Doppler Spectroscopy - Finding mass of an exoplanet

Question

An astronomer, T observed the spectrum of a distant star X. The spectral lines were observed to have tiny blueshifts and redshifts regularly with a period of 1.27 years. T concluded that the star is wobbling slightly due to a planet orbiting it. Let the estimated mass of the planet when

(a) eccentricity of the orbit e = 0 be k*10^23 kg
(b) and for e = 0.6 be x*10^22 kg.
Assume that there is only one major planet and the axis of rotation of the system is perpendicular to the line of sight.

Mass of Star = 23.86163 x 10^30 kg Time Period = 463.55 days Shift in wavelength = 1.7627 x 10^-5 nm

You have to find out 'k' and 'x'

I studied about the Doppler Spectroscopy method. I know that the relation $$ K = \left(\dfrac{2\pi G}{P_{orb}}\right) ^{1/3}\dfrac{M_p}{(M_{*}+M_p)^{2/3}}\dfrac{1}{\sqrt{1-e^2}} $$ holds where K = radial velocity half amplitude. How can I use the doppler shift given into this? Thanks.

Answer 1

To use the Doppler shift, you need to know by how much the light is blue/red-shifted.

Suppose you have a light curve where the wavelength varies between $\lambda_1$ and $\lambda_2$. Using these wavelengths, the Doppler shift will give you an interval of radial velocities. This gives you a value for K that you can plug into your last equation to derive $M_p$.