HW 3, Problem 6

  • The Hubble Space Telescope (HST) is on a circular, low-Earth orbit, at an elevation h = 600 km above the Earth’s surface. What is its orbital period in minutes (to within 0.1 minute)?
  • Kepler's third law $$ \frac{a^3}{P^2} = \frac{G M}{4 \pi^2} $$  
    $$ \frac{(6 \times 10^5+ 6.371 \times 10^8)^3}{P^2} = \frac{6.674 \times 10^{-11} 6 \times 10^{24}}{4 \pi^2} $$  
    $$ P = \frac{1}{60} \sqrt{\frac{(6 \times 10^{5} + 6.371 \times 10^8)^3 4 \pi^2}{6.674 \times 10^{-11} 5.972 \times 10^{24}}}~\mathrm{min} $$  
    $$ = 96.3~\mathrm{minutes} $$

HW 3, Problem 7

  • For an observer who sees HST pass through the zenith, how long is HST above the horizon during each orbit in minutes (to within 0.1 minutes)?
     
    (This problem is a bit more obtuse than I had intended, so I'll give everyone credit for this one and it won't be on the exam.)

PE1, Problem 12

  • Ceres is a dwarf planet orbiting in the Asteroid belt 2.77 AU from the Sun. Like all members of the Asteroid belt, it is periodically buffeted by the passage of Jupiter orbiting at 5.20 AU. Use MCeres = 9.43×1020 kilograms, RCeres = 4.77 × 105 meters, MJupiter = 1.90 × 1027 kilograms and RJupiter = 6.99 × 107 meters. You may assume that both Ceres and Jupiter follow circular orbits. Calculate the gravitational acceleration at the surface of Ceres due to its mass.
  •  
    $$ g = \frac{G m_{\mathrm{Ceres}}}{r_{\mathrm{Ceres}}^2} - \frac{G m_{\mathrm{Jupiter}}}{r_{\mathrm{J-C~orbit}}^2} $$  
    $$ \frac{6.67 \times 10^{-11} 9.43 \times 10^{20}} {(4.77 \times 10^{5})^2} - \frac{6.67 \times 10^{-11} 1.90 \times 10^{27}} {(7.78 \times 10^{11})^2} $$  
    $$ (2.76 \times 10^{-1} - 2.09 \times 10^{-7})~ \mathrm{m}/\mathrm{s}^2 $$

HW 3 Problem 1

  • From question 3.2 in Ryden and Peterson. The asteroid Eros is seen in opposition from the Earth once every 847 days. What is the sidereal orbital period of Eros in days (to within 0.1 days)?
  • $$ P_{\mathrm{Eros}}^{-1} = \frac{1}{365.24} - \frac{1}{847} $$