Question

A planet revolves around the sun in an elliptical orbit. If ${v_p}$ and ${v_a}$ are the velocities of the planet at the perigee and apogee respectively, then the eccentricity of the elliptical orbit is given by:
A) $\dfrac{{{v_p}}}{{{v_a}}}$
B) $\dfrac{{{v_a} - {v_p}}}{{{v_a} + {v_p}}}$
C) $\dfrac{{{v_p} + {v_a}}}{{{v_p} - {v_a}}}$
D) $\dfrac{{{v_p} - {v_a}}}{{{v_p} + {v_a}}}$

Answer 1

The eccentricity of an elliptical orbit is defined as the measure of the amount by which it deviates from a circle. It is obtained by dividing the distance between the focal points of the ellipse by the length of the major axis.
Formula used:
$\dfrac{{{v_p}}}{{{v_a}}} = \dfrac{{a\left( {1 + e} \right)}}{{a\left( {1 - e} \right)}}$

Complete answer:
Perigee is defined as the orbit of an artificial satellite or natural satellite at which the centre of the satellite is nearest to the centre of the planet.Similarly, Apogee is defined as the orbit of an artificial satellite or natural satellite at which the centre of the satellite is farthest to the centre of the planet.
The eccentricity of an elliptical orbit is known as the measure of the amount by which it deviates from a circle. In other words, it is a measure of how non – circular is the orbit of a body.
Mathematically it is given by,
$\dfrac{{{v_p}}}{{{v_a}}} = \dfrac{{a\left( {1 + e} \right)}}{{a\left( {1 - e} \right)}}$
Where ‘a’ is expressed as the semi – major axis of the ellipse. An eccentricity of zero is known to be a perfect circle.

For a better understanding of the concept, let us see a diagram of an eccentric orbit

Now, using the above-mentioned formula we get,
${v_p}\left( {1 - e} \right) = {v_a}\left( {1 + e} \right)$
$\Rightarrow {v_p} - e{v_p} = {v_a} + e{v_a}$
$\Rightarrow e\left( {{v_a} + {v_p}} \right) = {v_p} - {v_a}$
$\Rightarrow e = \dfrac{{{v_p} - {v_a}}}{{{v_a} + {v_p}}}$
Thus, the eccentricity of the elliptical orbit is given by $\dfrac{{{v_p} - {v_a}}}{{{v_p} + {v_a}}}$.

So, the correct answer is “Option D”.

Note:
These concepts are based on Kepler’s law of planetary motion. Kepler’s first law states that all the planets revolve around the sun in elliptical orbits with the sun at one focus. Kepler’s second law states that the line which joins the earth or any other planet to the sun sweeps out equal areas in equal intervals of time. Kepler’s third law states that the square of the time period of the planet is directly proportional to the cube of the semi - major axis of its orbit.