JEE Mains Physics · Gravitation

Kepler's Laws of Planetary Motion

Orbits are ellipses with the sun at a focus, the line to the sun sweeps equal areas in equal times, and T² is proportional to r³/M for the central mass M.

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Why this matters

Twenty-four PYQs, twenty-two of them multiple choice, and one from 2026. Seventeen use the third law, mostly as a ratio of two periods; seven use the second law or the shape of the orbit. Four are statement questions. Most of the ratio questions take one line once you write T ∝ √(r³/M).

Concept 1 of 2: Third law: period and orbit radius

From

GMm/r^{2} = m(2\pi/T)^{2}r

, the square of the period is proportional to the cube of the radius, divided by the mass of the body being orbited. The orbiting body's own mass drops out. Almost every question compares two orbits, so write the ratio and let the constants cancel.

Definition

$T^{2} = \dfrac{4\pi^{2}}{GM}r^{3}$ ; for an ellipse, r is the semi-major axis.
Same central mass: $\dfrac{T_2}{T_1} = \left(\dfrac{r_2}{r_1}\right)^{3/2}$ . Radius $\times 4 \to T \times 8$ ; radius $\times 9 \to T \times 27$ ; radius $\times 3 \to T \times 3\sqrt{3}$ .
Different central masses: $T \propto \sqrt{r^{3}/M}$ .
Small change: $\dfrac{\Delta T}{T} = \dfrac{3}{2}\dfrac{\Delta r}{r}$ .
Mass of the central body from a moon's orbit: $M = \dfrac{4\pi^{2}r^{3}}{GT^{2}}$ .
A force $F \propto r^{-n}$ gives $T^{2} \propto r^{n+1}$ ; the inverse square (n = 2) gives Kepler's $r^{3}$ .
Two coplanar satellites going the same way: at closest approach their relative angular speed is $\dfrac{|v_2 - v_1|}{r_2 - r_1}$ .

Kepler's third law

T^{2} = \frac{4\pi^{2}}{GM}r^{3} \qquad \frac{T_2}{T_1} = \left(\frac{r_2}{r_1}\right)^{3/2}\sqrt{\frac{M_1}{M_2}}

Worked example

A moon circles planet X, of mass M, at radius r once every 10 days. A moon of planet Y, of mass 9M, circles at radius 4r. Find its period.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

JEE Mains · 2026 · 2 Apr 2026 Shift 1 · Q5Moderate

Example 1 · Gravitation · Kepler's Laws of Planetary Motion

A planet (

P_{1}

) is moving around the star of mass 2 M in the orbit of radius R . Another planet (

P_{2}

) is moving around another star of mass 4 M in a orbit of radius

2R

. Ratio of time periods of revolution of

P_{2}

and

P_{1}

\_\_\_\_

The central mass matters; the planet's mass does not

T depends on the mass being orbited. A heavier planet in the same orbit has the same period. When two different stars or planets are involved, include √(1/M).

Height or radius?

Kepler's law uses the distance from the centre. A geostationary satellite 6R above the surface is at r = 7R.

Raise the ratio to the power 3/2, not 3

Halving the radius divides T by 2√2, not by 8. Square both sides only after you have written T² ∝ r³.

Drill 16 more on third law: period and orbit radius

Concept 2 of 2: First and second laws: ellipses and equal areas

Gravity points at the sun, so it exerts no torque about the sun and the planet's angular momentum stays constant. Equal areas in equal times is that statement in geometric form. The planet must therefore move fastest when it is closest to the sun and slowest when it is farthest.

Definition

First law: every orbit is an ellipse with the sun at one focus. The force is towards the sun, proportional to the product of the masses and to $1/r^{2}$ .
Second law: $\dfrac{dA}{dt} = \dfrac{L}{2m}$ , a constant.
At the nearest and farthest points the velocity is perpendicular to the radius, so $v_{\max}r_{\min} = v_{\min}r_{\max}$ .
The speed changes round the orbit; the areal velocity and the angular momentum do not.
Areal velocity is not proportional to the speed: it depends on the component of velocity perpendicular to r, times r.

Kepler's second law

\frac{dA}{dt} = \frac{L}{2m} = \text{constant} \qquad v_{\max}r_{\min} = v_{\min}r_{\max}

Worked example

The earth moves at 30.3 km/s when it is

1.47 \times 10^{11}

m from the sun, its nearest point. Find its speed at its farthest point,

1.52 \times 10^{11}

m away.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

JEE Mains · 2021 · Paper 5 · Q7Moderate

Example 2 · Gravitation · Kepler's Laws of Planetary Motion

The maximum and minimum distances of a comet from the Sun are

1.6 \times10^{12}\text{ }m

and

8.0 \times10^{10}\text{ }m

respectively. If the speed of the comet at the nearest point is

6 \times10^{4}{\text{ }ms}^{- 1}

, the speed at the farthest point is :

Fastest when nearest

A statement that a planet is slowest near the sun is false. Constant angular momentum means a smaller r needs a larger speed.

Constant areal velocity, not constant speed

The area swept per second is fixed; the distance travelled per second is not. 'The linear speed is constant' is the incorrect statement in this pair.

Drill 6 more on first and second laws: ellipses and equal areas

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (2)

Third law: period and orbit radius
Kepler's third law
$T^{2} = \frac{4\pi^{2}}{GM}r^{3} \qquad \frac{T_2}{T_1} = \left(\frac{r_2}{r_1}\right)^{3/2}\sqrt{\frac{M_1}{M_2}}$
First and second laws: ellipses and equal areas
Kepler's second law
$\frac{dA}{dt} = \frac{L}{2m} = \text{constant} \qquad v_{\max}r_{\min} = v_{\min}r_{\max}$

Watch out for (5)

The central mass matters; the planet's mass does not
→ Third law: period and orbit radius
Height or radius?
→ Third law: period and orbit radius
Raise the ratio to the power 3/2, not 3
→ Third law: period and orbit radius
Fastest when nearest
→ First and second laws: ellipses and equal areas
Constant areal velocity, not constant speed
→ First and second laws: ellipses and equal areas

Test yourself on Gravitation

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