Kepler's Laws of Planetary Motion

Why Kepler's laws matter

Before Newton, planetary motion was a puzzle of observation, not of force. Ptolemy's geocentric model packed circles within circles to fit the data; Copernicus restored the Sun to the centre but still insisted on circular orbits. The breakthrough was empirical. Tycho Brahe spent a lifetime recording the planets with the naked eye, and his assistant Johannes Kepler (1571–1640) distilled that data into three laws — without yet knowing why they held.

These laws were, in NCERT's words, "known to Newton and enabled him to make a great scientific leap in proposing his universal law of gravitation." Kepler described the what; Newton supplied the why. For NEET, Kepler's laws are both a standalone topic and the conceptual bridge into the universal law of gravitation. The law of areas in particular reappears in every elliptical-orbit problem as a disguised statement of angular-momentum conservation.

The three laws at a glance

NCERT states the three laws as follows. Read the structured cards first, then we treat each in turn.

A useful mnemonic for the order: orbits, areas, periods — shape, then speed, then size-versus-time. The first law fixes the geometry, the second governs the planet's speed at each point of that geometry, and the third ties the orbit's overall scale to its period.

Law of orbits — the ellipse

An ellipse is the closed curve traced by a point whose summed distances to two fixed points — the foci $F_1$ and $F_2$ — stay constant. NCERT describes the classic construction: fix the two ends of a string at $F_1$ and $F_2$, pull it taut with a pencil tip, and sweep the pencil around. For any point on the ellipse, the sum of the distances from the two foci is the same.

Join the two foci and extend the line until it meets the ellipse at P (the nearest point, called perihelion) and A (the farthest point, called aphelion). The midpoint O of PA is the centre of the ellipse, and the length $\mathrm{PO} = \mathrm{AO}$ is the semi-major axis $a$ — half the longest diameter. The Sun sits at one focus, not at the centre. The other focus is empty.

The "fatness" of an ellipse is measured by its eccentricity $e$, the ratio of the focus-to-centre distance to the semi-major axis. When $e = 0$ the two foci merge into a single point, the curve becomes a circle, and the semi-major axis becomes the radius. NCERT notes this directly: "For a circle, the two foci merge into one and the semi-major axis becomes the radius of the circle." Most planetary orbits have small eccentricity, so they look nearly circular, but the Sun is always at a focus, never at the geometric centre.

Law of areas and angular momentum

The second law states that the line joining a planet to the Sun sweeps out equal areas in equal times. Equivalently, the rate at which area is swept — the areal velocity $dA/dt$ — is constant. This came from the observation that planets move more slowly when far from the Sun and faster when near it. The deep reason is conservation of angular momentum, which holds for any central force: a force directed along the line joining the planet to the Sun.

Place the Sun at the origin and let the planet of mass $m$ have position $\vec r$ and momentum $\vec p$. In a small time interval $\Delta t$ the planet moves by $\vec v \,\Delta t$, and the triangular area swept by the radius vector is

$$\Delta A = \tfrac{1}{2}\,\lvert \vec r \times \vec v\,\Delta t \rvert.$$

Dividing by $\Delta t$ and using $\vec v = \vec p / m$,

$$\frac{\Delta A}{\Delta t} = \frac{1}{2m}\,\lvert \vec r \times \vec p \rvert = \frac{L}{2m},$$

where $\vec L = \vec r \times \vec p$ is the angular momentum. For a central force, $\vec L$ is constant in time (the torque $\vec r \times \vec F = 0$ because $\vec F$ is along $\vec r$). Therefore $dA/dt = L/2m$ is constant — which is precisely the law of areas. As NCERT puts it: "Gravitation is a central force and hence the law of areas follows."

The immediate physical consequence: a planet is fastest at perihelion (closest, smallest $r$) and slowest at aphelion (farthest, largest $r$). At these two points $\vec r$ and $\vec v$ are perpendicular, so $L = m\,r\,v$, and conservation gives $r_P v_P = r_A v_A$. Since $r_A > r_P$, it follows that $v_P > v_A$.

Law of periods — T² ∝ a³

The third law states that the square of a planet's orbital period $T$ is proportional to the cube of the semi-major axis $a$ of its orbit: $T^2 \propto a^3$. NCERT's Table 7.1 verifies it across all eight planets — the quotient $T^2/a^3$ is essentially the same constant for each, despite their wildly different distances and periods.

The near-constancy of the last column — every entry close to $3 \times 10^{-34}$ — is Kepler's third law made quantitative. The constant is shared by all bodies orbiting the same central mass, and as the next section shows, it is set entirely by that mass.

Deriving T² ∝ a³ for a circular orbit

For a circular orbit (the special case $e = 0$, with radius $a$), the third law drops out of Newton's gravitation in two lines. The gravitational pull supplies the centripetal force keeping the planet of mass $m$ on its circle around the Sun of mass $M$:

$$\frac{GMm}{a^2} = \frac{m v^2}{a}.$$

Cancel $m$ and solve for the orbital speed: $v^2 = \dfrac{GM}{a}$. The period is the circumference divided by this speed,

$$T = \frac{2\pi a}{v} = 2\pi a \sqrt{\frac{a}{GM}} = 2\pi \sqrt{\frac{a^3}{GM}}.$$

Squaring both sides isolates the law of periods:

$$\boxed{\,T^2 = \frac{4\pi^2}{GM}\,a^3\,} \qquad \Longrightarrow \qquad T^2 \propto a^3.$$

The proportionality constant $4\pi^2/GM$ depends only on the mass $M$ of the central body — not on the orbiting planet. This is why every planet around the Sun shares one value of $T^2/a^3$, and every satellite around the Earth shares a different value. NCERT derives exactly this form for Earth satellites, $T^2 = (4\pi^2/GM_E)(R_E + h)^3$, and notes it "is Kepler's law of periods, as applied to motion of satellites around the earth."

Worked example 1 — perihelion vs aphelion

Worked example 2 — weighing the Sun

Worked example 3 — locating Saturn from its period

These two exercises show the third law's two everyday uses: an absolute form ($M = 4\pi^2 a^3/GT^2$) when you want the central mass, and a ratio form ($a_2/a_1 = (T_2/T_1)^{2/3}$) when you compare two bodies orbiting the same centre. The ratio form is the faster route in NEET whenever a known reference orbit is supplied.