Planetary motion

Throughout history, people have tried to explain the cosmos and the motion of celestial bodies, with varying degrees of success.

ancient Greeks
The Earth is stationary at the centre of the universe (geocentric model). The stars move on the inside of a huge crystal sphere. The Sun, Moon, and stars have different motion, so they move on different spheres. There retrograde motion of Mars is baffling.
There are eight crystal spheres, each on its own axis.
Actually, there has to be 54 spheres.
The earth isn’t at the exact centre of the universe. It’s sightly off.
The Sun is at the centre of the universe (heliocentric model), and the Earth and other planets revolve around it. The motion of the Earth explains the apparent motion of the stars.
Galileo confirmed the heliocentric model using the telescope. He observed the moons of Jupiter and the phases of Venus. He still predicted circular orbits.
Brahe measured astronomical positions with great precision. He observed a comet that followed an elliptical path, not a circular one.

Kepler analyzed Brahe’s data. He made three conclusions, known as Kepler’s laws of planetary motion:

  1. The orbit of a planet is an ellipse with the Sun at one focus.
  2. The line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time.
  3. The period of the orbit is related to the mean radius by `K = r^3//T^2`, where `K` is a constant equal to 3.35 × 1018 m3/s2.

Finally, we come to Newton. Sir Isaac Newton discovered the universal law of gravitation,

`|vec F_"g"| = G(m_1m_2)/r^2`,

where `G` is a constant equal to 6.67 × 10−11 N·m2/kg2.

Sometimes, we will want to look at how gravitation force changes in terms of ratios. To do that, we can use the following equation:

`F_"g,2"/F_"g,1" = (m_2/m_1)(r_1/r_2)^2`.

Another type of problem involves whether or not an orbit is stable. For our purposes, an orbit is stable if and only if

`vec F_"net" = vec F_"g" = mvec a_"c"`.

Similar to a turning car, `F_"g" < F_"c"` means that the object with fly off into space. On the other hand, `F_"g" > F_"c"` means that the object will spiral inwards into the Sun (or whatever it’s orbiting), whereas excess friction with a car is no problem.