Question 8.2: Calculate the minimum wait time for initiating a return trip......
Calculate the minimum wait time for initiating a return trip from Mars to earth.
From Tables A.1 and A.2 we have
\begin{aligned}& R_{\text {earth }}=149.6 \times 10^6 km \\& R_{\text {Mars }}=227.9 \times 10^6 km \\& \mu_{\text {sun }}=132.71 \times 10^9 km ^3 / s ^2\end{aligned}According to Eqn (8.11), the time of flight from earth to Mars is
t_{12}=\cfrac{\pi}{\sqrt{\mu_{\text {sun }}}}\left(\cfrac{R_1+R_2}{2}\right)^{\frac{3}{2}} (8.11)
or
t_{12}=258.82 \text { days }From Eqn (3.9) and the orbital periods of earth and Mars (see Example 8.1 above) we obtain the mean motions of the earth and Mars.
n=\cfrac{2 \pi}{T} (3.9)
\begin{aligned}& n_{\text {earth }}=\cfrac{2 \pi}{365.26}=0.017202 rad / \text { day } \\\\& n_{\text {Mars }}=\cfrac{2 \pi}{687.99}=0.0091327 rad / \text { day }\end{aligned}The phase angle between earth and Mars when the spacecraft reaches Mars is given by Eqn (8.13).
\phi_f=\pi-n_1 t_{12} (8.13)
\phi_f=\pi-n_{\text {earth }} t_{12}=\pi-0.017202 \cdot 258.82=-1.3107 ( rad )Since n_{\text {earth }}>n_{\text {Mars }}, we choose Eqn (8.16) to find the wait time.
t_{\text {wait }}=\cfrac{-2 \phi_f-2 \pi N}{n_2-n_1} \quad\left(n_1>n_2\right) (8.16)
t_{\text {wait }}=\cfrac{-2 \phi_f-2 \pi N}{n_{\text {Mars }}-n_{\text {earth }}}=\cfrac{-2(-1.3107)-2 \pi N}{0.0091327-0.017202}=778.65 N-324.85 \text { (days) }N = 0 yields a negative value, which we cannot accept. Setting N = 1, we get
\boxed{t_{\text {wait }}=453.8 \text { days }}This is the minimum wait time. Obviously, we could set N = 2, 3, . to obtain longer wait times.
Table A.1 Astronomical Data for the Sun, the Planets, and the Moon
\begin{array}{|c|c|c|c|c|c|c|c|c|}\hline \text { Object } & \begin{array}{l}\text { Radius } \\\text { (km) }\end{array} & \text { Mass (kg) } & \begin{array}{l}\text { Sidereal } \\\text { Rotation } \\\text { Period }\end{array} & \begin{array}{l}\text { Inclination } \\\text { of Equator } \\\text { to Orbit } \\\text { Plane }\end{array} & \begin{array}{l}\text { Semimajor } \\\text { Axis of } \\\text { Orbit }( k m )\end{array} & \begin{array}{l}\text { Orbit } \\\text { Eccentricity }\end{array} & \begin{array}{l}\text { Inclination } \\\text { of Orbit } \\\text { to the } \\\text { Ecliptic } \\\text { Plane }\end{array} & \begin{array}{l}\text { Orbit } \\\text { Sidereal } \\\text { Period }\end{array} \\\hline \text { Sun } & 696,000 & 1.989 \times 10^{30} & 25.38 d & 7.25^{\circ} & – & – & – & – \\\hline \text { Mercury } & 2440 & 330.2 \times 10^{21} & 58.65 d & 0.01^{\circ} & 57.91 \times 10^6 & 0.2056 & 7.00^{\circ} & 87.97 d \\\hline \text { Venus } & 6052 & 4.869 \times 10^{24} & 243 d^* & 177.4^{\circ} & 108.2 \times 10^6 & 0.0067 & 3.39^{\circ} & 224.7 d \\\hline \text { Earth } & 6378 & 5.974 \times 10^{24} & 23.9345 h & 23.45^{\circ} & 149.6 \times 10^6 & 0.0167 & 0.00^{\circ} & 365.256 d \\\hline \text { (Moon) } & 1737 & 73.48 \times 10^{21} & 27.32 d & 6.68^{\circ} & 384.4 \times 10^3 & 0.0549 & 5.145^{\circ} & 27.322 d \\\hline \text { Mars } & 3396 & 641.9 \times 10^{21} & 24.62 h & 25.19^{\circ} & 227.9 \times 10^6 & 0.0935 & 1.850^{\circ} & 1.881 y \\\hline \text { Jupiter } & 71,490 & 1.899 \times 10^{27} & 9.925 h & 3.13^{\circ} & 778.6 \times 10^6 & 0.0489 & 1.304^{\circ} & 11.86 y \\\hline \text { Saturn } & 60,270 & 568.5 \times 10^{24} & 10.66 h & 26.73^{\circ} & 1.433 \times 10^9 & 0.0565 & 2.485^{\circ} & 29.46 y \\\hline \text { Uranus } & 25,560 & 86.83 \times 10^{24} & 17.24 h ^{*} & 97.77^{\circ} & 2.872 \times 10^9 & 0.0457 & 0.772^{\circ} & 84.01 y \\\hline \text { Neptune } & 24,760 & 102.4 \times 10^{24} & 16.11 h & 28.32^{\circ} & 4.495 \times 10^9 & 0.0113 & 1.769 & 164.8 y \\\hline \text { (Pluto) } & 1195 & 12.5 \times 10^{21} & 6.387 d ^* & 122.5^{\circ} & 5.870 \times 10^9 & 0.2444 & 17.16^{\circ} & 247.7 y \\\hline\end{array}*Retrograde
Table A.2 Gravitational Parameter (m) and Sphere of Influence (SOI) Radius for the Sun, the Planets, and the Moon
| Celestial Body | \mu\left( km ^3 / s ^2\right) | SOI Radius (km) |
| Sun | 132,712,000,000 | – |
| Mercury | 22,030 | 112,000 |
| Venus | 324,900 | 616,000 |
| Earth | 398,600 | 925,000 |
| Earth’s moon | 4903 | 66,100 |
| Mars | 42,828 | 577,000 |
| Jupiter | 126,686,000 | 48,200,000 |
| Saturn | 37,931,000 | 54,800,000 |
| Uranus | 5,794,000 | 51,800,000 |
| Neptune | 6,835,100 | 86,600,000 |
| Pluto | 830 | 3,080,000 |