Question

Prove Kepler's Second Law: Each ray from the sun to a planet sweeps out equal areas of the ellipse in equal times.

Short answer

Kepler's Second Law can be proved using the equation of an ellipse and the concepts of calculus and physics. The law states that the line connecting a planet to the Sun sweeps out equal areas during equal time intervals, which is synonymous with saying that the rate of change of area swept by a planet orbiting the sun is constant.

Step-by-step solution

Review the equation of an ellipse

An ellipse is defined by the set of points such that the sum of the distances from two foci to any point on the ellipse is constant. The semi-major and semi-minor axes of an ellipse are denoted as \(a\) and \(b\) respectively, the foci are a distance \(c = \sqrt{a^2 - b^2}\) from the center.

Define planet’s position and velocity vectors

A planet in an elliptical orbit around the Sun (located at a focus) can be generalized by two vectors: the position vector \(r\), where its magnitude is the distance between the planet and the Sun; and the velocity vector \(v\), where its magnitude is the rate of displacement of the planet along its orbit over time.

Define area swept by the planet

During a short time interval \(dt\), the area dA swept out by the line from the Sun to the planet is half of the parallelogram area formed by \(r\) and \(r+dr\), which is \(dA = 0.5 r dr sin(α)\), where \(α\) is the angle between \(r\) and \(r+dr\). This dA can be also expressed as \(dA = 0.5 r^2 dθ\), where \(dθ\) is the angle subtended by \(dr\) at the Sun. Hence \(dθ = dr sin(α) / r\)

Express dr using velocity

The vector \(dr\) represents a small displacement of the planet in time \(dt\), hence its magnitude can be expressed as \(dr = ds = v dt\), where \(v\) is the magnitude of the planet velocity vector \(v\) and \(ds\) is the displacement.

Substitute dr into the dA expression

By substituting \(dr = v dt\) and \(dθ = dr sin(α) / r\) into \(dA = 0.5 r dr sin(α)\) we can simplify the expression to \(dA = 0.5 r v dt\).

Constant Area with Constant Time

The area \(dA\) is swept in a time \(dt\), hence the rate at which the area is being swept is \(dA/dt = 0.5 r v = constant\), which is Kepler's Second Law, that states that the areas swept out by the planet in equal times are equal, regardless of where the planet is located in its orbit.