Evaluations of the Spacex Starship/Superheavy

Over the last few years,  I have done several reverse-engineering evaluations of the Spacex Starship/Superheavy design,  as it has evolved into what is being test-flown as of this writing.  All of those evaluations are documented in articles posted on this site.  Here is a list,  from latest to oldest.  I have included evaluations of some test flights in this list,  but not the descriptions of the analysis techniques that I used.  

UPDATE 5-21-21:  I added another reference (#18) where the reduced payload/reduced tanker flights option was explored,  done with the best rocket performance estimating spreadsheet yet.  

Update 6-7-21:  I added another reference (#19) where the details of the simplest,  most straightforward landing leg concept were explored.  This was a retractable variant of the Falcon-9 legs,  to get a wide stance,  but also with a fold-out pad area to achieve low bearing pressures.  

Update 6-15-21:  I added a second landing legs article as another reference.  This is again a retractable variant of the Falcon-9 legs,  but done with link bars to get vertical leg orientation with no bending.

Update 8-7-2022:  These estimates were made using my evaluations of the sea level and vacuum Raptor-1 engines,  assuming they could be operated at the full intended chamber pressure of 4400 psia = 303 bar.  As it turned out,  they never operated that high,  and fell short on thrust and specific impulse.  I have since evaluated the revised designs for Raptor-2,  which indeed achieve the intended pressure levels.  Specific impulses are very close to what I assumed for these analyses because the pressure was achieved,  and thrusts are a bit higher because the throat diameter is slightly larger.  As a result,  these analysis results for payload capability on the various missions are still very close in the ballpark.  The other uncertainties and assumptions are likely larger effects than the specific impulse differences,  which are slight. 

Ref.        date                      title [comments]

#1.         3-23-21                Third Spacex Tanker Study [found practical approach to refilling lunar missions]

#2.         3-21-21                Second Spacex Tanker Study [investigation of lunar mission refill feasibility]

#3.         3-17-21                Spacex Tanker Investigation [initial investigation of lunar mission refill;  includes capabilities to low circular orbit]

#4.         3-15-21                Reverse Engineering Estimates: Starship Lunar Landings [determining elliptic departure orbits to make lunar nearside landings feasible with 2021vehicle estimates]

#5.         3-9-21                  Reverse-Engineering Starship/Superheavy 2021 [estimated performance to low circular orbit with best 2021 vehicle data and techniques,  including booster fly-back]

#6.         12-10-20              Spacex Test Flight Results In Explosion

#7.         7-13-20                Non-Direct to the Moon with 2020 Starship [looking for ways to make lunar landings feasible with 2020 vehicle estimates but without penetrating into the Van Allen radiation belts]

#8.         7-5-20                  2020 Starship/Superheavy Estimates for the Moon [found to be infeasible from departure orbit that does not penetrate into the Van Allen radiation belts]

#9.         6-21-20                2020 Starship/Superheavy Estimates for Mars [found to be feasible from low circular orbit with 2020 vehicle estimates,  including faster transfer orbits]

#10.       5-25-20                2020 Reverse Engineering Estimates for   Starship/Superheavy [payload to low circular orbit with 2020 vehicle estimates and techniques,  including booster fly-back]

#11.       10-22-19              Reverse-Engineering the 2019 Version of The Spacex “Starship” / “Super Heavy” Design [first attempt at performance estimates to orbit,  with staging assumptions and crude booster fly-back,  using 2019 version of design;  updates included evaluations of overturn stability and soil bearing strength effects,  plus trips to Mars and to the moon,  plus proper landing engine choices]

#12.       9-26-19                Reverse-Engineered “Raptor” Engine Performance [evaluations of sea level and vacuum-bell engines,  complete with part-throttle vs full-throttle data based on chamber pressures]

#13.       9-16-19                Spacex “Starship” as a Ferry for Colonization Ships [an alternate use scenario]

#14.       2-4-19                  Designing Rough Field Capability Into the Spacex Starship [serious look at rough-field landing issues with 2019 version of the design]

#15.       4-17-18                Reverse Engineering the 2017 Version of the Spacex BFR [earlier look at the 2017 version of Starship with the projected 85-ton inert mass;  includes estimates to orbit and to Mars,  includes a look at rough-field landing issues and at the tanker issue;  a revisit of Ref. 16]

#16.       10-23-17              Reverse-Engineering the ITS/Second Stage Of the Spacex BFR/ITS System [an earlier look at only the 2017 version of the Starship second stage vehicle,  diameter reduction to 9 m]

#17.       10-2-16                Elon Musk Reveals His Plans for Mars [very first armchair look at the 12 m diameter Starship concept,  not a reverse-engineering evaluation;  some significant issues identified]

#18.        2-9-21                Rocket Vehicle Performance Spreadsheet [descriptive information that is essentially a user's manual for the spreadsheet,  plus the example of SS/SH to Mars for 2021 at max payload,  and at reduced payload for reduced on-orbit refill requirements]

#19.       6-7-21                  One Concept for Landing Legs [spreadsheet pad sizing and by-hand concept design analysis of a simple retractable landing leg concept of wide stance,  with large fold-out landing pads to reduce bearing pressure]

#20.        6-15-21                Landing Legs Concept 2  [legs fold out on two link bars to achieve vertical leg orientation with no bending,  slide-out pads that attach at their centers to legs]

These articles are most easily found by using the navigation tool on the left of the page.  Record the dates and titles you want to find on a piece of scrap paper.  Click on the year,  then click on the month,  then if need be on the title.  

Figure 1 -- Brief Summary of Results

Best of the Various Estimates

For a mission Starship to low circular Earth orbit,  the most realistic reverse-engineering estimates that I have are given in Ref. 5,  for a 9 m diameter system with inert masses of 120 metric tons for Starship and 180 metric tons for Superheavy,  as they are best understood in early 2021.  These include my best estimates of staging velocity and booster flyback,  which has some impact on payload to orbit.  The vehicle acceleration impacts of thrust levels are included,  along with my best available estimates of the landing after the belly-flop maneuver.

For a mission Starship to elliptical orbit for lunar landing missions,  the most realistic reverse-engineering estimates are given in Ref. 4,  which includes the analysis of direct landings on the lunar nearside.  This is for the Starship/Superheavy as it is understood in early 2021:  9 m diameter,  with 120 and 180 metric ton inert masses.  Thrust effects are included.  The lunar departure is from an elliptic orbit with its apogee inside the Van Allen radiation belts.  It must be fully-refilled at departure.

For missions from low circular Earth orbit to Mars (and back),  the best reverse-engineering estimates available are given in Ref. 9,  for the same 9 m diameter Starship of inert mass of 120 metric tons.  The data are as understood during 2020,  and could be updated for better precision,  although the older booster unknowns (such as inert mass) are not a major impact on overall results.  Both Hohmann min-energy transfer and faster transfer orbits were evaluated.  Payloads are smaller than could be lofted to low Earth orbit. That reduces the refill requirements (and the number of tanker flights) to depart.

Update 5-21-21:  Ref. 18 has the 2021 spreadsheet results for the Mars mission from LEO,  at full payload,  and for a reduced payload that reduces on-orbit refill requirements.  Scope includes both Hohmann transfer at average planetary distances,  and a 2-year "abort" orbit at average planetary distances.  These really are the best available estimates based on the best available data as the SS flight test program proceeds (up to SN-15).  

For tanker capabilities to low circular orbit,  the best reverse-engineering estimates are given in Ref. 4.  They include withholding adequate landing propellant allowances under different constraints than would apply to mission Starships.  Both rough-estimated “dedicated” tanker designs,  and “ordinary” Starships used as tankers,  are covered.  Estimates of tanker capabilities to the feasible lunar departure elliptic orbits are also given (these are quite low in comparison).  Thrust effects are included.

The capacities and mission strategy for tankers to support lunar missions are given in Ref. 1.  These reverse-engineering estimates minimize the number of tanker flights for a single mission to a lunar landing.   Thrust effects are included.  This best scenario does most of the refilling in low Earth orbit,  with the mission Starship and one refilled tanker sent to the elliptic lunar departure orbit. 

The best Raptor engine performance characteristics are given in Ref. 12.  Although those data reported are given in US customary units,  they are easily converted to metric using  4.44822 N = 1 lbf for thrust data.  There is no need to convert seconds of specific impulse. 

Scope includes both the sea level and vacuum-bell Raptor engines,  and full data at both 100% and 20% chamber pressures.   You may scale linearly between thrusts and specific impulses at 100% and 20% chamber pressure.  You may not scale linearly between vacuum and sea level performances.  The reference provides performance vs altitude lists for both sea level and vacuum-bell designs,  at both throttle settings. 

I don’t see much reason to keep updating these reverse-engineering estimates until after the Starship and Superheavy flight tests are completed,  and the system has begun flying to low circular orbit.  At that time,  the inert masses and the propellant capacities of the two stages will be known far more clearly,  as will the number of engines in the booster,  and the best practices for landing the upper stage spacecraft.  Not to mention many important details,  like the landing legs and the heat shield.

Beware the Unaddressed Issues

These estimates that I made do not address all the issues that must be resolved before this system can do any of the projected missions!  Some of those issues have been anticipated,  such as rough-field landing capability.  Others are becoming apparent during the early Starship flight tests!  All are potentially-fatal problems that must be resolved,  before the system can become operational.  The performances I have identified are certainly attractive enough to warrant the massive efforts needed to resolve those problems!

One should be aware that Spacex has taken a far different approach to that of NASA,  in designing its large lifter-to-space.  There are a whole host of risky new technologies embodied in the Starship/Superheavy design.  These are what must be included to get large capabilities along with low costs.  The NASA approach (as embodied with SLS) is very much lower-risk approach (a new combination of things that have been used before),  but that is already known to be inherently very expensive. 

What Spacex did with its Falcon launch vehicles lies sort-of “in the middle” of that high-risk/low-cost to low-risk/high-cost spectrum.  Those vehicles flew conventionally first,  then added new technologies one or two at a time to get reusability and lower costs.  This was successful,  although Spacex nearly went bankrupt at first,  learning how to fly and stage supersonic vehicles.  They are now the low-price leader in the launch industry,  and some of their competitors are beginning to move toward reusability as well.  However,  while leading the pack,  Spacex is also encountering problems to solve that no one has ever addressed before.

Problem to Solve:  Rough-Field Landing Capability

There are three separate aspects of this.  All three require effective solutions.  They are (1) overturn stability,  (2) safe bearing pressure exerted on soft substrates,  and (3) a telescoping capability to address surface contour roughness and shock absorption.  The landing legs we have seen so far on the "Starship" prototypes address none of these concerns adequately!

I covered much of this topic (the first two issues) in Ref. 11,  including some typical safe soil bearing pressure values,  and a guide to which soil in the list corresponds to the most common surfaces on the moon and Mars.  Refs. 14 and 15 also explore this topic. 

               Safe Bearing Pressures

The safe pressures in the Ref. 11 list are factored down from experimental soil failure values,  which factored-down allowable pressures prevent unintended penetration upon load application,  and also prevent settlement (compaction under the landing pad. 

Such compaction allows penetration downward) over long periods of time.  In the absence of actual test data,  you must use the low end of the range of values in the list as your landing leg design value.  You also need to factor-up the weight to be supported,  to represent the dynamic effects of touchdown,  and also the effects of coming down "crooked",  so that one leg hits first. 

The main lesson from the soil data is to use only 1 US ton/sq.ft = 0.1 MPa safe bearing pressure as a maximum tolerable value when sizing landing pad areas for the moon or Mars (or Earth,  for that matter).  That corresponds to loose fine sand or soft clay.  Even with the rocks dispersed (without adhesion) in the loose sand and dust of lunar and Martian regolith,  these regoliths would be very similar to loose fine sand on Earth,  because there is no adhesion between these sand and dust particles,  and no adhesion to the rocks. 

Further,  the soft tidal-flat muds adjacent to Spacex's landing pads at Boca Chica would be at most only that strong (0.1 MPa),  and very likely weaker still.  A prototype massing 120 metric tons,  and landing with 10-15 tons of propellant still aboard,  weighs about 1.3 MN here on Earth.  6 legs with "feet" 0.5 m x  0.5 m square would have about 1.5 sq.m total "pad" area.  The average static pressure upon the surface is then near 0.9 MPa (already 9 times too high),  and likely 2-to-3 times that (near 2 to near 3 MPa) during the dynamic transient of touchdown,  and for sure 6 times higher still (near 15 MPa) if one leg hits first. 

So,  a mudflat landing at less-than-0.1 MPa safe pressure (after missing the landing pad for any reason whatsoever) would clearly lead to disaster,  as the legs would penetrate deeply and unevenly,  leading to topple-over and explosion (see following)! 

               Static Stability

I also showed in Ref. 11 how the weight vector "hanging" from the vehicle center of gravity must fall within the footprint of the landing legs (as a polygon upon the ground),  even as the vehicle is tipped by sloping ground,  or leg penetration,  or boulders,  or contour roughness,  or anything at all.  That is the essence of static stability.  It is in all the elementary statics textbooks.

If the vehicle is statically unstable because its center of gravity height is too high relative to the stance (footprint polygon dimension),  the vehicle will (inevitably!!) topple over.  With topple-over of  a rocket,  an explosion is simply guaranteed. 

Note that the height to stance width ratio of all the successful landers on the moon and Mars has been in the vicinity of 1.  That ratio in the current Starship prototypes with 6 small legs mounted to the aft skirt is currently near 25/9 (not quite 3).  That is simply too high for rough field operation,  even for emergency landings on Earth.

Update 6-7-21:  Ref. 19 has the analysis results for a simple,  straightforward update to the Falcon-9 landing leg concept,  one that provides wide stance and large fold-out landing pads,  yet is hydraulically retractable.  Scope includes post-landing static bearing,  bearing of transient landing dynamics,  and pre-launch static bearing,  for each of 3 scenarios.  Those scenarios are Earth off-site abort,  lunar landings,  and Mars landings.  The basic forces and moments were determined with freebody diagram analysis.  

Update 6-15-21:  Ref. 20 has an alternative version of the fold-out landing legs held by two link bars,  such that the leg itself is vertical.  This eliminates bending loads on the leg, reducing it to compression.  A revised fold-out pad is given that could be used with either concept.  It eliminates the large pad hinge moment,  and greatly reduces bending loads in the pads themselves. 

               Uneven Surfaces and "Crooked" Touchdowns

Hydraulically-telescoping landing legs are required to overcome three problems:  (1) the elastic bounce-back of spring forces-only acting upon the vehicle,  (2) the anticipated dimension of surface roughness features (boulders,  gullies,  and the like,  which would be typically around a meter or so),  and (3) having to come down "crooked" so that one leg hits first,  for any reason at all (such as leaning into a gust of wind on Earth). 

If the legs have sufficient telescoping stroke,  then both problems (2) and (3) can be solved,  without raising the forces on any one leg too high.  If you have sufficient stroke,  then the leg that hits first need not support the entire weight of the vehicle,  which has to be factored-up by 2 to 3 for dynamic effects.  It will support something only a little more than its share of the total,  and only for a short transient. 

It takes hydraulic dissipation to provide the damping necessary to prevent problem 1.  Otherwise,  you will bounce around like a rubber toy,  which means you are completely out of control,  during touchdown.

Engine Bay Fires

Fires inside the engine bay during ascent were seen on engine bay camera footage and from external views during the test flights of SN-8,  SN-9,  and SN-10,  and in engine bay footage only  during SN-11 (which was flown in fog,  obscuring the view from the ground).  The only failure officially attributed to fires destroying engine wiring was SN-11.  These fires are clearly methane leaks burning with air. 

No ascent fires were seen in the engine bay footage of SN-15's first flight (it may be re-flown).  That footage is incomplete,  but enough could be seen to rule out ascent fires.  SN-15 is said to be a revised design from that of SN-8 through SN-11. 

Lesson learned:  Spacex appears to have stopped the methane leaks and fires seen during ascent,  with its SN-15-on design revisions. 

Engine-Out During Flip Maneuver

Low thrust during the flip to vertical and final touchdown was seen during SN-8 (attributed to low fuel pressure) and SN-10 (attributable to helium pressurant ingestion?).  An engine-out (or two) was seen during SN-9,  SN-11,  and SN-15 (planned 3-engine relight for flip,  only two actually seen to light).  There was insufficient altitude available to decelerate SN-8 and SN-9 at all,  leading to fatal crashes. 

SN-10 was damaged in landing,  and then blew up a few minutes after the landing,  probably attributable to the post-landing methane-air fire that was seen.  SN-11 was destroyed by a midair explosion above the landing pad,  said to be caused by an engine "hard start" (which is just code for an engine explosion upon attempted ignition).  SN-15 landed successfully,  and survived the post-landing methane-air fire. 

Lesson learned:  start the flip and deceleration early,  because lowered thrust is likely,  due to multiple causes.  You can always throttle back or shut an engine down,  if you have more thrust than you need.  But insufficient altitude for lower-than-expected thrust to actually decelerate the vehicle,  will always lead to a fatal outcome.

Post-Landing Fires

SN-10 landed hard and crushed its legs unevenly,  so that it ended up leaning,  almost toppling.  It also reportedly crushed part of its skirt.  It definitely had a post-landing methane-air fire for a few minutes,  then blew up.  SN-15 landed more softly,  had a post-landing methane-air fire for a few minutes,  which was put out,  so that the vehicle survived.  Are these fires methane leaks associated with engine shutdowns?  Nobody has said,  yet.  But fire exposure endangers hardware,  and damaged hardware will leak propellant.  That is the nature of rocketry.

Lesson needing to be learned:  Spacex must stop the post landing fires,  because (1) any LOX venting into the fire for any reason at all will inevitably cause an explosion, and/or (2) the heat from fire exposure may damage tanks and plumbing,  causing a subsequent oxygen leak and explosion.

Cross Winds and Gusts at Touchdown

While SN-15 landed successfully in its first flight and survived the post-touchdown fire,  it did land near the edge of its landing pad,  with one leg about a meter from the soft tidal flat mud.  It supposedly was blown there by a last-second gust of wind.  I have seen photos of crumpling in the landing legs that supposedly reflect shear force damage being slid sideways during touchdown.  I am not so sure about that,  myself,  seeing more evidence of compressional crumpling than anything else in those photos. 

That being said,  being blown sideways at touchdown is a serious topple-over risk.  The lateral friction force (between pads and surface) is located pretty much perpendicular to a moment arm from there to the center of gravity.  For significant friction (comparable to weight) that is a big overturn moment,  which in turn is a really strong argument for a wide landing leg stance,  beyond the arguments already listed above.

While a flight control might compensate by leaning the vehicle into the crosswind,  the magnitude of the wind overturning moment once touched down is also quite large:  it is a force applied at the lateral center of pressure,  which is just a diameter or so down from the center of gravity.  The couple created by the wind force (which more-or-less equals the pad lateral friction) has a large moment arm,  almost as large as the center of gravity height.  That is a large overturn moment.  This is another argument for a wide stance.  It applies to Earthly landings where wind forces can be large. 

Beyond that effect,  leaning into the crosswind guarantees an uneven touchdown onto the landing legs.  One pad will always hit first because of the off-angle attitude.  The strength and stroke of that leg has to be high enough to absorb the entire weight load of the vehicle,  factored-up for dynamics,  during the touchdown transient!  It has to stroke,  in order to absorb those forces hydraulically;  spring forces-only will not do the job,  because of bounce-back risks!  All the legs must be designed that way,  because you simply cannot predict which one will hit first. 

Lessons to be learned:  landing leg designs must have

               (1) sufficiently wide stance (center-of-gravity height / stance dimension near 1), 

               (2) pad area on each leg must be large enough not to exceed safe soil bearing pressures,  for a vehicle weight factored-up for dynamic effects and for any uneven effects not mitigated with hydraulic leg stroke, 

               (3) legs must absorb dynamic loads with hydraulic stroking in order to have the damping necessary to prevent bounce-back,  and

               (4) the stroke length is set by the greater of (a) expected surface roughness dimension or (b) the off-angle effect of coming down "crooked". 

Heat Shield Robustness

This evaluation cannot be done until Spacex is flying its prototypes to orbit,  or at least to near-orbital speeds,  requiring hypersonic entry and aerobraking.  I would anticipate potential difficulties in 3 areas:  (1) loss of heat shield tile or tiles,  (2) vortex scrubbing of windows,  and (3) over-the-nose jet reattachment due to lateral vortices. 

               Tile Loss

Unlike the Space Shuttle,  any burn-through due to a lost tile will very likely be directly into a cryogenic propellant tank.  That is because a bit over half the vehicle length is propellant tankage,  of a single-wall design.  Such a burn-through not only causes loss of any remaining propellant in the tank,  it also causes depressurization of the tank to ambient,  which at entry altitudes is essentially vacuum.  If the vehicles depends in part on tank pressurization to maintain strength against the high-angle air loads,  then it will likely break up. The same would apply to any pressurized cargo spaces,  and to any pressurized crew habitation spaces.  If the tile retention scheme is redundant,  likelihood of any tile loss is significantly reduced.

               Vortex Scrubbing

There is a line of flow separation somewhere along each side of the vehicle when at significant angle of attack.  In the wake zone just downstream of the separation line,  there is a vortex along each side.  These vortices are much weaker at low angle of attack,  and very strong indeed at high angle of attack (where the flow is mostly crossflow). 

Where the vortex contacts the aft skin,  there is a scrubbing action at higher and higher velocities as the vortex grows stronger.  That scrubbing action enhances heat transfer rates to the skin,  despite the low pressures (and densities) in the wake zone.  If there are windows touched by this scrubbing,  they could very easily overheat and fail.  To a lesser extent,  the same risk is true of exposed metal skins. 

               Nose Jet Reattachment

The same lateral vortices,  if very strong,  can induce material from beyond the far side of the wake to relocate close to the aft surface between them.  In effect,  they can cause a jet of air coming over the nose not to separate,  but to stay attached.  This jet flows along the dorsal line.  There is very strong scrubbing associated with this jet contacting the surface.  And,  if it hits any projecting protuberance (such as a windscreen),  the jet shocks to high-pressure flow on the protuberance.  The combination of high scrubbing velocity and locally high shocked density can produce catastrophic heat transfer rates. 

Figure 2 -- Flow Field Effects

The Space shuttle suffered from this effect with its flight deck cabin roof structure and windscreen projecting into that reattached dorsal jet of high energy air.  This risk precluded operating the Shuttle at higher angles of attack than 40 degrees.  Below 20 degrees,  there was no lee-side separation,  and flow simply came around the nose and struck the windscreen.  The shockdown caused catastrophic heating.  Thus the Space Shuttle was strictly limited to angles of attack between 20 and 40 degrees during entry.

The cross section shapes and planform shapes of the Spacex Starship and the NASA Space Shuttle are different.  The Starship may,  or may not,  experience the same lateral separation lines with vortices;  and if it does,  the angle limits will likely be somewhat different.  However,  this same flow pattern effect was seen experimentally,  within pretty much the same limits,  for a variety of candidate nose shape details,  when the Shuttle was being designed.   Thus,  the Starship may well suffer similar risks to the windows and bare metal skin on its lee-side surfaces.

Lesson(s) to be learned are still TBD.

On-Orbit Refilling

This evaluation cannot be done until Spacex is flying multiple prototypes to orbit,  and attempting refilling operations on-orbit.  I can anticipate potential difficulties in two areas:  (1) required high attitude accuracy for tail-to-tail docking and automatic connection of propellant plumbing,  and (2) massive attitude thruster propellant usage during the main engine cryogenic propellant transfers,  at micro-acceleration levels (thus requiring a long time to accomplish). 

Lesson(s) to be learned are still TBD.

Radiation Risks

This will become a problem once Spacex attempts flights of its prototypes outside low Earth orbit.  I can anticipate problems in 3 areas:  (1) radiation-hardening of flight controls and propulsion controls,  (2) a place for any crew or passengers to shelter from solar storm events during flights to off-Earth destinations,  as well as sheltering from Van Allen radiation belt exposures,  while in high elliptic orbit for lunar departures,  and (3) co-locating crew control stations and at least some of the radiation sheltering,  so that critical maneuvers can be conducted by the crew during severe exposures.

Lesson(s) to be learned are still TBD.

Final Comments

I started this write-up in March,  intending to complete it and post it during April.  However,  my laptop died,  and took with it a lot of data,  including this write-up.  I was able to recover the data,  so that finishing this write-up became possible.

This article is intended as guide to previous reverse-engineering analyses,  which have evolved over time.  I have identified for the reader which articles have the latest and greatest data,  to which mission scenario they apply,  and exactly how to quickly and easily reach them.   

I am hoping that some of Spacex's engineers are aware of me and what I have done.  I might be able to shorten their learning process a little;  the "school of hard knocks" can be very expensive indeed.  

Third Spacex Tanker Study

Update 3-26-21:  this third tanker study identified the best approach to refilling the lunar mission Starship on-orbit.  This approach maximizes tanker operations in low circular orbit,  and minimizes them to one refilled fly-along tanker sent to the elliptical departure orbit. 

-------------------------------

This article presents work that follows up on the lunar mission results already posted in Ref. 1 and Ref. 2.  The first was a Spacex tanker study aimed at refilling a lunar landing mission Starship directly in the elliptical departure orbit required to make a lunar landing mission feasible.  The second was a two-stage refilling operation,  with a full refill of the mission Starship in low circular orbit,  and a more modest top-off in the elliptic departure orbit,  from tankers sent directly there. Those orbits were defined in Ref. 3.  The reverse-engineering of vehicle performance characteristics to support this was presented in Ref. 4.

This third study conducts all refilling operations but one in low circular orbit,  where tanker capacities are the highest.  The mission Starship is fully refilled there,  and then sent to its elliptic departure orbit,  and topped off there.  All tankers are initially sent to low circular orbit,  where they fully refill the mission Starship,  and partially refill only one tanker!  That one tanker goes with the mission Starship to the elliptic departure orbit,  where it tops-off the mission vehicle before returning to Earth directly from that orbit.  This approach got the tanker flight requirement down to only one more tanker than would otherwise be required for a full mission Starship refill at max payload in low circular orbit!

Orbits Considered

For clarity,  the elliptic departure orbits being considered are 300 x 7000 km altitude,  and 300 x 10,000 km altitude.  The low circular orbit is 300 x 300 km altitude.  Lunar missions flown from the lower-apogee orbit can take a max 75 metric tons payload to the lunar surface,  with 0 tons payload returned to Earth.  From the higher orbit,  59 tons can be landed on the moon,  with 32 tons returned.  Bear in mind that these elliptic departure orbits penetrate deeply into the Van Allen radiation belts,  whose “base” is nominally about 1400 km altitude.  That definitely means that cargo must be radiation-hard,  and any crew/passengers must have a good radiation shelter,   in order to perform these lunar missions. 

Tanker Options Considered

As indicated in Ref. 1,  there are two possible tanker configurations:  (1) a dedicated tanker design with extra tankage volume in the forward spaces,  and (2) an ordinary cargo or crew Starship flow at zero payload,  so that there is unused propellant aboard upon arrival.  These are based upon comments made by Mr. Musk in his public pronouncements,  and presume the same inert vehicle mass of 120 metric tons as I used,  in reverse-engineering Starship/Superheavy performance in Ref. 4. 

Flown to the low circular orbit,  and withholding only a dry-tanks landing allowance of 9.02 tons propellant,  the deliverable-propellant capacities of these tanker options currently calculate as 232.00 metric tons for the dedicated design,  and 192.51 tons for the ordinary Starship flown at zero payload.   Those capacities are dramatically lower still,  when flown to higher-energy orbits!  They are max capacities:  you can always fly with less,  but never more! 

Calculations

The refill requirements from elliptic departure for the mission Starship were computed in Ref. 1,  and are repeated here in Figure 1 below.  That figure shows both the direct refill in elliptic,  and the two-step refill and top-off requirements.  It is quite easy to see the dramatic difference!  These refill requirements in elliptic orbit are substantially larger than the capacities of the tanker configurations,  when flown directly to the elliptic orbits.  That is exactly why the second study in Ref. 2 got the results that it did,  despite doing the initial full refill of the mission Starship in the low circular orbit.   

I did simple hand calculations with a calculator to determine the departure mass from low circular orbit for the one tanker that accompanies the mission Starship to its elliptic departure orbit.  Those are based on the top-off requirements of the mission vehicle in the elliptic departure orbit.  Those top-off requirements are shown on the right-hand side of Figure 2,  as well as the left side of Figure 1.  Note that due allowance was made for the landing reserve propellant,  while still carrying full deliverable payload. 

A Starship fully refilled in low circular orbit arrives in its lunar departure elliptical orbit only a little less than full,  despite it being early in the burn when you use lots of propellant for little effect.  That is because the mass ratio MR required for the orbit-changing burn is actually rather modest.  This was figured for 3 vacuum Raptor engines at full thrust in vacuum.  That’s enough to be an impulsive burn,  so the orbital mechanics delta-vee is the mass ratio-effective delta-vee for the maneuver (factor = 1.000).  Mass ratio data are given in the middle of Figure 2,  along with the propellant loads upon arrival that allow topping off the mission Starship,  while still retaining a landing reserve. 

The tanker propellant loads needed in low circular orbit are shown on the left of Figure 2.  The refill quantities are less than the propellant load by the landing reserve.  Depending upon which lunar departure orbit and which tanker version,  you add the tanker refill quantity to the mission Starship refill quantity,  to determine the total propellant that must be delivered by tankers to low circular orbit. 

The resulting procedure to run the lunar landing mission is simple.  Note that “ord” refers to an ordinary Starship used as a tanker by flying at zero payload,  and “ded” refers to a dedicated tanker design with extra tankage capacity in the forward spaces,  beyond the regular 1200 metric ton capacity.

Procedure:

Step 1. – Launch the mission Starship plus 7 ord (or 6 ded) tankers to low circular orbit.

Step 2. – Refill the mission Starship from the first 5 ded (or 6 ord) tankers,  leaving the 5th (or 6^th) with a partial load still aboard,  while always withholding landing reserve in all tankers.

Step 3. – Put that partial load in the next-to-last tanker into the last tanker,  to fill it with enough propellant to reach the elliptical departure orbit,  plus just enough to cover the mission Starship refill,  while still maintaining the tanker landing reserve.

Step 4. – The mission Starship plus that one partly-refilled tanker,  go to the departure elliptical orbit,  while the other (empty) tankers return to Earth using their landing reserves.  Their entry speed will be about 8 km/s.

Step 5. – The last tanker tops-off the mission Starship,  there in the elliptical departure orbit,  then returns to Earth using its landing reserve.  Entry speed will be about 9 km/s. 

Step 6. – The mission Starship,  being now fully refilled in its elliptic departure orbit,  can then carry out its lunar nearside landing mission,  including return to a free entry at Earth,  and landing with its landing reserve.  Its entry speed will be about 11 km/s.

The tanker and refill quantity data are summarized in Table 1 below,  just ahead of the figures.  All are at the end of this article.  How these lunar landing missions compare to others is shown in Figure 3. 

Remarks

The reader should bear in mind that these results are not from computer trajectory or orbital simulations run on a computer!  These are the results of simple hand calculations made from simple equations,  sometimes semi-automated with a spreadsheet,  and sometimes not.  For such,  the use of empirical “jigger factors” is necessary to get realistic results.  Those derive from experience in the field.

The reader should also bear in mind that the two-way mission to a lunar landing requires significantly more delta-vee than the one-way flight to Mars.  There is potentially propellant manufacture capability on Mars,  but not the moon.  This has an exponential effect on vehicle mass ratio. 

The assumptions in common here with all the references cited are:

#1. Vehicle “weight statement” is inert mass + payload mass + propellant mass = ignition mass

#2.  Vehicle inert mass presumed to be 120 metric tons,  regardless of type

#3. Engine performance factors are per Ref. 5,  converted to metric

#4. Effects on performance due to throttle setting are presumed linear,  from min to max

#5. Kinematic delta-vee values are factored-up “appropriately” to obtain mass ratio-effective values

#6. Delta vee factors are experiential judgements that account for gravity losses,  drag losses,  and any hover or divert effects during landings

#7. For purposes of eastward launch,  the surface circular velocity is used as the kinematic velocity to attain,  in order to include potential energy effects into the rocket equation calculation

#8. For purposes of eastward launch,  the surface circular velocity (or higher) gets factored by 1.10 to account for gravity and drag losses at 5% each

#9. If launch were polar,  the eastward velocity due to Earth’s rotation at the launch latitude would be added to the surface circular (or higher) velocity (add two rotation velocities if westward launch)

References (all are located on this site)

#1. G. W. Johnson,  Spacex Tanker Investigation,  dated 17 March 2021.

#2. G. W. Johnson,  Second Spacex Tanker Study,  dated 21 March 2021.

#3. G. W. Johnson,  Reverse-Engineering Estimates:  Starship Lunar Landings,  dated 15 March 2021.

#4. G. W. Johnson,  Reverse-Engineering Starship/Superheavy 2021,  dated 9 March 2021.

#5. G. W. Johnson,  Reverse-Engineered “Raptor” Engine Performance,  dated 26 September 2019.

The easiest way to locate references on this site is to use the navigation tool on left-of-page.  You will need the year and month it was posted,  and the title.  Click first on the year,  then on the month.  If need be,  click on the title.  It would be wise to scrawl all this reference date and title information onto a piece of scrap paper,  before starting that research.

               Table 1 – Summary of Third Study Results 

Figure 1 – Mission Starship Refill Requirements

 

Figure 2 – Tanker Refill Requirement to Go From Circular to Elliptic and Top-Off Mission Vehicle

Figure 3 – Comparing Lunar Missions to Other Possible Missions

Second Spacex Tanker Study

Update 3-26-21:  this second tanker study looked at refilling the lunar mission Starship in low circular orbit,  where tanker Starship capacities are high,  then topping it off after moving it to the departure orbit,  with more tankers sent directly to that orbit.  That reduced the number of required tankers,  but not by enough to be truly practical.  The strategy explored in the third study (also posted here) turned out to be the most effective one.

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In Ref. 1,  I defined two elliptical orbits from which a Spacex Starship could be sent fully-fueled on a lunar nearside landing mission,  be unrefueled on the moon,  and return to a free entry landing trajectory at Earth.  Payloads sendable to the moon this way are far smaller than payloads deliverable to low circular Earth orbit,  because payload capacity to the higher-energy orbit is reduced,  as well as the inherently-smaller payload capacity for the lunar flight. 

It makes sense to use the elliptical orbit at which Starship payload capacity to that orbit matches the Starship lunar mission capacity.  Some 75 metric tons can be sent to the moon,  with zero return payload,  from an elliptical orbit that has 300 x 7000 km altitudes.  Some 59 tons can be sent to the moon with a 32 ton return payload,  from an orbit that is 300 x 10,000 km altitudes.

Both elliptic orbits have apogee altitudes that penetrate well into the Van Allen radiation belts.  Thus cargoes delivered this way must be radiation-hard,  and any crew or passengers will require effective shelters from serious radiation exposure.  The “base” of the Van Allen radiation belts is considered to be about 1400 km altitude,  outside the South Atlantic Anomaly. 

In Ref. 2,  I determined the deliverable propellant quantities from two potential Spacex tanker configurations:  a dedicated tanker design with extra tankage volume in the forward spaces,  and an ordinary Starship flown fully fueled,  but at zero payload,  so that it arrives with significant unused propellant.  The scope of that study looked at the 300 x 300 km altitude low circular Earth orbit,  and the two possible elliptical lunar mission departure orbits that “sort-of” bound the problem. 

Deliverable tanker “payload” with either configuration was considerable in low circular orbit,  but a lot less when sent directly to either of the two elliptical orbits.  This is because the higher energy orbits require a whole lot more propellant just to be reached.  In all these scenarios,  the dedicated tanker design carries somewhat more deliverable propellant than the ordinary Starship flown at zero payload.

This basic information is summarized in Figure 1 below.  All figures are located at the end of this article.  The two elliptical lunar departure orbits and associated lunar payloads are in the upper part of the figure.  The tanker delivery capacities are shown for both tanker configurations,  at each of the three orbit locations,  in the lower part of the figure.  In each case,  the Starship/Superheavy tanker vehicle is flown directly to the target orbit location,  as is the cargo/passenger Starship/Superheavy vehicle. 

Two Waves Of Refueling On-Orbit

The small tanker propellant delivery capacities to elliptic orbit versus the large capacities to low circular orbit suggested a two-step refueling operation for the lunar-bound Starship:  a full-capacity refill in low circular orbit,  followed by moving it to the elliptic orbit,  with another refill there.  This would apply to either lunar mission scenario. It could be done with either tanker configuration.  Hopefully,  fewer tanker flights would be required than by direct one-step refueling in the elliptic departure orbit.

To support this two-step refueling operation,  I had to figure out what the refill requirements would be upon arrival in the target elliptic orbit,  and also the low circular refueling requirements when carrying the smaller lunar payloads.  The elliptic refill requirements I hand-calculated,  and these are summarized in Figure 2.  The low circular refuel requirements at lower payloads,  I figured with one of the Starship spreadsheet models.  Those results are Figure 3 for the 75 ton payload from the 7000 km apogee orbit,  and in Figure 4 for the 59 ton payload from the 10,000 km apogee orbit.

Calculating Numbers of Tanker Flights

Then I calculated the number of tanker flights required to carry out these operations (first the low circular refill,  then second the elliptic refuel after moving there).  In each operation,  the requisite tankers are flown directly to the orbit where the refill will take place. The limiting number of tanker flights is the refill requirement divided by tanker delivery capacity,  but the only such number that makes any sense is an integer!  You always hold back the tanker landing reserve from deliverable propellant. 

If there are decimals,  you must round up to the next-larger integer!  That is because you cannot carry more deliverable propellant than the max capacities these calculations have identified,  but you can always carry a little less than max capacity. 

Intermediate Results

The refill requirements in low circular orbit for the lunar-bound Starship are just about 1050 tons of propellant,  depending upon which mission and payload we are talking about.  The tanker capacities to low circular orbit are right at 200 tons per vehicle,  give or take,  depending upon whether the tanker is the dedicated design or the ordinary Starship flown as a tanker.  Those numbers and the rounded-up integer numbers of flights are given in Figure 5.  Note that the delivered tanker loads are just not very far from the max capacities of the tankers,  as also given in the figure.   We are looking at 5 dedicated tankers,  or 6 ordinary tankers,  to accomplish this,  for either lunar departure scenario.

The second refueling operation is illustrated in Figure 6.  The refill requirements are far smaller,  but then so are the max tanker capacities!  Again,  you round up the decimals to the next larger integer number of tankers.  Those results show 3 tankers (of either type) are needed for the 300 x 7000 km orbit that supports 75 ton lunar landings,  with 0 tons return payload.  The more demanding 300 x 10,000 km orbit requires 5 tankers of either type,  but that scenario supports a 32 ton return payload,  with 59 tons landed on the moon.  It is the higher-energy orbit.

Ultimate Results

A summary comparison of these results versus the direct staging in elliptical orbit,  and versus operations only to 300 x 300 km circular  are given in Figure 7.  That figure this summarizes the results of the first tanker study and this second one. 

The max payload deliverable to low circular orbit is quite a bit more at 171 metric tons.  That is why 7 ordinary or 6 dedicated tankers are needed to fully refill the Starship there.  That’s a Ref. 2 result.

Another Ref. 2 result is the single refueling operation conducted with Starship and tanker flights directly to the lunar departure elliptic orbit.  That is also indicated in the figure,  which shows some 21 ordinary or 19 dedicated tanker flights to refill the Starship carrying 75 tons to the moon from the 300 x 7000 km orbit.  The Starship carrying 59 tons from the 300 x 10,000 km orbit requires 17 ordinary or 15 dedicated tanker flights to fully refill.  Those high tanker flight numbers are not very attractive!

The two-refill operation approach analyzed in this article (with the first wave of tankers sent to low circular orbit,  and the second wave of tankers sent to the departure elliptic orbit) does indeed reduce the number of tanker flights.  For the 300 x 7000 km departure with 75 tons to the moon,  the first wave is 6 ordinary or 5 dedicated tankers to circular,  followed by the second wave of 3 ordinary or 3 dedicated tankers to the elliptical orbit,  for a total of 9 ordinary or 8 dedicated tankers to support the mission.  For the 300 x 10,000 km departure (59 tons to the moon),  the first wave is the same 6 ordinary or 5 dedicated tankers,  and the second wave is 5 ordinary or 5 dedicated,  for a total of 11 ordinary or 10 dedicated tankers to support that mission.  That is a substantial improvement,  but still unattractive.

Remarks

Readers need to be aware that these calculations I have made are not any sort of simulations run with any sort of computer programs.  These are the kind of calculations I would make,  if I sat down at the kitchen table with pencil,  paper,  and a pocket calculator (or even a slide rule).  These have been semi-automated with spreadsheet software,  but are essentially the same very simple calculations made with simple models,  plus the engineering art of selecting the right “jigger factors” to get realistic results. 

For those readers too young to know what a slide rule is,  see Ref. 3.

Plans

The improvement achieved with two-step refilling is attractive,  and the disparity between tanker capacities in low circular versus feasible lunar departure orbits is large.  That suggests refilling not only the lunar mission Starship in low Earth orbit,  but also 1 or maybe 2 tankers.  The mission Starship plus the 1 or 2 refilled tankers would then move to the elliptical departure orbit,  where those tankers would top-off the mission Starship.  As time and opportunity permits,  I will look at this method of employing these assets,  to see if the total number of supporting tanker flights can be reduced further.

References (all are located on this site)

#1. G. W. Johnson,  Reverse Engineering Estimates:  Starship Lunar Landings,  dated 15 March 2021.

#2. G. W. Johnson,  Spacex Tanker Investigation,  dated 17 March 2021.

#3. G. W. Johnson,  THIS Is a Slide Rule!,  dated 16 March 2019.

Figure 1 – Selected Results From the First Tanker Investigation

Figure 2 – Determining the Refueling Needs Moving From Circular to Elliptical With Lunar Payloads

Figure 3 – Determining the Refueling Needs in Circular with Lunar Payloads:  75 ton

Figure 4 – Determining the Refueling Needs in Circular with Lunar Payloads:  59 ton

Figure5 – Tankers Required to Refuel Lunar-Bound Starships in Circular Orbit (First Refuel)

Figure 6 – Tankers Required to Refuel Lunar-Bound Starships in Elliptical Orbit (Second Refuel)

Figure 7 – Comparison of First Investigation Results with Results of Second Study

 

Spacex Tanker Investigation

Update 3-26-21:  This first study looked at refueling a Starship sent directly to its elliptical departure orbit with tanker Starships also sent directly to the same orbit.  That turned out not to be the best way to do the required on-orbit refilling for lunar missions.  The third study (also posted here) turned out to be the best way. 

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This article is based on published comments by Elon Musk that (1) there will be a dedicated tanker design for on-orbit refueling operations with Starship,  but (2) initially the “tanker” will just be another Starship. 

In the first case (dedicated tanker design),  there will be additional propellant tankage volume in the forward spaces,  where the cargo bays and or passenger accommodations are located,  in the other designs.  The propellant in that extra tankage volume is the “payload mass” ferried to orbit.  It computes directly as any other payload in rocket equation and the 2-body orbital mechanics approximations to performance that I have been using for these estimates. 

For that second (initial) case,  the only thing that makes sense is the fly the ordinary Starship-as-tanker to orbit with zero payload but a full propellant load.  It will have unused propellant upon arrival to orbit,  in excess of that needed to land.  That excess can be transferred to the target Starship. 

For both tanker cases,  there is no need to support an abort to the surface with payload aboard,  because excess propellant mass can be vented to space prior to any deorbit burn.  Thus,  the tanker landing can be made to the “dry tanks” standard with zero payload,  in either case.  That reduces the propellant reserve for landing that must be withheld from any on-orbit refueling operation.

When I investigated lunar missions with the Starship in Ref. 1,  I looked at a variety of orbits,  from 300 x 300 km altitude circular,  through a variety of elliptic orbits all sharing the 300 km perigee altitude.  I found that the 300 x 7000 km altitude orbit,  and the 300 x 10,000 km altitude orbit,  might prove feasible for Starship lunar missions,  depending upon the payload to and from the moon.  The 300 x 300 km altitude circular is of general interest for all sorts of applications.  Those three orbits are the scope considered in this article.  This article also presumes that all refueling is done in the departure orbit.

General Operations In 300 x 300 km Circular (Excluding Lunar Landing Missions)

When I reverse-engineered orbital performance of the Starship/Superheavy in Ref. 2 (and to support Ref. 1),  I obtained payload capabilities and staging velocities for Starship/Superheavy,  under the assumption that the Starship landing reserve of propellant be sufficient to support an abort to surface with the full payload still aboard (a dry-tanks landing at full payload). 

That is not necessary for any Starship design used as a tanker.  Tankers can always vent propellant overboard prior to the deorbit burn,  down to only that required for a dry-tanks landing at zero payload. Which situation is exactly why the propellant-as-payload and landing reserve numbers for the tankers are different from those for the target ship,  to the same orbit.  Data are given in Table 1.

                              Table 1 – 300 x 300 km Altitude Results

Vehicle                               target                   dedicated            ordinary

Payload, m.ton                 171                       232                       192.506

Vstaging, km/s                 2.23                      2.21                      2.375

Land. Res., m.ton             21.876                 9.019                   9.019

Full refill req., m.ton       1178.124            --                             --

Tankers to refill                --                           5.08                      6.12

In the data table,  “target” means the Starship vehicle to be refueled on-orbit.  “Dedicated” means a dedicated tanker Starship design with extra tankage volume in the forward section.   “Ordinary” means another cargo or passenger Starship operated with zero payload in the forward section,  whose deliverable “payload” is merely the excess propellant not used reaching orbit,  less the landing reserve.  “Full refill” means filling to the capacity,  currently thought to be 1200 metric tons.  The number of “tankers to refill” is for a full refill,  and is the max refill requirement divided by the deliverable propellant from a tanker vehicle.  For planning purposes,  you should round up any decimals.

Not all missions flown from this orbit require a full refill to max capacity.  Mars missions can be flown from this orbit,  but lunar landing missions cannot. 

Lunar Landing Missions From 300 x 7000 km Elliptic Without Return Payload

Lunar landing missions are too demanding in terms of total delta-vee to be flown from the low circular orbit,  without any refueling on (or near) the moon.  The same is true of Mars missions,  although there are materials on Mars from which propellant can be made,  unlike the moon. 

Departing from a sufficiently-elongated elliptic orbit with a high-enough perigee velocity lowers the departure delta-vee sufficiently to make an unrefueled lunar landing mission feasible with a filled-to-capacity Starship,  at some reduced payload.  The higher-energy elliptic orbit also reduces the payload that can be carried up to orbit from Earth.  It makes sense that payload ferried up is also payload carried to the moon.  The elliptical orbit where these payloads are equal is preferred for the mission. 

See the data in Table 2,  bearing in mind that these results for the target vehicle were partially just read from a graph in Ref. 1,  not specifically supported by precise calculations.  Imprecision is inherent.

                              Table 2 – 300 x 7000 km Altitude Results

Vehicle                               target                   dedicated            ordinary

Payload, m.ton                 ~75                       83                         73.4

Return payload, m.ton     0                          --                             --

Vstaging, km/s                 ~2.23                    2.22                      2.275

Land. Res., m.ton             ~14.9                    9.019                   9.019

Full refill req., m.ton       ~1185.1               --                             --

Tankers to refill                --                           14.28                   16.15

Also,  the presumption here is that the tankers fly directly to the same elliptic departure orbit as the target vehicle.  No other scenario was considered at this time.  That does not mean that there are no other scenarios!  One possibility for future investigation is refueling in low circular orbit,  then moving the target vehicle to the departure orbit,  along with 1 or 2 more tankers to fully refill it just before departure.

Lunar Landing Missions From 300 x 10,000 km Elliptic With Max Return Payload

There is one other nuance to consider:  how much payload can be returned from the moon?  The 7000 km apogee altitude is associated with a higher payload to the moon,  but a zero return payload,  per Ref. 1.  The 10,000 km apogee altitude is associated with a lower payload deliverable to the moon,  but a nonzero return payload,  again per Ref. 1.  See the data in Table 3.  (Again,  the presumption here is that all of the tankers rendezvous with the target vehicle in the elliptic departure orbit.) 

                              Table 2 – 300 x 10,000 km Altitude Results

Vehicle                               target                   dedicated            ordinary

Payload, m.ton                 59                         64                         57.657

Return payload, m.ton   32                         --                             --

Vstaging, km/s                 2.23                      2.215                   2.263

Land. Res., m.ton             13.456                 9.019                   9.019

Full refill req., m.ton       1186.544            --                             --

Tankers to refill                --                           18.54                   20.58

“How I Did It” for the 2 Different Tanker Choices

These tanker versions of Starship and the “regular” cargo or passenger Starship all share exactly the same “model” of the Superheavy booster.  That booster’s flyback for recovery is modeled first,  and in reverse.  The touchdown is figured as a dry-tanks landing,  with engine cutoff just as the on-board propellant zeroes.  Then the entry burn is modeled,  and finally the boost-back burn.  That quantifies the propellant still on board at staging. 

The capacity less the propellant-still-on-board at staging,  is the booster propellant available to reach staging velocity.  The way I set this up,  you iterate the imposed staging velocity until the overall mission’s remaining propellant is a small positive fraction of a ton.  That result is also affected by the payload in the Starship weight statement.  The Superheavy booster sees the fully-loaded Starship upper stage as its “payload”,  which affects how much propellant is needed to reach staging velocity.

The Starship upper stage analysis begins with the landing.  For the cargo/passenger vehicle,  I require a dry-tanks landing with the full payload still aboard,  as a way to support an abort to surface without the chance to unload payload.  For the tankers,  the dry-tanks landing is made with zero payload aboard.  This is because with the dedicated tanker design,  the “payload” is propellant that can be dumped to space before the deorbit burn.  For the “ordinary Starship-as-tanker”,  loaded payload is zero.  The excess propellant in the regular tanks can be dumped to space before the deorbit burn.

You analyze the effects of the landing and the deorbit burn,  to determine the propellant reserve that must be aboard,  at the time of the deorbit burn.  That reserve is larger for the cargo/passenger vehicle,  and smaller for the tanker (of either kind). The propellant available for the ascent burn is capacity less the landing reserve. 

For the dedicated tanker design,  I use exactly the same analysis as for the cargo/passenger Starship.  You burn the available propellant,  and see what delta-vee that gets you.  You set the payload to that value which gets you the right delta-vee.  Your “payload” is the transferable propellant.

For the ordinary Starship-as-tanker,  you set payload to zero,  always.  The calculation finds out how much propellant is required to reach the correct delta-vee.  That is the required ascent propellant.  The capacity less the ascent propellant,  and less the landing reserve,  must be a non-negative number.  That is the unused propellant available for transfer while on-orbit. 

Future Plans

I want to look more closely at doing a full refill of the target vehicle in the low circular orbit,  then moving it and 1 (or 2) more tankers to the elliptic departure orbit.  That would be an attempt to reduce the number of tankers required to effect a full refill at departure for a lunar landing mission.  Scope would be for the 300 x 7000 km orbit with 75 tons to moon,  and 0 return payload,  plus the 300 x 10,000 km orbit with 59 tons to the moon and 32 tons returned.  That might have a large payoff,  given the large disparities in the number of tanker flights needed for the low circular orbit,  versus either of the lunar-departure elliptical orbit choices.

Final Comments

These numbers were created with essentially pencil-and-paper hand calculations for the rocket equation and the combination of 2-body solutions for the orbital mechanics.  The only way to make such calculations realistic is the judicious use of appropriate factors to increase the theoretical delta-vee requirements,  and a judicious approximation to combine 2-body orbital mechanics models in lieu of a real 3-D trajectory program in a computer.  That art is summarized in Refs. 3 and 4. 

These numbers were also created with the best available data regarding the propellant capacities and inert masses for the Starship and Superheavy stages,  and for the performance characteristics of the sea level and vacuum versions of the Raptor engine.  More of that was available on the Spacex website in prior years than is there now.  Much of this now has to be obtained from online sources like Wikipedia. 

I did my engine performance figures from information that used to be posted on the Spacex website,  and is no longer posted there.  They do still post propellant capacities for the stages,  but not the inert masses (and they never did).  Inert masses came from Wikipedia. 

The results are very sensitive to such data,  particularly the stage inert masses.  Increases in inert mass must come directly out of payload capability.  What I have is “in the ballpark”,  but still subject to change,  especially as Spacex has begun flight testing of the Starship vehicle,  and soon the Superheavy booster.

These results indicate some phenomenally-attractive capabilities with this basic vehicle design.  Those are enough to warrant solving both the known,  and the still-unknown,  problems with this design,  which problems will become more-and-more evident as flight testing proceeds.  Those include (but are not limited to) ullage/slosh problems inside un-bladdered cryogenic propellant tanks,  propellant leaks leading to engine bay fires,  and a totally-inadequate design concept for the landing legs (both stages,  but more especially the second stage). 

References (all are articles published on this site)

#1. G. W. Johnson,  Reverse-Engineering Estimates:  Starship Lunar Landings,  15 March 2021.

#2. G. W. Johnson,  Reverse-Engineering Starship/Superheavy 2021,  9 March 2021.

#3. G. W. Johnson,  Fundamentals of Elliptic Orbits,  5 March 2021.

#4. G. W. Johnson,  Back of the Envelope Rocket Propulsion Analysis,  23 August 2018.

Update 3-18-21:

In response to a comment,  I have added 6 figures that are the images to the spreadsheet worksheets that I used to model these vehicles.  The inert masses for both stages are there,  along with full engine performance figures,  and choices for number of engines and  thrust settings.  Figures 1-3 are for the ordinary Starship-as-tanker to 300x300 circular,  300x7000 elliptical,  and 300x10,000 elliptical.  Figures 4-6 are for the dedicated tanker design for 300x300,  300x7000,  and 300x10,000.  

Figure 1 -- Ordinary Starship as Tanker, 300 x 300 km Circular

Figure 2 -- Ordinary Starship as Tanker, 300 x 7000 km Elliptic

Figure 3 -- Ordinary Starship as Tanker, 300 x 10,000 km Elliptic

Figure 4 -- Dedicated Tanker Design, 300 x 300 km Circular

Figure 5 -- Dedicated Tanker Design, 300 x 7000 km Elliptic

Figure 6 -- Dedicated Tanker Design, 300 x 10,000 km Elliptic