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()
plot size

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)

Total running time of the script: ( 0 minutes 0.267 seconds)

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