Going Back to the Moon

The author has updated this posting as of 17 Oct 2025,  with revised and expanded conclusions and final remarks.  Nothing else has changed.

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This paper explores and compares two different approaches for returning crews to the moon,  this time to the poles,  instead of being limited to near-equatorial sites. 

About the Author

See Table 1.  The author is qualified by both training and experience to do this kind of study.  More details are provided in Appendix A,  for those interested.

                                             Table 1 – Education and Experience

BS         Aerospace Engineering          UT Austin          1972

MS        Aerospace Engineering           UT Austin          1974

PhD      General Engineering                KWU                   2000

20 years defense/weapons research/development/test/engineering

20 years mostly teaching, plus some civil engineering and aviation work

Now retired

How to Reach Polar-Capable Staging Orbits

We wish to return to moon,  but,  with bigger crews and more cargo,  and we wish to visit polar regions instead of just near-equatorial regions,  as with Apollo.  There are fundamentally two options:  staging from circular low lunar orbit (LLO),  or staging from the Gateway space station located in its “halo” orbit about the moon.  See Table 2.

Table 2 – Objectives

               Want to Return to the Moon,  But …

                              Visit instead the south polar region

                              Bigger crews and payloads

               Two Possible Classes of Mission …

                              Direct to LLO like Apollo,  but do it polar

                              Gateway-based from “halo” orbit (to polar)

Both options require an orbital plane change to reach the lunar polar regions,  since both the transfer trajectory to the moon,  and the Gateway halo orbit,  are fundamentally Earth-equatorial,  which inherently makes them lunar equatorial.   However,  in extended elliptic orbits,  the apoapsis speed of an extended ellipse is inherently low,  so that plane changes made at apoapsis cost very little in the way of the velocity-change requirement dV.  For no change in speed,  only a change in direction,  the plane change dV is computed as:

dV = 2*V*sin(a/2),  where a is the angle by which direction is changed

The Gateway halo orbit is already a very-extended ellipse: 70,000 km by 3000 km, center-to-center (CTC).  It is so greatly extended,  that it is fundamentally unstable long-term.   Making the 90-degree plane change at its apolune results in a very small plane-change dV indeed.  Apollo entered very low circular LLO directly,  the speed of which would require significant dV for all but a trivial plane change.  But,  if instead the entry was made to an extended ellipse (“elliptic capture”),  the same type of low-cost large-angle plane change could be obtained,  before circularizing.  Nominally,  these are 90 degree plane changes,  for which the dV is just apolune speed multiplied by the square root of 2.  See Figure 1. 

The “Hill Sphere” stability limit distance (a “fuzzy” value) about the moon is some 60,000 km CTC.  Beyond that distance,  the gravitational influences of the Earth are getting stronger than that of the moon,  making such an orbit about the moon long-term unstable.  Staying in such an orbit would require corrective propulsion periodically.  The Gateway station in its “halo” orbit suffers this problem.  The elliptic-capture modification to the Apollo trajectory selected for this study does not!   For more details,  see Appendix B. 

Figure 1 – Gateway “Halo” and Apollo-with-Elliptic-Capture Offer Polar Orbits Capability

Landings and Earth Return

From a polar LLO,  landing on the desired site is simple:  one merely times the deorbit burn to hit the desired landing site.  Ascent is just the reverse,  although the lander weight statement upon ascent is likely different from that during descent.  From the Gateway halo,  making a landing is not so simple.  There is the plane change at apolune,  followed by circularization at perilune.  From there the timed deorbit is similar to the LLO option,  except that the orbit altitude is much higher than that used for Apollo.  See Figure 2.  For more details,  see Appendix C.  The net effect is a higher dV than from LLO/polar.

Bear in mind that returning to Earth is pretty much the reverse of how one gets to the moon.  The basis is the Apollo transfer trajectory,  which orbital “three-body effects” warp into a lopsided figure-8 shape,  resulting in a retrograde pass behind the moon,  at a speed relative to the moon somewhat higher than lunar escape at that altitude.  Entry is possible into a retrograde orbit about the moon,  by making a deceleration burn.  The retrograde direction is no problem for landings,  since the moon’s rotation rate is so very slow. 

There are two possible options upon returning to Earth:  (1) direct entry,  descent,  and landing,  or (2) decelerating into Earth orbit for recovery and reuse without doing an entry. 

The figure also shows departures from Earth,  all of those being from 300 km altitude low Earth orbit (LEO) for this study.  LEO is usually considered to be 300-600 km.

Figure 2 – Making the Landings,  and Returning to Earth

Using the Rocket Equation Correctly When Weight Statements Change

The way the rocket equation was defined and derived,  each calculation is made with a mass ratio that must be associated with the one-and-only weight statement which produces that very mass ratio.   The dV’s for multiple burns that occur within that one weight statement may be summed for a single overall calculation,  but if the weight statement changes between burns,  those dV’s may not be summed!

If there are changes to the itemized buildup in the weight statement,  such as offloading payload between burns,  or jettisoning inert items between burns,  then those mass ratios are simply different.  In such cases,  one must do separate,  but linked,  rocket equation calculations,  for each different weight statement.      

See Figure 3 for guidance on how doing linked calculations relates to the weight statement buildups.  More details (including how to iterate) are given in Appendix D.  

Figure 3 – Doing Rocket Equation Calculations Correctly

The other major influence upon the rocket equation is the effective exhaust velocity,  defined as thrust divided by propellant massflow rate,  a specific impulse.  In order to relate to the mass ratios for the stage or vehicle,  that propellant massflow rate must be the value actually drawn from tankage,  not just the nozzle massflow rate!  If the engine cycle involves a dumped bleed flowrate to power the pumps,  those two flow rates are not equal!  The flow from the tanks is the sum of nozzle and dumped bleed flows!

Vehicle Concepts    

To get from LEO to polar region landings on the moon,  this study presumes 3 vehicle concepts to be sized.  These are (1) the lander module (LM),  (2) a “main craft” that comprises a command module (capsule) for the crew,  plus a propulsion and support service module,  all together designated CSM,  and (3) the departure stage or translunar injection (TLI) stage.   Each is the payload for the next,  in the order listed.

The CSM must push the LM to the moon,  but not back to Earth.  The TLI stage might be expendable,  or it might be recoverable into LEO for re-use.  For this study,  the capsule makes a direct free entry and landing on Earth,  while the service module is expendable,  very much like Apollo.  Also for this study,  the LM is single stage,  unlike Apollo,  and gets left in the polar circular orbit from which the landings are made.  If reused,  it is refueled there in that circular polar orbit,  with propellants brought from LEO on subsequent visits.

This study presumes that different propellants and inert mass fractions are used to size these 3 vehicles.  The lander and the TLI departure stage are both presumed to be high-specific impulse (Isp) propellants,  specifically oxygen-hydrogen (LOX-LH2).  This is necessary because the dV requirements are substantial,  for both.

The main craft (CSM) is presumed to be lower-Isp storable propellants,  since those velocity requirements are substantially lower than those of the other two vehicles.  That makes something like a SpaceX Dragon capsule,  fitted with an oversized trunk that is fully propulsive,  feasible as a possible candidate hardware item. 

Note that the lander payload is different for ascent than it was for descent.  For descent,  a 4 metric ton allowance was made for 4 crew plus all their suits,  supplies,  and personal equipment,  plus another 4 ton allowance for cargo delivered to,  and left upon,  the lunar surface.    For ascent,  the same 4 ton allowance for crew is used,  knowing that return samples effectively replace the mass of supplies used.

For purposes of this study,  a constant 10 metric ton mass for the crew capsule was used,  for a 4-person crew.  That really is “Dragon-class”. 

The TLI stage has both the CSM and the LM together as its payload for departure from LEO.  It has no payload at all for a recovery back into LEO upon return,  if it is to be reused. 

A key to understanding the lander concept illustrated was “rough field stability”.  Min leg span must significantly exceed center of gravity height.  Minimizing crew ladder height is also essential.   The rest of the propulsive items are just simple cylindrical stages.  Tanks are indicated,  as well as guidance and control (G&C) units,  power supplies for those G&C units,  plus power and life support for the crew as applicable. 

Low density propellants increase stage inert fraction.  So also does adding the insulation and equipment to achieve “long” stage life (multiple weeks) with highly-evaporative cryogenics like hydrogen.  Adding landing legs plus enclosed spaces for crew and cargo further increase inert fraction for the lander.  The extra power and life support supplies for the service module of the CSM increase its inert fraction,  despite the use of high-density,  storable propellants. See Figure 4.

Figure 4 – Vehicle Concepts,  As Used for this Study

Requirements and Results for Landers 

These vary strongly by orbital attitude and mission complexity.  Both descent and ascent dV values (and totals) are shown in Figure 5.  Be aware that for rocket equation analysis,  the summed descent and ascent dV’s cannot be used in a single calculation,  as the weight statements change for ascent vs descent.  Note that the general trend is higher dV requirements as polar circular orbital altitude increases.  The “halo” option is further complicated by the need to even reach such a polar circular orbit from the Gateway station.  These dV values include the increases for losses and budgets for maneuvering.

Figure 5 – Requirements for Landers by Mission

Lander sizing results are depicted in Figure 6.  These are single-stage landers,  potentially refuellable and reusable,  left in the circular polar orbit from which the final descent was initiated,  and to which the ascent returns.     

Note that lander mass about doubles from 100 km circular up to Gateway/halo energy levels.  That is the exponential relationship between dV requirements that do not double and the resulting mass ratios demanded of the vehicle.  These figures do reflect the different weight statements for ascent vs descent. 

Note also that max and min thrust levels were determined from appropriate LM acceleration values at ignition and at landing,  plus at takeoff.  These are merely appropriate gee levels applied to the corresponding Earth weights in the weight statements.  

Figure 6 – Results for Lander by Mission

Requirements and Results for Main Vehicle (CSM) by Mission

The dV requirements for this vehicle to get into,  and back out of,  the appropriate mission orbit are summarized in Figure 7.  These include budgets for course corrections and rendezvous as required by mission. 

Note that for the Apollo-type LLO cases,  the dV requirements actually decrease a little with increasing altitude,  despite the elliptic capture and plane change to polar.  This is the only vehicle case where that happened.  The rest of the vehicle cases show an increase in dV with increasing orbit altitude. 

The more complex CSM mission is getting all the way to Gateway station in the halo orbit,  from circular capture at the halo perilune altitude.  Circular capture is required to time entry onto the halo ellipse for Gateway rendezvous.  That drives the CSM mission dV requirement up greatly,  relative to the other LLO missions.  

Figure 7 – Requirements for Main Craft (CSM) by Mission

Main craft (CSM) sizing results are given in Figure 8.  These include weight breakouts for the LM and the service module,  as well as the CSM-LM cluster.  The command module (capsule) mass is a constant 10 metric tons. 

The more challenging halo mission doubles CSM+LM cluster mass over that of the 100 km circular polar mission.

Note that the inert fraction applied to the CSM sizing analysis is done without including the LM in the ignition mass.  That is unique among all these sizing calculations.  

Figure 8 – Results for Main Craft (CSM) by Mission

Requirements and Results for Departure             

As shown in Figure 9,  the dV is higher for recovery vs expendable,  but it applies to all the mission concepts.  Only the TLI stage payload mass varies by mission,  that being the CSM+LM cluster by mission.  Note that the recoverable dV is higher not only because of the orbital change dV,  but also budgets for course correction and a second rendezvous in LEO.  

Figure 9 – Requirements for Departure (TLI) Stage

The departure (TLI) stage results are given in Figure 10.  TLI stage mass almost doubles as the lunar orbit energy increases from 100 km LLO polar to the Gateway/halo.  Adding recovery back into LEO almost doubles TLI stage mass again.

Figure 10 – Departure Stage Results by Mission,  Expendable and Recoverable

Conclusions and Disclaimers 

The trends identified by this study are very clear (as listed in Table 3):

(#1) The dV requirements are mostly higher for halo vs LLO.  That implies better designs will be had using the modified Apollo polar LLO approaches.  Such will be smaller and lower in mass. 

(#2) The resulting halo-based vehicles ~2x heavier than LLO.  Smaller,  lighter vehicles enable simpler designs,  less costly to build,  and less costly to launch.

(#3) The lowest LLO is the most attractive,  but only use the lowest LLO altitude that is safe!  One does not want to “smack” into a lunar mountain by using too low an altitude,  either directly,  or because of small navigation errors that are simply inherent.  .  That puts the safe range of altitudes somewhere near,  or not very far above,  the Apollo altitude.  That would be 100 to 500 km.

    Table 3 – Conclusions

                 Trends are very clear:

                              Mission dV requirements are larger for halo

                              Halo propulsion items ~ 2 times heavier

                   Use the lowest LLO that is safe

                                Raising altitude increases masses

                                Results include using elliptic capture to polar

Disclaimers include both the generic Isp values and the “WAG” estimates of stage/vehicle inert fractions,  as listed in Table 4.  This study used only generic Isp values,  not real engine design performance.  It also used “WAG” guesses for vehicle inert mass fractions.  “WAG” literally means “wild-ass guess”.   While the numbers themselves may not be very good,  the trends of those numbers by mission really are good!

Of these two influences,  inert fractions are actually the stronger influence.  But,  they show up indirectly,  in the weight statements associated with the linked rocket equation calculations.  That half-hidden nature may be why most people do not recognize this issue,  the way that they do the explicitly-appearing Isp values.

Generally,  this study used a loaded stage basis to “model” structural.  The idea behind that was to provide more structural mass to push the heavier payloads at acceptable gees,  without risking compressive collapse of the stage.  The only exception was the CSM unit.

                      Table 4 – Disclaimers

                       No specific engine designs

                                   Generic Isp levels only

                        Loaded stage inert fractions = “WAG”

                                    Inert fraction effect > Isp effect

                                    Used loaded stage as basis with payload

                                    Structure to push very large payloads

Overall Conclusion

If we could do a “clean sheet of paper” program design,  not constrained by any existing spacecraft or launch rockets,  the answer is crystal clear:  return to the moon basing out of a low lunar orbit,  somewhere between 100 and 500 km altitude,  just oriented polar by using elliptic capture.   With elliptic capture-enabled low-cost plane change,  the entire surface of the moon is reachable with the smallest,  lightest vehicle designs.  

Overall:  go with an elliptic capture-modified Apollo-to-polar-LLO,  at an altitude between about 100 and 500 km.

Final Remarks

For those who have difficulty with the notion of doing burns at min orbital speed to drastically change the plane of an ellipse,  see Appendix E.  This is most easily done by using elliptic capture,  and then making the plane change at a high apoapsis.  You can circularize as needed,  after making the plane change.  The dV to capture elliptically,  plus the dV to circularize later,  is equal to the dV to directly into the circular orbit.  You are only “out” the small apoapsis dV to change the plane.

               assembly vs one mission,  one launch

In general,  the notion of “one mission,  one launch” is obsolete,  and has been,  ever since construction began on the International Space Station over 30 years ago.  There is simply no reason why vehicles of any size for any deep space missions,  cannot be assembled by docking smaller items together,  in LEO.  You just need a craft or a facility with remote manipulator arms to make it practical and relatively easy.  We already know that! 

               surviving rough field landings

We also have long known the main survivability requirement for rough-field landings on the moon.  The height of the center-of-gravity above the surface must be less than the minimum span dimension of the landing leg pads.  Some recent commercial landers have violated that criterion,  and every one of them has fallen over,  so far.  That pattern is quite real!  And it applies to Mars,  as well!

               longer “stage life” with cryogens

To get “stage life” in the 2 week range for these lunar missions with liquid hydrogen and the other cryogenics,  we already know the common tank bulkhead approach is wrong!  The powered “cryocooler” approach adds significant inert mass and a power requirement.  There is a notion in between those two extremes:  a thick layer of low thermal conductivity insulation wrapped around the cryogen tank,  topped with a very reflective thin foil to reflect away solar heating.  

It is basically an integral sunshade,  the “sunshade technique” already known to be effective with (more modestly) cooled infrared equipment in space.   This design approach needs to be investigated,  but seems likely to buy a stage life of a week or two,  maybe a little more,  even with hydrogen.  Not months,  but a week or two.   Hydrogen is the worst about evaporating away,  the others will last longer.  Getting a couple of weeks with something that simple,  is worth the extra inert weight of the insulation and foil reflector,  and the separate tank bulkheads. 

               adding reusability in steps

There is simply no valid reason why this Apollo-style approach,  as modified with elliptic capture to go polar,  cannot be used to send astronauts to the lunar south (or north) pole,  the equatorial regions,  or anything in between at high latitudes.  Initially,  this is done with a direct-entering space capsule and an expendable service module,  and perhaps an expendable trans-lunar injection stage. 

But with a single stage lander left in lunar orbit,  the first reusability step becomes bringing only the lander propellant,  not the full lander,  on subsequent missions.   The second step would be recovering and reusing the trans-lunar injection stage,  essentially becoming an Earth orbit space tug.  The third and most difficult step would be replacing the capsule and service module with a dedicated vehicle to be used for multiple trips to and from the moon,  recovered into LEO by the TLI stage that was turned into a tug. 

By the time those 3 steps are done,  the radiation protection and microgravity and life support issue solutions should be ready enough to support going to Mars by the same LEO-based tug assist means that were developed for going to the moon.

These things are suggested as alternatives to,  not a replacement for,  the SpaceX “Starship/Superheavy” approach.  We really do need more than one way to do these missions.

Appendix A – Further Details Regarding the Author

Short course materials by the author:

               “Orbits+” courses available via “New Mars” forums          

                              “Interplanetary transportation” topic,  “orbital mechanics traditional” thread

                              Links to free downloads:  orbits,  rocket performance,  and entry

                              spreadsheets,  plus lessons with problems and solutions

               Compressible Flow and Heat Transfer Basics

                              Contact author:  includes spreadsheets,  with problems and solutions

Contact data

               Email preferred:  gwj5886@gmail.com

               Blog site:  http://exrocketman.blogspot.com

               Youtube:  search for “exrocketman1”

               Also on LinkedIn

               Mars Society’s Newmars.com/forums/

Appendix B – Extra Orbital Calculation Details

The figure shows how the numbers were actually computed.  This was assisted by a spreadsheet doing classical 2-body elliptical orbit calculations.   Note that only data for 100 km LLO is shown.  The others were done by the same method,  just the numbers change.  Those were for 500 km and 1000 km altitudes for the LLO options,  and 1261.7 km altitude (3000 km CTC) for the Gateway/halo option.

Appendix C – Landing Calculation Details

The figure shows that calculations were made with a surface-grazing ellipse as the transfer from orbital altitude to the surface.  Only the case for 100 km is shown;  the others are similar,  just different  numbers.  The difference between circular and apolune speeds is the magnitude of the deorbit burn dV for descent,  which is also the magnitude of the circularization burn dV upon ascent. 

The perilune speed is the ideal dV to be “killed” for landing,  and to be supplied for starting the return ascent.  That has to be increased to cover losses,  of which there is no drag loss,  and the gravity loss is quite small. 

For descent,  the landing burn dV is factored up by 1.5 to cover the needs for hover and divert to avoid obstacles on the surface.  This was a serious effect during the Apollo-11 landing,  nearly causing a crash or an abort,  from landing propellant exhaustion.

For ascent,  there is an added dV budget to cover rendezvous with any craft left in orbit during the landing.  This is “guesstimated” as twice the magnitude of the deorbit burn. 

Appendix D – Linked Rocket Equation Calculations When Weight Statements Change

The example in the figure is for recovering the reusable lower stage of a launch rocket,  but the linked process is the same,  regardless of the application.        

The dV and Vex values associated with the two different weight statements are used to compute the resulting mass ratios.  There is also a stage or vehicle inert mass fraction,  defined as Winert/Wign.  The propellant fractions Wp/Wign are simply 1 – 1/MR,  for each mass ratio.  The sum of payload,  inert,  and propellant fractions must be 1,  by definition. 

One must assume an ignition weight for the first burn (or set of burns) calculation,  associated with the first weight statement.  From it,  the inert mass and burnout mass are easily computed as shown,  as is the propellant mass expended.

As shown,  you adjust the burnout mass from the first calculation to be the ignition mass in the second calculation,  by means of the differences between the first burn and second burn weight statements.  Then you do similar burnout mass and propellant-expended computations,  as shown.

When the inherently-iterative calculation is fully converged,  the second calculation burnout mass should equal the first calculation inert mass,  with due regard for any non-zero payload in the second calculation,  or any jettisoned-between-burns inert items.  One must iteratively adjust the first calculation ignition mass guess,  until this convergence is achieved.

Once convergence is achieved to within a tiny fraction of a percent,  the results can be used to finalize 3 representative weight statements.  The first is the as-built stage or vehicle weight statement,  including all the propellant for both sets of burns,  the initial inert mass,  and the initial payload mass.  

The second is for the first set of burns,  with the propellant for the second set of burns carried along as if it were additional payload.  It actually results in the same design mass ratio for that first set of burns,  and the same design dV estimate is obtained.

The third is for the second set of burns,  where the propellant for the first set of burns is zeroed.  The inert mass has been adjusted for any items jettisoned between the sets of burns,  and the payload has been adjusted to reflect anything offloaded between the sets of burns.  This weight statement then produces the design mass ratio and design dV for the second set of burns.

Appendix E --  Making Orbital Plane Changes

The illustration shows exactly how this is done and why it works.  But it is complicated. 

At the top is a depiction of an extended elliptical orbit about some central body (of interest here is the moon as the central body).  The plane of the ellipse is that of the page.  The side moving toward the central body is at the top of the diagram,  and the side moving away at its bottom. 

The author is coasting in that elliptical orbit in his spaceship,  currently located at its far point from the central body (the “apoapsis”,  or at the moon,  the “apolune”) In top left of the illustration,  you are viewing this ellipse end-on,  so that it appears to you as a vertical line in view A-A,  bottom left.  The author’s spaceship appears to you to be moving upward at its smallest orbital speed V.

Imagine now that author turns his spaceship and faces precisely the other way,  now looking downward in your view.  That’s the next image to the right of view A-A in the illustration.  While facing “downward”,  he is still moving “upward” at V,  in your view of him.

Now the author makes an engine burn,  adding a speed V in that downward direction,  which added to the upward V he originally had,  now makes precisely zero speed along the orbit!  Because he is no longer moving with respect to the central body,  there is no longer any vertically-oriented elliptical orbit;  it just goes away.  (The lesson here:  kill the speed at distance,  you “kill” the elliptical orbit!)  The author also now looks stationary in space to you!

Now look at the second image to the right of view A-A in the illustration.  The author now turns his spaceship to your view left,  which is perpendicular to the plane of the original elliptical orbit that has just gone away.   He makes another engine burn,  to add the same speed V in that new direction,  while still just as far away from the central body as he was.  He is now moving with speed V to view left instead of view up (the very same speed as at the apoapsis of the original ellipse),  and he is still the same distance away from the central body! 

That action inherently creates a new ellipse of the same dimensions as the original,  about that central body.  However its orientation in 3-D space is now horizontal in your view,  no longer vertical,  because the new speed was created to view left,  not the original view up.  (The lesson here:  create a speed at a distance from a central body,  and you create an elliptical orbit about it,  corresponding to that speed and direction at that distance.) 

This same thing could be done at the point of closest approach to the central body (the “periapsis”,  or at the moon,  the “perilune”).  It is just that the speed in the orbit is very much the highest there,  so that the rocket burns would have to create an awful lot of vehicle speed changes,  and it would cost enormous amounts of propellant.  So,  plane changes are always done at apoapsis,  where the orbit speed is the lowest,  and the propellant cost is less.  Using a very extended ellipse really helps save propellant,  because the apoapsis speeds are really low!

There is one more propellant savings you can have,  if you take advantage of the “magic” of vector math,  where the vectors have both length and direction,  and they literally add nose to tail graphically.   This is the image to the farthest-right of view A-A in the illustration.  The original view-upward V and the desired view-leftward V form an isosceles (equal-leg) triangle at the apoapsis of the original ellipse.  You simply draw a slanted line between the tips of the two velocity vectors. 

If you make that slanted line into a vector pointing left and down in your view,  that is the speed vector you must add (by burning your engine) to the original apoapsis speed vector to simultaneously “kill” your view-upward speed V and replace it with a view leftward speed of that same V.  That one burn is inherently less in magnitude than just adding the separate burns for two legs,  like we just did before. 

That is because the “slanty line” connecting the tips of the legs of that isosceles triangle is always shorter than the sum of the lengths of its two legs.  There is an algebra formula based on this isosceles triangle geometry that relates your required burn dV to the orbital speed V and the angle a by which you wish to turn the plane of the ellipse: 

dV = 2 * V * sin(a/2). 

For a = 90 degrees,  this simplifies as illustrated to

dV = 1.414*V,  where the 1.414 is really just a 4-digit approximation to the square root of 2. 

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