Question 8.7: Find the distance between earth and Mars at 12 h UT on Augus......
Find the distance between earth and Mars at 12 h UT on August 27, 2003. Use Algorithm 8.1.
Step-by-Step
Step 1:
According to Eqn (5.48), the Julian day number J_0 for midnight (0 h UT) of this date is
J_0=367 y-\operatorname{INT}\left\{\cfrac{7\left[y+\operatorname{INT}\left(\cfrac{m+9}{12}\right)\right]}{4}\right\}+\operatorname{INT}\left(\cfrac{275 m}{9}\right)+d+1,721,013.5 (5.48)
\begin{aligned}J_0 & =367 \cdot 2003- INT \left\{\cfrac{7\left[2003+ INT \left(\frac{8+9}{12}\right)\right]}{4}\right\}+ INT \left(\cfrac{275 \cdot 8}{9}\right)+27+1,721,013.5 \\\\& =735,101-3507+244+27+1,721,013.5 \\\\& =2,452,878.5\end{aligned}
At UT = 12, the Julian day number is
J D=2,452,878.5+\cfrac{12}{24}=2,452,879.0Step 2:
The number of Julian centuries between J2000 and this date is
Step 3:
Table 8.1 and Eqn (8.93b) yield the orbital elements of earth and Mars at 12 h UT on August 27, 2003.
Q=Q_0+\dot{Q} T_0 (8.93b)
Table 1
Step 4:
\begin{aligned}& h_{\text {earth }}=4.4451 \times 10^9 km ^2 / s \\& h_{\text {Mars }}=5.4760 \times 10^9 km ^2 / s\end{aligned}Step 5:
\begin{aligned}& \omega_{\text {earth }}=(\varpi-\Omega)_{\text {earth }}=102.95-0=102.95^{\circ} \\& \omega_{\text {Mars }}=(\varpi-\Omega)_{\text {Mars }}=336.07-49.549=286.52^{\circ} \\& M_{\text {earth }}=(L-\varpi)_{\text {earth }}=335.27-102.95=232.32^{\circ} \\& M_{\text {Mars }}=(L-\varpi)_{\text {Mars }}=334.51-336.07=-1.56^{\circ}\left(358.43^{\circ}\right)\end{aligned}
Step 6:
\begin{aligned}& E_{\text {earth }}-0.016710 \sin E_{\text {earth }}=232.32^{\circ}(\pi / 180) \Rightarrow E_{\text {earth }}=231.57^{\circ} \\& E_{\text {Mars }}-0.093397 \sin E_{\text {Mars }}=358.43^{\circ}(\pi / 180) \Rightarrow E_{\text {Mars }}=358.27^{\circ}\end{aligned}
Step 7:
\begin{aligned}& \theta_{\text {earth }}=2 \tan ^{-1}\left(\sqrt{\cfrac{1+0.016710}{1-0.016710}} \tan \cfrac{231.57^{\circ}}{2}\right)=-129.18^{\circ} \Rightarrow \theta_{\text {earth }}=230.82^{\circ} \\\\& \theta_{\text {Mars }}=2 \tan ^{-1}\left(\sqrt{\cfrac{1+0.093397}{1-0.093397}} \tan \cfrac{358.27^{\circ}}{2}\right)=-1.8998^{\circ} \Rightarrow \theta_{\text {Mars }}=358.10^{\circ}\end{aligned}
Step 8:
From Algorithm 4.5,
\begin{aligned}& R _{\text {earth }}=(135.59 \hat{ I }-66.803 \hat{J }-0.00056916 \hat{ K }) \times 10^6 ( km ) \\& V _{\text {earth }}=12.680 \hat{ I}+26.610 \hat{ J }-0.00022672 \hat{ K } ( km / s ) \\& R _{\text {Mars }}=(185.95 \hat{ I}-89.959 \hat{ J }-6.4534 \hat{ K }) \times 10^6 ( km ) \\& V _{\text {Mars }}=11.478 \hat{I}+23.881 \hat{ J }+0.21828 \hat{ K } ( km / s )\end{aligned}
The distance d between the two planets is therefore
d=\left\| R _{\text {Mars }}- R _{\text {earth }}\right\|
=\sqrt{(185.95-135.59)^2+[-89.959-(-66.803)]^2+(-6.4534-0.00056916)^2} \times 10^6
or
\boxed{d=55.80 \times 10^6 km}The positions of earth and Mars are illustrated in Figure 8.26. It is a rare event for Mars to be in opposition (lined up with earth on the same side of the sun) when Mars is at or near perihelion. The two planets had not been this close in recorded history.
Table 8.1 Planetary Orbital Elements and Their Centennial Rates
| \begin{aligned}& a( AU ) \\& \dot{a}( AU / Cy )\end{aligned} | \begin{aligned}& e \\& \dot{e}(1 / C y)\end{aligned} | \begin{aligned}& i\left({ }^{\circ}\right) \\& i\left({ }^{\circ} / Cy \right)\end{aligned} | \begin{aligned}& \Omega\left({ }^{\circ}\right) \\& \dot{\Omega}\left({ }^{\circ} / Cy \right)\end{aligned} | \begin{aligned}& \varpi\left(^{\circ}\right) \\& \dot{\varpi}\left({ }^{\circ} / Cy \right)\end{aligned} | \begin{aligned}& L\left({ }^{\circ}\right) \\& \dot{L}\left({ }^{\circ} / Cy \right)\end{aligned} | |
| Mercury | 0.38709927 | 0.20563593 | 7.00497902 | 48.33076593 | 77.45779628 | 252.25032350 |
| 0.00000037 | 0.00001906 | -0.00594749 | -0.12534081 | 0.16047689 | 149472.67411175 | |
| Venus | 0.72333566 | 0.00677672 | 3.39467605 | 76.67984255 | 131.6024672 | 181.97909950 |
| 0.00000390 | -0.00004107 | -0.0007889 | -0.27769418 | 0.00268329 | 58517.81538729 | |
| Earth | 1.00000261 | 0.01671123 | -0.00001531 | 0.0 | 102.93768193 | 100.46457166 |
| 0.00000562 | -0.00004392 | -0.01294668 | 0.0 | 0.32327364 | 35999.37244981 | |
| Mars | 1.52371034 | 0.09339410 | 1.84969142 | 49.55953891 | 23.94362959 | -4.55343205 |
| 0.0001847 | 0.00007882 | -0.00813131 | -0.29257343 | 0.44441088 | 19140.30268499 | |
| Jupiter | 5.20288700 | 0.04838624 | 1.30439695 | 100.47390909 | 14.72847983 | 34.39644501 |
| -0.00011607 | 0.00013253 | -0.00183714 | 0.20469106 | 0.21252668 | 3034.74612775 | |
| Satum | 9.53667594 | 0.05386179 | 2.48599187 | 113.66242448 | 92.59887831 | 49.95424423 |
| -0.00125060 | -0.00050991 | 0.00193609 | -0.28867794 | -0.41897216 | 1222.49362201 | |
| Uranus | 19.18916464 | 0.04725744 | 0.77263783 | 74.01692503 | 170.95427630 | 313.23810451 |
| -0.00196176 | -0.00004397 | -0.00242939 | 0.04240589 | 0.40805281 | 428.48202785 | |
| Neptune | 30.06992276 | 0.00859048 | 1.77004347 | 131.78422574 | 44.96476227 | -55.12002969 |
| 0.00026291 | 0.00005105 | 0.00035372 | -0.00508664 | -0.32241464 | 218.45945325 | |
| (Pluto) | 39.48211675 | 0.24882730 | 17.14001206 | 110.3039368 | 224.0689163 | 238.92903833 |
| -0.00031596 | 0.00005170 | 0.00004818 | -0.01183482 | -0.04062942 | 145.20780515 |
Source: From Standish et al. (2013). Used with permission.
Table 1
| a(km) | e | i\left({ }^{\circ}\right) | \Omega \left({ }^{\circ}\right) | \varpi\left({ }^{\circ}\right) | L\left({ }^{\circ}\right) | |
| Earth | 1.4960 \times 10^8 | 0.016710 | -0.00048816 | 0.0 | 102.95 | 335.27 |
| Mars | 2.2794 \times 10^8 | 0.093397 | 1.8494 | 49.549 | 336.07 | 334.51 |
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