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 crosscorrelation 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
earlytype 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 latetype galaxy can resemble that of an
earlytype galaxy, our selection includes spectra of bulges of nearby
latetype 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
chi^{2} = sum { [G  B * S]^{2} }
where G is the galaxy, S the star and B is the gaussian broadening
function (* denotes a convolution).
 The "Fourierfitting" 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 dispersionsthis first method
makes use of this:
chi^{2} = sum { [G~(k)  B~(k,sigma) S~(k)]^{2} /Var_{k}^{2}},
where G~, B~ and S~ are the Fourier Transforms of G, B and S,
respectively, and Var_{k}^{2} = sigma_{G}~^{2} + sigma_{S}~^{2} B~(k,sigma).
(Note that in Fourier space, the convolution is a multiplication.)
 The "Directfitting" 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 pixelspace based "Directfitting" method which minimizes
chi^{2} = sum { [G(n)  B(n,sigma) S(n)]^{2} /Var_{n}^{2}}.
Because the S/N of the SDSS spectra are relatively low, we assume that the
observed absorption line profiles in earlytype 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
40007000 Å, and then averaging the estimates provided by the
"Fourierfitting" and "Directfitting" 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 Fourierfitting
and Directfitting). The typical error is between
delta(logsigma) ~ 0.02 dex and 0.06 dex, depending on the
signaltonoise 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 38006170 Å and 39 lines
between 57809230 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 "Fourierfitting" method, and 7
principal component analysis (PCA) Eigentemplates used in the
"Directfitting" 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:
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 perpixel 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 spectro1D 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 earlytype 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: Sun Mar 14 20:42:15 CST 2004
