Calculating Luminosity Distances: Converting Angular Distances to Parsecs

[Radiohannah]

Hey

I'm getting very muddled with my units, and would really appreciate some clarity :-)

I have angular distances between galaxies at some redshift, in arcseconds

I want to calculate the distance in parsecs, taking into account the luminosity distance.


In the equation;

r = \frac{D_{L}}{(1+z)^{2}} \theta

I'm assuming that "r" in this will be my distance in parsecs.
"D_{L}" will be the luminosity distance.
and...\theta will be the angular distance (in arcseconds..?) between the galaxies.

What units would my luminosity distance have to be in, in order to calculate my "r" in parsecs?

I know that the equation for the luminosity distance is

D_{L} =(1+z)\frac{2c}{H_{0} } \frac{\Omega_{z + (\Omega - 2)[\sqrt{1+\Omega_{z}}-1]}}{\Omega^{2}(1+z)}

Does this give the correct units for "r" to be in parsecs? I am getting so confused!

Thank you

 

[Ken G]

To get r in parsecs, you need D in parsecs, and theta must be in radians (which is effectively unitless). I believe the r we are talking about is distance between the galaxies, not distance to the galaxies (which is already accommodated by D). To get D in parsecs, you need to use c in km/s, and H in km/s per Mpc, and then you have to convert H to km/s per pc, and your formula should show you that D ends up in parsecs. More likely, you want want r in Mpc, so then you just need D in Mpc, c in km/s, and H in km/s per Mpc-- those are standard units.

 

[Radiohannah]

That is very clear and helpful, Thank you!