Page:Advanced Automation for Space Missions.djvu/162

Appendix 4A: Lunar Supply Of A Low Earth Orbit Station: Derivation Of Formulas

The mass brought to LEO from the Moon is M_PL + M_LAN where M_PL is the mass of the payload of lunar soil and M_LAN is the mass of the LANDER system that carries it. The LANDER must have sufficient tankage to carry payload plus the propellant to lift off from the Moon (M_PR4), or to carry the hydrogen required on the Moon plus the propellant to carry the system to the Moon from the OTV (M_PR2+3 = M_PR2 + M_PR3 the propellant requirements for burns two and three), whichever is greater. The fact that δV₄ ~ δV₂ + δV₃ and that M_PL >> M_H where M_H is the mass of hydrogen carried to the Moon, makes it clear that the former tankage requirement is the more stringent. It has therefore been assumed that:

M_LAN = M_LS + aM_PL + BM_PR4

where M_LS is the mass of the LANDER structure and a and B are the tankage fractions for the payload and propellant, respectively. For all burns and for both the OTV and the LANDER B is assumed to be the same.

On the lunar surface prior to takeoff, the mass of the LANDER system is:

M_LAN + M_PL + M_PR4 = (M_PL + M_LAN)e^δV₄/c = (M_PL + M_LS+ aM_PL + BM_PR4)e^δV₄/c

Therefore,

$M_{PR_{4}}={\frac {K_{4}[(1+a)M_{PL}+M_{LS}]}{1-BK_{4}}}$

where c is exhaust velocity. Therefore,

K_n = e^δV_n/c-1

Since the OTV and LANDER are fueled at LEO, the only hydrogen carried to the Moon is that required in M_PR4. If M_H is defined as the mass of hydrogen carried to the lunar surface, then

$M_{H}=B_{H}M_{PR_{4}}=B_{H}K_{4}{\frac {(1+a)M_{PL}+M_{LS}}{1-BK_{4}}}$

where B_H is the hydrogen fraction in the propellant.

The mass landed on the Moon must be:

$M_{LAN}+M_{H}=M_{LS}+aM_{PL}+BM_{PR_{4}}+B_{H}M_{PR_{4}}={\frac {(B+B_{H})K_{4}[(1+a)M_{PL}+M_{LS}]}{1-BK_{4}}}+M_{LS}+aM_{PL}$

The payload for the OTV is therefore

(M_LAN + M_H)e^{δV₂₊₃/c}

where:

δV₂₊₃ = δV₂ + δV₃

M_PR2+3 = (M_LAN + M_H)(e^{δV₂₊₃/c} - 1) = K₂₊₃(M_LAN + M_H)

and

M_OTV = M_OS + BM_PR1

for M_OS defined as OTV structure mass.

The mass leaving LEO is therefore:

M_OTV + M_PR1 + (M_LAN + M_H)e^{δV_{2 + 3}/c}

where:

M_PR1 = [M_OTV + (M_LAN + M_H)e^{δV₂₊₃/c}](e^δV₁/c - 1)

= K₁[M_OS + BM_PR1 + (M_LAN + M_H)e^{δV₂₊₂/c}]

$={\frac {K_{1}[M_{OS}+(M_{LAN}+M_{H})e^{\delta V_{2+3}/c}]}{1-BK_{1}}}$ (1)