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 (aka luptitude)
• The PSF Magnitude (psfMag)
• The Fiber Magnitude (fiberMag)
• The Petrosian Magnitude (petroMag)
• The Model Magnitude (deVMag,expMag,modelMag) Important caveat; please read!
• The Reddening Correction (reddening)
• Which Magnitude should I use?
• Object Radial Profiles

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

• 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

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/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.

2. a pure exponential profile
I(r) = I₀exp(-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 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.

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

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 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. 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

The Reddening Correction

Which Magnitude should I use?

• 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.

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.

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

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