Galaxy Size Distributions

This example demonstrate how to sample sizes for early and late type galaxies in SkyPy.

Size-Magnitude Relation

In Shen et al. 2003 [1], the observed sizes, \(R\), of galaxies were shown to follow simple analytic relations as a function of their absolute magnitudes, \(M\). For early-type galaxies, their mean radius follows Equation 14:

\[\log_{10} (\bar{R}/{\rm kpc}) = -0.4aM + b,\]

with \(a\) and \(b\) fitting constants. Likewise, late-type galaxies follow Equation 15:

\[\log_{10}(\bar{R}/{\rm kpc})=-0.4\alpha M+ (\beta -\alpha)\log \left[1+10^{-0.4(M-M_0)}\right]+\gamma \, .\]

The dispersion on these relations is given by Equation 16:

\[\sigma_{ln R} = \sigma_2 + \frac{\sigma_1 - \sigma_2}{1 + 10^{-0.8(M - M_0)}}\]

where \(\alpha\), \(\beta\), \(\gamma\), \(\sigma_1\), \(\sigma_2\) and \(M_0\) are fitting parameters.

In SkyPy, we can sample physical sizes for each galaxy type from lognormal distributions, with median \(\bar{R}\) and width \(\sigma_{ln R}\), using the functions skypy.galaxies.morphology.early_type_lognormal_size() and skypy.galaxies.morphology.late_type_lognormal_size().

In this example, we simulate the sizes of galaxies with random magnitudes using the values for the parameters given in Shen et al. 2003 Table 1 [1] :

import numpy as np
import matplotlib.pyplot as plt
from skypy.galaxies.morphology import (early_type_lognormal_size,
                                       late_type_lognormal_size)

# Parameters for the late-type and early-type galaxies
alpha, beta, gamma = 0.21, 0.53, -1.31
a, b = 0.6, -4.63
M0 = -20.52
sigma1, sigma2 = 0.48, 0.25

# SkyPy late sample
M_late = np.random.uniform(-16, -24, size=10000)
R_late = late_type_lognormal_size(M_late, alpha, beta, gamma, M0, sigma1, sigma2).value

# SkyPy early sample
M_early = np.random.uniform(-18, -24, size=10000)
R_early = early_type_lognormal_size(M_early, a, b, M0, sigma1, sigma2).value

Validation against SDSS Data

Here we reproduce Figure 4 from [1], comparing our simulated galaxy sizes against observational data from SDSS. You can download the data files for early-type and late-type SDSS galaxies which have the following columns: magnitudes, median radius, minus error, and plus error.

# Load data from figure 4 in Shen et al 2003
sdss_early = np.loadtxt('Shen+03_early.txt')
sdss_late = np.loadtxt('Shen+03_late.txt')
error_late = (sdss_late[:, 2], sdss_late[:, 3])
error_early = (sdss_early[:, 2], sdss_early[:, 3])

# Bins for median radii
M_bins_late = np.arange(-16, -24.1, -0.5)
M_bins_early = np.arange(-18, -24.1, -0.5)

# Center bins
center_late = (M_bins_late[:-1] + M_bins_late[1:]) / 2
center_early = (M_bins_early[:-1] + M_bins_early[1:]) / 2

# Median sizes for SkyPy late- and early-type galaxies
R_bar_early = [np.median(R_early[(M_early <= Ma) & (M_early > Mb)])
               for Ma, Mb in zip(M_bins_early, M_bins_early[1:])]
R_bar_late = [np.median(R_late[(M_late <= Ma) & (M_late > Mb)])
              for Ma, Mb in zip(M_bins_late, M_bins_late[1:])]

# Plot
plt.plot(center_early, R_bar_early, 'r', label='SkyPy early')
plt.plot(center_late, R_bar_late, 'b', label='SkyPy late')

plt.errorbar(sdss_early[:, 0], sdss_early[:, 1], yerr=error_early, color='coral',
             marker='s', label='Shen+03 early', ls='none')
plt.errorbar(sdss_late[:, 0], sdss_late[:, 1], yerr=error_late, color='deepskyblue',
             marker='^', label='Shen+03 late', ls='none')

plt.ylim(5e-1, 2e1)
plt.xlim(-16, -24)
plt.xlabel('Magnitude $M$')
plt.ylabel('$R_{50,r} (kpc)$')
plt.legend(frameon=False)

plt.yscale('log')
plt.show()

References

[1] (1,2,3)

S. Shen, H.J. Mo, S.D.M. White, M.R. Blanton, G. Kauffmann, W. Voges, Brinkmann, I. Csabai, Mon. Not. Roy. Astron. Soc. 343, 978 (2003)