Bounding Analysis for Lunar Lander Designs

I did this as a "clean sheet" bounding analysis.  Friends I correspond with have asked repeatedly how a lunar base might be established,  and with what.  I know the most about Spacex rockets and capsules,  but actually fleshing out these designs could use anybody's existing equipment,  in whole or in part.  The challenge I now throw out to them is to design something within these limits.

Bounding Analysis for Lunar Lander Designs  

GWJ  3-18-17 completed 3-18-17

The scenario here is a lander delivered “neat” to lunar orbit as an unmanned item.  A crew will arrive separately to rendezvous with it in lunar orbit.  The plane of that orbit is presumed to be very close to the ecliptic.  Orbital direction is retrograde,  in accordance with the figure-eight patched-conic trajectory used during Apollo.  The delta-vee to land one-way is 1.68 km/s.  For design purposes,  a few percent higher is used to provide a little margin:  1.75 km/s. 

The lander is delivered “neat” to lunar orbit,  meaning the rocket that takes it to the moon must do the “burn” to put it into lunar orbit.  The total rocket design delta-vee from the surface of the Earth to do that is at most 12.4 km/s,  when the first 8 km/s to Earth orbit is factored for drag and gravity losses by 1.05.  This is very close to the surface launch for a more-or-less worst-case slow trajectory to Mars,  which is about 12.1 km/s,  factored the same way.  That way,  the tonnage sendable onto a Mars transfer trajectory is almost the same as what can be delivered into lunar orbit,  for our purposes here. 

Descent Design Requirements

Spacex lists on its website that its Falcon-Heavy can send 13.6 metric tons to Mars,  flown fully-expendably,  for about $90 M launch price.  This heavy-lift booster hasn’t yet flown,  but it should fly this year (2017).  Reducing that payload slightly for the slightly-higher delta-vee to lunar orbit,  call that a max payload to lunar orbit of an even 13 metric tons.  

For the descent stage,  ready to fire in lunar orbit,  we are looking at an ignition mass of 13,000 kg maximum,  and a required design delta-vee of 1.75 km/s.  Propellants should be storable,  since days to weeks,  even months,  in space (or on the moon) are contemplated.  With nozzles designed for vacuum,  and assuming NTO-MMH propellants,  a delivered Isp = 335 sec is quite realistic.  Engine thrust/weight ratio of 100 Newtons-of-thrust per Newton of engine Earth weight seems feasible.  

Thrust to ignition Earth weight ratio should just barely exceed lunar gravity’s pull,  so that plenty of thrust margin is available at burnout weight:  0.2 seems “reasonable”.  We’d like the vehicle acceleration at burnout to be less than 1 gee,  preferably under 0.5 gee,  to keep the ride from being too rough,  and to limit throttleability requirements to feasible values.   

The propellant tanks will need a sun-reflective surface and some insulation,  plus electric in-tank heaters,  on a single-hull tank.  That means the descent stage propellant tankage will not be quite as lightweight as that of an expendable booster.  Just considering the tankage alone,  a 95-5 split of propellant to tank masses seems reasonable to assume (Wp/Wt = 95/5 = 19). 

The rest of the stage structure must bear the thrusted flight maneuvering loads carrying as large a payload as possible,  plus incorporate a set of broad-span landing legs,  and some means of unloading large items (ramps,  crane,  etc.).  An inert structural fraction for the stage near 15% should cover all of this.  That fraction does not include tank inerts or engine hardware.  Those get figured separately,  and then added to determine an overall stage inert mass fraction. 

The objective here is to determine max payload mass within that ignition mass limitation.  That payload can be either (1) cargo delivered one-way,  or (2) an ascent vehicle carrying minimum crew and cargo weight.  They mass the same,  though. 

Sizing a “Clean-Sheet” Bound on the Descent Stage

Exhaust velocity is rather accurately estimated as Vex, km/s = 9.8067*(Isp, sec)/1000.  That and the design delta-vee value combine to determine stage mass ratio MR = exp(dV/Vex).  The required propellant fraction (of ignition mass) is Wp/Wig = 1 – 1/MR.  The corresponding fraction for tankage inerts is Wt/Wig = (Wp/Wig)/(Wp/Wt).  The corresponding engine inert fraction of ignition mass is We/Wig = (T/Wig)/(T/We).  The rest of the stage structural and equipment inerts is represented by the 15% figure.  These total together for the overall stage inert fraction. 

Payload fraction of ignition mass is just 1 minus the propellant fraction and minus the sum total inert fraction.  Once you specify an absolute payload mass,  that determines ignition mass,  from which all the component masses are determined by their fractions.  That finalizes the weight statement.  For this bounding exercise based on Falcon-Heavy delivery,  those results are in Figure 1.  

Figure 1 – Limits for Descent Stage,  One-Way,  Falcon-Heavy Delivery to Lunar Orbit

Payload is 5.372 metric tons.  This could be all cargo,  or it could represent a crewed ascent stage.  If cargo,  that’s $90M/5.372 metric tons = $16.8M per metric ton delivered to the surface of the moon.  Actually,  you design to a slightly-smaller payload mass,  because of all the uncertainties.  There is always the unexpected outcome,  when sizing vehicles like this “from scratch”.  The weight margins don’t have to be all that large,  because I already put that into the design delta-vee figures. 

Ascent Design Requirements

The same propellant and tankage choices are presumed.  The same engine T/We is assumed.  A slightly-higher T/Wig = 0.3 is assumed,  to accelerate “smartly upward” against lunar gravity.  Stage inert fractions can be lower since no unload equipment or landing legs are needed.  However,  these inerts are likely higher than a typical booster rocket (5%) because of the protective cabin surrounding the crew,  the docking hatch,  and the instruments and controls they must use.  I simply assumed 10%. 

This ascent stage must ascend to lunar orbit (requiring 1.68 km/s),  and also maneuver to rendezvous with the crew return craft left in lunar orbit.  It therefore needs more design delta-vee than the descent stage.  Call it 2.0 km/s,  for a kitty of 0.3 km/s to cover maneuvering and the unexpected. 

Its maximum ignition mass cannot exceed the descent stage payload capability of 5372 kg.  Prudence dictates very slightly less.  Call it 5360 kg for design-bounding purposes. 

Sizing a Clean-Sheet Ascent Design to Fit the Descent Stage

All the calculations and equations are basically the same as before.  I simply used the same spreadsheet with different numbers.  The results are given in Figure 2.  Deliverable “payload” is 2235 kg,  which would be suited crew plus a few of days of life support,  plus any samples sent back to Earth.

Figure 2 – Limits for Ascent Stage,  One-Way,  To Fit Descent Stage That Falcon-Heavy Can Deliver

For the sake of argument,  use 80 kg per person body weight,  and 120 kg for a surface EVA-capable pressure suit.  That’s 200 kg per person.  Set food,  water,  and breathing oxygen supplies to 100 kg to cover an unexpectedly-long rendezvous interval of several days.  That’s 300 kg allotted per person.  There’s “room” for 7 such masses in the payload. 

If this were 6 crew,  there’s room for around 300 kg of samples or return cargo.  If the crew is 5,  there’s room for about 600 kg of samples or return cargo,  and so forth.  But the point is,  there’s room for a much larger crew than Apollo had.  That’s partly the difference in technologically-achievable storable propellant performance,  and in structural technologies,  since the 1960’s.  The rest is landing without unknown obstacles in your path,  which is what happened on Apollo 11,  nearly depleting its propellant. 

How This Can Be Used

The one-way cargo-only variant can be used at $90M a shot to deliver 5.36 metric tons of cargo to the moon ($16.8 M/delivered metric ton).  Several could be sent to the same site.  Some of these could be the modules from which some sort of surface habitat could be assembled.  The rest could be the supplies,  equipment,  and surface rover vehicles needed to operate that base. 

The manned lander conforms to the same 5.36 metric ton weight limit.  If crew were 3,  then 1200 kg of surface supplies could go down with them.  If crew were 2,  then 1500 kg of cargo could ride down.  Reducing the ascent load just increases the rendezvous maneuver capability upon returning to lunar orbit,  a very beneficial safety factor.    

Say,  we sent 9 of these to the moon:  6 cargo-only landers and 3 landers with manned ascent stages,  each with a crew of 2 and 1500 kg of cargo on board.  That gives us three ascent vehicles on the lunar surface ready to use,  when the entire crew really only needs one to return.  Added safety,  that is. 

That’s a total of 32.16 tons delivered with the cargo landers,  and 4.5 more tons sent down with the manned landers,  for a total crew of 6.  Assume simply for the sake of argument that the surface habitat requires 20 tons.  We need to reserve 0.6 tons of supplies for the crew to ascend.  Assume two rovers,  each 1 ton.  Assume one electric backhoe-like device,  at 2 tons. 

36.66 tons total delivered cargo,  less 20 ton habitat,  4 tons for vehicles,  and 0.6 tons for ascent supplies,  leaves 12.06 tons allocatable for surface stay supplies and other equipment items.  At a nominally-assumed 10 kg life support per person per day for 3 months,  then about half that 12 tons is something other than life support supplies.  Also nominally,  3 months of life support supplies for a crew of 6 is pretty close to a lander’s deliverable payload at 5400 kg.  I tried to overestimate this requirement. 

Looks to me like there is very good potential for establishing a fairly substantial lunar experiment station,  temporarily occupied for a considerable time (at least 3 months).  This requires 9 Falcon-Heavy fully-expendable launches for the landers,  plus one more to send the crew in a crew Dragon (with its trunk modified to carry propellant,  something not addressed here),  for $900 M in launch costs.  If launch costs were 20% of the program that develops these vehicles and the surface equipment,  total program cost to put a small base temporarily on the moon would be in the ballpark of $4.5 B. 

Launching another cargo lander every 3 months or thereabouts brings the supplies to keep that base permanently occupied at crew size 6.  Maybe switch out crews yearly,  by adding a crewed Dragon to lunar orbit along with a fresh manned lander to take them down to the surface.  That’s a total of 6 Falcon-Heavy launches per year to maintain a continuous presence at the base.  That’s $540M per year to maintain the base after it is built,  plus the costs of keeping the necessary vehicles and equipment in production.  Development is complete,  so call launch costs ~50% of continuing program costs. 

About $4.5 B to establish a 3-month-capable,  6-man base on the moon,  and about $1B/year to keep it continuously manned and operating is just not very expensive as space ventures go!  This analysis is based on the use of a commercial heavy lift rocket that is far less expensive to use than NASA’s SLS,  and which will also be far more available for routine use multiple times per year,  than NASA’s SLS ever can. 

Blue Origin is also planning to get into this kind of lunar capability with its New Glenn rocket.  Between them and Spacex,  putting a base on the moon looks to be quite feasible and quite affordable.  This could provide the bootstrap start needed to begin doing something useful,  or for profit,  on the moon. 

Final Remarks

This kind of experiment station allows evaluation of low-gravity effects upon health versus the zero-gravity effects that we are familiar with in Earth orbit.  It allows a place to experiment with increasingly-capable recycling life support systems.  It allows a place to experiment with meteoroid and radiation protection by regolith cover.  It allows a place to experiment with ways and means to overcome contamination and wear issues with very-fine-but-sharp-edged dust particles.  All these are needed to visit Mars or the asteroids,  and are available on the moon “close by” in case of trouble. 

The same base allows experimentation with ways and means to dig and drill deep in a harsh environment.  It allows experimentation with the recovery of mineral resources.  It allows experimentation with how to establish roads under such conditions,  so that future long-distance surface transport becomes feasible.  These things are needed for establishing useful and prosperous industrial applications on the moon and Mars,  and to some extent the asteroids. 

This is the kind of thing we should have attempted to close-out Apollo,  had a useful lunar presence been the goal,  instead of “flags-and-footprints”.  It is still a good rationale for returning and doing something very much like what I described here,  as a first step. 

Addendum:  Crew Dragon Modified to Leave Lunar Orbit

The “design” trajectory to reach lunar orbit is pretty much the same as was used for Apollo decades ago.  A direct launch from Canaveral into low Earth orbit more or less eastward at low inclination (the part requiring factoring ideal delta-vee for gravity and drag losses),  followed by a burn to escape onto the lunar transfer trajectory,  and a final upper-stage burn to place the payload into a retrograde orbit about the moon.  The worst-case total rocket design delta-vee for this is just about 12.4 km/s (factored),  and worst-case 0.8 km/s to leave lunar orbit onto a trajectory home.  See Figure 3. 

If we stay under 13 thrown metric tons,  the Falcon-Heavy should have enough delta-vee capability to put that 13 tons into lunar orbit,  same as the lander designs just bounded above.  The problem is then leaving lunar orbit with enough propellant reserve to cover attitude control and a powered landing on land back on Earth (Spacex’s preferred mode).  Attitude control consumption should be modest,  but we might need around 0.5 km/s capability to land safely on Earth,  where capsule terminal fall velocities are only around half a Mach number.  0.8 + 0.5 + small change is close to 1.35-1.4 km/s delta vee capability demanded of the Super Dracos on crew Dragon.  It simply does not have that much capability without extra propellant added in the trunk,  and connected to the system in the capsule.

Figure 3 – Design Trajectory and Delta-Vee Requirements

Design Requirements for Modified Crew Dragon

Total delta-vee capability 1.35 km/s min,  1.40 preferred.  Maximum spacecraft mass at launch 13.0 metric tons.  Minimum crew 3.  I have a spreadsheet model already constructed for this purpose,  which I proceeded to run again for these exact numbers.  Masses for the dry weights of capsule and trunk (before modification) are my best guesses,  but their sum matches published data. 

The modification is to install more tanks of NTO-MMH propellants in the trunk,  to a maximum of the 3000 kg quoted cargo capacity for that trunk.  I estimated propellant-tank mass split as 95-5 or a 19:1 ratio,  same as for the landers.  I did not estimate volumes,  although there are 14 cubic meters available in the trunk for this. 

Results That Bound the Design

These are shown in Figure 4.  Payload mass is limited more by the 13.00 ton thrown weight than the 1.35-1.4 km/s delta-vee requirement.  That payload mass is 1760 kg. 

The per person allotment we used for the lander was 200 kg person-plus-suit,  and 100 kg of packed life support supplies.  The life support supplies are probably a bit of an overkill,  so 1760 kg ~ 1800 kg,  and 1800 kg / 300 kg/person is crew = 6 max.  Slightly less actually.  Call it no more than 5 crew at a time,  plus life support supplies,  and no more than about 150 kg of equipment or cargo in the capsule with them,  for the trip to the moon. 

Having the extra delta-vee means we can carry 6 crew,  even 7,   home.  That is a good safety bonus.  Crew Dragon is supposedly rated for the same cargo home as cargo Dragon (3000 kg),  so we are well within that limit. 

This was accomplished by adding 2800 kg propellants to the trunk,  which also adds 147 kg of tank inerts to the trunk inert weight.  That leaves a smidge for any extra plumbing before we hit the 3000 kg limit. 

The only remaining question is for how long a crew Dragon can be parked in lunar orbit before the crew that needs it must come back to it and fly it home.  There are limits to lifetime allowable parked in space.  Perhaps this can be made into some number of months to a year,  given some experience flying the capsule design in Earth orbit.  If a year,  then the lander plan given above is quite feasible just as it is laid out.  If not,  we’ll have to switch out crews on the moon at 6 months,  perhaps.

Figure 4 – Modifying Crew Dragon Into Lunar Orbit Dragon for Falcon-Heavy Launch

Final Remarks

With these two bounding analyses,  I have shown how it is possible to ship 13-ton lunar cargo and crew landers to the lunar orbit with Falcon-Heavy as the launch rocket.  I have also shown how it is possible to ship crews to lunar orbit with the same rocket and a 13 ton modified crew Dragon that has 2.8 extra tons of propellant in its trunk,  connected to the Super Draco thruster systems in the capsule. 

The cargo landers deliver slightly over 5.3 tons to the surface.  The crew landers have a 5.3 ton ascent stage that could carry as many as 6 crew back to lunar orbit. 

At only $17M/delivered ton,  building a practical small experiment station that is permanently occupied becomes easily possible,  at a price far below what was experienced doing the Apollo “flag-and-footprints” stuff during the cold war. 

What makes this feasible is a heavy lift rocket of adequate size to put 13-ton payloads into lunar orbit,  and at a commercial launcher’s far lower price.  This is true flying the rockets fully-expendably.  This capability should become available within the next 1-2 years. 

All that is needed from a vehicle development standpoint is the two versions of the lander designed to these bounding limits,  and then developed and made ready for use.  These share a common descent stage.  That should help lower costs and development time. 

Adding propellant capacity to crewed Dragon with tankage in the trunk is not so much development work,  more of a routine modification that can be tested all-up in Earth orbit,  to make it ready to use.
 

We’ll need a 2 or 3 seat open electric rover car that weighs no more than a ton.  Between the Apollo rovers and the recent Mars robot rovers,  this should not be a major development item.  

Development,  yes,  just not a “biggie”.  Same for a 2-ton electric front-end loader. 

The hardest nut to crack is a surface habitat that can be assembled from modules that fit within the 5 ton lander payload capacity,  and that can be erected by men on foot in spacesuits with hand tools.  The idea is to assemble it in an excavation done with the front end loader,  and then bury it at least partially with that front end loader. 

This is the kind of thing that could be done within 1 or 2 presidential terms,  which would net returns orders of magnitude greater than Apollo,  for costs orders of magnitude less than Apollo.