ASTR 109 HOMEWORK #3 (Hamilton) Solutions

1. Build a Planet.
• a) MERCURY 5430 kgm-3: made of roughly 1/2 iron and 1/2 rock.
VENUS 5250 kgm-3: made of roughly 1/2 iron and 1/2 rock.
EARTH 5520 kgm-3: made of roughly 1/2 iron and 1/2 rock.
MARS 3950 kgm-3: made of mostly rock with some iron.
JUPITER 1330 kgm-3: made of mostly water with some rock.
SATURN 690 kgm-3: not made of water, rock or iron! Its density is too low.
URANUS 1290 kgm-3: made of mostly water with some rock.
NEPTUNE 1640 kgm-3: made of mostly water with some rock. More rock than Uranus.
PLUTO 2030 kgm-3: made of roughly 1/2 rock and 1/2 water (ice).
• b) Saturn must be made of a less dense material, presumably gas.
• c) The giant planets have an even higher percentage of low density stuff than we assumed above. Saturn must be mostly gas. Jupiter and Uranus must also be mostly gas as well. Neptune must be mostly gas and water.
2. Kepler's Laws
• a) It takes about 1.7 years for Venus to lap the Earth. Mercury is even faster, lapping the Earth three times per year. This illustrates Kepler's third law: Planets that are closer to the Sun move faster than those that are further away.
• b) Low eccentricity ellipses look like circles that have been offset from the Sun.
• c) The asteroid orbits fastest at its closest approach to the Sun (its pericenter). The asteroid completes one orbit per year - after one orbit, the Earth and asteroid are both back to where they started.
3. Travel to Mars!
• a) Mars and Earth should be as far apart as possible on your diagram.
• b) Closest: 0.5 AU, Furthest 2.5 AU.
• c) Your curve should go half way around the Sun, increasing steadily in distance the whole way.
• d) Distance ~ 1/2 (π * 1AU + π * 1.5AU) = 1.25&pi AU~ 4AU.
• e) Earth's Period: 1 year. Mars' Period: 1.88 years. Flight time
= average of (1/2 Earth's Period + 1/2 Mars' Period)
= 1/2 * (1/2 year + 1.88/2 years)
= 0.72 years
= 8.6 months.
• f) When Mars is close, radio signals take 4 minutes, but when it is at its most distant, the signals take 20 minutes.