Sunday, July 5, 2020

How the Spreadsheet Works


The image shown is the image of the spreadsheet I have been using recently to evaluate multi-burn performance of rocket vehicles.  The specific one illustrated is that used for analyzing the Spacex “Starship” on a direct-landing moon mission,  with direct return.  It is the same spreadsheet layout as was used to analyze that vehicle on a two-way Mars mission.  That Mars article is 6-21-2020 “2020 Starship/Superheavy Estimates for Mars”.  The lunar trajectory delta-vee (dV) data came from 7-3-2020 “Cis-Lunar Orbits and Requirements”.  The moon mission analysis is 7-5-2020 “2020 Estimates for Spacex’s “Starship” to the Moon”.

The spreadsheet has a block of weight-statement reference data above the main calculation block.  These weight statements apply to the outbound and return legs of the mission.  Only the yellow highlighted items are inputs.  That is because the inert structural mass and loadable propellant mass do not change.  The dead-head payload mass does not have to be the same during the return as it was during the outbound voyage,  so it is an input.  The pre-calculated items are highlighted tan.

Return leg items D3 (an identifier),  E3 (inert structural mass in metric tons),  and G3 (propellant max load in metric tons) are set equal to outbound leg inputs D2,  E2,  and G2,  respectively.  The ultimate outbound ignition mass in H2 is the sum of E2 and F2 and G2.  Similarly,  the ultimate return ignition mass in H3 is the sum of E3 and F3 and G3.  The ultimate “dry tanks” masses in I2 and I3 are the sums of E2 and F2,  and E3 and F3,  respectively.  All masses are metric tons.

This vehicle has a mix of two different engines,  a vacuum design and a sea level design.  The sea level design has both sea level and vacuum performance,  the vacuum design has only vacuum performance.  The performance index is specific impulse (Isp) in seconds.  Values are input for the outbound leg,  and they do not change for the return leg.  Those values are merely set equal to the outbound values.

For the main calculation block,  row 6 contains the headings.  The input kinematic dV values (km/s) are in column C,  rows 7 through 12,  highlighted yellow.  In this analysis,  each leg of the voyage has a departure burn,  a course correction burn,  and a landing burn.  The “jigger factor” factors which multiply these dV values are located in column D,  rows 7 through 12.  All of these are actually user inputs,  though not all are highlighted yellow in this image.  The mass ratio-effective dV values (km/s) are in column E,  rows 7 through 12.  E7 is D7 multiplied by C7,  E8 is D8 times C8,  etc. 

The engine selections and performance values are given in columns F,  G,  and H,  rows 7 through 12.  The names and Isp values are actually user inputs and should be highlighted yellow,  although these are not in the image. The names should match the identifiers in row 1,  items J, K,  or L,  as appropriate.  The Isp values should match row 2 / J through L as appropriate for the outbound leg,  and row 3 / J through L as appropriate for the return leg.  For this model as illustrated,  the orbit departure and course correction burns,  plus the takeoff from the moon,  are all done with the vacuum engines.  The two landings require the thrust vectoring available only with the sea level engines,  despite their being used in vacuum on the moon,  and at sea level on Earth. The exhaust velocity values (km/s) in column H,  rows 7 through 12,  are all computed from the Isp data in column G.  That computation is Vex = 9.80667*Isp/1000.  It is done by rows:  H7 = G7*9.80667/1000.,  H8 = G8*9.80667/1000.,  etc.

It is the individual-burn ignition weights in column I,  rows 7 through 12,  that allow one to tailor this model between refueled or unrefueled missions.  For the outbound leg,  it is presumed the vehicle departs fully fueled from LEO.  Thus I7 must be set equal to H2.  The velocity ratios in column J are simply the factored dV data in column E divided by the exhaust velocity data in column H.  This is done row-by-row: J7 = E7/H7,  J8 = E8/H8,  etc.  The mass ratio data in column K for each burn are the base-e exponentials of the velocity ratio data in column J,  done row-by-row:  K7 = EXP(J7),  K8 = EXP(J8),  etc.  The mass at end-of-burn data is in column L.  This is the burn ignition mass in column I,  divided by the mass ratio data in column K,  done row-by-row:  L7 = I7/K7,  L8 = I8/K8,  etc.

The “trick” is in the details of how the burn ignition mass data in column I are computed.  For the unrefueled lunar mission shown,  the ignition mass for the burn at hand is the previous end-of-burn mass,  all the way down to row 12:  I8 = L7,  I9 = L8,  etc,  with one exception.  The ignition mass for the return leg launch must reflect the return payload,  not the outbound payload.  Thus I10 is not just L9,  but is I10 = L9 -F2 + F3,  which subtracts the outbound payload F2 from that weight statement,  and then adds in the return leg payload F3 from that weight statement.   

For the Mars mission,  which is presumed to be fully refueled on Mars before its return launch,  you set the launch burn ignition mass in I10 directly to the ignition mass H3 in the return leg weight statement:  I10 = H3.  Doing the “right thing” with return launch ignition mass is exactly how one refuels-or-not,  and what payload one carries.  That is the utility of having the weight statements as a closely-adjacent reference data block.

For each burn,  the change in mass from ignition to end-of-burn is literally the mass of propellant used to make that burn.  Propellant increments used are in column M,  done row-by-row:  M7 = I7 – L7,  M8 = I8 – L8,  etc.  What one wants to book-keep is propellant-remaining,  which would be the propellant mass you started with,  minus the sum of what you have used so far.  This is the blue-highlighted data in column N.  When that number goes negative,  you have tried to use more than what you had available.  The analysis becomes infeasible,  and you can tell exactly at what point in the mission this infeasibility sets in.  The most straightforward way to adjust this outcome is to adjust that leg’s payload in the corresponding weight statement.

For the lunar mission,  which is unrefueled on the moon,  you model the Earth departure burn in row 7 by setting propellant remaining N7 to what you originally started with G2 less what you just burned M7:  N7 = G2 - M7.  After that,  you just use the previous propellant remaining value and subtract your current usage from that:  N8 = N7 - M8,  N9 = N8 – M9,  etc,  all the way down to row 12. For the Mars mission,  you launch the return leg fully fueled,  so you set propellant remaining after the first burn N10 to what you start the leg with G3,  less what you just used M10:  N10 = G3 – M10.  Otherwise,  the recursion pattern is the same.

I put a little calculation block out to the right to investigate the sensitivity of these results to the inert mass in the weight statements.  The inert mass as it is in Q6 gets set to E2.  You set a lower revised inert mass in Q7.  The difference Q6-Q7 is the change in non-propellant mass.  Payload mass as it is in Q9 gets set to the weight statement value F2.  What the payload could become is its current value plus the difference in inert masses:  Q10 = Q8 + Q9.  These numbers apply to the outbound voyage,  and only have physical meaning if that outbound voyage shows as feasible (positive propellant-remaining values in column N, specifically N7 through N9).

If you need to model more than 3 burns per leg,  insert more rows. Remember,  the modeling controls for refueling-or-not,  and what payload you carry in each leg,  are in the initial-burn selections for ignition mass,  and for propellant-remaining, for each leg.  The modeling controls for what engine performance to use are in those columns.  The rest is nothing but a straight recursion of cell updates down each column.





Update 7-6-20:  This is what the spreadsheet looks like when I clean it up,  make it run two cases in the one worksheet,  and include all the instructions and notes upon it.  It is in effect its own user’s manual,  and a template for all sorts of analyses.  What one should do is copy this to another worksheet,  then edit that copy’s inputs to represent the analysis you want to run.  Keep this one as a template.

The first case is departure from circular low Earth orbit (LEO) at 300 km altitude with “Starship”,  to a direct landing that is an out-of-propellant stranding upon the moon.  You can tell by the negative numbers for propellant remaining for the return trip burns,  with only a small fraction of a ton remaining upon landing on the moon.  One iteratively inputs payloads until that criterion is satisfied:  fractional-ton propellant remaining at whatever condition is the end-of-mission.  The payload carriable in this scenario is considerable,  but the vehicle is lost:  this is a one-way trip!  Very unattractive for a vehicle design whose two major attractive characteristics are (1) its large payload size,  and (2) its low cost because of its reusability.

The second case departs from elliptic LEO (300 x 1400 km altitudes),  and is actually able to return all the way to direct Earth entry and landing.  You can tell by the positive fractional-ton remaining propellant after the Earth landing.  This is a two-way trip,  but the return payload is restricted to zero in order to maximize payload brought to the moon.  Unfortunately,  the delta-vee demanded of the round trip is just too high,  resulting in a rather trivial payload-to-the-moon capability.  Reducing the inert mass (120 metric tons) to goal levels (100 metric tons) does not change that outcome all that materially-much:  payload is still too small to be attractive.

The message from comparing these two cases is that the delta-vee requirement demanded of the design must reduce further still.  That is the biggest difference between the two missions shown.  Either this vehicle should deliver payload into lunar orbit instead of the surface,  or else a far more elliptical LEO is needed,  to decrease the departure delta-vee further.  That last would incur an apogee well into the dangerous Van Allen radiation belts,  while in elliptic LEO.  It also reduces the payload the vehicle can carry to LEO.




2020 Estimates for Spacex’s “Starship” to the Moon

Unlike the projected Mars missions,  Spacex has not revealed very much about how its “Starship” craft might go about travel to and from the moon. By implication,  this seems to be flights from low Earth orbit (LEO) to direct landings upon the moon,  and unrefueled flights from there back to Earth.  Those returns could be direct launches,  with or without stops in low lunar orbit (LLO),  to a direct aerobraking entry at Earth,  and the associated retropropulsive touchdown.


About the only complicating detail available for this is the possibility of using an elliptical LEO as the departure point,  to reduce the departure delta-vee (dV) somewhat.  This is based mostly upon informal comments made by Mr. Musk in his various presentations.  That gain would be at the cost of increasing the difficulty of reaching that elliptic LEO with the Starship/Superheavy vehicle.  In effect,  payload to such an orbit is reduced relative to that deliverable to circular LEO.  That is precisely because the elliptic perigee velocity is higher than circular. 

There is a very practical constraint on how far this option could be pushed:  beyond a certain eccentricity,  the apogee altitude of the elliptic LEO orbit falls within the dangerous radiation environment of the Van Allen radiation belts.  Running for the radiation shelter (whatever that is) every hour and a half is just not a practical option for a crewed mission.  That limits the elliptic LEO to about 900 miles (1400 km) apogee altitude,  and 300 km perigee altitude would be rather typical. 

The perigee velocity of a 300x1400 km elliptic LEO is 0.25 km/s faster than 300 km circular.  This reduces the departure dV by that same 0.25 km/s,  and it increases the second-stage burn dV by 0.25 km/s to reach that elliptic LEO vs circular. That reduces the payload carriable by “Starship” and its tankers to LEO,  while increasing the payload carriable by the refilled “Starship” to the moon.  It means you have to pay a price to send more payload to the moon:  lower payload to LEO and more tanker flights to refill.

What I do here in this article is investigate the one mission type (direct lunar landings,  with unrefueled return to a direct landing on Earth),  with the latest estimates that I have for Spacex’s vehicle.  This is comparable to the Mars mission study I did earlier,  in terms of methods and data.  The source documents for this are:

5-25-2020 “2020 Reverse Engineering Estimates for Starship/Superheavy”
7-3-2020 “Cis-Lunar Orbits and Requirements”
6-21-2020 “2020 Starship/Superheavy Estimates for Mars”
10-22-2019 “Reverse Engineering the 2019 Version of the Spacex Starship / Super Heavy Design”

What Is Covered In This Article

This article concerns only “Mission C” as described in the Cis-Lunar Orbits article.  There are other missions this vehicle might fly,  one of which is a dedicated lander per the recent NASA contract.  Mission C is the departure from LEO to a direct landing on the moon,  and an unrefueled return.  That return is presumed a direct escape from the moon,  aerobraking direct entry at Earth,  and a retropropulsive touchdown.  Whether one stops in low lunar orbit (LLO) or not makes no real difference.

There are multiple choices that affect the outcomes greatly,  all of which get considered herein.  One is elliptical vs circular LEO departure.  Another is the achieved inert structural mass of the Starship vehicle.  And the third is just how much of the descent dV onto the moon needs to be factored-up greatly,  to cover hover and divert requirements.  I cannot get the final answer,  but I can illustrate just how critical these issues are.  And they are.

Elliptic vs Circular LEO Departure

The orbital mechanics for this were already covered in detail in the Cis-Lunar Orbits article.  The net effects are:  (1) reduced departure dV by 0.25 km/s,  and (2) reduced payload to elliptic LEO because the final stage-two burnout velocity is 0.25 km/s faster. Handling the rocket equation estimates for the lunar trip with this effect is easy:  just change the departure dV.  Evaluating the elliptic LEO payload reduction requires determining the effect of the higher stage-two burnout velocity,  all else but payload being equal.  In effect,  you have to reduce payload to achieve the higher mass ratio,  and still be able to land.

I probably should have re-run my entire 2020-version spreadsheet model of the Starship/Superheavy vehicle to determine this,  but I chose a simpler estimate.  I think it gets about the same answer.  The higher dV at the same effective exhaust velocity results in a higher mass ratio for that burn,  which is a lower end-of-burn mass.  The difference in the two end-of-burn masses is the extra propellant burn required at full payload,  which must come out of the landing allowance.  I then assumed that same number also has to come out of payload carried,  in order not to burn up the landing allowance. 

My crude estimate says that change is about 19 metric tons.  Which reduces the 149 metric ton circular LEO payload estimate in the 2020 reverse-engineering article to some 130 metric tons to elliptic LEO.  It also uses up all the on-board propellant,  precluding any abort-to-surface landing,  before I reduce payload!  That rocket equation estimate is shown in Figure 1,  as a spreadsheet image.



Figure 1 – Effects of Elliptic vs Circular LEO,  and Sources of Inert Vehicle Mass

Inert Mass Trends

For the 2019 and earlier versions of Starship/Superheavy (under various names),  the Starship inert structural mass was given by Spacex as 85 metric tons.  That was before they switched to stainless steel construction,  and it was before they began to manufacture prototypes.  The propellant load for the stage was also given as 1100 metric tons. 

With the 2020 estimates,  both the stage inert is different,  and its propellant load is different.  The website now shows 1200 metric tons propellant load.  Musk’s presentation in front of a prototype at Boca Chica says the inert mass is 120 metric tons,  that his slide showing 85 tons is in error,  and that he would be delighted if they ever get the inert down to 100 tons.  See again Figure 1.

Because in vehicle developments,  there is usually growth,  not loss,  of inert mass,  I chose the current prototype inert mass of 120 metric tons as “baseline”,  with Musk’s preferred goal of 100 metric tons as a bounding optimistic estimate.

Retropropulsive Landings in Vacuum vs Atmospheres

The worst-case direct kinematic dV for landing direct upon the moon from the transfer ellipse is some 2.533 km/s,  and never much less than that.  Unlike aerobrake landings on Earth or at Mars,  all of this must come from the on-board propulsion.  In the past,  I have factored up the kinematic landing dV by 1.500 to cover hover and divert needs.  With aerobrake landings,  the propulsive kinematic dV is quite small,  since most of the deceleration occurs by hypersonic entry drag.  Such are typical of Earth and Mars,  although the trajectory details are wildly different,  due to the wildly-different surface densities.

For a vacuum landing (as on the moon),  the kinematic dV is quite large,  and is also likely one long,  continuous burn.  I suspect that factoring all of it by 1.5 is very likely “overkill”.  Factoring most of it by the lunar gravity loss factor of 1.008 seems reasonable.  Then factoring only a small,  terminal portion of it by 1.50 (or even 2.00) for hover and divert,  would thus result in a lower overall average factor applied to the kinematic dV.  How much of the dV to factor up by the higher factor is nothing but a guess.  I arbitrarily chose the last 0.25 km/s,  that being about the last 10% of the burn.  I chose to factor that portion by 2.00,  with the other 90% factored only by 1.008 for lunar gravity losses.  See Figure 2.




Figure 2 – Revising the Vacuum Landing Factor Applied to Kinematic Delta-Vee

That gives us two dV-factor values to explore,  the overkill 1.50 applied to the whole 2.533 km/s,  and the arbitrary 1.106 applied to the whole 2.533 km/s,  reflecting factor 2.00 on the last 0.25 km/s,  and factor 1.008 on the rest.  This turns out to be a very major effect.

Return Payload

There are two possibilities which sort-of-bound the results.  One is to carry the same payload both ways.  The other is to carry zero payload on the return flight.  Failing feasibility,  one can look at zero return payload with all propellant used in the landing,  thus stranding the vehicle upon the moon.  I did so.

Methods

I did this with a spreadsheet,  very similar to that used for the 2020 Mars estimates.  In point of fact,  it is another worksheet in that same spreadsheet file.  The difference is refueled versus unrefueled,  for the return flight.  The worksheet has two weight statements,  one for departure,  the other for return.  They have inputs for different payloads each way.  The difference is that for Mars,  the return flight is presumed to start with a full propellant load.  The lunar model does not:  it used the propellant still on board at landing as the propellant supply for the return trip.  This ignores evaporative losses.

You will note that for each burn in the list,  there is a selection of which engines to use,  along with their effective specific impulse.  That sets the exhaust velocity for the burn,  which leads to mass ratio from the rocket equation,  given an appropriately-factored kinematic dV.  In order to achieve thrust gimballing for landing control,  it is the three sea level Raptor engines that must be used for the lunar landing,  at their vacuum specific impulse levels. 

The same was true for the Mars landing in that study,  and it is true for all the Earth landings in all the studies,  just at sea level specific impulse in Earth’s atmosphere.  All the in-space burns are done with the vacuum Raptor engines.  Those would include LEO departure,  course corrections,  and lunar (or Mars) liftoff.  There is no Earth liftoff with only three Raptors (sea level or vacuum).  There is only a second-stage burn with three vacuum Raptors,  and at dV virtually perpendicular to the gravity vector.

There is one additional calculation done here for the lunar mission that was not done for the Mars mission.  That is a little block to the right of the main calculation block,  showing the change in payload for a revised inert mass.  The fundamental assumption here is that every ton of inert mass saved is an extra ton of carriable payload.  So each spreadsheet image reported here has the inert mass variation included. There is one image for circular LEO departure,  and another for elliptic LEO departure.  That leaves re-running the whole set to evaluate the effects of the factor applied to lunar landing dV,  and re-running another set to evaluate the effects of zero return payload.

Results Obtained

The baseline case is 120 metric tons inert,  with the lunar landing kinematic dV =2.533 km/s factored by 1.50.  There are calculation blocks for both circular and elliptic LEO departure on the worksheet page.  These are given separately in Figures 3 and 4.  The effects of reduced inert mass are also given,  as described above.  These are for the same payload carried both ways. 

This was disappointing in the extreme:  the vehicle simply cannot make the two-way trip from circular LEO,  or from elliptic LEO.  That is shown by the negative remaining propellant numbers for the return trip,  even at zero payload both ways.  This makes the inert mass reduction results shown in both figures irrelevant.  It is futile to show zero return propellant,  because the payloads are already zero.  That leaves only the landing dV factor as something practical to pursue.




Figure 3 – Circular LEO Departure,  Two-Way Payload,  “Overkill” Landing




Figure 4 – Elliptic LEO Departure,  Two-Way Payload,  “Overkill” Landing

I did look at a one-way flight to the moon,  stranded there with essentially no propellant remaining.  Under those circumstances,  the effects of inert mass reduction do apply,  and are given in the figures.  Again,  this is for the “overkill” factor upon the kinematic dV for lunar landing.  Figure 5 shows the spreadsheet images for the circular departure case,  and Figure 6 for the elliptic case.



Figure 5 – Circular LEO Departure,  One-Way Stranding,  “Overkill” Landing




Figure 6 – Elliptic LEO Departure,  One-Way Stranding,  “Overkill” Landing

Reducing the overall-average factor applied to the kinematic landing dV does dramatically improve these results,  but not enough to be truly attractive.  Circular departure is shown in Figure 7,  for only the one-way stranding on the moon.  It includes showing the reduction in inert mass.  No two-way trip is feasible,  even at zero return payload and zero payload to the moon. 



Figure 7 – One-Way Stranding is Feasible for Circular Departure at Reduced Landing Factor

The same data for elliptic departure at the same two-way payload is shown in Figure 8.  These results are not very attractive,  even at only 100 metric tons inert mass.  Deliverable payload is just pathetically low,  at either inert mass.



Figure 8 – Two-Way Payload for Elliptic LEO at Reduced Landing Factor


Figure 9 shows the elliptic departure results for reduced landing factor and for zero return payload.  These are still quite pathetic. 

 
Figure 9 – Results for Elliptic Departure,  Zero Return Payload,  and Reduced-Factor Landing

Evaluation of These Results

In no case was the payload deliverable to the moon in the least attractive,  except for the one-way strandings,  which are extremely unattractive because the vehicles cannot be re-used!  This is because the lunar landing dV is large,  no matter how we factor it,  quite unlike the Mars landing dV’s.  The sum of departure,  course correction,  and landing dV is just too large,  no matter how exactly it is figured.  That plus the departure,  course correction,  and Earth landing dV’s for the return are just far beyond any realistic values for mass ratio and exhaust velocity performance out of this vehicle.  What that really says is that the direct landing scenario is just the wrong mission for this vehicle to fly to the vicinity of the moon!  The “kicker” for this is the unrefueled return,  which is quite unlike the nominal Mars mission. 

Refueling this vehicle on the moon is simply not a practical option,  as there is no practical source of carbon for making methane out of local materials.  Further,  the best information we have (poor though it is) says that water is only available near the south pole of the moon,  and we do not yet understand how concentrated,  or pure,  this resource is.  Making Raptor engine propellants on the moon looks to be unlikely at this time!  For any reusability,  that means any lunar fights must return to Earth unrefueled.  And this analysis shows that cannot happen,  at any practical payload levels.

This vehicle,  as it is currently understood,  is just not well-suited for direct landings upon the moon.  The rocket equation and publicly-available numbers prove that beyond a shadow of a doubt,  no matter the assumptions made factoring landing kinematic dV values.  And,  while elliptic LEO departure offers some small improvement,  it is nowhere near enough to make a significant difference,  as long as crewed excursions into the Van Allen radiation belts are excluded for crew safety reasons (which they should be).

The only attractive deliverable payload numbers are for non-reusable one-way strandings upon the moon.  Thus as we currently understand it,  this design is only feasible for some other mission,  than direct landings upon the moon,  excepting only one-way strandings there.

Friday, July 3, 2020

Cis-Lunar Orbits and Requirements


There are a number of different concepts for space missions in cis-lunar space.  The oldest of these is Apollo,  which departed a low circular Earth orbit (circ LEO) onto a nominal transfer ellipse to the vicinity of the moon,  and then entered low circular orbit (LLO) about the moon in a retrograde direction (nominal altitude 60 miles = 100 km).  The three-body mechanics (Earth,  moon and spacecraft) of this process converted the nominal transfer ellipse into the lopsided figure-eight trajectory we all remember.  The Earth return was the reverse,  excepting for the free entry into Earth’s atmosphere and parachute ocean landing upon arrival. 

Getting into LEO from the surface of the Earth is a different problem that is intimately linked with the characteristics of the design being considered.  It is more of an atmospheric/exoatmospheric flight trajectory analysis,  than a simple orbital mechanics analysis.  Of particular impact are the staging velocity,  the stage mass ratios and propulsion characteristics,  and any hardware recovery schemes.  That problem is NOT considered here.

What I do in this article is approximate the three-body problem of Earth,  moon,  and spacecraft by simple coupled two-body problems that each solve as closed-form equations.  The three-body problem requires numerical solution on a computer,  and it generates the figure-eight trajectory,  if used to enter a retrograde lunar orbit.  The two-body Earth-spacecraft problem gives me a spacecraft velocity vector out at the moon,  measured with respect to the Earth.  The two-body Earth-moon problem gives me a velocity vector for the moon with respect to the Earth.  The two-body moon-spacecraft problem gives me a velocity vector,  with respect to the moon,  for the spacecraft in lunar orbit,  if applicable. 

The appropriate vector sum of these velocity vectors,  given a selection of just where and how I want to approach the moon,  gives me the spacecraft velocity vector with respect to moon,  presumed to be “far” from the moon.  At the appropriate distance from the moon,  lunar escape velocity is reduced below its surface value,  inversely proportional to the square root of distance from the moon’s center.  The “far” kinetic energy plus the escape velocity kinetic energy add to equal the “near” kinetic energy,  with all the ½-factors dividing out.  You solve that for the “near” velocity magnitude,  and its direction comes approximately from your approach selection.

If you are landing direct,  the “near” velocity is the kinematic velocity you have to “kill”, in order not to crash.  Appropriately factored-up to cover hover and divert needs,  that is the mass ratio-effective dV required to land.  Appropriately factored up for small gravity losses,  that same “near” velocity is the mass ratio-effective dV needed to escape from the moon onto the Earth return trajectory.  If you are instead going into orbit,  the difference between “near” velocity and orbit velocity is the kinematic dV required to arrive in orbit,  or depart from orbit.  The mass ratio-effective factor for that is just 1.000.

Basic Lunar Transfer Ellipse From Earth

The basic notions and numbers for a transfer ellipse to the distance of the moon’s orbit is shown in Figure 1.  This would be the path to any space station or other facility located ahead or behind the moon in its orbit,  as well as part of the basis to reach the moon itself using orbits similar to Apollo.  The variation in the exact numbers is due to the slight eccentricity of the moon’s orbit about the Earth.  Note the modest velocity of the moon in its orbit about the Earth (roughly 1 km/s).  This is due to its great distance from the Earth.  Once circularized into the moon’s orbit,  rendezvous with any sort of facility ahead of (or behind) the moon is trivial,  as long as the apogee of the transfer ellipse is centered upon it. 

Note also the factors quoted for the various burns.  These would be the appropriate factors for combined gravity and drag losses to convert the kinematic dV’s into mass ratio-effective dV’s for sizing or evaluating vehicles.  Using the unfactored kinematic dV’s in the rocket equation is a serious design mistake!  All figures are at the end of this article.

Approach to the Moon

Illustrated in Figure 2 are some of the details to reach the moon,  whether into low lunar orbit (LLO) or for a direct landing right off the transfer trajectory.  Also illustrated are the details to proceed from the moon to a location ahead of the moon in its orbit about the Earth.  The values for a location behind the moon in its orbit about the Earth would be similar.  The details of the orbits to rendezvous are not covered here,  and are not trivial,  as the period must be made different in order to rendezvous,  then must be made the same again.  These are measured on weeks to months;  such a path is therefore not recommended.

The figure-eight trajectory into a retrograde orbit about the moon takes advantage of the vector sum of the moon’s velocity about the Earth,  and the LLO velocity about the moon,  on the backside of the moon,  to reduce the dV into LLO significantly.  This figure-eight trajectory is a numerical solution to the three-body problem on a computer.  The calculations in this article are but approximations that work to get you a good approximation of the “right” answers. 

Note that the same LEO departure dV applies to any of the destinations,  because it is the same basic transfer ellipse to lunar orbit distance,  regardless.

Elliptic LEO Departure

It s possible to reduce the LEO departure dV somewhat by switching from a circular LEO to an elliptical LEO with a higher perigee velocity at the same altitude.  This is shown in Figure 3.  However,  there is a very serious limitation to how elliptical this LEO can actually be,  because of the apogee’s proximity to the Van Allen radiation belts,  if this is to be used for a crewed mission!  The nominal figure for the inner “edge” (not really a sharp boundary) of the radiation belts is some 900 miles = 1400 km altitude.  (The most notable exception to this is the so-called South Atlantic Anomaly,  where the radiation extends down to typical LEO altitudes.) 

Be that as it may,  the nominal max-eccentric elliptical LEO configuration is an ellipse of some 300 x 1400 km altitude.  This adds 0.25 km/s perigee velocity to the ellipse,  versus the circular orbit at 300 km.  That reduces departure dV by the same 0.25 km/s,  which saves propellant departing for the moon.  However,  this is not free!  It also adds the very same 0.25 km/s to the dV required of the second stage burn getting into this elliptic LEO.  You trade the one for the other!  That is inevitable! 

Whether you choose this option depends upon which burn is more critical for your vehicle design,  in all its propulsive detail:  getting to LEO or leaving it.

Trips to Lunar Orbit,  to LLO,  or to the Surface

A summary of these mass ratio-effective dV values to reach LLO or the lunar surface,  or a point on the moon’s orbit ahead (or behind) the moon,  is given in Figure 4.  The factors applied to kinematic dV are 5% gravity loss and 5% drag loss for Earth ascent,  the same values multiplied by 0.165 gee and 0.00 surface air density ratio for lunar ascent,  no losses at all for on-orbit operations in vacuum (factor 1.000),  and factor 1.500 applied to landing burn dV’s,  to cover anticipated hover and divert budgets.

One should note that lunar landing zones are limited to those directly underneath the LLO path.  It costs another significant burn,  to get a rather limited plane change in LLO,  since the orbital velocity magnitude is significant.  The amount of this plane-change delta-vee is indicated for a 10 degree plane change in the figure.  It applies both ways:  to descent,  and to ascent. 

The “Halo” Orbit Concept

There is another different mission concept that requires evaluation.  NASA has finalized its concept for the “Gateway” space station that is to orbit the moon.  That station is to be in a “halo” orbit about the moon,  meaning a very eccentric elliptical orbit about the moon.  Its radial distance from the center of the moon reportedly varies from 3000 km to some 70,000 km.  This is associated with a lower perilune velocity than LLO because of the higher perilune altitude,  and a very low apolune velocity indeed. 

However,  the plane and axis orientation of this orbit is fixed,  and the long axis of the halo ellipse is more-or-less radial to the Earth-moon axis twice a month,  with the apolune point facing the Earth once a month.  What that does is offer a once-a-month geometry for low dV to enter the station’s orbit,  at the cost of a very-limited launch window each month.  This is shown in Figure 5.  Bear in mind that the same basic transfer ellipse to the lunar vicinity applies. 

You enter this orbit at min dV when its perilune velocity is directed opposite that of the moon’s velocity about the Earth.  That minimizes the dV required to enter orbit,  to a surprisingly-low value.  But this geometry obtains for only 1 or 2 days out of each month.  Note further that unless the halo orbit is oriented radial to the transfer orbit,  with its perilune velocity vector in the transfer plane,  this advantage cannot be had! 

This halo orbit,  properly oriented,  is a location that NASA’s Space Launch System (SLS) can reach with its Orion capsule and service module (or things equally heavy),  in the block 1 configuration for SLS.  It cannot reach the same LLO orbit that Apollo-Saturn-5 reached in that program,  not with two-way capability,  the way Apollo did.  This bizarre highly-elliptical orbit for “Gateway” is so very clearly intended to give SLS/Orion block 1 a destination that it can actually reach.  But you pay two prices for that:  (1) the excursion to the surface has a higher delta-vee than that from LLO,  and (2) you cannot use this very low-cost orbit entry,  once there is a facility in this orbit to which you must rendezvous! 

Otherwise,  the recommended procedure,  for the halo orbit with a pre-existing facility,  is to enter the 3000 km radius lunar orbit,  from the transfer ellipse from Earth.  One waits in that 3000 km circular orbit until the alignment with the facility in the halo orbit is right,  then one does a burn to enter a 3000 x 70,000 km halo ellipse,  and then a final burn at its apolune,  to exactly match the required plane of the orbit with that of the pre-existing facility.  There is no other reliable means to enter the halo orbit,  and also rendezvous with a pre-existing facility there,  simultaneously.  This is shown in Figure 6. 

The path indicated in Figure 5 is only feasible for the first piece of hardware you send to the “halo” orbit.  After that,  you must rendezvous with what hardware is already there,  which means you must use the approach of Figure 6. There is no way around that difficulty,  without waiting weeks or months in high orbit about the moon.  That is because the period of the “halo” orbit is a little over a week. 

Getting to the Surface from the “Halo”

Shown in Figure 7 is the descent/ascent path to the lunar surface from the Gateway halo orbit.  You first make your plane-change burn at its apolune,  where this costs the least propellant.  You then have to do a dV burn at its perilune to put you onto a much smaller ellipse that grazes LLO.  Then you do another burn to put you into that LLO.  Then you wait for the landing zone to present itself in the appropriate geometry.  From there,  you do a landing burn,  which requires a high factor to cover hover and divert needs.  The kinematic landing burn dV is the surface circular orbit velocity,  to cover potential energy effects as well as the kinetic energy.  

The ascent uses the same concept in reverse,  except that the required factor for losses is much lower,  than for descent.  The rest is the reverse of the descent trip.  You must wait in LLO for the destination facility to be in the right place,  so that its position will coincide with yours,  at the halo orbit apolune.   This is driven by time constraints for a crewed mission,  since the period of the halo orbit is about a week,  while the period of the LLO orbit is only a couple of hours.

There is a distinct advantage to this wildly-eccentric “Gateway” orbit that staging out of LLO cannot duplicate:  its apolune velocity is quite low at 92 m/s!  That makes drastic plane changes quite inexpensive,  opening up the entire moon for exploration,  quite unlike Apollo!  Of course,  the most drastic plane change of all is 90 degrees.  If you budget for that,  plus some course correction,  plus the orbit changes and landing/takeoff burns I have identified,  then you have all the burns budgeted for your two-way trip from “Gateway” to anywhere on the surface,  and back.  

There is a very good reason for breaking the descent from the halo orbit into a smaller descent ellipse to LLO,  and then staging the landing out of LLO.  This occurs during the return trip,  when you must rendezvous with the space station in the halo orbit,  restricted to affordable plane changes at its apolune.  The short period of LLO lets you wait a short time for the “orbits to be right” to make your burn to the ascent ellipse,  which with another burn then puts you onto the halo ellipse at the correct time to make station rendezvous at its apolune,  where any drastic plane change gets affordably made in order to actually rendezvous. This trip is long enough at 3-4 days,  as it is.  It would be unwise to make it any longer by matching orbits involving the higher apolunes,  earlier in the process.

The total from “Gateway” to the surface and back,  including up to 90-degree plane changes,  is 5.50 km/s.  The total from LLO and back is 4.74 km/s a 10 degree plane change.   The difference (about 0.76 km/s) is what the performance shortfall of SLS/Orion block 1 has cost us,  by forcing staging out of the halo orbit instead of LLO.  Is the expanded plane change capability of the “Gateway” orbit really worth that cost?  Who really knows yet?

Organizing the Data into Missions

So,  the possible cis-lunar missions are:

Mission A:  circular LEO to lunar distance ahead of (or behind) the moon;  the return could be to LEO (A) or direct entry and landing on Earth (A1).  A2 would be departing from elliptical LEO instead of circular LEO.  There is no landing associated with this. A1+2 is direct landing on Earth plus elliptical LEO departure.

Mission B:  circular LEO to LLO,  plus the LLO-surface landing,  with or without any plane changes;  return could be to LEO (B) or direct entry and landing on Earth (B1).  B2 would be departing from elliptical LEO instead of circular LEO.  B1+2 is direct landing on Earth plus elliptical LEO departure.

Mission C:  circular LEO to direct to the lunar surface right off the transfer trajectory; return could be to LEO (C) or direct entry and landing (C1).  C2 would be departing from elliptical LEO instead of circular LEO.  C1+2 is landing direct on Earth plus elliptical LEO departure.  There is no separate landing associated with this,  because the mission is direct landing upon the moon.

Mission D:  circular LEO to 3000 km circular lunar orbit,  then enter a halo-orbit,  terminating in an apolune plane-change to rendezvous with  “Gateway”.  This trip leads to a landing that includes a 90-degree plane change.  Return to Earth could be to LEO (D) or direct entry and landing (D1).  D2 would be departing from elliptical LEO instead of circular LEO.  D1+2 includes both direct landing on Earth and elliptical LEO departure. 

The suffix-2 variation for all of these would be departing from a 300 x 1400 km elliptic LEO,  which subtracts 0.25 km/s each for the LEO departure burn and the LEO arrival burn,  but adds 0.25 km/s to the second stage burn needed to reach the elliptic LEO.

Factored dV’s for the suffix-1 option of direct entry and landing on Earth are quite vehicle design-dependent,  likely reducing total dV,  but at the expense of carrying the entry heat shield and landing apparatus to the moon and back.  An arbitrary 0.2 km/s mass ratio-effective landing burn dV is included as a ballpark guess for something under half a Mach number terminal velocity at very low altitude. 

Similarly,  there are educated-guess course-correction dV budgets included.  These are generally a percent or two of the largest velocity along the track,  but that is not a strict rule.  These are really just ballpark guesses. 

The separate landings are from LLO or from the wide “halo” orbit,  denoted as Landing A and Landing B. 

From LLO with Landing A,  there is a plane change burn (max 10 degrees),  factored 1.000,  and a descent burn,  factored 1.500 to cover hover and divert budgets,  and figured from the surface circular velocity to include potential energy effects.  Ascent is the reverse,  except the ascent burn factor is only 1.008,  reflecting low lunar gravity losses and no lunar drag losses.   It is entirely possible that factoring the entire kinematic descent delta-vee by 1.50 could well be overkill.  The portion of Earthly and Martian aerobraking descents that gets factored as the landing burn is far less than 100%.

From the “halo” orbit,  Landing B has more steps.  The first thing is the plane change (up to 90 degrees) at the “halo” apolune.  At the “halo” perilune,  we go directly to a descent ellipse that takes us to LLO altitude at its perilune.  At that LLO-altitude perilune,  we burn to enter circular LLO,  then wait until the geometry is “right” for the landing.  From there,  the landing is exactly like the LLO-start Landing A,  except that no plane change is required,  that already being done at the start of this journey. This process is reversed for the ascent,  with the waiting for the Gateway station position to be “right” for rendezvous,  being done in LLO with the shorter period (LLO is just under 2 hours,  while “halo” is just over 7 days).

Using These dV Data

Converting these dV data into vehicle weight statements and mass ratios requires knowledge of the selection,  thrust level,  and specific impulse performance of the propulsion used for each and every burn.  It is different for every design,  and design variation,  that you look at.  This has to be done burn-by-burn,  not from the overall dV total!  

Evaluating the performance of specific vehicles on any of these mission choices would be the topic of future articles,  not this one.  The point here is to list all the mass ratio-effective dV data for the missions burn-by-burn,  as a convenient reference for credible data. The tables below the figures contain the dV data for each individual burn.  Scope is the four mission types,  plus the two types of landings.  Broken out this way,  these become very convenient inputs for the engineering sizing of multiple vehicle concepts.

References

This article makes use of the basic data that was behind these two articles,  also on this site:  “Interplanetary Trajectories and Requirements” dated 21 November 2019,  and “Analysis of Space Mission Sensitivity to Assumptions”,  dated 2 January 2020.  I keep this in a big spreadsheet file.


 Figure 1 – The Transfer Ellipse From LEO to the Orbit of the Moon

 Figure 2 – Using the Transfer Ellipse to Reach the Moon Itself

Figure 3 -- Adjusting Delta-Vee with Departure from Elliptical LEO

 Figure 4 – Delta-Vees to Cis-Lunar Space

 Figure 5 – Special Case:  First-Time Lunar “Gateway” Extended-Elliptical “Halo” Ellipse About the Moon

 Figure 6 – Entry to the Halo Orbit for Rendezvous with a Facility Already There

 Figure 7 – Landing on the Moon from the “Halo” Orbit


 Table 2 – Mass Ratio-Effective Velocity Requirements for Low Lunar Orbit

 Table 3 – Mass Ratio-Effective Velocity Requirements for Direct Landing Upon the Lunar Surface

 Table 4 – Mass Ratio-Effective Velocity Requirements for Rendezvous with a “Halo” Orbit Facility

 Table 5 – Mass Ratio-Effective Velocity Requirements for Landing from LLO and “Halo”