Gravity as the geometry of an orbit
An orbit is continuous free fall: gravity bends a moving object’s path so that it repeatedly misses the central body. Spaceflight is therefore the controlled exchange of speed, direction and orbital energy.
Learning goals
- use force, field, potential and potential energy consistently;
- derive circular speed and period;
- interpret elliptical orbits, total energy and escape speed;
- explain transfer burns, the rocket equation and gravity assists;
- distinguish free-fall microgravity from absence of gravity.
1. Universal gravitation
Two point masses attract with $F=GMm/r^2$ along the line joining them. Outside a spherical body, its mass acts as if concentrated at its centre. The field strength is $g=GM/r^2$, the potential per unit mass is $\phi=-GM/r$, and the potential energy is $U=m\phi=-GMm/r$. Zero potential is chosen at infinity, so bound gravitational energy is negative.
| Quantity | Meaning | Unit |
|---|---|---|
| $g$ | force per unit test mass | N/kg |
| $\phi$ | potential energy per unit mass | J/kg |
| $U$ | energy of the two-body configuration | J |
2. Circular and elliptical orbits
For a circular orbit, gravity supplies centripetal acceleration: $GMm/r^2=mv^2/r$. Therefore $v_c=\sqrt{GM/r}$ and $T=2\pi\sqrt{r^3/(GM)}$. Higher circular orbits move more slowly but take longer to complete.
Kepler’s first law gives an ellipse with the central body at one focus; the second says equal areas are swept in equal times; the third gives $T^2\propto a^3$. The vis-viva equation $v^2=GM(2/r-1/a)$ applies anywhere on an ellipse of semi-major axis $a$.