Computing Exposure Times

To derive the exposure time to achieve a given signal-to-noise ratio, or to derive the signal-to-noise ratio you will achieve in a given exposure time for your source, there are four principal ingredients:

• Expected count rate from your source over some area (C).
• The area (in pixels) over which those counts are received (N_pix).
• Sky background (B_sky) in counts pixel^-1 sec^-1.
• The detector background, or dark current, (B_det) in counts sec^-1 pixel^-1 and the read noise (RN) in counts if using the CCD.

Detector and Sky Backgrounds provides the information you need to determine the sky-plus-detector background for your observation.

Calculating Exposure Times for a Given Signal-to-Noise

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

Where:

• C = the signal from the astronomical source in counts sec-1
• t = the integration time in seconds
• G = the gain (always 1 for the MAMAs and 1 or 4 for the CCD, depending on your choice of CCDGAIN)
• N_pix = the total number of detector pixels integrated over to achieve C
• B_sky = the sky background in counts sec-1 pixel-1
• B_det = the detector dark current in counts sec-1 pixel-1
• N_bin = the total number of on-chip binned pixels for the CCD (N_bin = BINAXIS1 x BINAXIS2 (see "Binning" on page 213)
• N_read = the number of CCD readouts
• RN = the read noise in electrons; = 0 for MAMA observations

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

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

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

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

Special Case-Spectroscopic CCD Observations at <2500 Å

In the optical, each photon generates a single electron (i.e., counts times the gain correspond to the total number of electrons). However, in the near UV, shortward of ~3200 Å there is a finite probability of creating more than one electron per UV photon (see Christensen, O., 1976, J. App. Phys. 47, 689). Theoretically, the quantum yield (Q, or the mean number of electrons generated per photon) is given by the energy of the photon divided by 3.65 eV, and ranges from Q=1.06 electrons for every UV photon at 3200 Å, to Q=1.89 electrons for every photon at 1800 Å. The actual electron yield of the STIS CCD has not been measured in the near UV.

The sensitivity plots correctly predict the number of electrons generated per UV photon. However, since multiple electrons are generated from a single photon, the signal-to-noise achieved in a given integration time is affected. The explicit expression is given by:

For observations which are not in the read-noise or dark-current limited regime, the effective signal-to-noise you should expect to achieve is then ~1/sqrt(Q) times the signal-to-noise ratio calculated directly from the sensitivities given in Chapter 13 ignoring this effect. This effect is negligible at 3000 Å but amounts to 40% at 1800 Å.