The “giant Mars rocket” proposed by Spacex has reduced in
size somewhat since its first reveal at the Guadalajara meeting. The term “BFR” is now beginning to refer to the
first stage of the two-stage system,
which flies back and lands for reuse.
The term “ITS” more properly applies to the reusable second stage, which apparently has two forms. Those are the cargo/passenger craft that goes
to destination after refilling on-orbit,
and a flyback tanker that provides the refill propellants on-orbit.

Data published by Spacex at the latest meeting indicate a cargo/passenger
vehicle that summarizes as given in Figure 1. Grossly, this is a 9 m diameter vehicle about 48 m
long, with a dry mass of about 85 metric
tons, and propellant tankage that holds
about 240 metric tons of liquid methane and 860 metric tons of liquid
oxygen. Stated payload weights are 150 metric
tons on ascent (and presumably to destination), and “typically” 50 metric tons
on return. Characteristics of the tanker
form are less clear, but it seemingly
has a lighter dry weight of about 50 metric tons.

Figure 1 -- Estimated Characteristics of ITS Per 2017 Revelations

These two versions presumably share the same ascent propellant
tankage and engine cluster. Those
engines include both sea level and vacuum expansion forms of the same Raptor
engine, with a nominal chamber pressure
of 250 bar, and deeply-throttleable to
20% thrust. The cluster has 4 vacuum
engines of 1900 kN thrust each at 375 sec vacuum specific impulse, and two sea level engines of 1700 kN thrust
each, and specific impulses of 356 sec
in vacuum and 330 sec at sea level. Exit
diameters are 1.3 m and 2.4 m for the sea level and vacuum forms, respectively.
(I did not correct sea level thrust to vacuum.)

I am presuming here that second stage operation during
launches to Earth orbit takes place in vacuum,
so I use the vacuum thrust data for both versions of the engine. Each type’s thrust is therefore associated
with a propellant flow rate via its specific impulse. Summing these gets a total full thrust and a
total propellant flow, and thus an
effective “average” vacuum specific impulse with all six engines running, for an effective exhaust velocity of about
3.5762 km/sec. That calculation
summarizes as follows, where effective
cluster specific impulse is total thrust divided by total flow rate (Figure 2).

Now, on the
assumption that both forms of the vehicle have the same ascent propellant tanks
and quantities (totaling 1100 metric tons of propellants), the following weight statement and delta-vee
table applies (Figure 3). For the
tanker, the first-listed payload of 150
tons is assumed from the cargo passenger version. The second is back-calculated from holding
tanker delta-vee capability to be the same as the heavier ascent form of the
cargo/passenger vehicle.

To do that, one finds
the required mass ratio from the delta-vee,
then solves the mass ratio build-up for the unknown payload:

Wpay =
[Wp – (MR – 1)Wdry] / (MR – 1)

What I find very interesting here is that Spacex seems to
have said it takes 6 tankers to fully refill an ITS on orbit for its voyage to
destination. If you look at the heavier
tanker that gets the same 6.2 km/sec delta-vee as the fully-loaded
cargo/passenger form, then 1100 metric
tons of propellant divided by an estimated 184.7 metric tons per tanker equals
5.956 (almost exactly 6) tankers required.
So the tanker at 50 tons dry weight seems to hold 1100 tons of ascent
propellant, and just about 185 more tons
of propellant-as-payload with which to refill a cargo/passenger ITS on orbit. It would appear this estimate is then just about
right. It does presume all 6 engines
running all of the time.

**Using BFR/ITR at Mars**

For a trip to Mars from low Earth orbit, the departure delta-vee for a Hohmann
minimum-energy orbit to Mars is around 3.71 km/sec at average orbital
conditions. For a direct entry without
stopping in Mars orbit, you let the
planet hit you from behind, as the
planet’s orbital velocity is faster than the transfer orbit’s aphelion
speed. Velocity at entry interface will
fall in the 6 km/sec range, and
aerodynamic drag kills most of that to about 0.7 km/s coming out of hypersonics
fairly deep in the Martian atmosphere.
Double or triple that for the landing burn: about 1.5-to-2 km/sec delta-vee requirement.

That’s crudely 5.21 to 5.71 km/sec delta-vee required to
make a direct landing on Mars, with just
almost 6.2 km/sec available. The
difference can be used to fly a somewhat higher-energy transfer orbit, for a shorter flight time than 8 months. Faster is possible if payload is reduced.

To return, the ITS is
refilled with in-situ propellant production on Mars. It will need around 6 km/sec delta-vee
capability to launch and escape directly, with enough energy to achieve the return
transfer orbit. We assume a direct entry
at Earth, which means in turn we run into
the planet from behind, since vehicle
perihelion velocity is higher than Earth’s orbital velocity.

It will be a very demanding entry interface speed (well
above 11 km/sec): this is what stresses
the heat shield, not entry at Mars. But,
the vehicle will come out of hypersonics at about the same 0.7 km/sec
moderately high in the atmosphere. It
will need at least 3 times that as the landing burn delta vee requirement, because the altitude is higher, and the gravity is stronger. Call it 2 km/sec as a “nice round number” to
assume.

The total delta-vee requirement to ascend from Mar’s surface
and achieve a direct transfer orbit and a powered landing on Earth is therefore
in the neighborhood of 8 km/sec. That is
just about what the ITS cargo/passenger version seems capable of, if restricted to about 50 metric tons return
payload. Again, that particular payload correspondence lends
confidence to these otherwise-guessed numbers.

It also points out how critical in-situ propellant
production will be for using this vehicle on Mars.

**Each launch from Mars requires 240 metric tons of locally-produced liquid methane, and 860 metric tons of locally-produced liquid oxygen. Launch opportunities are 26 months apart. Required production rates are thus 9.23 tons/month methane, and 33.08 tons/month oxygen, at a bare minimum, per launch.***Unless this vehicle is refilled locally with the full 1100 metric ton propellant load, it is stranded there!***BFR/ITS For the Moon**

Some have pointed out that this vehicle could also visit the
moon. To leave Earth orbit for the
moon, the delta-vee requirement about
3.29 km/sec. The delta-vee to arrive
into low lunar orbit is just about 0.8 km/sec,
or to land direct, about 2.5
km/sec. Those one-way totals are 4.09
km/sec to lunar orbit, and 5.79 km/sec
to land direct (remarkably close to the Mars value at min energy transfer).

To return by a direct departure from the lunar surface
requires about 2.5 km/s, or from orbit about
0.8 km/sec. Landing at Earth is largely
by aerodynamic braking, but requires
about a 2 km/sec landing burn.
Therefore, total delta-vee
requirements to return are 4.5 km/sec from the surface, or 2.8 km/sec from lunar orbit.

One could conclude that the ITS could ferry cargo to lunar
orbit and return entirely unrefilled, a
trip requiring total 6.89 km/sec delta-vee capability. This is not available at 150 metric tons of
payload, but it is available at
something a little larger than 50 tons. I get about 102 metric tons of payload.

The requirements to land and return entirely unrefilled
would be 10.29 km/sec, which is
out-of-reach even at only 50 tons payload.

*To use the ITS on the lunar surface will require propellant production on the moon, although likely at somewhat lower rates and quantities than at Mars.***Guessing Reusable Performance of BFR**

A related point: if
we presume the fully-loaded ITS uses essentially all of its 1100 tons of
propellant achieving low Earth orbit, we
can back-estimate the delta-vee that is actually available from its BFR first stage, even allowing for flyback. Earth orbit velocity is just about 8.0
km/sec. Allowing 5-10% gravity and drag
losses for a vertical ballistic trajectory,
the min total delta vee is about 8.4-8.8 km/sec. About 6.1 of that is from the ITS second
stage. The first stage need only supply
2.3-2.7 km/sec, which means the staging
velocity is just exoatmospheric at around 2.5 km/sec. It should easily be capable of ~5
km/sec, so the difference is for flyback
all the way to launch site, and
propulsive landing.

**Suborbital Intercontinental Travel**

Finally, there has
been some excited talk about using the BFR/ITS for suborbital high speed
transportation across intercontinental ranges here on Earth. That is a ballistic requirement similar to
that of an ICBM. The burnout velocity of
the typical ICBM is around 6.7 km/s. Allowing
5-10% margin for gravity and drag losses,
the delta-vee necessary to fly intercontinentally is 7 to 7.3
km/sec, plus for the ITS, about 2 km/sec for the landing burn. Total is thus 9 to 9.3 km/sec delta-vee.

This is way beyond the delta-vee capability of the ITS stage
alone, notwithstanding the fact that 4
of its 6 engines will not operate at sea level,
and even if they did, total 6-engine
thrust of the ITS stage (1100 kN) is less than its weight (1300 kN or more). But this delta-vee is within reach of the
two-stage BFR/ITS combination (6.2 to 7.9 km/sec ITS and ~2.5 km/sec BFR for
8.7 to 10.4 km/sec), and likely with a
little less payload than the 150 tons typical to Mars. Maybe something in the vicinity of 100 tons.

**Final Remarks**

These estimates are rough. I did not correct sea level thrust to vacuum for one thing, my delta vee requirements are approximate for another, and I did not explore the effects of using only the vacuum engines for higher specific impulse out in space.

Even so, these results are very intriguing. These calculations were made pencil-and-paper with a calculator. Nothing sophisticated.