Physics · Gravity and Orbits

Kepler's Third Law and Orbital Periods

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Wrap-up

What you learned

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Key takeaway

At fixed source mass, a larger circular orbit has both a longer circumference and a lower circular speed, so the period grows faster than radius alone.
Kepler's third law describes the circular-match case. If the speed factor moves away from 1.00, the path is no longer the circular orbit whose period the law describes.

Common misconception

A larger orbit takes longer only because the object has farther to go around, while the orbital speed stays about the same.

The path is longer, but that is only half of the story. The circular speed is also lower at larger radius because gravity is weaker there.

Carry the orbit-period idea forward

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Unbound thresholdGravity and Orbits

Escape Velocity

Launch outward from one bounded gravity source and see how source mass, launch radius, and total specific energy decide whether the object escapes or eventually returns.

Energy linkGravity and Orbits

Gravitational Potential and Potential Energy

See one source mass create a negative potential well, compare how potential and potential energy change with distance, and connect the downhill slope of phi to the gravitational field on the same live model.

Speed reviewGravity and Orbits

Circular Orbits and Orbital Speed

See why a circular orbit needs the right sideways speed, how gravity supplies the centripetal acceleration, and how source mass and radius together set orbital speed and period on one bounded live model.

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Reference and support

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Equation snapshotGravity from one source mass · Circular speedShow 2 equations

Gravity from one source mass
$g = \frac{M}{r ^{2}}$
In this bounded lab with displayed units using $G = 1$ , the inward gravitational acceleration is set by source mass and radius.
Circular speed
$v_{c} = \frac{M}{r}$
This is the sideways speed that makes gravity exactly match the turning requirement at the chosen radius.

Why it behaves this way

Explanation

Kepler's third law is the timing consequence of the same circular-orbit balance you met in orbital speed. A larger circular orbit takes longer not just because the path is longer, but also because the allowed circular speed is smaller at larger radius.

This lab keeps one source mass, one chosen reference orbit, and one live path on the same state. That lets you connect the period law to what the simulation is showing: gravity sets the circular speed, and that speed sets the time for one full circular orbit. The law applies only when the speed factor stays at the circular match.

Key ideas

01For circular orbits around one source mass, the period follows

T_{c} = 2 π r^{3} / M

in the displayed units with

G = 1

02At fixed source mass, a larger circular orbit has both a longer circumference and a lower circular speed, so the period grows faster than radius alone.

03Kepler's third law describes the circular-match case. If the speed factor moves away from 1.00, the path is no longer the circular orbit whose period the law describes.

Worked examples

Solve the live orbit period

Open examples when you want to see the same idea walked through step by step.

Frozen walkthrough

Step through the frozen example

Frozen walkthrough

Use the current source mass and reference orbit as evidence. First find the circular period for the chosen radius. Then compare the circular speed and period at

r

and

2 r

so you can see why a wider circular orbit takes much longer.

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Example 1 of 2

Frozen valuesUsing frozen parameters

For the current source mass and chosen circular orbit radius, what period should one full circular orbit take?

M

Source mass

4 kg

r

Reference orbit radius

1.4 m

1. Recall the circular speed at this radius

For the circular case, the same gravity-turning balance gives

v_{c} = M / r = 4/1.4

2. Turn that speed into one full-orbit time

A circular period is circumference divided by circular speed, so

T_{c} = 2 π r / v_{c} = 2 π r^{3} / M

3. Compute the live circular period

For this bench,

T_{c} = 5.2 s

, with

ω_{c} = 1.21 rad/s

and

T_{c}^{2} / r^{3} = 9.87 s^{2} / m^{3}

Circular period at this radius

T_{c} = 5.2 s

The period comes from the same live circular condition: one full orbit is the circumference divided by the allowed circular speed.

Quick test

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Accessibility

Open the text-first descriptions when you need the simulation and graph translated into words.

The simulation shows one fixed source mass at the center, a moving satellite launched from the right side of a chosen circular reference radius, and optional overlays for the dashed reference orbit, the live radius line, the tangent velocity vector, the inward gravity vector, and the trajectory trail.

Changing source mass, reference orbit radius, or speed factor updates the same orbit path, circular-speed readout, circular-period readout, law-ratio readout, and linked graphs together. Compare mode overlays a second setup without switching to a separate orbit model.

The displayed units use a bounded one-source gravity model with $G = 1$ . A minimum sample radius keeps the stage and graphs finite and readable while preserving the correct inward and inverse-square trends.

Graph summary

The radius-history graph shows whether the live path stays on the chosen circular reference radius or drifts inward or outward. The speed-history graph compares the actual speed with the circular speed required at the current radius.

The acceleration-balance graph compares inward gravity with the turning requirement $v^{2} / r$ on the same live path. Kepler's third law applies here only when those match in the circular case.

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Starter track

Step 4 of 5

Gravity and Orbits

Kepler's Third Law and Orbital Periods appears later in this track, so it is cleaner to start from the beginning first.

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