Starship On-orbit refueling - Options and Discussion

[Barley]

What I would like is an extremely lightweight drop tank (20 ton tank 1,200 ton prop???) mated to the LSS as it leaves LEO. I don't see that being feasible short term due to the logistics of getting it in orbit. Plus development, attachment and several other issues. Possibly not even long term as it might not make any sense with Lunar ISRU. Trying to not engage in too much handwavium engineering.

If it's extremely light weight, there is less need to actually drop the tank.  This becomes roughly equivalent to a stretched Starship with 2400 tonne capacity. 

I think it would be best to wait until there is a working Starship to see how far the design can be pushed before either scheme is attempted.  Note that the current F9 is almost double the mass of V9 v1.0.  This does not say that SS can do the same, but it doesn't say it can't

[redneck]

[/quote]
If it's extremely light weight, there is less need to actually drop the tank.  This becomes roughly equivalent to a stretched Starship with 2400 tonne capacity. 

I think it would be best to wait until there is a working Starship to see how far the design can be pushed before either scheme is attempted.  Note that the current F9 is almost double the mass of V9 v1.0.  This does not say that SS can do the same, but it doesn't say it can't
[/quote]

You triggered this vision of a tankerish Starship launched with extremely extended tanks that reaches orbit empty and with little payload due to extreme stretch. It is docked to the true payload and refueled with 2,400+ tons of propellant for the Lunar mission. A somewhat more feasible method than what I had suggested.

[LMT]

If a tanker had 12-hour turnaround, and it delivered 1/5 of a load, it could fill an inflatable depot for ~ 300 crews each synod, by itself. 

Otherwise, a fleet of tankers would be needed.  E.g., given a 14-day window and preloading of depot tankers at synod start, at least 46 tankers would be needed. 

Scale up ~ 10x for settlement cargo flights.

The obvious idea is to use the Starships themselves as the "depot." This eliminates the entire depot fleet, which is a massive savings.

Instead you just transfer the passengers at the last minute. You can use 500 passenger P2P-derived vehicles to do that. This further increases the efficiency of launching within that 14-day window, because each launch serves five Starships instead of one.

Delete depots, get taxis.

No, you wouldn't want to haul all depot hw (ruggedized ZBO and propellant transfer hw) on every ship, to Mars and back.  It cuts cargo, and of course requires duplicate crewed systems.  There's little gain -- and little margin for Mars return.

Passive depot inflatables would be a big win, but the efficient fleet depot requires creative thought.

Also, ZBO hw makes more sense on the notional LEO/Deimos tanker ISRU loop, where it eliminates not the depot, but Earth-launch tankers and their SH boosters.

[TheRadicalModerate]

If it's extremely light weight, there is less need to actually drop the tank.  This becomes roughly equivalent to a stretched Starship with 2400 tonne capacity. 

I think it would be best to wait until there is a working Starship to see how far the design can be pushed before either scheme is attempted.  Note that the current F9 is almost double the mass of V9 v1.0.  This does not say that SS can do the same, but it doesn't say it can't

You triggered this vision of a tankerish Starship launched with extremely extended tanks that reaches orbit empty and with little payload due to extreme stretch. It is docked to the true payload and refueled with 2,400+ tons of propellant for the Lunar mission. A somewhat more feasible method than what I had suggested.

There are sharply diminishing returns to adding more propellant.  Remember that more tankage means more dry mass, so that the mass ratio changes slower than you think as you add prop.  In addition, a Starship with a heavy payload already has a mass ratio of 5.4.  Even if your dry mass doesn't change at all (which is unlikely), doubling the propellant only adds 35% more delta-v.  And if the structural mass fraction stays above 6% (which is pretty aggressive for a vehicle like Starship, which, at 120t dry and 1200t prop, has an SMF of 9.09%), then doubling the prop only gets you 30%.

Instead of just blindly throwing prop at Starship, you need to ask what you're trying to achieve.  When you narrow it down to a variety of round-trip cislunar missions and one-way Mars missions, I haven't found a case where stretching the tanks past 1500t helps (although that extra 300t helps a lot, just because of how the delta-v budget to and from cislunar works out), nor have I found any cases where pusher/depot, tandem burns, or even drop tanks help very much.

[Greg Hullender]

There are sharply diminishing returns to adding more propellant. . . . Instead of just blindly throwing prop at Starship, you need to ask what you're trying to achieve.  When you narrow it down to a variety of round-trip cislunar missions and one-way Mars missions, I haven't found a case where stretching the tanks past 1500t helps (although that extra 300t helps a lot, just because of how the delta-v budget to and from cislunar works out), nor have I found any cases where pusher/depot, tandem burns, or even drop tanks help very much.

I find myself thinking that a special cargo "tug" with a solar-powered (or nuclear) ion drive that slowly chugs from LEO to (say) NRHO would be worth it at some point. Yeah, it's slow and not very energy-efficient, but if it could move a full depot in (say) a month or two, that'd probably be well worth it at some point, since, with specific impulse of ~5000s it wouldn't need to be refueled very often.

[LMT]

There are sharply diminishing returns to adding more propellant. . . . Instead of just blindly throwing prop at Starship, you need to ask what you're trying to achieve.  When you narrow it down to a variety of round-trip cislunar missions and one-way Mars missions, I haven't found a case where stretching the tanks past 1500t helps (although that extra 300t helps a lot, just because of how the delta-v budget to and from cislunar works out), nor have I found any cases where pusher/depot, tandem burns, or even drop tanks help very much.

I find myself thinking that a special cargo "tug" with a solar-powered (or nuclear) ion drive that slowly chugs from LEO to (say) NRHO would be worth it at some point. Yeah, it's slow and not very energy-efficient, but if it could move a full depot in (say) a month or two, that'd probably be well worth it at some point, since, with specific impulse of ~5000s it wouldn't need to be refueled very often.

At ~ 30 mN/kW, and allowing for LEO night and drag, a 2-month transfer of 1500 t would need > 80 MWe.

That's a square of PV > half a km on a side.

How many ion engines would you need?

[Greg Hullender]

How many ion engines would you need?

I'm told there's a quantity discount. :-)

[InterestedEngineer]

] nor have I found any cases where pusher/depot, tandem burns, or even drop tanks help very much.

You get free deltaV (~2x) with Oberth maneuvers, so getting fuel into a higher energy orbit (HEEO) effectively doubles Isp.

(or alternatively 2.72^2 = 7.4x more fuel).

Is why that maneuver is so valuable, and why ion drives are effectively obsolete with orbital refueling.

[TheRadicalModerate]

There are sharply diminishing returns to adding more propellant. . . . Instead of just blindly throwing prop at Starship, you need to ask what you're trying to achieve.  When you narrow it down to a variety of round-trip cislunar missions and one-way Mars missions, I haven't found a case where stretching the tanks past 1500t helps (although that extra 300t helps a lot, just because of how the delta-v budget to and from cislunar works out), nor have I found any cases where pusher/depot, tandem burns, or even drop tanks help very much.

I find myself thinking that a special cargo "tug" with a solar-powered (or nuclear) ion drive that slowly chugs from LEO to (say) NRHO would be worth it at some point. Yeah, it's slow and not very energy-efficient, but if it could move a full depot in (say) a month or two, that'd probably be well worth it at some point, since, with specific impulse of ~5000s it wouldn't need to be refueled very often.

This isn't going to happen any time soon.  For it to make any sense at all, the cadence would have to be high enough to warrant a major development effort to produce such a huge SEP tug.  And if the cadence is high, then you'll have a whole bunch of fleet costs with a whole bunch of tugs in transit.  Those fleet costs will likely swamp the cost of adding a few additional (again, high-cadence) lift tanker launches to move prop to cislunar via brute force.

Maybe to focus the discussion a bit, I think there are threeš major milestones in the use of tankers to handle cislunar operations:

1) Refueling for the Option A and B LSS.  This is important because it's the key to Artemis's continued existence.

2) Refueling for an LSS staged with crew out of LEO, and returning propulsively to LEO, with an F9/D2 or Starliner bringing the crew to/from LEO.  This is important because it does away with the use of SLS/Orion and therefore reduces the cost per mission by $2B-$3B, even before Starship is crew-certified for launch and EDL.

3) Refueling for a not-quite-vanilla Starship, which is capable of landing on the Moon and returning directly to EDL on Earth.  This is almost certainly SpaceX's endgame, but it requires that all-important crew-certification of launch and EDL.

Finding the proper conops for each of these requires balancing:

- Operational complexity
- Risk of refueling and extra RPODs
- The fact that refueling risk is reduced over time
- Tolerance of risk for crewed refueling operations
- Tolerance of risk for uncrewed operations involving spacecraft that will later be crewed (and are therefore expensive)
- Tolerance of risk for dumb tankers and depots
- And finally the cost (mostly in lift tanker launches) of the various conops. 

It is, to put it mildly, a rich trade space.

After fooling with these for a while, I think you need to follow these guidelines:

1) Option A and B missions have to be as simple as possible, and the LSS itself can't be refueled more than once per mission, nor can it be refueled with crew on board.

2) Staging LSS from LEO for eventual propulsive return requires two refuelings per mission (one in LEO and one... somewhere else).  The second refueling has two problems:
a) The crew is on board (unless you do weird stuff using the Gateway).
b) The refueling has to occur post-lunar-ascent, which makes abort contingency planning very dicey.  (There is an option to do pre-descent refueling, but the number of lift tankers needed almost doubles.  Carrying the prop to return to LEO down to the lunar surface and back is expensive.)

3) When crews are able to be launched and landed via Starship, there are two crewed refuelings instead of one (one in LEO, one at some higher energy), but pre-descent refueling becomes less expensive (since the Starship can return to EDL, rather than propulsive LEO), which gives you a lot more wiggle-room on abort contingencies.

It is possible that crew certification for Starship launch and EDL comes early enough that milestone #2 never needs to be dealt with.  I'd be surprised if that were true, but I've been surprised before.

_________________
šNote that I'm not talking about Mars, not because I don't think it'll happen, but rather because it's easy:  just refuel once in VLEO and you're good to go.

[TheRadicalModerate]

nor have I found any cases where pusher/depot, tandem burns, or even drop tanks help very much.

You get free deltaV (~2x) with Oberth maneuvers, so getting fuel into a higher energy orbit (HEEO) effectively doubles Isp.

(or alternatively 2.72^2 = 7.4x more fuel).

Is why that maneuver is so valuable, and why ion drives are effectively obsolete with orbital refueling.

Well... not really.

It's certainly true that if prop were magically to appear in a high-energy eccentric orbit, it's worth a lot more to climb up to that orbit to refuel than to refuel at low energy.  But prop doesn't magically appear, at least not until somebody can fling it off the Moon using super-cheap ISRU (aka science fiction).  You have to haul it up through the same gravity well as your target Starship.

This is not to say that there aren't missions enabled by refueling in HEEO; there are.  Getting a full tank of prop at high energy and high eccentricity means that you've got more delta-v to play with at the end of your mission, which is important.  But the cost of that mission, in terms of Starship lift tanker launches, doesn't change for using the HEEO.

Furthermore, if you're trying to get to some target orbit that requires Δv_target, and you boost up to an HEEO that requires expending Δv_heeo, then to get to the target from the HEEO still requires Δv_remainder = Δv_target - Δv_heeo.  It can be worse than this if you don't burn at perigee, but it can't be better.

This is all fully consistent with the Oberth Effect, but there's no free lunch, unless you're falling into a gravity well that you didn't pay to get out of in the first place.

My recommendation:  Use vis-viva and you'll never go wrong.

[InterestedEngineer]

nor have I found any cases where pusher/depot, tandem burns, or even drop tanks help very much.

You get free deltaV (~2x) with Oberth maneuvers, so getting fuel into a higher energy orbit (HEEO) effectively doubles Isp.

(or alternatively 2.72^2 = 7.4x more fuel).

Is why that maneuver is so valuable, and why ion drives are effectively obsolete with orbital refueling.

Well... not really.

It's certainly true that if prop were magically to appear in a high-energy eccentric orbit, it's worth a lot more to climb up to that orbit to refuel than to refuel at low energy.  But prop doesn't magically appear, at least not until somebody can fling it off the Moon using super-cheap ISRU (aka science fiction).  You have to haul it up through the same gravity well as your target Starship.

This is not to say that there aren't missions enabled by refueling in HEEO; there are.  Getting a full tank of prop at high energy and high eccentricity means that you've got more delta-v to play with at the end of your mission, which is important.  But the cost of that mission, in terms of Starship lift tanker launches, doesn't change for using the HEEO.

Furthermore, if you're trying to get to some target orbit that requires Δv_target, and you boost up to an HEEO that requires expending Δv_heeo, then to get to the target from the HEEO still requires Δv_remainder = Δv_target - Δv_heeo.  It can be worse than this if you don't burn at perigee, but it can't be better.

This is all fully consistent with the Oberth Effect, but there's no free lunch, unless you're falling into a gravity well that you didn't pay to get out of in the first place.

My recommendation:  Use vis-viva and you'll never go wrong.

The HEEO refuel only costs one additional fully refueled standard Starship (said Starship reusable).   You use up so little fuel in the Oberth burn you have enough left over to complete the entire rest of the Moon mission (full round trip).

Which is cheaper than developing a custom depot stuck in Lunar orbit.

That's 2400t of fuel to LEO.   How much do the other mission concepts cost in fuel to LEO?   Getting fuel to Lunar orbit isn't going to be cheap.

Granted, the "one starship round trip with HEEO refueling" mission concept does require a Starship capable of  Lunar landing and an aerobraked return to Earth.   Same requirements as Mars, IOTW (but at double the fuel cost).

So won't happen with crew until Starship is crew certified.

My bet is by 2035 this will be the standard way to get to the Moon.   All the other stuff (e.g. Artemis) is a place holder.

[tbellman]

I find myself thinking that a special cargo "tug" with a solar-powered (or nuclear) ion drive that slowly chugs from LEO to (say) NRHO would be worth it at some point. Yeah, it's slow and not very energy-efficient, but if it could move a full depot in (say) a month or two, that'd probably be well worth it at some point, since, with specific impulse of ~5000s it wouldn't need to be refueled very often.

In addition to the large solar panels as LMT pointed out, the propellant costs for your electrical engines are likely to be rather large.

Delta-v for trans-lunar injection from LEO is 3.2 km/s, if you are doing an impulsive maneuver (i.e. a short, high-thrust burn).  For a low-thrust continous burn over a two month period, we are talking about roughly double the Δv.  Let's be generous and say 6 km/s.  To accelerate a 1500 tonne craft by 6 km/s using engines with specific impulse of 5000 s, requires

        m_prop = m_final * (e^{Δv / (I_sp*g₀)} - 1) =
            = 1500 t * (e^{6 km/s / 49 km/s} - 1) ≈
            ≈ 195 tonne

propellant.

Most ion engines use xenon as propellant.  I only found xenon prices from 1999 in a quick search, but then xenon cost $1800/kg when buying in small quantites, so the propellant cost for a single such delivery would be about $350M.  And unfortunately we are unlikely to get a quantity discount, because 195 tonnes is several years worth of xenon production (we produce roughly 50 tonnes per year), so we are likely to drive up the world prices quite significantly.

Krypton is another viable propellant for ion engines.  SpaceX use that for Starlink, because it is so much cheaper than xenon.  The disadvantage is that you need more electrical power for the same thrust, but we can ignore that for this analysis.  Back in 1999, krypton cost $290/kg, so 195 tonnes would cost us $56M.  And krypton is much more abundant than xenon, so we might not affect the world prices much.  At least as long as we only do a very small handful of such 1500t deliveries every year.  We are still talking about maybe $20M in just the cost of krypton propellant, so even with krypton, electric propulsion might not be much of a win for this application...

[InterestedEngineer]

I find myself thinking that a special cargo "tug" with a solar-powered (or nuclear) ion drive that slowly chugs from LEO to (say) NRHO would be worth it at some point. Yeah, it's slow and not very energy-efficient, but if it could move a full depot in (say) a month or two, that'd probably be well worth it at some point, since, with specific impulse of ~5000s it wouldn't need to be refueled very often.

In addition to the large solar panels as LMT pointed out, the propellant costs for your electrical engines are likely to be rather large.

Delta-v for trans-lunar injection from LEO is 3.2 km/s, if you are doing an impulsive maneuver (i.e. a short, high-thrust burn).  For a low-thrust continous burn over a two month period, we are talking about roughly double the Δv.  Let's be generous and say 6 km/s.  To accelerate a 1500 tonne craft by 6 km/s using engines with specific impulse of 5000 s, requires

        m_prop = m_final * (e^{Δv / (I_sp*g₀)} - 1) =
            = 1500 t * (e^{6 km/s / 49 km/s} - 1) ≈
            ≈ 195 tonne

propellant.

Most ion engines use xenon as propellant.  I only found xenon prices from 1999 in a quick search, but then xenon cost $1800/kg when buying in small quantites, so the propellant cost for a single such delivery would be about $350M.  And unfortunately we are unlikely to get a quantity discount, because 195 tonnes is several years worth of xenon production (we produce roughly 50 tonnes per year), so we are likely to drive up the world prices quite significantly.

Krypton is another viable propellant for ion engines.  SpaceX use that for Starlink, because it is so much cheaper than xenon.  The disadvantage is that you need more electrical power for the same thrust, but we can ignore that for this analysis.  Back in 1999, krypton cost $290/kg, so 195 tonnes would cost us $56M.  And krypton is much more abundant than xenon, so we might not affect the world prices much.  At least as long as we only do a very small handful of such 1500t deliveries every year.  We are still talking about maybe $20M in just the cost of krypton propellant, so even with krypton, electric propulsion might not be much of a win for this application...

It's even worse.   The static mass of a modern X3 ion thruster is 561kg/engine (including the required energy source/sink), which can only supply 35M Newton-seconds of propellant momentum and 626GJ of energy over 45 days.   A Raptor 2, OTOH, only weighs twice that amount and supplies that energy over 10 minutes at 1.6G Newton-seconds of propellant momentum.   That's 20 times the momentum efficiency of the X3 thruster.

Most of the mass of an ion drive thruster isn't tossed out the back as fuel, it's a parasitic mass remaining when the fuel is exhausted.

In the era of refueled chemical rockets, ion drives are obsolete for any conceivable use beyond maintaining orbits.

https://docs.google.com/spreadsheets/d/1mrlKefLWzGxsYtBXWF92a0cXqsPZoKLGlPbQizKHI3g/edit#gid=0

[Greg Hullender]

It's even worse.   The static mass of a modern X3 ion thruster is 561kg/engine (including the required energy source/sink), which can only supply 35M Newton-seconds of propellant momentum and 626GJ of energy over 45 days.   A Raptor 2, OTOH, only weighs twice that amount and supplies that energy over 10 minutes at 1.6G Newton-seconds of propellant momentum.   That's 20 times the momentum efficiency of the X3 thruster.

Most of the mass of an ion drive thruster isn't tossed out the back as fuel, it's a parasitic mass remaining when the fuel is exhausted.

Thanks for the time and effort you guys put into this. I started to try to do this myself, but got frustrated when I couldn't easily find answers to questions like a) what's the best ion engine today b) what are its specific impulse and thrust c) what is the mass of that engine? d) what would the mass be of the power supply (whether solar or nuclear)?

Of course, nothing says that there couldn't be some breakthrough that made for a light-weight ion engine with a light-weight power supply, but, as TheRadicalModerate suggests above, we're trying to talk about things SpaceX might plausibly attempt in the next few years (next decade at the most). I'm now convinced that an ion tug isn't going to be feasible in that time frame--if it ever is. Thanks again for the demonstration.

In the era of refueled chemical rockets, ion drives are obsolete for any conceivable use beyond maintaining orbits.

Even if it's not useful for a tug, what about a high Δv mission--to Neptune, say? Do you really think ion drives have no use whatsoever?

[LMT]

...what about a high Δv mission--to Neptune, say? Do you really think ion drives have no use whatsoever?

If the objective is a crewed Starship mission to an outer planet, you have to get creative with propulsion.  Refuel at both ends, yes, and more than that.  Unfortunately, ion engines can't deliver the thrust needed for the big outer-planet maneuvers.

What's the best use of deployed film?  If PV film can't give the thrust, you can consider a dielectric reflector film instead, for laser propulsion.

Case:  a 2-year crewed mission to Callisto:

1.  A PV GW-scale laser array at the LEO depot boosts the ship to Jupiter.

2.  A comparable retrograde electrodynamic laser array at Jupiter drops the ship into Jovian orbit. 

3.  Land and refuel on icy Callisto.

4.  Reapply engines and lasers for Earth-return delta-v.

Basic delta-v calcs in the DE-STAR thread.

[TheRadicalModerate]

The HEEO refuel only costs one additional fully refueled standard Starship (said Starship reusable).   You use up so little fuel in the Oberth burn you have enough left over to complete the entire rest of the Moon mission (full round trip).

You use exactly the same amount of delta-v, irrespective of the energy of the HEEO.  If you're trying to get to LEO+3200 and you stop in an LEO+2500 HEEO, then that HEEO to LEO+3200 costs 700m/s.  It's simple addition, nothing more--as long as you always burn at perigee.  There are no Oberth freebies.  However, prop consumption will obviously increase if you refueled more than you needed to.

You'll always be the most prop-efficient if you refuel in the lowest energy orbit that still leaves you with enough prop to meet the remaining mission delta-v requirements.  Things get weird if no HEEO exists that leaves you with enough prop to finish the mission--that's where something like laddering comes in--but you need a pretty extreme conops for that to happen.

Don't get me wrong:  if you can't do the mission with a full tank from VLEO, then an HEEO of just enough energy is the most efficient thing, and will result in the lowest amount of prop needing to be delivered to orbit.  But prop minimization isn't the only constraint.

Which is cheaper than developing a custom depot stuck in Lunar orbit.

Who said anything about a custom depot?  If you need prop in cislunar and your logistics work out, you send a regular lift tanker, refuel, and the tanker returns to EDL.  For a crewed mission that relies on prop in cislunar to get home, you can't do quite this well, because it may be weeks before the actual mission shows up.  In that case, you don't need a custom depot but you may need a regular VLEO depot that hangs around forever, simply to manage the boil-off.

That's 2400t of fuel to LEO.   How much do the other mission concepts cost in fuel to LEO?   Getting fuel to Lunar orbit isn't going to be cheap.

Assuming a vanilla Starship can get 150t to LEO (which would make 2400t of prop roughly 16 tankers):

1) Option A 1200t LSS, refueling once in LEO+1420 HEEO (no refueling in VLEO): 9 tankers.  (The HEEO energy is limited by how high the LSS can boost itself after launch without refueling in VLEO; the assumption is that only one refueling RPOD will be allowed this early in the program.)

2) Option A 1500t LSS, refueling only in VLEO: 9 tankers.

3) Option B 1200t LSS already in NRHO, refueled by 1500t lift tanker: 10 tankers.

4) LSS staged with crew from VLEO, second refuel post-lunar-ascent, propulsive return to LEO: 15 tankers.  (For all of the remaining conops, note that the first refuel is always in VLEO.)

5) LSS staged crewed from VLEO, second refuel pre-lunar-descent, propulsive return to LEO: 28 tankers.  (No, really!  It's terrible!  And it requires more than one tanker to NRHO.  But that's a "Doctor, Doctor, it hurts when I did this" kind of problem.)

6) Launch/EDL crew-certified Starship with surface landing, second refuel in HEEO, direct EDL on return: 13 tankers.

7) Launch/EDL crew-certified Starship, second refuel pre-descent in LLO, direct EDL on return: 15 tankers

All of these have anywhere between 5t and 50t of non-crew deployable cargo, depending on how things even out to an integral number of lift tankers.  All get the tankers back to EDL.  They all assume LSS is 95t dry, crew module is 15t.  For EDL-capable lunar versions, Starship is 120t dry, crew module 15t.  Isp=378s everywhere.  FPR, boil-off, and fixed prop losses included.

Granted, the "one starship round trip with HEEO refueling" mission concept does require a Starship capable of  Lunar landing and an aerobraked return to Earth.   Same requirements as Mars, IOTW (but at double the fuel cost).

So won't happen with crew until Starship is crew certified.

My bet is by 2035 this will be the standard way to get to the Moon.   All the other stuff (e.g. Artemis) is a place holder.

I agree, although I wouldn't be surprised to see the second refueling in cislunar.  That's sub-optimal from a propellant efficiency standpoint, but I still think that HEEO for crewed missions comes with a lot of baggage that may make the cislunar version worth the extra 2 lift tanker launches.

And "placeholder" may turn out to be a dismissive word.  If you don't get the launch/EDL crew certification for ten years, it's not a placeholder; it's the early version of your conops.

[InterestedEngineer]

The HEEO refuel only costs one additional fully refueled standard Starship (said Starship reusable).   You use up so little fuel in the Oberth burn you have enough left over to complete the entire rest of the Moon mission (full round trip).

You use exactly the same amount of delta-v, irrespective of the energy of the HEEO.  If you're trying to get to LEO+3200 and you stop in an LEO+2500 HEEO, then that HEEO to LEO+3200 costs 700m/s.  It's simple addition, nothing more--as long as you always burn at perigee. 

You need to do C3 calculations, not deltaV calculations.

In the end, you end up with far less errors if you always using conservation of momentum and conservation of energy when performing conic section calculations.

[InterestedEngineer]

You'll always be the most prop-efficient if you refuel in the lowest energy orbit that still leaves you with enough prop to meet the remaining mission delta-v requirements.  Things get weird if no HEEO exists that leaves you with enough prop to finish the mission--that's where something like laddering comes in--but you need a pretty extreme conops for that to happen.

When fuel costs $30-80/kg, "prop efficient" is a minor consideration.

Completing a mission with the least cost, including development costs of custom components, is the most important metric.  Reliability is another important metric, and doing almost the same thing hundreds of times is far more reliable than almost any custom component.

[TheRadicalModerate]

The HEEO refuel only costs one additional fully refueled standard Starship (said Starship reusable).   You use up so little fuel in the Oberth burn you have enough left over to complete the entire rest of the Moon mission (full round trip).

You use exactly the same amount of delta-v, irrespective of the energy of the HEEO.  If you're trying to get to LEO+3200 and you stop in an LEO+2500 HEEO, then that HEEO to LEO+3200 costs 700m/s.  It's simple addition, nothing more--as long as you always burn at perigee. 

You need to do C3 calculations, not deltaV calculations.

In the end, you end up with far less errors if you always using conservation of momentum and conservation of energy when performing conic section calculations.

Spacecraft have a delta-v budget, not a C3 budget.  And vis-viva is conservation of energy.

v_peri˛ = μ(2/r_peri - 1/a)

Let's set μ=1 and start in a circular orbit, i.e., r=a.  All maneuvers will be performed at r_periapse=1, which will therefore stay the same (i.e., set to 1 in our normalized units), as r_apoapse increases.

If I prove that Δv_a=1→a=2 = Δv_a=1→a=1.5 + Δv_a=1.5→a=2, will you concede?

Here goes:

v_a=1˛ = (2 - 1) = 1, so v_peri,a=1 = 1
v_a=2˛ = (2 - 1/2) = 1.5, so v_peri,a=2 = 1.22
v_a=1.5˛ = (2 - 1/1.5) = 1.33, so v_peri,a=1.5 = 1.15

Δv_a=1→a=2 = v_peri,a=2 - v_peri,a=1 = 1.22 - 1 = 0.22.
Δv_a=1→a=1.5 = v_peri,a=1.5 - v_peri,a=1 = 1.15 - 1 = 0.15
Δv_a=1.5→Δv_a=2 = v_peri,a=2 = v_peri,a=1.5 = 1.22 - 1.15 = 0.07

Δv_a=1→a=1.5 + Δv_a=1.5→Δv_a=2 = 0.15 + 0.07 = 0.22 = Δv_a=1→a=2
QED

Again, all maneuvers are performed at periapse, where you'd get the maximum Oberth effects.

Things are different if you transition from an elliptical orbit to a hyperbolic one, but we're not doing that.

[Barley]

1) Option A and B missions have to be as simple as possible, and the LSS itself can't be refueled more than once per mission, nor can it be refueled with crew on board.

This is a silly constraint.  By the time a crew certified LSS leaves LEO there will be there will be an order of magnitude more experience refueling than launching SLS or reentering Orion or a great many other parts of the planned mission.  If that's not enough refueling experience, you can get a lot more fairly cheaply and quickly.  Since refueling is an essential part of any BLEO use of Starship you have to do that eventually, so the best plan is to expedite testing refueling and not place unneeded constraints on the eventual mission.