9.6 Estimating Exposure Times

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9.6.1 SNR Reached in a Given Exposure Time

To derive the exposure time to achieve a given SNR ratio, or to derive the SNR ratio achieved in a given exposure time, there are four principal ingredients:

  • Expected count rate C (e– s−1) from your source over some area.
  • The area (in pixels) over which those e– are received (Npix).
  • Sky background (Bsky) in e– s−1 pixel−1.
  • The detector background, or dark, (Bdet) in e– s−1 pixel−1 and the read noise (R) in e–. (Section 9.7 provides information for determining the sky-plus-detector background.)

The SNR ratio Σ achieved in exposure time t seconds, is given by

\Sigma = \frac{C t} {\sqrt{C t + N_{pix} \big( B_{sky} + B_{det} \big) t + N_{pix} N_{read} P + \frac{N_{pix}}{N_{bin}} N_{read} R^2}}

where:

  • C = the signal from the astronomical source in e− s−1. (Note that the raw output image uses DN, which will be equal to C/G, where G is the gain.)
  • Npix = the total number of detector pixels integrated over to achieve C.
  • Nbin = the number of detector pixels binned to one read-out pixel when on-chip binning is used.
  • Bsky = the sky background in e− s−1 pixel−1.
  • Bdet = the detector dark current in e− s−1 pixel−1.

  • R = the read noise in electrons; 3.1 e− for the UVIS channel and 12. e− for the IR channel (this is the effective read noise achieved by fitting the ramp of IR readouts, if close to the full sequence of 16 readouts is used).

  • Nread = the number of detector readouts.

  • P = the background added using the post-flash option (Section 6.9.2) in e− pixel-1

The above equation assumes the optimistic (but often realistic) condition that the background zero-point level under the object that is subtracted could be well known (from integrating over many pixels) but is still noisy in Npix and does not significantly contribute to the error budget; in crowded fields this may not be true. In general, C in the numerator should be the signal in Npix from the component to be measured (e.g., the point source in an image or the line emission in a spectrum), while C in the denominator is the total astronomical signal in Npix (e.g., including light from the underlying galaxy in the image or from the continuum in the spectrum).

9.6.2 Exposure Time to Reach a Given SNR

Observers normally take sufficiently long integrations that the read noise is not important. This condition is met when:

C t + N_{pix} \big( B_{sky} + B_{det} \big) t + N_{pix} N_{read} P > 2 \frac{N_{pix}}{N_{bin}} N_{read} R^2

In the regime where read noise is unimportant, and virtually all of the astronomical signal in Npix comes from the component being measured, the integration time to reach a given signal-to-noise ratio Σ is:

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

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

t = \frac{\Sigma^2}{C}

i.e., the usual result for Poisson statistics of \small{\Sigma=\sqrt{\mathrm{total\ counts}}}.
More generally, the required integration time to reach a given SNR ratio is:

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

9.6.3 Exposure Time Estimates for Red Targets in F850LP

At long wavelengths, ACS CCD observations are affected by a red halo due to light scattered off the CCD substrate; i.e. an increasing fraction of the light as a function of wavelength is scattered from the center of the PSF into the wings. This problem affects particularly the very broad z-band F850LP filter in ACS, for which the encircled energy depends on the underlying spectral energy distribution the most. This problem has not been seen in WFC3/UVIS observations, and so should not be a concern for those planning WFC3 observations.

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