I visited the Spacex website in April 2018 and recovered
from it the revised characteristics of the large BFR Mars vehicle, as presented during 2017. These characteristics are somewhat downsized
from those in the 2016 presentation given in Guadalajara. Thus this article supersedes the previous
article posted on this site 10-23-17,
titled “Reverse Engineering the ITS/Second Stage of the Spacex BFR/ITS
System”. Both articles share the search
keywords “launch”, “Mars”, and “space program”. The older article has been updated to
redirect readers here.

**Data From the Spacex Website**

What is listed includes a basic vehicle diameter of 9 meters
(both first stage booster and second stage spacecraft), a booster length of 58 m, and a spacecraft length of 48 m. The forward half of the crewed spacecraft
includes an unpressurized cargo bay, and
over 825 cubic meters of pressurized space for the crew and passengers. The rest is propellant tankage and
engines.

No data are given for the uncrewed cargo version, but it appears quite similar overall, with a giant clamshell door to open the
entire space corresponding to the crewed vehicle’s unpressurized and
pressurized payload spaces.

For the crewed vehicle,
payload mass is listed as 150 metric tons to low Earth orbit, and the same on to Mars with refilling
on-orbit. Return payload (presumably
from Mars) is listed as 50 tons, with
refilling done on Mars by in-situ propellant manufacture from local ice
deposits and atmospheric carbon dioxide.
The cargo deliverable to orbit by the uncrewed cargo version is also
said to be 150 tons.

The crewed spacecraft is listed as having an 85 ton inert
structure weight. Musk says the data
right now actually say 75 tons, but that
always grows as development proceeds.
Propellant loadout in the tanks is 240 tons of liquid methane, and 860 tons of liquid oxygen (1100
total), both superchilled to a higher
density than in normal usage. The listed
data imply similar numbers for the uncrewed cargo version.

Basic weight statements for the crewed vehicle (implied to
operable either manned or unmanned by the mission planning) would then be as
follows (all masses are metric tons):

__Item__

__at launch__

__ign. in LEO__

__ign. on Mars__

Payload 150 150 50

__Inert 85 85 85__

Dry 235 235 135

__Prop. 1100 1100 1100__

Ign. 1335 1335 1235

There are not enough
data given to establish a weight statement for the first stage booster, because no inert weight was given. Only the gross takeoff mass (for 150 tons of
payload) was given to be 4400 metric tons.
The inert fraction of the crewed spaceship is a bit over 6%. Falcon-9 first stages seem to be just under
5%.

__On the assumption__that the inert mass fraction of the BFR first stage is 5%, given the 1335 ton max weight of the second stage, then the first stage inert mass should be near 220 metric tons. Under that assumption, its weight statement would be approximately:

Payload
(second stage at max) 1335

__Inert 220__

Dry 1555

__Propellant 2845__

Ignition 4400

The first stage is listed as having some 31 sea level Raptor
engines. The second stage spacecraft is
listed as having 2 sea level Raptor engines of high gimballing capacity, and 4 vacuum Raptor engines of limited
gimballing capacity. The two versions differ
only in the expansion bell size, very
much like the Merlin engines used on the Falcon 9 and Heavy.

Design chamber pressure is listed as 250 bar, with growth eventually anticipated to 300
bar. The engines are said to be
throttleable from 20% to 100% of rated thrust.
The given specific impulses (Isp) apply to 100% thrust at 250 bar, and should decrease slightly as thrust is
throttled down. No data are given for
that Isp reduction. It is a relatively
minor effect at this level of analysis,
and is ignored in my reverse-engineering analysis here.

Data as listed for the individual Raptor engines are as
follows. The sea level design data as
presented do not report vacuum thrust,
but an exit diameter is given, so
that it may be calculated using sea level air pressure of 101.325 KPa. I did this in my reverse-engineering
analysis. Only vacuum performance is
listed for the vacuum design. Listed
data are for one engine.

The data on the site claim a vacuum engine exit diameter too
big to fit a 9 m diameter vehicle; that
has to be a typographical error! Because
the vacuum design is never operated with backpressure (excepting 6 mbar on
Mars, essentially negligible), the lack of a reliable vacuum exit diameter
does not impact my estimates.

__Version__

__sea level__

__vacuum__

Vacuum Isp,
s 356 375

Vacuum
thrust, KN not given 1900

Sea level
Isp, s 330 not
given

Sea level
thrust, KN 1700 not given

Exit dia, m 1.3 12.4
(typo !!!)

The presentation as given on the Spacex website does show
the reentry sequence as currently envisioned at Mars. This shows direct entry from an
interplanetary trajectory, with an entry
interface speed up to about 7.5 km/s.
Initially the vehicle is flown inverted for downlift, presumably to keep it from “bouncing off” the
Martian atmosphere while traveling faster than Martian escape speed. Later in the entry trajectory, the vehicle rolls upright for uplift, in
order to shape the trajectory.

By the time the hypersonics are ending at about 0.7 km/s
(local Mach 3) speeds, the vehicle is
within 4-5 km of the surface, and
climbing upward towards about 10 km altitude.
It reverses to tail first for the retropropulsive burn, all the way to landing. The presentation claims max 5 gees
deceleration during entry sequence while hypersonic.

No data is given at all for the return entry at Earth. Presumably,
this would also be direct from the interplanetary trajectory, with entry interface speeds somewhere in the
vicinity of 17 km/s. This will be a very
high-gee entry, as return from the moon
with Apollo was 11 gees at 11 km/s entry interface speed. 12-15 gees (or more) is nothing but a gut-feel
guess.

The data on the site show 4 landing legs whose span is 3 or
4 times less than the length of the crewed second stage vehicle. Cargo is lowered to the ground from the cargo
bay door with a crane. People would
leave the vehicle by the same means, and
reverse the crane to return to it.

**Organizing the Analysis**

The Spacex website presentation claims refilling on orbit
from uncrewed tanker vehicles. The
illustrations show variously 4 to 5 such tankers, but no data are given. Propellant transfer is by tail-to-tail
docking, with thruster-induced
microgravity driving propellant flow from one vehicle to the other. It is unclear whether the tanker is an
unmanned crewed vehicle or the cargo vehicle,
and it is unclear whether propellant is loaded as cargo, or is leftover in the tanks after flying to
orbit with no payload.

The tanker performance problem is

__not__analyzed here.
The analyses here were made using simple rocket equation
estimates modified with realistic “jigger factors” for gravity and drag losses. Because of the long times in interplanetary
flight with cryogenic propellants, other
“jigger factors” get included to account for boiloff effects and midcourse
correction budgets. And because
retropropulsive landings must have a margin to adjust touchdown if obstacles
are encountered, yet another “jigger
factor” must be included to model the need to hover and/or redirect touchdown
laterally some distance away.

For Earth launch with slender, “clean” vehicle shapes, something like 2.5% gravity loss and 2.5%
drag loss provide realistic first estimates.
The two add to 5%, which gets
incorporated into a “jigger factor” of 1.05,
that multiplies the kinematic delta-vee requirement, making it suitably larger to cover gravity
and drag losses in the simple rocket equation.
At Mars, gravity is weaker and
the air much thinner, so I reduce the
gravity percentage by a factor of 0.384 (Mars surface gravity in gees), and the drag percentage by a factor of 0.007
(Mars surface density ratioed to Earth standard). In retropropulsion, drag helps rather than hinders, so the two percentages subtract instead of
adding.

I assumed a factor of 1.1 increase on the already “jigger-factored”
propellant weight (out of the rocket equation for that burn) to include a “kitty”
for minor midcourse corrections, and the
same for long-term propellant boiloff effects in transit. I also assumed a hover/redirect factor of 1.5, to multiply the same already-factored propellant
weight, during landing burns.

These calculations can be made by hand, but are very conveniently entered into a
spreadsheet, for rapid changes and
refinements. That is what I did for this
analysis. Its results were incorporated
into a series of figures.

**First Stage Problem**

The first stage problem has but one payload, a weight statement based on the 5% inert
fraction assumption, and suitable sea
level and vacuum engine performance data for the sea level Raptor engine design. The website presentation does not give any
indication of the staging speed or altitude,
only depictions showing the trajectory bent over almost level by the
time staging occurs.

This is complicated by the need to fly the booster back to
launch site, and land it there, very much like the Falcon-9 first stages. Therefore,
beyond just reaching staging speed against realistic gravity and drag
losses, enough propellant must remain on
board after staging, for the stage (with
no payload) to more-than-“kill” its flight velocity with a boostback burn, ease the entry heating with a
shock-penetrating entry burn, and conduct
the final touchdown burn.

I made the “reasonable” guesses of allowing 0.99 km/s worth
of delta-vee for the touchdown burn, an
arbitrary 0.10 km/s delta vee for the entry burn, and a boostback delta-vee equal to, or slightly exceeding, the speed at staging. I assumed staging altitude to be essentially
in vacuum, so that the average of the
sea level and vacuum Isp’s could be used to represent average booster
performance.

Staging speed was an input assumed value, since it was not specified in the
presentations on the website. I
iteratively modified this value until the 3 flyback burns all had reasonable
delta-vee values. The boostback kinematic
delta-vee is the staging speed. It is
figured for the ascent weight statement, “jiggered” by factor 1.05, and then plugged into the rocket equation for
a mass ratio less than what the stage provides overall. Dividing launch mass by that ratio gets the
burnout mass, and the difference is the ascent
boost propellant mass burned to reach staging speed.

Deleting the payload mass (second stage) gives a new weight
statement for booster inert and the mass of propellant still on board. It corresponds to a certain delta-vee from
the rocket equation, “jigger-factored”
down by 1.025, since there is no drag
outside the atmosphere. Subtracting 0.99
km/s for the touchdown burn, and 0.10
km/s for the entry burn, leaves the
delta-vee benefit available for the boostback burn. I adjusted my assumed staging speed until my
boostback delta-vee equaled or slightly exceeded this staging speed value.

At launch, the sea
level thrust of the 31 sea level Raptor engines totals to 5375 metric
tons-force (force in KN/9.805). The
website data lists 5400 tons of thrust,
close enough. The thrust/weight
ratio minus unity gives the kinematic acceleration straight upwards, in gees.
At staging, one must correct to
vacuum thrust of the sea level engines,
and since the trajectory is nearly level, the thrust/weight ratio is the pathwise
kinematic acceleration in gees. These
values can be factored by the throttle percentage expressed as a fraction, if needed.
I bounded the burnout accelerations by calculating values at 100% and
20% thrust.

Those results are summarized in Figure 1.

Figure 1 – Results for the Booster Problem with Flyback Recovery

**Going to Mars After Refilling in Earth Orbit**

This is not quite straightforward, because you cannot use all your propellant to
go to Mars. You must have enough
propellant still on board after departure,
to enable the landing on Mars. That
means you analyze the Mars landing first,
and then the departure burn from Earth orbit.

You have to analyze from dry tanks at touchdown on Mars back
to the landing burn ignition conditions,
complete with Mars retropropulsion gravity/drag effects, for the min propellant needed to land. Then you scale that amount of propellant up
with the hover/redirect factor on your delta-vee for a realistic propellant
budget to land from the rocket equation.

Then you scale that realistic landing budget up with the
boiloff factor and the midcourse factor to find the actual reserve propellant
that you must still have on board after the departure burn. That is what sets your weight statement for
the departure burn.

These results for the Mars landing are given in Figure 2.

The departure burn from Earth orbit sees an Earth gravity
loss, but no drag loss. The usable delta-vee from it adds to the
Earth orbit velocity for a velocity Vdep at that Earth orbit location. A little farther from Earth, you figure a velocity-at-infinity as Vinf =
(Vdep^2 + Vesc^2)^0.5. Then you add
Earth’s orbital speed to that value for the vehicle speed with respect to the
sun Vwrt sun.

For a min energy Hohmann-type transfer ellipse orbit there
is a perigee velocity, which varies with
planetary positioning along their ellipses.
I used the worst case (highest) value.
The vehicle velocity with respect to the sun Vwrt sun, minus the Hohmann transfer perigee speed
Vper-HOH, is the margin you have, that might be used to fly a faster
trajectory. Hohmann transfer is about
8.5 months one way.

These results for the departure from Earth orbit are given
in Figure 3.

Figure 3 – Results for the Analysis of Departure from Earth
Orbit

**Returning From Mars**

Very similarly to the trip outbound to Mars, one must analyze first the landing on
Earth, to define the propellant needed
to land there, jigger that up for the
transit, and have that quantity in
reserve after the Mars departure burn. So
you analyze the Earth landing first,
then the departure from Mars,
which is a direct ascent into the interplanetary trajectory.

I used the same basic end-of-hypersonics at 0.7 km/s (local
Mach 3) as my kinematic definition of min landing delta-vee. This just happens at higher altitude, and in much thicker air, on Earth.
Unlike Mars, it is quite feasible
to fall a long way in the transonic/low-supersonic flight speed range, before reversing vehicle orientation to tail-first
for the touchdown burn.

You “jigger-up” this min delta vee by a gravity-drag factor (near
1 in retropropulsion on Earth) and by the hover/redirect factor (I used the
same factor 1.5 for this). That much
larger delta-vee goes through the rocket equation for a mass ratio from dry
tanks at touchdown. This leads to a
realistic landing propellant budget. Then
you factor that up for boiloff and midcourse,
to find the reserve propellant that must be on board after the Mars
departure burn.

Figure 4 – Results for the Earth Landing Analysis

This Earth landing propellant budget at Mars departure then
adjusts the weight statement of the refilled craft upon Mars, in addition to the stated reduction in return
payload. This is a direct
departure: the weight statement and
choice of engines finds the ideal delta-vee,
adjusted downward by the Mars gravity-drag factor to a realistic
delta-vee. We are not stopping in Mars
orbit, this realistic delta-vee becomes
the speed of the vehicle near Mars. It
is adjusted using Mars escape velocity to find the Vinf in the vicinity of
Mars. That is in turn subtracted from
Mars’s orbital velocity to find the velocity with respect to the sun. This is compared to the Hohmann min energy
transfer orbit’s apogee velocity, to
determine any margin available for a faster trip home.

These numbers indicate little or no potential for a fast
return home. The results are given in
Figure 5.

Figure 5 – Results of the Mars Departure Analysis

**Conclusions**

Using Spacex’s own data
plus some reasonable assumptions regarding gravity and drag losses, and hover requirements for retropropulsive
landings, and for boiloff and midcourse
budgets, I calculated performances
estimates, for the big Spacex Mars
vehicle as presented in 2017, that are not very far at all from what is claimed
in their 2017 presentation.

I show some potential for a slightly-faster trajectory to
Mars than min energy Hohmann transfer. I
show very little, essentially zero, potential for a faster return trajectory from
Mars, compared to min energy Hohmann
transfer.

To get these data, I
had to assume that the inert mass fraction of the BFR first stage is 5%, and I had to assume that the BFR stages at
just about 2.55 km/s flight speed,
outside the sensible atmosphere,
and already almost level.

I used the reported engine performance data for the sea
level and vacuum forms of the Raptor engine,
operating at 250 bar chamber pressure at full thrust. If the chamber pressure can be raised closer
to 300 bar (as Spacex wants), some of
the faster-trip performance shortfalls ease.
I did not analyze these data to quantify that effect.

Gravity and drag loss effects upon ideal rocket equation
delta vee are assumed at 2.5% each here on Earth, and ratioed down by 0.384 for gravity, and 0.007 for drag, at Mars.
Propellant quantities coming from the rocket equation mass ratio and
appropriate weight statements get ratioed by an assumed factor of 1.5 for
retropropulsion hover/redirect effects,
by a factor of 1.1 for boiloff effects in transit, and by a factor of 1.1 for budgeting
midcourse correction propellant.

To correct sea level thrust of a sea level Raptor engine to
vacuum conditions, add a force equal to the
exit area multiplied by Earth sea level air pressure. This does not affect rocket equation
results, but it does affect vehicle
acceleration-capability calculations.
These help you choose which engines to burn, and what levels of throttling to use.

Spacex has posted data for anticipated Mars entry from the
interplanetary trajectory, but not for
Earth entry from the interplanetary trajectory.
It is peak entry deceleration gees during the return to Earth that is very
probably the highest gee requirement for occupants to endure. This is not something controllable with
engine thrust, as there is no propellant
available to budget for this purpose.

Earth entry deceleration from Mars (on a direct entry from
the interplanetary trajectory) is quite likely to be far more severe than the
Apollo 11-gee peak deceleration coming back from the moon at an entry interface
speed of 11 km/s. Coming back direct
from Mars, entry interface speed is
likely to be in the vicinity of 17 km/s.
The crew simply must be very physically-fit to endure this. This is a serious issue yet to be addressed in
the Spacex presentations.

Landing stability of a relatively tall and narrow
vehicle, on unprepared rough ground on
Mars, is not addressed here. This is another serious issue yet to be
addressed in the Spacex presentations.

__Update 4-18-18 artificial gravity__:
For Spacex’s mission plan with its BFR vehicle, the health risk for high gee entry occurs at
Earth return. It cannot be avoided. The occupants so exposed will have endured
months-to-years of low Mars gravity (0.384 gee), followed by about 8 to 8.5 months exposure to
zero-gee on the transit home. They are very
unlikely to be physically fit for an 11+ gee entry, even if Mars gravity is found to be fully
therapeutic for microgravity diseases.

The support for that assertion comes from years of orbital
experiences at zero gee. Astronauts
exposed to zero gee for times on the order of 6 months to a year have proven to
be fit enough to endure a 4 gee ride down from Earth orbit, with an entry speed of 8 km/s. We have absolutely nothing to point at, to support the assumption that higher gee
levels are safely endurable in that physical state! Coming from Mars is a faster entry at about
17 km/s than from the moon, and that was
an 11 gee ride at 11 km/s.

Given that risk,
artificial gravity for at least the voyage home seems prudent. This could possibly be accomplished by having
two ships make the return voyage together.
Taking advantage of the refilling plans and procedures, dock the two ships tail-to-tail during the
long coasting transit. Spin them up
end-over-end with the attitude thrusters.
At the nominal 4 rpm spin limit, near-Earth
gee levels are obtained as shown in Figure 5.

Figure 5 – Obtaining Spin Gravity in Two BFR Ships Docked
Tail-to-Tail

The 4 rpm limit is a “fuzzy” limit. If a very slightly-higher spin rate (maybe 4.6
or 4.7 rpm) is tolerable to the balance organs in the middle ear, then very near full Earth gravity can be
simulated in the occupied decks. This is
also shown in Figure 5 as the data in parentheses. This is an artifact of the size of the BFR
vehicle. Achieved gee level is
proportional to spin radius, and to spin
rate squared. A nominal reference point
is 1 gee at 56 m radius and 4 rpm.

There are two inconvenient design issues with this
notion, but they are not “show-stoppers”.
One is the reversed directions for up
and down, sitting on the landing legs
versus spinning for artificial gravity. What
were floors become ceilings, and vice
versa. All the interior equipment and
appurtenances will have to be reversible physically, or “double-ended” if not.

The other is the solar panel fans that Spacex shows for
powering these vehicles with electricity.
The presentations show them deployed near the tail of the vehicle while
in space (free fall). These will have to
be strong enough to deploy properly, and
stay in position, while exposed to low
levels of effective gravity while spinning.
As shown in Figure 5, these
levels of gravity will be less than lunar gravity. Capture of solar energy for conversion to
electricity will usually be intermittent,
at the spin rate.

These are design inconveniences, to be sure.
But in comparison to losing occupants due to heart failure at high entry
gee, just minutes from returning to
Earth, these are minor
inconveniences. I am fond of reminding
people that “there is nothing as expensive as a dead crew”.

__Update 4-19-18 landing stability etc__:
Landing stability on Mars is associated with overturning
issues, landing pad penetration into
soil, and the perturbing dynamics of
trying to land among rocks. There are
also issues with the jet blast flinging debris where it is not wanted, although these only arise with subsequent
ships landing near the first ship or any other structures or equipment already
there.

From an analysis standpoint,
there are static effects and dynamic effects. From a mission standpoint, there is a landing at low weight, and a takeoff at high weight. For the takeoff, there are few, if any dynamic effects to worry about.

**Landing Statics**

This topic divides into static overturn stability, and the soil bearing pressure underneath the
landing leg pads. Static overturn
stability simply requires that the weight vector fall within the polygon
defined by the landing pads, no matter
how off-angle the ship sits, such as on
inclined ground. As shown in Figure 6
below, this isn’t much of an issue for inclinations
oriented directly toward a pad, as
illustrated.

The center of gravity position in the figure is only a
guess, but a realistic one. The span pad-to-pad is only a guess, but also a realistic one. The ground could incline some 18 degrees directly
toward a pad, and still be stable, as shown in the figure. If the inclination is directly between two
pads, the lateral distance is 70% of the
value shown, for a max inclination angle
of about 13 degrees.

13 degrees is a rather steep local slope on terrain chosen
to be flat. We can conclude that the
otherwise tall BFR spaceship is at relatively lower risk of simple static
overturn, as long as small localized
hazards like a pad coming down in a dry stream gully can be avoided. That might be very challenging to satisfy in
a robotic landing, though. Hopefully the available stroke in each
landing leg exceeds the roughly 2 m shown in the figure. That should take care of most of the
localized roughness hazards. Stroke rate
capability should be comparable to ship speed just as it touches down.

At landing, with the
full 150 ton payload and 85 tons of inert,
the ship at “dry tanks” masses 235 metric tons. If the landing is “perfect” and does not use
any of the hover/redirect allowance built into the propellant budget, there might still be something like 53 tons
of propellant on board at touchdown,
bringing the as-landed mass to 288 tons as a maximum. At Mars 0.384 gee, the corresponding weight on the landing legs
is 1084 KN.

If evenly distributed among the 4 landing legs, and if the total pad area is 10 sq.m as shown
in the figure, then the bearing pressure
exerted upon the soil after landing is 108-109 KPa. If all the propellant is used, the bearing pressure is the lesser 88.5 KPa shown
in the figure.

The figure shows typical soil bearing pressure capabilities
for two soils that might be like soils that could be encountered on the plains
of Mars. One is soft fine sand, like many deserts with sand dunes on
Earth, capable of from 100-200 Kpa. The other is more like desert hardpan on
Earth, with lots of gravel mixed into coarse
sand and relatively-compacted: some
380-480 KPa. Excluding dynamic
effects, even the soft fine sand seems
capable of supporting the low weight of the as-landed ship.

**Landing Dynamics**

If there is some inclination, it will tend to throw some of the ship’s
weight toward the downslope pad or pads.
Landing impact dynamics could possibly double the static forces on a
short transient. If we double the static
bearing pressures, this should typically
“cover” the landing impact dynamics and any small inclination effects. For a ship with residual propellants on
board, the doubled static bearing
pressures are in the 220 KPa class.

That rules out soft fine sand by considerable margin. Any landing site must be desert hardpan or
better in terms of soil bearing strength,
or else the landing pads had better total far more than 10 square meters
of bearing area (something difficult to achieve). Thus it would pay to select a landing site
already visited by an earlier probe or rover,
whose visit could verify soil type and estimated strength.

Once the equipment is in place to prepare hard-paved landing
sites ahead of time, this restriction
loosens.

About the worst conceivable landing dynamics event is for
one pad to touch down on a boulder, and
then slip off during the touchdown,
leaving the vehicle temporarily unsupported on that side. If that happens to be downslope, the vehicle will start to topple that
way, while the leg strokes to reach the
actual surface.

If one assumes a realistic 5 degree local slope down toward
the pad that hits and slips off a 1 m boulder,
then for the max landed weight of 1084 KN (as calculated above), a side force at the center of gravity of
about 95 KN acts to topple the vehicle.
This is on an effective moment arm of 22 m, using the surface as the coordinate
reference. That torque is 2090 KN-m =
2.09 million N-m.

Approximating the vehicle as a solid bar 48 m long of mass
288 metric tons, its moment of inertia
is roughly (1/12) m L^2 = 55.3 million kg-sq.m.
The resulting angular acceleration is something like 0.0378 rad/sq.sec
or 2.16 deg/sq.sec.

Ignoring the recovering stroke rate of the slipped leg, the vehicle could rotate through about 5
degrees for that pad to actually strike the real surface. Adding the inclination of 5 degrees, that’s 10 degrees out-of-plumb, within the 18 degree limit for one pad
directly downslope, and also within the
limit for two pads downslope.

Now, in this
transient as the slipped pad hit surface,
the vehicle is already moving,
and is going to take time to stop moving once the pad is on solid ground
to resist. Crudely speaking, the time for the pad to strike the surface is
near 2.15 sec, and the ship will be
moving at about 4.6 deg/s. The pad
support force has to stop this motion in its 1 remaining m of stroke.

That pad force will be in the neighborhood of 279 KN, just ratioed from the disturbing force by the
ratio of moment arms, and acting upon
about 2.5 sq.m for one pad. The vehicle
will move another 5 degrees during this deceleration transient.

At the end of this transient, the vehicle is about 15 degrees
out-of-plumb, dangerously close to the
18 degree limit directly toward 1 pad,
and beyond the 13 degree limit directly between two pads. The bearing pressure under the pad exerting
the restoring force is in the neighborhood of 112 KPa plus the allocated static
weight pressure for the vehicle at rest (109 KPa): a total near 221 KPa.

From an inclination standpoint during this transient, the vehicle is dangerously close to toppling
over, even if the soil is infinitely
strong and hard. If the soil resembles
packed coarse sand with gravel, it is
theoretically strong enough to resist serious compaction under the pad exerting
the restoring force, but whatever
compaction does occur allows the vehicle to incline just that much more. If the soil resembles more soft fine sand in
strength, the restoring pad will inevitably
dig in, without being able to exert
enough restoring force, and so the
vehicle will indeed topple over and be destroyed.

The toppling risk is high if a pad strikes an obstruction of
any significant size (in this example, a
boulder 1 m in dimension). Things of
this size are difficult to observe by remote sensing from orbit. The risk situation is quite similar to the
boulder field encountered during the Apollo 11 moon landing, and requires a hover and redirect action away
from the threat.

The BFR vehicle obviously needs some sort of “see-and-avoid”
capability at touchdown. This would be
direct vision with a human pilot, or
something very sophisticated indeed built-in for a robotic landing. If the vehicle were not so tall, the moment arm of the disturbing force would
not amplify the toppling effect so much.
That is why the Apollo LM and all the Mars landers have had a
height/span ratio under 1. That number
is 3+ for this vehicle.

Figure 6 – Data to Support Statics and Dynamic Calculations

**Takeoff Statics (Only)**

At takeoff, payload
is reduced to 50 tons, inert is the same
85 tons, and the refilled propellant
load is the maximum 1100 metric tons. The
vehicle mass at takeoff is then 1235 metric tons. At Mars 0.384 gee, the vehicle weight is near 4650 KN. Evenly distributed onto a nominal total of 10
sq.m of pad area, the exerted bearing
pressure is 465 KPa.

The soil upon which it rests had better not be similar to
fine soft sand, or the landing legs will
sink deeply into the soil as propellant is loaded.

This could lead to landing leg damage as the ship
launches, or even prevent its
launch, given the high extraction forces
trying to pull out deeply-embedded pads.
If the soil properties vary on a 5 m scale, the legs could embed unevenly. That could risk toppling over with propellants
on board, leading to a huge explosion.

If the soil resembles gravel embedded in packed coarse
sand, the static bearing pressure at
launch falls within the range of soil bearing strengths, before the safety factor of 2 is
applied. That means it is very likely
that the pads will sink a bit into the soil,
although not likely enough to run the embedding risk. If the soil properties vary on a 5 m
scale, this sink-in will be uneven, adding to the vehicle inclination.

Given those outcomes,
it would be wise to design larger pads on the landing legs, or use more than the 4 legs depicted in the
website presentations. This size and
shape of vehicle is more safely operated from a very level, hard-paved pad, or from level, flat solid rock.

I do recommend using the coarse sand/gravel soil strength as
representative of the sand/rock mix we see on Mars. The minimum Earthly strength of that material
is about 380 KPa, with around 480 KPa as
the max strength. I also recommend using
at least a factor of 2 upon exerted pad pressures to model inclination and
sudden-impact effects on landing. Best
practice would apply that to takeoff as well,
although perhaps a factor nearer 1.1 might be adequate.

The landing leg and pad depictions on the website are very
generic and lack detail. I don’t believe
this part of the vehicle has yet received much in the way of design
attention. I presumed 10 sq.m of pad
area, and it is really not enough. This part of the design needs such
attention, as it will be difficult to
incorporate 14+ sq.m of flat pad area on the ends of 4 landing legs, and still stow these away successfully for
hypersonic entry aerobrake events. If
circular, that 14 sq.m of pad area
corresponds to pads about 2.1 m in diameter.
Even my 10 sq.m analysis implies circular pads 1.8 m diameter.

**Debris Flung By Jet Blast**

The force exerted on the surface by the rocket streams as
the vehicle touches down is just about equal to the engine thrust. At landing on Mars, this would be the vacuum thrust of two sea
level Raptor engines. This is about 3670
KN. It would be exerted over an area on
the ground comparable to the exit area of the engines, or maybe a little more. That would be something in the vicinity of 3
sq.m. That effective average pressure is
near 1220 KPa.

That pressure is so far above the soil bearing strength, even for coarse sand with gravel, that the sand will get flung as a supersonic
sandstorm, and rocks of substantial size
are going to get torn loose and flung with considerable force, albeit at a rather low launch angle.

For a 10 cm rock, the
cross section area is about 0.00785 sq.m.
The jet blast pressure acting on that area produces a force in the
vicinity of 9.6 KN. At specific gravity
2.5, such a rock should mass something
in the vicinity of 1.3 kg, and on Mars
would weigh something like 4.9N. The
force to weight ratio is huge at nearly 2000 to 1.

Using impulse and momentum,
with a wild-guessed action time on the order of 0.01 sec, the thrown velocity of this rock would be on
the order of 75 m/s. If the elevation
angle were 10 degrees, the range at
which the rock comes down would be around 0.5 km. These crude estimates could easily be too
conservative.

**Loss of a Ship Leading to Explosion**

If a ship should topple over and explode, or crash and explode, large debris will be flung with incredible
force. Based partly on half the speed of
the military fragment impact tests, the
velocity of such debris might be in the vicinity of 1.2 km/s. It will leave the scene at all elevation
angles from zero to straight up. In the
low gravity on Mars, such debris flung at 45 degrees will travel over 350 km.

It will be almost impossible to protect from debris flung
nearly vertically, which will come down
close to the site. It would be possible
to intercept the low-angle debris with an earthen embankment bulldozed around
the landing pad. That might protect
neighboring ships on adjacent pads, or
base buildings erected nearby, acting as
a shadow shield. Based on what I have
seen published about the base to eventually be constructed with these BFR ships, I don’t think this issue has yet been
considered at all. Yet, such an explosion is inevitable over the long
haul, even if very rare.

**Conclusions and Recommendations**

It is probably too late in the design process to adjust
vehicle length and diameter to a shorter,
fatter proportion. That means the
landing pads must exceed about 14 sq.m total area in actual contact with the
ground, if my soil strength estimates
have any reality at all.

I strongly recommend to Spacex that they start looking
closely at these issues of landing and takeoff statics, and the landing dynamics. From what has been published so far, I have to conclude that they have not
addressed these particular issues in any detail, yet.

I would definitely,
and very strongly, recommend that
BFR landing sites be at least 1 km away (preferably 2+ km) from other grounded
ships or any other structures or equipment,
to mitigate the flung rock hazard.
This would be true until such time as hard paved landing pads can be
constructed. “Hard” means materials with
compressive and shear strengths both exceeding about 1500 KPa = 1.5 MPa. Such will automatically satisfy needed
bearing strength.

I would also recommend to Spacex that they begin considering
the possible effects of a ship explosion upon adjacent ships, base buildings, and other nearby equipment. Berms around hard-paved pads are highly
recommended. Such high-velocity debris
will travel a long way in Mars’s low gravity,
and some of these pieces will be quite large. Such events may actually be exceedingly
rare, but the results are unacceptably
catastrophic, no matter how rare. Probability x cost is NOT the way to judge
this.

Figure 7 – Assumptions Made to Model the Tanker Problem Two
Ways

__Update 4-21-18 speculations on the tanker issue__:
This update presents very speculative numbers I ran trying
to understand the tanker vehicle issue.
This includes both the design of the tanker, and how many are needed to fully refill a
crewed BFR second stage vehicle in Earth orbit.

Spacex presents on its website estimated weights data for
the crewed vehicle, and not for the
cargo-only unmanned vehicle (or the tanker).
The implication is that the cargo vehicle overall characteristics are
similar to the crewed vehicle. There is
no clue given as to the identity or characteristics of the tanker.

It would make sense that the same engine section and
propellant tank section would be used in both (or all three), just with different forward section structures, although with the same heat shield. That is the “justification” for using the
same inert weights, propellant
weights, and payload weights for crewed
and cargo vehicles in the original portion of this article. In particular, the inert weights are likely not to be the
same, although they are likely to be crudely
similar.

Here in this update,
I extend that inert weight assumption to any potential third
configuration that would serve as a dedicated tanker. It would have additional tanks holding 150
tons of propellant mounted in the forward section, and plumbed into the other tanks. The other competing idea is to fly a crewed
BFR unmanned, or a cargo BFR, but both with zero payload on board. If fully loaded with propellant in the tanks, either of these would arrive on orbit with
considerable unused propellant beyond the landing budget (effectively the
“tanker load”).

All the assumptions and engine performance data are the same
as I already used in the original portion of this article. The same basic mission analysis consideration
applies: hold enough propellant in
reserve to land. The difference here is
that the landing is made with zero payload on board, which reduces the necessary budget for
landing propellant. Plus, no budgets need be maintained for long-term
boiloff effects, or for deep space midcourse maneuvers. Thus the reduced figure of 32.6 tons of
landing propellant will suffice to land at zero payload, even with factor 1.5 on the landing delta-vee
to cover any “hover and redirect” effects.

Similar to what I did in the original portion of the
article, we start with the landing to
determine that landing propellant budget.
With all three designs sharing the same overall weight statement, all three then share the same dry-tanks
weight at landing, and thus the same landing
propellant budget. Then we apply that
reserve to each of the three tanker configuration candidates to determine how
much deliverable propellant they can provide on orbit. Assumptions are shown in Figure 7, results in Figure 8.

Figure 8 – Results Obtained Modeling the Tanker Problem Two
Ways

Applicable overall weight statements (metric tons) are:

crewed cargo ded. tanker

inert 85 85 85
(#1)

__payload 0 (#3) 0 (#3) 150 (#2)__

b.o. 85 85 235

__propellant 1100 1100 1100__

ign 1185 1185 1335

notes:

#1. incl
extra tanks

#2. payload
is propellant

#3. capable of 150 tons, flown here at 0

The resulting as-flown weight statements are:

Crewed cargo ded.tanker

Ign 1185 1185 1335

__Asc.prop 925.6 925.6 1042.8__

b.o. 259.4 259.4 292.2

__less del.prop 141.8 141.8 174.6 (#1)__

land.ign. 117.6 117.6 117.6

__land.prop 32.6 32.6 32.6__

b.o. (inert) 85 85 85

notes:

#1. 141.8
left in main tanks excl. landing prop.

#1. 24.6 left in main tanks excl. land., plus 150 in payload tanks

The “dedicated tanker” configuration with extra tanks
holding 150 tons in the forward section is the most efficient tanker of the
three candidates. It takes 6.3 of these
to completely refill a crewed BFR vehicle in low Earth orbit, using that 150 tons plus the excess in the main
tanks beyond the landing budget. The
downside of this design is a third configuration to account fir in
manufacturing. It wouldn’t take much
design refinement to eliminate the “.3” and get to 6 tanker flights. The target
is 183-184 tons delivered to low Earth orbit.
The design (as crude as it is) delivers 175.

Provided that the crewed and cargo vehicles share the same
inert masses, then if flown at zero
payload, they arrive with enough excess
propellant in the main tanks beyond the landing budget, to enable complete refilling of a crewed
vehicle in low Earth orbit, if flown
some 7.8 times. This approach is less
efficient from a number of flights standpoint,
but allows the advantage of having to account for only two
configurations in the manufacturing effort. It is probably not possible to get enough design
refinement to eliminate the “.8” and get to 7 tanker flights. 8 flights is likely realistic.

The “real” numbers are going to be different, once revealed by Spacex, because these configurations will not all
share the same inert weight as was assumed here. The “flown-at-zero-payload” potential is
close enough to the “dedicated tanker” potential that perhaps Spacex should
investigate this possibility closer, before “freezing” the crewed and cargo designs, in spite of the inefficiency of flying with
such large volumes of empty space on board.

Note also in the results figure that gee loads have been
held to very tolerable values with very simple choices of engines and throttle
settings. In particular, the landing settings have been chosen to
enable either a 1-engine or 2-engine touchdown from the very same point in the descent
trajectory. You plan on flying as
2-engine, but if one fails to
ignite, you just immediately double the
thrust setting on the remaining engine. The
throttle margins are there to support that.

Nice analysis Gary. I've been pondering using the BFS for a Titan mission. If loaded in a HEEO, like they propose for the Lunar missions, then it'd easily launch to Titan on a 2.8 year transfer orbit and aero-capture/direct descent with an entry speed of 7 km/s - all without needing a full tank. Details are on my blog: http://crowlspace.com/

ReplyDeleteYour discussion of atmosphere models for EDL is very pertinent. Thanks for the good NASA reference.

Glad you liked it, and the Justus & Braun reference. Be aware that Mars density varies by a factor of 2 with season and location from their model atmosphere, unlike Earth. The Titan atmosphere model is based on one sample: Huygens. -- GW

DeleteMars's variability I get, thanks to the amount of CO2 that freezes out periodically. Titan, since its main atmospheric component is non-condensible throughout the atmospheric column and at all latitudes, should be more stable. Yet it's bit of a knife edge balance I've read. Titan's close to having the whole lot condense out - if its albedo rose just a bit more, then the N2 would collapse into liquid.

Delete