Question

You are observing a spacecraft moving in a circular orbit of radius 100,000 km around a distant planet. You happen to be located in the plane of the spacecraft’s orbit. You find that the spacecraft’s radio signal varies periodically in wavelength between 2.99964 m and 3.00036 m. Assuming that the radio is broadcasting at a constant wavelength, what is the mass of the planet?

Answer 1

To solve this problem we will apply the concepts related to centripetal acceleration, which will be the same - by balance - to the force of gravity on the body. To find this acceleration we must first find the orbital velocity through the Doppler formulas for the given periodic signals. In this way:

`v_(o) = c (\frac{\lambda_(max)-\bar{\lambda}}{\bar{\lambda}}})`

Here,

`v_(o) =` Orbital Velocity

`\lambda_(max) =` Maximal Wavelength

`\bar{\lambda}} =` Average Wavelength

c = Speed of light

Replacing with our values we have that,

`v_(o) = (3*10^5) ((3.00036-3)/(3))`

Note that the average signal is 3.000000m

`v_o = 36 km/s`

Now using the definition about centripetal acceleration we have,

`a_c = (v^2)/(r)`

Here,

v = Orbit Velocity

r = Radius of Orbit

Replacing with our values,

`a = ((36km/s)^2)/(100000km)`

`a= 0.01296km/s^2`

`a = 12.96m/s^2`

Applying Newton's equation for acceleration due to gravity,

`a =(GM)/(r^2)`

Here,

G = Universal gravitational constant

M = Mass of the planet

r = Orbit

The acceleration due to gravity is the same as the previous centripetal acceleration by equilibrium, then rearranging to find the mass we have,

`M = (ar^2)/(G)`

`M = ((12.96)(100000000)^2)/( 6.67*10^(-11))`

`M = 1.943028*10^(27)kg`

Therefore the mass of the planet is
`1.943028*10^(27)kg`