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Transformations between SDSS magnitudes and UBVRcIc

There have been several efforts in calculating transformation equations between ugriz (or u'g'r'i'z') and UBVRcIc. Here, we focus on seven of the most current efforts:

There are currently no transformation equations explicitly for galaxies, but Jester al.'s (2005) and Lupton's (2005) transformation equations for stars should also provide reasonable results for normal galaxies (i.e., galaxies without strong emission lines).

Caveat: Note that these transformation equations are for the SDSS ugriz (u'g'r'i'z') magnitudes as measured, not for SDSS ugriz (u'g'r'i'z') corrected for AB offsets. If you need AB ugriz magnitudes, please remember to convert from SDSS ugriz to AB ugriz using AB offsets described at this URL).

At the end of this webpage, we estimate the ugriz colors of Vega and the Sun.

Jester et al. (2005)

The following transformation equations were extracted from Table 1 of Jester et al. (2005) and are generally useful for stars and for quasars. The transformation equations for z<=2.1 quasars is based upon synthetic photometry of an updated version of the quasar composite spectrum of Vanden Berk et al. (2001) using DR1 data as well as the red and reddened quasar composites for Richards et al. (2003). The transformations for stars were derived from the Smith et al. (2002) u'g'r'i'z' photometry of Landolt stars, suitably transformed from the USNO-1.0m u'g'r'i'z' system to the SDSS 2.5m ugriz system via the u'g'r'i'z'-to-ugriz transformations.

The transformation equations for stars supercede those of Fukugita et al.(1996) and Smith et al. (2002).


UBVRcIc -> ugriz
================

Quasars at z <= 2.1 (synthetic)    
        Transformation                RMS residual
    u-g    =    1.25*(U-B)   + 1.02      0.03
    g-r    =    0.93*(B-V)   - 0.06      0.09
    r-i    =    0.90*(Rc-Ic) - 0.20      0.07
    r-z    =    1.20*(Rc-Ic) - 0.20      0.18
    g      =    V + 0.74*(B-V) - 0.07    0.02
    r      =    V - 0.19*(B-V) - 0.02    0.08


Stars with Rc-Ic < 1.15 and U-B < 0
        Transformation                RMS residual
    u-g    =    1.28*(U-B)   + 1.14      0.05
    g-r    =    1.09*(B-V)   - 0.23      0.04
    r-i    =    0.98*(Rc-Ic) - 0.22      0.01
    r-z    =    1.69*(Rc-Ic) - 0.42      0.03
    g      =    V + 0.64*(B-V) - 0.13    0.01
    r      =    V - 0.46*(B-V) + 0.11    0.03


All stars with Rc-Ic < 1.15
        Transformation                RMS residual
    u-g    =    1.28*(U-B)   + 1.13      0.06
    g-r    =    1.02*(B-V)   - 0.22      0.04
    r-i    =    0.91*(Rc-Ic) - 0.20      0.03
    r-z    =    1.72*(Rc-Ic) - 0.41      0.03
    g      =    V + 0.60*(B-V) - 0.12    0.02
    r      =    V - 0.42*(B-V) + 0.11    0.03



ugriz -> UBVRcIc
================

Quasars at z <= 2.1 (synthetic)
        Transformation                RMS residual
    U-B    =    0.75*(u-g) - 0.81        0.03 
    B-V    =    0.62*(g-r) + 0.15        0.07
    V-R    =    0.38*(r-i) + 0.27        0.09
    Rc-Ic  =    0.72*(r-i) + 0.27        0.06
    B      =    g + 0.17*(u-g) + 0.11    0.03
    V      =    g - 0.52*(g-r) - 0.03    0.05


Stars with Rc-Ic < 1.15 and U-B < 0
        Transformation                RMS residual     
    U-B    =    0.77*(u-g) - 0.88        0.04
    B-V    =    0.90*(g-r) + 0.21        0.03
    V-R    =    0.96*(r-i) + 0.21        0.02
    Rc-Ic  =    1.02*(r-i) + 0.21        0.01
    B      =    g + 0.33*(g-r) + 0.20    0.02
    V      =    g - 0.58*(g-r) - 0.01    0.02


All stars with Rc-Ic < 1.15     
        Transformation                RMS residual
    U-B    =    0.78*(u-g) - 0.88        0.05
    B-V    =    0.98*(g-r) + 0.22        0.04
    V-R    =    1.09*(r-i) + 0.22        0.03
    Rc-Ic  =    1.00*(r-i) + 0.21        0.01
    B      =    g + 0.39*(g-r) + 0.21    0.03
    V      =    g - 0.59*(g-r) - 0.01    0.01

Jordi et al. (2006)

The following transformation equations were extracted from Table 3 of Jordi et al. (2006) and are generally useful for stars. They are derived from comparing Stetson's extension of the Landolt standard stars with the corresponding SDSS DR4 photometry. The equations including the Johnson U band are based on the comparison of Landolt's original standard stars and the SDSS DR4.


UBVRcIc -> ugriz
================

        Transformation                   
    u-g   =     (0.750 ± 0.050)*(U-B)  + (0.770 ± 0.070)*(B-V) + (0.720 ± 0.040)
    g-V   =     (0.630 ± 0.002)*(B-V)  - (0.124 ± 0.002)
    g-B   =     (-0.370 ± 0.002)*(B-V) - (0.124 ± 0.002)
    g-r   =     (1.646 ± 0.008)*(V-R)  - (0.139 ± 0.004)
    g-i   =     (1.481 ± 0.004)*(V-I)  - (0.536 ± 0.004) if V-I <= 1.8
    g-i   =     (0.83 ± 0.01)*(V-I)    + (0.60 ± 0.03)   if V-I >  1.8
    r-i   =     (1.007 ± 0.005)*(R-I)  - (0.236 ± 0.003) 
    r-z   =     (1.584 ± 0.008)*(R-I)  - (0.386 ± 0.005)
    r-R   =     (0.267 ± 0.005)*(V-R)  + (0.088 ± 0.003) if V-R <= 0.93
    r-R   =     (0.77 ± 0.04)*(V-R)    - (0.37 ± 0.04)   if V-R >  0.93 
    i-I   =     (0.247 ± 0.003)*(R-I)  + (0.329 ± 0.002)



ugriz -> UBVRcIc
================

        Transformation                   
    U-B   =     (0.79 ± 0.02)*(u-g)    - (0.93 ± 0.02)
    U-B   =     (0.52 ± 0.06)*(u-g)    + (0.53 ± 0.09)*(g-r) - (0.82 ± 0.04)
    B-g   =     (0.175 ± 0.002)*(u-g)  + (0.150 ± 0.003)
    B-g   =     (0.313 ± 0.003)*(g-r)  + (0.219 ± 0.002)
    V-g   =     (-0.565 ± 0.001)*(g-r) - (0.016 ± 0.001)
    V-I   =     (0.675 ± 0.002)*(g-i)  + (0.364 ± 0.002) if  g-i <= 2.1
    V-I   =     (1.11 ± 0.02)*(g-i)    - (0.52 ± 0.05)   if  g-i >  2.1
    R-r   =     (-0.153 ± 0.003)*(r-i) - (0.117 ± 0.003)
    R-I   =     (0.930 ± 0.005)*(r-i)  + (0.259 ± 0.002)
    I-i   =     (-0.386 ± 0.004)*(i-z) - (0.397 ± 0.001)

The following transformation equations were extracted from Table 4 of Jordi et al. (2006) and are generally useful for Population I and metal-poor Population II stars, respectively. The transformations for the Population II stars are derived from comparing Stetson fields around Draco, NGC 2419 and NGC 7078 with their SDSS DR4 photometry. The transformations for the Population I stars are derived from the Stetson extension of Landolt's equatorial fields compared with the SDSS DR4 photometry. The transformation equation for Population II stars including the SDSS (i-z)-color is not calculated, because of the small number of stars.

    
BVRcIc -> griz
================

        Transformation for Population I stars:                
    g-V   =     (0.634 ± 0.002)*(B-V)  - (0.127 ± 0.002)
    g-B   =     (-0.366 ± 0.002)*(B-V) - (0.126 ± 0.002)
    g-r   =     (1.599 ± 0.009)*(V-R)  - (0.106 ± 0.006)
    g-i   =     (1.474 ± 0.004)*(V-I)  - (0.518 ± 0.005) if V-I <= 1.8
    g-i   =     (0.83 ± 0.01)*(V-I)    + (0.62 ± 0.03)   if V-I >  1.8
    r-i   =     (0.988 ± 0.006)*(R-I)  - (0.221 ± 0.004)
    r-z   =     (1.568 ± 0.009)*(R-I)  - (0.370 ± 0.006)
    r-R   =     (0.275 ± 0.006)*(V-R)  + (0.086 ± 0.004) if V-R <= 0.93 
    r-R   =     (0.71 ± 0.05)*(V-R)    - (0.31 ± 0.05)   if V-R >  0.93
    i-I   =     (0.251 ± 0.003)*(R-I)  + (0.325 ± 0.002) 
          
        Transformation for metal-poor Population II stars:                
    g-V   =     (0.596 ± 0.009)*(B-V)  - (0.148 ± 0.007) 
    g-B   =     (-0.401 ± 0.009)*(B-V) - (0.145 ± 0.006)
    g-r   =     (1.72 ± 0.02)*(V-R)    - (0.198 ± 0.007)
    g-i   =     (1.48 ± 0.01)*(V-I)    - (0.57 ± 0.01)   if V-I <= 1.8
    r-i   =     (1.06 ± 0.02)*(R-I)    - (0.30 ± 0.01)  
    r-z   =     (1.60 ± 0.06)*(R-I)    - (0.46 ± 0.03)  
    r-R   =     (0.34 ± 0.02)*(V-R)    + (0.015 ± 0.008) if V-R <= 0.93 
    i-I   =     (0.21 ± 0.02)*(R-I)    + (0.34 ± 0.01)



griz -> BVRcIc
================

        Transformation for Population I stars:          
    B-g   =     (0.163 ± 0.002)*(u-g)  + (0.170 ± 0.004)
    B-g   =     (0.312 ± 0.003)*(g-r)  + (0.219 ± 0.002)
    V-g   =     (-0.573 ± 0.002)*(g-r) - (0.016 ± 0.002)
    V-I   =     (0.671 ± 0.002)*(g-i)  + (0.359 ± 0.002) if g-i <= 2.1
    V-I   =     (1.12 ± 0.02)*(g-i)    - (0.53 ± 0.06)   if g-i >  2.1
    R-r   =     (-0.257 ± 0.004)*(r-i) + (0.152 ± 0.002)
    R-I   =     (0.977 ± 0.006)*(r-i)  + (0.234 ± 0.003)
    I-i   =     (-0.409 ± 0.006)*(i-z) - (0.394 ± 0.002)
    

        Transformation for metal-poor Population II stars:                
    B-g   =     (0.20 ± 0.01)*(u-g)    + (0.15 ± 0.01)
    B-g   =     (0.349 ± 0.009)*(g-r)  + (0.245 ± 0.006)
    V-g   =     (-0.569 ± 0.007)*(g-r) + (0.021 ± 0.004)
    V-I   =     (0.674 ± 0.005)*(g-i)  + (0.406 ± 0.004) if g-i <= 2.1
    R-r   =     (-0.25 ± 0.02)*(r-i)   - (0.119 ± 0.005)
    R-I   =     (0.80 ± 0.02)*(r-i)    + (0.317 ± 0.004)

             

Karaali, Bilir, and Tuncel (2005)

These transformations appeared in Karaali, Bilir, and Tuncel (2005). They are based on Landolt (1992) UBV data for 224 stars in the color range 0.3 < B-V < 1.1 with SDSS ugr photometry from the CASU INT Wide Field Survey. An improvement over previous SDSS<->UBVRcIc transformations is the use of two colors in each equation, which is particularly helpful for the u-g transformation.


UBVRcIc -> ugriz
================

Stars with  0.3 < B-V < 1.1
    u-g    =    0.779*(U-B) + 0.755*(B-V)  + 0.801
    g-r    =    1.023*(B-V) + 0.016*(U-B)  - 0.187


ugriz -> UBVRcIc
================

Stars with  0.3 < B-V < 1.1
    B-V    =    0.992*(g-r) - 0.0199*(u-g) + 0.202

Bilir, Karaali, and Tuncel (2005)

These transformation equations appeared in Bilir, Karaali, and Tuncel (2005, AN 326, 321). They are based upon 195 dwarf stars that have both ugriz photometry and Landolt UBV photometry.


UBVRcIc -> ugriz
================

Dwarf (Main Sequence) Stars
    g-r    =    1.124*(B-V) - 0.252
    r-i    =    1.040*(R-I) - 0.224
    g      =    V + 0.634*(B-V) - 0.108

West, Walkowicz, and Hawley (2005)

These transformation equations appeared in West, Walkowicz, and Hawley (2005, PASP 117, 706). They are based upon photometry of M and L dwarf stars from SDSS Data Release 3.


UBVRcIc -> ugriz
================

M0-L0 Dwarfs, 0.67 <= r-i <= 2.01
       Transformation                      RMS residual     
    r-i    =    -2.69 + 2.29*(V-Ic)           0.05
                      - 0.28*(V-Ic)**2

M0-L0 Dwarfs, 0.37 <= i-z <= 1.84
        Transformation                     RMS residual
    i-z    =    -20.6 + 26.0*(Ic-Ks)          0.10
                      - 11.7*(Ic-Ks)**2
                      - 2.30*(Ic-Ks)**3
                      - 0.17*(Ic-Ks)**4

Rodgers et al. (2005)

These equations are from Rodgers et al. (2005, AJ, submitted). They are based upon a set of main sequence stars from the Smith et al. (2002) u'g'r'i'z' standard star network that also have Landolt UBVRcIc photometry. Note that these equations, strictly speaking, transform from UBVRcIc to u'g'r'i'z' and not to ugriz. The transformation from u'g'r'i'z' to ugriz, however, is rather small. Note also, as with the Karaali, Bilir, and Tuncel (2005) transformations listed above, two colors are used in the u'-g' and g'-r' equations to improve the fits. The use of two colors in the fits is especially useful for u'-g', which is strongly affected by the Balmer discontinuity.


UBVRcIc -> u'g'r'i'z'
=====================

Main Sequence Stars
    u'-g'  =    1.101(+/-0.004)*(U-B) + 0.358(+/-0.004)*(B-V) + 0.971
    g'-r'  =    0.278(+/-0.016)*(B-V) + 1.321(+/-0.030)*(V-Rc) - 0.219
    r'-i'  =    1.070(+/-0.009)*(Rc-Ic) - 0.228
    r'-z'  =    1.607(+/-0.012)*(Rc-Ic) - 0.371

Lupton (2005)

These equations that Robert Lupton derived by matching DR4 photometry to Peter Stetson's published photometry for stars.


Stars

   B = u - 0.8116*(u - g) + 0.1313;  sigma = 0.0095
   B = g + 0.3130*(g - r) + 0.2271;  sigma = 0.0107

   V = g - 0.2906*(u - g) + 0.0885;  sigma = 0.0129
   V = g - 0.5784*(g - r) - 0.0038;  sigma = 0.0054

   R = r - 0.1837*(g - r) - 0.0971;  sigma = 0.0106
   R = r - 0.2936*(r - i) - 0.1439;  sigma = 0.0072

   I = r - 1.2444*(r - i) - 0.3820;  sigma = 0.0078
   I = i - 0.3780*(i - z)  -0.3974;  sigma = 0.0063

Here is the CAS SQL query Robert used to perform the matchup of DR4 photometry with Stetson's:


  select
     dbo.fSDSS(P.objId) as ID, name,
     S.B, S.Berr, S.V, S.Verr , S.R, S.Rerr, S.I, S.Ierr,
     psfMag_u, psfMagErr_u, psfMag_g, psfMagErr_g,
     psfMag_r, psfMagErr_r, psfMag_i, psfMagErr_i, psfMag_z, psfMagErr_z,
     case when 0 = (flags_u & 0x800d00000000000) and status_u = 0 then 1 else 0 end as good_u,
     case when 0 = (flags_g & 0x800d00000000000) and status_g = 0 then 1 else 0 end as good_g,
     case when 0 = (flags_r & 0x800d00000000000) and status_r = 0 then 1 else 0 end as good_r,
     case when 0 = (flags_i & 0x800d00000000000) and status_i = 0 then 1 else 0 end as good_i,
     case when 0 = (flags_z & 0x800d00000000000) and status_z = 0 then 1 else 0 end as good_z
  from
     stetson as S
  join star as P on S.objId = P.objId
  join field as F on P.fieldId = F.fieldId
  where
     0 = (flags & 0x40006)

Estimates for the ugriz Colors of Vega and the Sun

Assuming V=+0.03 and U-B = B-V = V-Rc = Rc-Ic = 0.00, we find for the A0V star Vega the following:

    g   = -0.08  (+/-0.03)
    u-g = +1.02  (+/-0.08)
    g-r = -0.25  (+/-0.03)
    r-i = -0.23  (+/-0.02)
    i-z = -0.17  (+/-0.02)

where we used the Bilir, Karaali, and Tuncel (2005) transformation for g and the Rodgers et al. (2005) transformations (plus the u'g'r'i'z'-to-ugriz transformations) for the u-g, g-r, r-i, and i-z colors. The error bars in parentheses are rough estimates of the systematic errors based upon the different values that different sets of transformation equations yield.

Assuming M(V)=+4.82, U-B=+0.195, B-V=+0.650, V-Rc=+0.36, and Rc-Ic=+0.32, we find for the Sun the following:

    M(g)= +5.12  (+/-0.02)
    u-g = +1.43  (+/-0.05)
    g-r = +0.44  (+/-0.02)
    r-i = +0.11  (+/-0.02)
    i-z = +0.03  (+/-0.02)

where, again, we used the Bilir, Karaali, and Tuncel (2005) transformation for g and the Rodgers et al. (2005) transformations (plus the u'g'r'i'z'-to-ugriz transformations) for the u-g, g-r, r-i, and i-z colors. As above, the error bars in parentheses are rough estimates of the systematic errors based upon the different values that different sets of transformation equations yield.


This page was last modified on $Date: 2008/10/24 11:20:48 $ (UT).