Question 8.5: After a Hohmann transfer from earth to Mars, calculate (a) t......
After a Hohmann transfer from earth to Mars, calculate
(a) the minimum delta-v required to place a spacecraft in orbit with a period of 7 h.
(b) the periapsis radius.
(c) the aiming radius.
(d) the angle between periapsis and Mars’ velocity vector.
The following data are required from Tables A.1 and A.2:
\begin{aligned}\mu_{\text {sun }} & =1.327 \times 10^{11} km ^3 / s ^2 \\\mu_{\text {Mars }} & =42,830 km ^3 / s ^2 \\R_{\text {earth }} & =149.6 \times 10^6 km \\R_{\text {Mars }} & =227.9 \times 10^6 km \\r_{\text {Mars }} & =3396 km\end{aligned}(a) The hyperbolic excess speed is found using Eqn (8.4),
\Delta V_A=V_2-V_A^{(ν)}=\sqrt{\cfrac{\mu_{\text {sun }}}{R_2}}\left(1-\sqrt{\cfrac{2 R_1}{R_1+R_2}}\right) (8.4)
We can use Eqn (2.83) to express the semimajor axis a of the capture orbit in terms of its period T,
\boxed{T=\cfrac{2 \pi}{\sqrt{\mu}} a^{\frac{3}{2}}} (2.83)
a=\left(\cfrac{T \sqrt{\mu_{ Mars }}}{2 \pi}\right)^{\frac{2}{3}}Substituting T = 7.3600 s yields
a=\left(\cfrac{25,200 \sqrt{42,830}}{2 \pi}\right)^{\frac{2}{3}}=8832 kmFrom Eqn (2.73) we obtain
r_p=a(1-e) (2.73)
a=\cfrac{r_p}{1-e}On substituting the optimal periapsis radius, Eqn (8.67), this becomes
r_p=\cfrac{2 \mu_2}{ν_{\infty}^2} \cfrac{1-e}{1+e} (8.67)
a=\cfrac{2 \mu_{\text {Mars }}}{ν_{\infty}^2} \cfrac{1}{1+e}from which
e=\cfrac{2 \mu_{\text {Mars }}}{a ν_{\infty}^2}-1=\cfrac{2 \cdot 42,830}{8832 \cdot 2.648^2}-1=0.3833Thus, using Eqn (8.70), we find
\Delta ν=ν_{\infty} \sqrt{\cfrac{1-e}{2}}=2.648 \sqrt{\cfrac{1-0.3833}{2}}=1.470 km / s(b) From Eqn (8.66), we obtain the periapse radius
\cfrac{ d ^2}{ d \xi^2} \cfrac{\Delta ν}{ν_{\infty}}=\cfrac{\sqrt{2}}{64} \cfrac{(1+e)^3}{(1-e)^{\frac{3}{2}}} (8.66)
r_p=\cfrac{2 \mu_{\text {Mars }}}{ν_{\infty}^2} \cfrac{1-e}{1+e}=\cfrac{2 \cdot 42,830}{2.648^2} \cfrac{1-0.3833}{1+0.3833}=\boxed{5447 km}(c) The aiming radius is given by Eqn (8.71),
\Delta=2 \sqrt{2} \cfrac{\sqrt{1-e}}{1+e} \cfrac{\mu_2}{ν_{\infty}^2}=\sqrt{\cfrac{2}{1-e}} r_p (8.71)
\Delta=r_p \sqrt{\cfrac{2}{1-e}}=5447 \sqrt{\cfrac{2}{1-0.3833}}=\boxed{9809 km}(d) Using Eqn (8.43), we get the angle to periapsis
\beta=\cos ^{-1}\left(\cfrac{1}{e}\right)=\cos ^{-1}\left(\cfrac{1}{1+\frac{r_p ν_{\infty}^2}{\mu_1}}\right) (8.43)
\beta=\cos ^{-1}\left(\cfrac{1}{1+\frac{r_p ν_{\infty}^2}{\mu_{\text {Mars }}}}\right)=\cos ^{-1}\left(\cfrac{1}{1+\frac{5447 \cdot 2.648^2}{42,830}}\right)=\boxed{58.09^{\circ}}Mars, the approach hyperbola, and the capture orbit are shown to scale in Figure 8.17. The approach could also be made from the dark side of the planet instead of the sunlit side. The approach hyperbola and capture ellipse would be the mirror image of that shown, as is the case in Figure 8.12.
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 ^{\star} & 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 (μ) 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 |
