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!