exhaust velocity Vex is, and is measured in km/s to match the other speeds. See Figure 1.
Figure 1 -- Image of
the “SSTO” Worksheet
The plots respond automatically to changes in the
yellow-highlighted user inputs for a spread of Isp values, and a required velocity increment capability “rq
dV”, km/s. That last is the end of all burns speed upon reaching
orbit, with drag and gravity losses
added to it, plus a small budget to
cover anything else. The input Isp range
covers whatever ascent-averaged Isp might obtain, for any given propellant combination and
engine technology. Generally
speaking, the user need not change the
Isp spread inputs as shown.
For a given ascent-averaged Isp, and a given vehicle inert fraction, the corresponding payload fraction can be
read right off the plot to about 2 significant figures!
There is an aid for figuring vehicle weight statements, located top right. It has yellow-highlighted inputs for vehicle
inert and payload fractions. This takes
two forms: you know a known
(yellow-highlighted) liftoff mass, or
you know a (yellow-highlighted) delivered payload mass. It generates the correspond weight
statement, either way, whichever is chosen.
To bottom right is an aid for determining the “rq dV” needed
as an input for the main worksheet calculation.
There is a yellow-highlighted input for the actual speed at entry into
orbit, plus two yellow-highlighted
inputs for the percentages of that speed,
that are the gravity and drag losses.
There is one other yellow-highlighted input for a small budget to cover
anything else, such as landing burns and
rendezvous burns. It sums the 4
components to create the “rq dV” value,
which is essentially the speed needed, factored up to cover all the loss and budget
items. That is what is used for the
rocket equation-based calculation for design mass ratio MR.
Figure 2 below shows a part of the “SSTO” worksheet
and the plots it generates. This image
has been annotated to indicate where the appropriate bands of ascent-averaged
Isp are, corresponding to LOX-LCH4 and
LOX-LH2 propulsion. LOX is liquid
oxygen. LCH4 is liquid methane. LH2 is liquid hydrogen.
Note that there is no indicated feasibility (positive
payload fraction) for inert fractions much above 5%. That low fraction corresponds to an
expendable stage design. And note that
the payload fraction with LOX-LCH4 is substantially less than that with
LOX-LH2. What that really says is
that you actually can build an SSTO to reach low Earth orbit, but only as an expendable, and it cannot be truly competitive unless
you use LOX-LH2 propulsion.
Figure 3 below is an image of the “TSTO”
worksheet, which creates plots of
payload fraction vs stage Isp,
parametric on stage inert fraction,
for both stages. Top left are the
yellow-highlighted user inputs, which
include orbit entry speed and staging speed,
plus the losses to be added to each stage’s effective dV requirement.
The plots and the data are generated automatically from these inputs.
The first stage must ascend through the atmosphere, needing an ascent-averaged Isp for a sea
level-capable engine design, while the
second stage makes its burns essentially exoatmospherically, using a vacuum-capable engine.
Figure 2 – Example Results of the “SSTO” Worksheet
Figure 3 – Image of the “TSTO” Worksheet
There are two plots,
one for each stage. They are both
plots of stage payload fraction vs a range of stage average Isp, parametric on stage inert fraction.
User instructions are given on the page. There was NOT room to include stage and
overall weight statements. Those were
included as a separate worksheet “TSTO wts”.
The example TSTO here is a crude approximation of the
earlier block 1 or block 2 versions of SpaceX’s “Starship/Superheavy” vehicle, that is still in experimental development
flight test, as of this writing. Figure 4 is an image of a png file
containing the two plots from the “TSTO” worksheet, annotated to reflect the vehicle
characteristics also sketched. The old
Windows “Paintbrush 2-D” software was used to generate this png file.
Figure 4 – Example Results of the “TSTO” Worksheet
For each stage, the
payload fraction is read from the plot at the appropriate Isp and inert
fraction. The best way to do this is to
sketch-in the appropriate curve for the inert fraction. Then read up from the appropriate Isp until
you hit that inert curve. Then read
across left to the payload fraction scale,
to estimate the payload fraction.
Once you have the appropriate payload fraction for each stage (along
with its inert fraction), you can go to
worksheet “TSTO wts” and run weight statements for each stage.
Figure 5 shows an image of the “TSTO wts”
worksheet. The user inputs are the
yellow-highlighted items. They are the
payload and inert fractions for each of the two stages, plus an estimate of the lift-off mass. This worksheet creates no plots. This worksheet also creates a sort of overall
weight statement for the entire two-stage vehicle, and computes its overall payload
fraction, that being delivered payload
divided by liftoff mass. User instructions
are listed on the page. If you know
payload instead, just iterate lift-off
mass until you get the payload you desire.
Figure 5 – Image of the “TSTO wts” Worksheet
For the “Starship/Superheavy” example already shown in Figure
4 above, the results are indicated
just below in Figure 6. That is
an image of just the weight statement calculation blocks, annotated.
It is amazing how close these results are, to the actual test vehicles flown so far, given just how crude these input data values
are, that were used here.
Figure 6 -- Example
Results of the “TSTO wts” Worksheet
One could actually run a sort of sensitivity study with this
worksheet and the “TSTO” worksheet together.
One would do this by varying the inert fraction somewhat, and seeing how the rest of the results are
affected. You must do this on both
worksheets together in sequence, because
inert fraction affects payload fraction,
with the propellant fraction set by Isp and required dV. Note that 1 = payload fraction + inert
fraction + propellant fraction.
Critical Issues NOT Addressed By This Level of Analysis
(#1) As indicated in the first paragraph, stage inert fractions are merely
assumed! Verifying these requires more
detailed design analysis of the inerts build-up for an actual vehicle design
concept. The more realistic the assumed
values are, the more realistic the
results.
(#2) Ascent-averaged and vacuum Isp values are simply
assumed in this analysis. To verify
those values, one must rough out the
actual nozzle designs, and engine cycle
characteristics, for his propellant
combination. Better assumed values are
better results.
(#3) In this analysis there is no attempt to size thrust
requirements, and determine the
numbers of engines, their thrust
ratings, and their turndown ratios, for each stage. If the engines will not fit behind the
stage, Isp and dV do not matter, the design is infeasible.
How to Obtain this Spreadsheet File (“stage
studies.xlsx”)
To obtain a copy of this spreadsheet, either contact the author directly by
email, or go to the Mars Society’s New
Mars forums, for the links to a free
download of this, and many other related
things. That website is https://newmars.com/forums/. Once there,
scroll down to the Acheron Labs section,
and select the “Interplanetary Transportation” topic. Within that topic, select the thread titled “orbital mechanics
class traditional”. The links to the
online drop box are in those postings. Those
downloads are free! This particular spreadsheet
is one of the items stored under “lesson 8B”,
post 45.
About the Author
The author is well-qualified to create spreadsheets like
this, or those lessons on the
forums, and much more. He obtained BS and MS degrees in aerospace
engineering long ago at UT Austin, and
then went to work in the defense industry for 20 years, doing mainly rocket and ramjet work, mostly as pencil-and-paper engineering, starting in the slide rule days (see Figure
7 below). He even did some
industrial launch vehicle work while still a graduate student. His majors were aerodynamics, thermodynamics, and propulsion. He obtained a terminal doctorate late in life,
in general engineering.
That aerospace/defense career was cut short by the enormous
defense industry drawdown after the fall of the Soviet Union. The author had to do something else for a
living, being thrown onto the job market
along with nearly 2 million other defense engineers, from an industry that could no longer employ
any of them.
His second 20-year career was mostly teaching, at all levels from public school to
university. This had some civil
engineering and aviation work mixed in with the teaching. He taught mathematics, physics,
and engineering at a variety of institutions. Those included Bosqueville and McGregor high
schools, Minnesota State
University, Baylor University, Texas State Technical College, and McLennan Community College.
The author is now long-retired. He still builds and sells custom farm
implements that he invented, which kill
prickly pear cactus out of farm and ranch pastures, without pickup and disposal, and without chemicals. He also still occasionally consults in topics
like ramjet propulsion.
Figure 7 -- Old-Time
Engineering Design Tools
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Update (same day) 5-18-2026:
The author has added another worksheet to the “stage
studies.xlsx” spreadsheet file, that
does both SSTO point sizing and TSTO sizing versus staging velocity. These are still only bounding calculations, and are no better than the quality of the Isp
and inert fraction values assumed.
These TSTO trends with staging velocity are of interest toward
design, but bear in mind that these values
are “down in the weeds” relative to the effects of both engine Isp values and
stage inert fraction values! There is
still no assessment of stage thrust requirements, or whether the engines would actually fit
behind the stages.
The user instructions for this added worksheet are on the
worksheet. As set up, it analyzes multiple cases. An overall view of this worksheet and the
plots it automatically generates from these cases is given in Figure 8 below. A more
close-up view of the user-input portion of the worksheet is given in Figure 9
below, which is more useful for dealing
with the inputs.
Looking at Figure 9,
the top left yellow inputs are for SSTO,
as 1-shot and as reusable. Those
calculations are simpler, and the
results are highlighted blue in that same little block.
Just below it are the inputs block for the TSTO cases versus
stage velocity Vstg. Vstg is not an input, it is a wide range of values “built-in” to
the worksheet. Although,
they could be changed (those are highlighted green). The cases themselves are the calculation blocks
that stretch far to the right on the worksheet, something more apparent in Figure 8.
The bottom block of yellow inputs are the names and identifying
values for the various cases this worksheet is set up to compute. These are for the different possible stage
propellant selections, and whether the
stage is reusable or not.
The main inputs the user should be concerned with, are the engine Isp values, the stage inert fractions, and the mission dV components that add up to
stage dV requirements. Thos dV values
are currently set up for low circular orbit at 300 km altitude, and low inclination.
The first of the several calculation blocks that stretch to
the right is where the user inputs for speeds,
losses, and budgets are recombined
into the 1-shot and the reusable stage dV values versus staging speed. A plot of stage dV requirements versus
staging speed is one of the plots generated automatically. The other plots, also generated automatically, compare various cases in the format of overall
payload fraction versus staging speed.
Figure 8 -- Overall View of the Added Worksheet “Both”
Figure 9 -- More
Closeup View of the Input Portion of the Added Worksheet “Both”
Watch this site for a future posting that deals with the results
generated by this added worksheet.
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