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,
firstorder 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]*(XX_zp) + k_u*X
g'_inst = g' + a_g + b_g*[(g'r')(g'r')_zp] + c_g*[(g'r')(g'r')_zp]*(XX_zp) + k_g*X
r'_inst = r' + a_r + b_r*[(r'i')(r'i')_zp] + c_r*[(r'i')(r'i')_zp]*(XX_zp) + k_r*X
i'_inst = i' + a_i + b_i*[(r'i')(r'i')_zp] + c_i*[(r'i')(r'i')_zp]*(XX_zp) + k_i*X
z'_inst = z' + a_z + b_z*[(i'z')(i'z')_zp] + c_z*[(i'z')(i'z')_zp]*(XX_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 firstorder 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; darkrunbydarkrun values can be found at
this URL.
(In standard processing of the prefilter change data, the b terms
are linearly interpolated from the darkrunbydarkrun 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 welldetermined 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 nonSDSS 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
