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

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

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

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

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

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

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

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

That is __not__ necessary for any Starship design used as
a tanker. Tankers can always vent
propellant overboard prior to the deorbit burn,
down to only that required for a dry-tanks landing at zero payload. ** Which
situation is exactly why the propellant-as-payload and landing reserve numbers
for the tankers are different from those for the target ship, to the same orbit.** Data are given in Table 1.

Table
1 – 300 x 300 km Altitude Results

__Vehicle__ __target__ __dedicated__ __ordinary__

Payload, m.ton 171 232 192.506

Vstaging, km/s 2.23 2.21 2.375

Land. Res., m.ton 21.876 9.019 9.019

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

Tankers to refill -- 5.08 6.12

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

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

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

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

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

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

Table
2 – 300 x 7000 km Altitude Results

__Vehicle__ __target__ __dedicated__ __ordinary__

Payload, m.ton ~75 83 73.4

Return payload, m.ton 0 -- --

Vstaging, km/s ~2.23 2.22 2.275

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

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

Tankers to refill -- 14.28 16.15

** Also, the
presumption here is that the tankers fly directly to the same elliptic
departure orbit as the target vehicle.**
No other scenario was considered at this time. That does

__not__mean that there are no other scenarios! One possibility for future investigation is refueling in low circular orbit, then moving the target vehicle to the departure orbit, along with 1 or 2 more tankers to fully refill it just before departure.

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

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

Table
2 – 300 x 10,000 km Altitude Results

__Vehicle__ __target__ __dedicated__ __ordinary__

Payload, m.ton 59 64 57.657

Return payload, m.ton 32 -- --

Vstaging, km/s 2.23 2.215 2.263

Land. Res., m.ton 13.456 9.019 9.019

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

Tankers to refill -- 18.54 20.58

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

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

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

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

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

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

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

**Future Plans**

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

**Final Comments**

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

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

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

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

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

**References (all are articles published on this site)**

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

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

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

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

__Update 3-18-21__:

some of your numbers are similar to mine but others not; to reconcile discrepancies it would be handy to know assumptions like dry mass of starship, how many engines, Isp, etc you used in your calculations.

ReplyDeletehere are the parameters i use in my sim: https://imgur.com/a/JNkWoBn

i am setting "dry mass" for these as including propellant reserve, boost back and landing for super heavy, and EDL for starship. i have trouble, though, both with getting super heavy back to the launch pad (because it is quite far down range), and with starship getting to orbit at all on fewer than five engines. the plan for approximately 300 km altitude circular orbit for refueling seems pretty good. i would refine that to make sure the orbital tanker flies precisely over the launch site once per day; in addition i would set inclination = 36.758 deg and adjust altitude to account for earth's rotation and precession of the orbital plane. this would give two launch opportunities per day for rendezvous with the tanker, one in 3 hours, launching to the northeast, one in 9 hours, launching to the southeast.

I think the answers to all your questions are in the 3-18-21 update that I added. -- GW

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