Question 8.10: In Example 8.8, calculate the delta-v required to place the ......
In Example 8.8, calculate the delta-v required to place the spacecraft in an elliptical capture orbit around Mars with a periapsis altitude of 300 km and a period of 48 h. Sketch the approach hyperbola.
From Tables A.1 and A.2, we know that
\begin{aligned}& r_{\text {Mars }}=3380 km \\& \mu_{\text {Mars }}=42,830 km ^3 / s ^2\end{aligned}The radius to periapsis of the arrival hyperbola is the radius of Mars plus the periapsis of the elliptical capture orbit,
r_p=3380+300=3680 kmAccording to Eqn (8.40) and Eqn (b) of Example 8.8, the speed of the spacecraft at periapsis of the arrival hyperbola is
ν_p=\cfrac{h}{r_p}=\sqrt{ν_{\infty}^2+\cfrac{2 \mu_1}{r_p}} (8.40)
\left.v_p\right)_{\text {hyp }}=\sqrt{\left.\left[v_{\infty}\right)_{\text {Arrival }}\right]^2+\cfrac{2 \mu_{\text {Mars }}}{r_p}}=\sqrt{2.8852^2+\cfrac{2 \cdot 42,830}{3680}}=5.621 km / sTo find the speed \left.v_p\right)_{ ell } at periapsis of the capture ellipse, we use the required period (48 h) to determine the ellipse’s semimajor axis, using Eqn (2.83),
\boxed{T=\cfrac{2 \pi}{\sqrt{\mu}} a^{\frac{3}{2}}} (2.83)
a_{ ell }=\left(\cfrac{T \sqrt{\mu_{\text {Mars }}}}{2 \pi}\right)^{\frac{3}{2}}=\left(\cfrac{48 \cdot 3600 \cdot \sqrt{42,830}}{2 \pi}\right)^{\frac{3}{2}}=31,880 kmFrom Eqn (2.73), we obtain
r_p=a(1-e) (2.73)
e_{\text {ell }}=1-\cfrac{r_p}{a_{\text {ell }}}=1-\cfrac{3680}{31,880}=0.8846Then Eqn (8.59) yields
\left.ν_p\right)_{\text {capture }}=\sqrt{\cfrac{\mu_2(1+e)}{r_p}} (8.59)
\left.v_p\right)_{ ell }=\sqrt{\cfrac{\mu_{\text {Mars }}}{r_p}\left(1+e_{ ell }\right)}=\sqrt{\cfrac{42,830}{3680}(1+0.8846)}=4.683 km / sHence, the delta-v requirement is
\left.\left.\Delta v=v_p\right)_{\text {hyp }}-v_p\right)_{\text {ell }}=\boxed{0.9382 km / s}The eccentricity of the approach hyperbola is given by Eqn (8.38),
e=1+\cfrac{r_p ν_{\infty}^2}{\mu_1} (8.38)
e=1+\cfrac{r_p v_{\infty}^2}{\mu_{\text {Mars }}}=1+\cfrac{3680 \cdot 2.8851^2}{42,830}=1.715Assuming that the capture ellipse is a polar orbit of Mars, then the approach hyperbola is as illustrated in Figure 8.30. Note that Mars’ equatorial plane is inclined 25° to the plane of its orbit around the sun. Furthermore, the vernal equinox of Mars lies at an angle of 85° from that of the earth.
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 (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 |
