9.3 Computing Exposure Times
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.
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. The actual output signal from a CCD is C/G where G is the gain. You must remember to multiply by G to compute photon events in the raw CCD images.
- G is the gain in units of electrons per count. 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 encircled energy fraction is calculated 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 available in AstroConda.
To quantify this new 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 |
O | 0.76 | 0.74 |
M | 0.74 | 0.72 |
L | 0.70 | 0.69 |
T | 0.65 | 0.64 |
The ETC results are off by 3% (O star), 2% (M star), 2% (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.
-
ACS Instrument Handbook
- • Acknowledgments
- • Chapter 1: Introduction
- Chapter 2: Considerations and Changes After SM4
- Chapter 3: ACS Capabilities, Design and Operations
- Chapter 4: Detector Performance
- Chapter 5: Imaging
- Chapter 6: Polarimetry, Coronagraphy, Prism and Grism Spectroscopy
-
Chapter 7: Observing Techniques
- • 7.1 Designing an ACS Observing Proposal
- • 7.2 SBC Bright Object Protection
- • 7.3 Operating Modes
- • 7.4 Patterns and Dithering
- • 7.5 A Road Map for Optimizing Observations
- • 7.6 CCD Gain Selection
- • 7.7 ACS Apertures
- • 7.8 Specifying Orientation on the Sky
- • 7.9 Parallel Observations
- • 7.10 Pointing Stability for Moving Targets
- Chapter 8: Overheads and Orbit-Time Determination
- Chapter 9: Exposure-Time Calculations
-
Chapter 10: Imaging Reference Material
- • 10.1 Introduction
- • 10.2 Using the Information in this Chapter
-
10.3 Throughputs and Correction Tables
- • WFC F435W
- • WFC F475W
- • WFC F502N
- • WFC F550M
- • WFC F555W
- • WFC F606W
- • WFC F625W
- • WFC F658N
- • WFC F660N
- • WFC F775W
- • WFC F814W
- • WFC F850LP
- • WFC G800L
- • WFC CLEAR
- • HRC F220W
- • HRC F250W
- • HRC F330W
- • HRC F344N
- • HRC F435W
- • HRC F475W
- • HRC F502N
- • HRC F550M
- • HRC F555W
- • HRC F606W
- • HRC F625W
- • HRC F658N
- • HRC F660N
- • HRC F775W
- • HRC F814W
- • HRC F850LP
- • HRC F892N
- • HRC G800L
- • HRC PR200L
- • HRC CLEAR
- • SBC F115LP
- • SBC F122M
- • SBC F125LP
- • SBC F140LP
- • SBC F150LP
- • SBC F165LP
- • SBC PR110L
- • SBC PR130L
- • 10.4 Geometric Distortion in ACS
- • Glossary