# Estimating diameter of red-shifted galaxy

Q: Estimate the diameter in kpc of the red-shift=6.56 galaxy at the moment when it emitted light. We're assuming that early galaxies with similar red-shifts have the same number of stars to galaxies as in the present state of the universe.

I've taken a look at the small angle formula a = s / d where a = angular size, d = distance and s = galaxy true size (diameter). But for that I need the angular size and distance. I'm not sure if if there is a way to calculate that using the information given in the question. Other than the given information, I have the average density and scale-factor as well.

mfb
Mentor
Average density of what?

You need something more than the redshift value, because there are galaxies of different size. The problem statement mentions "early galaxies with similar red-shifts", which suggests to compare this galaxy to some other galaxies at that redshift.

I have the average density of the z= 6.56 galaxy. The question says that all galaxies around the time of the red-shift = 6.56 have similar masses and number of stars to galaxies in the present universe. I'm assuming that is supposed to make the question easier and overall I'll need less variables to estimate the diameter, but I'm not sure what to do next.

mfb
Mentor
Okay, so you know the density, and you can take the mass and number of stars from a current galaxy. That alone should be sufficient to get the volume, and if you assume that the shape is the same you can also get the diameter.

If you put in a wavelength it can give you the distance to the galaxy. I also found out that c * z = v is the velocity of the galaxy with a red-shift wavelength z. So I now I have the velocity of the galaxy as well. I calculated distance using v = H0 x d OR d = v / H0 and I also have its distance in Mpc. Not sure what to do at this point. I've looked at a lot of equations now and am confused.

mfb
Mentor
I also found out that c * z = v is the velocity of the galaxy with a red-shift wavelength z.
There is no unique meaningful definition of a velocity for galaxies far away from us.
So I now I have the velocity of the galaxy as well. I calculated distance using v = H0 x d OR d = v / H0 and I also have its distance in Mpc.
That does not work, neither for the current nor for the past distance.

If I understand the problem correctly, you are overthinking this massively. You have the mass and the density, calculate the volume.

Ok I finally got the value I was looking for. I really was overthinking it for no reason. The final distance in Kpc worked out to be the correct one. Thanks.