Closing thoughts

1. Kepler's three laws for planet orbits. In particular, we used the third law to relate the period P of a planet to the radius a of its orbit: if we measure P in years and a in AU then
   

2. Newton's three laws of motion. In particular, the second law is

3. Newton's law of gravity:
   

4. Observations of a moon going around a planet or a spacecraft going near it, together with the law of gravity, can be used to determine the mass of the planet or other body.
5. From the mass M of a planet or moon and its radius r, we can compute its density. If we measure everything relative to the earth, the formula is simple:
   
   where M_E is the mass of the earth, where R_E is the radius of the earth, and (density)_E is the density of the earth. Then the density tells us something what the planet is made out of.
6. From the mass and radius, we can also comute the square of the escape velocity from the planet's surface:
   
   This can tell us how well the planet could hold an atmosphere.
7. Radiometric dating is based on the decay of radioactive nuclei. If nucleus A decays to nucleus B with a half life T, then after a time T, half of the A nuclei are left; after a time 2T, 1/4 of the A nuclei are left etc..
8. Gravity pulls everything together.
9. Gas pressure can resist the pull of gravity and stabalize a ball of gas.
   • The hotter the gas, the bigger the pressure.
   • Compression of the gas makes it hotter.
   • In a star, nuclear reactions make it very hot inside.
10. Stuff in orbit around a central heavy body is also stable, as long as the orbits don't intersect
    • If there is lots of stuff, the way to do this is to have each piece orbiting in its own circular orbit in a disk.
11. In a self-gravitating ball of fluid, the dense stuff sinks, the light stuff floats.
12. Tidal forces are often important. Consider, say, a moon of Jupiter. The gravitational force from Jupiter on the near side of the moon is greater than average, while the force on the far side is less than average. This difference is the tidal force, which tends to distort the shape of the moon.
13. Heat can move from one place to another by convection, conduction, or radiation.
    • Convection is important for the motions in the atmosphere of, for instance, the earth and Jupiter.
    • Convection also transfers heat from the middle of the earth and the surface, thus generating plate tectonics.
    • Radiation transfers energy from the sun to the planets.
    • The earth radiates away the energy it gets from the sun. The so-called greenhouse gases interfere with this. (The same holds for Venus.)
14. The rotation of the earth has a big effect on the motion of the atmosphere, bending the path of air clockwise in the northern hemisphere. The same holds for Jupiter.
15. Planets get magnetic fields when they have convection currents in an electrically conducting fluid and the currents are bent by the rotation effect. (Eg. the iron core of Earth, the liquid metallic hydrogen center of Jupiter.)

1. Astronomers didn't know if the same rules that apply on Earth should apply in the heavens.
   • They guessed ``no.''
   • In any case, they didn't have too much right about the laws on Earth.
   • This situation didn't change until Galileo and Newton and their contemporaries.
2. The simple questions of where things are and how they are moving were central.
3. Astronomers didn't know whether to put the earth at the center or the sun at the center.
   • They mostly guessed ``earth-centered.''
   • This situation didn't change until Copernicus.

1. the rotation of the earth once per day
2. the orbital motion of the earth about the sun once per year
3. the tilt of the earth's rotation axis relative to the plane of the ecliptic
4. the motion of the planets about the sun, with periods less than a year for Mercury and Venus and with periods of greater than a year for Mars, Jupiter, and Saturn (and Uranus, Neptune and Pluto).

1. We can think of viewing the sky once per day when the same star is overhead.
   • Then we think of the celestial sphere with the stars fixed on it.
   • We can plot the positions of the sun, moon, and planets on the celestial sphere.
   • The sun, moon, and planets move on the celestial sphere as the earth orbits the sun, the moon orbits the earth, and the planets orbit the sun.
2. Based on the orbits of Earth and a planet about the sun, we can plot the path of the planet relative to the earth. This is pretty straightforward for Mercury or Venus, a but more subtle for Mars, Jupiter, or Saturn. (We make the simplifying assumption that the orbits are all circles.)

There is a famous quote of Isaac Newton in a letter to Robert Hooke, February 5, 1675:

If I have seen further [than certain other men], it is by standing upon the shoulders of giants.

Surely Ptolemy was one of those giants.