Appendix 1 - Estimating exposure times, approximate zero-points, colour terms, etc.

  1. Calculation of Signal-to-Noise Ratios
  2. f/3.3 Direct
    1. Sensitivities
    2. Photometric zero-points & Colour terms
    3. Extinction Coefficients  
  3. f/1 LBL
    1. Sensitivities
    2. Photometric zero-points & Colour terms
    3. Extinction Coefficients  
  4. f/8 Cassegrain

An on-line calculator for Direct Imaging is now available
at http://www.aao.gov.au/cgi-bin/pfcalc.pl


Introduction The Telescope & Optics The Detectors
The Imaging Cameras An Imaging Cookbook The Data you Take Away
Exposure times OFFSET_RUN files CCD Windows Data Catalogs
On-line Reduction Filters Flat-fields Blank Fields Orientation Shutters

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1. Calculation of Signal-to-Noise Ratios

The signal to noise ratio for a given exposure can be calculated from

where
is the total number of counts in the object,
is the number of pixels covered by the object (i.e. the size of the photometrickaperture adopted if the object is a star),

is the number of counts per pixel from the sky background, and
is the CCD readout noise in electrons.

For broadband imaging, where even the shortest exposures through a broad band filter are sky limited, this simplifies to:

For an exposure of time t seconds, let

Nobj = O  x  t

Nsky
= S x  t

Then the above expression simplifies to

S/N = O/sqrt(O+Npix x S) x sqrt(t)

Example: for the f/1 + Thomson,  a star with R=23 has an expected total count rate of about 8.8 photons/second, ie. O = 8.8 and, for a dark sky, S = 72 photons/second/pixel in R (see 3.1 below). Assuming the seeing is 1.5 arcsec FWHM and the we do photometry over 9 pixels (ie a roughly 3x3 arcsec aperture, so Npix=9), we have S/N = 6.0 after 300 seconds. From this, we can estimate that a five minute exposure with the Thomson  CCD in 1.5 arcsec seeing will reach a limiting magnitude (5 detection) slightly fainter than 23 in R.

An on-line calculator  is available for performing these calculations.

Typical sky brightness (new moon) at Siding Spring
Passband U B V R I
Mag/sq.arcsec 21.2 22.5 21.5 20.8 19.3

2. f/3.3 + Tektronix CCD

2.1 Sensitivities

Typical object and sky countrates for the TEK chip for observations at the zenith with the chip at 200K ("warm"). These have been used to calculate the ratio of the noise to the signal (ie 1/SNR) purely due to photon counting for a 300s exposure in 1.5" seeing in a 3" aperture.

U B V R I
O = OBJECT PHOTONS PER SECOND for 22.5 mag star.
(summed over the entire image. Chip at 200K )
 5.7 59.7 33.5 38.1 17.6
S = SKY PHOTONS PER PIXEL PER SECOND
(TEK Scale = 0.391"/pix. Chip at 200K )
Dark 1.7 9 9 20 42
6-day Moon   28 17 28 46
Bright   270 138 112 113
PHOTON-COUNTING UNCERTAINTIES for 22.5 mag star,
300s
exposure in 1.5" seeing. using a 3"diameter aperture
Dark   2% 4% 5% 15%
6-day Moon   4% 5% 6% 15%
Bright   11% 14% 11% 24%

An on-line calculator  is available for performing these calculations.

2.2 Photometric Zero-points and Colour terms (UBVRI)

The following BVR results were obtained by Gary Da Costa, after the reduction of a night of Service Imaging on 26 Oct 1992. A total of 36 standards in 7 fields (some observed more than once) have gone into this determination. The standard stars ranged in colour from B-V=-0.3 to 1.9. The TEK chip was used WARM and all these data were obtained in SPEED FAST.

The U and I results are derived from the previous  RCA CCD photometry, and have been scaled by the appropriate QEs and gains. They should only be used for VERY rough estimation.

It must be emphasised that the magnitude zero-points and extinction coefficients provided here are specific to the night on which the observations were carried out. The extinction coefficients in particular  for this night were quite large, and should not be considered as representative.  On the other hand, quick-look photometry can probably be estimated to ~20% using these numbers.

Note that these numbers are for FAST readout. Observations at NORMAL will have twice as many ADU/e, so the zero-points will be increased by 0.753 magnitudes.

For u,b,v,r,i = instrumental magnitudes = 2.5*log(signal in ADU/time)
I(std) ~ 23.8 - 0.06*Sec(ZA) - i
This is a rough estimate based on RCA CCD photometry
R(std) = 24.81 + 0.04(V-R) - 0.26*Sec(ZA) - r  
The formal uncertainties in the zero-point and colour slope are ±0.005 and ±0.01 mag respectively.
The RMS deviation from this relation is ±0.016 mag.
V(std) = 24.63             - 0.30*Sec(ZA) - b
The formal uncertainties in the zero-point and colour slope are ±0.005 and ±0.01 mag respectively.
The RMS deviation from this relation is ±0.011 mag
B(std) = 25.02 + 0.12(B-V) -(0.43-0.02(B-V))*Sec(ZA) - v
The formal uncertainties in the zero-point and colour slope are ±0.004 and ±0.01 mag respectively.
The RMS deviation from this relation is ±0.016 mag
U(std) ~ 22.5 - 0.5*Sec(ZA) - u
This rough estimate is based on RCA CCD photometry, however observations were performed with the TEK CCD with the CuSO4 U-filter in the SLOW speed on 6 Sep 1997 confirm this rough estimate. The zero-point at speed slow is 24.0 (compared with the 22.5 at FAST).

For what its worth, here are the corresponding equations for the RCA CCD from the 1989 CCD User's Guide. (The approximate zero-points for U and I with the TEK shown above are derived from these given the known  relative QE of the RCA and TEK chips (see the AAO Observer's Guide, Chapter 6).

I(std) = 23.83 + 0.032(R-I) - 0.063*Sec(ZA) - i
or 150e-/s for I=20
R(std) = 24.81 + 0.015(V-R) - 0.109*Sec(ZA) - r  
or 360e-/s for R=20
V(std) = 24.70             - 0.154*Sec(ZA) - b
or 310e-/s for V=20
B(std) = 24.94 + 0.177(B-V) -(0.263-0.026(B-V))*Sec(ZA) - v
or 350e-/s for B=20
U(std) = 21.48 + 0.139(U-B) - 0.502*Sec(ZA) - u
or 12e/s for U=20

Notice that the extinctions for the RCA are different - this mostly reflects intrinsic night-to-night differences in extinction, rather than the two chips (slightly) different QE curves.

2.3 Extinction

'Standard' useful values for the extinction at the Siding Spring site are being sought. In their absence, the solutions shown in the f/3.3 and f/1 sections above and below can be used as a very rough guide.

 

 

3. f/1 + Thomson CCD

3.1 Sensitivities

Typical object and sky countrates for the Red Thomson chip at f/1, based on the standard star observations of 01 January 1992.

V R I
O = OBJECT PHOTONS PER SECOND for 22.5 mag star
(summed over the entire image)
   9.2 10.1 6.2
S = SKY PHOTONS PER PIXEL PER SECOND
(Thompson Scale = 0.98"/pix)
Dark 22 46 113
6-day Moon 42 64 123
Bright 337 257 304

From these rates, it is estimated that in dark time a 5 minute exposure in 1.5" seeing will reach a limiting magnitude (5 sigma detection) of R = 23.2. With the f/1 system, even the shortest exposures through the broadband filters are sky limited, due to the the large pixels and the low read-noise of the Thomson CCD. Stars brighter than 13th magnitude saturate in less than 5s, which is the absolute minimum exposure time set by the shutter speed.

An on-line calculator  is available for performing these calculations.

3.2 Photometric Zero-points and Colour terms

The following information is based on the reduction of a single night's data in January 1992, and a single night's data in March 1992. As such they do not represent the final word on f/1 performance, but should provide a rough guide. Observers who have derived more colour terms and zero-points for the f/1 system are asked to communicate these to cgt@aaoepp.aao.gov.au, so that these can be distributed to the wider community.

Standards due to Landolt (1992, AJ, 104, 340) and Graham (1982, PASP, 94, 244) were used together in this reduction. These two systems (the `Kron-Cousins' of Landolt, and the `Cousins' system of Graham are actually slightly different. However, these differences are fairly small, and unlikely to be of much concern to a 4-m imaging observer - see Bessell (1995, PASP, 107, 672) and references therein for more information on different photometric systems. In this reduction the two sets of data have been used as published without further conversion. It must be emphasised that the magnitude zero-points and extinction coefficients provided here are specific to the night on which the observations were carried out, and should not really be used as reprasentative

Standards over a colour range of V-R=-0.2 to 1.0, or V-I=-0.4 to 2.0, or R-I=-0.2 to 1.0 were used. Over this range a measureable colour term was seen in the V passband, though there is evidence to indicate a smaller colour term was present in I. No significant colour term was seen in R.

If we use the subscript "s" is used to denote a standard magnitude/colour, and "i" is used to denote an observed "instrumental" magnitude, corrected for exposure time and atmospheric extinction, and "r" is used to denote the residual between the two in the sense observed - standard.)

Then the following solution was derived on 01 Jan 1992. (These are for instrumental magnitudes as -2.5log(ADU/s) in speed NONASTRO, Gain=5e-/ADU).

Vs = Vi - (0.324 ± 0.014)*Airmass - (-0.079 ± 0.014)*(V-R)s + 31.0 - (7.503±0.007)
Vs = Vi - (0.324 ± 0.014)*Airmass - (-0.041 ± 0.008)*(V-I)s + 31.0 - (7.503±0.007)
or  93 e/s for V=20
Rs = Ri - (0.260 ± 0.020)*Airmass + 31.0 - (7.473±0.019)
or 101e/s for R=20
Is = Ii - (0.246 ± 0.020)*Airmass + 31.0 - (8.007±0.023)
or 62 e/s for I=20

If a colour term in I was solved for, I got slopes of I/V-I: -0.027±0.009 mag/mag, and I/R-I: -0.055±0.020. these did not seem to me to be sufficiently well determined to make including them in the solution for this particular night's data worthwhile. However, further experience with the f/1 system may improve the significance of their estimation.

The solution derived on 28 March 1992 was poorer than that shown above, but confirmed the general colour term trends and zero-points shown above.

3.3 Extinction

'Standard' useful values for the extinction at the Siding Spring site are being sought. In their absence, the solutions shown in the f/3.3 and f/1 sections above can be used as a very rough guide.

4. f/8 Cassegrain

The total eficiency for or f/8 Cassegrain imaging at one of the auxiliary focii is about 20% less than that for the same detector at Prime Focus, due to the two extra mirrors in the light path.  The sensitivies seen at f/3.3 can, therefore, be used as a guide to f/8 performance - though the sky-counts must be scaled for the different pixel scale, and the object- and sky-counts scaled for the differing QEs of the TEK and Thomson (the most used chip at f/8).

An on-line calculator  is available for performing these calculations.


Introduction The Telescope & Optics The Detectors
The Imaging Cameras An Imaging Cookbook The Data you Take Away
Exposure times OFFSET_RUN files CCD Windows Data Catalogs
On-line Reduction Filters Flat-fields Blank Fields Orientation Shutters

Back Contents Next

Last Contents Next

This Page maintained by : Chris Tinney (cgt@aaoepp.aao.gov.au)
This Page last updated:  7 Sep 1997, by Chris Tinney