Kepler's Laws of Planetary Motion
On this page
Direct answer
Kepler's three laws describe how any body moves under an inverse-square gravitational pull. The law of orbits: every planet moves in an ellipse with the sun at one focus. The law of areas: the line joining planet to sun sweeps equal areas in equal times — fastest at perihelion, slowest at aphelion — a direct consequence of angular momentum conservation, since gravity exerts no torque about the sun. The law of periods: the square of the period is proportional to the cube of the semi-major axis, T^2 = k a^3, with the same k = 4π^2/GM for every body orbiting the same central mass. Together they let NEET convert one observation into every other quantity.
What you must remember
- Law of orbits: ellipses, not circles — the circle is merely the special case of zero eccentricity; the sun occupies one focus, not the centre.
- Law of areas: areal velocity dA/dt = L/2m is constant; angular momentum conservation makes this Kepler's second law a one-line consequence of τ = 0 for a central force.
- Law of periods: T^2 ∝ a^3; for bodies around the sun, T^2/a^3 is identical for every planet — the numerical table NCERT prints to verify it is a favourite source of assertion-reason statements.
- Speed contrast: v_perihelion > v_aphelion; the ratio follows from conservation of mvr with the two different distances.
- Universal scope: the laws hold for satellites around the earth and moons around planets, with k = 4π^2/(GM of the central body) changing accordingly.
- Ellipse vocabulary: semi-major axis a, eccentricity e; perihelion = a(1 - e), aphelion = a(1 + e) from the focus.
- Historical anchor: published between 1609 and 1619 from Tycho Brahe's observations, before Newton — laws first, theory after.
A transit between orbits, worked
Take a clean computation first. A planet's mean distance from its star is four times the earth-sun distance: by T^2 ∝ a^3 in units of years and astronomical units, T = 4^(3/2) = 8 years. Now the elliptical layer: if the orbit has eccentricity 0.5, the perihelion distance is a(1 - e) = 4(0.5) = 2 AU and the aphelion 6 AU. Angular momentum conservation links the speeds at those extremes: v_p × 2 = v_a × 6, so the planet moves three times faster at perihelion than at aphelion — the law of areas made quantitative. For a satellite version, replace AU-years with the earth: a geostationary orbit has a = about 42,000 km and T = 24 h, and any other circular radius follows from (T1/T2)^2 = (r1/r2)^3 — how one actually computes, for instance, the period of a near-surface orbit of about 85 minutes without touching G.
Where the questions slip past memory
NEET's commonest device here is the mismatched pair: quoting "T^2 ∝ R^3" while forgetting that the proportionality constant belongs to the central body, so earth-satellite and sun-planet data cannot be mixed in one ratio — a trap dressed up as a data-comparison question. The second device is the second law's cause: options offer "conservation of energy" alongside "conservation of angular momentum"; only the latter is correct, because energy conservation fixes the speeds at perihelion and aphelion only through the potential-energy difference, whereas the equal-areas statement is purely the torque-free consequence about the sun. Third, the focus confusion: the sun sits at a focus, so the planet's distance from the sun varies between a(1 - e) and a(1 + e), never a itself — questions on "closest and farthest distance" punish those who read a as the distance. And when a question quietly gives the nearest and farthest distances, remember a is their arithmetic mean.
Frequently asked questions
What does Kepler's second law physically follow from?
Gravity is a central force exerting no torque about the sun, so angular momentum mvr is conserved and the areal velocity dA/dt = L/2m stays constant.
How is Kepler's third law used for earth satellites?
As (T1/T2)^2 = (r1/r2)^3 with the earth as the common central body — knowing one satellite's period and radius delivers any other's.
At which point of the orbit is a planet fastest?
At perihelion, the closest approach, because the constant areal velocity forces a larger linear speed when the radius is smaller.
Is the sun at the centre of a planetary ellipse?
No — it occupies one focus; the centre of the ellipse is empty, and the planet's sun-distance oscillates between a(1 - e) and a(1 + e).
Do Kepler's laws apply to systems other than planets around the sun?
Yes — any inverse-square central field: artificial satellites and the moon around the earth, or moons around Jupiter, each with its own k = 4π^2/GM.