The Mars rocket hopper design rough-out was done using the
course materials and tools for the “Orbit Basics +” course offered on the New
Mars forums. There is an “orbit basics”
spreadsheet that does elliptical orbit 2-body calculations for either the __two-endpoints
case__ or the __R-V-q observation case__.
That tool’s R-V-q option can create suborbital trajectories, which was done for the rocket hopper. The spreadsheet calculates speeds V at
periapsis, apoapsis, and at any one user-input radius. **See Fig.
1.**

Figure 1 – The Two Cases Handled by the Orbit Basics
Spreadsheet

To use this tool for the rocket hopper, the most effective way was to define an exit
(and by symmetry entry) point at the edge of Mars’s atmosphere, and investigate various speeds V and exit
angles relative to local horizontal. Not
every combination is allowable, only
certain values produce survivable peak heating and peak deceleration gees, and also a feasible end-of-hypersonics
altitude, for a direct rocket-braked
landing. In fact, many combinations produced instead a surface
impact while still quite hypersonic, in
Mars’s thin atmosphere.

The symmetry of the exposed portion of the ellipse makes the
V and angle “a” values the same for exit and entry, at the entry interface altitude. That is exactly how the suborbital trajectory
analysis links directly to the hypersonic entry analysis.

For the launch speed required of the hopper, we need the speed along the orbit at the
surface of the planet. We need to be
moving that fast at just about the same angle “a”, at the end of the launch burn. That is the theoretical dV^{o} value, which needs to be factored up by about 1.02
to cover gravity and drag losses on Mars.
The factored-up launch dV is the mass ratio-effective value needed for
proper use in the rocket equation. **See
Fig. 2.**

Figure 2 – Using the R-V-q Option in the Orbit Basics
Spreadsheet for Suborbital Trajectories

The ”Orbits +” course covers launch, entry,
descent-and-landing, use of the
rocket equation, and estimating real
engine performance, as well as 2-body
orbital mechanics of elliptical orbits. For the rocket hopper, both __entry__ and __descent-and-landing
apply__, using the methods and tools
that are part of the course. The direct
rocket-braked landing is so simple, it
can be estimated from hand calculations.

The entry analysis is a 2-D Cartesian simplified analysis
dating to about 1953, and attributed to
H. Julian Allen. It was used in the
1950’s for estimating entry conditions for ICBM and IRBM warheads. It was declassified by the mid-1960’s, and then taught in engineering school
classes. Entry is presumed to happen
along a straight line trajectory at a fixed entry angle. The range is a crude estimate that you must
wrap around the curved surface of the planet.
The constant angle you have to presume is relative to local
horizontal, as you move around the curve
of the planet’s surface.

These crude estimates get you “into the ballpark” only! There is no substitute for a real digital
trajectory program in polar coordinates,
but you do have to expend the significant efforts to construct the model
to run in it. *At this stage of the
game, that is very inconvenient, since the model to be input changes
drastically as you iterate configurations.
Hence the need for a quicker ballpark estimate.*

There is a lesson in the “Orbits +” course that deals with
using the simplified entry analysis as a spreadsheet model of the entry process. That spreadsheet is supplied as part of the
course materials.

In reality, there is
significant trajectory “droop” after the peak deceleration gees point, that the simplified analysis does not account
for. I merely presume the local angle
has increased to 45 degrees down, by the
Mach 3 end-of-hypersonics point, when I
do the rocket-braking by-hand calculations.

There is also a lesson in the “Orbits +” course that deals
with multiple ways to land after the hypersonics are over. There is no spreadsheet, but all the calculation equations are there
to estimate any of these things by hand.
For the thin atmosphere of Mars,
from inevitable very low end-of-hypersonics altitudes with multi-ton
vehicles, there really is only direct
rocket braking as a feasible thing to do.

There is no time to deploy a chute, much less get any deceleration from it, plus there are no chute designs capable of
surviving opening at Mach 3. Even the
ringsail chute designs used for probes at Mars have a maximum opening speed of
Mach 2.5, and slower-still is preferred
as more reliable.

Direct rocket braking is actually the simplest case, and easily figured with nothing more than the
simple kinematics of a high school-level physics course. **See Fig. 3.**

Figure 3 – The Entry Model,
Plus Descent-and-Landing for Direct Rocket Braking

The vehicle layout and dimensions, plus its weight statement, are essentially custom hand
calculations, the suite of which is
different for each different configuration class. I started with three configurations, but only one gave me the low ballistic
coefficient that the entry analyses said I must have. I included wide-stance folding landing legs
for rough-field operations.
Clearly, there are a lot of
considerations to address. I created a
custom spreadsheet to estimate all these quantities rapidly, since I had to iterate multiple times before
identifying a feasible solution.

The “Orbits +” course has a lesson on vehicle layout, and a spreadsheet by which to set the weight
statement, but that spreadsheet was not
really suitable for this very specialized suborbital vehicle, especially since it must enter the
atmosphere, and also do that entry dead-broadside
to get the necessary lower ballistic coefficient. It is critical to select the correct diameter
for this kind of vehicle, so that the
lengths are in the correct range, and
those results must be compatible and consistent with the seating arrangements
in the passenger cabin. That’s why I did
it as a custom calculation, and why I
created my own spreadsheet for that purpose.
**See Fig. 4.**

Figure 4 – Downselecting to One Configuration for Vehicle
Layout

All of this is aimed at using the rocket equation to relate
vehicle weight statement to its velocity-increment (dV) performance
capability. The spreadsheet in the
lesson on vehicle sizing of the “Orbits +” course does exactly that, in a spreadsheet that is supplied as part of
the course materials. Since I did the
hopper with a custom layout sheet, I had
to include this rocket equation stuff in it.

The classic rocket equation dV = Vex LN(MR) uses the vehicle
weight statement (from a vehicle layout process) to determine mass ratio MR =
Wign/Wbo, and an estimate of engine Isp
to determine the effective exhaust velocity Vex = Isp * gc. It then gives you the performance estimate
dV, which must cover the mission needs
plus any gravity and drag losses, or
other considerations, such as hover and
divert during landings.

__There is a restriction on this__: you may sum the dV values estimated for all
the mission burns into an overall mission dV,
__only__ if the weight statement __does not change between burns__. That means the payload and inert masses do
not change, and the only propellant mass
changes are those for the burns. Failing that restriction, you have a slightly different weight
statement each time one of those items changes.
You must do a separate rocket equation calculation for only the burn
associated with each slightly-different weight statement. This hopper does not change its weight
statement between burns!

For sizing vehicles,
the reverse process is what we really want to do, for which the rocket equation rearranges to
MR = exp(dV/Vex). The engine Isp
estimate gets us a Vex as before. The
mission dV is as before. __The layout
gets us a payload mass and an estimate of vehicle inert mass fraction__. We use the rocket equation in reverse with
the mission dV and the engine Vex to determine the MR that is required.

This MR result determines the propellant mass fraction = 1 –
1/MR. The payload fraction is 1 –
propellant fraction – inert fraction.
Payload divided by payload fraction is the ignition mass, ignition mass times the inert fraction is the
inert mass, and propellant fraction
times ignition mass is the propellant mass.
Payload plus inert is burnout mass,
and burnout plus propellant is ignition mass. __In effect,
we are finding the vehicle weight statement from mission dV and engine
performance to complete the vehicle layout process__. **See Fig. 5.**

Figure 5 – Using the Rocket Equation Properly to Size
Vehicles to Missions

Clearly, an accurate
estimate of expected engine performance (as Isp or Vex) is crucial to the
results! There are a lot of references
out there that list tables of Isp versus propellant combinations. Just picking one right out of such tables is
a serious error! That is because engine
Isp depends at least as much on the nozzle expansion characteristics, as it does the propellant combination. The expansion in the table is rarely the one
you want to use, and nozzle efficiency
effects are __never included__ in those tables.

These things are all functions of the chamber pressure, as measured at the nozzle entrance. __The chamber pressure value used in the
tables is rarely the value you want to use__.

Finally, Isp is
directly affected by the engine cycle (through the dumped bleed gas fraction), which those tables __never include__. You can easily be 10%-or-more wrong just
pulling values out of those tables. Due
to the exponential nature of the rocket equation, that error in Isp can lead to fatal errors in
your vehicle results for mass ratio and weight statement.

Thrust is often represented in terms of chamber pressure as
F_{th} = C_{F} Pc At.
Isp is thrust divided by flow rate,
__but it has to be the flow rate drawn from the tanks to be consistent
with the rocket equation__. Flow rate
from tanks = flow rate through nozzle + flow dumped overboard. The flow rate through the nozzle relates to
chamber pressure and c*-velocity as Pc C_{D} At g_{c} /
c*. And c* is a weak power function of
Pc, where the exponent is usually in the
vicinity of 0.01. The specific heat
ratio of most rocket gases is in the vicinity of 1.20. **See Fig. 6,** for which the __only__ propellant
combination-related item is c*.

Figure 6 – How Engine Performance Must Really Be Estimated
for a Specific Design

You are not totally free to set an arbitrary expansion ratio
Ae/At! It does not matter whether your
nozzle is a “sea level” design or a ”vacuum-adapted” design, any engine that is to be tested in the open
air at sea level on Earth must not be allowed to flow-separate, because that risks destruction of at least
the nozzle exit bell! __Testing into a
vacuum tank is extremely expensive__!

For any given expanded pressure in the exit plane, there is a value of the ambient atmospheric
“back pressure” Pback that is “too much”,
causing flow separation. That
level is denoted Psep, and it is easily
estimated from the nozzle expansion pressure ratio: Psep/Pc = (1.5 Pe/Pc)^{0.8333}, which is an entirely empirical correlation
developed for conical nozzles, and is
only slightly conservative for curved bells.

For a “sea level” nozzle design, you want predicted Psep = sea level
barometric, at some part-throttle
Pc. That way, you can test in the open air for all power
settings that high, or higher. The same is true of “vacuum-adapted”
designs, unless you give up testing in
the open air! But even then, the engine and its nozzle still have to fit
within the allotted space behind the stage.

The “Orbits +” course has a lesson on this topic, plus a spreadsheet tool that does all these things. It includes a database of c* and r-value data
versus several propellant combinations,
as functions of Pc.

**Updated 11-21-2023:**
These very same methods were used to compute revised data for the upgraded
Mars rocket “hopper” that could also serve as a personnel taxi to low Mars
orbit.

The original suborbital rocket “hopper” design summary was
posted on this site as “Rocket Hopper for Mars Planetary Transportation”, dated 1 November 2023. The upgraded “hopper” that can also serve as
an orbital taxi is posted on this site as “Upgraded Rocket Hopper as Orbital
Taxi”, dated 21 November 2023.

There is a completely unrelated posting that deals with
long-distance bulk freight transport on the surface of Mars. That one is “Surface Freight Transport on
Mars”, dated 4 November 2023.

The final landing choice not described here is the lifting
pull-up proposed by SpaceX for landing its Starship vehicle on Mars. That is distinct from direct rocket
braking, and from parachute-assisted
descents, which require terminal rocket
braking on Mars. It is covered in the
entry, descent, and landing lesson of the “orbits +” course
materials.

I did not examine that choice for any of these rocket
“hopper” designs, because I did not
believe that my cylindrical layout has the mild-supersonic lift/drag ratio
necessary to execute an aerodynamic pull-up,
even at very low altitudes on Mars.
I don’t really believe SpaceX’s Starship can do that either, but that would be another study.

To access the “orbits +” course materials, which includes the spreadsheets, go to the Mars Society’s New Mars forums
online. Go to the “Acheron Labs”
section, “interplanetary transportation”
topic. On about the second page of the
list of conversation threads, ** look
for the “orbital mechanics class traditional” thread.** The course materials are actually posted elsewhere
online,

__but the links to each class session’s materials are in posts 3-to-21 of that thread__.

You will have to download the Excel spreadsheet files to
make them functional. The classes have a
sort of lecture session (numbered) and a problem-working session (numbered with
a “B” suffix). These are available as
Powerpoint slide sets and as pdf documents that are basically the
traditional-style textbooks. I recommend
you download the pdf textbooks, because
all the explanations are in there. They
would be partly missing in the slide sets.