# Mass of Planet Using Radius and Doppler Effect

## Homework Statement

Imagine you are observing a spacecraft moving in a circular orbit of radius 128,000 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 normally, at a constant wavelength, what is the mass of the planet?

## Homework Equations

$$M= \displaystyle{\frac{rv^2}{G}}; \space where \space G= 6.67\times10^{-11} \space m^3 kg^{-1} s^{-2}, \space r \space is \space km, \space and \space v \ is \space km/s$$

## The Attempt at a Solution

Well, as we have a change in wavelength 2.99964 m and 3.00036 m respectively, the original signal should equal 3.00000m. With the formula from my textbook ( "Astronomy" 6th edition by Chaisson and McMillan, page 63), $$\frac{apparent\space \lambda}{true \space \lambda} -1 = speed \space in \space c$$ Then I multiply it by c and convert meters to kilometers and get$$\approx 36 km/s.$$

I input r and G as $$G= 6.67\times10^{-11} \space m^3 kg^{-1} s^{-2}, \space r= 128, 000 km.$$

So: $$M= \displaystyle{\frac{(128000km)*(36 km/s)^2}{6.67\times 10^{-11}\space m^3 kg^{-1} s^{-2}}} = 2.48\times10^{18} kg.$$

When I input this answer into the website in which we do our homework by, it gives me a lousy red X. I'm sure I messed up, because I was expecting a planet approximately in the 10^20-28 kg range.

Regardless, I've been stuck on this for a bit. Help is much appreciated.

Sincerely,

Nikos

phyzguy
Check your units, I think you are not converting km to m.

Simon Bridge
Homework Helper
possibly you are using different values for your constants, or you have rounded off differently?
ah - your value for G has length in meters.

Hey there,

Aha! I missed that my constant was in meters.

Honest mistake. Thanks a lot.