3d distance between the Galaxies inside the galaxy cluster

Question

I am currently working on the dynamics of the galaxy cluster, so i am trying to get the distance between the galaxies inside the galaxy cluster from its centre. As a input i have RA , DEC and Z (Redshift of the galaxies and its centre). my approach is something like this

1. First of all i start with the commoving distance of the galaxies and Cluster for a given z (Galaxies redshift) and z_cl (Clusters redshift) .
2. then i get the Cartesian position of each object in 3D space
3. Once i have the Cartesian of the objects, i use the standard distance formula to compute 3-D Distances between them the python Code is like this

`#  comoving distances
D_cl=cosmos.comoving_distance(z).value #  z for each galaxy
D_clus=cosmos.comoving_distance(z_cl).value # z_cl for Cluster's centre

def get_x_y_z(ra, dec, D):
    phi   = ( ra*180/np.pi   - 180 ) * np.pi / 180.
    theta = (dec*180/np.pi + 90 ) * np.pi / 180.
    xx = D * np.cos( phi) * np.sin( theta )
    yy = D * np.sin( phi) * np.sin( theta )
    zz = D * np.cos( theta )
    return xx, yy, zz

# get 3D Cartesian positions of the sub haloes
xx, yy, zz = get_x_y_z(ra, dec, D_cl)

# get 3D Cartesian positions of the cluster
xx_cl, yy_cl, zz_cl = get_x_y_z(ra_cl, dec_cl, D_clus)

# array of distances between sub haloes and the cluster : 
distances = np.sqrt((xx_cl-xx)**2 + (yy_cl-yy)**2 + (zz_cl-zz)**2)
distances.min(),distances.max()`

Another Approach using astropy

c2 = SkyCoord(ra*180/np.pi *u.deg, dec*180/np.pi *u.deg, distance=D_clus*u.Mpc, frame='icrs')
distance_3d = c1.separation_3d(c2)
distance_3d

So after using this method i am getting a wrong value's (Almost 70 Mpc for a given redshift therefore i would like to know what can i change or adapt to have right measurement of the 3d distances or my method is wrong. As you can see i am going to use this distance r in order to calculate the jeans solution. where i need two distances r and R.

• R is the projected separation of the galaxies from clusters centre

• r is the actual 3d distance of the galaxies from clusters centre

$$v_g(r) = -\frac{1}{\pi}\int_r^\infty \frac{d \Sigma}{dr} \frac{dR}{\sqrt{R^2-r^2}}$$

Answer 1

You can't calculate the distances to galaxies in a cluster like that to any useful precision.

The mean redshift will give you an estimate of the mean distance of the cluster, but the individual galaxies have perturbations around this value to that reflect their dynamics within the cluster rather than cosmological redshift. These "peculiar velocities" with respect to the cluster mean can be as large as 1000 km/s (RMS), which leads to distance error bars of order 15 Mpc compared with a typical cluster diameter of a few Mpc. And of course, some small fraction of deviations could be several times this RMS error.

In general, I do not think there is any reliable way to estimate the 3D positions of cluster galaxies with respect to the cluster centre for distant clusters. For closer clusters (Virgo) there is the possibility of using some of the other distance ladder indicators (Cepheids, globular clusters, most luminous red giants, fundamental plane, Tully-Fisher etc.) to get the individual galaxy distances with a precision somewhat better than the cluster diameter (e.g. Gavazzi et al. 1999; Solanes et al. 2002).