Definition: Newton's Law of Gravitation

The gravitational force between two objects $m_1$ and $m_2$ is equal to $$F=\frac{Gm_1m_2}{r^2}$$The gravitational force is always attractive. The gravitational potential energy between two objects, with $U=0$ at infinite distance, is therefore $$U=-\int_r^\infty \frac{Gm_1m_2}{r^2}dr=-\frac{Gm_1m_2}{r}$$

Theorem: Shell theorem

The gravitational acceleration within a hollow, uniformly dense spherical shell is always 0, while the gravitational acceleration outside the shell is as if all the mass of the shell were concentrated at its center. A natural consequence is that when standing inside a solid planet with spherical symmetry, you only feel the gravitational force due to the portion of the planet underneath you.

Idea: Cosmic speeds

The first cosmic speed $v_1$ is defined as the speed necessary for a planet at a distance $r$ from a star $M$ to move in a circular orbit around the star. The second cosmic speed $v_2$ is defined as the necessary speed for the planet to escape the star, i.e. move an infinite distance away from the star, starting from a distance $r$. It's helpful to memorize these speeds so you don't need to rederive them every time you use them. $$v_1=\sqrt{\frac{GM}{r}}\qquad v_2=\sqrt{\frac{2GM}{r}}$$

Theorem: Conservation laws

The total energy of an orbiting planet $m$ around a star $M$ with semimajor axis length $a$ is conserved, and is equal to $$E=-\frac{GMm}{2a}$$If we choose the star as our origin, there exists no torque acting on the planet, so angular momentum is also conserved, which implies that $$vr_\perp=\text{const}$$

Definition: Kepler's laws

Kepler's laws define the motion of closed orbits. Consider a planet orbiting a star with mass $M$. Kepler's law states that

1. The planet's orbit traces a conic section (ellipse, parabola, or hyperbola) with the star at a focus.
2. By drawing lines from the planet to the star, the planet sweeps out equal areas in its orbit per unit time. This statement is equivalent to conservation of angular momentum.
3. For a planet in a closed elliptical orbit of semimajor axis length $a$, the period of the orbit is $$T=2\pi \sqrt{\frac{a^3}{GM}}$$

Tip: Reduced Mass

One way to deal with two body problems is with \highlight{reduced mass}. The separation between the two bodies, $r(t)$ is the equivalent to the problem with a mass of $$M = m_1+m_2$$ in the center and an orbiting $$\mu = \frac{m_1m_2}{m_1+m_2}.$$

Tip: Effective Mass

Another way is to place an effective mass that causes the same force, $$m_{\text{eff}} = \frac{m_2^3}{(m_1 + m_2)^2},$$ with $m_1$ going around. This is particularly useful for looking at the actual trajectory of one of the masses.