Here's a little study I recently did of using the SpaceX Interplanetary Transport System (ITS) for missions to the Moon.

I first had a look of using ITS directly to Moon (after filling up with 1950 t of methane/LOX propellant from five tanker flights).

For the landing burn on Earth, looking at the YouTube technical broadcasts the burn was about 25 seconds for JCSat 16 and 31 seconds for SpX-9. Lets take the average of 28 seconds. We'll also assume Th = 14 seconds at full thrust and Tl = 14 seconds at minimum thrust of 55% as given below

http://spaceflight101.com/spacerockets/falcon-9-ft/This results in a landing delta-V of 719 m/s. Data is below showing how I calculated this. This results in a negative cargo mass of -36.7 t, which means Lunar Direct will not work with ITS.

One idea I had was to use Lunar orbit propellant transfer. On the first flight, we bring cargo to the surface, but reach Lunar orbit with no propellant left in the tank. That brings a huge 180.6 t to the Lunar surface!

Waiting in Lunar orbit is another ITS, which transfers 107.2 t of propellant to the returning ITS. The second ITS then lands on the Moon bringing 105.1 t of cargo! The process then repeats. This works because we don't bring down and up again the return propellant from Lunar orbit, which is a huge penalty.

Attached is my Pascal program used to work out the payload.

Falcon 9 landing Delta-V calculation

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ms = 22.2 t

ves = g*282 = 2765.48 m/s

Fs = 845 kN

Rp = Fs/ves = 0.306 t/s

Th = 14 s

Tl = 14 s

mp = (Th + Tl*0.55)*Rp = 6.63 t

margin = 2%

We need to solve these two equations where mpr is the residual propellant.

dv = ves*ln(1+mp/(ms+mpr))

margin*dv = ves*ln(1+mpr/ms)

margin*ln(1+mp/(ms+mpr)) = ln(1+mpr/ms)

mpr = ms*(exp(margin*ln(1+mp/(ms+mpr)))-1)

This is a non-linear equation which we solve iteratively. This gives

mpr = 0.116 t

dv = ves*ln(1+mp/(ms+mpr)) = 719 m/s.

Check second equation

margin*dv = 14.4 m/s

ves*ln(1+mpr/ms) = 2765.48*ln(1+0.116/22.2) = 14.4 m/s

ITS Direct Cargo Calculation

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Delta-V values are

dv1 = 3180 m/s (TLI)

dv2 = 960 m/s (LOI)

dv3 = 28 m/s (PDI)

dv4 = 2041 m/s (Lunar Descent)

dv5 = 1850 m/s (Lunar Ascent)

dv6 = 1169 m/s (TEI)

dv7 = 719 m/s (Earth landing)

marginl = 2%

margine = 1%

ve = g*382 = 3746.14 m/s (Raptor exhaust speed)

ves = g*334 = 3275.42 m/s (Sea Level exhaust speed)

ms = 150 t (ITS empty mass)

mp = 1950 t (propellant mass)

dvm = (1+margine)*(dv1+dv2+dv3+dv4) = 6271 m/s (delta-V to Lunar surface)

dve = (1+margine)*(dv5+dv6) = 3049 m/s (delta-V back to Earth)

dvl = (1+marginl)*dv7 = 734 m/s (delta-V landing on Earth)

Need to solve these three equations where mc is the cargo mass and mpe is the propellant to get to Earth and mpl is the landing propellant.

dvm = ve*ln(1+(mp-mpe-mpl)/(ms+mpe+mpl+mc))

dve = ve*ln(1+mpe/(ms+mpl)))

dvl = ves*ln(1+mpl/ms))

Solving the equations from last to first gives

mpl = ms*(exp(dvl/ves)-1) = 37.7 t

mpe = (ms+mpl)*(exp(dve/ve)-1) = 235.9 t

mpm = mp-mpe-mpl = 1676.5 t

mc = mpm/(exp(dvm/ve)-1)-ms-mpl-mpe = -36.7 t

Unfortunately, the negative value for mc means that this won't work.

ITS Lunar Orbit Propellant Transfer First Flight

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dvm = (1+margine)*(dv1+dv2+dv3+dv4) = 6271 m/s (delta-V to Lunar surface)

dva = (1+margine)*dv5 = 1869 m/s (delta-V to Lunar orbit)

Need to solve these two equations where mc is the cargo mass and mpa is the propellant to get to Lunar orbit.

dvm = ve*ln(1+(mp-mpa)/(ms+mpa+mc))

dva = ve*ln(1+mpa/ms))

Solving the equations from last to first gives

mpa = ms*(exp(dva/ve)-1) = 97.0 t

mpm = mp-mpa = 1853.0 t

mc = mpm/(exp(dvm/ve)-1)-ms-mpa = 180.6 t

Check

dvm = 3746.14*ln(1+1853.0/(150+97.0+180.6)) = 6271 m/s

dve = 3746.14*ln(1+97.0/150)) = 1869 m/s

ITS Lunar Orbit Propellant Transfer Subsequent Flights

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dvo = (1+margine)*(dv1+dv2) = 4181 m/s (delta-V to Lunar orbit)

dvd = (1+margine)*(dv3+dv4) = 2090 m/s (delta-V to Lunar surface)

dva = (1+margine)*dv5 = 1869 m/s (delta-V to Lunar orbit)

dvt = (1+margine)*dv6 = 1181 m/s (delta-V to Earth)

dvl = (1+marginl)*dv7 = 734 m/s (delta-V landing on Earth)

dvo = ve*ln(1+(mp-mpd-mpa-mpt-mpl)/(ms+mpd+mpa+mpt+mpl+mc))

dvd = ve*ln(1+mpd/(ms+mpa+mc))

dva = ve*ln(1+mpa/ms)

dvt = ve*ln(1+mpt/(ms+mpl))

dvl = ves*ln(1+mpl/ms))

From before we have

mpl = 37.7 t

mpa = 97.0 t

The fourth equation gives us

mpt = (ms+mpl)*(exp(dvt/ve)-1) = 69.5 t

The second equation gives

ms+mpa+mc = mpd/(exp(dvd/ve)-1)

which we subsitute into the first equation

mp-mpd-mpa-mpt-mpl = (exp(dvo/ve)-1)*(mpd+mpt+mpl+mpd/(exp(dvd/ve)-1))

Let

ao = exp(dvo/ve)-1

ad = exp(dvd/ve)-1

Then

mp-mpa-mpt-mpl = mpd+ao*(mpd(1+1/ad)+mpt+mpl)

mpd = (mp-mpa-mpt-mpl-ao*(mpt+mpl))/(1+ao*(1+1/ad))

mpd = (mp-mpa-(1+ao)*(mpt+mpl))/(1+ao*(1+1/ad)) = 262.9 t

mpo = mp-mpd-mpa-mpt-mpl = 1482.8 t

From the second equation we get

mc = mpd/ad-ms-mpa = 105.1 t

Check

dvo = 3746.14*ln(1+1482.8/(150+262.9+97.0+69.5+37.7+105.1)) = 4181 m/s

dvd = 3746.14*ln(1+262.9/(150+97.0+105.1)) = 2090 m/s

dvt = 3746.14*ln(1+69.5/(150+37.7) = 1181 m/s

Transfer propellant = mpt+mpl = 107.2 t

NOTE See also the following thread

https://forum.nasaspaceflight.com/index.php?topic=41298.0