Transformations between SDSS magnitudes and other systems

Introduction

There have been several efforts to calculate transformations between ugriz (or u′g′r′i′z′) and UBVRcIc. Here, we summarize seven such efforts. We note that any such transformation relies on knowledge of the absolute calibration, and is inherently uncertain.

Applicable Objects

Reference
Applicable Objects
Jester et al. (2005)
Stars and redshift ≤ 2.1 quasars
Jordi et al. (2006)
Stars, including for Population I and metal-poor Population II stars
Karaali, Bilir & Tunçel (2005)
Stars
Bilir, Karaali & Tunçel (2005)
Dwarf Stars
West, Walkowicz & Hawley (2005)
M and L dwarf stars
Rodgers et al. (2006)
Main Sequence Stars
Lupton (2005)
Stars

Caveats

There are two important caveats to note.

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

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 redshift ≤ 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 supersede 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
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Jordi et al. (2005)

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)
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Karaali, Bilir & Tunçel (2005)

These transformations appeared in Karaali, Bilir & Tunçel (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
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Bilir, Karaali & Tunçel (2005)

These transformation equations appeared in Bilir, Karaali & Tunçel (2005). 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
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West, Walkowicz & Hawley (2005)

These transformation equations appeared in West, Walkowicz & Hawley (2005). 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
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Rodgers et al. (2006)

These equations are from Rodgers et al. (2006). 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 & Tunçel (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'  =    (1.000 ± 0.006)*(Rc-Ic) - 0.212
    r'-z'  =    (1.567 ± 0.020)*(Rc-Ic) - 0.365
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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);
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