Kepler's First Law (Law of Ellipses)

Planets move in elliptical orbits with the Sun at one focus.
Equation:

r(\theta) = \frac{a(1 - e^2)}{1 + e \cos\theta}

Where:

• r(θ) = distance from the planet to the Sun at angle θθ.

• a = semi-major axis of the ellipse.

• e = eccentricity of the orbit.

Kepler's Second Law (Law of Areas)

A line joining a planet to the Sun sweeps out equal areas in equal times.
Equation:

\frac{dA}{dt} = \frac{L}{2m} \quad \text{(constant)}

Where:

• dA/dt​ = rate of area swept.

• L = angular momentum.

• m = mass of the planet.

Alternatively, in terms of angular velocity:

r^2 \frac{d\theta}{dt} = \text{constant}

Kepler's Third Law (Law of Periods)

The square of the orbital period TT is proportional to the cube of the semi-major axis aa.
Equation:

T^2 = \frac{4\pi^2}{G(M + m)} a^3

Where:

• T = orbital period.

• G = gravitational constant.

• M = mass of the central body (e.g., the Sun).

• m = mass of the orbiting body (e.g., a planet).

For M≫m, this simplifies to:

T^2 \propto a^3 \quad \text{(in astronomical units)}

Simplified Form for the Solar System

When using astronomical units (AU), solar masses (M⊙​), and years (T):

T^2 = a^3

This is the classic form of Kepler’s Third Law in the solar system context.

All this calculated by a former Musketeer who was almost killed by an accordion.

Staring off with the expression of a man who just discovered the orbital eccentricity of Mars and is wondering if anyone remembered to pay the candle tax. No money, no flair, no court composer backing him — just brutal math, a handmade telescope, and maybe a borrowed quill.