I am trying to determine approximately how much fuel it would take to launch a bi-propellant rocket from Mars and Earth such that they escape each planet's respective gravity. Unfortunately, my orbital mechanics is a little rusty, and the solution I'm getting does not make much sense. Here's what I've been trying - hopefully someone can point out my error:(adsbygoogle = window.adsbygoogle || []).push({});

Assumptions:

Earth escape velocity (Vesc,e) = 11 200 m/s

Mars escape velocity (Vesc,m) = 5 027 m/s

Gravity of Earth (ge) = 9.81 m/s2

Gravity of Mars (gm) = 3.71 m/s2

Specific Impulse of a bipropellant liquid rocket (Isp) = 450 s

Dry mass of rocket (structure + payload) = Mdry

Mass of rocket fuel = Mfuel

Wet mass of rocket = Mwet

Okay, using the Rocket Equation we have

deltaV = Vexhaust * ln(Mwet/Mdry) = Vexhaust * ln((Mdry + Mfuel)/Mdry)

Therefore, deltaV/Vexhaust = ln(Mdry + Mfuel)/Mdry)

And, e(deltaV/Vexhaust) = e(ln(Mdry + Mfuel)/Mdry)) = (Mdry + Mfuel)/Mdry)

Mdry * e(deltaV/Vexhaust) - Mdry = Mfuel

Therefore, Mfuel = Mdry(e(deltaV/Vexhaust) - 1)

If Vexhaust = Isp * g then Mfuel = Mdry(e(deltaV/Isp*g) - 1)

Assuming that the deltav equals the escape velocity, the fuel required to escape Earth gravity is

Mfuel,earth = Mdry(e(11200/(450*9.81)) - 1) = 5.8965 * Mdry

and,

Mfuel,mars = Mdry(e(5027/(450*3.71)) - 1) = 7.1828 * Mdry

This would seem to say that more fuel is required to escape Mars than is needed to escape Earth, which seems completely incorrect. Can someone please point out to me where I've gone wrong?

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# How to calculate mass of fuel needed to escape Earth gravity vs. Mars gravity

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