Redshift of non-comoving galaxy

[wabbit]

Homework Statement

In a flat FRW universe, for a source moving radially at velocity v (at emission time) relative to the local comoving frame, what is the redshift observed by a comoving observer?

Homework Equations

$c=1$
Proper time to cosmological time ratio $\frac{d\tau}{dt}=\sqrt{1-v^2}$
Redshift of comoving source $\frac{\lambda_{obs}}{\lambda_{em}}=\frac{a(t_{obs})}{a(t_{em})}$
Note : Abusing terminology here, the "redshifts" here are quoted as $S=z+1$ intead of $z$.

The Attempt at a Solution

Based of the infinitesimal motion of the source and on light paths, I get $$\frac{\lambda_{obs}}{\lambda_{em}^{com}}=(1+v)\frac{a(t_{obs})}{a(t_{em})}$$
This is the redshift between $\lambda_{em}^{com}$ measured at emission time in a comoving frame, and $\lambda_{obs}$ measured by the comoving receiver.
And combining this with $\frac{\lambda_{em}^{com}}{\lambda_{em}}=\frac{1}{\sqrt{1-v^2}}$, the total redshift is
$$\frac{\lambda_{obs}}{\lambda_{em}}=\sqrt{\frac{1+v}{1-v}}\frac{a(t_{obs})}{a(t_{em})}$$

Can someone confirm if this correct, and if not point to the error? This is not exactly homework, rather a calculation I did since I wondered about that case, and I didn't find the relevant formula online to check against.

Thanks

 

[mfb]

I think you can split the problem in two parts. Find the frequency or wavelength observed by a comoving observer at (nearly) the same place, then apply the usual scale factor for distant observers. And the result is exactly what you got if the object is moving away.

 

[wabbit]

Thanks - Indeed, I didn't see it this way, the result is just a Doppler shift times an expansion shift, that's a much better way to get it than what I did !