Understanding the Ratio of Orbital Velocities for Satellites

When satellites revolve around the Earth in circular orbits, their speed, known as orbital velocity, depends on the mass of the Earth and the radius of their orbit. The masses of the satellites themselves do not directly affect their orbital velocity in a given orbit.

Formula for Orbital Velocity

The orbital velocity \( v \) of a satellite in a circular orbit around a planet (like Earth) is given by the formula:

\( v = \sqrt{\dfrac{GM}{r}} \)

Where:

• \( G \) is the universal gravitational constant.
• \( M \) is the mass of the central body (Earth, in this case).
• \( r \) is the radius of the circular orbit.

Notice that the mass of the satellite itself (\( m_1 \) or \( m_2 \)) does not appear in this formula. This means the orbital velocity depends only on the mass of the planet and the distance from the center of the planet to the satellite.

Calculating the Ratio of Orbital Velocities

We have two satellites, satellite 1 with mass \( m_1 \) in an orbit of radius \( r_1 \), and satellite 2 with mass \( m_2 \) in an orbit of radius \( r_2 \). Using the orbital velocity formula:

The orbital velocity of satellite 1 is \( v_1 = \sqrt{\dfrac{GM}{r_1}} \).

The orbital velocity of satellite 2 is \( v_2 = \sqrt{\dfrac{GM}{r_2}} \).

To find the ratio of their orbital velocities, \( \dfrac{v_1}{v_2} \), we divide the expression for \( v_1 \) by the expression for \( v_2 \):

\( \dfrac{v_1}{v_2} = \dfrac{\sqrt{\dfrac{GM}{r_1}}}{\sqrt{\dfrac{GM}{r_2}}} \)

We can combine the square roots:

\( \dfrac{v_1}{v_2} = \sqrt{\dfrac{\dfrac{GM}{r_1}}{\dfrac{GM}{r_2}}} \)

Now, simplify the fraction inside the square root. Dividing by a fraction is the same as multiplying by its reciprocal:

\( \dfrac{v_1}{v_2} = \sqrt{\dfrac{GM}{r_1} \times \dfrac{r_2}{GM}} \)

The \( GM \) terms cancel out:

\( \dfrac{v_1}{v_2} = \sqrt{\dfrac{\cancel{GM}}{r_1} \times \dfrac{r_2}{\cancel{GM}}} \)

\( \dfrac{v_1}{v_2} = \sqrt{\dfrac{r_2}{r_1}} \)

Therefore, the ratio of their orbital velocities \( \dfrac{v_1}{v_2} \) is equal to the square root of the inverse ratio of their orbital radii, \( \sqrt{\dfrac{r_2}{r_1}} \). This calculation confirms how the orbital velocity ratio depends only on the radii.

This step-by-step process shows how the ratio of orbital velocities is derived directly from the fundamental formula.