There are a number of different concepts for space missions
in cis-lunar space. The oldest of these
is Apollo, which departed a low circular
Earth orbit (circ LEO) onto a nominal transfer ellipse to the vicinity of the
moon, and then entered low circular
orbit (LLO) about the moon in a retrograde direction (nominal altitude 60 miles
= 100 km). The three-body mechanics
(Earth, moon and spacecraft) of this process
converted the nominal transfer ellipse into the lopsided figure-eight
trajectory we all remember. The Earth
return was the reverse, excepting for
the free entry into Earth’s atmosphere and parachute ocean landing upon
arrival.
Getting into LEO from the surface of the Earth is a
different problem that is intimately linked with the characteristics of the
design being considered. It is more of
an atmospheric/exoatmospheric flight trajectory analysis, than a simple orbital mechanics analysis. Of particular impact are the staging
velocity, the stage mass ratios and
propulsion characteristics, and any
hardware recovery schemes. That problem
is NOT considered here.
What I do in this article is approximate the three-body
problem of Earth, moon, and spacecraft by simple coupled two-body
problems that each solve as closed-form equations. The three-body problem requires numerical
solution on a computer, and it generates
the figure-eight trajectory, if used to
enter a retrograde lunar orbit. The
two-body Earth-spacecraft problem gives me a spacecraft velocity vector out at
the moon, measured with respect to the
Earth. The two-body Earth-moon problem
gives me a velocity vector for the moon with respect to the Earth. The two-body moon-spacecraft problem gives me
a velocity vector, with respect to the
moon, for the spacecraft in lunar orbit, if applicable.
The appropriate vector sum of these velocity vectors, given a selection of just where and how I
want to approach the moon, gives me the
spacecraft velocity vector with respect to moon, presumed to be “far” from the moon. At the appropriate distance from the
moon, lunar escape velocity is reduced
below its surface value, inversely
proportional to the square root of distance from the moon’s center. The “far” kinetic energy plus the escape velocity
kinetic energy add to equal the “near” kinetic energy, with all the ½-factors dividing out. You solve that for the “near” velocity
magnitude, and its direction comes approximately
from your approach selection.
If you are landing direct,
the “near” velocity is the kinematic velocity you have to “kill”, in
order not to crash. Appropriately
factored-up to cover hover and divert needs,
that is the mass ratio-effective dV required to land. Appropriately factored up for small gravity
losses, that same “near” velocity is the
mass ratio-effective dV needed to escape from the moon onto the Earth return
trajectory. If you are instead going
into orbit, the difference between
“near” velocity and orbit velocity is the kinematic dV required to arrive in
orbit, or depart from orbit. The mass ratio-effective factor for that is
just 1.000.
Basic Lunar Transfer Ellipse From Earth
The basic notions and numbers for a transfer ellipse to the
distance of the moon’s orbit is shown in Figure 1. This would be the path to any space station
or other facility located ahead or behind the moon in its orbit, as well as part of the basis to reach the
moon itself using orbits similar to Apollo.
The variation in the exact numbers is due to the slight eccentricity of
the moon’s orbit about the Earth. Note
the modest velocity of the moon in its orbit about the Earth (roughly 1 km/s). This is due to its great distance from the
Earth. Once circularized into the moon’s
orbit, rendezvous with any sort of facility
ahead of (or behind) the moon is trivial,
as long as the apogee of the transfer ellipse is centered upon it.
Note also the factors quoted for the various burns. These would be the appropriate factors for
combined gravity and drag losses to convert the kinematic dV’s into mass
ratio-effective dV’s for sizing or evaluating vehicles. Using the unfactored kinematic dV’s in the
rocket equation is a serious design mistake! All figures are at the end of this
article.
Approach to the Moon
Illustrated in Figure 2 are some of the
details to reach the moon, whether into
low lunar orbit (LLO) or for a direct landing right off the transfer trajectory. Also illustrated are the details to proceed
from the moon to a location ahead of the moon in its orbit about the
Earth. The values for a location behind
the moon in its orbit about the Earth would be similar. The details of the orbits to rendezvous are not
covered here, and are not trivial, as the period must be made different in order
to rendezvous, then must be made the
same again. These are measured on weeks
to months; such a path is therefore not
recommended.
The figure-eight trajectory into a retrograde orbit about
the moon takes advantage of the vector sum of the moon’s velocity about the Earth, and the LLO velocity about the moon, on the backside of the moon, to reduce the dV into LLO significantly. This figure-eight trajectory is a numerical
solution to the three-body problem on a computer. The calculations in this article are
but approximations that work to get you a good approximation of the “right”
answers.
Note that the same LEO departure dV applies to any of
the destinations, because it is the same
basic transfer ellipse to lunar orbit distance,
regardless.
Elliptic LEO Departure
It s possible to reduce the LEO departure dV somewhat by
switching from a circular LEO to an elliptical LEO with a higher perigee
velocity at the same altitude. This
is shown in Figure 3.
However, there is a very
serious limitation to how elliptical this LEO can actually be, because of the apogee’s proximity to the Van
Allen radiation belts, if this is to be
used for a crewed mission! The
nominal figure for the inner “edge” (not really a sharp boundary) of the
radiation belts is some 900 miles = 1400 km altitude. (The most notable exception to this is the
so-called South Atlantic Anomaly, where
the radiation extends down to typical LEO altitudes.)
Be that as it may,
the nominal max-eccentric elliptical LEO configuration is an ellipse of some
300 x 1400 km altitude. This adds 0.25
km/s perigee velocity to the ellipse, versus the circular orbit at 300 km. That reduces departure dV by the same
0.25 km/s, which saves
propellant departing for the moon.
However, this is not
free! It also adds the very same
0.25 km/s to the dV required of the second stage burn getting into this
elliptic LEO. You trade
the one for the other! That
is inevitable!
Whether you choose this option depends upon which burn is
more critical for your vehicle design,
in all its propulsive detail: getting
to LEO or leaving it.
Trips to Lunar Orbit,
to LLO, or to the Surface
A summary of these mass ratio-effective dV values to reach
LLO or the lunar surface, or a point on
the moon’s orbit ahead (or behind) the moon,
is given in Figure 4.
The factors applied to kinematic dV are 5% gravity loss and 5% drag loss
for Earth ascent, the same values
multiplied by 0.165 gee and 0.00 surface air density ratio for lunar
ascent, no losses at all for on-orbit
operations in vacuum (factor 1.000), and
factor 1.500 applied to landing burn dV’s, to cover anticipated hover and divert budgets.
One should note that lunar landing zones are limited to
those directly underneath the LLO path.
It costs another significant burn, to get a rather limited plane change in
LLO, since the orbital velocity
magnitude is significant. The amount of
this plane-change delta-vee is indicated for a 10 degree plane change in the
figure. It applies both ways: to descent,
and to ascent.
The “Halo” Orbit Concept
There is another different mission concept that requires
evaluation. NASA has finalized its
concept for the “Gateway” space station that is to orbit the moon. That station is to be in a “halo” orbit about
the moon, meaning a very eccentric
elliptical orbit about the moon. Its
radial distance from the center of the moon reportedly varies from 3000 km to
some 70,000 km. This is associated with
a lower perilune velocity than LLO because of the higher perilune altitude, and a very low apolune velocity indeed.
However, the
plane and axis orientation of this orbit is fixed, and the long axis of the halo ellipse is
more-or-less radial to the Earth-moon axis twice a month, with the apolune point facing the Earth once
a month. What that does is offer
a once-a-month geometry for low dV to enter the station’s orbit, at the cost of a very-limited launch window
each month. This is shown in
Figure 5. Bear in mind that the
same basic transfer ellipse to the lunar vicinity applies.
You enter this orbit at min dV when its perilune velocity is
directed opposite that of the moon’s velocity about the Earth. That minimizes the dV required to enter
orbit, to a surprisingly-low value. But this geometry obtains for only 1 or
2 days out of each month. Note
further that unless the halo orbit is oriented radial to the transfer orbit, with its perilune velocity vector in the
transfer plane, this advantage cannot
be had!
This halo orbit,
properly oriented, is a location
that NASA’s Space Launch System (SLS) can reach with its Orion capsule and
service module (or things equally heavy),
in the block 1 configuration for SLS.
It cannot reach the same LLO orbit that Apollo-Saturn-5 reached in
that program, not with two-way
capability, the way Apollo did. This bizarre highly-elliptical orbit for
“Gateway” is so very clearly intended to give SLS/Orion block 1 a destination
that it can actually reach. But you
pay two prices for that: (1)
the excursion to the surface has a higher delta-vee than that from LLO, and (2) you cannot use this very low-cost
orbit entry, once there is a facility in
this orbit to which you must rendezvous!
Otherwise, the
recommended procedure, for the halo
orbit with a pre-existing facility, is to enter the 3000 km radius
lunar orbit, from the transfer ellipse
from Earth. One waits in that 3000 km
circular orbit until the alignment with the facility in the halo orbit is
right, then one does a burn to enter a
3000 x 70,000 km halo ellipse, and then
a final burn at its apolune, to exactly match
the required plane of the orbit with that of the pre-existing facility. There is no other reliable means to
enter the halo orbit, and also
rendezvous with a pre-existing facility there,
simultaneously. This is shown
in Figure 6.
The path indicated in Figure 5 is only feasible for
the first piece of hardware you send to the “halo” orbit. After that,
you must rendezvous with what hardware is already there, which means you must use the approach of
Figure 6. There is no way around that difficulty, without waiting weeks or months in high orbit
about the moon. That is because the
period of the “halo” orbit is a little over a week.
Getting to the Surface from the “Halo”
Shown in Figure 7 is the descent/ascent path
to the lunar surface from the Gateway halo orbit. You first make your plane-change burn at its
apolune, where this costs the least
propellant. You then have to do a dV
burn at its perilune to put you onto a much smaller ellipse that grazes LLO. Then you do another burn to put you into that
LLO. Then you wait for the landing
zone to present itself in the appropriate geometry. From there,
you do a landing burn, which
requires a high factor to cover hover and divert needs. The kinematic landing burn dV is the surface
circular orbit velocity, to cover
potential energy effects as well as the kinetic energy.
The ascent uses the same concept in reverse, except that the required factor for losses is
much lower, than for descent. The rest is the reverse of the descent trip. You must wait in LLO for the destination
facility to be in the right place, so
that its position will coincide with yours,
at the halo orbit apolune. This
is driven by time constraints for a crewed mission, since the period of the halo orbit is about a
week, while the period of the LLO orbit
is only a couple of hours.
There is a distinct advantage to this wildly-eccentric
“Gateway” orbit that staging out of LLO cannot duplicate: its apolune velocity is quite low at 92 m/s! That makes drastic plane changes quite
inexpensive, opening up the entire moon
for exploration, quite unlike Apollo! Of course,
the most drastic plane change of all is 90 degrees. If you budget for that, plus some course correction, plus the orbit changes and landing/takeoff
burns I have identified, then you have
all the burns budgeted for your two-way trip from “Gateway” to anywhere on the
surface, and back.
There is a very good reason for breaking the descent from
the halo orbit into a smaller descent ellipse to LLO, and then staging the landing out of LLO. This occurs during the return trip, when you must rendezvous with the space
station in the halo orbit, restricted to
affordable plane changes at its apolune.
The short period of LLO lets you wait a short time for the “orbits
to be right” to make your burn to the ascent ellipse, which with another burn then puts you
onto the halo ellipse at the correct time to make station rendezvous at its
apolune, where any drastic
plane change gets affordably made in order to actually rendezvous. This
trip is long enough at 3-4 days, as it
is. It would be unwise to
make it any longer by matching orbits involving the higher apolunes, earlier in the process.
The total from “Gateway” to the surface and back, including up to 90-degree plane changes, is 5.50 km/s.
The total from LLO and back is 4.74 km/s a 10 degree plane change. The difference (about 0.76 km/s) is what the
performance shortfall of SLS/Orion block 1 has cost us, by forcing staging out of the halo orbit
instead of LLO. Is the expanded plane
change capability of the “Gateway” orbit really worth that cost? Who really knows yet?
Organizing the Data into Missions
So, the possible
cis-lunar missions are:
Mission A: circular
LEO to lunar distance ahead of (or behind) the moon; the return could be to
LEO (A) or direct entry and landing on Earth (A1). A2 would be departing from elliptical LEO
instead of circular LEO. There is no
landing associated with this. A1+2 is direct landing on Earth plus elliptical
LEO departure.
Mission B: circular
LEO to LLO, plus the LLO-surface landing, with or without any plane changes; return could be to LEO (B) or direct entry and
landing on Earth (B1). B2 would be
departing from elliptical LEO instead of circular LEO. B1+2 is direct landing on Earth plus
elliptical LEO departure.
Mission C: circular
LEO to direct to the lunar surface right off the transfer trajectory; return
could be to LEO (C) or direct entry and landing (C1). C2 would be departing from elliptical LEO
instead of circular LEO. C1+2 is landing
direct on Earth plus elliptical LEO departure.
There is no separate landing associated with this, because the mission is direct landing upon
the moon.
Mission D: circular
LEO to 3000 km circular lunar orbit,
then enter a halo-orbit,
terminating in an apolune plane-change to rendezvous with “Gateway”.
This trip leads to a landing that includes a 90-degree plane change. Return to Earth could be to LEO (D) or direct
entry and landing (D1). D2 would be
departing from elliptical LEO instead of circular LEO. D1+2 includes both direct landing on Earth
and elliptical LEO departure.
The suffix-2 variation for all of these would be departing
from a 300 x 1400 km elliptic LEO, which
subtracts 0.25 km/s each for the LEO departure burn and the LEO arrival
burn, but adds 0.25 km/s to the second
stage burn needed to reach the elliptic LEO.
Factored dV’s for the suffix-1 option of direct entry and
landing on Earth are quite vehicle design-dependent, likely reducing total dV, but at the expense of carrying the entry heat
shield and landing apparatus to the moon and back. An arbitrary 0.2 km/s mass
ratio-effective landing burn dV is included as a ballpark guess for something
under half a Mach number terminal velocity at very low altitude.
Similarly, there are
educated-guess course-correction dV budgets included. These are generally a percent or two of the
largest velocity along the track, but
that is not a strict rule. These are
really just ballpark guesses.
The separate landings are from LLO or from the wide “halo”
orbit, denoted as Landing A and Landing
B.
From LLO with Landing A, there is a plane change burn (max 10
degrees), factored 1.000, and a descent burn, factored 1.500 to cover hover and divert
budgets, and figured from the surface
circular velocity to include potential energy effects. Ascent is the reverse, except the ascent burn factor is only 1.008, reflecting low lunar gravity losses and no
lunar drag losses. It is entirely
possible that factoring the entire kinematic descent delta-vee by 1.50 could
well be overkill. The portion of Earthly
and Martian aerobraking descents that gets factored as the landing burn is far
less than 100%.
From the “halo” orbit,
Landing B has more steps.
The first thing is the plane change (up to 90 degrees) at the “halo”
apolune. At the “halo” perilune, we go directly to a descent ellipse that
takes us to LLO altitude at its perilune.
At that LLO-altitude perilune, we
burn to enter circular LLO, then wait
until the geometry is “right” for the landing.
From there, the landing is
exactly like the LLO-start Landing A,
except that no plane change is required,
that already being done at the start of this journey. This process is
reversed for the ascent, with the
waiting for the Gateway station position to be “right” for rendezvous, being done in LLO with the shorter period (LLO
is just under 2 hours, while “halo” is
just over 7 days).
Using These dV Data
Converting these dV data into vehicle weight statements and
mass ratios requires knowledge of the selection, thrust level,
and specific impulse performance of the propulsion used for each and
every burn. It is different for every
design, and design variation, that you look at. This has to be done burn-by-burn, not from the overall dV total!
Evaluating
the performance of specific vehicles on any of these mission choices would be
the topic of future articles, not this
one. The point here is to list all the mass
ratio-effective dV data for the missions burn-by-burn, as a convenient reference for credible data.
The tables below the figures contain the dV data for each individual burn. Scope is the four mission types, plus the two types of landings. Broken out this way, these become very convenient inputs for the
engineering sizing of multiple vehicle concepts.
References
This article makes use of the basic data that was behind
these two articles, also on this
site: “Interplanetary Trajectories and
Requirements” dated 21 November 2019,
and “Analysis of Space Mission Sensitivity to Assumptions”, dated 2 January 2020. I keep this in a big spreadsheet file.
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