Question 8.1: Calculate the synodic period of Mars relative to that of the......
Calculate the synodic period of Mars relative to that of the earth.
Step-by-Step
In Table A.1 we find the orbital periods of earth and Mars:
\left.T_{\text {earth }}=365.26 \text { days ( } 1 \text { year }\right)T_{\text {Mars }}=1 \text { year } 321.73 \text { days }=687.99 \text { days }
Hence,
T_{\text {syn }}=\cfrac{T_{\text {earth }} T_{\text {Mars }}}{\left|T_{\text {earth }}-T_{\text {Mars }}\right|}=\cfrac{365.26 \times 687.99}{|365.26-687.99|}=\boxed{777.9 \text { days }}These are earth days (1 day = 24 h). Therefore it takes 2.13 years for a given configuration of Mars relative to the earth to occur again.
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
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