9.3 Computing Exposure Times

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To derive the exposure time to achieve a given signal-to-noise ratio, or to derive the signal to noise ratio in a given exposure time, there are five principal ingredients:

  • Expected counts, C, from your source over some area.
  • The area (in pixels) over which those counts are received, Npix.
  • Sky background, Bsky, in counts/pixel/second.
  • The detector background, Bdet, or dark rate in units of counts/second/pixel.
  • The readnoise, R, in counts of the CCD.

Section 9.4 provides the information for determining the sky-plus-detector background. Signal of one electron is equivalent to one count.

9.3.1 Calculating Exposure Times for a Given Signal-to-Noise

The signal-to-noise ratio, Σ, is given by:

(1) \Sigma = \frac{Ct}{\sqrt{Ct + N_{\mathrm{pix}}(B_{\mathrm{pix}} + B_{\mathrm{det}} ) t + N_{\mathrm{pix}}N_{\mathrm{read}} R^2}}

where:

  • C is the signal from the astronomical source in counts/second, or electrons/second, from the CCD. 
  • G is the gain in units of electrons per data number (DN). It is always 1 for the SBC and 0.5, 1, 1.4, or 2 for the WFC after SM4, depending on GAIN. For archival purposes, gains prior to SM4 for WFC and HRC were ~1, 2, 4, or 8.
  • Npix is the total number of detector pixels integrated over to achieve C.
  • Bsky is the sky background in counts/second/pixel.
  • Bdet is the detector dark current in counts/second/pixel.
  • R is the readnoise in electrons; it is equal to zero electrons for SBC observations. See Table 4.1 for WFC after SM4. For archival purposes, see Table 4.2 and Table 4.3 for WFC and HRC prior to SM4.
  • Nread is the number of CCD readouts.
  • t is the integration time in seconds.

This equation assumes the optimistic (and often realistic) condition that the background level under the object is sufficiently well known (and subtracted) to not significantly contribute; in crowded fields this may not be true.

Observers using the CCD normally take sufficiently long integrations that the CCD readnoise is not important. This condition is met when:

(2) Ct + N_{\mathrm{pix}} (B_{\mathrm{sky}} + B_{\mathrm{det}}) t > 2N_{\mathrm{pix}} N_{\mathrm{read}} R^2

For the CCD in the regime where readnoise is not important and for all SBC observations, the integration time to reach a signal-to-noise ratio Σ, is given by:

(3) t = \frac{\Sigma^2 [C + N_{\mathrm{pix}} (B_{\mathrm{sky}} + B_{\mathrm{det}})]}{C^2}

If your source count rate is much brighter than the sky plus detector backgrounds, then this expression reduces further to:

(4) t = \frac{\Sigma^2}{C}

i.e., the usual result for Poisson statistics of \Sigma = \sqrt{\mathrm{total\ counts}} .

More generally, the required integration time to reach a signal-to-noise ratio Σ is given by:

(5) t = \frac{\Sigma^2 [C + N_{\mathrm{pix}} (B_{\mathrm{sky}} + B_{\mathrm{det}})] + \sqrt{\Sigma^4[C + N_{\mathrm{pix}} (B_{\mathrm{sky}} + B_{\mathrm{det}})]^2 + 4\Sigma^2 C^2 [N_{\mathrm{pix}} N_{\mathrm{read}} R^2]}}{2C^2}

9.3.2 Exposure Time Estimates for Red Targets in F850LP

At wavelengths greater than 7500 Å (HRC) and about 9000 Å (WFC), ACS CCD observations are affected by a red halo due to light scattered off the CCD substrate. An increasing fraction of the light as a function of wavelength is scattered from the center of the PSF into the wings. This problem particularly affects the very broad z-band F850LP filter, for which the encircled energy mostly depends on the underlying spectral energy distribution. The ETC finds the encircled energy fraction at the effective wavelength, which takes into account the source spectral distribution. This fraction is then multiplied by the source counts. (The effective wavelength is the weighted average of the system throughput AND source flux distribution integrated over wavelength). However, this does not account for the variation in enclosed energy with wavelength.

As a consequence, in order to obtain correct estimated count rates for red targets, observers are advised to use the pysynphot package.

To quantify this pysynphot capability, we compare the ETC results with pysynphot for a set of different spectral energy distributions and the observation mode WFC/F850LP. In Table 9.3, the spectral type is listed in the first column. The fraction of light with respect to the total integrated to infinity is listed in the other two columns, for the ETC and pysynphot calculations respectively. These values are derived for a 0.2 arcsecond radius aperture for the ETC calculations and pysynphot.

Table 9.3: Encircled energy comparison for WFC/F850LP.

Spectral type

ETC

pysynphot

O5V (Kurucz model)

0.78

0.75

M2V (Kurucz model)

0.76

0.73

L3 (Phoenix model) 

0.73

0.71

T2 (Phoenix model)

0.67

0.66


The ETC results are off by 4% (O star), 4% (M star), 3% (L star), and 1% (T star). If this small effect is relevant to particular observations, then the pysynphot software package can be used. Further information about filter F850LP can be found in Sirianni, M. et al. 2005, PASP, 117, 1049, Bohlin 2016, AJ, 152, 60, and ACS ISR 2020-08.



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