6.8 Exposure Time Examples

Here are a few simple examples to illustrate how an integration time may be computed for point sources and diffuse sources. The flux values given here are for illustrative purposes only; you need to check the flux values if you are planning your own observations of one of these targets. Also note that the examples in this section have not been updated to take into account the latest revised values for throughputs and other detector parameters, and so the numerical results presented here will differ from the more up-to-date ETC calculations.

6.8.1 Spectroscopy of Diffuse Source (M86)

We want to observe M86, an elliptical galaxy in Virgo, using the G750M grating at a central wavelength setting of λ_c = 6768 Å, the CCD detector and the 52X0.2 arcsec slit. Our aim is to calculate the Hα count rate in the central region of M86 and the expected signal-to-noise ratio per resolution element for an exposure time of 1 hour. M86 has an inhomogeneous surface brightness distribution in Hα, and the line is well resolved with this grating. Let us consider a region with an Hα surface brightness of I_λ = 1.16 × 10^–15 erg/s/cm²/Å/arcsec² (note the unit—it is not the entire Hα flux but the flux per unit wavelength interval) and a continuum surface brightness I_λ = 2.32 × 10^–16 erg/s/cm²/Å/arcsec². To derive the Hα and continuum count rates from the source we use the formula from Chapter 13:

• $//<![CDATA[ \begin{array}{l}S^d_\lambda\end{array} //]]>$ = 1.14 × 10¹³ counts/s/pix_λ/pix_s per incident erg/s/cm²/Å/arcsec².
• N_λpix = 4 and N_spix = 2 (1 resolution element).

Using the equation given by the Diffuse Source calculation, we get the count rate C = 0.106 counts/s in Hα and C = 0.021 counts/s in the continuum. Since we are interested in the properties of the Hα line, the Hα counts constitute the signal, while both the Hα counts and the continuum counts contribute to the noise.

The sky background is negligible in comparison to the source, but the dark current (4.5 × 10^–3 count/s/pix × 8 pixels = 0.020 count/s) and the read noise squared (29 e–/pix × 8 pixels × 3 reads = 465 counts, for CR-SPLIT=3) are important here. Substituting the numbers into the equation for signal-to-noise, we get:

 To increase our signal-to-noise or decrease our exposure time, we can consider using on-chip binning. Let us bin 2 pixels in the spatial direction so that N_bin = 2. To allow adequate sampling of our new binned pixels, we leave N_λpix = 4, but set N_spix = 4, so N_pix = 16 and then C = 0.212 for Hα and C = 0.254 for the sum of Hα and continuum. To compute the time to achieve a signal-to-noise of 12 using this configuration, we use the full expression for the exposure time given in Section 6.4.1, generalized to treat the line counts (for signal) and total counts (for noise) separately, and determine that roughly 35 minutes are needed in this configuration:

6.8.2 Spectroscopy of Solar-Analog Star P041-C

We wish to study the shape of the continuum spectrum of the solar-analog star P041-C from the near-infrared (NIR) to the NUV. We wish to obtain spectroscopy with the CCD detector covering the entire useful spectral range from 2000 Å to 10,300 Å with gratings G230LB, G430L, and G750L. Since we require accurate photometry, we use the wide 52X0.5 slit. The goal is to reach a signal-to-noise ratio of 25 in the NUV (at 2300 Å), 100 in the blue, and 280 in the red. P041-C has V = 12.0.

The fluxes of P041-C at the desired wavelengths obtained from a spectrum of the Sun scaled from V = –26.75 to V = 12.0, are available here.

G230LB

We illustrate the calculation of the exposure time for the G230LB grating. P041-C is found to have a flux of 1.7 × 10^–15 erg/s/cm²/Å at 2300 Å.
We get the following values for G230LB from Chapter 13:

• $//<![CDATA[ \begin{array}{l}S^p_{2300}\end{array} //]]>$ = 1.7 × 10¹⁴ counts/s/pix_λ per incident erg/s/cm²/Å;
• T_A = 0.86 for the aperture throughput, taken from Chapter 13;
• ε_f  = ~0.8;
• N_spix =3, since ~80% of the point source light is encircled within 3 pixels;
• N_λpix = 2, since two pixels resolve the LSF.

Using the equation in Section 6.2.1, we calculate a point source count rate of C = 0.34 counts/s over N_pix = 6 pixels for GAIN=1.

The source count rates can be compared with the background and detector dark current rates. We’ll make the approximation that the background and detector rates are negligibly small for this setting; therefore we can neglect their contributions. Since we are aiming for a signal-to-noise ratio of 25, we can estimate that we must obtain 625 counts minimum. The read noise squared (~350 over 6 pixels for 2 readouts) must therefore be taken into account. Finally, since we are observing with the CCD in the NUV, we must correct for the effect of the multiple-electron process (see Section 6.4). This will cause the exposure time to be scaled approximately by Q, where Q is ~1.5 at 2300 Å. Using the STIS ETC, we estimate the required time for S/N = 25 is ~3560 seconds. To check that we indeed get S/N = 25, we use the formula, in Section 6.4.

G750L and G430L

Exposure times for the two remaining wavelength settings can be calculated directly as time = signal-to-noise²/C since the read noise, detector background, and sky background are negligible. As above, 3 pixels are taken to contain 80% of the flux. The results are summarized in Table 6.6.

Table 6.6: Low Resolution Spectroscopy of Solar Analog Star.

6.8.3 Extended Source with Flux in cgs Units (NGC 6543): Imaging and Spectroscopy

Let us consider NGC 6543, the Cat’s Eye planetary nebula, where the aim is to use the CCD to image using the [O II] filter, and to do spectroscopy both in the visible and in the UV.

Imaging

The aim is to get a signal-to-noise ratio of 30 using the [O II] filter. We know that NGC 6543 is about 6 times fainter in [O II] than in Hβ, and its total flux at [O II] 3727 Å is ~4.4 × 10^–11 erg/s/cm² contained within 1 Å. Since the radius of the object is about 10 arcseconds, the average [O II] surface brightness is about 1.4 × 10^–13 erg/s/cm²/arcsec²/Å.

We take:

• $//<![CDATA[ \begin{array}{l}S^d_\lambda\end{array} //]]>$ = 6.7 × 10¹¹ counts/s/pix/Å per incident erg/s/cm²/Å/arcsec² as given in Chapter 14.
• We take N_pix = 4 × 4 = 16, since a resolution element has radius of two pixels (see Chapter 14).

To calculate the count rate we use the equation in Section 6.2.1 for diffuse sources and determine a per-pixel count rate of 0.094 counts/s/pix or a count rate C = 1.5 counts/s over 16 pixels. The background and the dark current can be neglected. To get a signal-to-noise of 30 we need ~10³ counts, so the read noise can also be neglected and we can use the simplified expression to calculate exposure time (see Section 6.4.1). We obtain 10³ counts in ~667 seconds. To allow post-observation removal of cosmic rays we use CR-SPLIT=2. We note that in each ~330 second exposure we predict a mean of ~31 counts/pix, and thus we are safely within the limits of the CCD full well.

Diffuse Source Spectroscopy in the Visible and Ultraviolet Regions

In the visible, the aim is to get a signal-to-noise of about 100 at λ = 4861 Å, with the G430M grating at a central wavelength setting of λ_c = 4961 Å, the CCD detector, and the 52X0.1 arcsecond slit. In the UV, the aim is to get a signal-to-noise ratio of about 20 at the C IV ~1550 Å line with the G140M grating at a central wavelength setting of λ_c = 1550 and the FUV-MAMA detector. To increase our signal-to-noise ratio in the UV, we use the 52X0.2 arcsecond slit for the G140M spectroscopic observations. 

Visible Region

NGC 6543 has an average Hβ surface brightness of S(Ηβ) ~ 8.37 × 10^–13 erg/s/cm²/Å/arcsec² at 4861 Å and has a radius of about 10 arcseconds.

We take from Chapter 13:

• $//<![CDATA[ \begin{array}{l}S^d_\lambda\end{array} //]]>$ = 1.62 × 10¹² counts/s/pix_λ/pix_s per incident erg/s/cm²/Å/arcsec² for G430M;
• N_λpix = N_spix =2 since 2 pixels resolves the LSF and PSF;

Using the equation for diffuse sources in Section 6.2, we derive a per-pixel count rate of 1.4 counts/s/pix and a count rate integrated over the four pixels of C = 5.4 counts/s at 4861 Å from the astronomical source. The sky background and the detector background are much lower. To allow cosmic ray removal in post-observation data processing, we use CR-SPLIT=3. To achieve a signal-to-noise of 100, we require a total of roughly 10,000 counts, so read noise should be negligible, even over 4 pixels and with NREAD=3. We calculate the time required to achieve signal-to-noise of 100, using the simplified equation Section 6.4.1, and determine that we require roughly 30 minutes:

At a count rate of ~1 counts/s/pix for 600 seconds per CR-SPLIT exposure, we are in no danger of hitting the CCD full-well limit.

Ultraviolet Region

The C IV flux of NGC 6543 is ~2.5 × 10^–12 erg/s/cm²/arcsec² spread over ~1 Å. The line, with a FWHM ~ 0.4 Å, will be well resolved in the G140M configuration using the 52X0.2 slit.

We take from Chapter 13:

• $//<![CDATA[ \begin{array}{l}S^d_\lambda\end{array} //]]>$ = 5.15 × 10⁹ counts/s/pix_λ/pix_s per incident erg/s/cm²/Å­/arcsec² for G140M at λ = 1550 Å using the 0.2 arcsecond wide slit.
• We take N_λpix = N_spix = 8, since the line emission is spread over the ~8 pixels of the slit width in dispersion, and we are willing to integrate flux along the slit to improve the signal-to-noise ratio.

Using the equation for diffuse sources in Section 6.2.1, we determine a per-pixel peak count rate of ~0.013 counts/s/pix and a count rate over the 64 pixels of C = 0.82 counts/s at 1550 Å from the astronomical source. The sky and detector backgrounds are still negligible, and the read noise is zero for the MAMA detector so we can use the simplified equation for exposure time, see Section 6.4.1. We determine that we require ~7 minutes:

 We are well below the MAMA local linearity limit of 50 counts/s/pix. Even assuming the nebula evenly illuminates the full 28 arcseconds of the long slit, we are well below the global absolute and linearity limits, since the flux from the nebula is concentrated in the C IV emission line. Then the global count rate, if the source fully fills the slit in the spatial direction, is given roughly by (0.015 × 8 × 1024) $//<![CDATA[ \begin{array}{l}\ll\end{array} //]]>$200,000 counts/s. Finally, we are well below the MAMA 16 bit buffer limit of a maximum of 65,536 counts/pix integrated over the exposure duration.

6.8.4 Echelle Spectroscopy of a Bright Star, Reddened LMC Star

The aim here is to do high-resolution echelle spectroscopy of an O5 star in the LMC at 2500 Å, using the E230H grating at a central wavelength of λ_c = 2513 Å and using the 0.2X0.09 arcsecond slit. The aim is to get a signal-to-noise ratio of about 50 from photon statistics. We will assume that the exact UV flux of the star is unknown and we need to estimate it from the optical data. This calculation of the stellar flux at 2500 Å involves 2 steps:

1. Calculation of the dereddened flux at 5500 Å.
2. Calculation of predicted flux at 2500 Å taking reddening with standard extinction and stellar models into account.

Dereddened Magnitude and Prediction of 2500 Å Flux

We assume that it is an O5 V star with V = 11.6 and B – V is –0.09. The expected B – V value from such a star is –0.35, so that E(B – V) = 0.26 mag.

We assume all the extinction to be due to the LMC, and use the appropriate extinction law (Koornneef and Code, ApJ, 247, 860, 1981). The total visual extinction is then R × E(B – V) = 3.1 × 0.26 = 0.82, leading to an unreddened magnitude of V₀ = 10.78. The corresponding flux at 5500 Å (using the standard zero point where V = 0 corresponds to F(5500 Å) = 3.55 × 10^–9 erg/s/cm²) is F(5500 Å) = 1.73 × 10^–13 erg/s/cm²/Å.

The model atmosphere of Kurucz predicts F(2500 Å)/F(5500 Å) = 17.2 for an O5 star, which leads to a flux of F(2500 Å) = 2.98 × 10^–12 erg/s/cm²/Å at 2500 Å for the unreddened star. Reddening will diminish this flux by a factor of 10^{–0.4xA(2500 Å)}, where the absorption at 2500 Å can be determined from the extinction curve; the result in this case is ~0.3. Thus the predicted flux of this star at 2500 Å is 9.0 × 10^–13 erg/s/cm²/Å.

Exposure Time Calculation

We take from Chapter 13:

• $//<![CDATA[ \begin{array}{l}S^d_\lambda\end{array} //]]>$ = 2.9 × 10¹¹ counts/s/pix_λ per incident erg/s/cm²/Å for E230H;

• T_A = 0.659 for the aperture throughput;
• ε_f = 0.8 for the encircled energy;
• N_λpix = 2, since two pixels resolve the LSF;
• N_spix = 3, since 80% of the point source light is encircled by 3 pixels.

Using the equation for point sources in Section 6.2, we determine a total count rate from the star of C = 0.3 counts/s over 6 pixels. From Chapter 13 we see that ~22 percent of the point source flux will be contained within the peak pixel. Thus the peak per pixel count rate will be approximately 0.3 × 0.22/(0.8 × 2) = 0.045 counts/s/pix and well within the local linear counting regime. We can use the information that we register ~0.3 counts/s for every two pixels in the dispersion direction to estimate the global count rate (over the entire detector) as follows. Each order contains ~1024 pixels, and the E230H grating at the central wavelength setting of 2513 Å covers 33 orders (see Chapter 13). A rough estimate of the global count rate is thus ~33 × 512 × 0.3/0.8 ~6400 counts/s and we are well within the linear range.

To calculate the integration time, we can ignore both the sky background and the detector dark current which are several orders of magnitude fainter than the source. To achieve a signal-to-noise ratio of 50, we then require ~2500 counts which would take a total of ~2.3 hours. Fortunately, this is a CVZ target!

6.8.5 Imaging a Faint Stellar Source

Consider a case where the aim is to image a faint (V = 28), A-type star with the clear filter and the CCD detector. We want to calculate the integration time required to achieve a signal-to-noise ratio of 5. The count rate from the source is 0.113 counts/s distributed over about 25 pixels using the information in Chapter 14. If we assume the background to be “typical high” (Table 6.3), the count rate due to the background integrated over the bandpass is ~0.15 counts/s/pix or 3.8 counts/s in 25 pixels (and the detector dark rate is 35 times lower). We will need to be able to robustly distinguish cosmic rays if we are looking for faint sources, so we will use CR-SPLIT=4. We use the STIS ETC to estimate the required exposure time to be 8548 seconds. To reproduce the numbers given by the ETC, we use the equation in Section 6.4.1:

Alternately, we could have requested LOW-SKY (see Section 6.5.2), since these observations are sky-background limited. In that case the sky background integrated over the bandpass produces ~0.035 counts/s/pixto which we add the detector dark current to get a total background of 0.039 counts/s/pix. Using the full equation for exposure time again, we then determine that we require only ~60 minutes. This option is preferable to perform this experiment. To check the S/N, we use the equation in Section 6.4.1:

6.8.6 Time-Tag Observations of a Flare Star (AU Mic)

Suppose the aim is to do TIME-TAG observations of a flare star such as AU Mic, in the hydrogen Lyman-α 1216 Å line (see Section 11.1.3). We wish to observe it with the G140M grating, the MAMA detector and a 0.2 arcsecond slit. AU Mic has V = 8.75, the intensity of its Ly-α line is about 6 (±3) × 10^–12 erg/s/cm²/Å, and the width (FWHM) of the line is about 0.7 (±0.2) Å. We will assume that during bursts, the flux might vary by a factor of 10, so that the line flux may be up to 60 × 10^–12 erg/cm²/s/Å.

AU Mic is an M star and its UV continuum is weak and can be neglected.

We use from Chapter 13:

• $//<![CDATA[ \begin{array}{l}S^d_\lambda\end{array} //]]>$ = 2.30 x 10¹² counts/s/pix_λ per incident erg/s/cm²/Å;
• Aperture throughput T_A = 0.6;
• Encircled energy ε_f = 0.8;
• N_spix = 10;
• Derive N_λpix = 14 since the line FWHM is ~0.7 Å and the dispersive plate scale for G140M is 0.05 Å/pix;

Plugging these values into the point source equation in Section 6.2.1, we get C = 927 counts/s over 10 × 14 pixels, or ~1160 counts/s from the source during a burst (taking ε_f = 1.0). This is well below the MAMA TIME-TAG global linearity limit of 30,000 counts/s and the continuous observing limit of 26,000 count/s. The line is spread over 14 pixels in dispersion and roughly only 10% of the flux in the dispersion direction falls in the peak pixel; thus the peak per-pixel count rate, P_cr, is roughly 927/(14 × 10) = 7 counts/s/pix, and we are not near the MAMA local linearity limit.

For a TIME-TAG exposure, we need to determine our maximum allowed total observation time, which is given by 6.0 × 10⁷/C seconds or roughly 1079 minutes = 18 hours. For Phase II only, we will also need to compute the value of the BUFFER-TIME parameter, which is the time in seconds to reach 2 × 10⁶ counts, in this case 2157 seconds (= 2 x 10⁶/927).