# VIMOS Exposure Time Calculator

VIMOS Exposure Time Calculator

User Manual

## Title page

 Prepared By: A. Zanichelli Approved by: O. Le Fevre Revised by: B. Garilli

# Version Information

• Version 6.2.0 (August 30, 2016)
The new sky model has been updated with adjusted ozone and CO2 abundances and minor changes in the moonlight model.
The outer-scale parameter L0 in the seeing/IQ model was changed. This has changed the size of the S/N reference area and consequently many numerical results, in particular at long wavelengths.

• Version 6.1.2 (July 4, 2016)
Imaging: The new advanced sky model has been enabled in the imaging ETC.
Spectroscopy: The new sky model was already enabled in the previous version.

• Version 6.1.1 (February 29, 2016)
Bug fix for VIMOS IFU/MOS Spectroscopy: The atmospheric extinction was overestimated (applied twice) in the previous version.

• Version 6.0.0 (February 26, 2015) A unified convention for seeing and image quality applied in all ETCs offered from CfP P96.
Spectroscopy ETC: A new advanced sky model was enabled.

• Version 5.0.0 (July 1, 2013)
All ETC version numbers have been aligned at 5.0.0 as the ETC system infrastructure was re-factored and installed on a new web server. Due to implementation changes, some numerical results may differ slightly from previous versions. Please report significant discrepancies to usd-help@eso.org

# INTRODUCTION

This document is intended to provide information and help for the VIMOS Exposure Time Calculator, which can be operated to simulate observations both in imaging and in spectroscopic (MOS and IFU) modes.
The general features of VIMOS ETC as well as the basic formulae used to compute S/N or Exposure Time are described in Section 2, while a description of input parameters and their meaning is given in Section 3. The computations performed by the code are listed in Sections 4 and 5. A description of ETC output is given in Section 6.

Some general conventions used in this document:
• Magnitude for extended sources is always intended in mag/arcsec2 (except for extended source integral photometry where it is the magnitude over the aperture ).
• Pointlike sources are approximated with a gaussian of FWHM = ImageQualityFWHM. We assume that the total flux from the source falls inside a circle of radius = ImageQualityFWHM.
• The terms "source" or "object" are equivalent: they identify the astronomical target of the observations.

Some conventions specific for spectroscopy:
• MOS and IFU: S/N is always intended at the central grism wavelength.
• MOS and IFU: X direction = spatial direction (= along the slit = slitLength).
• MOS and IFU: Y direction = dispersion direction (= across the slit = slitWidth).
• MOS: Extended sources: we assume the major axis always lies along the slit.
• MOS: Extended sources: for De Vaucouleurs and Exponential profile, the parameter "Projected Semi-Major Axis" is intended as the effective radius in arcseconds, along the spatial direction.
• MOS: Extended sources: if you give S/N and ask for Exposure Time, remind that this S/N is intended as what you want to measure over 2*(projected semi-major axis) (see Section 2).
• IFU: when we speak about no. of fibers fully covering the source, we refer to the input microlens array.

# 2. ETC GENERAL FEATURES

The ETC works by simulating four main "components" of an observation: the source, the atmosphere, the telescope, and the instrument.
Spectral energy distribution, redshift, magnitude, and surface brightness profile characterize the simulated source. A sky spectrum, magnitude, and extinction as a function of wavelength, airmass, ImageQualityFWHM, describe the atmosphere.
A spectral energy distribution is described with a bag of optical rays, sampling the spectrum in lambda. Each optical ray is thus characterized by wavelength, intensity, and two directional angles (Alpha, Gamma). These angles are referred respectively to Y and X axis of the optical system.
The simulated telescope is characterized by its effective area, and transmission as a function of wavelength.
The simulated instrument is characterized by:
1. Slit or slit+IFU for spectroscopy.
2. Grism + order sorting filter for spectroscopy, or filters if in imaging mode.
3. Lens.
4. Detector.

A filter is described by a transmission curve as a function of wavelength. A grism is described by its angular dispersion, central wavelength and transmission curve. A further transmission curve is used to describe the order-sorting filter. The lens component describes the sum of all the lenses present in the instrument, by means of system focal length (telescope + instrument) and an "overall" (i.e. due to all the lenses + folding mirror of the instrument) transmission curve.
Grism and lens are the two components, which perform geometrical transformations of the coordinates of the input optical ray. That is, if the ray enters the instrument with directional angles inAlpha=0, inGamma=0, along Y and X respectively, after grism and lens these angles will be different from 0 and the light is dispersed onto the CCD.
The detector (CCD) is characterized by its responsive quantum efficiency, gain, readout noise, dark current, saturation, plate scale.

For what concerns the instrument optical components, with the exception of the IFU lenses optical transmission and the Intermediate resolution grism, numbers/curves used for this version of the ETC are measured values, stored in the ETC Calibration Database. For what concerns the CCD, we used values listed in the EEV 44-82 CCD Test Report.
In the following subsections we briefly describe how the ETC computes the S/N or Exposure Time in the three observing modes.

## 2.1 BASIC FORMULAE

Here we report the formulae used for S/N or Exposure Time calculations. For details on how these formulae are applied to the different source geometries and observing modes, refer to the next Sections.

1) Given the S/N, evaluate the Exposure Time:
Evaluate the time for 1 exposure:

2) Given the Exposure time, evaluate the S/N:
Evaluate S/N for 1 exposure of t seconds:

Where:
S = count rate from the source (e- / s).
B = count rate from the sky (e- / s).
ff = flat-field accuracy.
F = ff 2(S + B) 2
NDC = detector noise due to dark current (e- / s).
RON 2 = (ron 2 ) * n_pix
ron = detector readout noise in e-
n_pix = number of pixels. See next Sections for different meanings depending on observing mode.
t = exposure time in seconds.
S/N = Signal to Noise Ratio.

## 2.2 IMAGING ETC

For what concerns S/N or Exposure Time calculations in imaging mode, three options are given: pointlike sources, extended sources and integral photometry. For pointlike sources, the S/N is evaluated over the PSF area. For extended sources, the S/N is computed per pixel and the source surface brightness is assumed to be uniform. The third option, called "Integral Photometry", evaluates the S/N over the selected aperture area.

## 2.3 SPECTROSCOPIC ETC

The S/N or Exposure Time you get from the spectroscopic ETC is NOT over one resolution element along dispersion, BUT over just 1 pixel, that is a dispersion element.

### 2.3.1 MOS

The ETC for MOS spectroscopy supports pointlike and extended sources. Extended sources can be characterized by a uniform surface brightness, or by a De Vaucouleurs or Exponential law.
Some VERY IMPORTANT points to be kept in mind by the user are:

Pointlike sources : given the exposure time, the evaluate S/N is the one you would get over 1 pixel in the dispersion direction and npsf pixels in the spatial direction, where npsf is the number of pixels in (ImageQualityFWHM) arcsecs.

• This means that: if you give a S/N and you want the correspondent exposure time, the ETC interprets the S/N as the one obtained taking into account signal, sky and detector noise over npsf pixels.

• You can always evaluate also the S/N over 1 dispersion element (Y) * 1 pixel (X) by using Intsat output parameter (see Section 4.3.7)

Extended sources:given the exposure time, the evaluated S/N is the one you would get over 1 pixel in the dispersion direction and n_ext pixels (in the X direction). Where: n_ext is the number of pixels inside 2*projected semi-major axis (= 2*r_eff for De Vaucouleurs or Exponential profiles, see Sec. 3.2)

• This means that: if you give a S/N and you want the exposure time requested to reach it, the ETC interprets the S/N as the one obtained taking into account signal, sky,... over n_ext pixels. However, also the S/N AT THE CENTRAL PIXEL in the spatial profile is given in the numerical output.

### 2.3.2 IFU

The two options are again pointlike or extended sources, but their meaning is different from MOS.

Pointlike sources: here we consider integral spectroscopy, i.e. the signal from all fibers covering the source is summed to obtain a single spectrum. The number of fibers on the source, nfib, is computed as the ratio between the source area and the area of 1 microlens. This number is an approximation, as it is not possible to take into account the actual microlens geometrical disposition on the source.

Extended sources: this option is intended as bidimensional spectroscopy, that is we consider the spectrum from just 1 fiber. The S/N is thus computed summing signal, sky, etc. over fib_pix pixels.

• The resolution element in the spatial direction is set by the fiber diameter, i.e. the number of pixels in 1 fiber diameter, fib_pix, is computed and the sky signal (in e-/sec/pixel) is summed over (nfib * fib_pix) pixels.

# 3. ETC INPUT

In this section a description of the input parameters you are requested to provide in the ETC input form is reported. More details on how they are used in the computations are found in Section 4. The ETC input form is divided in various parts, some of them different when in imaging or in spectroscopic observing modes.

## 3.1 INPUT SPECTRUM

Here you can select the spectral energy distribution of the source you want to observe. The options are: flat spectrum, blackbody, template SED from list, or user-defined spectral energy distribution. This part of the input form is the same for each observing mode.

Flat spectrum: constant flux at each lambda.

BlackBody: if you select this option, you must provide the temperature in K.

A template SED. The flux scale is of no interest, as the SED will be scaled to the desired magnitude.
Template SEDs from list do not include intrinsic galactic absorption.

In this section of the input form you can also set the source reshift. Redshifting is applied to template distributions .

## 3.2 SPATIAL DISTRIBUTION

Here you can choose source geometry. Different input parameters are foreseen in the three observing modes.
Surface brightness profiles are currently allowed only for MOS spectroscopy.

### 3.2.1 IMAGING

The three possibilities are: pointlike (seeing limited) sources, extended sources, and integral photometry.

POINTLIKE SOURCES:
For pointlike sources the user must provide:
1. Total magnitude in one of the associated broad band filters. It is assumed that the total light coming from the source falls inside a circle of radius = ImageQualityFWHM.
S/N : Exposure Time are computed over the whole PSF area.

EXTENDED SOURCES:
For extended sources the user must provide:
1. Mean surface brightness in mag/arcsecs 2 again in one of the associated filters.
S/N : Exposure Time are computed over one pixel.

EXTENDED SOURCES : INTEGRAL PHOTOMETRY:
The user must provide:
1. Aperture magnitude (mag) they want to reach.
S/N : Exposure Time are computed over the aperture area.

### 3.2.2 MOS SPECTROSCOPY

Here the two possibilities are: pointlike or extended sources. Different input parameters are expected in the two cases.

POINTLIKE SOURCES:
Requested parameters are the same as for pointlike sources in the imaging case.

EXTENDED SOURCES:
Requested parameters are:
1. Magnitude (mag/arcsecs 2) in one of the associated broad band filters. For De Vaucouleurs and Exponential profiles, this is intended as the central surface brightness.
2. Surface brightness distribution.
3. Projected semi-major axis.
Three surface brightness distributions are provided: uniform, i.e. a constant surface brightness; De Vaucouleurs profile to represent elliptical galaxies; Exponential law to represent spirals.
Values of surface brightness and effective radius / projected semi-major axis are intended outside the atmosphere, i.e. before applying ImageQualityFWHM PSF convolution and atmospheric extinction.
It is assumed that the source major axis lies along the slit, and the profile is computed only along the spatial direction (i.e. slit length).
De Vaucouleurs and Exponential profiles:
In both cases, the given magnitude is intended as the CENTRAL surface brightness, and the projected semi-major axis is intended as the effective radius along the slit. These profiles are normalized to 1 at the central pixel and are then convoluted with the PSF before computations.
Subsampling of pixels is applied in the computation of the spatial profile.
Uniform surface brightness profile:
The intensity is set to 1 for (:projected radius) < x < (+projected radius) and to zero elsewhere.
Also this profile is convoluted with the PSF, so to smooth the distribution at the edges.

NOTE: no cosmology is applied to the surface brightness profiles, i.e. what you give is what you observe, irrespective of source redshift. We have chosen this approach because we think it is handier for the user to scale surface brightness and radius with his preferred cosmology, than obliging him/her to use our idea of universe.

### 3.2.3 IFU SPECTROSCOPY

Source geometries for IFU spectroscopy: pointlike or extended. Parameters have the same meaning as the ones in the corresponding case of the imaging mode. See Section 4.3.6 and following for a description of the meaning of source geometries in IFU spectroscopy (different from MOS).

## 3.3 ATMOSPHERE

This part of the input form is common to all the observing modes.
Requested parameters:
• Airmass.
• Seeing.

### Sky Conditions

#### Sky Model

Since ETC version 6.0.0, the sky background radiance and transmission model is based on the Cerro Paranal advanced sky model.

#### Seeing and Image Quality

Since version 6.0.0, 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.

Seeing statistics:

The Paranal seeing statistics is based on the so-called UT seeing measurements obtained from the UT1 Cassegrain Shack-Hartmann wavefront sensor used for active optics.

The measurements are deconvolved in order to represent the seeing outside the dome (i.e. they are corrected for the instrument+telescope resolution).

The La Silla seeing statistics is based on the DIMM FWHM measurements corrected for the instrumental resolution.

These data come from http://www.eso.org/gen-fac/pubs/astclim/paranal/seeing/singcumul.html

#### Airmass

The airmass of the observed target. The airmass must be ≥ 1

#### Moon Phase

Number of days from New Moon.

## 3.4 OPTICAL PATH

### 3.4.1 IMAGING

Requested parameters:
1. Filter.
3. Binning.

### 3.4.2 MOS SPECTROSCOPY

Requested parameters:
1. Grism name.
2. Slit dimensions (length = spatial direction, width = dispersion direction) in arcsec.
4. Binning.

The various order sorting filters are automatically loaded once you have selected the grism.
Choosing a long slit requires more computational time for the source profile (which is computed along the whole slit), especially for extended sources, for which both the PSF and the surface brightness profiles must be evaluated, and convolution is applied.

### 3.4.3 IFU SPECTROSCOPY

Requested parameters:
1. Grism name.
2. IFU spatial sampling factor.
4. Binning.

## 3.5 OBSERVATION

This part of the input form is common to all the observing modes.
Here the user can choose between:
1. Given the S/N (intended as the one per pixel at grism central wavelength for the spectroscopic case), compute the requested Exposure Time.
2. Given the Exposure Time, compute the S/N (intended as the S/N per pixel one would reach at grism central wavelength for the spectroscopic case).

Other parameters are:
3) Fractional flat-field accuracy.This is currently set to 0.0 until empirical values are known
4) Optional: S/N or Exposure Time range.

Note that for spectroscopy the reference wavelength is always the grism central wavelength.
Ranges for exposure time and S/N are used to evaluate S/N as a function of exposure time: see the graphical outputs below. If you leave the fractional flat-field accuracy to zero, this term is not taken into account in S/N computations (see the formulae in Section 2.1).

# 4. ETC COMPUTATIONS

In this section we describe the various computations made by the ETC. The first step, common both to imaging and spectroscopic modes, consists in initializing the model. Different computations performed by the various observing modes are described in Section 4.2 and following.

## 4.1 INITIALIZING THE MODEL

a) Instrument setup: starting from your choice of instrument setup, a "simulated" instrument is built with the proper optical components.

c) Source: according to the user choice, a source spectrum is built. Redshift the spectrum (only for User Defined and From List SEDs).

d) Scale source template spectrum accordingly to magnitude

e) Scale sky spectrum accordingly to sky brightness. This is done by scaling a template spectra of the sky (including emission lines) to a certain magnitude.

Maths:
1) Redshifting (optional).
For each lambda in the input spectrum do:

LambdaNew = (LambdaOld * Redshift) + LambdaOld
IntensityNew = IntensityOld / (1 + Redshift)

Where LambdaOld, IntensityOld are rest frame.

2) Magnitude scaling. Once selected the band and zero-point f do:

logFlux = -0.4 * Magnitude + Log 10 (f)
Flux = 10 (logFlux)

Then, scale the source spectrum so to have this flux at the filter central lambda. The same operation is done for the sky spectrum.

NOTE: because of some problems in the WucSpectralTarget class that handles the input flux distributions, except for Flat Spectrum currently the scaling is done at the filter central lambda; integration and normalization under the filter "profile" is not implemented yet.

## 4.2 IMAGING MODE: TRANSFORMING THE MODEL

### 4.2.1 System efficiency

a) Create system efficiency for source: atmosphere + telescope + instrument
b) Create system efficiency for sky: telescope + instrument

Instrument overall efficiency is evaluated as a function of lambda. Its intensity is the combination of the efficiencies of each single component and its wavelength range is the resultant bandpass of the instrument for that observational setup.

Maths:
1) A tmospheric extinction (source only).
Atmospheric extinction is applied at each lambda in the source spectrum, by means of the formula:

Intensity(Lambda) = Intensity(Lambda) * 10 (-0.4 * Extinction(lambda) * Airmass)

Components acting (also) like filters are: telescope mirror, photometric filters, lenses, detector (QE). For each component acting like a filter, do:

Efficiency(Lambda) = Efficiency(Lambda) * Table(Lambda)

Where, for each component, "Table" is its tabulated transmission. Tabulated values are read and transformed to a continuous function 1D, to interpolate missing values.

### 4.2.2 Going to Counts: Effective Area and Detector

a) Conversion from erg/sec/cm 2/A to photo-electrons/sec is performed at each lambda in the spectrum.
b) Wavelength integration is done over the sky and source spectrum.
c) Pointlike sources and integral photometry: source and sky signals are multiplied by the telescope effective area. Sky signal is multiplied by the number of pixels in a PSF area as well.
d) Extended sources: source and sky signals are multiplied by the telescope effective area and the pixel size.

Units are now: e-/sec for pointlike sources and integral photometry options.
e-/sec/pixel for extended sources.
Source and sky counts are then used to apply the formulae in Section 2.1 for S/N and Exposure Time calculation.

Maths:
1) Conversion to photons. At each wavelength in the spectrum do:

Intensity(Lambda) = Intensity(Lambda) * Lambda / (h * c)

Where:
c = 2.9979 x 10 18 [A/s]
h = 6.6262 x 10 -27 [erg s]

2) Effective area and Detector.

- Pointlike sources:
source: Intensity = Intensity * LensSurface
sky: Intensity = Intensity * LensSurface * arcsecs 2 * npsf
Where npsf = (π * ImageQualityFWHM 2 ) / (arcsecs 2 )

- Extended sources:
source and sky: Intensity = Intensity * LensSurface * arcsecs 2

- Integral photometry:
source: Intensity = Intensity * LensSurface
sky: Intensity = Intensity * LensSurface * arcsecs 2 * n_aper
Where n_aper = (π * aperture_radius 2 ) / arcsecs 2

## 4.3 SPECTROSCOPIC MODE: TRANSFORMING THE MODEL

In this Section we describe the computations performed by the spectroscopic ETC. Unless explicitly specified in Section headings, the following steps are applied both to IFU and MOS modes.
Starting from the source and sky spectra normalized to observed magnitude, the first two steps are the evaluation of the overall system efficiency and of the dispersion relation.

### 4.3.1 System efficiency

See Section 4.2.1. Here the components acting like filters are: telescope mirror, grism and order sorting filter, lens, detector (QE).
For IFU spectroscopy, the microlens+fiber etc. etc. are considered as well. When the user selects IFU high resolution (spatial sampling 0.3 arcsecs per fiber), the focal elongator transmission is also included.

### 4.3.2 Dispersion relation

This step evaluates the geometrical transformations, i.e. grism and lens. These transformations act only on directional angles and not on intensity. Look at each "photon" in the spectrum as an optical ray entering the system with directional angles inAlpha =0; inGamma = 0 (Alpha = atan (Y), Gamma = atan (X).
Gamma angle (X direction) is not affected by dispersion/geometrical transformations in this model.

a) Determine dispersion relation for the given grism: its angular dispersion and lambda_c are used to determine the number of nm/pixels at each pixel on the detector. It is a linear relation.

Maths:
1) Grism. The grism is treated as a dispersing element.
Each optical ray enters the system with (inAlpha=0,inGamma=0) directional angles.
For each lambda in the spectrum, do:

outAlpha = inAlpha + ((Lambda - Lambda_c) * mDisp)
(outGamma = inGamma)

Where Lambda_c = grism central wavelength and mDisp is the grism angular dispersion.
Angular dispersion for each grism is evaluated as:

A = 1 / (f camera * P)

Where P is the reciprocal linear dispersion and f camera the Camera focal length.

2) Lens. The optical ray now enters the lens with inAlpha ≠ 0.
For each lambda do:

y = tan(inAlpha) * fy
outAlpha = atan(y)
(outGamma = inGamma)

Where fy is the focal length of the instrument, given by:

fy = (pixel size [cm] ) / (plate scale [radians] )

### 4.3.3 Apply efficiency and dispersion relation to source and sky spectra

System efficiency and dispersion relation are applied to the source spectrum: we obtain a new spectrum, with now intensity expressed as a function of pixel. The same holds true for the sky spectrum.

Maths:
1) Transformed intensity.
From the dispersion relation we know skyProjection = dispersion at central pixel, i.e. the number of Angstroms per pixel at the central position (which : given the dispersing characteristics of this model - corresponds to the grism central lambda)
At each pixel yp along dispersion direction do:

yp = pixel_size * (mw - Lambda_c) / skyProjection
factor = (Efficiency(mw)) * dispRelation(yp)
Intensity(yp) = Intensity(Lambda) * factor

Where yp is the pixel coordinate, pixel_size is the linear size of the pixel, mw is the wavelength corresponding to the pixel yp (due to dispersion), Lambda_c is the grism central wavelength.
The dispersing element is defined as the range in lambda covered by one pixel at Lambda_c, thus this range is affected by the pixel size which, in turns, is affected by the detector binning.
By means of the term "factor" we integrate the signal over one dispersing element.

### 4.3.4 Slit PSF

We need now to build the slit PSFs for source and sky: these are used for convolution of source and sky spectra along the dispersion direction. Refer to these as the PSFs along the dispersion direction (= slit PSFs). Later we will define the PSFs along the spatial direction for spectroscopy.

a) Source: a gaussian profile with radius = ImageQualityFWHM is defined from (- slitWidth/2) to (+ slitWidth/2) and its area is normalized to unity.

b) Sky: a "flat" profile is built from (- slitWidth/2) to (+ slitWidth/2). The transmission of this profile is the same at each position, and the area is again normalized to unity.

c) Convolution of the spectra with their associated slit PSFs is done

For IFU, the slit width is set equal to the IFU pseudoslit size along the dispersion direction, which is about 1".

Maths:
1) Slit PSF for source and sky.
For each sampling interval i across the slit do:

x(i) = slit_start + arcsecs * i

Define the gaussian sigma: s_sigma = ImageQualityFWHM / 2.35482
PSF_source(i) = [ exp(- x 2 / (2 * s_sigma 2 )) / (sqrt(2 * π * s_sigma) ] / norm1

PSF_sky(i) = [1 / (2 * n_sample + 1) ] / norm2

Where n_sample is the number of sampling intervals across slitWidth, norm1 and norm2 are the area normalization factors, and arcsecs is the plate scale (0.205").

### 4.3.5 MOS: going to Counts

In the following subsections we describe the computations necessary to evaluate source and sky counts in MOS observing mode.

#### 4.3.5.1 Slit losses : Pointlike sources only

Slit losses are computed by integrating the slit PSF over the slitWidth.

Maths:
1) Slit transmission.

slit_trans = erf [ (slitWidth / 2) / sqrt(2.) * s_sigma ];

The erf function comes from Math Library. The output parameter "Slit Losses" is (1 - slit_trans).

#### 4.3.5.2 Effective Area and Detector

a) Conversion from erg/sec to photo-electrons/sec is performed.
b) Pointlike sources: counts coming from the source are integrated over the telescope effective area and multiplied by slit losses.
c) Extended sources: counts coming from the source are integrated over the telescope effective area, over the solid angle of the source (slitWidth), and over pixel size in arcsec. We are assuming that in the dispersion direction the source dimension is greater or equal to the slitWidth.

Maths:
1) Conversion to photons. At each pixel yp do:

Intensity(yp) = Intensity(yp) * mw /(h * c)

Where mw is the lambda corresponding to pixel yp, and c and h are defined in Section 4.2.2.

2) Integration.

- Pointlike sources . At each pixel do:
Intensity = Intensity * LensSurface * slit_trans

- Extended sources . At each pixel do:
Intensity = Intensity * LensSurface * slitWidth * arcsecs

Where lensSurface is same as in Section 4.2.2.

At this point, we have: source and sky spectra as seen through VIMOS. These are 1D spectra, with lambda sampled at each pixel along the CCD.
Units: (e-/sec) for pointlike sources;
(e-/sec/pixel) for extended sources.
In MOS spectroscopic case, before evaluating S/N we need a further step in the spatial direction.

#### 4.3.5.3 Spatial Direction

Pointlike sources
The flux we see at each lambda in the spectrum is the TOTAL one (we assume the total light inside a circle of radius ImageQualityFWHM). The "spatial" PSF (i.e. along X direction) is computed along the slit, and its area is normalized to unity. This is necessary not only to generate the simulated image, but especially because - as the sky is given as [e-/sec pixel] - we need to know how many pixels are inside the 2*ImageQualityFWHM length, to compute sky signal inside a PSF diameter. The computations for spatial PSF are similar to those for the slit PSF, except that now we use slitLength instead of slitWidth.

Extended sources
Hereafter: SBP = surface brightness profile.
a) Take the SBP curve, symmetric around the central pixel and sampled at each pixel along the slit, i.e. define it each 0.205" from (- slitLength / 2) to (+ slitLength / 2). The slit center coincides with the profile center.
b) Apply pixel subsampling (we use subsampling factor = 11).
c) Normalize the SBP so that its intensity over the central pixel is =1.
d) Take the spatial PSF and subsample it.
e) Convolution of the SBP with the PSF along the slit.
f) Rebin the SBP so to have again 1 sampling per pixel.

Now we have the final spatial profile: its value at the central pixel is no more equal to 1, due to convolution.
Here we determine also the output parameter "Intensity factor (spatial summing)" and the central pixel intensity factor that will be used to evaluate S/N over the central pixel. See below for details.

NOTE: convolution is applied also to uniform brightness profile, in order to smooth the SBP at its edges along the slit.

Maths:
1) Profiles.
For each x position along the slit do:

De Vaucouleurs: Intensity[i] = exp( (-7.67) * [ (x[i]) / r_eff) 0.25 ]

Exponential: Intensity[i] = exp ( -1.6783 * [ (x[i]) / r_eff) ]

Uniform: if (x[i] >= -r_eff or x[i] <= r_eff) Intensity[i] = 1.
If (x[i] < -r_eff or x[i] > r_eff) Intensity[i] = 0.

Where r_eff is the effective radius in arcsecs. The number of points, i, is defined by slitLength and subsampling factor.

#### 4.3.5.4 Counts

Pointlike sources
As already said, all the flux we see at each lambda in the spectrum comes from the TOTAL source (we assume the total light inside a circle of radius ImageQualityFWHM), i.e. we already have source counts. The sky counts (in e-/sec/pixels) are multiplied by the number of pixels in a PSF diameter, to get the total sky contribution.

Extended sources
Source and sky signal must be summed over 2*r_eff along the slit.
a) The SBP is the observed (i.e. transformed by the instrument + atmosphere etc. etc.) spatial intensity distribution of the source. If we sum the profile values from -r_eff to +r_eff, we know the fraction of flux falling inside (-r_eff, +r_eff): this number is the "Source: Intensity factor" parameter in the numerical output for extended sources.
Having the source spectrum, which is still normalized to the central surface brightness outside atmosphere, if we multiply the spectral intensities at each lambda by this "intensity factor", we have the (e-/sec) source counts integrated over (-r_eff, +r_eff) along the slit.
Sky counts are multiplied by the number of pixels in (-r_eff, +r_eff). This number is referred as " Sky n. of pixels (spatial summing )" in the numerical output.

### 4.3.6 IFU: going to Counts

In the following subsections we describe the computations that are done to evaluate source and sky counts in the IFU observing mode. In IFU observing mode, source and sky counts are computed in a different way with respect to MOS mode.
After entering the IFU head, each source is seen as a pointlike one, and what "dominates" from the IFU head onward is the fiber shape. What enters the spectrograph is the total light collected by one microlens (extended sources) or by all the microlenses covering the source (pointlike sources) and each source is "seen" as a pointlike one.
When dealing with extended sources and sky, the signal entering the IFU head is first integrated over a fiber area and no more referred to square arcsecs. For pointlike sources, we again assume the total light inside a circle of radius ImageQualityFWHM and compute the number of fibers receiving signal from the source. Both for sky and source, the flux at each lambda in the spectrum is the TOTAL one coming from 1 fiber.

#### 4.3.6.1 Effective Area and Detector

a) Conversion from erg/sec to photo-electrons/sec is performed.
We consider the microlens area as a square one.
b) Pointlike sources: counts coming from the source are integrated over the telescope effective area. It is assumed that the signal from all the fibers covering the source is summed to obtain a single spectrum. The sky signal (mag/square arcsec) is first integrated over the area of 1 microlens, and then multiplied by the number of fibers covering the pointlike source.
c) Extended sources: counts coming from the source are integrated over the telescope effective area and over 1 fiber area. The same is done for sky counts: in this case N fib is defaulted to 1.

Maths:
1) Conversion to photons. At each pixel yp do:

Intensity(yp) = Intensity(yp) * mw /(h * c)

Where mw is the lambda corresponding to pixel yp, and c and h are defined in Section 4.2.2.

2) Integration.

- Fiber area:
Afib = (spatial_sampling) 2

- Number of fibers covering the source:
Nfib = (PI * (seing) 2 ) / A fib

Where the IFU spatial sampling can be 0.33" or 0.66".

- Pointlike sources . At each pixel do:
source: Intensity = Intensity * LensSurface
sky: Intensity = Intensity * LensSurface * A fib * N fib

- Extended sources . At each pixel do
source: Intensity = Intensity * LensSurface * A fib
sky: Intensity = Intensity * LensSurface * A fib

Where LensSurface is as in Section 4.2.2.

At this point, we have: source and sky spectra as seen through VIMOS. These are 1D spectra, with lambda sampled at each pixel along the CCD.
Units: (e-/sec) both for pointlike and extended sources.

#### 4.3.6.2 Spatial Direction

As mentioned above, after having entered the IFU head, we loose track of the source morphology, that is the signal is a total one (integrated over a fiber area when needed) and no more referred to square arcsecs. The flux we see at each lambda in the spectrum is the TOTAL one (we assume the total light inside a circle of radius ImageQualityFWHM). The "spatial" PSF (i.e. along X direction) is computed along the slit, and its area is normalized to unity. This is necessary to generate the simulated image. The computations for spatial PSF are similar to those for the MOS slit PSF, except that now slitLength is fixed by the characteristics of IFU slits. Each fiber "produces" on the IFU mask a pseudoslit whose dimensions (slitWidth and slitLength) are equivalent to about 1". Thus, CCD for one IFU pseudoslit we fixed the number of pixels in the spatial direction on the to be 5.

### 4.3.7 S/N and Exposure Time computations

Evaluation of S/N or Exposure Time is done by means of the formulae in Section 2.1.
As already said, for MOS extended sources the ETC output gives both the S/N over (2 * r_eff) and the S/N over the central pixel at central lambda. For pointlikes is possible to recover the S/N over the central pixel at central lambda as well. This is done computing source signal at central pixel by means of the parameter "Max. intensity at central wavelength (source+sky)" . This number is the total number of electrons falling at the central pixel and it is computed to check saturation.

Maths:

3) The total signal at central pixel (source + sky) for pointlike sources is evaluated as:

IntSat = sourceSignal(centralPixel) * erf[ 2.35482*0.5 / (psf * sqrt(2)) ]+
+ skySignal(centralPixel) )

Where psf is the number of pixels corresponding to the ImageQualityFWHM value, and centralPixel is the one at the slit center = SBP center.

# 5. IMAGE SIMULATION

This tool generates a simulated image of the source, as it would be seen when observed with VIMOS. In direct imaging the image size is set to (10 * ImageQualityFWHM)/arcsecs pixels; in MOS and IFU the image size is equal to 4096 pixels in Y and to slitLength arcsecs in X (plus 10 overscan pixels at both edges).

## 5.1 DIRECT IMAGING

a) 2D simulation in direct imaging is provided for pointlike sources only.
b) A two-dimensional gaussian is generated, with FWHM equal to the ImageQualityFWHM, and subsampling of pixels is applied both in X and Y dimension to better approximate the flux at each pixel. The area below the gaussian is normalized to unity and scaled to the total signal coming from the source.
c) At each pixel, sky signal is summed.
d) A bias level of 237ADU * 1.86(e-/ADU) is added to the 2D image. This numerical value is from a bias frame obtained with the EEV 44-82 CCD provisionally assigned to VIMOS.
e) Noise terms are added on the resultant image, and FITS file is written.

## 5.2 MOS AND IFU SPECTROSCOPY

a) The resultant source and sky spectra from the ETC are transformed to a two-dimensional one by applying, along the spatial direction: The size along spatial direction is set by slitLength plus an overscan region of 10 pixels on both sides.
b) MOS: a PSF profile for pointlike sources or a user-selected profile for extended ones (see Section. 4.3.5.3) are used. A flat profile (normalized to unity inside the slit size and to zero outside) for sky.
c) IFU: sky and source signal are summed and a PSF profile is applied, since what rules the spatial shape is the fiber profile (see Section 4.3.6).

NOTE: currently the ESO library routine that handles the CCD is not able to read the proper value of CCD gain (gain is set = 1). The intensity on the images is thus in e- and not in ADU.

## 5.3 NOISE GENERATION

Noise terms are added to the simulated image in the following way:
Poissonian noise is added for (sky+source) at each pixel, by randomly generating a Poissonian deviate with mean equal to (sky+source).
Gaussian readout noise is also added at each pixel by randomly generating a gaussian deviate of zero mean, which is multiplied by the gaussian readout noise.

NOTE: noise pattern repeats from one frame to the other, so it is not recommended to run the ETC many times with the aim to sum many exposures in order to reproduce a true long observation.

# 6. ETC OUTPUT

The ETC output consists of a summary of input parameters, some numerical outputs giving basically the requested "numbers" at the reference lambda and, optionally, the graphs/images you selected in the input form.

## 6.1 NUMERICAL OUTPUT

After a brief summary of your input parameters, the ETC shows you the computational results in the form of some numbers. All the quantities relevant for the computations are printed out. For what concerns spectroscopy, all these values are referred to the grism central wavelength.

### 6.1.1 IMAGING

General
1. Plate Scale of the CCD.
2. Sky Background per pixel = signal from the sky in the given (or evaluated) exposure time, integrated over 1 pixel.
3. Detector parameters: read-out noise, dark current, saturation level.
4. Exposure time / Signal to Noise as computed starting from your inputs. See Section 2.2 to check how S/N is computed for different source geometries.

Pointlike sources
1. Number of pixels in the PSF area ImageQualityFWHM 2 arcsecs: Source signal in the PSF area.
2. Peak pixel value (source + sky).
3. S/N at PSF central pixel.

Extended sources
1. Source signal per pixel.
2. Peak pixel value (source + sky).

Extended sources - Integral Photometry
1. Number of pixels in the aperture area = number of pixels in pi * (aperture_radius 2): the sky signal per pixel is multiplied by this factor when computing the S/N or Exposure Time.
2. Source counts in the aperture area (object only).

### 6.1.2 SPECTROSCOPY

General
1. Central wavelength and dispersion of the selected grism, as computed by the ETC.
2. Plate scale of the instrument CCD.
3. Total efficiency at central lambda: two values are given, the first one including atmospheric extinction (for the source), and the second one excluding atmospheric extinction (for the sky).
4. Sky background level at central wavelength = signal from the sky in the given (or evaluated) exposure time. This number is computed by integrating counts over a dispersion element along Y and over 1 pixel along X.
5. Maximum intensity at central wavelength: this is the (source + sky) signal detected over 1 resolution element and 1 pixel along X. If this value is greater than the saturation level, is will be marked with " ***SATURATION***".
6. Detector parameters: read-out noise, dark current, saturation level.
7. Exposure time / Signal to Noise as computed starting from your inputs and referred to grism central wavelength. See Section 2.3 to check how S/N is computed for different source geometries and spectroscopic modes.

MOS specific
Pointlike sources:
1. Slit losses = fraction of light coming from the source that is lost due to the selected ImageQualityFWHM and slit width.
2. Total object signal at central wavelength = signal from the source in the given (or evaluated) exposure time. This number is computed by integrating counts over a dispersion element along Y and over (ImageQualityFWHM) / (Plate scale) pixels along the slit.
3. PSF extension = number of pixels in 2*FWHM of the image_quality profile: the sky signal is multiplied by this factor when computing the S/N or Exposure Time.

Extended sources:
1. Total object signal at central wavelength = signal from the source, summed over (2* radius) / (Plate scale) pixels and 1 dispersing element.
2. Object signal at central wavelength = source signal over 1 pixel at the center of the spatial profile and 1 dispersing element.
3. S/N at central wavelength = signal to noise over 1 pixel (i.e. not over 2*radius in the spatial direction)
4. Source: intensity factor (spatial summing) ): see Section 4.3.5.4.
5. Sky: no. of pixels (spatial summing) = number of pixels in 2*radius arcseconds. Sky signal is summed over this number of pixels for the evaluation of exposure time or S/N.

IFU specific
Pointlike sources:
1. Total object signal at central wavelength = signal from the source in the given (or evaluated) exposure time. This number is computed by integrating counts over a dispersion element along Y and refers to the total signal summed over nfibs fibers (see Section 4.3.6.1).
2. Number of fibers fully covering the source: see Section 4.3.6.1.
3. Fiber diameter in pixels.

Extended sources:
1. Object signal at central wavelength = signal from the source in the given (or evaluated) exposure time. This number is computed by integrating counts from 1 fiber over a dispersion element along Y.
2. Fiber diameter projection onto the detector in pixels.

## 6.2 GRAPHICAL OUTPUT

Possible graphical outputs are:
1. Resultant source spectrum as a function of lambda (1-dimensional).
2. Sky spectrum: same as source (spectroscopy only).
3. Resultant spectrum: sum of source + sky spectrum (spectroscopy only).
4. Total system efficiency (for source, i.e. including atmospheric extinction).
5. S/N as a function of lambda, i.e. S/N along the spectrum (spectroscopy).
6. S/N as a function of exposure time. To get this graph, you must specify the desired range of exposure time or S/N in the input form.
7. S/N as a function of ImageQualityFWHM (pointlike sources in imaging mode only).

The graphs are interactive Java Applets. A link to a summary of the available Java commands is available in the output page.
Simulated images in FITS format are available as well. For the spectroscopic modes, three images are generated: a "true" source + sky image, with noise terms added, plus two images (one for sky and one for source) with no noise terms added.
In direct imaging, simulated images are generated for pointlike sources only.

YOU ARE STRONGLY RECOMMENDED TO MAKE EXTENSIVE USE OF THE GRAPHICAL OPTION N. 5 WHEN USING THE ETC IN SPECTROSCOPIC MODE:
The ETC computes the S/N or Exposure Time at the grism central wavelength. It is possible that the selected source SED has absorption or emission features at that lambda, or that one feature lies at lambda_c due to the redshifting of the spectrum (or some sky feature...). Moreover, as already said, the current sky spectral template has strong absorption features that sensibly affect the resultant S/N spectrum when using, for instance, the Low Resolution Red grism.
This can cause an overestimate/underestimate of the S/N or Exposure Time: in such cases, the numerical output does not represent the "mean" S/N or Exposure Time for that observation and can be misleading.

 Send comments and questions to usd-help@eso.org