Q.Using Kepler's second law (the law of areas), explain why a planet moves fastest when it is closest to the Sun (at perihelion) and slowest when it is farthest from the Sun (at aphelion).

Kepler's first law fixes the shape/geometry of planetary orbits; his second law is a statement of conservation of angular momentum.

Law of orbits (Kepler's first law): Every planet revolves around the Sun in an elliptical orbit, with the Sun situated at one of the two foci of the ellipse (not at the centre). The degree of ellipticity varies from planet to planet, but the Sun always occupies a focus.

Law of equal areas (Kepler's second law): The line segment (radius vector) joining the Sun to a planet sweeps out equal areas in equal intervals of time, i.e. the areal velocity $\dfrac{dA}{dt}$ of the planet is constant throughout its orbit. As a consequence, the planet moves fastest when it is closest to the Sun (perihelion) and slowest when farthest (aphelion). This law is a direct consequence of the conservation of angular momentum, since the gravitational force on the planet acts along the radius vector (a central force), so it exerts zero torque about the Sun and the planet's angular momentum $L = mvr\sin\theta$ stays constant — and $\dfrac{dA}{dt} = \dfrac{L}{2m}$ is therefore also constant.

Kepler's three laws: (1) elliptical orbits with the Sun at a focus, (2) equal areas swept in equal times, (3) T^2 proportional to a^3.

1. Law of Orbits: Every planet revolves around the Sun in an elliptical orbit, with the Sun located at one of the two foci of the ellipse (not at the centre).

2. Law of Areas: The line segment joining a planet to the Sun (the radius vector) sweeps out equal areas in equal intervals of time. This means the planet moves faster when it is closer to the Sun (near perihelion) and slower when farther away (near aphelion). This law is a direct consequence of the conservation of angular momentum, since the Sun's gravitational force on the planet is a central force (always along the line joining them), producing zero torque about the Sun.

3. Law of Periods: The square of the time period T of revolution of a planet around the Sun is directly proportional to the cube of the semi-major axis a of its elliptical orbit: T^2 is proportional to a^3, i.e. T^2/a^3 is the same constant for every planet in the solar system. This follows from Newton's law of gravitation combined with the requirement of centripetal force for (approximately) circular orbits: GMm/r^2 = momega^2r, giving T^2 = (4*pi^2/GM)*r^3.