The FUSE Observer's Guide Appendix C: Signal-to-noise Calculations and Examples Version 2.0, May 10, 2000

WARNING!!! Some information in this document may be dated!!

This appendix was formed from material cut out of the FUSE Observer's Guide, ver. 1.2. It contains extra discussion that was deemed "optional" for Cycle 2 observers and beyond, but may still be of interest. Some of this information has been duplicated and updated in the FUSE Observer's Guide, Version 2.0. For Any material shown here and in Version 2.0, the information in the main document is more current and should be used.

Outdated material in this appendix will be updated only as time permits.

Below, we describe the essential steps involved in computing exposure times for FUSE, so that the reader can better understand how the Exposure Time Calculator works and extend these concepts if necessary to more complicated situations.

These are the recommended steps for estimating exposure times:

• Determine the wavelength of interest, the flux of the source at that wavelength, the desired resolution, and the desired S/N.
• Ensure that the source does not violate the bright limit anywhere in the 900-1200 Å band (see FOG Section 3.5.2).
• Select an aperture from FOG Table 2.3-1 .
• Determine which segments contribute to the spectrum at the wavelength of interest (use FOG Table 2.4.2-1).
• Determine the total effective area for the detector segments which will be used (use FOG Table 2.4.2-1).
• Calculate the number of events per second per spectral bin from the source. The number of counts per second per resolution element for a source of flux F_lambda (erg cm^-2 s^-1 Å^-1) is

  

  where f is the aperture throughput, A_eff is the effective area, and Delta lambda is the width of a single resolution element. In addition, the web-based Exposure Time Calculator assumes that only 90% of the photons will actually be recovered (~10% of the photons are spread by astigmatism into a halo surrounding the spectrum).

• Determine the dark count rate per second per spectral bin. The dark count varies as a function of wavelength because the astigmatic heights of the spectra vary with wavelength. The dark count rate is

  

  where D_R is the expected rate per square mm, h_astig is the astigmatic height of the spectrum at the wavelength of interest, delta lambda is the desired resolution, and d is the dispersion in Å/mm (d is 1.03 Å mm^-1 for SiC and 1.12 Å mm ^-1 for LiF). D_R is expected to be 0.5-1.0 count cm^-2 s^-1 (0.005-0.01 count mm^-2 s^-1). The astigmatic height (in mm) of the spectra is shown in FOG Figure 2.5.1-1. Note that the dark count rate in the LiF and SiC channels is different at the same wavelength, so the dark rate must be computed for each channel.

• Determine the count rate contribution from scattered light. The scattered light properties of the grating are expected to be excellent. The light scattered from an emission line into 1 Å of the nearby spectrum is expected to be less than 10^-5 of the total light in the line. The most important airglow line is Lyman alpha (1215.67 Å). At the orbital height of FUSE, the Lyman alpha airglow is expected to be a few kR during the night part of the orbit and around 20 kR during the day (kR=kilo Rayleighs; one Rayleigh is equal to 10⁶/4pi photons/steradian). The amount of light scattered into a resolution element of width delta lambda can be calculated using

  

  where A is the airglow rate (in R), alpha_aperture is the angular area of the aperture in steradians (1 steradian = 4.2 × 10¹⁰ arcsec²), and A_eff is the effective area at 1216 Å. The effective areas of single SiC and LiF channels at 1216 Å are 10.0 and 15.0 cm², respectively.

• The integration time to reach a given S/N ratio is

  

Fixed pattern noise due to the multifiber boundaries is expected to be aproximately 10% of the signal and can probably be reduced to 3% using lab measured flat fields. Therefore, observations of bright objects will have a S/N of ~25-30.

Example 1: Bright point source. In this case, the dark count and scattered light can be assumed to be negligible. Consider a star with a flux of 3.0 × 10^-12 ergs cm^-2 s^-1 Å^-1 at 1030 Å. Suppose that we wish to obtain a S/N=30, R=20000 spectrum at 1030.00 Å using the 1.25" aperture (67% transmission). At 1030 Å, R=20000 corresponds to a spectral bin width of 0.052 Å. Table 2.4.2-1 shows that 1030 Å falls on four detector segments and Figure 2.4.3-1 shows that all four segments have a resolution greater than or equal to the required 20000. The effective area at 1030.00 Å using all four segments is 61.6 cm² (see Table 2.4.2-1). If the data pipeline uses all photons inside the 90% astigmatic height, the source rate is 0.298 events per second per spectral bin. Assuming the dark and scattered rates are negligible, an exposure time of 3025 seconds is necessary to obtain a S/N ratio of 30:1.

Example 2: Faint point source. In this case the dark count and scattered light must be included. Consider a QSO with a flux of 2.0 × 10^-14 ergs cm^-2 s^-1 Å^-1 at 1070 Å. Suppose that we wish to obtain a S/N=10, R=3000 spectrum at 1070.00 Å using the 4.0" aperture (98% transmission). At 1070 Å, R=3000 corresponds to a spectral bin width of 0.357 Å. The selected wavelength falls on four detector segments and the effective area is 69.8 cm². The astigmatic heights of the LiF and SiC channels at 1070 Å are 0.405 mm and 0.351 mm respectively (see Figure 2.5.1-1).

The source rate is 2.37 × 10^-2 events per second per spectral bin. The physical areas of single LiF and SiC resolution elements are 0.405mm × 0.357Å/1.12Å/mm = 0.129 mm² and 0.351mm × 0.357Å/1.03Å/mm = 0.121 mm², respectively. Since 2 SiC and 2 LiF channels are in use, and assuming the detector dark rate is 1 count cm^-2 s^-1, the total dark count rate is 5.01 × 10^-3 events per second per 0.357 Å bin. Assuming the observations are equally split between the day and night portions of the orbit, the average Lyman-alpha airglow will be 10000 Rayleighs. The aperture has an angular area of 1.88x10^-9 steradians, and the scattered light rate will be 2.4 × 10^-4 events per second per spectral bin if 90% of the photons are used by the pipeline. To achieve S/N=10, an exposure of 5166 seconds is needed.

Example 3: Very faint point source. In this case the limiting factor is the uncertainty in the detector background (see Section 2.5.1). Consider a QSO with a flux of 2.0 × 10^-15 ergs cm^-2 s^-1 Å^-1 at 1150 Å. Figure 2.5.1-2 shows that the background flux at 1150 Å is equivalent to 4.0 × 10^-15 ergs cm^-2 s^-1 Å^-1. Because the source is fainter than the background, the limiting factor is not the Poisson noise but rather the uncertainty in the background. The 1 sigma uncertainty in the background should be about 10% of the background level (Section 2.5.1), or 4.0 × 10^-16 ergs cm^-2 s^-1 Å^-1. At best, we will be able to achieve a S/N=5 observation at any spectral resolution. An exposure time of approximately 16700 seconds will be required to reach a S/N=5 (see Example 2). A longer observation will not result in higher S/N ratio.

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