All Labs

Gravity & Orbits Lab

Set a central mass and orbital parameters, then watch the animated orbit unfold. Record period, velocity, and energy data to verify Kepler's Third Law and explore how eccentricity affects orbital motion.

Guided Experiment: Verifying Kepler's Third Law

If you keep the central mass constant and change the semi-major axis, how do you expect T and T^2/a^3 to change? What should the ratio equal?

Write your hypothesis in the Lab Report panel, then click Next.

Orbit Visualization

OrbiterVelocityPeriapsisApoapsis

Controls

Central Mass (M)1.989e+30 kg
10²² kg2×10³⁰ kg
Semi-Major Axis (a)1.496e+11 m
10⁶ m5×10¹² m
Eccentricity (e)0.017
0 (circle)0.99 (elongated)
Orbiter Mass (m)5.972e+24 kg
1 kg10²⁷ kg
Animation Speed1.0×

Results

T2=4π2GM a3⇒(3.155e+7)2=4π2G⋅1.989e+30⋅(1.496e+11)3T^2 = \frac{4\pi^2}{GM}\,a^3 \quad\Rightarrow\quad (3.155e+7)^2 = \frac{4\pi^2}{G \cdot 1.989e+30} \cdot (1.496e+11)^3
Kepler Ratio Verification
T²/a³ = 2.974e-19vs4π²/(GM) = 2.974e-19
Error: 0.000000%
Period (T)
365.22 days
3.155e+7 s
Orbital Velocity
3.029e+4 m/s
Escape Velocity
4.248e+4 m/s
Periapsis
1.471e+11 m
Apoapsis
1.521e+11 m
Current Radius
1.471e+11 m
Total Energy
-2.650e+33 J
Potential Energy
-5.389e+33 J
Kinetic Energy
2.740e+33 J
Nearly circular orbit (e = 0.0167) — Circular velocity at a: 2.979e+4 m/s

Data Table

(0 rows)
#TrialSystemM(kg)a(m)eT(s)T²(s²)a³(m³)v_orbit(m/s)v_esc(m/s)
0 / 500
0 / 500
0 / 500

Reference Guide

Newton's Law of Gravitation

Every object attracts every other object with a force proportional to the product of their masses and inversely proportional to the square of the distance between them.

F=GMmr2F = \frac{GMm}{r^2}

G = 6.674 × 10⁻¹¹ N m²/kg² is the gravitational constant. M is the central body mass, m is the orbiter mass, and r is the distance between their centers.

Kepler's Third Law

The square of the orbital period is proportional to the cube of the semi-major axis, with the proportionality constant depending only on the central mass.

T2=4π2GM a3T^2 = \frac{4\pi^2}{GM}\,a^3

This means T²/a³ is the same for all orbits around the same body, regardless of eccentricity or orbiter mass. Earth, Mars, and Halley's Comet all share the same ratio when orbiting the Sun.

Vis-Viva Equation

The vis-viva equation gives the orbital speed at any point in an elliptical orbit as a function of the current distance r and the semi-major axis a.

v2=GM(2r−1a)v^2 = GM\left(\frac{2}{r} - \frac{1}{a}\right)

For a circular orbit (r = a), this simplifies to v = sqrt(GM/a). At periapsis the orbiter moves fastest; at apoapsis it moves slowest.

Orbital Energy

A bound orbit has negative total energy. The total mechanical energy depends only on the semi-major axis, not the eccentricity.

E=−GMm2a,U=−GMmr,K=12mv2E = -\frac{GMm}{2a}, \quad U = -\frac{GMm}{r}, \quad K = \frac{1}{2}mv^2

The escape velocity from distance r is v_esc = sqrt(2GM/r), which is sqrt(2) times the circular velocity at that radius. Any faster and the object escapes the gravitational pull.