gravity - About binary stars and calculating velocity, period and radius of their orbit

I saw somewhere about being able to measure the velocity, period and radius of a binary star orbit by looking at red shift and blue shift.

I understand it but can someone give me an example of calculations etc done to calculate the velocity, period and radius of a binary stars in orbit?

Answer

You measure the velocity (relative the Earth frame) by observing the red- or blue-shift of easily identified lines in the spectrum, once you have more than a full cycle (preferably several cycles) you:

• Subtract off the contributions from the Earth (known) orbital velocity around the sun


• Find the period of the orbit just by looking at the peak-to-peak time of the cycle


• Remove the mean relative velocity, and find longitudinal velocity in the remote system center of mass frame as a function of time. You can also get the longitudinal size of the orbit by integrating the velocity


• Compute the reduced mass and reduced radius of the orbit from Kepler's laws

In the limit of one massive and one light body the reduced mass is the mass of the heavy partner.

• If you can make these observations for both bodies you can get both masses.