Elliptical Orbit of a Planet around the Sun

Introduction
The figure shows the elliptical orbit of a planet with mass m around the sun. The shaded area SCD is twice the shaded area SAB. We need to explain the time taken by the planet to move from point C to D (t1) and from point A to B (t2).

Explanation

1. Elliptical Orbit
- An elliptical orbit is the path followed by a planet or any other celestial body when it revolves around another body under the influence of gravitational force.
- In an elliptical orbit, the planet moves in an elongated circle, with the sun located at one of the foci of the ellipse.

2. Shaded Areas
- The shaded area SCD represents the area swept by the line joining the planet to the sun as the planet moves from point C to D.
- The shaded area SAB represents the area swept by the line joining the planet to the sun as the planet moves from point A to B.

3. Kepler's Second Law
- Kepler's Second Law states that a line joining a planet and the sun sweeps out equal areas in equal intervals of time.
- This means that the rate at which the shaded area is swept by the planet's line from C to D is equal to the rate at which the shaded area is swept from A to B.
- Since the shaded area SCD is twice the shaded area SAB, the time taken to sweep the shaded area SCD (t1) is twice the time taken to sweep the shaded area SAB (t2).

Conclusion
In conclusion, the time taken by the planet to move from point C to D (t1) is twice the time taken to move from point A to B (t2) in an elliptical orbit around the sun. This is due to Kepler's Second Law, which states that equal areas are swept out in equal intervals of time. The shaded areas SCD and SAB represent the areas swept by the line joining the planet to the sun, and their ratio determines the ratio of the time taken.

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