Algorithms: Velocity dispersion measurements

The observed velocity dispersion sigma is the result of the superposition of many individual stellar spectra, each of which has been Doppler shifted because of the star's motion within the galaxy. Therefore, it can be determined by analyzing the integrated spectrum of the whole galaxy - the galaxy integrated spectrum will be similar to the spectrum of the stars which dominate the light of the galaxy, but with broader absorption lines due to the motions of the stars. The velocity dispersion is a fundamental parameter because it is an observable which better quantifies the potential well of a galaxy.

Selection criteria

Estimating velocity dispersions for galaxies which have integrated spectra which are dominated by multiple components showing different stellar populations and different kinematics (e.g. bulge and disk components) is complex. Therefore, the SDSS estimates the velocity dispersion only for spheroidal systems whose spectra are dominated by the light of red giant stars. With this in mind, we have selected galaxies which satisfy the following criteria:

• classified as galaxy (specClass EQ 'SPEC_GALAXY')
• redshift obtained from cross-correlation with template (zStat EQ 'XCORR_HIC')
• no warnings from the spectroscopic pipeline (zWarning AND ('Z_WARNING_NO_SPEC' OR 'Z_WARNING_NO_BLUE' OR 'Z_WARNING_NO_RED' OR 'Z_WARNING_LOC') EQ 0)
• PCA classification (eClass LT -0.02) typical of early-type galaxy spectra (Connolly & Szalay 1999)
• redshift < 0.4

Because the aperture of an SDSS spectroscopic fiber (3 arcsec) samples only the inner parts of nearby galaxies, and because the spectrum of the bulge of a nearby late-type galaxy can resemble that of an early-type galaxy, our selection includes spectra of bulges of nearby late-type galaxies. Note that weak emission lines, such as H_alpha and/or O II, could still be present in the selected spectra.

Method

A number of objective and accurate methods for making velocity dispersion measurements have been developed (Sargent et al. 1977; Tonry & Davis 1979; Franx, Illingworth & Heckman 1989; Bender 1990; Rix & White 1992). These methods are all based on a comparison between the spectrum of the galaxy whose velocity dispersion is to be determined, and a fiducial spectral template. This can either be the spectrum of an appropriate star, with spectral lines unresolved at the spectra resolution being used, or a combination of different stellar types, or a high S/N spectrum of a galaxy with known velocity dispersion.

Since different methods can give significantly different results, thereby introducing systematic biases especially for low S/N spectra, we decided to use two different techniques for measuring the velocity dispersion. Both methods find the minimum of

  chi2 = sum { [G - B * S]2 }

1. The "Fourier-fitting" method (Sargent et al. 1977; Tonry & Davis 1979; Franx, Illingworth & Heckman 1989; van der Marel & Franx 1993). Because a galaxy's spectrum is that of a mix of stars convolved with the distribution of velocities within the galaxy, Fourier space is the natural choice to estimate the velocity dispersions---this first method makes use of this:

    chi2  = sum { [G~(k) - B~(k,sigma) S~(k)]2 /Vark2},

   where G~, B~ and S~ are the Fourier Transforms of G, B and S, respectively, and Var_k² = sigma_G~² + sigma_S~² B~(k,sigma). (Note that in Fourier space, the convolution is a multiplication.)
2. The "Direct-fitting" method (Burbidge, Burbidge & Fish 1961; Rix & White 1992). Although the Fourier space seems to be the natural choice to estimate the velocity dispersions, there are several advantages to treating the problem entirely in pixel space. In particular, the effects of noise are much more easily incorporated in the pixel-space based "Direct-fitting" method which minimizes

    chi2 = sum { [G(n) - B(n,sigma) S(n)]2 /Varn2}.

   Because the S/N of the SDSS spectra are relatively low, we assume that the observed absorption line profiles in early-type galaxies are Gaussian.

It is well known that the two methods have their own particular biases, so we carried out numerical simulations to calibrate these biases. In our simulations, we chose a template stellar spectrum measured at high S/N, broadened it using a Gaussian with rms sigma_input, added Gaussian noise, and compared the input velocity dispersion with the measured output value. The first broadening allows us to test how well the methods work as a function of velocity dispersion, and the addition of noise allows us to test how well the methods work as a function of S/N. Our simulations show that the systematic errors on the velocity dispersion measurements appear to be smaller than ~ 3% but estimates of low velocity dispersions (sigma< 100 km s^-1) are more biased (~ 5%).

Measurements

The SDSS uses 32 K and G giant stars in M67 as stellar templates. The velocity dispersion templates are available for download below. The SDSS velocity dispersion estimates are obtained by fitting the restframe wavelength range 4000-7000 Å, and then averaging the estimates provided by the "Fourier-fitting" and "Direct-fitting" methods. The error on the final value of the velocity dispersion is determined by adding in quadrature the errors on the two estimates (i.e., the Fourier-fitting and Direct-fitting). The typical error is between delta(logsigma) ~ 0.02 dex and 0.06 dex, depending on the signal-to-noise of the spectra. The scatter computed from repeated observations is ~ 0.04 dex, consistent with the amplitude of the errors on the measurements.

Estimates of sigma are limited by the instrumental dispersion and resolution. The instrumental dispersion of the SDSS spectrograph is 69 km s^-1 per pixel, and the resolution is ~ 90 km s^-1. In addition, the instrumental dispersion may vary from pixel to pixel, and this can affect measurements of sigma. These variations are estimated for each fiber by using arc lamp spectra (upto 16 lines in the range 3800-6170 Å and 39 lines between 5780-9230 Å). A simple linear fit provides a good description of these variations. This is true for almost all fibers, and allows us to remove the bias such variations may introduce when estimating galaxy velocity dispersions.

Velocity dispersion templates for download

We offer the velocity dispersion templates for download here. There are two sets of templates, 32 template stars (giant stars in M67) which are used in the "Fourier-fitting" method, and 7 principal component analysis (PCA) Eigentemplates used in the "Direct-fitting" method. We offer the templates in two formats:

• An IDL save file veldisptemplates.idl. Save the file, and type restore, 'veldisptemplates.idl' at the IDL prompt. The file includes:

        EIG             DOUBLE    = Array[1944, 7]  -> flux of the PCA template
        EIGLAMBDA       FLOAT     = Array[1944]     -> wavelength of the PCA template
        STARFLUX        FLOAT     = Array[3918, 32] -> flux of the 32 M67 giant stars
        STARSIG         FLOAT     = Array[3918, 32] -> fluxerr
        STARWAVE        FLOAT     = Array[3918, 32] -> wavelength
        

• Two fits tables with the templates as a function of wavelength, one for each set:
  • eigtemplates.fits, the PCA Eigentemplates
  • startemplates.fits, the stellar templates

Caveats

The velocity dispersion measurements distributed with SDSS spectra use template spectra convolved to a maximum sigma of 420 km/s. Therefore, velocity dispersion sigma > 420 km/s are not reliable and must not be used. There is a postscript file showing the quality of velocity dispersion error estimates.

We recommend the user to not use SDSS velocity dispersion measurements for:

• spectra with median per-pixel S/N< 10
• velocity dispersion estimates smaller than about 70 km s^-1 given the typical S/N and the instrumental resolution of the SDSS spectra

Also note that the velocity dispersion measurements output by the SDSS spectro-1D pipeline are not corrected to a standard relative circular aperture. (The SDSS spectra measure the light within a fixed aperture of radius 1.5 arcsec. Therefore, the estimated velocity dispersions of more distant galaxies are affected by the motions of stars at larger physical radii than for similar galaxies which are nearby. If the velocity dispersions of early-type galaxies decrease with radius, then the estimated velocity dispersions (using a fixed aperture) of more distant galaxies will be systematically smaller than those of similar galaxies nearby.)