** Update 3-26-21: ** this second tanker study looked at refilling the lunar mission Starship in low circular orbit, where tanker Starship capacities are high, then topping it off after moving it to the departure orbit, with more tankers sent directly to that orbit. That reduced the number of required tankers, but not by enough to be truly practical. The strategy explored in the third study (also posted here) turned out to be the most effective one.

-------------------------------------

In Ref. 1, I defined
two elliptical orbits from which a Spacex Starship could be sent fully-fueled
on a lunar nearside landing mission, be unrefueled
on the moon, and return to a free entry
landing trajectory at Earth. ** Payloads
sendable to the moon this way are far smaller than payloads deliverable to low
circular Earth orbit,** because
payload capacity to the higher-energy orbit is reduced, as well as the inherently-smaller payload
capacity for the lunar flight.

It makes sense to use the elliptical orbit at which Starship
payload capacity to that orbit matches the Starship lunar mission
capacity. Some 75 metric tons can be
sent to the moon, with zero return
payload, from an elliptical orbit that
has 300 x 7000 km altitudes. Some 59
tons can be sent to the moon __with a 32 ton return payload__, from an orbit that is 300 x 10,000 km
altitudes.

Both elliptic orbits have apogee altitudes that penetrate
well into the Van Allen radiation belts.
Thus cargoes delivered this way must be radiation-hard, and any crew or passengers will require
effective shelters from serious radiation exposure. The “base” of the Van Allen radiation belts
is considered to be about 1400 km altitude,
outside the South Atlantic Anomaly.

In Ref. 2, I
determined the deliverable propellant quantities from two potential Spacex
tanker configurations: a dedicated
tanker design with extra tankage volume in the forward spaces, and an ordinary Starship flown fully fueled, but at zero payload, so that it arrives with significant unused
propellant. The scope of that study
looked at the 300 x 300 km altitude low circular Earth orbit, and the two possible elliptical lunar mission
departure orbits that “sort-of” bound the problem.

Deliverable tanker “payload” with either configuration was
considerable in low circular orbit, but
a lot less when sent directly to either of the two elliptical orbits. This is because the higher energy orbits require
a whole lot more propellant just to be reached.
In all these scenarios, the
dedicated tanker design carries somewhat more deliverable propellant than the
ordinary Starship flown at zero payload.

This basic information is summarized in Figure 1 below. All figures are located at the end of this
article. The two elliptical lunar
departure orbits and associated lunar payloads are in the upper part of the
figure. The tanker delivery capacities
are shown for both tanker configurations,
at each of the three orbit locations,
in the lower part of the figure.
In each case, the
Starship/Superheavy tanker vehicle is flown directly to the target orbit
location, as is the cargo/passenger
Starship/Superheavy vehicle.

**Two Waves Of Refueling On-Orbit**

The small tanker propellant delivery capacities to elliptic
orbit versus the large capacities to low circular orbit suggested a two-step
refueling operation for the lunar-bound Starship: a full-capacity refill in low circular orbit, followed by moving it to the elliptic orbit, with another refill there. This would apply to either lunar mission
scenario. It could be done with either tanker configuration. Hopefully,
fewer tanker flights would be required than by direct one-step refueling
in the elliptic departure orbit.

To support this two-step refueling operation, I had to figure out what the refill
requirements would be upon arrival in the target elliptic orbit, and also the low circular refueling
requirements when carrying the smaller lunar payloads. The elliptic refill requirements I
hand-calculated, and these are
summarized in Figure 2. The low circular
refuel requirements at lower payloads, I
figured with one of the Starship spreadsheet models. Those results are Figure 3 for the 75 ton
payload from the 7000 km apogee orbit,
and in Figure 4 for the 59 ton payload from the 10,000 km apogee orbit.

**Calculating Numbers of Tanker Flights**

Then I calculated the number of tanker flights required to
carry out these operations (first the low circular refill, then second the elliptic refuel after moving
there). ** In each operation, the requisite tankers are flown directly to
the orbit where the refill will take place.** The limiting number of
tanker flights is the refill requirement divided by tanker delivery
capacity, but the only such number that
makes any sense is an integer! You
always hold back the tanker landing reserve from deliverable propellant.

** If there are decimals,
you must round up to the next-larger integer!** That is because you cannot carry more deliverable
propellant than the max capacities these calculations have identified, but you can always carry a little less than
max capacity.

**Intermediate Results**

The refill requirements in low circular orbit for the
lunar-bound Starship are just about 1050 tons of propellant, depending upon which mission and payload we
are talking about. The tanker capacities
to low circular orbit are right at 200 tons per vehicle, give or take,
depending upon whether the tanker is the dedicated design or the
ordinary Starship flown as a tanker.
Those numbers and the rounded-up integer numbers of flights are given in
Figure 5. Note that the delivered tanker
loads are just not very far from the max capacities of the tankers, as also given in the figure. We are looking at 5 dedicated tankers, or 6 ordinary tankers, to accomplish this, for either lunar departure scenario.

The second refueling operation is illustrated in Figure
6. The refill requirements are far
smaller, but then so are the max tanker
capacities! Again, you round up the decimals to the next larger
integer number of tankers. Those results
show 3 tankers (of either type) are needed for the 300 x 7000 km orbit that
supports 75 ton lunar landings, with 0
tons return payload. The more demanding
300 x 10,000 km orbit requires 5 tankers of either type, but that scenario supports a 32 ton return
payload, with 59 tons landed on the
moon. It is the higher-energy orbit.

**Ultimate Results**

A summary comparison of these results versus the direct
staging in elliptical orbit, and versus
operations only to 300 x 300 km circular
are given in Figure 7. *That
figure this summarizes the results of the first tanker study and this second
one. *

The max payload deliverable to low circular orbit is quite a
bit more at 171 metric tons. That is why
7 ordinary or 6 dedicated tankers are needed to fully refill the Starship
there. That’s a Ref. 2 result.

Another Ref. 2 result is the single refueling operation
conducted with Starship and tanker flights directly to the lunar departure
elliptic orbit. That is also indicated
in the figure, which shows some 21
ordinary or 19 dedicated tanker flights to refill the Starship carrying 75 tons
to the moon from the 300 x 7000 km orbit.
The Starship carrying 59 tons from the 300 x 10,000 km orbit requires 17
ordinary or 15 dedicated tanker flights to fully refill. Those high tanker flight numbers are not very
attractive!

** The two-refill operation approach analyzed in this
article (with the first wave of tankers sent to low circular orbit, and the second wave of tankers sent to the
departure elliptic orbit) does indeed reduce the number of tanker flights.** For the 300 x 7000 km departure with 75 tons
to the moon, the first wave is 6
ordinary or 5 dedicated tankers to circular,
followed by the second wave of 3 ordinary or 3 dedicated tankers to the
elliptical orbit, for a total of 9
ordinary or 8 dedicated tankers to support the mission. For the 300 x 10,000 km departure (59 tons to
the moon), the first wave is the same 6
ordinary or 5 dedicated tankers, and the
second wave is 5 ordinary or 5 dedicated,
for a total of 11 ordinary or 10 dedicated tankers to support that
mission. That is a substantial
improvement, but still unattractive.

**Remarks**

Readers need to be aware that these calculations I have made
are __not__ any sort of simulations run with any sort of computer
programs. ** These are the kind of
calculations I would make, if I sat down
at the kitchen table with pencil,
paper, and a pocket calculator
(or even a slide rule).** These
have been semi-automated with spreadsheet software, but are essentially the same very simple
calculations made with simple models,
plus the engineering art of selecting the right “jigger factors” to get
realistic results.

For those readers too young to know what a slide rule
is, see Ref. 3.

**Plans**

The improvement achieved with two-step refilling is
attractive, and the disparity between
tanker capacities in low circular versus feasible lunar departure orbits is
large. That suggests refilling not only
the lunar mission Starship in low Earth orbit,
but also 1 or maybe 2 tankers.
The mission Starship plus the 1 or 2 refilled tankers would then move to
the elliptical departure orbit, where those
tankers would top-off the mission Starship.
As time and opportunity permits,
I will look at this method of employing these assets, to see if the total number of supporting
tanker flights can be reduced further.

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

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

#2. G. W. Johnson,
Spacex Tanker Investigation,
dated 17 March 2021.

#3. G. W. Johnson,
THIS Is a Slide Rule!, dated 16
March 2019.

Figure 1 – Selected Results From the First Tanker
Investigation

Figure 2 – Determining the Refueling Needs Moving From
Circular to Elliptical With Lunar Payloads

Figure 3 – Determining the Refueling Needs in Circular with
Lunar Payloads: 75 ton

Figure 4 – Determining the Refueling Needs in Circular with
Lunar Payloads: 59 ton

Figure5 – Tankers Required to Refuel Lunar-Bound Starships
in Circular Orbit (First Refuel)

Figure 6 – Tankers Required to Refuel Lunar-Bound Starships
in Elliptical Orbit (Second Refuel)

Figure 7 – Comparison of First Investigation Results with
Results of Second Study

## No comments:

## Post a Comment