Homework 8 due March 8, 2004

  1. Using the Starry Night software, switch off the daylight and set your latitude to 90o (North Pole). Center and lock on Jupiter and zoom in until the four Galilean moons (from close to far they are Io, Europa, Ganymeade, and Callisto). are plainly visible. Note the positions of the moons. Step forward in increments of 1 hour and calculate the periods of all four moons. Then, using Kepler’s third law, calculate the distances from Jupiter of Europa, Ganymeade, and Callisto in terms of Io’s distance. (For example Ganymeade is ___ Io distances from Jupiter). You do not need to know Jupiter’s mass or Io’s distance in km (or AU) to do this. All you need to assume is that the mass of the moons are small compared to the mass of Jupiter. Do your results agree roughly with what you see on Starry Night?

    2. I find that the period of Io is roughly 42 hours, Europa's period is roughly 85 hours, Ganymede's period is roughly 171 hours, and Callisto's period is roughly 400 hours.

    I am going to compare the period of Europa to the period of Io using Kepler's third law.

    PIo2=(4*p2/GMJupiter)*R3Io Jupiter

    PEuropa2=(4*p2/GMJupiter)*R3Europa Jupiter

    Note that we have treated the mass of the moons as insignificant in comparison to the mass of Jupiter. Now we divide the first equation by the second to arrive at

    (PIo/PEuropa)2=(RIo-Jupiter/REuropa-Jupiter)3.

    This implies that Europa is 1.6 times as far from Jupiter as Io is. A similar calculation shows that Ganymede is 2.5 Io distances from Jupiter and Callisto is 4.5 Io distances away. If you look at the orbits of the moons, these numbers seem to be correct.

  2. A satellite is said to be in a "geosynchronous" orbit if it appears always to remain over the exact same spot on Earth.

    3. a). What is the period of this orbit?

    24 hours=1 day

    b). At what distance from the center of the Earth must such a satellite be placed into orbit? Express this distance as a fraction of the earth-moon (center to center) distance.

      Psatellite2=(4*p2/GMEarth)*R3satellite-Earth

      Pmoon2=(4*p2/GMEarth)*R3moon-earth

      Note that we have treated the mass of the moon and the satellite as insignificant in comparison to the mass of the Earth. Now we divide the second equation by the first to arrive at

      (Pmoon/Psatellite)2=(Rmoon-earth/Rsatellite-earth)3

    Using 27.3 days for the period of the moon, we find that the moon-earth distance is 9x the satellite-earth distance, or equivalently, the satellite's distance to the earth is 1/9 the moon's distance to the earth.

    c). Explain why the orbit must be in the plane of the Earth’s equator.

    Note: There is only one orbit which is geosynchronous. Often one will read an article in the news about how crowded this orbit has become. Almost all communications satellites are in this orbit.

  3. Suppose you have discovered an alien solar system in which a planet circles a star once every three years at an average distance of 9 AU. How does the mass of this star compare with that of our Sun? (Assume the planet’s mass is very small compared to the Sun’s).

    Compare the alien system with that of the earth orbitting the Sun.

        Pplanet2=(4*p2/GMStar)*R3planet

        Pearth2=(4*p2/GMSun)*R3Earth

        Note that we have treated the mass of the planet and the earth as insignificant in comparison to the mass of the star or the sun. Now we divide the first equation by the second to arrive at

        (PPlanet/PEarth)2=(Msun/Mstar)*(Rplanet/Rearth)3

        (3)2=(Msun/Mstar)*(9)3

        This implies that the star is 81 times the mass of the sun.