6.4 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 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 (Npix).
  • Sky background (Bsky) in counts/pix/s;
  • The detector background, or dark current, (Bdet) in counts/s/pix, and the read noise (RN) in counts, if using the CCD.

As of Cycle 25, the ETC has a trio of Dark Rate choices for the CCD. These choices; low (top), medium (default) and high (bottom), correspond to location on the detector and allow a more precise estimate of the dark current for a particular CCD configuration. We would like to emphasize that, while the ETC can only reflect an average of the dark current rate, the STIS FUV dark current rate in particular exhibits tremendous variations with position on the detector (due to the infamous glow region) and with the time during which the High Voltage (HV) has been on. These effects are described in detail in Section 7.5.2, in particular in Figure 7.21 and Figure 7.22. STIS FUV MAMA users whose observations are sensitive to dark current (e.g., faint targets) are strongly encouraged to read the corresponding documentation to assess the feasibility of their observations and better constrain the exposure time needed to achieve the required accuracy.

Section 6.5 provides the information you need to determine the sky-plus-detector background for your observation.

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

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

StoN=\frac{CtG}{\sqrt{CtG+N_{pix}\big(B_{sky}+B_{det}\big) Gt+ \big(N_{pix}\big/N_{bin}\big) \big(N_{read}RN^2\big)}}

where:

  • C = the signal from the astronomical source in counts/s;
  • 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);
  • Npix = the total number of detector pixels integrated over to achieve C;
  • Bsky = the sky background in counts/s/ pix;
  • Bdet = the detector dark current in counts/s/ pix;
  • Nbin = the total number of on-chip binned pixels for the CCD = BINAXIS1×BINAXIS2 (see "Binning" );
  • Nread = the number of CCD readouts (Note for the ETC, the number of CCD readouts is equal to the number of CR-SPLITs);
  • 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:

CtG+N_{pix}\big(B_{sky}+B_{det}\big) Gt\gg 2\big(N_{pix}\big/N_{bin}\big) N_{read}RN^2~.

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:

t=\frac{(StoN)^2 \big(CG+N_{pix}G\big[B_{sky}+B_{det}\big]\big)} {C^2G^2}~.

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

t=\frac{(StoN)^2}{CG}~.

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

\begin{eqnarray*} t&=&\frac{(StoN)^2 \big(CG+N_{pix}G\big[B_{sky}+B_{det}\big]\big)} {2C^2G^2}\\ \\ &&+\frac{\sqrt{ (StoN)^4 \big(CG+N_{pix}G\big[B_{sky}+B_{det}\big]\big)^2 +4(StoN)^2C^2G^2\big(\big(N_{pix}\big/N_{bin}\big) N_{read}RN^2\big)}} {2C^2G^2} \end{eqnarray*}~.

Special Case: Spectroscopic CCD Observations at λ < 2500 Å

In the optical, each photon generates a single electron (i.e., counts × the gain correspond to the total number of electrons). However, in the NUV, shortward of ~3200 Å, there is a finite probability of creating more than one electron per ultraviolet (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 NUV.

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:

StoN=\frac{Q^{-1}CtG} {\sqrt{Q^{-1} \big(C+N_{pix}B_{sky}\big) Gt+N_{pix}B_{det}Gt+ \big(N_{pix}\big/N_{bin}\big) N_{read}RN^2}}~.

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 \small{\sim1/\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.

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