Measures of Flux and Magnitude

In SDSS-III/IV, we express all fluxes in terms of nanomaggies, which are a convenient linear unit. For example, quantities labeled petroFlux, psfFlux, etc. are (unless otherwise stated) in these units. In each case, there is a corresponding asinh magnitude, such as petroMag, psfMag etc., explained further below.

A "maggy" is the flux f of the source relative to the standard source f₀ (which defines the zeropoint of the magnitude scale). Therefore, a "nanomaggy" is 10^-9 times a maggy. To relate these quantities to standard magnitudes, an object with flux f given in nMgy has a Pogson magnitude:

m = [22.5 mag] - 2.5 log₁₀ f.

Note that magnitudes listed in the SDSS catalog, however, are not standard Pogson magnitudes, but asinh magnitudes.

The standard source for each SDSS band is close to but not exactly the AB source (3631 Jy), meaning that a nanomaggy is approximately 3.631×10^-6 Jy. However, our current understanding is that the absolute calibration of the SDSS system has some percent-level offsets relative to AB, discussed in detail in the section on AB calibration.

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 report a magnitude 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/ln(10) * [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 in maggies; it is given in the table of asinh softening parameters below (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. For details on converting asinh magnitudes to other flux measures, see converting counts to magnitudes.

Asinh Softening Parameters

Filter	b	Zero-flux Magnitude [m(f/f0 = 0)]	m(f/f0 = 10b)
u	1.4×10-10	24.63	22.12
g	0.9×10-10	25.11	22.60
r	1.2×10-10	24.80	22.29
i	1.8×10-10	24.36	21.85
z	7.4×10-10	22.83	20.32

Fiber magnitudes reflect the flux contained within the aperture of an spectroscopic fiber in each band. In the case of fiberMag we assume an aperture appropriate to the SDSS spectrograph (3′′ in diameter). In the case of fiber2Mag we assume an aperture appropriate to the BOSS spectrograph (2′′ in diameter).

• 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.

There are several "model" magnitudes associated with each catalog object. First, there are devMag and expMag magnitudes, associated with de Vaucouleurs and exponential model fits, as explained more fully below. These magnitudes are calculated from independent models in each band. Second, there is modelMag, which uses the better of the two fits in the r-band as a matched aperture to calculate the flux in all bands. For extended objects, modelMag usually provides the best available SDSS colors.

Just as the PSF magnitudes are optimal measures of the fluxes of stars, the optimal measure of the flux of a galaxy uses 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 de Vaucouleurs profile: I(r) = I₀ 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).
2. A pure exponential profile: I(r) = I₀ 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:

• rdeV and rExp, the effective radii of the models;
• abDeV and abExp, the axis ratio of the best fit models;
• phiDeV and phiExp, the position angles of the ellipticity (in degrees East of North).
• deVlnL and explnL, 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 roughly 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 de Vaucouleurs) 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 obviously aperture-dependent. For faint galaxies, the model colors have appreciably higher signal-to-noise ratio than do the Petrosian colors.

For the current version of the photometric pipeline, the model magnitude is 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.

Due to the way in which model fits are carried out, there is some weak discretization of model parameters, including rExp and rDeV but especially phiDeV and phiDeV.

The code 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. From this we calculate and store a composite flux as follows:

Fcomposite = fracDeV FdeV + (1 - fracDeV) Fexp,

where FdeV and Fexp are the de Vaucouleurs and exponential fluxes (not magnitudes) of the object in question. The magnitude derived from Fcomposite 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).

There is 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 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.

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. We do star-galaxy separation using the difference between cmodel and PSF magnitudes, with the threshold at the same value (0.145 magnitudes).

Please note that while cModelMag is a good total flux indicator, it is based on the independent model fits (devMag and expMag) in each bandpass. For that reason, it generally does not yield as a high signal-to-noise color as the aperture-matched modelMag values do. For this reason, photometric redshift algorithms and other color-sensitive procedures are usually better off relying on modelMag.

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 consistently distinct 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. (2001) and 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 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. (2001), 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 de Vaucouleurs 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.

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.

Galactic extinction corrections in magnitudes at the position of each object, extinction, are computed following Schlegel, Finkbeiner & Davis (1998). These corrections are not applied to the magnitudes in the databases unless otherwise noted; if desired, they must be explicitly applied by the user.

For DR13 and beyond, the conversions we use from E(B-V) to total extinction A_λ, are tabulated in Table 6 of Schlafly and Finkbeiner (2011). They assume a 7000 K source and R_V = 3.1. Previous data releases used a different conversion (tabulated in Table 22 of the EDR Paper).

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.

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 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, we continue to recommend using the model (not the cmodel) magnitudes. 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, without fully modeling and accounting for the resulting aperture corrections.

Finally, we note that azimuthally-averaged radial profiles are also provided, as described below in the Radial Profiles section, 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.

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 nanomaggies per square arcsec, as described above. The number of annuli for which there is a measurable signal is listed as nprof (in CAS, in the photoObjAll table). The mean surface brightness is listed as profMean, and the error is listed as profErr (in CAS, both of these values are stored separately in the photoProfile table). The 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