MIT OpenCourseWare http://ocw.mit.edu 16.346 Astrodynamics Fall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.Exercises 05 An extraordinary alignment of the planets occurred on June 17, 1991. Try the Solar System Simulator on the internet at (space.jpl.nasa.gov) 1. Plot the positions of the four planets: Venus, Earth, Mars and Jupiter for this date corresponding to Julian day number 2,448,424. For convenience, assume the planets are coplanar. Use the tabulated data for the epoch 1960 January 1.5 ephemeris time corresponding to Julian day number 2,436,935: Planet Ω deg deg deg e a a.u. i deg Mercury 47.85714 76.83309 222.62165 0.205627 0.387099 7.00399 Venus 76.31972 131.00831 174.29431 0.006793 0.723322 3.39423 Earth 102.25253 100.15815 0.016726 1.000000 0.0 Mars 49.24903 335.32269 258.76729 0.093368 1.523691 1.84991 Jupiter 100.04444 13.67823 259.83112 0.048435 5.202803 1.30536 Saturn 113.30747 92.26447 280.67135 0.055682 9.538843 2.48991 Uranus 73.79630 170.01083 141.30496 0.047209 19.181951 0.77306 Neptune 131.33980 44.27395 216.94090 0.008575 30.057779 1.77375 See also (http://ssd.jpl.nasa.gov/) for the Planetary Orbital Elements for which the epoch is J2000 or 2000 January 1.5 2. Show that the equations for the position and velocity vectors in reference coordinates for zero inclination angle are r = r cos L i + r sin L i x y v = − µh(sin L + e sin ) i x + µh(cos L + e cos ) i y where L = + f is the true longitude and =Ω+ ω is the longitude of pericenter. 3. Verify the mean anomalies, expressed in radians, of the planets for the given date are: MV =1.573746963 ME =2.81988346 MM =3.212448431 MJ =2.108102726 Use the following tabulated values for the mean motion n: nV =1.602131◦/day nE =0.985609◦/day nM =0.524033◦/day nJ =0.083091◦/day4. The position and velocity vectors of a spacecraft are 1 r = −√2(35 i +4 i +3i )x y z 1 v = − 25√2(5 i x +4 i y +3i z ) with units so chosen that the gravitational constant µ is unity. Determine the param-eter, the eccentricity, the longitude of the ascending node, the argument of pericenter, the inclination and the eccentric anomaly of the orbit. 5. The position vector of a spacecraft is r =3 i +4 i x y If the eccentricity vector is e = 21 i z , find the parameter of the orbit and the eccentric anomaly of the spacecraft. 6. A transit of Venus across the sun took place on June 8, 2004. The last time it happened was 121 years ago. Calculate the positions of the Earth and Venus on the day of the most recent transit. Here again use the Solar System Simulator on the internet at (space.jpl.nasa.gov). Euler is buried in St. Petersburg (Petrograd), Russia in a cemetery with all the famous Russians. He is there because he was a protege of the Empress Catherine the
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