Photometric Flux Calibration

• Introduction
• Overview of the Photometric Calibration in SDSS
• Assessment of Photometric Calibration
• Converting counts to magnitudes
• Converting SDSS magnitudes to AB magnitudes

Introduction

The objective of the photometric calibration process is to tie the SDSS imaging data to an AB magnitude system, and specifically to the "natural system" of the 2.5m telescope defined by the photon-weighted effective wavelengths of each combination of SDSS filter, CCD response, telescope transmission, and atmospheric transmission at a reference airmass of 1.3 as measured at APO (see transmission curves for SDSS 2.5m telescope).

The calibration process ultimately involves combining data from three telescopes: the USNO 40-in on which our primary standards were first measured, the SDSS Photometric Telescope (or PT) , and the SDSS 2.5m telescope. At the beginning of the survey it was expected that there would be a single u'g'r'i'z' system. However, in the course of processing the SDSS data, the unpleasant discovery was made that the filters in the 2.5m telescope have significantly different effective wavelengths from the filters in the PT and at the USNO. These differences have been traced to the fact that the short-pass interference films on the 2.5-meter camera live in the same vacuum as the detectors, and the resulting dehydration of the films decreases their effective refractive index. This results in blueward shifts of the red edges of the filters by about 2.5 percent of the cutoff wavelength, and consequent shifts of the effective wavelengths of order half that. The USNO filters are in ambient air, and the hydration of the films exhibits small temperature shifts; the PT filters are kept in stable very dry air and are in a condition about halfway between ambient and the very stable vacuum state. The rather subtle differences between these systems are describable by simple linear transformations with small color terms for stars of not-too-extreme color, but of course cannot be so transformed for very cool objects or objects with complex spectra. Since standardization is done with stars, this is not a fundamental problem, once the transformations are well understood.

It is these subtle issues that gave rise to our somewhat awkward nomenclature for the different magnitude systems:

• magnitudes in the the USNO 40-in system are primed (u'g'r'i'z')
• magnitudes in the SDSS 2.5m system are unprimed (ugriz)
• magnitudes in the PT system only exist internally within the Monitor Telescope Pipeline (mtpipe) and have no official designation.

Previous reductions of the data, including that used in the EDR, were based on inconsistent photometric equations; this is why we referred to the 2.5m photometry with asterisks: u*g*r*i*z*. With the DR1, the photometric equations are properly self-consistent, and we can now remove the stars, and refer to u g r i z photometry with the 2.5m.

Overview of the Photometric Calibration in SDSS

The photometric calibration of the SDSS imaging data is a multi-step process, due to the fact that the images from the 2.5m telescope saturate at approximately r = 14, fainter than typical spectrophotometric standards, combined with the fact that observing efficiency would be greatly impacted if the 2.5m needed to interrupt its routine scanning in order to observe separate calibration fields.

The first step involved setting up a primary standard star network of 158 stars distributed around the Northern sky. These stars were selected from a variety of sources and span a range in color, airmass, and right ascension. They were observed repeatedly over a period of two years using the US Naval Observatory 40-in telescope located in Flagstaff, Arizona. These observations are tied to an absolute flux system by the single F0 subdwarf star BD+17_4708, whose absolute fluxes in SDSS filters are taken from Fukugita et al. 1996 As noted above, the photometric system defined by these stars is called the u'g'r'i'z' system. You can look at the table containing the calibrated magnitudes for these standard stars.

Most of these primary standards have brightnesses in the range r = 8 - 13, and would saturate the 2.5-meter telescope's imaging camera in normal operations. Therefore, a set of 1520 41.5x41.5 arcmin² transfer fields, called secondary patches, have been positioned throughout the survey area. These secondary patches are observed with the PT; their size is set by the field of view of the PT camera. These secondary patches are grouped into sets of four. Each set spans the full set of 12 scan lines of a survey stripe along the width of the stripe, and the sets are spaced along the length of a stripe at roughly 15 degree intervals. The patches are observed by the PT in parallel with observations of the primary standards and processed using the Monitor Telescope Pipeline (mtpipe). The patches are first calibrated to the USNO 40-in u'g'r'i'z' system and then transformed to the 2.5m ugriz system; both initial calibration to the u'g'r'i'z' system and the transformation to the ugriz system occur within mtpipe. The ugriz-calibrated patches are then used to calibrate the 2.5-meter's imaging data via the Final Calibrations Pipeline (nfcalib).

Monitor Telescope Pipeline

The PT has two main functions: it measures the atmospheric extinction on each clear night based on observations of primary standards at a variety of airmasses, and it calibrates secondary patches in order to determine the photometric zeropoint of the 2.5m imaging scans. The extinction must be measured on each night the 2.5m is scanning, but the corresponding secondary patches can be observed on any photometric night, and need not be coincident with the image scans that they will calibrate.

The Monitor Telescope Pipeline (mtpipe), so called for historical reasons, processes the PT data. It performs three basic functions:

1. it bias subtracts and flatfields the images, and performs aperture photometry;
2. it identifies primary standards in the primary standard star fields and computes a transformation from the aperture photometry to the primary standard star u'g'r'i'z' system;
3. it applies the photometric solution to the stars in the secondary patch fields, yielding u'g'r'i'z'-calibrated patch star magnitudes, and then transforms these u'g'r'i'z' magnitudes into the SDSS 2.5m ugriz system.

There is a document describing photometric equations used in items 2 and 3 in detail .

The Final Calibration Pipeline

The final calibration pipeline (nfcalib) works much like mtpipe, computing the transformation between psf photometry (or other photometry) as observed by the 2.5m telescope and the final SDSS photometric system. The pipeline matches stars between a camera column of 2.5m data and an overlapping secondary patch. Each camera column of 2.5m data is calibrated individually. There are of order 100 stars in each patch in the appropriate color and magnitude range in the overlap.

The transformation equations are a simplified form of those used by mtpipe. Since mtpipe delivers patch stars already calibrated to the 2.5m ugriz system, the nfcalib transformation equations have the following form:
m_{filter_inst}(2.5m) = m_filter(patch) + a_filter + k_filterX,
where, for a given filter, m_{filter_inst}(2.5m) is the instrumental magnitude of the star in the 2.5m data [-2.5 log10(counts/exptime)], m_filter(patch) is the magnitude of the same star in the PT secondary patch, a_filter is the photometric zeropoint, k_filter is the first-order extinction coefficient, and X is the airmass of the 2.5m observation. The extinction coefficient is taken from PT observations on the same night, linearly interpolated in time when multiple extinction determinations are available. (Generally, however, mtpipe calculates only a single k_filter per filter per night, so linear interpolation is usually unnecessary.) A single zeropoint a_filter is computed for each filter from stars on all patches that overlap a given CCD in a given run. Observations are weighted by their estimated errors, and sigma-clipping is used to reject outliers. At one time it was thought that a time dependent zero point might be needed to account for the fact that the 2.5m camera and corrector lenses rotate relative to the telescope mirrors and optical structure; however, it now appears that any variations in throughput are small compared to inherent fluctuations in the calibration of the patches themselves. The statistical error in the zeropoint is usually constrained to be less than 1.35 percent in u and z and 0.9 percent in gri.

Assessment of Photometric Calibration

With Data Release 1 (DR1), we now routinely meet our requirements of photometric uniformity of 2% in r, g-r, and r-i and of 3% in u-g and i-z (rms).

This is a substantial improvement over the photometric uniformity achieved in the Early Data Release (EDR), where the corresponding values were approximately 5% in r, g-r, and r-i and 5% in u-g and i-z.

The improvements between the photometric calibration of the EDR and the DR1 can be traced primarily to the use of more robust and consistent photometric equations by mtpipe and nfcalib and to improvements to the PSF-fitting algorithm and flatfield methodology in the Photometric Pipeline (photo).

Note that this photometric uniformity is measured based upon relatively bright stars which are no redder than M0; hence, these measures do not include effects of the u band red leak (see caveats below) or the model magnitude bug.

How to go from Counts in the fpC file to Calibrated ugriz magnitudes?

First, note that all SDSS images ("corrected frames", fpC*.fit files) have a "soft bias" of 1000 data numbers (DN) added so they can be stored as unsigned integer. Secondly, since DR1, the sky has not been subtracted from the corrected frames, but is stored in the header keyword sky in units of DN. The tsObj*.fit and related tables and the CAS database store sky values as surface brightness (maggies/sq. arcsec., where one maggie corresponds to 0 magnitudes/sq. arcsec).

Asinh and Pogson magnitudes

All calibrated magnitudes in the photometric catalogs are given not as conventional Pogson astronomical magnitudes, but as asinh magnitudes. We show how to obtain both kinds of magnitudes from observed count rates from the SDSS images and vice versa.

See further down for conversion of SDSS magnitudes to physical fluxes. For both kinds of magnitudes, there are two ways to obtain the zeropoint information for the conversion.

1. A little slower, but gives the final calibration and works for all data releases

   Here you first need the following information from the tsField files:

                  aa = zeropoint
                  kk = extinction coefficient
           and    airmass
   

   To get a calibrated magnitude, you first need to determine the extinction-corrected ratio of the observed count rate to the zero-point count rate:

   • Convert the observed number of counts to a count rate using the exposure time exptime = 53.907456 sec,
   • correct counts for atmospheric extinction using the extinction coefficient kk and the airmass, and
   • divide by the zero-point count rate, which is given by f0 = 10-0.4*aa counts/second both for asinh and conventional magnitudes.

   In a single step,

   f/f0 = counts/exptime * 100.4*(aa + kk * airmass)

   Then, calculate either the conventional ("Pogson") or the SDSS asinh magnitude from f/f0:

   Pogson

   mag = -2.5 * log10(f/f0)
error(mag) = 2.5 / ln(10) * error(counts) / counts
   To get the error on the counts, see the note on computing count errors below.

   asinh

   mag = -(2.5/ln(10))*[asinh((f/f0)/2b)+ln(b)]
error(mag) = 2.5 / ln(10) * error(counts)/exptime * 1/2b *
        100.4*(aa + kk * airmass) / sqrt(1 + [(f/f0)/2b]2),
   where b is the softening parameter for the photometric band in question and is given in the table of b coefficients below (for details on the asinh magnitudes, see the paper by Lupton, Gunn, and Szalay 1999 [AJ 118, 1406]).

   asinh Softening Parameters (b coefficients)

   Band	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

   Note: These values of the softening parameter b are set to be approximately 1-sigma of the sky noise; thus, only low signal-to-noise ratio measurements are affected by the difference between asinh and Pogson magnitudes. The final column gives the asinh magnitude associated with an object for which f/f₀ = 10b; the difference between Pogson and asinh magnitudes is less than 1% for objects brighter than this.

   The calibrated asinh magnitudes are given in the tsObj files. To obtain counts from an asinh magnitude, you first need to work out f/f0 by inverting the asinh relation above. You can then determine the number of counts from f/f0 using the zero-point, extinction coefficient, airmass, and exposure time.

   The equations above are exact for DR1 and later releases. Strictly speaking, for EDR photometry, the corrected counts should include a color term cc*(color-color0)*(X-X0) (cf. equation 15 in section 4.5 in the EDR paper), but it turns out that generally, cc*(color-color0)*(X-X0) < 0.01 mag and the color term can be neglected. Hence the calibration looks identical for EDR and DR1.

2. Faster magnitudes via "flux20"

   The "flux20" keyword in the header of the corrected frames (fpC files) approximately gives the net number of counts for a 20th mag object. So instead of using the zeropoint and airmass correction term from the tsField file, you can determine the corrected zero-point flux as

   f/f0 = counts/(108 * flux20)

   Then proceed with the calculation of a magnitude from f/f0 as above.

   The relation is only approximate because the final calibration information (provided by nfcalib) is not available at the time the corrected frames are generated. We expect the error here (compared to the final calibrated magnitude) to be of order 0.1 mag or so, as estimated from a couple of test cases we have tried out.

   Note the counts measured by photo for each object are given in the fpObjc files, as e.g., "psfcounts", "petrocounts", etc.

On a related note, in DR1 one can also use relations similar to the above to estimate the sky level in magnitudes per sq. arcsec (1 pixel = 0.396 arcsec). Either use the header keyword "sky" in the fpC files, or remember to first subtract "softbias" (= 1000) from the raw background counts in the fpC files. Note the sky level is also given in the tsField files. This note only applies to the DR1 and later data releases. Note also that the calibrated sky brightnesses reported in the tsField values have been corrected for atmospheric extinction.

Computing errors on counts (converting counts to photo-electrons)

The fpC (corrected frames) and fpObjc (object tables with counts for each object instead of magnitudes) files report counts (or "data numbers", DN). However, it is the number of photo-electrons which is really counted by the CCD detectors and which therefore obeys Poisson statistics. The number of photo-electrons is related to the number of counts through the gain (which is really an inverse gain):

              photo-electrons = counts * gain

Thus, the error in DN is given by the following expression:

              error(counts) = sqrt([counts+sky]/gain + Npix*(dark_variance+skyErr)),

Conversion from SDSS ugriz magnitudes to AB ugriz magnitudes

The SDSS photometry is intended to be on the AB system (Oke & Gunn 1983), by which a magnitude 0 object should have the same counts as a source of F_nu = 3631 Jy. However, this is known not to be exactly true, such that the photometric zeropoints are slightly off the AB standard. We continue to work to pin down these shifts. Our present estimate, based on comparison to the STIS standards of Bohlin, Dickinson, & Calzetti~(2001) and confirmed by SDSS photometry and spectroscopy of fainter hot white dwarfs, is that the u band zeropoint is in error by 0.04 mag, u_AB = u_SDSS - 0.04 mag, and that g, r, and i are close to AB. These statements are certainly not precise to better than 0.01 mag; in addition, they depend critically on the system response of the SDSS 2.5-meter, which was measured by Doi et al. (2004, in preparation). The z band zeropoint is not as certain at this time, but there is mild evidence that it may be shifted by about 0.02 mag in the sense z_AB = z_SDSS + 0.02 mag. The large shift in the u band was expected because the adopted magnitude of the SDSS standard BD+17 in Fukugita et al.(1996) was computed at zero airmass, thereby making the assumed u response bluer than that of the USNO system response.

We intend to give a fuller report on the SDSS zeropoints, with uncertainties, in the near future. Note that our relative photometry is quite a bit better than these numbers would imply; repeat observations show that our calibrations are better than 2%.

Conversion from SDSS ugriz magnitudes to physical fluxes

As explained in the preceding section, the SDSS system is nearly an AB system. Assuming you know the correction from SDSS zeropoints to AB zeropoints (see above), you can turn the AB magnitudes into a flux density using the AB zeropoint flux density. The AB system is defined such that every filter has a zero-point flux density of 3631 Jy (1 Jy = 1 Jansky = 10^-26 W Hz^-1 m^-2 = 10^-23 erg s^-1 Hz^-1 cm^-2).

To obtain a flux density from SDSS data, you need to work out f/f0 (e.g. from the asinh magnitudes in the tsObj files by using the inverse of the relations given above). This number is then the also the object's flux density, expressed as fraction of the AB zeropoint flux density. Therefore, the conversion to flux density is

S = 3631 Jy * f/f0.

Then you need to apply the correction for the zeropoint offset between the SDSS system and the AB system. See the description of SDSS to AB conversion above.

Caveats

The u filter has a natural red leak around 7100 Å which is supposed to be blocked by an interference coating. However, under the vacuum in the camera, the wavelength cutoff of the interference coating has shifted redward (see the discussion in the EDR paper), allowing some of this red leak through. The extent of this contamination is different for each camera column. It is not completely clear if the effect is deterministic; there is some evidence that it is variable from one run to another with very similar conditions in a given camera column. Roughly speaking, however, this is a 0.02 magnitude effect in the u magnitudes for mid-K stars (and galaxies of similar color), increasing to 0.06 magnitude for M0 stars (r-i ~ 0.5), 0.2 magnitude at r-i ~ 1.2, and 0.3 magnitude at r-i = 1.5. There is a large dispersion in the red leak for the redder stars, caused by three effects:

• The differences in the detailed red leak response from column to column, beating with the complex red spectra of these objects.
• The almost certain time variability of the red leak.
• The red-leak images on the u chips are out of focus and are not centered at the same place as the u image because of lateral color in the optics and differential refraction - this means that the fraction of the red-leak flux recovered by the PSF fitting depends on the amount of centroid displacement.

To make matters even more complicated, this is a detector effect. This means that it is not the real i and z which drive the excess, but the instrumental colors (i.e., including the effects of atmospheric extinction), so the leak is worse at high airmass, when the true ultraviolet flux is heavily absorbed but the infrared flux is relatively unaffected. Given these complications, we cannot recommend a specific correction to the u-band magnitudes of red stars, and warn the user of these data about over-interpreting results on colors involving the u band for stars later than K.