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. 

Upgraded Rocket Hopper as Orbit Taxi

This article is about modifying a pre-existing design rough-out for a suborbital Mars rocket “hopper”,  into a design also capable of operating as a low Mars orbit personnel taxi.  That original rocket hopper design rough-out is covered in the article titled “Rocket Hopper for Mars Planetary Transportation”,  dated 1 November 2023,  and posted on this site.

 

               The Problem

Started with a suborbital “hopper”

               10 persons aboard on p-suits

               Short-term life support plus small luggage

Could it also serve as a low orbit taxi?

               Same payload

 

As indicated in the table just above,  I started with the earlier design rough-out that was only a suborbital “hopper”.  The idea was to carry 10 persons as the payload.  Although the cabin is pressurized,  these persons ride in pressure suits for a safety backup.  There are limited supplies of oxygen and drinking water,  plus minimal snack foods,  for up to a few hours’ ride.  A small luggage allowance was included.  The same payload would be carried to any low orbit destination.

As indicated in Figure 1 just below,  the suborbital trajectory is actually an ellipse in polar coordinates,  one with its periapsis inside the planet.  The vehicle launches into a gravity turn that reaches a suitable velocity and path angle at the entry interface altitude,  coasting from there. 

The best place to do a course correction is the apoapsis outside the sensible atmosphere,  where speeds are lowest and directions are easiest to change.  The entry conditions mirror the exit conditions,  with no burn.  The landing is a direct rocket-braked descent from the end-of-hypersonics point at local Mach 3 (about 0.7 km/s speed). 45 degrees of trajectory “droop” along a straight-line path is presumed.  I factored-up the speed to “kill” by 2,  to budget the final landing mass ratio-effective delta-vee (dV).

As illustrated in Figure 2 just below,  I used a surface-grazing ellipse as the initial transfer trajectory to the 300 km nominal low orbit altitude.  Like the long-range suborbital mission,  the vehicle launches into a gravity turn,  putting it onto the proper path at the entry interface altitude,  at end of launch burn.  Only the path angle is different,  being a lot smaller.  The entry point after de-orbiting is the mirror image. 

There is a small burn at apoapsis to raise the periapsis to the entry interface altitude,  with a period shorter than the target low circular orbit altitude.  This ellipse is the parking orbit in which to “chase” any target in the low circular orbit.  Once synchronized,  there is another small burn to circularize,  followed by a traverse to rendezvous,  plus a budget to actually dock.  Deorbiting is another small burn,  back onto the surface-grazing ellipse that guarantees entry.  The direct rocket-braked landing is identical to that of the long suborbital trajectory,  except that,  as it turned out,  the end-of-hypersonics altitude is higher,  coming back from orbit at the lower entry angle. 

Figure 1 – Suborbital Missions,  Longest-Range Shown  

Figure 2 – The Orbital Mission,  Including “Chase”,  Rendezvous,  and Docking  

To accommodate the more demanding mission,  I resized the candidate LOX-LCH4 engine design,  and revised the inert masses upward a little.  Entry conditions forced me to increase the diameter and length a little,  in order to keep the entry ballistic coefficient down to tolerable values.  The original rough-out had two sets of tanks:  mains and headers.   The landing and course correction propellant was in the headers,  with the launch propellant in the mains.  

This became 3 sets of tanks and two different engine designs.  The launch-and-landing main engines stayed about the same at 30,000 lb thrust,  each of 4,  drawing from the mains for launch and headers for landing.  I was able to increase the expansion ratio and specific impulse a little bit.  See Figure 3 for the basic layout revisions.

But course correction suborbitally,  and all the orbital maneuvering,  rendezvous,  and docking,  really needed much lower thrust levels.  So I sized some lower-pressure,  pressure-fed engines of only 550 lb thrust,  each of 4.  These used a small third set of 800 psi pressurized propellant tanks,  plus a supply of dry nitrogen gas at 2200 psi to power this,  in one of two options examined.  

Figure 3 – Revised Internal Layout at Larger Diameter and Length  

Because the inert fraction increased a bit,  I resized the expansion of the main engines to increase specific impulse a bit,  to compensate as much as possible.  The original “hopper” main engines had an expansion ratio sized for incipient separation at 67% chamber pressure,  if fired in the open air at sea level on Earth.  I raised that to 80%.  See Figure 4 just below.

The idea was to enable easy and relatively inexpensive development testing on Earth.  The change was small,  but every little bit helps.  These being full flow cycle,  turbo-pumped engines of significant thrust level,  I did not want to complicate things by adding vacuum bell extensions that were not regeneratively cooled.  These do not push the state of the art very hard,  being only 2000 psia chamber pressure.

Figure 4 – Reworked Main Engine Design for Slightly Higher Expansion 

The original “hopper” design study convinced me that I did not need the large main engine thrust levels to do course corrections on the suborbital missions,  or orbital maneuvering,  rendezvous,  and docking,  on the orbital mission.  I kept the redundancy of 4 engines,  but sized for crudely only 0.1 gee of vehicle acceleration,  once exoatmospheric. 

Since the propellant quantities would be small,  the simplification of a pressure-fed design would be beneficial.  Alternatively,  since the engines were small,  they could be fed by electric-driven positive-displacement pumps.  Either way,  I picked a simple conical bell shape,  two-piece,  with a bell extension that was not regeneratively cooled,  as shown in Figure 5 just below.  Development testing on Earth could be done without the extension,  but operations on Mars or in space would use the bell extension.  This was not a throttleable design.

I ran numbers both ways for the propellant feed to the maneuvering engines.  I did not like the pumping power required for the positive-displacement pumped option.  It implied very heavy batteries,  even for the modest propellant quantities.  Regulated constant inert gas pressure on the propellant tanks turned out to be the better option.  These used a small third set of 800 psi pressurized propellant tanks,  plus a supply of dry nitrogen gas at 2200 psi to power this.  The chamber pressures were low enough to keep the pressures fairly modest on the tanks,  so that at small size,  they were not that big an inert weight penalty.  See Figure 6 just below. 

There were many false starts and iteration cycles to achieve all of this,  none of which is covered here.  The result is summarized in the unavoidably-busy figure,  Figure 7 just below,  which includes a weight statement that also displays mass fractions.  

Figure 5 – Smaller Maneuvering Engines as Sized   

Figure 6 – Of the Options,  Pressurized Tanks Seemed Best  

Figure 7 – Summary Data for the Final Rough-Out Design 

Of interest would be the various tank volumes.  Bear in mind these are fully filled for the mission to low circular Mars orbit at only low inclinations eastward,  and also fully-filled for the long-range suborbital mission (at low or high inclinations).  The headers and maneuvering tanks are always fully-filled,  but the mains are only partially filled for the shorter-range suborbital missions,  so that entry mass is not too big.

Suborbital ranges from 9400 to just under 500 km were examined in this study.   Their entry angles turned out to be a strong function of the suborbital mission ranges.  All of those suborbital entry angles were considerably steeper than the return from the orbital mission.  They were determined by feasible altitudes at end-of-hypersonics,  and by feasible peak entry gee values.

Figure 8 just below shows the final plots I got of various flight data during entry and final descent.  The Suborbital trajectories form trends,  and the orbital data fall way off those trends.  Upper left is end-of-hypersonics altitude and entry angle vs entry speed.  Upper right are the peak heating values during entry.  Lower left are the trends of peak entry gee,  and average gee during the final rocket-braked landings.  There is a numbered key relating the missions to each data point in each plot.  No gee level exceeds 4.5,  and no stagnation heating level exceeds 12.5 W/cm².  

Figure 8 – Data for Entry,  Descent,  and Landing (E, D, & L)  

Once again in Figure 9 just below,  the suborbital data for surface temperatures also form trends versus entry speed,  with the orbital data falling far off of those trend lines,  plus a numbered mission key.  There are trends for surface temperatures at the stagnation line,  temperatures for its lateral surfaces where flow is still attached,  and temperatures for leeward separated wake zone surfaces.  These were figured for thermal re-radiation exactly balancing convective-only input,  as figured for a “dark” highly-emissive surface,  of thermal emissivity 0.8 as representative. 

Note that with the exception of only the longest-range suborbital mission,  all the rest of these data are under 1600 F,  and would permit exposed-metal construction of 316L stainless steel,  or of Inconel X-750,  or something in that same class!  And that even includes the return from the orbital mission!  Because of the stagnation zone temperature approaching 2000 F on the longest-range suborbital mission,  there needs to be some minimal heat protection in and near the stagnation zone.

In Figure 10 just below,  the format for surface pressures is similar:  trends of suborbital surface pressure vs entry speed at stagnation,  at lateral sides,  and in the separated leeside wake.  The orbital data again fall far off the trend lines.  There is a numbered key to relate missions to individual data points.  Note that no mission,  not even the longest-range suborbital,  exceeds 0.19 atmosphere anywhere.  That would be about 2.79 psi,  very modest indeed. 

Given the hot material strengths reported as part of Figure 9,  that means even a fragile extreme-low-density alumino-silicate ceramic composite could serve as heat protection.  So could ceramic fabric blanket or quilt-type materials,  if they survive wind shear.   Even a thin sheet of 2000 F-capable metal overlying mineral wool insulation would serve,  mounted only locally near the stagnation line. 

Conclusion:  the “hopper” could easily be designed to also serve as a low orbit taxi!

Figure 9 – Trends of Surface Temperatures vs Entry Speed  

Figure 10 – Trends of Surface Pressures vs Entry Speed  

Update 11-22-2023:  The following Figure 11 illustrates exactly why the surface emissivity must be high (a very dark or black surface color,  with a dull finish).  There are exposed metallic construction solutions and a refractory solution with simple alumino-silicate ceramics,  especially away from the stagnation zone,  if emissivity is high.  There ae only ablative solutions available if emissivity is low.

Figure 11 --  More Detailed Hopper/Taxi Heating Data

Surface Freight Transport on Mars

I have seen many notions for surface transport discussed on the New Mars forums,  mostly in the planetary transportation topic.  Nearly all of these suffer from the very serious downside of requiring the emplacement of significant infrastructure on Mars,  something very effort-intensive and very expensive,  here on Earth.  With transport costs to Mars being “astronomical” for the foreseeable future,  that is a fatal requirement for anything we might consider on Mars.

What one really needs to do first is determine a very fundamental characteristic:  is the cargo time-sensitive,  or is it not?  Transport of people over long distances is a very time-sensitive problem,  for which the solution is some sort of flight,  as Figure 1 suggests.  Most bulk freight is not time-sensitive at all,  and can travel very slowly,  similar to scheduled rail freight here on Earth.  Considering the expensive infrastructure issue,  what we need is a train that does not need any tracks. 

That small portion of freight to be transported that is time-sensitive is likely to be medicines and similar:  small packages that can fly with the people.

Figure 1 – Time-Insensitive Freight Should Go Slow,  On The Surface

The critical thing here is to minimize the infrastructure we must emplace to get any of these transportation ideas going.  The more you have to build,  the more it is going to cost,  for both the efforts,  and the materials,  plus the equipment with which to build it.  That’s true here on Earth,  which is why many of these ideas have never really been widely built,  even here.  On Mars,  it will cost a lot more still,  because of the interplanetary transport costs,  plus the development costs of re-making the materials and processes into those that will actually work on Mars.  As indicated in Figure 2,  about the only thing we have available that would actually work on Mars,  too,  would be graded dirt roads,  with “truck trains” moving on them.  True rail would cost less to run,  being lower friction,  but we have no way to manufacture steels on Mars,  including the extreme-cold adapted steels necessary there,  and no way to make the ties,  whether from steel,  concrete,  or wood.

Figure 2 – The Real Requirement is That There Be Almost No Infrastructure to Emplace

So,  it is the slow-moving “truck train” hauling the time-insensitive freight on graded dirt roads that we need for Mars.  This is really just a tractor pulling a string of freight wagons,  but internal combustion engines as we know them here on Earth are “right out”.  You not only have to carry the fuel,  but also the oxygen with which to burn it,  which considerably outweighs the fuel.  And unless you want to completely redesign the engines to handle 500-1000 C higher flame temperatures,  you will also have to carry diluent gas,  which in turn far outweighs the oxygen.  So,  in any practical sense,  you are looking at electric-powered tractors.

These can be solar electric powered,  however.  On a sunny day here on Earth at lower latitudes,  there is roughly a horsepower’s worth of energy per square yard falling on the ground.  At Mars,  there is about half that.  Call it 300 W per square meter as a rule-of-thumb.  Solar electric conversion efficiencies are around 20%,  so about 60 W per square meter of collector surface could be had,  for much of the day.  The trick is then to fit the freight wagons with solar panels on their roofs,  all connected electrically to power the tractor.  You just go very slow to stay within the power your solar collectors can generate.  But,  most bulk freight is very time-insensitive,  so who cares?

This thing doesn’t need a crew,  it can be self-driving between the spoil berms created from grading the road.  If programmed with a keep-right feature,  traffic on these roads can be two-way.  The basic characteristics and features are summarized in Figures 3 and 4. 

Figure 3 – The “Truck Train” Concept

Figure 4 – This Is How It Is Controlled

Now we need to verify feasibility with some numbers.  Trucks and train cars here on Earth are about 1/3 structure and 2/3 payload,  sometimes ¼-3/4.  Call it 30-70,  as a rule of thumb.  For a 100 metric ton loaded freight wagon,  you are looking at 70 tons of payload,  and 30 tons of chassis,  wheels,  drawbars,  containing-structure for the freight,  and solar panels on the roof. 

These things could use the very same giant rubber tires we use for off-road construction and mining work here on Earth,  but we may need to heat them slightly,  to prevent their cracking in the cold on Mars.  That can be done.  Tires on relatively smooth,  firm dirt have crudely 10 times the friction coefficient of tires on a paved road.  Using the Mars weight of a 100-ton loaded freight wagon,  augmented for climbing a 10% grade,  the drawbar power (drag x speed) is 20 KW at 0.33 m/s,  and quite a bit higher as you go faster.  There’s room for at least around 300-ish square meters of collectors atop each wagon.  Which in turn is why you select the lower of the speeds given in Figure 5.  

Figure 5 – This Is Why It Works (By the Numbers)

You can pull many freight wagons with a big powerful tractor.  Numbers are given in the figure for 10 and 100 wagons.  Power at the tractor is between 0.2 to 2 MW for these numbers.  That’s about like the power of a big mine loader as currently built here on Earth (somewhere near 1000 HP).  Which in turn means the same electric tractor that can pull the “truck train” can also be the road grader! You just power it differently.

The kind of thing I have in mind as the basis for designing the Mars tractor is pictured in Figure 6.  That is a big mine loader vehicle.  These are very tough,  very powerful machines.  That is the very thing we need for doing road grading,  and pulling heavy trains of freight wagons.  We just have to make it work on Mars,  in the cold and the near-vacuum.  That is why electric propulsion is preferred,  and a pressurized operator’s cabin is required. 

Figure 6 – The Mars Tractor Is a Known Revision of Something Like This That We Already Build

The basic notion is illustrated in Figure 7.  Something similar to a big mine loader is the design basis.  You remove the dump hopper and replace it with a battery bank and solar panels.  You remove the diesel power plant and replace it with the appropriate electric drive motors and gearing.   You replace the operator cab with a pressurized operator’s cabin for use on Mars. 

Rigged with a blade,  and powered by a nuclear generator aboard a shielded trailer,  you operate it manned for grading the road.  Unmanned and self-driving,  you pull the “truck train” with it,  powered by the solar panels on the freight wagons.

It will have to be manned for grading the road.  Even here on Earth,  “dirt work” is something that has so far proven to be not-automatable,  or it already would have been automated.  That’s slow multi-pass work,  with the operator’s judgement and skill showing up,  as knowing when the work is done “right” by the appearance of what he has done.  This aspect will be the same on Mars as it is here.

It does not have to be manned to pull the “truck train” on the finished roadway.  You remove the blade,  and delete the nuclear power trailer.  However,  I would leave the pressurized cabin in place and rigged with supplies,  for the off-chance unforeseen event that would require a human driver.  

Figure 7 – Converting a Mine Loader Design into a Mars Tractor Design That Does Two Jobs

The only real trouble is shipping such large pieces of equipment to Mars.  These will have to be shipped as individual components,  and assembled there on the planet Mars. That will be true until some very large interplanetary vehicles indeed,  eventually become available.  But it can be done,  with what we know and have available,  right now!

Rocket Hopper for Mars Planetary Transportation

The future presence of bases and settlements upon Mars brings the need for transportation of freight and people about the planet.  A little thought reveals the two categories,  freight and people,  are fundamentally different in their requirements.  Most freight is not time-sensitive,  while people are. 

Freight not time-sensitive needs to go by slow surface transportation,  without emplacing expensive and effort-intensive infrastructure to make that possible.   Such infrastructure is often very expensive here,  and the costs there are likely to be quite catastrophic.  Freight needs something similar to rail transport here,  but without the tracks.  A robot “truck train” on a graded dirt road is the answer.

People are time-sensitive,  and need to fly in order to cover large distances rapidly.  But physics in an atmosphere so thin,  that it is first cousin to the vacuum of space,  rules out air transport by airplanes or helicopters as we know them on Earth.  It also rules out any form of “lighter-than-air” transport,  since buoyant lift forces are proportional to differences in densities,  and those are so vanishingly-low on Mars.  That pretty much leaves suborbital rocket travel (the “rocket hopper”),  the topic of this document.  The same could be used for that tiny portion of freight that really is time-sensitive.

Suborbital Trajectories

I used my “orbit basics” spreadsheet,  “R V q orbits” worksheet,  to model suborbital trajectories about the planet,  using ellipses that lie mostly within the planet.  It was necessary to sharply limit the trajectory angle below local horizontal at entry interface altitude (135 km for Mars),  in order to limit peak entry deceleration gees and peak heating,  and to achieve altitudes from which rocket-braked landings were feasible in terms of timelines and deceleration gees.  The results I was able to obtain are shown in Figure 1.  The longest-range of these trajectories is very nearly antipodal (10,637 km). 

Figure 1 – Suborbital Trajectories

Entry,  Descent,  and Landing

I used the “entry spreadsheet” spreadsheet file,  worksheet “Mars variations”,  to compute estimates for the worst case (highest entry speed) entry trajectory,  using the old 1953-vintage analysis used for warheads by H. Julian Allen and his colleagues.  This stuff was declassified and taught in engineering schools in the late 1960’s.  It is not the most accurate thing to use,  but it gets you “well into the ballpark” with very simple,  essentially-by-hand,  calculations.  These are easily put into a spreadsheet.

The highest speed at entry is the most challenging,  since all the entry angles are about 15 degrees,  so that analysis is depicted in Figure 2.  It was crucial to get the hypersonic ballistic coefficient of the hopper vehicle down under 200 kg/sq.m,  and preferably under 180 kg/sq.m.  It proved feasible with the final design to get well under that requirement with the value shown in the figure,  although most design approaches will fail in that respect!  Getting the heating down to something easily handled required the largest-possible “nose radius”.  The value shown was dictated by the design approach taken here,  and is quite easily handled,  although it is too much for metal exposed at the stagnation zone. 

Figure 2 – Estimates for Entry Conditions

The hypersonic aerobraking is largely over once the vehicle has decelerated to about local Mach 3,  which is roughly 0.7 km/s speed in the Martian atmosphere.  Slower than Mach 3,  a blunt object is no longer hypersonic,  and the assumption of a constant ballistic coefficient fails.  It would be folly to continue the hypersonic estimate past that point,  for that very reason. 

Not included in the approximate analysis is the effect of trajectory “droop” to steeper angles due to gravity.  That mostly happens after the peak deceleration point,  which is actually rather close to the end-of-hypersonics (Mach 3) point.  Peak heating occurred slightly earlier.  For purposes of “reasonable approximation”,  I just used the hypersonic endpoint altitude as obtained,  but I presumed the trajectory was headed about 45 degrees downward at the Mach 3 point. 

Ignoring the effect of the potential energy associated with the Mach 3 point altitude,  it is the same Mach 3 point speed of 0.7 km/s that we have to “kill” with last-second rocket braking,  regardless of that altitude.  Appropriately factored for losses and maneuvering,  that is the rocket braking delta-vee (dV) that we must have.  I used factor 2,  and would never use less than 1.5 under any circumstances,  in order to account for losses,  plus hover and divert requirements.  The Mach 3 point altitude essentially determines how much time we have left before surface impact,  which sets the required average gees,  and thus the thrust required for any particular vehicle mass. 

I used simple high school-level geometry and physics/kinematics to establish the average deceleration gees during the rocket-braking landing.  This is a very simple hand calculation,  illustrated in Figure 3.  The 6 km altitude translates to an 8.5 km path length down a straight line at 45 degrees.  At an undecelerated constant 700 m/s,  we are about 12.1 sec from impact,  as shown.  Thus there is far too little time available to deploy a chute,  much less expect any deceleration from it. 

If we rocket-brake decelerate to zero,  the average velocity down the path is only 350 m/s,  and we have 24.3 sec to touchdown,  as shown.  The change in speed is the 700 m/s.  The change in time is the 24.3 sec.  Their ratio is the average deceleration required,  which is 28.8 m/sec²,  or some 2.94 standard gees.

Figure 3 – Hand Calculations for the Rocket-Braked Landing

Therefore,  we are looking at roughly a 3-gee rocket-braked landing,  with the gees felt over an interval 24-25 sec long.  That is easily handled by persons not trained in resisting gees,  if seated,  even more so if reclined.  Roller coaster riders endure worse all the time.

The hypersonic aerobrake peak gees fall in the 6.6 to 6.7 range,  but again for a short interval above 5 gees that is only around 20-25 sec long,  as well.  That is more difficult,  but it is endurable,  even by untrained persons as long as they are fully physically fit,  as also experienced by some roller coaster riders.  It is anticipated that passengers will be riding while wearing some sort of pressure suits.  It would help if these pressure suits had “gee-suit” features as well.  Otherwise,  some passengers might temporarily faint,  if sitting up.  Any crew must be trained to endure such gees,  and they must be wearing suits with “gee-suit” features.

“Rocket Hopper” Vehicle Design Concept and Estimates

I looked at 3 classes of possible design configurations trying to meet the requirements of low hypersonic ballistic coefficient and large “nose radius” simultaneously.  Only one approach satisfied those needs,  and still offered ways to mount landing legs for rough-field operations,  plus a simple unobstructed engine bay.  That was the cylindrical stack depicted in Figure 4,  but flown dead-broadside to the oncoming stream during entry! 

If the cylinder L/D ratio falls in the 4-to-6 range,  that is enough larger blockage area to greatly-reduce the hypersonic ballistic coefficient,  despite the low hypersonic drag coefficient of the cylinder shape.  That shape was required to keep tank construction lightweight.

As for the tanks,  these are main tanks that are integral components of the vehicle airframe,  but they also contain header tanks.  As the numbers worked out,  about 15% of the tank volume is in the headers,  for course correction and rocket-braking,  with the other 85% in the main tank volume for launch. 

Figure 4 – Sketch Layout and Characteristics of the “Rocket Hopper” Design Concept

As shown,  the smallest item was the engine bay,  and the largest item the cabin in which people ride.  The tanks are stacked in the middle,  so that center-of-gravity travel is not very large as the propellants burn off.  The figure shows the layout,  a weight statement,  seating arrangements,  the basic trajectory-related notions,  and some entry heat protection numbers. 

Most but not quite all of the surface of this craft could be exposed metal construction,  if something like a 316L stainless steel or an Inconel X-750 is used.   Only near the stagnation line on the windward side is something more heat-resistant required.  That could be a strip of ceramic tiles of some kind.  Even low-density alumino-silicate ceramics could be used,  if blackened for high emissivity,  as peak entry pressures are actually quite low.

Peak entry pressures are easily rough-estimated by simple hand calculations at the peak deceleration gee point in the hypersonic entry trajectory.  If you know the mass at entry,  the peak gees acting upon that mass give you the peak force decelerating the vehicle.  Dividing that force by the frontal blockage area gives you the average pressure applied to that area.  The peak is at the stagnation zone,  crudely twice the average value.

To run some of these numbers,  I did create a custom “rocket hopper” spreadsheet.  It has multiple worksheets,  of which two are relevant here.  Worksheet “veh” is set up to make the calculations illustrated in Figure 5.  Worksheet “sections” is set up to make the calculations shown in Figure 6. 

In worksheet “veh”,  user inputs are yellow,  significant outputs are blue,  and things requiring iteration or verification are green.  (The same color code applies to all my spreadsheets.)  It works in terms of mass ratio-effective dV,  which is the end-of-burn V multiplied by an appropriate factor to account for gravity and drag losses (about 1.02 on Mars).  

Figure 5 – Numbers Run for the Design Concept,  Part 1

The 3.6 km/s orbit speed at launch multiplied by that factor is the 3.672 km/s dV shown.  The course correction budget is 8% of the orbit apoapsis speed,  which is where corrections should be made.  For the highest-speed case,  that is the 0.215 km/s shown.  And at factor 2,  the 1.4 km/s dV is what the rocket-braking burn must be capable of,  the same for all cases. 

I used a launch liftoff 1.5 gees as required for initiating good ascent kinematics,  same as here on Earth.  0.1 gee for course corrections is probably just a lower limit.  I used 3.5 gees at landing to get some margin over the average 3 gees determined above.

The specific impulse input of 352 sec shown is justified by the analysis given below.  The sum of the dV values is the total dV to be delivered for the mission,  which sets vehicle mass ratio and propellant mass fraction.  1 – propellant fraction – inert fraction is the available payload fraction.

There is an “inert mass fraction build-out” block shown.  We can argue about the component inputs,  but their sum is likely “in the ballpark” no matter exactly what inputs one uses.  The same is true of the size payload block.  You are looking at the weights for a person,  his pressure suit,  a few hours worth of oxygen,  water,  and food,  plus some luggage.  It’ll be around 0.2-0.25 metric tons per person,  almost no matter what,  when you sum it up. That and how many people are aboard,  sets the payload mass,  which ultimately sets the weight statement.

The “heat shield re-radiation” block presumes convective heating is balanced by re-radiation from hot exposed surface materials.  It uses the peak heating from the entry analysis,  plus good guesses for surface emissivity and the effective temperature of the surroundings that must receive that re-radiated heat.  A highly-emissive surface is typically 0.8,  while low is 0.2.  400 R for the surroundings is -60 F.

The “run weight statement and size thrusts” block does exactly that.  Payload mass divided by payload fraction is ignition mass.  Ignition mass time inert fraction is inert mass,  and times propellant fraction is propellant mass.  Payload plus inert is burnout,  and burnout plus propellant is ignition.  Each burn has a dV that sets its mass ratio.  That in turn sets start and end-of-burn masses,  the difference being propellant used for that burn.  The sum of those propellant masses used must equal the total propellant mass already figured. 

There is a “check pressure on heat shield” block that uses the hypersonic ballistic coefficient and the hypersonic drag coefficient as inputs,  plus the initial mass at entry,  and the peak entry gees.  It computes the mass per unit blockage area from the ballistic coefficient and drag coefficient,  then the peak entry force from that and the entry mass and peak entry gees.  It then divides force by blockage area for average pressure,  and doubles that for the stagnation pressure estimate,  reported in a variety of units of measure.

The ”heating other locations” block gets you equilibrium temperatures in degrees F for stagnation,  “typical” lateral,  and separated wake zone locations.  The stagnation peak heating is reduced by a factor of 3 for “typical” lateral surfaces,  and by a factor of 10 for separated wake zone surfaces.

The only other block shows launch dV available as a function of percent max propellant loaded on board.  The course correction and landing dV values are not changed.  This would be useful trying to relate range to propellant load required.

Figure 6 – Numbers Run for the Design Concept,  Part 2

The “sections” worksheet works out the proportions of the engine bay,  tankage,  and cabin sections of the ship,  plus some other pertinent results.  It needs the r-ratio (oxidizer/fuel fuel mass ratio) for the engine,  and the specific gravities of the fuel and oxidizer materials.  It also needs as inputs the mass at start of entry (after course corrections from the weight statement),  and a hypersonic drag coefficient in crossflow for the presumed vehicle shape,  which in this case is a circular cylinder.

It also needs as inputs the propellant masses for each burn.  It works out from these the masses of oxidizer and fuel for each burn,  and their volumes.  This is done in an untitled block top center of the page.

There is an “engine resize” block that takes the engine characteristics modeled elsewhere,  and rescales them to the correct thrust size.  Those inputs are the modeled vacuum thrust,  the required vacuum thrust,  the throat and exit diameter sizes,  and the effective average half-angle of the supersonic expansion bell. 

There is a “tanks figured on totals,  with headers inside TBD” block.  It has vehicle diameter and a sort of interstage length between the tanks as inputs.  It works out all the lengths and L/D ratios,  and requires the “right” vehicle diameter to keep a full sphere as the smaller tank,  as well as to be consistent with the seating arrangements in the “cabin” block.    The seating is an input number per level,  and number of levels,  consistent with the total number of people.  There has to be room for seating at the input diameter in the “tanks” block,  and the input seat pitch is the spacing between levels.  Note how the empty main tank shells protect the propellants in the header tanks from the effects of entry heating.

The “engine bay” block takes the resized dimensions of the engine (overall rough estimate of length,  and the exit diameter,  and uses these with inputs for number of engines and the spacing requirements for gimballing,  to estimate min dimensions of the engine bay.  Its diameter should never exceed the input diameter in the “tanks” block.

From there,  the “overall vehicle” block puts together these results into estimates for the length and L/D ratio (which ought to be in the 4 to 6 range) of the entire vehicle stack,  and with inputs for entry mass and hypersonic drag coefficient,  estimates the broadside-entry ballistic coefficient. Too low an L/D will get you too low a blockage area and too high a ballistic coefficient.  Too high an L/D is a topple-over risk,  or at least a risk of bigger,  heavier landing legs.

Creating the Re-Sizable Engine Data

I used the “rocket nozzle” spreadsheet file,  worksheet “rocket noz”,  to rough-size a suitable engine and calculate a reliable performance estimate for it.  This was actually one of the first things I did.  This engine burns liquid methane fuel with liquid oxygen oxidizer,  similar to SpaceX’s Raptor and Blue Origin’s BE-4.  These propellants are thought to be manufacturable in situ on Mars.  This entire design study assumes that to be true.

I chose not to push the state of the art,  given the troubles SpaceX had getting to current Raptor-2 performance levels.  This engine is only sized for a chamber total pressure delivered to the nozzle entrance of 2000 psia,  and only a 3:1 pressure turndown ratio,  although this analysis does presume a full-flow cycle with no dumped bleed gas.  The nozzle is assumed to be an 18-8 degree curved bell,  with a throat discharge coefficient of 0.995.  Both c* and r are presumed functions Pc,  using the data shown in Figure 7 as the inputs.  Similar data for other propellant combinations are in the “prop comb” worksheet. 

Figure 7 – The Resizable Engine For This Study

The nozzle expansion was designed at full Pc = 2000 psia,  expanded to 5.97 psia at its exit plane.  This gave us a nozzle right at incipient separation when operated at 2/3 Pc,  at sea level on Earth,  allowing easy open-air testing on Earth.  It is unseparated for that power setting (or higher),  but cannot be operated at lower settings at sea level,  because the nozzle will separate. 

Thrust was sized at an arbitrary 10,000 lb in vacuum at full Pc.  Performance in the near-vacuum of Mars’s atmosphere is indistinguishable from true vacuum performance,  so the vacuum data were used for this study.  There is very little specific impulse variation from 1/3 to full thrust,  reflecting pretty much only the variation of c* with Pc. I used the 2/3 power value of 352 sec as “typical” of operation at any throttle setting.  Thrust and flow rates scale with throat area,  dimensions scale with throat diameter.

Final Note

Most of the spreadsheets used here are part of the course materials that I created for orbits and vehicle sizing.  Only the custom spreadsheet used for vehicle characteristics as calculated here,  is not part of those course materials.  Those course materials are available on the New Mars forums,  in the "interplanetary transportation" topic,  "orbit mechanics class traditional" thread.