This page needs to be proofread.
The amount of material lifted off the Earth is:
![{\displaystyle =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\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/16b27eab6571482bc5803d3d5a757f984256f865)
![{\displaystyle =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\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/53336005a8f73c684a43f8b514a32506b928b67a)
![{\displaystyle =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\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bba31b55fcd1ed72b8524d4ee0032a5a883db67a)
If we define A, b, and C as follows:
![{\displaystyle 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\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/898d7d4c0511d43c6de6390b29738bb2f7d7d456)
![{\displaystyle 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\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f283be11e03a7ef833b2ae84768c3498110f056c)
From MPL is taken material sufficient to replace the nonhydrogen part of the fuel supply. The amount of payload left over is P, hence:
![{\displaystyle =P+(1+B_{H})\left[{\frac {M_{H_{lift}}}{B_{H}}}-M_{PR_{4}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ddb1a0d23762f4df982ea403c940e301a041a88a)
![{\displaystyle =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\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ad2fb655385a1985d509dc9830504af4ed41e53e)
![{\displaystyle 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)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4abf00072e4cad737c8d5bad17872c9641121316)
and
![{\displaystyle 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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/188c787d2e0dba71a011b715fb7093c019ab7de1)
![{\displaystyle ={\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]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4c3affcbb40f0b953c67707007d0f4a1e5545b78)
