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Algorithms: Velocity dispersion measurements

The velocity dispersion measurement has been changed to use only the direct-fitting method in DR6.

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 < 0) 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 Halpha and/or O II, could still be present in the selected spectra.


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.

From DR1 to DR5, SDSS velocity dispersions were the average of a result from direct-fitting and Fourier methods for measuring the velocity dispersion. From DR6, only the "Direct-fitting" method (Burbidge, Burbidge & Fish 1961; Rix & White 1992) is used. It finds the minimum of

 chi2 = sum
{ [G - B * S]2 }
where G is the galaxy, S the star and B is the gaussian broadening function (* denotes a convolution). 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
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.

Correction of biases in veloctiy dispersions

Due to changes in the spectroscopic reductions from the EDR to later releases, a bias appeared in the veloctiy dispersion values available in the CAS. As shown in Bernardi (2007; Appendix A), σ values in the DR5 do not match the values used by Bernardi et al. (2003; B03). The difference is small but systematic, with spectro1d DR5 larger than B03 at σ ≤ 150 km/s. A similar bias is seen when comparing spectro1d DR5 with measurements from the literature (using the HyperLeda database; Paturel et al. 2003). Simulations similar to those in B03 show that the discrepancy results from the fact that the Fourier-fitting method is biased by ∼ 15% at low dispersions (~100 km/s), whereas the direct-fitting method is not. We therefore use the direct-fitting method only in DR6. Below, we show comparisons of the spectro1d DR6 velocity dispersions with those from B03, DR5 and the specBS measurements. There is good agreement between spectro1d DR6 and B03 (rms scatter ~ 7.5%), and between spectro1d DR6 and specBS (rms scatter ~ 6.5%), whereas spectro1d DR5 is clearly biased high at σ ≤ 150 km/s. The agreement between spectro1d DR6 and specBS is not surprising, since both are now based only on the direct-fitting method. The specBS measurements tend to be slightly smaller than DR6 at σ ≤ 100 km/s; specBS is similarly smaller than HyperLeda, whereas DR6 agrees with HyperLeda at these low dispersions.

Top panels: velocity dispersion measurements from B03 (left), DR5 (middle) and specBS (right) versus the spectro1d DR6 values for the sample of elliptical galaxies used in Bernardi et al. (2003a). Bottom panels: The ratio of DR6 values to the other three samples (i.e. B03, DR5, and specBS) versus the mean value (e.g. left panel σ = ( σDR6 + σB03 )/2). The median value at each value of σ is shown as the red curve.

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 were used in the "Fourier-fitting" method prior to DR6, 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:


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. The figure below shows the quality of velocity dispersion error estimates.

Error distribution of the velocity dispersion measurements from spectro1d DR6 (thin black solid line), spectro1d DR5 (dotted red line), specBS (dashed blue line), and B03 (dotted-dashed green line). The thick solid line was obtained by comparing repeated measurements.

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.)

Last modified: Thu Jul 12 21:59:52 CEST 2007