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.

Pythagoras

There are eight crystal spheres, each on its own axis.

Aristotle

Actually, there has to be 54 spheres.

Ptolemy

The earth isn’t at the exact centre of the universe. It’s sightly off.

Copernicus

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

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

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 $K=r^3/T^2\text{,}$ 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,

$\left|{\overrightarrow{F}}_{\text{g}}\right|=G\frac{m_1{m}_2}{r^2}\text{,}$

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:

$\frac{F_{\text{g,2}}}{F_{\text{g,1}}}=\left(\frac{m_2}{m_1}\right){\left(\frac{r_1}{r_2}\right)}^2\text{.}$

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

${\overrightarrow{F}}_{\text{net}}={\overrightarrow{F}}_{\text{g}}=m{\overrightarrow{a}}_{\text{c}}\text{.}$

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