Page:Advanced Automation for Space Missions.djvu/163



The amount of material lifted off the Earth is:


 * $$M_{H_{lift}} = M_H +B_H (M_{PR_1} + M_{PR_{2+3}})$$
 * $$= B_H(M_{PR_1} + M_{PR_{2+3}} + M_{PR_4})$$
 * $$= B_H \left\{ \frac{K_1}{1-BK_1} \left[ M_{OS} + \left( M_{LAN} + M_H \right) \left( K_{2+3} + 1 \right) \right] + K_{2+3} \left( M_{LAN} + M_H \right) + M_{PR_4} \right\}$$
 * $$= B_H \left\{ \frac{K_1}{1-BK_1} M_{OS} + \left[ \frac{K_1 \left( K_{2+3} + 1\right) } { 1-BK_1} + K_{2+3} \right] \times \left( M_{LAN} + M_H \right) + M_{PR_4} \right\}$$
 * $$= B_H \left\{ \frac{K_1}{1-BK_1} M_{OS} + \left[ \frac{K_1 \left( K_{2+3} + 1\right) } { 1-BK_1} + K_{2+3} \right] \times \left[ \frac{ \left( B + B_H \right) K_4 \left[ \left( 1+a \right) M_{PL} + M_{LS} \right] } { 1 - BK_4 } + M_{LS} + aM_{PL} \right] + \frac{ K_4} {1-BK_4} \left[ \left( 1+a \right) M_{PL} + M_{LS} \right] \right\}$$

If we define A, b, and C as follows:


 * $$A \equiv B_H \left\{ \left[ \frac{ K_1 \left( K_{2+3} + 1 \right) } { 1-BK_1} + K_{2+3} \right] \left[ \frac{ \left(B+B_H\right)K_4 \left(1+a\right) } { 1-BK_4} + a \right] + \frac{K_4 \left(1+a\right) } {1-BK_4} \right\}$$
 * $$b \equiv \frac{B_H K_1} {1 - BK_1}$$
 * $$C \equiv B_H \left\{ \left[ \frac{ K_1 \left( K_{2+3} + 1 \right) } { 1-BK_1} + K_{2+3} \right] \left[ \frac{ \left(B+B_H\right)K_4} { 1-BK_4} + 1 \right] + \frac{K_4 } {1-BK_4} \right\}$$
 * $$M_{H_{lift}} = AM_{PL} + bM_{OS} + CM_{LS}$$

From MPL is taken material sufficient to replace the nonhydrogen part of the fuel supply. The amount of payload left over is P, hence:


 * $$M_{PL} = P + (1-B_H) (M_{PR_1} + M_{PR_{2+3}})$$
 * $$= P+(1+B_H) \left[ \frac{M_{H_{lift}}} {B_H} - M_{PR_4} \right]$$
 * $$= P+ \frac{1+B_H} {B_H} \left\{ M_{H_{lift}} - \frac{B_H K_4} {1-BK_4} [(1+a)M_{PL} + M_{LS}] \right\}$$
 * $$M_{PL} = \frac{ P + \frac{1-B_H}{B_H} \left[ M_{H_{lift}} - \frac{ B_H K_4} {1-BK_4} M_{LS} \right]} {1 + \frac{1-B_H} {1-BK_4} K_4 (1+a)}$$

and


 * $$M_{H_{lift}} = A \frac{ P + \frac{1-B_H}{B_H} \left[ M_{H_{lift}} - \frac{ B_H K_4 M_{LS}} {1-BK_4} \right]} {1 + \frac{1-B_H} {1-BK_4} K_4 (1+a)} + bM_{OS} + CM_{LS}$$
 * $$\frac{dM_{H_{lift}}}{dP} = A \frac{ 1 + \left(\frac{1-B_H} {B_H}\right) \left(\frac{dM_{H_{lift}}}{dP}\right) } {1+ \frac{1-B_H} {1-BK_4} K_4 (1+a) }$$
 * $$= \frac{ \frac{A} {1+ \frac{1-B_H} {1-BK_4} K_4 (1+a) } } {1- \left[ \frac{1-B_H} {B_H} \right] \left[\frac{A} {1+ \frac{1-B_H} {1-BK_4} K_4 (1+a)} \right] }$$
 * $$= \frac{A} {1 + \frac{1-B_H} {1-BK_4} K_4 (1+a) - \frac{1-B_H} {BH} A}$$