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

**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.***shown in Figure 1.*
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**

**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**

*Illustrated in Figure 2*__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.

**However,***This is shown in Figure 3.***there is a very serious limitation to how elliptical this LEO can actually be**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.)*, because of the apogee’s proximity to the Van Allen radiation belts, if this is to be used for a crewed mission!*
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

**, which saves propellant departing for the moon. However, this is***reduces departure dV by the same 0.25 km/s*__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***That is inevitable!**__You trade the one for the other__!
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

**. 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.***given in Figure 4*__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.

**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.**

*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.***Bear in mind that the same basic transfer ellipse to the lunar vicinity applies.**

*This is shown in Figure 5.*
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.

**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.***It cannot reach the same LLO orbit that Apollo-Saturn-5 reached in that program, not with two-way capability, the way Apollo did.*__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.

**. This is***There is no other reliable means to enter the halo orbit, and also rendezvous with a pre-existing facility there, simultaneously***.***shown in Figure 6***After that, you**

*The path indicated in Figure 5 is only feasible for the first piece of hardware you send to the “halo” orbit.*__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”**

**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.**

*Shown in Figure 7*__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!

**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.***That makes drastic plane changes quite inexpensive, opening up the entire moon for exploration, quite unlike Apollo!*
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.

**, which with another burn***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***, where***then puts you onto the halo ellipse at the correct time to make station rendezvous at its apolune***in order to actually rendezvous.***any drastic plane change gets affordably made**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 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.**

*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.***Who really knows yet?**

*Is the expanded plane change capability of the “Gateway” orbit really worth that cost?***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.

**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.**

*The tables below the figures contain the dV data for each individual burn.***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.

Figure 1 – The Transfer Ellipse From LEO to the Orbit of the Moon

Figure 4 – Delta-Vees to Cis-Lunar Space

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