Kepler's Laws

What just happened?

The planet orbits along an ellipse, and the Sun sits at one of its foci — not at the center. Close to the Sun (perihelion) it races fastest, far away (aphelion) it slows down. The two shaded sectors have equal areas, because the planet sweeps them out in the same time.

First law (orbit shape):

The orbit of every planet is an ellipse with the Sun at one of its foci. The eccentricity $e$ tells you how flattened the ellipse is: $e=0$ is a circle, $e\to 1$ is a very elongated ellipse.

Third law (period and orbit size):

The square of the orbital period is proportional to the cube of the semi-major axis. The same constant holds for all planets of a given system — which is why more distant planets orbit more slowly. In units where $a$ is in AU and $T$ in years, this gives $T=a^{3/2}$.

Second law (conservation of angular momentum):

The areal velocity (the area swept out per unit time) is constant — another way of writing conservation of angular momentum. When $r$ decreases, the angular velocity must increase, so at perihelion the planet races fastest. This ties in with the law of universal gravitation: it is the Sun's gravity that bends the path into an ellipse and sustains the motion.

Why this matters for your exam:

The third law $T^2/a^3=\text{const}$ solves "compare the periods/distances of two planets or satellites" problems, while the second law explains why the speed changes along the orbit. The trick: write the constant for one body and set it equal to the other.

Remember: an orbit is an ellipse with the Sun at a focus, $T^2/a^3$ is constant for the whole system, and the planet is fastest at perihelion.