## Friday, July 3, 2020

### Cis-Lunar Orbits and Requirements

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.

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

Figure 2 – Using the Transfer Ellipse to Reach the Moon Itself

Figure 3 -- Adjusting Delta-Vee with Departure from Elliptical LEO

Figure 4 – Delta-Vees to Cis-Lunar Space

Figure 5 – Special Case:  First-Time Lunar “Gateway” Extended-Elliptical “Halo” Ellipse About the Moon

Figure 6 – Entry to the Halo Orbit for Rendezvous with a Facility Already There

Figure 7 – Landing on the Moon from the “Halo” Orbit

Table 2 – Mass Ratio-Effective Velocity Requirements for Low Lunar Orbit

Table 3 – Mass Ratio-Effective Velocity Requirements for Direct Landing Upon the Lunar Surface

Table 4 – Mass Ratio-Effective Velocity Requirements for Rendezvous with a “Halo” Orbit Facility

Table 5 – Mass Ratio-Effective Velocity Requirements for Landing from LLO and “Halo”