Magnitude systems ( tips for flux, magnitude, error and how to convert)

Hello, everybody! 

If you still get trouble to deal with the flux, magnitude, error of the catalog you get (Mostly, the catalog are created by SExtractor).  This file must be helpful for you. Just give you all the equations about how the photometric system works. 

If You still have question about these, please leave me a message, I will be quite pleasant to help you. You will be the nest astronomer! 

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Magnitude systems

STMAG

ZEROPT_second = -2.5 * log₁₀(PHOTFLAM) + PHOTZPT
ZEROPT_exptime = -2.5 * log₁₀(PHOTFLAM) + PHOTZPT + 2.5 * log₁₀(exptime)

PHOTZPT = -21.1 (always)
m_stmag = -2.5 * log₁₀(counts/exptime) + ZEROPT_exptime
m_stmag = -2.5 * log₁₀(counts/sec) + ZEROPT_seconds
m_stmag = -2.5 * log₁₀( PHOTFLAM * DN / EXPTIME) + PHOTZPT

AB mag

m_AB = -2.5 log₁₀( Flux_nu ) - 48.594
where flux_nu is in ergs cm^-2 s^-1 HZ^-1
m_AB = -2.5 log₁₀( Flux_lambda ) - 2.402 - 5.0 * log₁₀( lambda )
where flux_lambda is in ergs cm^-2 s_-1 A_-1
and lambda in A
F_nu = f_lambda * lambda² / c

flam = 3x10^18 * fnu / lam^2
fnu = (lam^2 * flam) / 3x10^18

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Converting an image to flux

flux = DN * PHOTFLAM / exposure_time
where flux is in erg cm^-2s^-1A^-1

flux = DN * PHOTFLAM / exposure_time * PHOTPLAM**2 / 2.9972E18 * 1E+23
where flux is in Jy

JANSKY

1 Jy = 1.0 * 10^-26 (W/m²/Hz) = 1.0E-23 erg/s/cm²/Hz

FWHM

        seeing_fwhm = 1.22 * (photplam / 24000000000.0 ) * 206264.806247
 seeing_fwhm = math.sqrt( (seeing_fwhm * seeing_fwhm) + (pixscale[chip] * pixscale[chip])  )
 24000000000.0 = size of the mirror
 206264.806247 = 180 * 3600. / pi


        $diameter=2.4e10; #diameter of the telescope

 $arc_rad = 206264.8; #number of arcsec/rad

 $rayleigh=1.22*$photplam * $arc_rad / $diameter; #1.22 * lambda/D

 $resolution  = sqrt( ($rayleigh**2) + ($pixscale**2))

CVORES

cvosres = (( photplam / 10.0E10 ) / 2.4 ) * (2.063 * (10**5)) cvonyqr = cvosres / pixscale[chip] cvosres = cvosres / 3600.0 The Nyquist ratio (spatial_resolution/spatial_sample). Values less than 2.5 are undersampled.

equations

# [Y ABnu] = -2.5 * log([X ergs/cm^2/s/Hz]) - 48.594 
# [Y ABnu] = -2.5 * log([X ergs/cm^2/s/A]) - 2.402 - 5.0 * log(lambda A) 
# [Y ABnu] = -2.5 * log([X W/m^2/Hz]) - 56.094 
# [Y ABnu] = -2.5 * log([X photon/cm^2/s/A]) + 16.852 - 2.5 * log(lambda A) 
# [Y ABnu] = -2.5 * log([X photon/cm^2/s/um]) + 16.852 - 2.5 * log(lambda um)
[Y Jy] = 1.0E+23 * [X erg/cm^2/s/Hz]
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Daniel Durand, April 2004

---

Alberto Micol, September 2004

Photometric systems: zeropoints and formulae

Generic: m = -2.5 * log10( DN/EXPTIME ) + ZEROPOINT

ST sys:  mst = -2.5 * log10( Flambda ) - 21.10

AB sys:  mAB = -2.5 * log10( Fnu ) - 48.60

where Flambda is in erg/cm2/sec/A
  and Fnu        in erg/cm2/sec/Hz

Flambda = countRate * PHOTFLAM

Fnu     = Flambda * lambda2 / c
                   = countRate * PHOTFLAM * PHOTPLAM2 / c

where:

 - counteRate is the number of counts diveded by the exptime

 - PHOTFLAM is the flux of a source with constant flux per unit wavelength
            (in erg/cm2/sec/A) which produces a count rate of 1DN/sec.
   Note: Obviously a change of GAIN will have repercussions on PHOTFLAM

 - c must be given in A/sec, that is: 2.99792E+18 A/sec
     since PHOTPLAM is in A, and PHOTFLAM in erg/cm2/sec/A.

Hence,

 mst = -2.5 * log10( countRate * PHOTFLAM ) - 21.10
     = -2.5 * log10( countRate ) -2.5*log10( PHOTFLAM ) - 21.10

     = -2.5 * log10( count ) -2.5*log10( PHOTFLAM ) - 21.10 + 2.5 * log10( EXPTIME )


 mAB = -2.5 * log10( countRate * PHOTFLAM * PHOTPLAM2 / c ) - 48.60
     = -2.5 * log10( countRate ) -2.5*log10(PHOTFLAM*PHOTPLAM2/c) - 48.60

     = -2.5 * log10( count ) -2.5*log10(PHOTFLAM*PHOTPLAM2/c) - 48.60 + 2.5*log10( EXPTIME )

 

Effective Gain

The effective gain is to be used within PIPE only when running SExtractor,
that is, when the noise of the image must be computed.

The effective gain is g for the noise of an single image.

The effective gain is g for the noise of an coadded image (pure sum).

The effective gain is N * g for the noise of an averaged image.

   That's because the poissonian noise can be computed only from the Total number of electrons,

   not from the average number of electrons; and
   the total number of electrons is N * g * the number of counts

    of an averaged image.


All the formulae to do with gain

FLUX COMPUTATION

For a source which has 1DN/sec in an image with gain g,
the flux is PHOTFLAM, where PHOTFLAM is to be scaled to g7 or g15.

Flux(erg/cm2/sec/A) = NDN/exptime * PHOTFLAM(g=g7 or g15)

For an averaged image, whose members are images at g=g7,
the flux is computed this way:

Flux = Total Ne / g7 / Total Exptime * PHOTFLAM(g7)

     = N * g7 * Mp / g7 / Total Exptime * PHOTFLAM(g7)

     = N/Total Exptime * Mp * PHOTFLAM(g7)

     = Mp / AVGExpTime * PHOTFLAM(g7)


For a summed image, whose members are images at g=g7,
the flux is computed this way:

Flux = Total Ne / g7 / Total Exptime * PHOTFLAM(g7)

     = g7 * Tp / g7 / Total Exptime * PHOTFLAM(g7)

     =  Tp / Total Exptime * PHOTFLAM(g7)


SExtractor outputs always FLUXes as counts;
hence,
a SExtractor run onto an averaged image, will return the average counts;
a SExtractor run on a total image will return the total number of counts.

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PS:   This is from an quite old document, but the idea will not change.

From:

http://www.cadc-ccda.hia-iha.nrc-cnrc.gc.ca/hst/wfpc2/wfpc2_r2_processing.html