Page:Advanced Automation for Space Missions.djvu/162



The mass brought to LEO from the Moon is MPL + MLAN where MPL is the mass of the payload of lunar soil and MLAN 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 (MPR4), or to carry the hydrogen required on the Moon plus the propellant to carry the system to the Moon from the OTV (MPR2+3</SUB> = M<SUB>PR<SUB>2</SUB></SUB> + M<SUB>PR<SUB>3</SUB></SUB> the propellant requirements for burns two and three), whichever is greater. The fact that δV<SUB>4</SUB> ~ δV<SUB>2</SUB> + δV<SUB>3</SUB> and that M<SUB>PL</SUB> &gt;&gt; M<SUB>H</SUB> where M<SUB>H</SUB> 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<SUB>LAN</SUB> = M<SUB>LS</SUB> + aM<SUB>PL</SUB> + BM<SUB>PR<SUB>4</SUB></SUB>

where M<SUB>LS</SUB> 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<SUB>LAN</SUB> + M<SUB>PL</SUB> + M<SUB>PR<SUB>4</SUB></SUB> = (M<SUB>PL</SUB> + M<SUB>LAN</SUB>)e<SUP>δV<SUB>4</SUB>/c</SUP> = (M<SUB>PL</SUB> + M<SUB>LS</SUB>+ aM<SUB>PL</SUB> + BM<SUB>PR<SUB>4</SUB></SUB>)e<SUP>δV<SUB>4</SUB>/c</SUP>

Therefore,


 * $$M_{PR_4} = \frac{K_4 [ ( 1 + a ) M_{PL} + M_{LS} ] } { 1-BK_4 }$$

where c is exhaust velocity. Therefore,


 * K<SUB>n</SUB> = e<SUP>δV<SUB>n</SUB>/c</SUP>-1

Since the OTV and LANDER are fueled at LEO, the only hydrogen carried to the Moon is that required in M<SUB>PR<SUB>4</SUB></SUB>. If M<SUB>H</SUB> 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<SUB>H</SUB> 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<SUB>LAN</SUB> + M<SUB>H</SUB>)e<SUP>δV<SUB>2+3</SUB>/c</SUP>

where:


 * δV<SUB>2+3</SUB> = δV<SUB>2</SUB> + δV<SUB>3</SUB>


 * M<SUB>PR<SUB>2+3</SUB></SUB> = (M<SUB>LAN</SUB> + M<SUB>H</SUB>)(e<SUP>δV<SUB>2+3</SUB>/c</SUP> - 1) = K<SUB>2+3</SUB>(M<SUB>LAN</SUB> + M<SUB>H</SUB>)

and


 * M<SUB>OTV</SUB> = M<SUB>OS</SUB> + BM<SUB>PR<SUB>1</SUB></SUB>

for M<SUB>OS</SUB> defined as OTV structure mass.

The mass leaving LEO is therefore:


 * M<SUB>OTV</SUB> + M<SUB>PR<SUB>1</SUB></SUB> + (M<SUB>LAN</SUB> + M<SUB>H</SUB>)e<SUP>δV<SUB>2 + 3</SUB>/c</SUP>

where:


 * M<SUB>PR<SUB>1</SUB></SUB> = [M<SUB>OTV</SUB> + (M<SUB>LAN</SUB> + M<SUB>H</SUB>)e<SUP>δV<SUB>2+3</SUB>/c</SUP>](e<SUP>δV<SUB>1</SUB>/c</SUP> - 1)
 * = K<SUB>1</SUB>[M<SUB>OS</SUB> + BM<SUB>PR<SUB>1</SUB></SUB> + (M<SUB>LAN</SUB> + M<SUB>H</SUB>)e<SUP>δV<SUB>2+2</SUB>/c</SUP>]
 * $$ = \frac{K_1 [M_{OS} + (M_{LAN} + M_H) e^{\delta V_{2+3}/c}]} {1 - BK_1} $$ (1)