Expert: Philip A. Stahl - 2/23/2008

Question
Can you please help with this problem?

Barnard's star has a proper motion of 10.31"/ yr. and a parallax angle p = 0.552". The H-alpha absorption line in its spectrum is observed to have a wavelength of 6560.44 Angstrom.

a) Determine the radial velocity of the star
b) Determine the transverse velocity.
c) Compute the speed of Barnard's star through space.

Answer
Hello,

This is a fairly straightforward problem, so I will just point you in the right direction to get you started. Believe it or not, most of your work in the problem will be in ensuring that all your units are consistent.

The radial velocity can simply be found by using the principle of the Doppler shift of light. You already have the observed wavelength of the H-alpha line, so you need to find the difference between that and the normal position H-alpha line. To save you having to look it up, I will simply tell you, it is 6562.81 Angstrom.

Call the difference in wavelengths "delta L" - then the ratio of this to the normal wavelength should be in the same ratio as the radial velocity is to the velocity of light. (Mind units here, and be aware that 1 A = 10^-8 cm = 10^-10 m)

The transverse velocity meanwhile is computed from the product of the distance, d, of the star and its proper motion.

To find d, you have to use the basic parallax relationship:

d =  1/p"

and the result appears in parsecs.

The trick now is to convert this into units which will be recognizable as velocity (m/s). Thus, you will need to get d (parsecs) into m (meters). (Don't forget you also have to get the proper motion in radians/sec!)

The speed of the star through space is simply the *vector sum* of the two velocities, radial and transverse.