Bear in mind that I am no expert in orbital mechanics, and I have no computer codes to analyze 3-body problems. But I do understand the basics of 2-body problems, especially elliptical orbits, fairly well. There are simple formulas for those. That is the basis for the discussion topic here.
It is one thing to build and fuel a large craft in low Earth
orbit. It is quite another to propel it
from there onto an interplanetary trajectory.
The velocity requirement is to reach a speed somewhat over escape, from only circular orbit speed.
The math says that escape speed is larger than circular
orbit speed by a factor of the square root of 2, or about 1.414. For Earth,
that is about 11 km/s for escape, and about 8 km/s for circular
orbit, with the velocity requirement
being the difference, at about 3
km/s. To depart, we actually need a bit
more than that 3 km/s difference,
because we actually need to exceed escape speed.
By using a space tug and an extended elliptical orbit about
the Earth, that velocity requirement on
the departing spacecraft can be substantially reduced. The space tug is only reusable if it ends up remaining
in that extended ellipse, from which it
can then return to low circular orbit.
Geometrically, the
situation is symmetrical, so that this
concept also works for arrivals from interplanetary trajectories. But the sequence of events that must
happen is not symmetrical, so the
arrival trajectory plan cannot be the exact mirror reverse of the departure
trajectory plan.
The example here is for Earth, using low Earth orbit as the basis, as this is the most easily reached orbit for
launches from the surface. But the same
arguments and math apply to departures and arrivals at any planetary body! Only the numbers are different.
Let us start with the arrival scenario depicted as
sequential sketches in Figure 1.
The arriving craft is coming in on a hyperbolic path with respect to the
Earth, with its perigee specifically located
at low Earth circular orbit altitude,
and a speed at perigee somewhat larger than Earth escape speed at that same
altitude.
As the arriving craft reaches its hyperbolic perigee, it must impulsively make a modest speed
reduction (delta-vee, or dV) to slow to
a speed just under Earth escape speed at that altitude. That puts it onto an extended elliptical
orbit about the Earth, instead of
continuing on into deep space.
The space tug that is going to retrieve this craft from that
extended elliptical orbit is not already on that ellipse, and in fact cannot be on that ellipse! It must be somewhere on the low circular
orbit with the short period of about 90 minutes. The extended ellipse has a much longer period
near 4 (or even 5) days!
Because the timing of the craft’s arrival into that extended
ellipse, relative to where the tug is
located on that low circular orbit, is
not something that can be controlled,
the tug could be “anywhere” around that circular orbit! It will pass the ellipse perigee point
multiple times, while the arriving craft
makes one circuit about the extended ellipse.
There will be one tug circuit about the circular orbit where
it and the arriving craft are both very near each other at the ellipse
perigee, at the same time! That is where the tug must fire its
propulsion to accelerate impulsively to ellipse perigee speed, thus matching both position and velocity with
the arriving craft. The closer the orbit
period ratio is to an integer, the
closer together this rendezvousing pair of vehicles will be, and the lower the required rendezvous dV
budget.
Actual rendezvous and docking take a finite interval that is
not trivial! The two can dock, but by the time that is accomplished, they will no longer be anywhere near the
ellipse perigee point! The docked pair
must complete another circuit about the ellipse. When they reach ellipse perigee the second
time, the tug can decelerate the docked
pair into low circular orbit, precisely
because the perigee altitude is preserved by making burns there, and not anywhere else along the orbit.
From there, normal
rendezvous procedures can be used to reach any desired orbital station or
facility.
Figure 1 – The Arrival Scenario
The departure scenario sequence of sketches is given
in Figure 2. The difference in the event sequence is that
the departing craft and the tug are docked together when this scenario
starts! They must only wait in
circular orbit until the geometry lines up with the intended departure
path, and then the tug fires to bring
the docked pair to the perigee speed of the extended ellipse.
They undock as that burn terminates, and the departing craft immediately makes its
modest burn to reach the intended departure speed. Meanwhile,
the tug just coasts one circuit about the ellipse. When it reaches perigee, it can decelerate back into low circular
orbit.
From there, standard
rendezvous procedures can return the tug to any desired orbital station or
facility. Starting out docked together
is precisely what provides the proper timing of the departure events
sequence, that is inherently lacking
during the arrival sequence! Undocking
and moving away a short distance, for
departure burn safety, is something that
only takes several seconds, not the
several minutes to an hour or so, for close-in
rendezvous and docking.
Figure 2 – The Departure Scenario
For the sake of argument,
presume the following data:
Vcirc = 7.8 km/s
Vesc = 11.0 km/s
Vper = 10.9 km/s
Vdep = 11.5 km/s = Varr
Those values produce the following
departure results:
Laden tug dV = Vper – Vcirc = 3.1
km/s
Craft departure dV = Vdep – Vper =
0.6 km/s (unassisted this is 3.7 km/s)
Unladen tug back to circular dV = Vper – Vcirc = 3.1 km/s
They produce similar results values for the arrival scenario, reflecting the geometric symmetry:
Craft arrival dV = Varr – Vper =
0.6 km/s (unassisted this is 3.7 km/s)
Unladen tug onto ellipse dV = Vper
– Vcirc = 3.1 km/s
Laden tug back onto circular dV = Vper – Vcirc = 3.1 km/s
To these figures one should add some dV budgets for
rendezvous and docking for the tug,
likely near 0.2 km/s for each such maneuver. For departure, there is likely only one such maneuver, conducted unladen, as the tug returns to the appropriate
facility in low orbit. For the arrival
scenario, there are likely two such
maneuvers: one unladen to rendezvous and
dock with the craft in the extended ellipse,
the other laden to rendezvous (and dock) the docked pair with the
appropriate facility in low orbit. This
reflects the decided asymmetry of the events sequences and circumstances, for arrival versus departure.
For the departure scenario,
the tug sees these dV requirements in the listed order:
Laden onto ellipse dV = 3.1
km/s (this is the total, and it is done first!)
Unladen back to circular dV = 3.1 km/s
Unladen rendezvous and dock dV = 0.2 km/s
Total unladen dV
= 3.3 km/s (second)
For the arrival scenario,
the tug sees these dV requirements in the listed order:
Unladen onto ellipse dV = 3.1 km/s
Unladen rendezvous and dock dV = 0.2 km/s
Total unladen dV
= 3.3 km/s (first)
Laden back onto circular dV = 3.1 km/s
Laden rendezvous and dock dV = 0.2 km/s
Total laden dV
= 3.3 km/s (second)
Either way, the
interplanetary craft sees the same departure and arrival dV requirements: some 0.6 km/s for both of these
scenarios. The weight statements for
these cannot be the same, so you cannot
simply sum the dV’s, as neither scenario
(as described) includes operations, any
staging, or any refueling, at the destination. All of those are set by the overall mission
design.
As for the tug, the
laden vs unladen weight statements are drastically different, so you cannot sum the laden and unladen total
dV’s for one simple rocket equation calculation. Those must be two separate but linked rocket
equation sizing calculations. And
the order in which the events occur controls the linkage between the
calculations! That linkage is via
the weight statements.
The asymmetry of the events sequence also shows up in how
many times the craft and the tug must each go around the ellipse.
For the arrival sequence,
the craft will go around the ellipse either 0 (unlikely) or 1 time, before the tug can enter the ellipse with
it. Then the docked pair go around the
ellipse a second time, for either 1 (unlikely)
or 2 (likely) total ellipse circuits on arrival for the craft, and just 1 for the tug.
In contrast, for the
departure sequence, the craft never goes
around the ellipse at all, and the tug
must go around once.
What if the departure/arrival speed requirement is
higher?
A higher departure/arrival speed requirement than the
nominal 11.5 km/s discussed above, just
means the difference between those arrival/departure speeds from 11.5
km/s, simply adds directly to the
interplanetary craft dV requirement.
There is no change to the tug dV requirements, because the actual departure/arrival speed is
not part of its dV requirement calculations.
Those only figure into the interplanetary craft dV requirements.
What would affect both is electing a different extended
ellipse from the 10.9 km/s perigee speed used for this discussion. For that reason, it is recommended that for any given mission
to be designed, first you pick a low
circular orbit and get its speed and period.
You don’t really need escape speed at that altitude, except as a check value (arrival/departure
must be higher than that).
Then pick an extended ellipse such that its perigee is at the
low circular altitude, and its period is
an integer multiple of the circular period.
Then get its perigee and apogee speeds.
I would recommend apogee altitudes almost out to the moon, in order to get perigee speeds up as close as
you can, to escape at perigee
altitude. That is the way to reduce (as
far as is possible) the dV requirement on the interplanetary craft.
Analyze separately what your near-Earth departure/arrival
speed requirement is, for the interplanetary
mission that you want to fly. The
formulas for the dV’s to be figured, are
given in the text right after Figure 2 above.
Final Remarks
The topic here has been propulsive arrivals and
departures, assisted by a space
tug. For arrivals only, there is also the possibility of the craft
aerobraking in a pass down in the atmosphere,
such that it is on an extended ellipse as it leaves the atmosphere. However,
this is not without propulsive burn requirements on the part of the
arriving craft, so it is no
“freebie”.
First, at the ellipse
apogee, the craft will have to make a
small burn posigrade to lift the ellipse perigee up out of the atmosphere and
yet further to circular orbit altitude.
This is to avoid an unintended entry on the next perigee pass.
Second, the craft
will have to make some sort of modest burn on the next raised perigee
pass, to get the “right” apogee, such that the ellipse period is an integer
multiple of low circular orbit period.
Third, bear in mind
that peak heating starts before peak deceleration gees in any sort of
entry, and the gees required here are
significant, meaning the pass has to
go deep in the atmosphere. In turn, that means the peak heating on this type of a
deceleration pass will be of similar magnitude to that of a direct entry! And that means the vehicle must be fairly
compact, have no parallel-mounted
nacelles or other structures, and must
be protected by what amounts to a fully-capable direct entry heat shield. Those are very restrictive design
requirements!
Once these orbital adjustments are made by the arriving
craft, then the tug rendezvous-and-assist
back to circular can proceed, exactly as
described above. (There is no such
aerobraking thing for departure,
though.)
For more details about aerobraking deceleration and elliptic capture, please see my article “Elliptic Capture” on this site, dated 1 October 2024.
Update 12-8-2024:
There’s no real difference between the results for the
presumed data used above in this article,
and more exact data obtained from analyzing the elliptical orbits
problem, in this case automated with the
“orbit basics.xlsx” spreadsheet that I created for 2-body ellipse problems.
Figure A is an image of a portion of the spreadsheet
results, showing both low circular and a
particular extended ellipse. Figure B
is a not-to-scale sketch of those orbital results, explicitly showing the important velocity
data. Figure C has a table of
computed delta-velocity (dV) data from these results, for an arbitrary set of departure speeds. All figures are at the end of this
update.
The first lesson here is that presumed data in the
article give almost exactly the same dV to get back and forth between the
circular and extended ellipse orbits:
3.103 versus 3.1 km/s. Therefore
the above article’s results are actually quite accurate.
The second lesson here is that it is not very hard to
determine an extended ellipse whose period is an integer multiple of the period
of the low circular orbit. Iterative
apogee distance inputs are easy with the spreadsheet. One can make this as accurate as one
desires. To 4 or 5 significant figures, that period ratio as an approximation to an
integer, should be more than accurate
enough to make rendezvous much easier. Clearly, there are other similar solutions at
144:1, 146:1, and so forth.
Take your pick, just get the
period ratio as close to an integer as you can.
The third lesson here is dramatically pointed out by
the third figure: the dV between
circular and elliptic is always constant,
regardless of the hyperbolic departure velocity needed by the interplanetary
craft. That is because neither the
circular nor the elliptical orbit are changing,
only the required hyperbolic departure speed. The dV to be supplied by the craft is low if
the required departure speed is lower. If
that departure speed is higher, that craft
dV then gets larger. The difference in departure
speeds simply adds to the craft’s dV requirement, not the tug’s.
All that really means is that tug assist is more
valuable for min energy Hohmann travel, and
a bit less valuable for higher-energy interplanetary trajectories. But, if you try to obtain more of the total dV from
the tug, you almost immediately push
it beyond escape speed, where it is
lost in space instead of returning via the ellipse to be reusable! This fourth important lesson shows up,
in the small difference between ellipse
perigee speed and local escape speed at that altitude. For these data as illustrated here, that difference is only 0.1 km/s = 100 m/s!
It is only for purposes of lunar missions that the tug can
supply essentially all of the departure speed requirement, precisely because the required departure
speed is just barely less than escape speed!
That puts the craft at essentially near-zero speed at that ellipse
apogee, very near the distance of the
moon’s orbit. The moon essentially “runs
over” the craft from behind at its orbital speed, which is very nearly 1 km/s. The gravitational interaction from there is a
3-body problem (Earth, moon, and craft),
which in Apollo’s case led to a “figure-8” trajectory into a retrograde
low lunar orbit, with the circular entry
burn occurring behind the moon as viewed from Earth.
3-body problems cannot be done pencil-and-paper, or with pencil-and-paper calculations
automated in a spreadsheet. They are
done only with finite-difference computer programs. That 3-body figure-8 trajectory was
essentially the only real computer analysis done during the Apollo
program. The rest was mostly slide rule (or
desktop calculator) work. Pencil-and-paper stuff!
Figure A – Two Results Excerpts From the Spreadsheet, for Circular and Extended Ellipse
Figure B – Sketch Showing the Spreadsheet-Calculated Orbits
and the Relevant Velocity Data
Figure C – Computed Delta-Velocity (dV) Data
Update 12-15-2024: this update explores the relative merit of
various possible tug propulsion systems for the tug-assisted departure
scenario only. The orbital scenario
is 300 km altitude eastward circular low Earth orbit (LEO), with an extended departure ellipse. That departure ellipse has the same perigee
altitude of 300 km, and apogee radial
distance near 361,000 km, for a period
near 5 days. Its perigee speed is near
Earth escape speed at that altitude.
I looked at the storable propellant combination nitrogen
tetroxide (NTO) and monomethyl hydrazine (MMH),
at liquid oxygen (LOX) and liquid methane (LCH4), LOX and liquid hydrogen (LH2), and at a version of the 1970’s-vintage NERVA
nuclear thermal rocket that used LH2. I
looked at 3 chemical engines, of 6.5
metric tons-force thrust each, on the
chemical rocket tug stages, with 3
burning laden and only 1 burning unladen,
and a stage inert fraction (laden) of 5% on all 3 versions. For the nuclear stage, I looked at only 1 engine, of some 13 metric tons-force thrust, and a laden stage inert fraction of 15%, reflecting the far lower engine thrust/weight
ratio of the nuclear engine.
I found all 4 possibilities to be feasible, with the desirability ranking in the same
order as specific impulse (Isp). That is
because the one-way delta-vee (dV) velocity requirements were at or below the
effective exhaust velocity values for these various engines. The best are the LOX-LH2 chemical, and the LH2-based nuclear. Using nuclear in Earth orbit raises all sorts
of technical safety and political issues,
that the chemical stage propulsions do not share. But, “you
get what you pay for”: take the nuclear
risks, and get the superior nuclear
performance.
In any event, this
kind of tug needs a “stage life” equal to,
or preferably significantly exceeding,
the elliptical orbital period of about 5 days. We simply cannot allow cryogenic propellant
evaporation to over-pressure tug stage tankage on that time scale. The storable option is not afflicted by this
design constraint, although it has a
propellant freezing constraint when shadowed.
The choice of payload mass to be pushed to near-escape
speeds by the tug was arbitrary: 10
metric tons. Its form is irrelevant in
these calculations. This could be almost
anything assembled by docking modules together (and fueling them) in LEO. That presumes an appropriate assembly and
refueling facility exists in LEO! That
is the critical enabling factor to make tug-assisted interplanetary departures
a feasible and practical thing to attempt!
The benefits of doing so are self-evidently obvious (greatly-reduced dV
requirements on the interplanetary vehicle design)!
I had created a custom spreadsheet file named “space
tugs.xlsx” to look at these issues in a convenient way, using the rocket equation with vehicle
thrust/weight acceleration constraints.
For this update, I added a
worksheet named “compare dep” to specifically examine elliptic departure onto
interplanetary trajectories, with tugs
using a variety of propellant and engine selections. These results are approximate, but very definitely well “inside the
ballpark” of what is achievable.
Figure I shows the spreadsheet image of the results I
got, for a storables tug. Figure II shows what I obtained for a
LOX-methane tug. Figure III shows
the results for a LOX-LH2 tug. And for
completeness, I examined a nuclear
thermal tug using LH2, based on the
1970’s-vintage NERVA results. Those are
given in Figure IV. Finally, there is a summary figure showing the salient
results for direct comparison; that is Figure
V. All figures are at the end of
this update. The best two are the
LOX-LH2 and the nuclear, but all are
quite feasible for interplanetary elliptic departures.
Figure I – Chemical Tug NTO-MMH Option
Figure II – Chemical Tug LOX-LCH4 Option
Figure III – Chemical Tug LOX-LH2 Option
Figure IV – NERVA LH2 Nuclear Thermal Tug Option
Figure V – Propellant Option Comparison (Departures Only)
Update 12-17-2024: I added an analysis extension to the “compare
dep” worksheet of the “space tug stuff.xlsx” spreadsheet file, that I have been using for these space tug
studies. That extension takes the
as-sized tug intended for hyperbolic departure missions, and simply uses it for hyperbolic arrival
missions, at full propellant load and a reduced
arrival payload. Hyperbolic arrival
increases the velocity requirement slightly because of a rendezvous budget not
needed for departures, but more
importantly it uses more propellant in the unladen phase of flight because that
occurs first, not last, as in the departure scenario.
I ran the same 4 propellant combinations, as for the departure propellant study in the
second update. Those are NTO-MMH
storables, LOX-LCH4 and LOX-LH2
cryogenics, and a NERVA-type nuclear
thermal option that uses cryogenic LH2.
I kept the same number and choices of engines, which tended to under-accelerate the unladen chemical
tugs at full propellant load. Going to 2
engines instead of only 1 would alleviate that,
getting a more efficient impulsive burn.
However, I did not specifically
analyze that issue in this study. It
did not affect the calculated arrival payload outcomes, which do presume an
efficient impulsive burn.
Images of the added arrival analysis blocks, directly from the spreadsheet worksheet, are included in Figures X and Y below.
The first one has the images for NTO-MMH
and LOX-LCH4. The second one has the
images for LOX-LH2 and NERVA. These are
provided to show the same full detail, matching that provided in update 2 above. However,
it is the summary in Figure Z below that provides the real results
with full clarity.
The payload fraction is deceptive: it reflects a reduced payload relative to a
reduced ignition mass (in turn because of the reduced payload). The real tale is told by the
propellant/payload ratio, which reflects
the critical refueling logistics that are likely to control decision-making. If that is more than about 3 or 4 to 1, the propellant combination is probably becoming
rather unattractive, in terms of those propellant
supplying effort logistics.
I did not include the ratio of arrival to departure payload
capability, but since all the departure
payloads were the arbitrary 10 metric tons,
just divide the arrival payloads by 10 to get that ratio. If that ratio gets much below 0.3 to 0.4, the attractiveness is also reduced, unless circumstances should prove to be that
arriving craft are just inherently very much lower in mass.
Overall, including
the departure studies in the second update,
any of the four propellant combinations studied could serve as the basis
for a departure tug design. The best two
by far were the LOX-LH2 chemical and the LH2-based nuclear, but the propellant/payload ratios were low
enough that any of the four combinations could usefully serve. That’s not so true for arrivals: the best two again are the LOX-LH2 and
nuclear, but the propellant/payload
ratios are starting to look rather unattractive for the NTO-MMH and even the
LOX-LCH4. Arrival is simply the
more-demanding mission, by a wide margin, because the unladen burns come first.
It would be easy to point at an inherent characteristic of
the rocket equation, as to why this
outcome obtains, if the velocity
requirements could be summed into a single burn for these scenarios. They cannot be, because the laden and unladen weight
statements are so different!
Nevertheless, that fundamental
concept still applies: there is a knee
in the curve of deliverable rocket delta-vee versus added propellant mass, that is not so apparent if you just look at
mass ratio versus added propellant. (One
is adding tankage mass to the inert mass in this “adding-propellant”
process, and also holding payload mass
constant.) Where the delta vee produced is
smaller than, to about the same size as,
about 1.5 times the effective exhaust
velocity, adding propellant has a rapid
effect to increase delta-vee capability.
Where the delta vee is larger than about 1.5 times the exhaust velocity, adding a lot of propellant has little effect
on delta-vee capability. The curve has
flattened! See Figure W below.
Key here is that effective exhaust velocity is proportional
to specific impulse (mostly set by the propellant choice, and somewhat by the engine technology used), while the mission itself sets the delta-vee
required. The more-demanding the
mission, the harder that pushes you to
select the higher specific impulse propulsion.
Simple as that!
Propellant/payload reflects that truth fairly reliably, for any sort of tug stage scenario.
We could get around that difficulty by the long-established
approach of staging, but that makes
full reusability very difficult to achieve!
For a fully-reusable tug-assist system,
we really need the tugs to be one-stage.
That is what is required for a long-term sustainable system to mount
interplanetary missions. That
sustainability is pretty much going to be driven by the logistics of supplying
the propellants on-orbit to the tugs, and for the interplanetary vehicles that they
push as payloads. Simple as that!
Figure X – Spreadsheet Images for NTO-MMH and LOX-LCH4 Tugs
Used for Arrivals
Figure Y – Spreadsheet Images for LOX-LH2 and NERVA Tugs
Used for Arrivals
Figure Z – Retrievable Payload Reduces When Departure-Sized
Tugs Are Used For Arrivals
Figure W – “Knee-in-the-Curve” Effects with the Rocket Equation