SDSS Photometric Equations

The version of the photometric equations used since DR1 for calibrating Secondary Patches observed by the SDSS Photometric Telescope (or PT) come in two parts:

equations for calibrating PT instrumental magnitudes to u'g'r'i'z' magnitudes using the u'g'r'i'z' primary standard star network, and
equations for transforming u'g'r'i'z' magnitudes to magnitudes in the ugriz SDSS 2.5m "natural" system.

We describe each set of equations in turn below. Further details of these two sets of equations can be found in the Photometry White Paper by Gunn, Hogg, Finkbeiner, and Schlegel at this URL

Calibrating instrumental magnitudes to the u'g'r'i'z' system

Each night's observations of the u'g'r'i'z' primary standard stars are fit to the following equations to determine values of the photometric zeropoints, first-order extinctions, and (optionally) the instrumental and atmospheric color terms:

  u'_inst = u' + a_u + b_u*[(u'-g')-(u'-g')_zp] + c_u*[(u'-g')-(u'-g')_zp]*(X-X_zp) + k_u*X
  g'_inst = g' + a_g + b_g*[(g'-r')-(g'-r')_zp] + c_g*[(g'-r')-(g'-r')_zp]*(X-X_zp) + k_g*X
  r'_inst = r' + a_r + b_r*[(r'-i')-(r'-i')_zp] + c_r*[(r'-i')-(r'-i')_zp]*(X-X_zp) + k_r*X
  i'_inst = i' + a_i + b_i*[(r'-i')-(r'-i')_zp] + c_i*[(r'-i')-(r'-i')_zp]*(X-X_zp) + k_i*X
  z'_inst = z' + a_z + b_z*[(i'-z')-(i'-z')_zp] + c_z*[(i'-z')-(i'-z')_zp]*(X-X_zp) + k_z*X

Furthermore, these equations are then applied to the secondary patch stars to convert their instrumental magnitudes to calibrated u'g'r'i'z' magnitudes. This step is performed iteratively until the calibrated u'g'r'i'z' magnitudes have converged. The variables are defined as follows:

  u'_inst     |  These are the instrumental magnitudes --
  g'_inst     |     -2.5log(counts/sec)
  r'_inst     |  -- in each of the u'g'r'i'z' filters.
  i'_inst     |
  z'_inst     |

  u'          |
  g'          |  The standard magnitudes for the standard star
  r'          |  in each of the u'g'r'i'z' filters.
  i'          |
  z'          |

  a_u         |
  a_g         |  These are the photometric zeropoints  
  a_r         |  in each of the u'g'r'i'z' filters.
  a_i         |
  a_z         |

  b_u         |
  b_g         |  These are the instrumental color term coefficients
  b_r         |  for each of the u'g'r'i'z' filters.
  b_i         |  
  b_z         |  

  c_u         |
  c_g         |  These are the atmospheric colors term coeffiecients)
  c_r         |  (aka, second order extinctions)
  c_i         |  for each of the u'g'r'i'z' filters.
  c_z         |

  (u'-g')     |  
  (g'-r')     |  The standard colors of the standard star.
  (r'-i')     |  
  (i'-z')     |  

  (u'-g')_zp  |  
  (g'-r')_zp  |  The calibrated zeropoint -- or "crossing" -- colors 
  (r'-i')_zp  |  for the instrumental and atmospheric color terms (see below).
  (i'-z')_zp  |  

  X           |  The airmass of the observation.

  X_zp        |  The zeropoint airmass for the atmospheric color term
              |  (see below).

  k_u         |
  k_g         |  The first-order extinction coefficients 
  k_r         |  for each of the u'g'r'i'z' filters.
  k_i         |
  k_z         |

The zeropoint -- or "crossing" -- colors are DEFINED to have the following values:

  (u'-g')_zp = 1.39     |  the values were defined
  (g'-r')_zp = 0.53     |  in a manner so that they
  (r'-i')_zp = 0.21     |  would be close to the mean
  (i'-z')_zp = 0.09     |  colors of the u'g'r'i'z' 
                        |  standard stars under the 
                        |  color cuts described below

The zeropoint airmass is DEFINED to be

  X_zp = 1.30           |  this is roughly the expected 
                        |  mean airmass of the SDSS

The b term coefficients for the new PT filter set (installed August 2001) are:

  b_u =  0.001
  b_g = -0.041
  b_r =  0.009
  b_i =  0.010
  b_z =  0.002

These b terms coefficents showed seasonal variability for the previous PT filter set; darkrun-by-darkrun values can be found at this URL. (In standard processing of the pre-filter change data, the b terms are linearly interpolated from the darkrun-by-darkrun values in this table to the night of the PT observation.) The c term coefficients are typically quite small (|c| < 0.02); for the purposes of SDSS calibration, they have been set to zero exactly for all five filters:

  c_u =  0.000
  c_g =  0.000
  c_r =  0.000
  c_i =  0.000
  c_z =  0.000

For SDSS calibration, the following color cuts are also imposed:

   0.70 <= u'-g' <= 2.70 &&
   0.15 <= g'-r' <= 1.20 && 
  -0.10 <= r'-i' <= 0.60 && 
  -0.20 <= i'-z' <= 0.40

The redward cuts are imposed because the u'g'r'i'z' standard star system is not well-determined for stars redder than about M0. The blueward cuts are imposed in order to avoid stars with large Balmer discontinuities and very cool stars which have blue colors due to molecular absorption. In general non-SDSS practice, it is generally safe to calibrate blueward of the blue limits; matters fall apart, however, much more quickly beyond the redward cuts.

Converting from u'g'r'i'z' magnitudes to SDSS 2.5m ugriz magnitudes

To convert from u'g'r'i'z' to 2.5m ugriz:

  u(2.5m) = u' - b25(u)*( (u'-g')-(u'-g')_zp ) + zpOffset25(u)
  g(2.5m) = g' - b25(g)*( (g'-r')-(g'-r')_zp ) + zpOffset25(g)
  r(2.5m) = r' - b25(r)*( (r'-i')-(r'-i')_zp ) + zpOffset25(r)
  i(2.5m) = i' - b25(i)*( (r'-i')-(r'-i')_zp ) + zpOffset25(i)
  z(2.5m) = z' - b25(z)*( (i'-z')-(i'-z')_zp ) + zpOffset25(z)

where

   b25(u) =  0.000
   b25(g) = -0.060
   b25(r) = -0.035
   b25(i) = -0.041
   b25(z) =  0.030

  (u'-g')_zp = 1.39  |
  (g'-r')_zp = 0.53  | these have the same values
  (r'-i')_zp = 0.21  |      as in the previous eqn.
  (i'-z')_zp = 0.09  |

Fortunately, all the zpOffset25's are

  zpOffset25(u) = 0.000
  zpOffset25(g) = 0.000
  zpOffset25(r) = 0.000
  zpOffset25(i) = 0.000
  zpOffset25(z) = 0.000

Converting from SDSS 2.5m ugriz magnitudes to AB magnitudes

See the separate section on conversion from SDSS to AB magnitudes.

Last modified: Wed Jun 15 20:34:01 CDT 2005