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**Application of the Doppler effect and Kepler's 3rd law**

## Homework Statement

(*This isn't the exact wording from a textbook, just as I had copied it down)

An eclipsing binary star system containing stars A and B in concentric orbits (about their common centre of mass) has it's maximum red- and blueshifts recorded.

The rest wavelength for the stars is 6000Å.

For star A, max-red occurs at 6001.3Å and max-blue at 6000.3Å.

For star B, max-red occurs at 6002.3Å and max-blue at 5999.3Å.

It takes 8.75 days to go from max-red to max-blue, and the stars orbit in the same period.

Find the following:

1. the radial velocity of the stars common centre of mass

2. the radial velocities of each star

3. the masses of each star

## Homework Equations

Doppler Effect equation (for light):

[tex]v_{source} = c \frac{\lambda_{observed} - \lambda_{rest}}{\lambda_{rest}}[/tex]

Newton's form of Kepler's 3rd law:

[tex]\frac{A^3}{T^2} = \frac{G}{4\pi^2} (M_A + M_B)[/tex]

where A is the mean distance of separation (semimajor axis), T the orbital period, G the gravitational constant, and M the masses of each star. If A, T and M are expressed in astronomical units, years and solar masses respectively, [tex]\frac{G}{4\pi^2}[/tex] is unity.

## The Attempt at a Solution

A very simple diagram of either orbit, would have the position of max-red at one end as the star recedes, and max-blue on the other end as the star approaches. The observer's line of sight would pass through the orbits' common centre.

1. To determine the shift for the centre of mass, I took the central (mean) wavelength for the stars:

[tex]\lambda_c = \frac{6000.3Å+6001.3Å}{2} = 6000.8Å[/tex]

Then the radial velocity for the centre of mass (G)

[tex]v_G = 3 \cdot 10^5 \frac{6000.8Å-6000Å}{6000Å} = +40km/s [/tex]

The positive implying it is receding from the observer.

2. For star A I compared wavelength at max-blue with the central wavelength,

[tex]v_A = 3 \cdot 10^5 \frac{6000.8Å-6000.3Å}{6000Å} = 25km/s [/tex]

And for star B using max-red

[tex]v_B = 3 \cdot 10^5 \frac{6000.8Å-5999.3Å}{6000Å} = 75km/s [/tex]

(these are the speeds of the stars relative to the centre of mass)

*I'm not entirely certain of my

*methods*above, but those are the final answers that were given.

3. Not sure how to relate their masses or find their separation, all I can gather from the question is both stars have a period of 17.5 days.

thanks

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