Calculating Basic Properties from SDSS Observables

The purpose of this page is to summarize the ways that we calculate the stellar properties of a galaxy using optical broadband data, with a specific focus on the Sloan Digital Sky Survey (SDSS). This is a rather extensive topic, so check back here as I update the webpage.

The outline here is two fold. First I'll try to go over the observables we have access to, first in general and then specific to SDSS. Then I will discuss derivation of stellar parameters from these observables, both in general and specific to our work.

The Observables:

General Photometric Observables

What sort of things can we observe about a galaxy? From photometry (pictures) we can observe a galaxy's brightness, its size, its shape, its morphology, and its color. From spectroscopy we can measure the strength and shifts of spectral lines. If you don't have spectra, you still can extract spectral information by comparing information in different bands, which is the idea behind the SED fitting done in Huang et al. 2012.

There are three main ways we measure magnitudes (see Elmegreen 4.1 p.61).

• Total Magnitudes: The "total" integrated light of a galaxy. This means that we fit a model profile to the galaxy (de Vauc, Sersic, exponential, etc.), and then integrate that profile out to infinity, or some cutoff (e.g. when the value decays by e⁸).
• Metric magnitudes: The integrated light measured within some aperature, say 17 kpc or 2 arcminutes.
• Isophotal magnitudes: The integrated light measured out to some limiting surface brightness, i.e. out to a variable aperture determined by some surface brightness (recall that surface brightness is defined as luminosity/area, usually reported in magnitudes per square arcsecond).

Surface brightness profiles for a sample of large elliptical galaxies. On this plot a metric magnitude corresponds to a vertical slice (See the dashed vertical line) whereas an isophotal magnitude would correspond to a horizontal slice. A total magnitude would correspond to an integral of all of the flux under the curve, extrapolated out to some radius, usually either infinity or eight exponential scale hights. (Image credit: Haarsma et al. 2010)

It is also worth remembering that there are several important corrections that we need to do to the magnitudes we observe before we can do any calculations:

• Milky Way Extinction (due to absorption and scattering, often called a dust correction)
• Internal extinction
• Redshift corrections (K-correction)
• For anything requiring absolute quantities, you need to determine the distance (see Mike's page) and then convert your observed quantity to an absolute quantity.

Specific SDSS Observables

SDSS reports many different observables, but some are more useful than others for our purposes.

As SDSS keeps releasing new data it takes some work to keep up with their ever changing links. Still, here is a list of the current, SDSS8 references.

NOTE: SDSS9 is coming out in July of this year (2012), so stay posted. I think the main change for SDSS9 is that they are adding some spectrascopic data for the anti-virgo (ie, our fall) sky. This will be completed by SDSS12, which is scheduled for a couple of years from now.

• For SDSS 8, you can access the full list of available photometric parameters at the schema browser.
• Under their algorithms link they have an excellent explanation of their different magnitudes on the page called

Here are the parameters we currently are using:

• Inclinations: SDSS provides a measure of the ratio of the major to minor axis, a/b, as found using the radius defined by their exponential fit to the galaxy. We currently use their measurement of this ratio done in the r band (In their jargon the parameter is called expAB_r). So to get the inclination we want to use 1-arccos(expAB_r).
• Masses We currently derive these from modelmagnitudes, because they give us the most consistancy between bands for calculation of colors. In dr7 this is the best of a de Vaucouleurs fit and a pure exponential fit. New in DR8 is a linear combo of these.
• Star Formation rates These also come out of the SED fits done to the modelmags.

How to Calculate Basic Properties:

There are two categories of approaches to determining galactic properties.

Direct tracers

One approach is to derive equations that directly connect observables to fundamental properties, like using a mass-luminosity relationship to determine mass from luminosity. Here we make the assumption that we know how an observable traces a fundamental property.

Spectral Energy Distribution (SED) fitting

A second approach is to simulate spectra of galaxies based on different physical properties, and then to try to match the simulated spectra to the observed spectra, or to observed broadband magnitudes. Here we make assumptions about the imprint that different physical properties leave on a galaxy's spectrum.

Here's the procedure:

• start with a library of realistic SEDs
• compare your data with these SEDs
• calculate Χ², i.e. numerically determine how well each of these template SEDs fits your data.
• Create a probability distribution based on the Χ² fits to narrow in on the most probable parameters.

The most difficult part of this procedure is developing an appropriate library of SED templates that cover all of the variables which leave their imprint on the spectrum. These include the mass, metallicity, initial mass function (IMF), and the star formation history, and the age of the galaxy.

Things we would like to calculate

Here are some specific things that we can calculate:

• Inclinations

  We expect that all spiral galaxies should be approximately shaped like a disk. Thus, we can calculate the angle that we are viewing the disk, or the inclination angle, defined such that a galaxy viewed face on had an inclination of 0 degrees and viewed edge on has an inclination of 90 degrees. To first order, we can do this using geometry taking the inverse cosine of the ratio of the shortest axis (the minor axis) to the the longest axis (the major axis, perpendicular to the minor axis). I recommend drawing yourself a picture, it really helps.

  Often astronomers use two corrections to this calculation, one taking into account the fact that galaxies have some inherent width, and the other taking into account other systematic errors associated with optical observations. If you are interested, check out this webpage as a starting point.

• Masses

  Perhaps the most common approach to calculating stellar masses is to assume some mass to light ratio based off of the properties of your galaxy, and then directly convert the observed luminosity into a mass.

  Another way to do it is to determine your mass to light ratio based off of your SED fit, i.e. the stellar mass M* comes directly out of your model of the stellar population.

• Star Formation rates

  We can get an estimate of the star formation rate from the color and the luminosity of the galaxy if we make a few assumptions about the galaxy. The color of the galaxy will be determined by its initial mass function, its metallicity, and its current rate of star formation/its age. If we make some assumptions about the IMF and the metal content of the galaxy, we can then use the galactic luminosity to determine the star formation rate (see Kennicutt 1998a; Salim et al. 2007).