This paper explores and compares two different approaches for
returning crews to the moon, this time
to the poles, instead of being limited
to near-equatorial sites.
About the Author
See Table 1.
The author is qualified by both training and experience to do this kind
of study. More details are provided in Appendix
A, for those interested.
Table
1 – Education and Experience
BS Aerospace
Engineering UT Austin 1972
MS Aerospace
Engineering UT Austin 1974
PhD General Engineering KWU 2000
20 years defense/weapons
research/development/test/engineering
20 years mostly teaching, plus some
civil engineering and aviation work
Now retired
How to Reach Polar-Capable Staging Orbits
We wish to return to moon,
but, with bigger crews and more
cargo, and we wish to visit polar regions
instead of just near-equatorial regions,
as with Apollo. There are
fundamentally two options: staging from
circular low lunar orbit (LLO), or
staging from the Gateway space station located in its “halo” orbit about the
moon. See Table 2.
Table 2 – Objectives
Want to
Return to the Moon, But …
Visit
instead the south polar region
Bigger
crews and payloads
Two
Possible Classes of Mission …
Direct
to LLO like Apollo, but do it polar
Gateway-based
from “halo” orbit (to polar)
Both options require an orbital plane change to reach the lunar
polar regions, since both the transfer
trajectory to the moon, and the Gateway
halo orbit, are fundamentally
Earth-equatorial, which inherently makes
them lunar equatorial. However, in extended elliptic orbits, the apoapsis speed of an extended ellipse is inherently
low, so that plane changes made at
apoapsis cost very little in the way of the velocity-change requirement
dV. For no change in speed, only a change in direction, the plane change dV is computed as:
dV = 2*V*sin(a/2), where a is the angle by which direction is
changed
The Gateway halo orbit is already a very-extended ellipse:
70,000 km by 3000 km, center-to-center (CTC).
It is so greatly extended, that
it is fundamentally unstable long-term.
Making the 90-degree plane change at its apolune results in a very small
plane-change dV indeed. Apollo entered very
low circular LLO directly, the speed of
which would require significant dV for all but a trivial plane change. But, if instead the entry was made to an extended
ellipse (“elliptic capture”), the same
type of low-cost large-angle plane change could be obtained, before circularizing. Nominally,
these are 90 degree plane changes,
for which the dV is just apolune speed multiplied by the square root of
2. See Figure 1.
The “Hill Sphere” stability limit distance (a “fuzzy” value) about the moon is some 60,000 km CTC. Beyond that distance, the gravitational influences of the Earth are getting stronger than that of the moon, making such an orbit about the moon long-term unstable. Staying in such an orbit would require corrective propulsion periodically. The Gateway station in its “halo” orbit suffers this problem. The elliptic-capture modification to the Apollo trajectory selected for this study does not! For more details, see Appendix B.
Figure 1 – Gateway “Halo” and Apollo-with-Elliptic-Capture
Offer Polar Orbits Capability
Landings and Earth Return
From a polar LLO, landing on the desired site is simple: one merely times the deorbit burn to hit the
desired landing site. Ascent is just the
reverse, although the lander weight
statement upon ascent is likely different from that during descent. From the Gateway halo, making a landing is not so simple. There is the plane change at apolune, followed by circularization at perilune. From there the timed deorbit is similar to
the LLO option, except that the orbit
altitude is much higher than that used for Apollo. See Figure 2. For more details, see Appendix C. The net effect is a higher dV than from
LLO/polar.
Bear in mind that returning to Earth is pretty much the
reverse of how one gets to the moon. The
basis is the Apollo transfer trajectory,
which orbital “three-body effects” warp into a lopsided figure-8
shape, resulting in a retrograde pass
behind the moon, at a speed relative to
the moon somewhat higher than lunar escape at that altitude. Entry is possible into a retrograde orbit
about the moon, by making a deceleration
burn. The retrograde direction is no
problem for landings, since the moon’s
rotation rate is so very slow.
There are two possible options upon returning to Earth: (1) direct entry, descent,
and landing, or (2) decelerating
into Earth orbit for recovery and reuse without doing an entry.
The figure also shows departures from Earth, all of those being from 300 km altitude low
Earth orbit (LEO) for this study. LEO is
usually considered to be 300-600 km.
Figure 2 – Making the
Landings, and Returning to Earth
Using the Rocket Equation
Correctly When Weight Statements Change
The way the rocket equation was defined and derived, each calculation is made with a mass ratio that must be associated with the one-and-only weight statement which produces that very mass ratio. The dV’s for multiple burns that occur within that one weight statement may be summed for a single overall calculation, but if the weight statement changes between burns, those dV’s may not be summed!
If there are changes to the itemized buildup in the weight statement, such as offloading payload between burns, or jettisoning inert items between burns, then those mass ratios are simply different. In such cases, one must do separate, but linked, rocket equation calculations, for each different weight statement.
See Figure 3 for guidance on how doing linked calculations relates to the weight statement buildups. More details (including how to iterate) are given in Appendix D.
Figure 3 – Doing Rocket Equation
Calculations Correctly
The other major influence upon the rocket equation is the effective exhaust velocity, defined as thrust divided by propellant massflow rate, a specific impulse. In order to relate to the mass ratios for the stage or vehicle, that propellant massflow rate must be the value actually drawn from tankage, not just the nozzle massflow rate! If the engine cycle involves a dumped bleed flowrate to power the pumps, those two flow rates are not equal! The flow from the tanks is the sum of nozzle and dumped bleed flows!
Vehicle Concepts
To get from LEO to polar region landings on the moon, this study presumes 3 vehicle concepts to be
sized. These are (1) the lander module
(LM), (2) a “main craft” that comprises
a command module (capsule) for the crew,
plus a propulsion and support service module, all together designated CSM, and (3) the departure stage or translunar
injection (TLI) stage. Each is the payload for the next, in the order listed.
The CSM must push the LM to the moon, but not back to Earth. The TLI stage might be expendable, or it might be recoverable into LEO for re-use. For this study, the capsule makes a direct free entry and
landing on Earth, while the service
module is expendable, very much like
Apollo. Also for this study, the LM is single stage, unlike Apollo, and gets left in the polar circular orbit
from which the landings are made. If
reused, it is refueled there in that
circular polar orbit, with propellants brought
from LEO on subsequent visits.
This study presumes that different propellants and inert
mass fractions are used to size these 3 vehicles. The lander and the TLI departure stage are
both presumed to be high-specific impulse (Isp) propellants, specifically oxygen-hydrogen (LOX-LH2). This is necessary because the dV requirements
are substantial, for both.
The main craft (CSM) is presumed to be lower-Isp storable
propellants, since those velocity
requirements are substantially lower than those of the other two vehicles. That makes something like a SpaceX Dragon
capsule, fitted with an oversized trunk
that is fully propulsive, feasible as a possible
candidate hardware item.
Note that the lander payload is different for ascent than it
was for descent. For descent, a 4 metric ton allowance was made for 4 crew
plus all their suits, supplies, and personal equipment, plus another 4 ton allowance for cargo
delivered to, and left upon, the lunar surface. For ascent,
the same 4 ton allowance for crew is used, knowing that return samples effectively replace
the mass of supplies used.
For purposes of this study,
a constant 10 metric ton mass for the crew capsule was used, for a 4-person crew. That really is “Dragon-class”.
The TLI stage has both the CSM and the LM together as its
payload for departure from LEO. It has
no payload at all for a recovery back into LEO upon return, if it is to be reused.
A key to understanding the lander concept illustrated was “rough
field stability”. Min leg span must
significantly exceed center of gravity height.
Minimizing crew ladder height is also essential. The rest of the propulsive items are just
simple cylindrical stages. Tanks are
indicated, as well as guidance and
control (G&C) units, power supplies
for those G&C units, plus power and
life support for the crew as applicable.
Low density propellants increase stage inert fraction. So also does adding the insulation and
equipment to achieve “long” stage life (multiple weeks) with highly-evaporative
cryogenics like hydrogen. Adding landing
legs plus enclosed spaces for crew and cargo further increase inert fraction
for the lander. The extra power and life
support supplies for the service module of the CSM increase its inert
fraction, despite the use of
high-density, storable propellants. See Figure
4.
Figure 4 – Vehicle Concepts,
As Used for this Study
Requirements and Results for Landers
These vary strongly by orbital attitude and mission complexity. Both
descent and ascent dV values (and totals) are shown in Figure 5. Be aware that for rocket equation
analysis, the summed descent and ascent
dV’s cannot be used in a single calculation,
as the weight statements change for ascent vs descent. Note that the general trend is higher dV
requirements as polar circular orbital altitude increases. The “halo” option is further complicated by
the need to even reach such a polar circular orbit from the Gateway station. These dV values include the increases for
losses and budgets for maneuvering.
Figure 5 – Requirements for Landers by Mission
Lander sizing results are depicted in Figure 6. These are single-stage landers, potentially refuellable and reusable, left in the circular polar orbit from which
the final descent was initiated, and to
which the ascent returns.
Note that lander mass about doubles from 100 km circular up to Gateway/halo energy levels. That is the exponential relationship between dV requirements that do not double and the resulting mass ratios demanded of the vehicle. These figures do reflect the different weight statements for ascent vs descent.
Note also that max and min thrust levels were determined from appropriate LM acceleration values at ignition and at landing, plus at takeoff. These are merely appropriate gee levels applied to the corresponding Earth weights in the weight statements.
Figure 6 – Results for Lander by Mission
Requirements and Results for Main Vehicle (CSM) by
Mission
The dV requirements for this vehicle to get into, and back out of, the appropriate mission orbit are summarized
in Figure 7. These include
budgets for course corrections and rendezvous as required by mission.
Note that for the Apollo-type LLO cases, the dV requirements actually decrease a
little with increasing altitude, despite
the elliptic capture and plane change to polar.
This is the only vehicle case where that happened. The rest of the vehicle cases show an
increase in dV with increasing orbit altitude.
The more complex CSM mission is getting all the way to
Gateway station in the halo orbit, from
circular capture at the halo perilune altitude.
Circular capture is required to time entry onto the halo ellipse for
Gateway rendezvous. That drives the CSM
mission dV requirement up greatly,
relative to the other LLO missions.
Figure 7 – Requirements for Main Craft (CSM) by Mission
Main craft (CSM) sizing results are given in Figure 8. These include weight breakouts for the LM and
the service module, as well as the
CSM-LM cluster. The command module
(capsule) mass is a constant 10 metric tons.
The more challenging halo mission doubles CSM+LM cluster mass
over that of the 100 km circular polar mission.
Note that the inert fraction applied to the CSM sizing
analysis is done without including the LM in the ignition mass. That is unique among all these sizing
calculations.
Figure 8 – Results for Main Craft (CSM) by Mission
Requirements and Results for Departure
As shown in Figure 9,
the dV is higher for recovery vs expendable, but it applies to all the mission
concepts. Only the TLI stage payload
mass varies by mission, that being the
CSM+LM cluster by mission. Note that the
recoverable dV is higher not only because of the orbital change dV, but also budgets for course correction and a
second rendezvous in LEO.
Figure 9 – Requirements for Departure (TLI) Stage
The departure (TLI) stage results are given in Figure 10. TLI stage mass almost doubles as the lunar orbit
energy increases from 100 km LLO polar to the Gateway/halo. Adding recovery back into LEO almost
doubles TLI stage mass again.
Figure 10 – Departure Stage Results by Mission, Expendable and Recoverable
Conclusions and Disclaimers
The trends identified by this study are very clear (as
listed in Table 3):
(#1) The dV requirements are mostly
higher for halo vs LLO. That implies
better designs will be had using the modified Apollo polar LLO approaches.
(#2) The resulting halo-based
vehicles ~2x heavier than LLO.
Smaller, lighter vehicles enable
simpler designs, less costly to
build, and less costly to launch.
(#3) The lowest
LLO is the most attractive, but only use
the lowest LLO altitude that is safe!
One does not want to “smack” into a lunar mountain by using too low an
altitude, either directly, or because of small navigation errors that
are simply inherent.
Final Remarks
For those who have difficulty with the notion of doing burns
at min orbital speed to drastically change the plane of an ellipse, see Appendix E.
Table 3 –
Conclusions
Trends
are very clear:
Mission
dV requirements are larger for halo
Halo
propulsion items ~ 2 times heavier
Use
the lowest LLO that is safe
Raising
altitude increases masses
Results
include using elliptic capture to polar
Disclaimers include both the generic Isp values and the
“WAG” estimates of stage/vehicle inert fractions, as listed in Table 4. This study used only generic Isp values, not real engine design performance. It also used “WAG” guesses for vehicle inert
mass fractions. “WAG” literally means
“wild-ass guess”. While the numbers themselves may not be
very good, the trends of those
numbers by mission really are good!
Of these two influences,
inert fractions are actually the stronger influence. But,
they show up indirectly, in the
weight statements associated with the linked rocket equation calculations. That half-hidden nature may be why most
people do not recognize this issue, the
way that they do the explicitly-appearing Isp values.
Generally, this study
used a loaded stage basis to “model” structural. The idea behind that was to provide more
structural mass to push the heavier payloads at acceptable gees, without risking compressive collapse of the
stage. The only exception was the CSM
unit.
Table
4 – Disclaimers
No
specific engine designs
Generic
Isp levels only
Loaded
stage inert fractions = “WAG”
Inert
fraction effect > Isp effect
Used
loaded stage as basis with payload
Structure
to push very large payloads
Overall: go with an elliptic capture-modified
Apollo-to-polar-LLO, at an altitude
between about 100 and 500 km.
Appendix A – Further Details Regarding the Author
Short course materials by the author:
“Orbits+”
courses available via “New Mars” forums
“Interplanetary
transportation” topic, “orbital
mechanics traditional” thread
Links
to free downloads: orbits, rocket performance, and entry
spreadsheets, plus lessons with problems and solutions
Compressible
Flow and Heat Transfer Basics
Contact
author: includes spreadsheets, with problems and solutions
Contact data
Email
preferred: gwj5886@gmail.com
Blog
site: http://exrocketman.blogspot.com
Youtube: search for “exrocketman1”
Also on
LinkedIn
Appendix B – Extra Orbital Calculation Details
The figure shows how the numbers were actually
computed. This was assisted by a
spreadsheet doing classical 2-body elliptical orbit calculations. Note that only data for 100 km LLO is
shown. The others were done by the same
method, just the numbers change. Those were for 500 km and 1000 km altitudes
for the LLO options, and 1261.7 km
altitude (3000 km CTC) for the Gateway/halo option.
Appendix C – Landing Calculation Details
The figure shows that calculations were made with a
surface-grazing ellipse as the transfer from orbital altitude to the
surface. Only the case for 100 km is shown; the others are similar, just different numbers.
The difference between circular and apolune speeds is the magnitude of
the deorbit burn dV for descent, which
is also the magnitude of the circularization burn dV upon ascent.
The perilune speed is the ideal dV to be “killed” for
landing, and to be supplied for starting
the return ascent. That has to be
increased to cover losses, of which
there is no drag loss, and the gravity
loss is quite small.
For descent, the
landing burn dV is factored up by 1.5 to cover the needs for hover and divert
to avoid obstacles on the surface. This
was a serious effect during the Apollo-11 landing, nearly causing a crash or an abort, from landing propellant exhaustion.
For ascent, there is
an added dV budget to cover rendezvous with any craft left in orbit during the
landing. This is “guesstimated” as twice
the magnitude of the deorbit burn.
Appendix D – Linked Rocket Equation Calculations When
Weight Statements Change
The example in the figure is for recovering the reusable
lower stage of a launch rocket, but the
linked process is the same, regardless
of the application.
The dV and Vex values associated with the two different
weight statements are used to compute the resulting mass ratios. There is also a stage or vehicle inert mass
fraction, defined as Winert/Wign. The propellant fractions Wp/Wign are simply 1
– 1/MR, for each mass ratio. The sum of payload, inert,
and propellant fractions must be 1,
by definition.
One must assume an ignition weight for the first burn (or
set of burns) calculation, associated
with the first weight statement. From
it, the inert mass and burnout mass are
easily computed as shown, as is the
propellant mass expended.
As shown, you adjust
the burnout mass from the first calculation to be the ignition mass in the
second calculation, by means of the
differences between the first burn and second burn weight statements. Then you do similar burnout mass and
propellant-expended computations, as
shown.
When the inherently-iterative calculation is fully
converged, the second calculation
burnout mass should equal the first calculation inert mass, with due regard for any non-zero payload in
the second calculation, or any
jettisoned-between-burns inert items.
One must iteratively adjust the first calculation ignition mass
guess, until this convergence is
achieved.
Once convergence is achieved to within a tiny fraction of a
percent, the results can be used to
finalize 3 representative weight statements.
The first is the as-built stage or vehicle weight statement, including all the propellant for both sets of
burns, the initial inert mass, and the initial payload mass.
The second is for the first set of burns, with the propellant for the second set of
burns carried along as if it were additional payload. It actually results in the same design mass
ratio for that first set of burns, and
the same design dV estimate is obtained.
The third is for the second set of burns, where the propellant for the first set of
burns is zeroed. The inert mass has been
adjusted for any items jettisoned between the sets of burns, and the payload has been adjusted to reflect
anything offloaded between the sets of burns.
This weight statement then produces the design mass ratio and design dV
for the second set of burns.
Appendix E --
Making Orbital Plane Changes
The illustration shows exactly how this is done and why it
works. But it is complicated.
At the top is a depiction of an extended elliptical orbit
about some central body (of interest here is the moon as the central
body). The plane of the ellipse is that
of the page. The side moving toward the
central body is at the top of the diagram,
and the side moving away at its bottom.
The author is coasting in that elliptical orbit in his
spaceship, currently located at its far
point from the central body (the “apoapsis”,
or at the moon, the “apolune”) In
top left of the illustration, you are
viewing this ellipse end-on, so that it
appears to you as a vertical line in view A-A,
bottom left. The author’s
spaceship appears to you to be moving upward at its smallest orbital speed V.
Imagine now that author turns his spaceship and faces
precisely the other way, now looking
downward in your view. That’s the next
image to the right of view A-A in the illustration. While facing “downward”, he is still moving “upward” at V, in your view of him.
Now the author makes an engine burn, adding a speed V in that downward
direction, which added to the upward V
he originally had, now makes
precisely zero speed along the orbit!
Because he is no longer moving with respect to the central
body, there is no longer any
vertically-oriented elliptical orbit; it
just goes away. (The lesson
here: kill the speed at distance, you “kill” the elliptical orbit!) The author also now looks stationary in space
to you!
Now look at the second image to the right of view A-A in the
illustration. The author now turns his
spaceship to your view left, which is
perpendicular to the plane of the original elliptical orbit that has just gone
away. He makes another engine burn,
to add the same speed V in that new
direction, while still just as far away
from the central body as he was. He is
now moving with speed V to view left instead of view up (the very same speed
as at the apoapsis of the original ellipse),
and he is still the same distance away from the central body!
That action inherently creates a new ellipse of the
same dimensions as the original, about
that central body. However its
orientation in 3-D space is now horizontal in your view, no longer vertical, because the new speed was created to view
left, not the original view up. (The lesson here: create a speed at a distance from a central
body, and you create an elliptical orbit
about it, corresponding to that speed
and direction at that distance.)
This same thing could be done at the point of closest
approach to the central body (the “periapsis”,
or at the moon, the
“perilune”). It is just that the speed
in the orbit is very much the highest there,
so that the rocket burns would have to create an awful lot of vehicle
speed changes, and it would cost enormous
amounts of propellant. So, plane changes are always done at
apoapsis, where the orbit speed is the
lowest, and the propellant cost is less. Using a very extended ellipse really helps save
propellant, because the apoapsis speeds
are really low!
There is one more propellant savings you can have, if you take advantage of the “magic” of
vector math, where the vectors have both
length and direction, and they literally
add nose to tail graphically. This is
the image to the farthest-right of view A-A in the illustration. The original view-upward V and the desired
view-leftward V form an isosceles (equal-leg) triangle at the apoapsis of the
original ellipse. You simply draw a slanted
line between the tips of the two velocity vectors.
If you make that slanted line into a vector pointing left
and down in your view, that is the speed
vector you must add (by burning your engine) to the original apoapsis speed
vector to simultaneously “kill” your view-upward speed V and replace
it with a view leftward speed of that same V. That one burn is inherently less in magnitude
than just adding the separate burns for two legs, like we just did before.
That is because the “slanty line” connecting the tips of the
legs of that isosceles triangle is always shorter than the sum of the lengths
of its two legs. There is an algebra
formula based on this isosceles triangle geometry that relates your required
burn dV to the orbital speed V and the angle a by which you wish to turn the
plane of the ellipse:
dV = 2 * V * sin(a/2).
For a = 90 degrees,
this simplifies as illustrated to
dV = 1.414*V, where
the 1.414 is really just a 4-digit approximation to the square root of 2.
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