Galaxy properties

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Blanton et al. (2003) present the joint bivariate distributions of a number of optical galaxy properties, such as color, luminosity, surface brightness, profile shape, and local overdensity. This paper provides the replacement for the galaxy luminosity function as a function of type given in Blanton et al (2001). Here we present a slightly more pedagogical presentation of the results of that paper. All of the bitmapped images found on this page, plus bitmapped images of the figures in the paper, can be accessed directly. In addition, the data comprising the figures in the paper is publicly available.

First let us review the properties we are going to be considering:

  1. broadband colors: u-g, g-r, r-i, and i-z
  2. half light surface brightness in the i-band: μ
  3. best fit seeing-corrected Sersic index: n
  4. absolute magnitude: M
  5. local overdensity smoothed on 8 Mpc scales: δ

The SDSS spectroscopic sample is flux limited. Thus the volume over which we can detect low luminosity galaxies is smaller than the volume over which we can detect high luminosity galaxies. In order to calculate the number density contribution of each galaxy, we must calculate the maximum volume over which it is observable, known as Vmax. We weight each galaxy by the inverse of Vmax.

For example, consider the luminosity function. The raw distribution of galaxy luminosities in the sample, which does not reflect the actual abundance of galaxies in the universe, is:

Figure 1: The distribution of SDSS galaxy absolute magnitudes in the 0.1i band (for h=1). Note that the sample we are using is for redshifts 0.02 < z < 0.22; the complete lack of galaxies less luminous than -17 is due to the lower redshift limit.
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Figure 1: The distribution of SDSS galaxy absolute magnitudes in the 0.1i band (for h=1). Note that the sample we are using is for redshifts 0.02 < z < 0.22; the complete lack of galaxies less luminous than -17 is due to the lower redshift limit.

However, when we weight by the inverse of Vmax, the lower luminosity galaxies are weighted more strongly than the higher luminosity galaxies, because we can observe the low luminosity galaxies over a much smaller volume. Thus, the resulting distribution, which better represents the distribution of galaxy luminosities in the universe, is:

Figure 2: The luminosity function of SDSS galaxies in the 0.1i band (for h=1), from a sample with redshifts 0.02 < z < 0.22; the complete lack of galaxies less luminous than -17 is due to the lower redshift limit. The y-axis scale is linear.
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Figure 2: The luminosity function of SDSS galaxies in the 0.1i band (for h=1), from a sample with redshifts 0.02 < z < 0.22; the complete lack of galaxies less luminous than -17 is due to the lower redshift limit. The y-axis scale is linear.

This is very close to a Schechter function, though you should note the deviation at around -19. Actually calculating the number density does matter, in this case quite a bit for the luminosity distribution; it will matter for the color distribution and other properties as well, however, since those properties do correlate with luminosity.

Now consider two properties, for example g-r color and luminosity. We can calculate the number density distribution of either quantity separately, or we can calculate the joint number density distribution. To visualize the latter, we use a greyscale and contours. For example:

Figure 3: The distribution of color and absolute magnitude for SDSS galaxies (inverse Vmax weighted, and for h=1), from a sample with redshifts 0.02 < z < 0.22. The upper left and lower right show the distribution of each property separately, and the other corners show the biariate distribution (and are just mirror images of each other). The greyscale is proportional to the square root of the number density. Note the characteristic division into the red and blue sequences.
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Figure 3: The distribution of color and absolute magnitude for SDSS galaxies (inverse Vmax weighted, and for h=1), from a sample with redshifts 0.02 < z < 0.22. The upper left and lower right show the distribution of each property separately, and the other corners show the biariate distribution (and are just mirror images of each other). The greyscale is proportional to the square root of the number density. Note the characteristic division into the red and blue sequences.

Here we show the one-dimensional distributions on the diagonal and the bivariate distributions on the off-diagonals. The greyscale is a square root stretch and the contours enclose 33%, 68%, 95% and 99% of the number density (from high to low). Several scientifically interesting features appear, including the clear red and blue sequences. It is bivariate plots like this that form the basis of the results in Blanton et al (2003).

We obviously can make this bivariate plot using an arbitary number of rows and columns. First let us consider the bivariate distributions among all of the optical colors:

Figure 4: Similar to Figure 3, but for the joint distribution of galaxy colors.
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Figure 4: Similar to Figure 3, but for the joint distribution of galaxy colors.

Two populations are clearly visible. In addition, it is clear that all of the colors are very correlated. We can make this case even more convincing by making a "conditional" version of the plot. That is, we take every column of pixels in each panel and normalize it so it has the same integral. The plot becomes an image of the probability of the parameter on the y-axis given a value of the parameter on the x-axis. For the colors the conditional version of the plot is:

Figure 5: Similar to Figure 4, but now showing the conditional distribution of each galaxy color on the others (that is, every column within each panel is normalized separatly, and the lines are not density contours but the quartiles within each column).
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Figure 5: Similar to Figure 4, but now showing the conditional distribution of each galaxy color on the others (that is, every column within each panel is normalized separatly, and the lines are not density contours but the quartiles within each column).

The lines in this case represent the quartiles (weighted by inverse Vmax of course) in each column of pixels. Clearly all of the colors are extremely strongly correlated. In particular, g-r is strongly predictive of the other colors. In what follows, therefore, we will not consider the other colors and only retain g-r in the plots.

Let us press on and consider the bivariate relationships between color, surface brightness, Sersic index, and i-band absolute magnitude:

Figure 6: The joint distribution of galaxy color, half-light surface brightness, Sersic index and absolute magnitude, shown similarly to Figure 4.
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Figure 6: The joint distribution of galaxy color, half-light surface brightness, Sersic index and absolute magnitude, shown similarly to Figure 4.

The details of this plot are discussed in Blanton et al. (2003). For now just note that the relationships are complex and interesting. Many of the trends have been known for many years, but here we see them at high signal to noise.

This result is fine for understanding the bivariate distributions, but can we go to higher dimensionality? Of course we can, by plotting the distribution after restricting the distribution to only a range of possible values of each quantity. For example:

  1. Dependence on absolute magnitude M
  2. Dependence on color g-r
  3. Dependence on surface brightness μ
  4. Dependence on Sersic index n

We can take one step further and add an estimate of the local overdensity as a variable. Now, this is a low signal-to-noise measurement, and in addition, the signal-to-noise is a function of luminosity. So we will only consider the distribution of density conditional on the other properties. The resulting distribution looks like:

Figure 7: The conditional distribution of galaxy color, half-light surface brightness, Sersic index, absolute magnitude and local density, as a function of the first four properties, shown similarly to Figure 5.
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Figure 7: The conditional distribution of galaxy color, half-light surface brightness, Sersic index, absolute magnitude and local density, as a function of the first four properties, shown similarly to Figure 5.

It is notable that the strongest apparent dependence of density is on luminosity. We perform a more detailed analysis of this in Hogg et al. (2003).

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