Measures of flux and magnitudes
This page provides detailed descriptions of various
measures of magnitude and related outputs of the photometry
pipelines. We also provide discussion of some methodology. For details
of the Photo pipeline processing please read the corresponding EDR paper section. There
are also separate pages describing the creation of flatfields and the photometric flux calibration.
The SDSS asinh magnitude system
Magnitudes within the SDSS are expressed as inverse hyperbolic sine
(or ``asinh'') magnitudes, described in detail by Lupton, Gunn, & Szalay (1999). They are sometimes
referred to informally as luptitudes . The transformation
from linear flux measurements to asinh magnitudes is designed to be
virtually identical to the standard astronomical magnitude at high
signaltonoise ratio, but to behave reasonably at low signaltonoise
ratio and even at negative values of flux, where the logarithm in the
Pogson magnitude
fails. This allows us to measure a flux even in the absence of a
formal detection; we quote no upper limits in our photometry.
The asinh magnitudes are characterized by a softening parameter
b, the typical 1sigma noise of the sky in a PSF aperture in
1'' seeing. The relation between detected flux f and asinh
magnitude m is:
m=2.5/ln(10) * [asinh(2b f/f_{0}) + ln(b)].
Here, f_{0} is given by the classical
zero point of the magnitude scale, i.e., f_{0} is the
flux of an object with conventional magnitude of zero. The
quantity b is measured relative to f_{0},
and thus is dimensionless; it is given in the table of asinh softening
parameters (Table 21 in the EDR
paper), along with the asinh magnitude associated with a zero flux
object. The table also lists the flux corresponding to
10f_{0}, above which the asinh magnitude and the
traditional logarithmic magnitude differ by less than 1% in flux. For
details on converting asinh magnitudes to other flux measures, see converting counts to magnitudes.
Fiber magnitudes
The flux contained within the aperture of a spectroscopic fiber (3" in
diameter) is calculated in each band.
Notes
 For children
of deblended galaxies,
some of the pixels within a 1.5" radius may belong to other
children; we now measure the flux of the parent at the position of the
child; this properly reflects the amount of light which the
spectrograph will see.
 Images are convolved to 2" seeing before fiberMags are
measured. This also makes the fiber magnitudes closer to what is seen
by the spectrograph.
Model magnitudes
The computation of model magnitudes in the DR1 and EDR processing
had a serious bug which implied that model magnitudes from the EDR and
DR1 should not be used for scientific analysis. The imaging data have
all been processed through a new version of the SDSS imaging pipeline,
that most importantly fixes an error in the model fits to each object.
The result is that the model magnitude is now a good proxy for point
spread function (PSF) magnitude for point sources, and Petrosian
magnitude (which have larger errors than model magnitude) for extended
sources.
Just as the PSF magnitudes are optimal
measures of the fluxes of stars, the optimal measure of the flux of a
galaxy would use a matched galaxy model. With this in mind, the code
fits two models to the twodimensional image of each object in each
band:
 A pure deVaucouleurs profile
I(r) = I_{0} exp{7.67 [(r/r_{e})^{1/4}]}
(truncated beyond 7r_{e} to smoothly go to zero at
8r_{e}, and with some softening within
r=r_{e}/50).
 A pure exponential profile
I(r) = I_{0} exp(1.68r/r_{e})
(truncated beyond 3r_{e} to smoothly go to zero
at 4r_{e}.
Each model has an arbitrary axis ratio and position angle.
Although for large objects it is possible and even desirable to fit
more complicated models (e.g., bulge plus disk), the computational
expense to compute them is not justified for the majority of the
detected objects. The models are convolved with a doubleGaussian fit
to the PSF, which is provided by psp.
Residuals between the doubleGaussian and the full KL PSF model are
added on for just the central PSF component of the image. These
fitting procedures yield the quantities
 r_deV and
r_exp, the effective radii of the
models;
 ab_deV and ab_exp, the axis ratio of the best fit models;
 phi_deV and phi_exp, the position angles of the ellipticity (in
degrees East of North).
 deV_L and exp_L, the likelihoods associated with each
model from the chisquared fit;
 deVMag and expMag, the
total magnitudes associated with each fit.
Note that these quantities correctly model the effects of the PSF.
Errors for each of the last two quantities (which are based only on
photon statistics) are also reported. We apply aperture corrections
to make these model magnitudes equal the PSF magnitudes in the case of
an unresolved object.
Cmodel magnitudes
The code now also takes the best fit exponential and de Vaucouleurs
fits in each band and asks for the linear combination of the two that
best fits the image. The coefficient (clipped between zero and one)
of the de Vaucouleurs term is stored in the quantity fracDeV in the CAS. (In the flat files of the DAS,
this parameter is misleadingly termed fracPSF.) This allows us to define a composite
flux:
F_{composite} = fracDeV
F_{deV} + (1  fracDeV) F_{exp},
where F_{deV} and
F_{exp} are the de Vaucouleurs and exponential
fluxes (not magnitudes) of the object in question. The
magnitude derived from F_{composite}
is referred to below as the cmodel magnitude (as distinct from
the model magnitude, which is based on the betterfitting of the
exponential and de Vaucouleurs models in the r band).
In order to measure unbiased colors of galaxies, we measure their
flux through equivalent apertures in all bands. We choose the model
(exponential or deVaucouleurs) of higher likelihood in the r
filter, and apply that model (i.e., allowing only the amplitude to
vary) in the other bands after convolving with the appropriate PSF in
each band. The resulting magnitudes are termed modelMag. The resulting estimate of galaxy color
will be unbiased in the absence of color gradients. Systematic
differences from Petrosian colors are in fact often seen due to color
gradients, in which case the concept of a global galaxy color is
somewhat ambiguous. For faint galaxies, the model colors have
appreciably higher signaltonoise ratio than do the Petrosian colors.
There is now excellent agreement between cmodel and Petrosian magnitudes of galaxies, and
cmodel and PSF magnitudes of stars. Cmodel and Petrosian magnitudes are not expected to
be identical, of course; as Strauss
et al. (2002) describe, the Petrosian aperture excludes the outer
parts of galaxy profiles, especially for elliptical galaxies. As a
consequence, there is an offset of 0.050.1 mag between cmodel and Petrosian magnitudes of bright galaxies,
depending on the photometric bandpass and the type of galaxy. The rms
scatter between model and Petrosian magnitudes at the bright end is
now between 0.05 and 0.08 magnitudes, depending on bandpass; the
scatter between cmodel and Petrosian
magnitudes for galaxies is smaller, 0.03 to 0.05 magnitudes. For
comparison, the code that was used in the EDR and DR1 had scatters of
0.1 mag and greater, with much more significant offsets.
The cmodel and PSF magnitudes of stars are
in good agreement (they are forced to be identical in the mean by
aperture corrections, as was true in older versions of the pipeline).
The rms scatter between model and PSF magnitudes for stars is much
reduced, going from 0.03 mag to 0.02 magnitudes, the exact values
depending on bandpass. In the EDR and DR1, stargalaxy separation was
based on the difference between model and PSF magnitudes. We now do
stargalaxy separation using the difference between cmodel and PSF magnitudes, with the threshold at
the same value (0.145 magnitudes).
Due to the way in which model fits are carried out, there is some
weak discretization of model parameters, especially r_exp and r_deV. This is
yet to be fixed. Two other issues (negative axis ratios, and bad model
mags for bright objects) have been fixed since the EDR.
Petrosian magnitudes
Stored as petroMag. For galaxy photometry,
measuring flux is more difficult than for stars, because galaxies do
not all have the same radial surface brightness profile, and have no
sharp edges. In order to avoid biases, we wish to measure a constant
fraction of the total light, independent of the position and distance
of the object. To satisfy these requirements, the SDSS has adopted a
modified form of the Petrosian (1976) system, measuring galaxy fluxes
within a circular aperture whose radius is defined by the shape of the
azimuthally averaged light profile.
We define the ``Petrosian ratio'' R_{P} at a
radius r from the center of an object to be the ratio of the
local surface brightness in an annulus at r to the mean
surface brightness within r, as described by Blanton et al. 2001a, Yasuda et al. 2001:
where
I(r) is the azimuthally averaged surface brightness profile.
The Petrosian radius r_{P} is defined as
the radius at which R_{P}(r_{P}) equals some
specified value R_{P,lim}, set to 0.2 in our
case. The Petrosian flux in any band is then defined as the flux
within a certain number N_{P} (equal to 2.0 in
our case) of r Petrosian radii: In the SDSS
fiveband photometry, the aperture in all bands is set by the profile
of the galaxy in the r band alone. This procedure ensures
that the color measured by comparing the Petrosian flux
F_{P} in different bands is measured through a
consistent aperture.
The aperture 2r_{P} is large enough to contain nearly all of
the flux for typical galaxy profiles, but small enough that the sky noise in
F_{P} is small. Thus, even substantial errors in
r_{P} cause only
small errors in the Petrosian flux (typical statistical errors near
the spectroscopic flux limit of r ~17.7 are < 5%),
although these errors are correlated.
The Petrosian radius in each band is the parameter petroRad, and the Petrosian magnitude in each band
(calculated, remember, using only petroRad
for the r band) is the parameter petroMag.
In practice, there are a number of complications associated with
this definition, because noise, substructure, and the finite size of
objects can cause objects to have no Petrosian radius, or more than
one. Those with more than one are flagged as MANYPETRO; the largest one is used. Those with
none have NOPETRO set. Most commonly, these
objects are faint (r > 20.5 or so); the Petrosian ratio
becomes unmeasurable before dropping to the limiting value of 0.2;
these have PETROFAINT set and have their
``Petrosian radii'' set to the default value of the larger of 3" or
the outermost measured point in the radial profile. Finally, a galaxy
with a bright stellar nucleus, such as a Seyfert galaxy, can have a
Petrosian radius set by the nucleus alone; in this case, the Petrosian
flux misses most of the extended light of the object. This happens
quite rarely, but one dramatic example in the EDR data is the Seyfert
galaxy NGC 7603 = Arp 092, at RA(2000) = 23:18:56.6, Dec(2000) =
+00:14:38.
How well does the Petrosian magnitude perform as a reliable and
complete measure of galaxy flux? Theoretically, the Petrosian
magnitudes defined here should recover essentially all of the flux of
an exponential galaxy profile and about 80% of the flux for a de
Vaucouleurs profile. As shown by Blanton et al. (2001a), this fraction
is fairly constant with axis ratio, while as galaxies become smaller
(due to worse seeing or greater distance) the fraction of light
recovered becomes closer to that fraction measured for a typical PSF,
about 95% in the case of the SDSS. This implies that the fraction of
flux measured for exponential profiles decreases while the fraction of
flux measured for deVaucouleurs profiles increases as a function of
distance. However, for galaxies in the spectroscopic sample
(r<17.7), these effects are small; the Petrosian radius
measured by frames is extraordinarily
constant in physical size as a function of redshift.
PSF magnitudes
Stored as psfMag. For isolated stars,
which are welldescribed by the point spread function (PSF), the
optimal measure of the total flux is determined by fitting a PSF model
to the object. In practice, we do this by syncshifting the image of
a star so that it is exactly centered on a pixel, and then fitting a
Gaussian model of the PSF to it. This fit is carried out on the local
PSF KL model at each position as well; the difference between the two
is then a local aperture correction, which gives a corrected PSF
magnitude. Finally, we use bright stars to determine a further
aperture correction to a radius of 7.4'' as a function of seeing, and
apply this to each frame based on its seeing. This involved procedure
is necessary to take into account the full variation of the PSF across
the field, including the low signaltonoise ratio wings.
Empirically, this reduces the seeingdependence of the photometry to
below 0.02 mag for seeing as poor as 2''. The resulting magnitude is
stored in the quantity psfMag. The flag
PSF_FLUX_INTERP warns that the PSF photometry
might be suspect. The flag BAD_COUNTS_ERROR
warns that because of interpolated pixels, the error may be
underestimated.
The Reddening Correction
Reddening corrections in magnitudes at the position of each object,
reddening, are computed following Schlegel,
Finkbeiner & Davis (1998). These corrections are not applied
to the magnitudes in the databases. Conversions from E(BV)
to total extinction A_{lambda}, assuming a
z=0 elliptical galaxy spectral energy distribution, are
tabulated in Table 22 of the EDR Paper.
Which Magnitude should I use?
Faced with this array of different magnitude measurements to choose
from, which one is appropriate in which circumstances? We cannot give
any guarantees of what is appropriate for the science you
want to do, but here we present some examples, where we use the
general guideline that one usually wants to maximize some combination
of signaltonoise ratio, fraction of the total flux included, and
freedom from systematic variations with observing conditions and
distance.
Given the excellent agreement between cmodel magnitudes (see cmodel
magnitudes above) and PSF magnitudes for point sources, and
between cmodel magnitudes and Petrosian
magnitudes (albeit with intrinsic offsets due to aperture corrections)
for galaxies, the cmodel magnitude is now an
adequate proxy to use as a universal magnitude for all types of
objects. As it is approximately a matched aperture to a galaxy, it
has the great advantage over Petrosian magnitudes, in particular, of
having close to optimal noise properties.
Example magnitude usage
 Photometry of Bright Stars: If the objects are bright enough,
add up all the flux from the profile profMean and generate a large aperture
magnitude. We recommend using the first 7 annuli.
 Photometry of Distant Quasars: These will be unresolved,
and therefore have images consistent with the PSF. For this reason,
psfMag is unbiased and optimal.
 Colors of Stars: Again, these objects are unresolved, and psfMag is the optimal measure of their
brightness.
 Photometry of Nearby Galaxies: Galaxies bright enough to
be included in our spectroscopic sample have relatively high
signaltonoise ratio measurements of their Petrosian magnitudes. Since
these magnitudes are modelindependent and yield a large fraction of
the total flux, roughly constant with redshift, petroMag
is the measurement of choice for such objects. In fact, the noise
properties of Petrosian magnitudes remain good to r=20
or so.
 Photometry of Galaxies: Under most conditions, the cmodel magnitude is
now a reliable estimate of the galaxy flux. In addition, this
magnitude account for the effects of local seeing and thus are
less dependent on local seeing variations.
 Colors of Galaxies: For measuring colors of
extended objects, we continue to recommend using
the model (not the cmodel) magnitudes; the
colors of galaxies were almost completely unaffected by the DR1
software error. The model magnitude is calculated using the
bestfit parameters in the r band, and applies it to
all other bands; the light is therefore measured consistently
through the same aperture in all bands.
Of course, it would not be appropriate to study the
evolution of galaxies and their colors by using Petrosian magnitudes
for bright galaxies, and model magnitudes for faint galaxies.
Finally, we note that azimuthallyaveraged radial profiles are also
provided, as described below, and can
easily be used to create circular aperture magnitudes of any desired
type. For instance, to study a large dynamic range of galaxy fluxes,
one could measure alternative Petrosian magnitudes with parameters
tuned such that the Petrosian flux includes a small fraction of the
total flux, but yields higher signaltonoise ratio measurements at
faint magnitudes.
Radial Profiles
The frames pipeline extracts an
azimuthallyaveraged radial surface brightness profile. In the
catalogs, it is given as the average surface brightness in a series of
annuli. This quantity is in units of ``maggies'' per square arcsec,
where a maggie is a linear measure of flux; one maggie has an
AB magnitude of 0 (thus a surface brightness of 20 mag/square arcsec
corresponds to 10^{8} maggies per square arcsec). The number
of annuli for which there is a measurable signal is listed as nprof, the mean surface brightness is listed as
profMean, and the error is listed as profErr. This error includes both photon noise,
and the smallscale ``bumpiness'' in the counts as a function of
azimuthal angle.
When converting the profMean values to a local surface
brightness, it is not the best approach to assign the mean
surface brightness to some radius within the annulus and then linearly
interpolate between radial bins. Do not use smoothing
splines, as they will not go through the points in the cumulative
profile and thus (obviously) will not conserve flux. What frames
does, e.g., in determining the Petrosian ratio, is to fit a taut spline to the
cumulative profile and then differentiate that spline fit,
after transforming both the radii and cumulative profiles with asinh
functions. We recommend doing the same here.
The annuli used are:
Aperture  Radius (pixels)  Radius (arcsec)  Area (pixels) 
1  0.56  0.23  1 
2  1.69  0.68  9 
3  2.58  1.03  21 
4  4.41  1.76  61 
5  7.51  3.00  177 
6  11.58  4.63  421 
7  18.58  7.43  1085 
8  28.55  11.42  2561 
9  45.50  18.20  6505 
10  70.15  28.20  15619 
11  110.50  44.21  38381 
12  172.50  69.00  93475 
13  269.50  107.81  228207 
14  420.50  168.20  555525 
15  657.50  263.00  1358149 
Last modified: Thu Feb 26 15:40:19 CST 2004
