# Summary

This online help provides some information and help for the MUSE Exposure Time Calculator. The ETC basic mode is the calculation of the reachable signal to noise ratio, considering:
• a target;
• some sources of noise;
• some instrumental characteristics.
The calculation of the signal to noise ratio involves the calculation of the signal produced by the target, and the noise produced by all the noise sources considered. By reversing the equations, the exposure time needed to reach a certain signal to noise ratio can be computed. An additional parameter: the number of detector integration time, is taken into account.

## Sources considered

Five sources are considered for the computation of the Signal to Noise Ratio (SNR), only the first one contributing to the 'signal' term of the SNR, and all of them contributing to the 'noise' term:
• the target: defined by a spectrum (spectral distribution) and a shape (spatial distribution). The target is the source which produces the 'signal' that the user wants to detect
• a background source: like the target, is defined by a spectrum and a spatial distribution, both of them independant from those of the target. This background source contributes only to the 'noise' term
• the sky: defined by a spectrum, is constant within the field of view. Contributes only to the 'noise' term
• dark current: defined by a single value. Contributes only to the 'noise' term
• readout noise: defined by a single value. Contributes only to the 'noise' term

## Signal to noise ratio obtained

For each spectral pixel, the ratio between the signal of the target (in electrons) and the square root of the sum of the noise sources (in electrons too) is the signal to noise ratio.

## Outputs

The ETC provides inline values (summary of input parameters, SNR for a particular wavelength...), downloadable .dat files (number of electrons per pixel, SNR at the various wavelengths...), and downloadable .fits files (target and background source spatial shapes, spatial PSF, result of their convolution).

# 1. Target and Background source

The target and the background source are characterized by the same parameters: the only differences are 1) that the background source contribution can be disabled, and 2) that the contribution of the background source contributes only to the noise term of the SNR.

## 1.1 Spatial distribution

The geometry of the target will affect the signal to noise. In order to simulate a broad range of source types, the available luminosity profiles are:
• Extended source: infinitely uniform luminosity (no parameter)
• Extended source with constant luminosity profile and finite radius (1 parameter: the diameter in arcsec)
• Extended sources: Sersic profiles [RD01] (2 parameters):
• the radius re of the source within which 50% of the energy is emitted (in arcsec)
• an exponent m giving the strength of the decrease along the profile
The template profile is I(r) = I(r0)e-b(r/re)1/m. The parameter b derives from the value of m [RD01], and I(r0) is calculated within the ETC as a normalization factor.
• Pointlike source (no parameter)

## 1.2 Distance

Except for the infinitely extended case, a shift parameter can be provided: in both the target and the background source cases, this value specifies at which distance from the center of the target or background source the SNR must be computed.

## 1.3 Spectral distribution

The user can use different kinds of spectral distribution:
• a continuum profile (no parameter);
• a template spectrum from a library of sources (stellar, nebular and galactic) (2 parameters: which spectrum template to use, and redshift z)
• a blackbody profile with a given temperature (1 parameter: blackbody temperature)
• a single emission line, defined by a Gaussian profile centered on a user-defined wavelength, characterised by its full-width at half-maximum width (3 parameters: wavelength in angstrom, fwhm in angstrom, flux in erg/s/cm2). Note that the fwhm will be forced to be at least 2 pixels = 2.5 Angstrom.
• the user can also upload his own spectrum, as a simple ascii file (2 parameters: link to the file, redshift z). The ETC will interpret the ascii file assuming that:
• file name must have the extension "dat" or FITS
• each data row contains two values: a wavelength in nm, and a flux in unit proportional to erg/nm (to be opposed to magnitudes, or photons/Hz for example)
• the rows are sorted by increasing wavelengths. The sampling doesn't need to be constant
The ETC reads the uploaded spectrum, shifts it according to the redshift, and resamples it at the MUSE spectral sampling using a linear interpolation. One of the outputs provided by the ETC is the uploaded spectrum in physical units after interpretation by the ETC: the user is required to check it.
Except for the emission line case, the user must also provide a magnitude in one of the proposed filters (Johnson-Cousins B, V, Rc, Ic or Sloan g', r', i' , z', [RD02] and [RD03]), so that the ETC can normalize the spectrum. The magnitude is supposed to be:
• per arcsec2 if the spatial shape is 'extended'
• the integral photometry if the spatial shape is 'pointlike' or 'Sersic'
The combinations are summarized in the following table:
 Extended source: constant luminosity Extended source: Sersic profile Pointlike source Emission line erg/s/cm2/arcsec2 erg/s/cm2: integral brightness erg/s/cm2: integral brightness Continuum source magnitude/asec2 magnitude: integral brightness magnitude: integral brightness Blackbody magnitude/asec2 magnitude: integral brightness magnitude: integral brightness Template spectrum magnitude/asec2 magnitude: integral brightness magnitude: integral brightness

# 2. Sky Conditions

## Seeing and Image Quality

### Turbulence Category (T category)

With the advent of instruments using new adaptive optics (AO) modes, new turbulence parameters need to be taken into account in order to properly schedule observations and ensure that their science goals are achieved. These parameters include the coherence time and the fraction of turbulence taking place in the atmospheric ground layer, in addition to the seeing. Starting from Period 105, the turbulence constraints are standardised to the turbulence conditions required by all instruments and modes, whether they are seeing-limited or AO-assisted.

The handling of atmospheric constraints thus changes for both Phase 1 (proposal preparation) and Phase 2 (OB preparation). In Phase 1, the seven current seeing categories are replaced by seven turbulence categories for all instruments. Each category can be defined by other parameters than a pure seeing threshold, depending on the instrument. For all instruments, all categories share the same statistical probability of realisation, which is key for an accurate time allocation process. In Phase 2, the image quality will still be the only applicable constraint for seeing-limited modes, whereas the same turbulence category as for Phase 1 will be used for diffraction-limited modes.

Users are encouraged to read the general description of these changes for Phase 1 and Phase 2 on the Observing Conditions webpage, as well as instrument User Manuals for specifics per instrument.

### Seeing and Image Quality

The definitions of seeing and image quality used in the ETC follow the ones given in Martinez, Kolb, Sarazin, Tokovinin (2010, The Messenger 141, 5) originally provided by Tokovinin (2002, PASP 114, 1156) but corrected by Kolb (ESO Technical Report #12):

Seeing is an inherent property of the atmospheric turbulence, which is independent of the telescope that is observing through the atmosphere; Image Quality (IQ), defined as the full width at half maximum (FWHM) of long-exposure stellar images, is a property of the images obtained in the focal plane of an instrument mounted on a telescope observing through the atmosphere.

The IQ defines the S/N reference area for non-AO point sources in the ETC.

With the seeing consistently defined as the atmospheric PSF FWHM outside the telescope at zenith at 500 nm, the ETC models the IQ PSF as a gaussian, considering the gauss-approximated transfer functions of the atmosphere, telescope and instrument, with s=seeing, λ=wavelength, x=airmass and D=telescope diameter:

Image Quality
 $${ $$\mathit{FWHM}_{\text{IQ}} = \sqrt{\mathit{FWHM}_{\text{atm}}^2(\mathit{s},x,\lambda)+\mathit{FWHM}_{\text{tel}}^2(\mathit{D},\lambda)+\mathit{FWHM}_{\text{ins}}^2(\lambda)}$$ }$$

For fibre-fed instruments, the instrument transfer function is not applied.

The diffraction limited PSF FWHM for the telescope with diameter D at observing wavelength λ is modeled as:
 \begin{aligned} \mathit{FWHM}_{\text{tel}} & = 1.028 \frac{\lambda}{D} \text{, } & \text{ with } \lambda \text{ and D in the same unit}\\ & = 0.000212 \frac{\lambda}{D} \text{arcsec, } & \text{ with } \lambda \text{ in nm and D in m}. \end{aligned}
For point sources and non-AO instrument modes, the atmospheric PSF FWHM with the given seeing $$s$$ (arcsec), airmass $$x$$ and wavelength $${ \lambda }$$ (nm) is modeled as a gaussian profile with:
 $${\mathit{FWHM}}_{\text{atm}}(\mathit{s},\mathit{x},\lambda) = \mathit{s} \cdot x^{0.6} \cdot (\frac {\lambda} {500})^{-0.2} \cdot \sqrt{[1+F_{\text{Kolb}} \cdot 2.183 \cdot ({r_0}/L_{0})^{0.356})]}$$ Note: The model sets $${ \mathit{FWHM}}_{\text{atm}}$$=0 if the argument of the square root becomes negative $${ [1+F_{\text{Kolb}} \cdot 2.183 \cdot ({r_0}/L_{0})^{0.356}] < 0 }$$ , which happens when the Fried parameter $${ {r_0} }$$ reaches its threshold of $${ r_{\text{t}} = L_{0} \cdot [1/(2.183 \cdot F_{\text{Kolb}})]^{1/0.356}}$$. For the VLT and $${ L_{0} = 46m}$$ , this corresponds to $${ r_{\text{t}} = 5.4m}$$.
$${ L_{0} }$$ is the wave-front outer-scale. We have adopted a value of $${ L_{0} }$$=46m (van den Ancker et al. 2016, Proceedings of the SPIE, Volume 9910, 111).

$$F_{\text{Kolb}}$$ is the Kolb factor (ESO Technical Report #12):
 $$F_{\text{Kolb}} = \frac {1}{1+300 {\text{ }} D/L_{0}}-1$$ For the VLT and $${ L_{0} }$$=46m, this corresponds to $$F_{\text{Kolb}} = -$$0.981644.
$${r_0}$$ is the Fried parameter at the requested seeing $$s$$, wavelength $${ \lambda }$$ and airmass $$x$$:
 $$r_0 = 0.100 \cdot s^{-1} \cdot (\frac{\lambda}{500})^{1.2} \cdot x^{-0.6} \text{ m, } \text{ } \text{ } \text{ } \text{ with } s \text{ in arcsec } \text{and } \lambda \text{ in nm.}$$

For AO-modes, a model of the AO-corrected PSF is used instead.

#### Line-Of-Sight Seeing

In Narrow Field Mode (NFM) with AO, the line-of-sight seeing (LOS Seeing) is defined as:

$$$$\mathit{LOS\, Seeing} = s \cdot x^{0.6}$$$$

## Sky Model

The sky background model is based on the Cerro Paranal Advanced Sky Model, also for instruments at la Silla, except for the different altitude above sea level. The observatory coordinates are automatically assigned for a given instrument.

Since version P101, the ETCs include a dynamic almanac widget to facilitate assignment of accurate sky model parameters for a given target position and time of observation. The sky radiation model includes the following components: scattered moonlight, scattered starlight, zodiacal light, thermal emission by telescope and instrument, molecular emission of the lower atmosphere, emission lines of the upper atmosphere and airglow continuum.

Alternatively, the almanac mode can be overridden to allow manual assignment of airmass and moon phase. In that case, the sky model will use fixed typical values for all remaining parameters (which can be seen in the output page by enabling the check box "show skymodel details").

The almanac is updated dynamically by a service on the ETC web server, without the need to manually update the web application.

Notes about the algorithms, resources and references for the almanac are available here

### Almanac Usage

Hovering the mouse over an input element in the almanac normally displays a pop-up "tooltip" with a short description.

### Time

The upper left part of the almanac box refers to the date and time of observation.
This can be done with a UT time or a MJD. A date/time picker widget will appear when the UT input field is clicked, but the UT can also be assigned manually. In any case, the UT and MJD fields are dynamically coupled to be mutually consistent.

The two +/- buttons can be used to step forward or backward in time by the indicated step and unit per click. The buttons can be held down to step continuously until released.

The third of night corresponding to the currently selected time is indicated. This is an input parameter to the airglow component in the sky model. Twilight levels (civil, nautical and astronomical) referring to the sun altitude ranges are also indicated in the dynamic text. These levels refer to the sun altitude:

• Astronomical Twilight −18° ≥ alt < −12°
• Nautical Twilight −12° ≥ alt < −6°
• Civil Twilight −6° ≥ alt < 0°

### Target

The target equatorial coordinates RA and dec can be assigned manually in the two input fields or automatically using the SIMBAD resolver to retrieve the coordinates.
If the lookup is successful, an "info" link will open a window in which the raw SIMBAD response can be inspected.
The units can be toggled between decimal degrees and hh:mm:ss [00:00:00 - 23:59:59.999] for RA and dd:mm:ss (or dd mm ss) for dec. A whitespace can be used as separator instead of a colon.

### Output Table

The table dynamically displays the output from the server back-end service, including temporal and spatial coordinates for the target, Moon and Sun. The bold-faced numbers indicate the parameters normally relevant in the phase 1 proposal for optical instruments. The numbers appear in red color if they are out of the range supported by the sky model.

### Visiblity Plot

The chart dynamically shows the altitude and equivalent airmass as function of time for the moon and target, centered on midnight for the currently selected date.
The green line, which refers to the currently selected time, can be dragged left and right to change the time, dynamically coupled with the sections in the Time section.

A more advanced version of the almanac is included in our SkyCalc web application, which provides more input and output options.

# 3. Instrument setup

• The user must select which instrument setting to use. The "extended" mode indicates that the wavelengths 465-480nm will be computed and observed by MUSE, but induce that in the real exposures the 2nd order of the blue will be seen at the red wavelengths, effect which is not taken into account in the ETC.
• The user can choose to obtain the signal to noise ratio which would result from coaddition of the neighbouring spectral and/or spatial pixels. The options of coadding 1*1, 2*2, 3*3, 4*4, 5*5, 10*10, 30*30, 60*60, 100*100 spatial pixels, and/or coadding 1, 2, 3, 4, 5, 10, 20, 30, 40, 50, 100, 200, 400, 800, 1600, 3200 spectral pixels, are available. This does not involve any different readout mode of the CCD, only a software coaddition of the neighbouring pixels once read.
When the user requests the SNR in nspat*nspat spatial pixels:
• the SNR is computed and displayed only in those nspat*nspat spatial pixels;
• the values in the least intense and most intense spatial pixel among those nspat*nspat are stored, in order to check the problem of saturation.
When the user requests the SNR in nspec spectral pixels:
• the SNR is computed and displayed both per spectral pixel and per spectral bin;
• the SNR per spectral bin is computed by convolving the signal and noise terms with a 'sliding' window of width nspec: therefore, the SNR computed still has the same number of values as the SNR computed per spectral pixel: this choice was made to allow the user to see which are the relevant wavelengths to combine together, but of course there is a correlation between the value in one spectral pixel and the value in the neighbouring one, the stronger correlation the larger nspec is.

# 4. Observation setup

• A reference wavelength must be provided:
• on the one hand, the text results will refer to this wavelength, and those results will be displayed on the graphs;
• on the other hand, if the user requests a specific SNR (see below), the Detector Integration Time (DIT) or Number of DIT (NDIT) will be determined at this wavelength, and those DIT and NDIT will be used to compute the SNR for the whole spectral band.
• The wavelength range to be plotted must be provided.
• The user has to provide 2 of the 3 following parameters: SNR (at the reference wavelength), number of exposures (NDIT), exposure time of an individual exposure (DIT).
• if DIT and NDIT are provided, the SNR is computed at every wavelengths;
• if SNR is provided, firstly the DIT or NDIT required at the reference wavelength is computed, and secondly the SNR at every wavelength is computed assuming the required and computed DIT and NDIT.
The ETC does not take into account instrument and telescope overheads.

# 5. Outputs

The ETC provides:
• a display of the formula used to compute the SNR and a summary of the values (in electrons) of the various contributions
• a summary of the selected configuration
• a summary of the computations at the reference wavelength

## 5.1 Graphic outputs

The following spectra can be displayed:
• target, in electrons, both per spectral pixel and bin;
• background source, in electrons, both per spectral pixel and bin;
• target input spectrum in physical unit;
• background source input spectrum in physical unit;
• target ensquared energy ;
• background source ensquared energy;
• sky emission and absorption;
• instrument transmission;
• SNR per spectral pixel and bin;
• saturation checks (see below).
All those spectra are 1) displayed as a .gif file, 2) can be displayed as an ascii file, 3) can be displayed via a java environment. Additionnaly, the value at the reference wavelength is shown on every plot.

## 5.2 Graphic outputs: about the saturation check ("peak pixels values")

• the sky spectrum is the same in every spatial pixel;
• if there was only the target, only the most intense spatial pixels among the coadded ones would have to be checked in order to ensure that the CCD is not saturated.
• but since there is a background source, and since its angular position with respect to the target is not known, two limits (values as a function of wavelength) are computed in the ETC:
• one which considers the worst configuration: the most intense spatial pixel due to the target is overimposed to the most intense spatial pixel due to the background source;
• one which considers the best configuration (target and background source 'far' from each other): the most intense spatial pixel due to the target is overimposed to the least intense spatial pixel due to the background source.
Note that since we want to compute the saturation limit, using the least intense spatial pixel due to the target and the least intense spatial pixel due to the background source would only result in giving the (useless) lowest value the user can expect.
=> if the two curves are below the saturation limit of the CCD, then it is sure that no saturation will happen. If the two curves are above the saturation limit, then it is sure that saturation will happen. If the saturation limit is in between the two curves, then saturation limit might happen, depending on the configuration between the two sources. Note that if there is no background source or if the background source is spatially homogeneous within the coadded spatial pixels, then the two curves will overlap.

## 5.3 .fits outputs

• the spatial PSF at the requested wavelength;
• the unconvolved target and background source;
• the convolved target and background source.
Except for the PSF, the images have a second layer onto which a patch displaying where the observed region is is present.

# Misc

## References

[RD01] "Stellar systems folowing the R1/m luminosity law", L. Ciotti, 1991, A&A 249:99-106
[RD02] "UBVRI Photometry II: the Cousins VRI system", M. S. Bessel 1979, PASP, v91 , No 543, pp589-607
[RD03] "The Sloan Digital Sky Survey Photometric System", M. Fukugita et. al. 1996, The Astronomical Journal, v.11 n.4:1748:1756

# Version Information

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