For any given vehicle design, what one assumes for mission delta-vees, vehicle weight statements, course corrections, and landing burn requirements greatly affects
the payload that can be carried. The
effect is exponential: variation in
required mass ratio with changes in delta-vee and exhaust velocity.
This analysis looks at trips from low Earth orbit to direct
entry at Mars, and for the return, a direct launch from Mars to a direct entry
at Earth. The scope is min-energy
Hohmann transfer plus 3 faster trajectories (see ref. 1).
The vehicle under analysis is the 2019 version of the Spacex
“Starship” design, as described in ref.
2. The most significant items about that
vehicle model are the inert mass and the maximum propellant load. For this study, the vehicle is presumed fully loaded with
propellant at Earth departure, and at
Mars departure. See also Figure 1. Evaporative losses are ignored.
Since a prototype has yet to fly, the design target inert mass of 120 metric
tons is presumed as baseline.
Uncertainty demands that inert mass growth be investigated. To that end,
the average of that design target and the 200 metric ton inert mass of
the so-called “Mark 1 prototype” (that average is some 160 metric tons) is used
to explore that effect.
As currently proposed,
the vehicle has six engines.
Three are the sea level version of the “Raptor” engine design, and the other three are vacuum versions of
the same engine design (basically just a larger expansion bell). I have already reverse-engineered
fairly-realistic performance for these in ref. 3. Because of the smaller bells, the sea level engines gimbal significantly, while the vacuum engines cannot. Thus it is the sea level engines that
must be used to land on Mars as well as Earth:
gimballing is required for vehicle attitude control.
Analysis Process
As shown in Figure 2,
the analysis process is not a simple single-operation calculation. The vehicle model provides a weight statement
and engine performance. The mission has
delta-vee requirements for departure,
course correction, and
landing, which must be appropriated
factored (in order to get mass ratio-effective values). There are two sets of analysis: the outbound leg from Earth to Mars, and the return leg from Mars to Earth.
Each leg analyzes 3 burns.
Earth departure, and course
correction are done with the vacuum “Raptor” engines, while the landing on Mars is done with the
sea level “Raptors” to obtain the necessary gimballing. Mars departure and course correction are done
with the vacuum “Raptor” engines (Mars atmospheric pressure is essentially
vacuum). The Earth landing is done with
the sea level “Raptors” to get the gimballing and to get the atmospheric
backpressure capability.
This analysis is best done in a spreadsheet, which then responds instantly to changes in one
of the constants (like an inert mass or a delta-vee). That is what I did here.
Referring again to Figure 2,
for each burn, there is an
appropriate vehicle ignition mass. At
departure, it is the ignition mass from
the weight statement. For each subsequent
burn, it is the previous burn’s burnout
mass. Each burn’s burnout mass is its
ignition mass divided by the required mass ratio for that burn, in turn figured from that burn’s delta vee
and the appropriate exhaust velocity.
For each burn, the
change in vehicle mass from ignition to burnout is the propellant mass used for
that burn. For the first burn, the propellant remaining (after the burn) is
the initial propellant load minus the propellant mass used for that burn. For the subsequent burns, propellant remaining is the previous value of
propellant remaining, minus the
propellant used for that burn.
After the final burn,
the propellant remaining cannot be a negative number! If it is,
one reduces the payload number originally input, and does all the calculations again. If this done in a spreadsheet, this update is automatic! Ideally,
the propellant remaining should be exactly zero, but for estimating purposes here, a small positive fraction of a ton (out of 1200
tons) is “close enough”.
Thus it is payload that is determined in this
analysis. This particular input
(payload mass) is revised iteratively until the final burn’s remaining-propellant
estimate is essentially zero. That is
the maximum payload value feasible for the mission case.
Orbits and the Associated Delta Vees
As indicated in ref. 2,
I have looked at a Hohmann min energy transfer orbit, and 3 faster transfers with shorter flight
times. All of these are transfer
ellipses with their perihelions located at Earth’s orbit. For Hohmann transfer, the apohelion is at Mars’s orbit. For the faster transfers, apohelion is increasingly far beyond Mars’s
orbit. Why this is so is explained in
the reference. See Figures 3 and 4.
Note that the overall period of the transfer orbit is
important for abort purposes. If the
period is an exact integer multiple of one Earth year, then Earth will be at the orbit perihelion
point simultaneously with anything traveling along that entire transfer
orbit. This offers the possibility of
aborting the direct entry and descent at Mars,
if conditions happen to be bad when the encounter happens. Otherwise,
the spacecraft is committed to entry and descent, no matter what.
The cases examined in ref. 1 were all computed for Earth and
Mars at their average distances from the sun.
The larger transfer ellipse with the longer period occurs when both
Earth and Mars are at their farthest distances from the sun. This leads to larger delta vees to reach
transfer perihelion velocity for the trip to Mars, and larger velocity on the transfer orbit for
the trip back to Earth.
Ref 1 has the required velocities and delta-vees, but the most pertinent data are repeated
here:
Transfer E.depdV, km/s trip time, days M. Vint, km/s
Hohmann 3.659 259 5.69
2-yr abort 4.347 128 7.40
No abort 4.859 110 7.36
3-yr abort 5.223 102 6.53
Transfer M.depdV, km/s trip time, days E.Vint, km/s
Hohmann 5.800 259 11.57
2-yr abort 7.548 128 12.26
No abort 7.509 110 12.77
3-yr abort 6.653 102 13.14
I did not examine the worst cases for all the transfer
orbits in ref. 1, but I do have the increase in perihelion velocity for the worst
case Earth departure on a Hohmann transfer for Mars: 0.20 km/s higher than average. I also have the increase in apohelion
velocity for the worst case Mars departure on a Hohmann transfer for
Earth: 0.16 km/s higher than average.
I cheated here: I
used those worst-case Hohmann increases for all the faster trajectories as
well. That’s not “right”, but it should be close enough to see the
relative size of the effect of worst case over average conditions. I also used the same additive changes on the
entry velocities.
Because of the precision trajectory requirements for direct
entry while moving above planetary escape speed, some sort of course correction burn or
burns will simply be required.
With this kind of analysis, I
have no way to evaluate that need.
So I just guessed: 0.5 km/s
delta-vee capability in terms of propellant reserves.
Because this is just a guess, I did not run any sensitivity analysis on
it. However, the delta-vee budget proposed here is factor
2.5 larger than the difference average-to-worst-case for the trip to Mars, which suggests it is “plenty”. It is about factor 3 larger than the
difference average-to-worst-case for the return trip to Earth. You can get a qualitative sense of this
effect from examining that average-vs-worst case effect.
Propellant Budgets for Direct Landings
With this vehicle (or just about any other vehicle), entry must be made at a shallow angle
relative to local horizontal. Down lift
is required to avoid bouncing off the atmosphere, since entry interface speed Vint
exceeds planetary escape speed. This is
true at both Mars and at Earth. Once
speed has dropped to about orbit speed,
the vehicle must roll to up lift,
to keep the trajectory from too-quickly steepening downward.
The hypersonics end at roughly local Mach 3 speeds, which is around 0.7-1 km/s velocity, near 5 km altitude on Mars, and near 45 km altitude on Earth which has about the same air pressure. Up to that point, entry at Mars and Earth look very much alike, excepting the altitude. After that point they diverge sharply, as illustrated in Figure 5.
The descent and landing at Earth require the ship to
decelerate to transonic speed, then pull
up to a 90-degree angle of attack (AOA, measured
relative to the wind vector). Thus, as the trajectory rapidly steepens to
vertical, the ship executes a broadside
“belly-flop” rather like a skydiver.
At low altitude where the air is much denser, the terminal speed in the “belly-flop” will
be well subsonic. I assumed 0.5
Mach, but that might be a little conservative. This is the point where AOA increases to 180
degrees (tail-first), and the landing
engines get ignited. From there, touchdown is retropropulsive.
The landing on Mars is quite different. The ship comes out of hypersonics very close
to the surface, still at high AOA and
still very supersonic. From there, the ship must pitch to higher AOA and pull
up, actually ascending back toward 5 km
altitude. This ascent is energy
management: speed drops rapidly as
altitude increases. It’s not quite a
“tail slide” maneuver, but it is similar
to one.
At the local peak altitude,
the ship is moving at about local Mach 1, and pitches to tail first attitude, igniting the landing engines. From there,
touchdown is retropropulsive. The
Martian “air” at the surface is very thin indeed, as the figure indicates. It may be that thrust is required to assist
lift toward bending the trajectory upward:
the engines would have to be ignited earlier, and at higher speed, as indicated in the figure. Whether this is necessary is just not yet
known.
The low point preceding the local pull-up is at some
supersonic speed; I just assumed about
local Mach 1.5, as indicated in the
figure. That would correspond to a factor
1.5 larger landing delta-vee requirement,
implying a larger landing propellant budget.
In either case, I also
use an “eyeball” factor of 1.5 upon the kinematic landing delta-vee, to cover gravity loss effects, maneuver requirements, and any hover or near-hover to divert
laterally to avoid obstacles.
So, for purposes of
this sensitivity analysis, the Earth
landing is not of much interest, but the
Mars landing is. The sensitivity
analysis looks at the effects of Mach 1.5-sized vs Mach 1-sized touchdown
delta-vee.
Analysis Results
The scope of the sensitivity analysis is illustrated in Figure
6. As indicated earlier, the orbital delta-vee increases worst-case-vs-average,
for Hohmann transfer, were applied additively to the departure
delta-vees for the faster trajectories.
No attempt was made to vary the course correction budgets. Growth in vehicle design inert mass was
examined. An increase in the Mars touchdown
delta-vee was examined. Nothing else was
considered.
The results start with the worst vs average orbital
delta-vee sensitivity. These
results are given in Figure 7. These are
the plots from the spreadsheet, copied
and pasted into the figure. There are 4
such plots in the figure: the top two
are for the outbound journey Earth to Mars.
The bottom two are for the return journey Mars to Earth. Results for all 4 transfer orbit cases are
shown simultaneously by using trip time as the abscissa.
Each has 4 data points:
these are for the Hohmann transfer at 259 days flight time, the 2-year abort orbit at 128 days, the non-abort orbit at 110 days, and the 3-year abort at 102 days. Be aware that the curves are probably not
really straight between the Hohmann orbit and the 2-year abort orbit. I did not run enough fast transfer cases
in ref. 1 to get a smooth curve here.
The most significant thing in the left hand figure for the
outbound trip is the about-40 ton loss of max payload between average and worst
case for the Hohmann transfer. This is a
lot less than the about-130 ton payload loss using the 2-year abort orbit
instead of Hohmann transfer, or the about-210
ton payload loss for using the 3-year abort orbit.
The average-vs-worst-case deficits are somewhat similar on
the faster orbits. The Mars entry
interface velocity trend in the right-hand figure is obviously very
nonlinear. Yet, all the calculated values fall below the
entry velocity from low Earth orbit (LEO).
Any heat shield capable of serving for return from LEO will serve this
Mars entry purpose, which would be the
governing case if the trip were one-way only.
There’s only a small change in entry speeds for average-vs-worst orbit case
in this estimated analysis.
The return voyage has trends shaped quite differently. For Hohmann transfer, the worst-vs-average payload loss is about 20
tons. The deficits on the faster orbits
should be similar. The deficit for using
the 2-year abort orbit instead of Hohmann is far larger at about 110 tons, and that’s from a small return payload to
begin with.
In the right hand Earth entry interface speed plot, the blue and orange curves in the entry
interface plot fall only slightly apart.
Note that all the entry velocities are much higher than the
just-below-escape speed seen with Apollo returning from the moon. The faster transfer orbits, and even the Hohmann transfer, are substantially more demanding than a lunar
return entry. It is clearly the
direct-entry Earth return that will size the heat shield design!
Figure 7 – Sensitivity to Worst-Case Orbital Distances vs
Averages
The sensitivities to the need for a thrusted pull-up
on Mars are given in Figure 9.
This follows the same format as Figures 7 and 8. Bear in mind that the nominal design lights
the engines for touchdown at about Mach 1 speed. For this analysis, the engines are ignited earlier, at about Mach 1.5 flight speed, to assist lift in pulling up to the Mach 1 “flip”,
to tail-first attitude. That makes the landing delta vee about 1.5
times larger. (Note that each case is also
factored up by 1.5 further, to cover any
maneuver / hover needs for the touchdown.)
What the figure shows is about the same 40-ton payload loss
on the voyage to Mars to cover the increased landing propellant requirement for
the Hohmann transfer. Effects on the
faster transfers are similar. This trend
is comparable to the worst-case orbit losses.
The return payload is entirely unaffected, as the landing occurs prior to refueling and
loading for the trip home.
Both the Earth and Mars entry interface velocities are
unaffected by this Mars thrusted pull-up scenario. The orange and blue curves fall right on top
of each other.
Final Remarks
#1. These
results are only approximate!
Real 3-body orbital analysis, and
real entry-trajectory lifting flight dynamics models, must be used to get better answers. Nevertheless,
the trends are quite clear from this approximate analysis.
#2. Flying on
faster transfer orbits will cost a lot of payload capability, on both the outbound voyage, and the return voyage. This effect is much worse on the return
voyage, where the allowable payload
is just inherently smaller.
#3. The effects of
worst-case orbital positions-relative-to-average, of Mars and Earth, have a significant effect on payload, but it is only half or less the effect of
choosing faster transfer orbits.
#4. The effect of
vehicle inert mass growth from the design target of 120 metric tons to an
arbitrary but realistic 160 metric tons is comparable to the effect of
worst-case vs average orbits on the outbound voyage. However it has catastrophic effects on
the return voyage! This is
enough to prevent faster-than-Hohmann transfers on the voyage home, for this vehicle model.
#5. The effects of
needing a thrusted pull-up for the Mars landing is comparable to the effects of
worst-case orbit distances on the outbound voyage. This has no effects upon the return voyage.
#6. It is the
direct Earth entry velocity that will design the vehicle heat shield for any
vehicle capable of making the return.
This is substantially more challenging than was the return from the
moon. For deliberately-designed one-way
vehicles to Mars, the heat shield design
requirements are comparable to entry from low Earth orbit.
#7. My personal
opinions are that thrusted pull-up will be needed, along with the need to fly when Earth and
Mars orbital distances are worst-case,
plus there will be a little inert mass growth (say by 20 metric tons to
140 metric tons vehicle inert mass). That
kind of thing is the proper design point for this vehicle, not the most rosy projections! Estimated performance data for this design
case (at 140 metric ton inert mass) are in Figure 10 (same basic format as
Figures 7, 8, and 9). Note that two of the faster transfers home
are precluded. The feasible one has a
very small max payload value compared to Hohmann transfer.
Figure 10 – Performance for Worst Orbits, Thrusted Pull-Up, and Some Inert Mass Growth
#8. Bear in mind that
the rather high max allowable payload figures feasible to Mars for
Hohmann transfer are incompatible with what can be aboard “Starship” for launch
to low Earth orbit. The payloads
for the faster transfers to Mars look more like what can be ferried up to
LEO. That suggests that a faster
transfer to Mars is most compatible with the projected “Starship” / “Super
Heavy” system design characteristics, as
these were evaluated in references 2 and 3.
#9. Bear also in
mind that a faster transfer orbit to Mars ought to include abort
capability, in case conditions
at arrival prove too bad to attempt the landing. There is simply not the propellant
available to enter orbit and wait for better conditions. Thus life support supplies must be carried to
last the entire period of the transfer orbit, and a full-capability heat shield for direct
Earth entry must be used.
#10. The fast
transfer home need not be limited by abort capability. It can be a different transfer orbit than the
outbound trip. Surprisingly, the shapes of the plotted curves suggest that
something faster than the “3-year abort” orbit could be used for the return
home.
#11. Given a way to
combine two payloads to LEO into one “Starship” by cargo transfer operations on
orbit, then (and only then) the very
large payloads to Mars indicated for Hohmann transfer become feasible. Like on-orbit cryogenic refueling, this on-orbit cargo transfer capability does
not yet exist, not even as a concept
(on-orbit refueling at least exists as a concept).
References:
#1. G. W. Johnson,
“Interplanetary Trajectories and Requirements”, posted 21 November 2019, this site.
#2. G. W. Johnson,
“Reverse-Engineering the 2019 Version of the Spacex “Starship”/ ”Super
Heavy” Design”, posted 22 October
2019, this site.
#3. G. W. Johnson,
“Reverse-Engineered “Raptor” Engine Performance”, posted 26 September 2019, this site.
Navigating the Site
Use the navigation tool on the left. Click on the year, then on the month, finally on the title.
Alternatively, use
one of the search keywords listed as “labels” (“Mars” or “space program”) at
the bottom of the article to see only those articles bearing the same search
keyword. Then use the navigation tool on
the left, if scrolling down is still too
inconvenient.
Once again, thanks for bringing a dose of reality to these discussions.
ReplyDelete