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jhochstein
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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:
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?
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?