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 flat-fields and the photometric flux calibration.
The asinh magnitude
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
signal-to-noise ratio, but to behave reasonably at low signal-to-noise
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 1-sigma 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/ln10)*[asinh((f/f0)/2b)+ln(b)].
Here, f0 is given by the classical zero
point of the magnitude scale, i.e., f0 is the flux
of an object with conventional magnitude of zero. The
quantity b is measured relative to f0,
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 10f0, above which the
asinh magnitude and the traditional logarithmic magnitude differ by
less than 1% in flux.
Fiber magnitudes
The flux contained within the aperture of a spectroscopic fiber
(3" in diameter) is calculated in each band and stored in
fiberMag.
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. This was not true in the EDR.
- Images are now convolved to 2" seeing before fiberMags are measured. This also makes the fiber magnitudes closer to what is seen by the spectrograph. This was not true in the EDR.
Model magnitudes
Important Note:Comparing the model (i.e., exponential and de Vaucouleurs fits) and Petrosian magnitudes of bright galaxies shows a systematic offset of about 0.2 magnitudes (in the sense that the model magnitudes are brighter). This turns out to be due to a bug in the way the PSF was convolved with the models (this bug affected the model magnitudes even when they were fit only to the central 4.4" radius of each object). This caused problems for very small objects (i.e., close to being unresolved). The code forces model and PSF magnitudes of unresolved objects to be the same in the mean by application of an aperture correction, which then gets applied to all objects. The net result is that the model magnitudes are fine for unresolved objects, but systematically offset for galaxies brighter than at least 20th mag. Therefore, model magnitudes should NOT be used. See the section on which magnitudes to use for workarounds.
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:
1. a pure deVaucouleurs profile:
I(r) = I0exp{-7.67[(r/re)1/4]}
(truncated beyond 7re to smoothly go to zero at 8re, and with some softening within r=re/50.
2. a pure exponential profile
I(r) = I0exp(-1.68r/re)
(truncated beyond 3re to smoothly go to zero at 4re.
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 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. We apply
aperture corrections to make these model magnitudes equal the PSF
magnitudes in the case of an unresolved object.
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 signal-to-noise ratio than do
the Petrosian colors.
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.
Caveat: At bright magnitudes (r <~ 18), model magnitudes
may not be a robust means to select objects by flux.
For example, model magnitudes in target
and best
imaging may often differ significantly because
a different type of profile (deVaucouleurs or exponential) was
deemed the better fit in target vs. best.
Instead, to select samples by flux, one should typically use
Petrosian magnitudes
for galaxies and psf magnitudes
for stars and distant quasars. However, model colors
are in general robust and may be used to select galaxy samples by color.
Please also refer to the SDSS
target selection algorithms for examples.
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'' RP 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 rP is defined as the radius
at which
RP(rP) equals some specified value
RP,lim, set to 0.2 in our case. The
Petrosian flux in any band is then defined as the flux within a
certain number NP (equal to 2.0 in our 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
FP in different bands is measured through a
consistent aperture.
The aperture 2rP is large enough to contain nearly all of
the flux for typical galaxy profiles, but small enough that the sky noise in
FP is small. Thus, even substantial errors in
rP 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 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, we do 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, 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 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.
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(B-V) to total extinction Alambda, 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 signal-to-noise ratio,
fraction of the total flux included, and freedom from systematic
variations with observing conditions and distance. We note that due to the problem with model magnitudes, the following caveats should be observed:
- Colors derived from model magnitudes are almost completely
insensitive to the bug. Model magnitudes remain the optimal
quantities to use for the colors of extended objects, especially at
the faint end.
- For photometry of unresolved sources, one should use PSF
magnitudes.
- For photometry of resolved sources, one should use Petrosian
magnitudes, especially at the bright end.
- The scale sizes derived from the model fits are systematically
wrong. For exponential fits, the effective radii are systematically
0.15" too large in the present code (almost independent of r_e
itself), while for the de Vaucouleurs fits, they are roughly 25% too
large (almost independent of r_e itself, for r_e > 2 arcsec). These
correction factors depend on seeing to some level, of course.
Again, it is worth emphasizing that the model magnitudes in DR1 are
appreciably better than those used in the EDR. We are reprocessing
all the DR1 data with a version of the photometric pipeline that
addresses this remaining bug, and will release the results to the
public through this website as soon as they are ready.
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
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 Distant Galaxies: For the faintest galaxies,
estimates of the Petrosian magnitudes become very noisy. Under these
conditions, the
modelMag is usually
a more reliable estimate of the
galaxy flux. In addition, these magnitudes account for the effects of local
seeing and thus are less dependent on local seeing variations. This
property also is desirable for the faintest, smallest objects. The
model colors for galaxies are also unbiased, as mentioned above.
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 azimuthally-averaged 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 signal-to-noise ratio measurements at faint
magnitudes.
Radial Profiles
The frames pipeline extracts an azimuthally-averaged 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
small-scale ``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: Tue May 27 14:49:01 CDT 2003
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