Question

A star lying 30 pc from the Sun has proper motion of 0.5 ′′/yr . A) What is the star's transverse velocity? B)If the star's spectral lines are observed to be redshifted by 0.01 percent, calculate the magnitude of its three-dimensional velocity relative to the Sun.

Answer 1

proper motion = 0.5"/yr = 1/7200 degree /yr
= 1.210 X 10^-6 Radian.
traverse velocity = 30pc X 1.210 X 10^-6 Radian
= 3.63 X 10^-5 pc/yr
= 1.183 X 10^-4 LY/yr
= [1.183 X 10^-4 c] km
= 35518.38 km.
Red shift = 0.01% = 10^-4 = z
since
1+z = √[(1+(v/c))/(1 - (v/c))] = 1.0001
[(1+(v/c))/(1 - (v/c))] = 1.00020001
v/c = (1.00020001 - 1) / (1.00020001+1)
= 0.000099995
=> v = 0.000099995 c = 29977.751 m/s.
There is no 3-D velocity unless you are contemplating a scientific breakthrough of a discovery; it is linear that implies 1 -D (one dimensional) along the line of sight.
Shift if it is 'Red' is moving away from us & if 'Blue' it is closing on us.