# 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 `UBVR`_{c}`I`_{c}. 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.

- 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). - 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

## Jordi *et al.* (2005)

*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 & Tunçel (2005)

`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`

_{c}

`I`

_{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

## Bilir, Karaali & Tunçel (2005)

`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 & Hawley (2005)

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.* (2006)

*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

`UBVR`

_{c}

`I`

_{c}photometry. Note that these equations, strictly speaking, transform from

`UBVR`

_{c}

`I`

_{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 & 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

## Lupton (2005)

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);