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,

$v_{o} = (3*10^5) (\frac{3.00036-3}{3})$

Note that the average signal is 3.000000m

$v_o = 36 km/s$

Now using the definition about centripetal acceleration we have,

$a_c = \frac{v^2}{r}$

Here,

v = Orbit Velocity

r = Radius of Orbit

Replacing with our values,

$a = \frac{(36km/s)^2}{100000km}$

$a= 0.01296km/s^2$

$a = 12.96m/s^2$

Applying Newton's equation for acceleration due to gravity,

$a =\frac{GM}{r^2}$

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,

$M = \frac{ar^2}{G}$

$M = \frac{(12.96)(100000000)^2}{ 6.67*10^{-11}}$

$M = 1.943028*10^{27}kg$

Therefore the mass of the planet is $1.943028*10^{27}kg$