Transformations between SDSS magnitudes and UBVR_cI_c

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

• Jester al. (2005), who derived transformation equations for stars and for z<=2.1 quasars,
• Kiraali, Bilir, and Tuncel (2005), who derived transformation equations for stars,
• Bilir, Karaali, and Tuncel (2005), who derived transformation equations for dwarf stars, and
• West, Walkowicz, and Hawley (2005), who derived transformation equations for M and L dwarf stars,
• Rodgers et al. (2005), who derived transformation equations for main sequence stars.
• Lupton (2005), who derived transformation equations for stars.

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.

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

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<->UBVR_cI_c 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

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*(B-V) - 0.224
    g      =    V + 0.634*(B-V) - 0.108

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

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 UBVR_cI_c photometry. Note that these equations, strictly speaking, transform from UBVR_cI_c 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

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)

    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-R_c=+0.36, and R_c-I_c=+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.