# ETCL: an Exposure Time Calculator for Lhires

The Excel file ETCL.xls is a spreadsheet for calculating the stellar magnitude attainable, for a given S/N ratio and resolving power, as a function of the various instrumental parameters (telescope aperture, slit width, detector QE, etc.)  Click here to download the file

As shown below, enter the telescope parameters and the estimated observing conditions. Here we see the data for a Celestron 11 (280mm diameter and focal length 2800mm). The central obstruction is the ratio between the diameter of the secondary mirror and the primary mirror. for a refractor, this value will be zero. A value of 0.92 for the optical transmission is typical for Schmitt-Cassegrain telescopes.

The seeing must be in seconds of arc. A value equal to 4 is fairly typical for a low-altitude sight. This parameter also includes any guiding errors on the spectrograph slit. The atmospheric transmission is for a sea-level site with the star 45° above the horizon.  For a site at an altitude of 3000 metres you can adopt a transmission value of about 0.9.

The magnitude of the sky background used here is 17 per square arcsecond, a value typically found in the suburbs and built-up areas. For darker countryside skies a value of 20 can be used. Note, however, that these values have very little effect on the final calculated results since a spectrograph such as LhiresIII is generally insensitive to such changes.

The total exposure time must be in seconds.  In the example below it is 3600 (1 hour). The number of individual exposures is also required. Here, we have 12, signifying an exposure time per separate exposure equal to 300s (12 x 300 = 3600s).

The next parameters describe the spectrograph configuration itself. As a rule, they should not be changed. The only exception is the grating dispersion, when a different grating is used. The example below is for a grating with 2400 grooves per mm. The red efficiency of this grating is taken to be 0.25 around the H-alpha line and the efficiency in the green is estimated to be 0.40.

If you want to simulate the behaviour of Lhires III with a 1200 lpm grating, enter this value in the cell "Number of Grooves per mm". Note that the efficiency of this grating is higher, of the order of 0.62. This value can also be adopted for the mechanically ruled gratings of 600, 300 and 150 grooves/mm.

The entrance slit width is 25 microns, a typical value for Lhires III. You can, however, choose a narrower slit width (20 μm) or a larger width (say 30 μm). The slit width directly affects the spectral resolving power achievable and also the overall optical throughput Ts (0.059 in this example, or about 6%).

The parameters FWHMc and FWHMo are the Full Width at Half Maximum of the image of a point source produced by the collimator and objective respectively and represent degrees of uncertainty owing to optical aberrations.

The reference wavelength is the wavelength at the centre of the spectral range that is recorded. In this example it is the H alpha line at 6563 Angstrom.

In the section "Star" you enter the three following parameters for the object to be observed:

the visual magnitude, the effective temperature and a bolometric correction factor. for example, for a B3V type star, you must enter a temperature of 28000 K and a bolometric correction of -2.5 to simulate the spectral flux of the this star outside the atmosphere. Values for these two parameters for several spectral types is provided in the following table:

 Type Te BC Type Te BC O5V 51000 -4.20 F2V 7700 -0.09 O7V 46000 -3.90 F5V 7000 -0.06 O9V 39000 -3.40 F8V 6700 -0.06 B1V 30000 -2.70 G2V 6300 -0.08 B3V 28000 -2.50 G5V 6200 -0.09 B5V 25000 -2.20 G8V 5900 -0.10 B8V 19500 -1.60 K0V 5500 -0.15 A1V 15000 -1.00 K4V 4900 -0.38 A3V 13500 -0.80 K7V 4000 -0.78 A5V 11000 -0.45 M2V 3100 -1.60 F0V 8300 -0.15 M4V 3050 1.70

ETCL uses a black-body model to simulate stellar flux. Several examples of spectra calculated from the Kurucz database (in the red, for a 10th magnitude star) and the black-body equivalent determined by ETCL (in blue). The overall agreement in line shape and the absolute values of the continuum between the two sets of data is reassuring. (Note that ETCL does not simulate actual spectral lines).

The "Camera" parameters depend on the type of CCD camera being used. In this example it is for an Audine camera with a Kodak KAF-0400 chip. Note that the sensor QE has to be adjusted as a function of the wavelength range covered.

The table below shows the variation in quantum efficiency with wavelength for the KAF-0400 CCD sensor:

 Lambda RQE (%) 4000 28 4500 38 5000 38 5500 55 6000 54 6500 54 7000 54 7500 42 8000 34 8500 30 9000 20 10000 10

The section "Acquisition and Pre-processing" applies to the settings used during the acquisition of the spectrum (binning in the dispersion axis direction and binning perpendicular to the dispersion axis direction). Also indicated is the width of the spectrum in pixels along the transverse direction and whether all the light flux signal received is used for creating the spectral profile (the parameter k).  In this example, k = 1 (all the light signal is used), no binning is employed and the total spectrum width is 12 pixels.

The "Results" section gives the parameters that depend on the wavelength selected (in this case H-alpha). Of particular note here are the values of the spectral dispersion (0.115 A/pixel), the grating angle (52°) and the spectral range covered by the detector (6519 to 6607 A).

The spreadsheet also determines the number of light photons incident from the object at the top of the atmosphere and the overall efficiency of the spectrograph (taking into account the detector response).  The final efficiency here is 2.1%, which may seem very low but is quite characteristic and typical in spectroscopy .

Finally, at the end of this spreadsheet is the calculated value for the resolving power and the overall signal-to-noise ratio. The latter is given for a sampling corresponding to the pixel size of the recorded spectrum, and sampling that is equal to the achievable resolving power.  It is usually this latter parameter that gives a good indication of the "sensitivity" of the spectrograph. Here, the S/N is equal to around 100 for a  B0V star of magnitude 6.2.

We are also given (bottom right hand side of the spreadsheet) the split in the amount of noise coming from the signal (the star) and that from the detector. For this particular example, a fairly weak star, detector noise tends to dominate, which means that we could probably gain in sensitivity by using a camera generating less electronic noise for this example.