HomeConcept libraryPhysicsMechanicsGravity and Orbits

Physics MechanicsIntermediateStarter track

Concept module

Kepler's Third Law and Orbital Periods

Compare circular orbits around one source mass and see why larger orbits take longer: the path is longer, the circular speed is lower, and the same live model makes the period law visible without hiding the gravity-speed link.

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Starter track

Step 4 of 50 / 5 complete

Gravity and Orbits

Earlier steps still set up Kepler's Third Law and Orbital Periods.

1. Gravitational Fields2. Gravitational Potential and Potential Energy3. Circular Orbits and Orbital Speed4. Kepler's Third Law and Orbital Periods+1 more steps

Previous step: Circular Orbits and Orbital Speed.

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Why it behaves this way

Explanation

Kepler's third law is the timing view of the same circular-orbit balance you already met in the speed story. Around one source mass, larger circular orbits take longer because they have farther to travel and because the allowed circular speed is lower at larger radius.

This bounded lab keeps one source mass, one chosen reference orbit, one speed factor, one live path, and the linked radius, speed, and acceleration-balance graphs tied to the same state. That makes the period law honest: it is not a disconnected astronomy rule, but the time consequence of gravity setting the circular speed.

Key ideas

01For circular orbits around one source mass, the period follows $T_c = 2\pi\sqrt{r^3/M}$ 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 the radius alone.

03Kepler's third law describes the circular-match case. If the speed factor moves away from 1.00, the path stops being the circular orbit whose period the law is talking about.

Frozen walkthrough

Step through the frozen example

Frozen walkthrough

Use the same source mass, reference orbit radius, and speed factor already on screen. These examples stay tied to the live bench instead of jumping to a detached answer key.

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View plans

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. Start from the circular speed

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

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

2. Turn 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}

Current circular period

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.

Kepler checkpoint

Keep the same source mass and the circular match. If the chosen orbit radius doubles, should the period double, grow by less than double, or grow by more than double?

Make a prediction before you reveal the next step.

Answer before you try the inner and outer circular presets: is the longer year caused only by more distance around, or by both more distance and lower circular speed?

Check your reasoning against the live bench.

It grows by more than double.

A larger circular orbit is not just a bigger circle. The allowed circular speed is also lower at larger radius, so the period scales as

r^{3/2}

rather than directly as

r

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.

That combination is why the period grows strongly with radius: the object travels farther and it does so more slowly.

Quick test

Reasoning

Question 1 of 4

Answer from the linked orbit state, not from detached memorization. The aim is to read why larger circular orbits take longer and when the law does not apply.

Around the same source mass, the circular orbit radius doubles while the path stays circular. What must be true?

Use the live bench to test the result before moving on.

Accessibility

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 compares the live radius with the chosen circular reference radius over time so you can tell whether the path is still the circular case.

The speed-history graph compares the live speed with the circular speed required at the current radius, and the acceleration-balance graph compares gravity with the turning requirement $v^{2} / r$ on that same live path.

Keep the orbit-timing story moving

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Unbound thresholdMechanics