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The Heliocentric Model: Kepler's Laws

The motions of the planets, as they seemingly wander against the background of the stars, have been a puzzles since the dawn of history. Johannes Kepler (1571 - 1630), after a lifetime of study, worked out the empirical laws that govern these motions.

Tycho Brahe (1546 - 1601), the last of the great astronomers to make observations without the help of a telescope, compiled the extensive data from which Kepler was able to derive the three laws of planetary motion that now bear his name.

Later, Newton (1642 - 1727) showed that his law of gravitation leads to Kepler's empirical laws.

The Law of Orbits

All planets move in elliptical orbits, with the sun at one focus.
Figure, shows a planet, of mass m, moving in such an orbit around the Sun, whose mass is M. We assume that M >> m, so that the center of mass of the planet-Sun system is virtually at the centre of the sun.
A planet of mass m moving in an elliptical orbit around the sun. The Sun, of mass M, is at one focus F of the eclipse. The other, or 'empty' focus is F'. Each focus is a distance 'ea' from the centre, e being the eccentricity of the ellipse. The semi major axis 'a' of the ellipse, the perihelion (nearest the Sun) distance Rp, and the aphelion (farthest from the Sun) distance Ra are also shown in the above figure.

So that 'ea' is the distance from the center of the ellipse to either focus F or F'. An eccentricity of zero corresponds to a circle, in which the two foci merge to a single central point. The eccentricities of the planetary orbits are not large. The eccentricity of Earth's orbit is only 0.0167.

The Law of areas


A line that connects a planet to the Sun sweeps out equal areas in equal times.

Qualitatively, this second law tells us that the planet will move most slowly when it is farthest from the Sun and most rapidly when it is nearest to the Sun.

As it turns out, Kepler's second law is totally equivalent to the law of conservation of angular momentum.

The area of the shaded wedge in figure closely approximates the area swept out in time Δt by a line connecting the Sun and the planet, which are separated by a distance r.

The area ΔA of the wedge is approximately the area of a triangle with base r Δθ and height r. 
Thus ΔA r2Δθ.
In time Δt, the line r connecting the planet to the sun (of mass M) sweeps through an angle Δθ, sweeping out an area ΔA.
The linear momentum P of the planet and its components.
More exact as Δt (hence Δθ ) approaches zero.
The instantaneous rate at which area is being swept out is then

'ω' is the angular speed of the rotating line connecting the Sun and planet.
Figure, shows the linear momentum P of the planet, along with the components.
The magnitude of the angular momentum L of the planet about the Sun is given by the product of r and the component of p perpendicular to r, or
L = rP = (r) (mv ) = (r) (mω r) = mωr2
where we have replaced v with its equivalent ω r.

Eliminating r2ω
If dA/dt is constant, as Kepler said it is, then L must also be constant. The angular momentum is conserved. So Kepler's second law is indeed equivalent to the law of conservation of angular momentum.

The law of periods

The Square of the period of any planet is proportional to the cube of the semi major axis of its orbit.

Consider a circular orbit with radius r (the radius is equivalent to the semi major axis of an ellipse).                                         
A planet of mars m moving around the sun is a circular orbit of radius r.
Applying Newton's second law, F = ma, to the orbiting planet,
= (m) (ω 2r)

If we replace ω with 2π/T, where T is the period of moon, we obtain Kepler's third law
T2 =
This Law predicts that the ratio T2/a3 has essentially the same value for every planetary orbit around a given massive body.

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