Assume you want to determine the distance to some distant open cluster of stars

Question

Assume you want to determine the distance to some distant open cluster of stars in the Milky Way. After observing the 100 stars of the cluster for several years, you have the following data: a) On average, the stars have a sideways angular motion (perpendicular to the line of sight) of 1 mu rad/yr = 10^-6 radians per year b) All these transverse velocity vectors seem to converge to a single point which is located 0.2 radians (11.46 degrees) away from the apparent position of the cluster. c) The average redshift of the light from these stars is z = 0.001, i.e., their wave lengths are stretched by 0.1% relative to the same light from a source at rest. Explain how you can combine this information to find the distance to the observed cluster, and give the numerical answer for the distance (don?t forget appropriate units). Note: I made all this numbers up randomly - this is just for illustration. Here is the formula for (relativisitic) Doppler shift: lambda obs/lambda emitted = (z +1) = root 1 + v/c/1 - v/c (v is the velocity

Explanation / Answer

z + 1 = sqrt((1+(v/c))/ (1 - (v/c)) )

Given z = 0.001

1.001 =  sqrt((1+(v/c))/ (1 - (v/c)) )

solving above equation we get

v = 0.0009995 c

v = 0.0009995 x 3 x 10^8 = 299850 m/s

Time taken = angle / speed of movement = 0.2 / (10^-6) = 200000 years

Distance = v x time taken = 299850 x 200000 x 365 x 24 x 60 x 60 = 1.891 x 1018 m

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Answer

Distance =  1.891 x 1018 m

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