Mass of Planet Using Radius and Doppler Effect

[Linuxkid]

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

[tex] 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
[/tex]

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), [tex] \frac{apparent\space \lambda}{true \space \lambda} -1 = speed \space in \space c[/tex] Then I multiply it by c and convert meters to kilometers and get[tex] \approx 36 km/s. [/tex]

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


So: [tex] 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. [/tex]


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.





Nikos

 

[phyzguy]

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

 

[Simon Bridge]

possibly you are using different values for your constants, or you have rounded off differently?
[edit]ah - your value for G has length in meters.

 

[Linuxkid]

Hey there,


Aha! I missed that my constant was in meters.



Honest mistake. Thanks a lot.

 

[Nickie]

, your approach to solving this problem is correct. However, there are a few things to consider in order to arrive at the correct answer.

Firstly, the equation you have used to calculate the mass of the planet assumes that the spacecraft is in a circular orbit. However, the problem statement does not explicitly state that the orbit is circular, so we cannot make that assumption. Instead, we can use the more general equation for orbital motion:

v^2=GM(2/r-1/a)

where v is the orbital velocity, G is the gravitational constant, M is the mass of the planet, r is the radius of the spacecraft's orbit, and a is the semi-major axis of the orbit.

Secondly, the wavelength change of the radio signal is due to the Doppler effect, which is caused by the relative motion between the spacecraft and the observer (in this case, you). This means that the velocity you have calculated (36 km/s) is the relative velocity between the spacecraft and you, not the orbital velocity of the spacecraft. In order to find the orbital velocity, we need to use the equation for the Doppler effect:

\frac{\Delta \lambda}{\lambda} = \frac{v}{c}

where \Delta \lambda is the change in wavelength, \lambda is the original wavelength, v is the relative velocity, and c is the speed of light.

Now, we can use these equations to solve for the mass of the planet. First, we need to convert the given radius of the spacecraft's orbit from kilometers to meters:

r = 128,000 km = 128,000,000 m

Next, we can plug the given values into the equations for orbital motion and the Doppler effect:

v^2 = (6.67 \times 10^{-11} m^3 kg^{-1} s^{-2}) (M) (2/128,000,000 m - 1/a)

\frac{\Delta \lambda}{\lambda} = \frac{v}{c}

We can rearrange the first equation to solve for the mass of the planet:

M = \frac{v^2 (128,000,000 m - a)}{2 (6.67 \times 10^{-11} m^3 kg^{-1} s^{-2})}

Then, we can substitute the value for v from the second equation: