This article picks up where Ref. 1 left off, looking at direct lunar landings using a
Starship refueled (fully) in Earth orbit.
That might require a lot of tanker flights, something __not__ investigated here or in
Ref. 1. The methods and assumptions
mostly conform to those given in Ref. 2.
A brief list of selected orbital data is given in Fig. 1, which includes the results initially
determined for Ref.1. All figures are at
the end of this article.

I chose to analyze a min-energy Hohmann transfer to the
vicinity of the moon, instead of the
Apollo figure-eight orbit. The Apollo
trajectory, while a slightly-shorter
transfer time, is more suited to
entering low retrograde orbit about the moon.
The Hohmann trajectory is closer to the real 3-body result for a direct
landing on the lunar near side. The trajectory
and related data are given in Fig. 2.
The variations in lunar distance from Earth are included, labeled as “close”, “average”,
and “far”.

The actual mission uses a slightly-different return
trajectory from the moon. It is a
min-energy Hohmann transfer ellipse, but
it is one that __grazes the Earth’s surface__ at perigee. That ensures a direct atmospheric entry and
landing upon return to Earth. Entry
interface speeds are very near 11 km/s.
The rough locations for midcourse and terminal course corrections are
shown with red stars in Fig. 3.

The course correction burn delta-vees are figured from cross
ranges at remaining-distances, and
speeds, using the small angle
approximation. These are in Fig. 4, and totaled into outbound and return course
correction budgets for delta-vee. Approximately, dV = (V) (cross range)/ (remaining distance).

Fig. 5 shows the basic approaches to and from the moon, viewed with respect to the moon, plus where the gravitational loss factor
comes from. There is no drag loss factor
at the airless moon. This stuff gets
used figuring the factored (mass ratio-effective) delta-vees to land, and to take off.

The landing and takeoff details have to include engine
selections, because for the touchdown
and liftoff immediately adjacent to the moon’s surface, the thrust vector control (TVC) capability
afforded by the sea level Raptor engines is simply required! The bulk of these burns can be accomplished
with the vacuum Raptor engines, farther
from the surface, at higher specific
impulse. ** This means there must be
a shift from vacuum to sea level engines near the end of the landing burn, and a similar shift from sea level to vacuum
engines after the start of the takeoff burn.** See Fig. 6. I

__guessed__“the last 0.5 km/s”.

Fig. 7 contains a multitude of useful data regarding burn
delta-vees. The departure burn from
Earth orbit onto the transfer trajectory is not only figured for “close”, “average”,
and “far” lunar distances, but
also for multiple choices of elliptical orbits about the Earth. The landing burns and the takeoff burns at
the moon depend only upon lunar distances, and not upon choice of Earth
elliptic orbit. Delta-vees are totaled
in this figure as a measure of how demanding the mission is, although with different engine specific
impulses, these are not values one can
use to design vehicle mass fractions.

The “right” way to estimate performance is with a __burn
record spreadsheet__ as illustrated by the image in Fig. 8. This one has inputs for the vehicle weight
statement that include separate entries for outbound and return payload. The relevant engine performances for the two
engine types are included as inputs (yellow-highlighted), complete with max and min power setting data. A linear model is computed between max and
min settings, for both thrust and
specific impulse.

There are 4 burns listed for the outbound journey, and 4 more for the return journey. There is an Earth orbit departure burn, starting from max propellant load, a burn representing the 2 course
corrections, the vacuum-engine portion
of the landing burn, and the sea level-engine
portion of the landing burn. For the
return, there is takeoff using the sea
level engines, then departure using the
vacuum engines, a burn representing the
2 course corrections, and the terminal
landing burn after atmospheric aerobraking entry, with the sea level engines. There are inputs for each of these 8 burns, for the user’s selection of how many engines
and what throttle setting to use (highlighted yellow).

Significant information or outputs are highlighted blue or
green. Lots of data are simply
built-in, including the worst-case
delta-vees related to lunar distance effects.
That means the payloads indicated can be carried, or perhaps very slightly more, but never less. The departure delta-vee associated with the
elliptical orbit about the Earth is the main variable here, besides outbound and return payload. That departure input is also highlighted
yellow, located top right of
worksheet. Adjacent to it are records of
the various inputs appropriate to the various elliptic orbits, plus a results record for the payloads
obtained with each of these departures.

The way this worksheet functions is a __user iteration of
outbound and return payloads__ (and the number of engines and their thrust
settings), until the propellant
remaining at end of mission is a nonzero fraction of a ton. Thrust setting affect specific impulse, but it and the number of engines affects
thrust/weight far more. These
thrust/weights need to meet the green-highlighted requirements shown.

I took the accumulated payload results (in the form of zero
return payload) vs elliptic orbit apogee altitude, and combined them with the payloads
deliverable to those same orbits from Ref. 1,
and cross-plotted them below the vehicle performance analysis in this
worksheet. All of that is shown in the
figure. The curves cross at about 7000
km apogee altitude, at about 75 tons
delivered (with zero return payload).
The 10,000 km apogee altitude is associated with 59 tons delivered to
orbit. Using those 59 tons as the
outbound payload to the moon from that 300x10,000 km orbit, I found there was just enough propellant to
bring 32 tons of payload back from the moon,
and then to land with it.

The 59 ton outbound with 32 ton return from 10,000 km apogee is
depicted in Figure 9. The 75 ton
outbound with 0 ton return from about 7000 km apogee is depicted in Fig.
10. The latter is more approximate, being just read from the cross plot
graph, but it is “in the ballpark”. *Thus it would seem to be feasible to
deliver significant tonnages to the moon with this Starship/Superheavy
system, but not the “100+ tons”
advertised on the Spacex website for low Earth orbit and Mars.*__The lunar mission is more demanding than
the Mars mission__, __because there
is no refueling on the moon for the return__.

Regardless, the 7000-to-10,000
km apogee altitude is well within the Van Allen radiation belts! *The cargo delivered from orbits like
this must be radiation-hardened. Any
crew (or passengers) will certainly require an effective radiation shelter
somewhere aboard the vehicle. *

** I did not investigate the tanker problem: on-orbit refueling of Starship to the 1200
metric ton max propellant load!**
Be aware that

__many__tanker flights will be required, especially since far smaller payloads (or transferable propellant) can be delivered to these high-apogee elliptic orbits.

**References (all located on this site)**

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

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

Figure 1 – Summary List of Selected Data for Elliptic Orbits About the Earth

Figure 2 – Hohmann Min-Energy Transfer to the Moon for
Direct Landings

Figure 3 – Actual Lunar Transfer and Return Orbit Data

Figure 4 – Figuring the Course Correction Budgets

Figure 5 – Direct Landing Approaches and Departures at the
Moon

Figure 6 – Working Out Engine Selections and Delta-Vees for
Lunar Landing and Takeoff

Figure 7 – Departure and Arrival Delta-Vees as a Function of
Earth Orbit Choice and Lunar Distance

Figure 8 – Spreadsheet Image of Performance Estimator with
Orbit Results Plot

Figure 9 – Pictorial Results for Direct Lunar Landing Mission with Return Payload

Figure 10 – Pictorial Results for Direct Lunar Landing Mission with No Return Payload

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