1. The gravitational field is assumed to be a function of the altitude and varies according to an inverse square law g(h) = g0(h=(Re+h))2. The atmospheric characteristics (temperature, pressure, density and speed of sound) follow the US Standard Atmosphere 1976 model up to 1000 km.
2. Within the control vector c, the angle of attack is discretized into 9 variables or elements for the first air-breathing phase, c(1:9) 2 [0;30] deg and 8 elements for the second rocket phase, c(10:17) 2 [ 5;30] deg. The throttle control setting for the first phase is preset to δT= 1, while the second phase is discretized into 9 elements c(18:26) 2 [0:5;1]. The last 2 design variables define the duration of first and second phase respectively, c(27;28) 2 [80;1800] s. The equality constraints on final states are converted into inequality constraints such that hf2[80;82] km, vf2[7:34;7:40] km/s, and γf2[ 2;2] deg.
3. In general, the level of uncertainty in atmospheric modeling increases with altitude. Specifically with the US 1976 Standard Atmosphere model used here, below 32 km the model is well known and identical to the Standard Atmosphere of the International Civil Aviation Organization (ICAO), however above that, and more notably above the ozone layer at 86 km the model becomes less accurate when compared with other complex models based on more recent experimental data.18Other key contributors to uncertainty are the stochastic nature of the radiation pattern from the Sun on the Earth’s surface and differences between geographic locations. A certain level of error is unavoidable when creating an averaged, globally applicable atmospheric model.
4. Final mass m, kg 56846 (full) (clean) (clean) 51091 86158 29103 1028 7370 250 29 -0.1 2.7 1941 56465 1293 Table 2: Statistics of path constraints.