Kepler’s Laws — friendly notes for quick revision

Around 1600, Johannes Kepler studied Tycho Brahe’s superb observations and spotted three clear patterns in planetary motion. These patterns—now called Kepler’s laws—explain how every planet goes around the Sun and set the stage for Newton’s law of gravitation. :contentReference[oaicite:0]{index=0}

1 · Understanding an ellipse

  • An ellipse is a closed curve with two special points called foci. For any point T on the curve, the sum \(TF_1 + TF_2\) stays the same. :contentReference[oaicite:1]{index=1}
  • Join the foci and extend the line until it meets the ellipse at P (closest point = perihelion) and A (farthest point = aphelion). Half the distance \(PA\) is the semi-major axis \(a\). :contentReference[oaicite:2]{index=2}

2 · Kepler’s three laws

Law of Orbits

Planets move in elliptical orbits with the Sun at one focus. The circle is just the special case where the two foci merge. :contentReference[oaicite:3]{index=3}

Law of Areas

Draw a line from the planet to the Sun. In equal times this line sweeps out equal areas. In other words, a planet moves faster when it is near perihelion and slower near aphelion. Quantitatively, \[ \frac{\Delta A}{\Delta t}=\frac{L}{2m}, \] where \(L = \mathbf r \times \mathbf p\) is the planet’s angular momentum and \(m\) its mass. The equality shows that the sweep-rate stays constant because \(L\) stays constant for a central force like gravity.

Law of Periods

The square of a planet’s time period \(T\) is proportional to the cube of the semi-major axis: \[ T^{2}\; \propto\; a^{3}. \] For every planet from Mercury to Neptune, the ratio \(T^{2}/a^{3}\) sits close to \(3.0\times10^{-34}\,\text{y}^{2}\,\text{m}^{-3}\). :contentReference[oaicite:5]{index=5}

3 · Speed at perihelion and aphelion

Because angular momentum stays the same, \[ m\,r_{p}\,v_{p}=m\,r_{A}\,v_{A}\quad\Longrightarrow\quad \frac{v_{p}}{v_{A}}=\frac{r_{A}}{r_{p}}. \] So the closer the planet is to the Sun, the faster it moves along its path. :contentReference[oaicite:6]{index=6}

4 · Why the area law works

Gravity pulls straight toward the Sun, so it never twists the planet’s motion. That straight pull keeps angular momentum fixed, and a fixed \(L\) automatically gives the constant area-sweep rate above. :contentReference[oaicite:7]{index=7}

High-yield ideas for NEET

  1. Relation \(T^{2}\propto a^{3}\)—questions often ask you to compare periods or predict missing data. :contentReference[oaicite:8]{index=8}
  2. Equal-area law—apply conservation of angular momentum to find relative speeds or travel times.
  3. Ellipse geometry—know terms like perihelion, aphelion, foci, and semi-major axis. :contentReference[oaicite:10]{index=10}
  4. Speed ratio \(v_{p}/v_{A}=r_{A}/r_{p}\)—quick shortcut for perihelion/aphelion speed problems. :contentReference[oaicite:11]{index=11}
  5. Central force & angular momentum link—connects to both planetary motion and satellite questions. :contentReference[oaicite:12]{index=12}

Keep these points in mind, practice with sample questions, and you’ll breeze through related NEET problems!