Question 8.4: A spacecraft is launched on a mission to Mars starting from ......
A spacecraft is launched on a mission to Mars starting from a 300 km circular parking orbit. Calculate (a) the delta-v required, (b) the location of perigee of the departure hyperbola, and (c) the amount of propellant required as a percentage of the spacecraft mass before the delta-v burn, assuming a specific impulse of 300 s.
From Tables A.1 and A.2, we obtain the gravitational parameters for the sun and the earth,
\begin{aligned}& \mu_{\text {sun }}=1.327 \times 10^{11} km ^3 / s ^2 \\& \mu_{\text {earth }}=398,600 km ^3 / s ^2\end{aligned}and the orbital radii of the earth and Mars,
\begin{aligned}& R_{\text {earth }}=149.6 \times 10^6 km \\& R_{\text {Mars }}=227.9 \times 10^6 km\end{aligned}(a) According to Eqn (8.35), the hyperbolic excess speed is
ν_{\infty}=\sqrt{\cfrac{\mu_{ sun }}{R_1}}\left(\sqrt{\cfrac{2 R_2}{R_1+R_2}}-1\right) (8.35)
ν_{\infty}=\sqrt{\cfrac{\mu_{\text {sun }}}{R_{\text {earth }}}}\left(\sqrt{\cfrac{2 R_{\text {Mars }}}{R_{\text {earth }}+R_{\text {Mars }}}}-1\right)=\sqrt{\frac{1.327 \times 10^{11}}{149.6 \times 10^6}}\left(\sqrt{\cfrac{2\left(227.9 \times 10^6\right)}{149.6 \times 10^6+227.9 \times 10^6}}-1\right)from which
ν_{\infty}=2.943 km / sThe speed of the spacecraft in its 300 km circular parking orbit is given by Eqn (8.41),
ν_c=\sqrt{\cfrac{\mu_1}{r_p}} (8.41)
ν_c=\sqrt{\cfrac{\mu_{\text {earth }}}{r_{\text {earth }}+300}}=\sqrt{\cfrac{398,600}{6678}}=7.726 km / sFinally, we use Eqn (8.42) to calculate the delta-v required to step up to the departure hyperbola.
(b) Perigee of the departure hyperbola, relative to the earth’s orbital velocity vector, is found using Eqn (8.43),
\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)
\begin{aligned}& \beta=\cos ^{-1}\left\lgroup \cfrac{1}{1+\frac{r_p ν_{\infty}^2}{\mu_{\text {earth }}}}\right\rgroup =\cos ^{-1}\left\lgroup\cfrac{1}{1+\frac{6678 \cdot 2.943^2}{368,600}}\right\rgroup \\\\&\boxed{ \beta=29.16^{\circ} }\end{aligned}Figure 8.12 shows that the perigee can be located on either the sunlit or the dark side of the earth. It is likely that the parking orbit would be a prograde orbit (west to east), which would place the burnout point on the dark side.
(c) From Eqn (6.1), we have
Substituting \Delta ν=3.590 km / s , I_{s p}=300 s , \text { and } g _o=9.81 \times 10^{-3} km / s ^2, this yields
\boxed{\cfrac{\Delta m}{m}=0.705}That is, prior to the delta-v maneuver, over 70% of the spacecraft mass must be propellant.
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 |
