Orbital Mechanics

Orbital Mechanics

Kepler and Newton gave us the laws that govern how objects orbit one another — from moons around planets to stars around black holes.

Kepler's Third Law

The orbital period $T$ of a body on a circular (or elliptical) orbit with semi-major axis $a$ around a central mass $M$ is:

$T^2 = \frac{4\pi^2 a^3}{GM}$

Solving for $T$:

$T = 2\pi \sqrt{\frac{a^3}{GM}}$

For Earth orbiting the Sun ($a = 1.496 \times 10^{11}$ m), this gives $T \approx 3.156 \times 10^7$ s — one year.

Circular Orbital Velocity

For a circular orbit, gravity provides exactly the centripetal force needed:

$v_c = \sqrt{\frac{GM}{r}}$

Earth's orbital velocity is approximately 29.8 km/s.

Orbital Energy

The total mechanical energy of a bound orbit (kinetic + potential) is:

$E = -\frac{GMm}{2a}$

Negative energy means the orbit is bound. The more negative (smaller $a$), the more tightly bound the orbit.

Your Task

Implement three functions. Use $G = 6.674 \times 10^{-11}$ m³ kg⁻¹ s⁻², defined inside each function.

• orbital_period_s(a_m, M_kg) — orbital period in seconds (Kepler's third law)
• circular_velocity_m_s(M_kg, r_m) — circular orbital speed in m/s
• orbital_energy_J(M_kg, m_kg, a_m) — total orbital energy in joules