Orbital Mechanics Calculator

Calculate orbital parameters using Kepler's laws and the vis-viva equation. Compute escape velocities for any solar system body, and plan Hohmann transfer orbits between circular orbits. All three modes include interactive SVG orbit diagrams and step-by-step formulas with KaTeX rendering.

Orbit Visualization

Controls

Mode

Presets

Central Body

Semi-major Axis a (m)

Eccentricity e (0 to <1)

Results

Period T

1.54 hr

Periapsis r_p

6766.9 km

Apoapsis r_a

6775.1 km

v at Periapsis

7.677 km/s

v at Apoapsis

7.668 km/s

Specific Energy

-2.943e+7 J/kg

Kepler's Third Law

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

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

T = 2\pi\sqrt{\frac{(6771000)^3}{(6.674 \times 10^{-11})(5.972e+24)}}

T = 5545.0579 \text{ s} \approx 92.4176 \text{ min}

Vis-Viva Equation (velocity at periapsis)

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

v = \sqrt{(6.674 \times 10^{-11})(5.972e+24)\left(\frac{2}{6766937.4} - \frac{1}{6771000}\right)}

v = 7676.9228 \text{ m/s} \approx 7.6769 \text{ km/s}

Reference Guide

Kepler's Laws

First Law — Planets move in ellipses with the Sun at one focus.

Second Law — A line from the Sun to a planet sweeps out equal areas in equal times. Planets move faster at periapsis and slower at apoapsis.

Third Law — The square of the orbital period is proportional to the cube of the semi-major axis.

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

For the same central body, doubling $a$ multiplies the period by $2\sqrt{2} \approx 2.83$ .

Orbital Velocity (Vis-Viva)

The vis-viva equation gives the speed of an orbiting body at any point in its orbit, connecting position $r$ and semi-major axis $a$ .

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

For a circular orbit where $r = a$ , this simplifies to $v = \sqrt{GM/r}$ . At periapsis the body is fastest; at apoapsis it is slowest.

This equation also applies to hyperbolic escape trajectories when $a < 0$ .

Escape Velocity

The minimum speed needed to escape a body's gravitational pull without further propulsion. Setting total orbital energy to zero gives

v_{\text{esc}} = \sqrt{\frac{2GM}{R}}

Escape velocity is exactly $\sqrt{2}$ times the circular orbital velocity at the same radius. For Earth's surface, $v_{\text{esc}} \approx 11.2$ km/s.

Escape velocity depends only on mass and distance, not on the direction of launch or the mass of the escaping object.

Hohmann Transfer Orbits

The Hohmann transfer is the most fuel-efficient two-burn maneuver for moving between two coplanar circular orbits. It uses an elliptical transfer orbit that is tangent to both the initial and target orbits.

Burn 1 (departure) — Accelerate at the initial orbit to enter the transfer ellipse.

Burn 2 (arrival) — Accelerate at the target orbit to circularize.

\Delta v_{\text{total}} = \Delta v_1 + \Delta v_2

The transfer time is half the period of the transfer ellipse. Earth to Mars via Hohmann takes approximately 259 days.