SDSS DR6 Photometry
Measures of SDSS flux, magnitudes, and diameters
(Adopted, with thanks, from
the SDSS DR6 photometry documentation pages. Latest revision: 19 July 2017)
This page provides detailed descriptions of various measures of magnitude and
related outputs (e.g., diameter) of the SDSS photometry pipelines.
The SDSS asinh magnitude system
All magnitudes within the SDSS are expressed as inverse hyperbolic sine
(or “asinh”) magnitudes, described in detail by Lupton, Gunn, & Szalay (1999) and in the
SDSS DR6 algorithms
page. They are sometimes referred to informally as luptitudes. The
transformation from linear flux density measurements to asinh magnitudes is designed
to be virtually identical to the standard astronomical magnitude at high
signal-to-noise ratio, but to behave reasonably at low signal-to-noise
ratio and even at negative values of flux density, where the logarithm in the
Pogson magnitude fails. This allows SDSS to report a flux density
even in the absence of a formal detection; they quote no upper limits
in their photometry.
The asinh magnitudes are characterized by a softening parameter
b, the typical 1-sigma noise of the sky in a PSF aperture in
1" seeing. The relation between flux density f and asinh
magnitude m is
m=[-2.5/ln(10)][asinh((f/f_{0})/(2b)) + ln(b)]
or
f=2bf_{0} sinh(m/[-2.5/ln(10)] - ln(b)).
Here,
f_{0} is the zero point of the magnitude scale, i.e.,
f_{0} is the flux density of an object with magnitude of zero, given in
the table below. The quantity
b is measured relative to
f_{0}, and thus is dimensionless.
The corresponding flux density uncertainty is
df= abs[(fdm/[-2.5/ln(10)])/tanh((m/[-2.5/ln(10)])-ln(b))],
where
dm is the uncertainty in SDSS magnitude
m.
IMPORTANT:
The DR5 and DR6 magnitudes, fluxes, and flux uncertainties were updated in the 2012 September NED release, to
correct three issues:
- The u band CModel magnitudes and z band PSF and Model magnitudes were restored to original SDSS values, removing
zero-point corrections mzpc of -0.04 to the u band CModel magnitudes and +0.02 to the z band PSF and
Model magnitudes. These corrections are instead applied to the flux densities, via the f_{0}
values in the table below. This has the effect of restoring the original zero-flux density magnitudes, but also
means that the u and z band magnitudes are not strictly on the AB system.
- The correct softening parameters b are now applied, which has the effect of shifting the u, g, i, and z
zero-flux magnitudes back to their correct values.
- The flux density uncertainties are now calculated using the formula above, from the derivative of the
formula for flux density. This gives more accurate uncertainties near zero flux density, and ensures that
these uncertainties are positive.
The
b and
f_{0} parameter values needed to compute SDSS flux densities
f from
SDSS magnitudes
m are given in the table below. The
mzpc values have already been applied
to the
f_{0} values, as described above, and
should not be used to correct the SDSS
magnitudes
m. The asinh magnitude associated with a zero flux density object is given for reference only.
The magnitude corresponding to 10
f_{0}, above which the asinh magnitude and the traditional logarithmic
magnitude differ by less than 1% in flux density, is also given. Note that this is 2.51 magnitudes brighter
than the zero-flux-density magnitude.
asinh Softening Parameters and Zero Points (b and f_{0})
Band | b | f_{0}(Jy) | mzpc | Zero-Flux-Density Magnitude [m(f/f_{0} = 0)] | m(f/f_{0} = 10b) |
u | 1.4 × 10^{-10} | 3767. | -0.04 (CModel magnitudes only) | 24.63 | 22.12 |
g | 0.9 × 10^{-10} | 3631. | 0.00 | 25.11 | 22.60 |
r | 1.2 × 10^{-10} | 3631. | 0.00 | 24.80 | 22.29 |
i | 1.8 × 10^{-10} | 3631. | 0.00 | 24.36 | 21.85 |
z | 7.4 × 10^{-10} | 3565. | 0.02 (PSF and Model magnitudes only) | 22.83 | 20.32 |
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; the flux of the parent is measured 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. For DR2 and later,
all the imaging data (including EDR and DR1) have
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 two-dimensional 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 double-Gaussian fit
to the PSF, which is provided by psp.
Residuals between the double-Gaussian 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 (half-light) 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 chi-squared 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. The SDSS algorithms 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 SDSS 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 better-fitting of the
exponential and de Vaucouleurs models in the
r band).
In order to measure unbiased colors of galaxies, the algorithms measure their
flux through equivalent apertures in all bands. The algorithms 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 signal-to-noise 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 0.05-0.1 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, star-galaxy separation was
based on the difference between model and PSF magnitudes. The algorithms
now do star-galaxy 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
magnitudes for bright objects) have been fixed since the EDR.
Petrosian magnitudes
This is 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, it is necessary 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.
The SDSS defines 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 this
case. The Petrosian flux in any band is then defined as the flux
within a certain number N_{P} (equal to 2.0 in
this case) of r Petrosian radii:
In the SDSS
five-band 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. (All of these
flags associated with the SDSS photometry are displayed on NED's
photometry pages for each SDSS object.)
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
These are stored as psfMag. For isolated stars,
which are well-described 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, SDSS does this by sync-shifting 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, SDSS uses bright stars to determine a further
aperture correction to a radius of 7.4" as a function of seeing, and
applies 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 signal-to-noise ratio wings.
Empirically, this reduces the seeing-dependence 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
under-estimated.
Which Magnitude should I use?
NED has chosen to display the PSF, Model, CModel, and Petrosian
magnitudes for the SDSS objects that have them (not all objects
have all of these magnitudes given in the SDSS data tables).
Here is more information from the SDSS documentation concerning
the choice of magnitude and the associated diameters.
Faced with this array of different magnitude measurements to choose
from, which one is appropriate in which circumstances? There are no
guarantees regarding what is appropriate for the science you
want to do, but here are some examples, using the
general guideline that one usually wants to maximize some combination
of signal-to-noise 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. It is recommended to use 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 the spectroscopic sample have relatively high
signal-to-noise ratio measurements of their Petrosian magnitudes. Since
these magnitudes are model-independent 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, it is recommended that you use
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
best-fit 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.