For those that think their geek quotient a while back was way too low and those that really want to objectively evaluate the Mars by way of the Moon approach:

Eq1: (the rocket equation) is (M+P)/M=e^(dV/C) where:

M=dry mass
P=propellant mass
dV=delta V
C=exhaust velocity

Using the equations below I calculated some interesting rough numbers.

Mars Direct from LEO (total dV=4.6 km/s) would require a dry mass to payload mass ratio of about 1.2 and a fuel to payload mass ratio of 2.2 using a H2/O2 fuel mixture with an ISP of 453 and 0.1 lbs of rocket mass beyond the payload per pound of fuel.

A single vessel Mars mission from LEO (total dV=10.2 km/s) would require a dry mass to payload mass ratio of about 9.5 and a fuel to payload mass ratio of 85.3 using the same fuel values. Even with 2 launches you still have about 1 fourth the dry mass and about 1/20 the fuel mass. That huge fuel mass difference also means a very big difference in volume. That increase in volue would probbaly make aerobreaking more difficult or impossible increasing the deltaV above 10.2.

Take into acount the difficulty of getting that mass into LEO and it should be easy to see why Mars Direct is far easier than launching a fully fueled mission from Earth.

One way to the Moon from LEO (total dV=5.9 km/s) would require a dry mass to payload mass ratio of about 1.4 and a fuel to payload mass ratio of 3.9 using the H2/O2 fuel mixture values above.

A single vessel Moon mission from LEO (total dV=8.3 km/s) would require a dry mass to payload mass ratio of about 2.2 and a fuel to payload mass ratio of 12.2 using H2/O2 fuel mixture values above showing one way to the Moon is harder than Mars but a single vessel 2-way trip is much easier. Apollo used lower ratios because it broke the mission into smaller steps. An approach used in a different way to develope Mars Direct.

Increase the deltaV just slightly (slightly above 10.6) and a 2-way mission to Mars becomes impossible with H2/O2. If the ratio of extra mass per pound of fuel drops to the 0.06 for Saturn V's kerosene first stage a deltaV up to 12.7 is possible. Plug in the ISP for kerosene (303) and the possible deltaV drops to slightly above 8.4 making a single vessel two way trip to Mars impossible even with an Rp=0.06.

The Math

Solving for P in Eq1 you get

Eq2: P=M*(Rm-1) where Rm is e^(dV/C) or the fueled mass to dry mass ratio.

Since the total dry mass is the science payload (let's call it Ms) + the mass needed to carry and use the fuel (Mp), M = Mp+Ms. Ms would also include life support and living quarters etc for a crewed mission. Basically whatever you want to send somewhere, but excluding the mass just associated with getting you there.

The ratio of Mp to P should be roughly a constant I'll call Rp (based on educated guess and Apollo data). M = Rp*P+Ms is the equation demonstrating the relationship between total dry mass, fuel mass, and payload mass with this approximation.

Plugging this equation for M back in to Eq2 we get the approximate equation:

P=(Rp*P+Ms)*(Rm-1)

Solving for P again we get

Eq3: P = Ms*(Rm-1)/[1-Rp*(Rm-1)]

M would then be Ms + Rp*Ms*(Rm-1)/[1-Rp*(Rm-1)] which simplifies to

Eg4: M = Ms/[1-Rp*(Rm-1)]

As Rm or Rp increase the denominator shrinks and the dry mass and fuel mass grow large compared to the desired payload. They go to infinity then becomes a negative number as either increases further.

Regards,
Mark (Engineers Break It Down) Eby