Question

A planet revolves around the sun in an elliptical orbit. If ${\mathrm{V}}_{\mathrm{p}}$ and ${\mathrm{V}}_{\mathrm{a}}$ are the velocities of the planet at the perigee and apogee respectively, then the eccentricity of the elliptical orbit is given by:

Answer 1

Step 1. Given data:

It is given that the ${\mathrm{V}}_{\mathrm{p}}$ and ${\mathrm{V}}_{\mathrm{a}}$ are the velocities of the planet at the perigee and apogee respectively.

We have to find the eccentricity of the elliptical orbit.

Step 2. Formula to be used

• The point in the orbit of an object (such as a satellite) orbiting the earth that is nearest to the center of the earth is known as perigee.

The velocity of the planet at perigee is,

${\mathrm{V}}_{\mathrm{p}}=\mathrm{a}\left(1-\mathrm{e}\right)$

Here, ${\mathrm{V}}_{\mathrm{p}}$ is the velocity of the planet at perigee, $\mathrm{a}$ is the semi-major axis which is half of the greatest width of the ellipse and $\mathrm{e}$ is the eccentricity of the elliptical orbit.

• The point in the orbit of an object (such as a satellite) orbiting the earth that is at the greatest distance from the center of the earth is known as apogee.

The velocity of the planet at the apogee is,

${\mathrm{V}}_{\mathrm{A}}=\mathrm{a}\left(1+\mathrm{e}\right)$

Step 3. Determine the eccentricity of the elliptical orbit.

We have to find the eccentricity of the elliptical orbit.

So,

The ratio of the velocities is,

$\frac{{\mathrm{V}}_{\mathrm{P}}}{{\mathrm{V}}_{\mathrm{A}}}=\frac{\mathrm{a}\left(1-\mathrm{e}\right)}{\mathrm{a}\left(1+\mathrm{e}\right)}$

Cancel the common term, we get,

$\frac{{\mathrm{V}}_{\mathrm{P}}}{{\mathrm{V}}_{\mathrm{A}}}=\frac{\left(1-\mathrm{e}\right)}{\left(1+\mathrm{e}\right)}$

${\mathrm{V}}_{\mathrm{P}}\left(1+\mathrm{e}\right)={\mathrm{V}}_{\mathrm{A}}\left(1-\mathrm{e}\right)$

${\mathrm{V}}_{\mathrm{P}}+{\mathrm{eV}}_{\mathrm{P}}={\mathrm{V}}_{\mathrm{A}}-{\mathrm{eV}}_{\mathrm{A}}$

$\mathrm{e}\left({\mathrm{V}}_{\mathrm{P}}+{\mathrm{V}}_{\mathrm{A}}\right)={\mathrm{V}}_{\mathrm{A}}-V_{\mathrm{P}}$

So, the eccentricity is,

$\mathrm{e}=\frac{{\mathrm{V}}_{\mathrm{a}}-{\mathrm{V}}_{\mathrm{P}}}{{\mathrm{V}}_{\mathrm{P}}+{\mathrm{V}}_{\mathrm{a}}}$

Hence, the eccentricity of the elliptical orbit is given by $\frac{{\mathrm{V}}_{\mathrm{a}}-{\mathrm{V}}_{\mathrm{P}}}{{\mathrm{V}}_{\mathrm{P}}+{\mathrm{V}}_{\mathrm{a}}}$.