Q.State Kepler's laws of planetary motion.

Kepler's laws (law of orbits, law of areas, law of periods) describe how planets orbit the sun.

Johannes Kepler, from Tycho Brahe's observational data, formulated three laws describing planetary motion: (1) Law of orbits — every planet moves in an elliptical orbit with the sun at one focus. (2) Law of areas — the line joining the planet and the sun sweeps out equal areas in equal time intervals. (3) Law of periods — the square of a planet's orbital period is proportional to the cube of the semi-major axis of its orbit. Newt …

Kepler's laws of planetary motion (formulated before Newton) describe how planets orbit the Sun.

Kepler's three laws state:

1. Planets move in elliptical orbits with the Sun at one focus.
2. The line joining a planet to the Sun sweeps equal areas in equal times (areal velocity is constant).
3. The square of a planet's orbital period is proportional to the cube of its orbit's semi-major axis (T^2 ∝ a^3). …

By Kepler's first law, every planet moves around the sun in an elliptical orbit, with the sun at one focus.

Kepler's first law of planetary motion (the "Law of Orbits") states that 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, and not in a perfect circle. A circle is only a special (limiting) case of an ellipse where both foci coincide; real …

Kepler's law of areas means a planet speeds up near the Sun and slows down far from it; velocity is maximum at the point of closest approach — here, point B.

Step 1 — Kepler's second law.

The line joining the Sun and a planet sweeps out equal areas in equal intervals of time: $\frac{dA}{dt} = \text{constant}$. This is a consequence of conservation of angular momentum about the Sun, since no external torque acts on the Sun-planet system (gravity is a central force).

Step 2 — Relating area-sweep rate to speed.

For a given small time interval, the swept area is (approximately) $\tfrac{1}{2} r v \,dt$, where $r$ is the Sun-planet distance and $v$ is the planet's speed perpendicular to $r$. Since $\frac{dA}{dt}$ is constant, $r v$ must be constant along the orbit — so where $r$ is small, $v$ must be large, and vice versa.

Step 3 — Apply to the figure. …