A distant star emitted a light with wavelength lambda 486 n

A distant star emitted a light with wavelength lambda = 486 nm when it was 52 mpc away from us. when the light reached us, the distance has expanded to S3 Mpc. What is the new wavelength?

Solution

let the star is moving at speed V away from earth.

So velocity of light in earths frame is,

C\'=(C-V)/(1-C*V/C^2)=(C-V)/(1-V/C)=C

V=1Mpc/(52Mpc/C)=C/52

So the frequency observed by the earth will be smaller and the wavelength will be greater.

The new frequency is,

f\'=f*sqrt[(1-V/C)/(1+V/C)]

so,V/C=1/52

l\'=l*sqrt[(1+V/C)/(1-V/C)]=495.4 nm

The wavelength is 495.4 nm