Question

⭐derivativon of first law of Kepler (law of orbit)

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Answer 1

Case 1. e = 0.

Then


The orbit is a circle.

Now let


Then



Convert to Cartesian coordinates:



We now obtain Case 2: If e = 1, the orbit is a parabola.

Also if e > 1, then we get Case 3: If e > 1, the orbit is a hyperbola.

Now assume . Complete the square to get




Let , . Then distance from the center to the ellipse to the focus is c where .





So . So (0,0) is a focus of the ellipse. Note that the eccentricity of the ellipse is

.

Answer 2

there is no derivation as such for the first law of Kepler..
the first law states that each planet revolves around the sun in an elliptical Orbit
A planet orbits the Sun in an ellipse, with the Sun at one focus of the ellipse


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