How the Suborbital “Hopper” Calculations Were Made and with What

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.