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Physics

Kepler's Third Law

The orbital period depends on the cube of the semi-major axis and the central body’s mass.

T = 2 π \frac{a ^{3}}{GM}

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Source & review

Difficulty

Intermediate

Read time

5 min

Prerequisites

Basic algebra, consistent units, and the stated model assumptions

Review statusNeeds review

Source

OpenStax, College Physics 2e, §6.6 (Paul Peter Urone and Roger Hinrichs)

OpenFormulaGraph equation reference; FormuLab adaptation details and license are provided below.

Adapted from OpenFormulaGraph, under CC BY 4.0.

FormuLab adapted wording, notation, applicability conditions and topic placement, and expanded the numerical check into a worked example.

Definition

Understanding the core concept

The orbital period depends on the cube of the semi-major axis and the central body’s mass. For a circular orbit, the semi-major axis is the orbital radius.

Variables & Units

Understanding each component

Symbol	Meaning	Units
$T$	Period (orbital)	$s$
$a$	Orbital semi-major axis	$m$
$G$	Gravitational Constant	$N \cdot m^{2} /k g^{2}$
$M$	Mass (central body)	$kg$

Real-World Applications

Where this formula is used in practice

Orbital timing

Estimate the period of a bound satellite orbit.

Orbit comparison

Relate different semi-major axes to orbital periods.

Worked Example

Step-by-step calculation with real numbers

Problem

Find the orbital period of a small Earth satellite with semi-major axis 6.771 × 10⁶ m using the listed constants.

Given

a = 6771000 m

G = 6.674 \times 1 0^{- 11} N \cdot m^{2} /k g^{2}

M = 5.972 \times 1 0^{24} kg

Solution

T = 2 π \frac{a ^{3}}{GM}

T = 2 \cdot π \cdot \frac{( 6771000 m ) ^{3}}{( ( 6.674 \times 1 0 ^{- 11} N \cdot m ^{2} /k g ^{2} ) \cdot ( 5.972 \times 1 0 ^{24} kg ) )}

T \approx 5545 s

Final Answer

T \approx 5545 s

Formula Guide

Info

DifficultyIntermediate

Read Time5 min

PrerequisitesBasic algebra, consistent units, and the stated model assumptions

Tools

Source & review

StatusNeeds review

Last reviewedPending

Source

OpenStax, College Physics 2e, §6.6 (Paul Peter Urone and Roger Hinrichs)

OpenFormulaGraph equation reference; FormuLab adaptation details and license are provided below.

Adapted from OpenFormulaGraph, under CC BY 4.0.

FormuLab adapted wording, notation, applicability conditions and topic placement, and expanded the numerical check into a worked example.

Assumptions

• Use a bound Newtonian two-body orbit with satellite mass negligible compared with the central mass.
• For comparable masses, replace M by the sum of both masses.

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Kepler's Third Law

Orbital timing

Orbit comparison

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