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
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.
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 ordersorting 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 4482 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.
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.
S
= count rate from the source (e / s).
B
= count rate from the sky (e / s).
ff
= flatfield accuracy.
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.
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.
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.
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.
The two options are again pointlike or extended sources, but their meaning is
different from MOS.
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
userdefined spectral energy distribution. This part of the input form is the
same for each observing mode.
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.
For
pointlike sources the user must provide:
 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.
For
extended sources the user must provide:
 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:
 Aperture
magnitude (mag) they want to reach.
 Aperture
radius (arcsecs).
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:
 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.
 Surface
brightness distribution.
 Projected
semimajor 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 semimajor 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 semimajor 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:
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 longexposure 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 nonAO 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 gaussapproximated transfer functions of the atmosphere, telescope and
instrument, with s=seeing, λ=wavelength, x=airmass and D=telescope diameter:
Image Quality
\(
{
\begin{equation}
\mathit{FWHM}_{\text{IQ}} = \sqrt{\mathit{FWHM}_{\text{atm}}^2(\mathit{s},x,\lambda)+\mathit{FWHM}_{\text{tel}}^2(\mathit{D},\lambda)+\mathit{FWHM}_{\text{ins}}^2(\lambda)}
\end{equation}
}
\)

For fibrefed 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:
\(
\begin{equation}
\begin{aligned}
\mathit{FWHM}_{\text{tel}} & = 1.028 \frac{\lambda}{D} \text{, } & \text{ with } \lambda \text{ and D in the same unit}\\
& = 0.000212 \frac{\lambda}{D} \text{arcsec, } & \text{ with } \lambda \text{ in nm and D in m}.
\end{aligned}
\end{equation}
\)

For point sources and nonAO 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 wavefront outerscale. 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
AOmodes, a model of the AOcorrected PSF is used instead.
Seeing statistics:
The Paranal seeing statistics is based on the socalled UT seeing measurements obtained from the UT1 Cassegrain ShackHartmann 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/genfac/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:
 Filter.
 Detector
read mode slow.
 Binning.
3.4.2 MOS
SPECTROSCOPY
Requested
parameters:
 Grism
name.
 Slit
dimensions (length = spatial direction, width = dispersion direction) in arcsec.
 Detector
read mode slow.
 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:
 Grism
name.
 IFU
spatial sampling factor.
 Detector
read mode slow.
 Binning.
3.5 OBSERVATION
This
part of the input form is common to all the observing modes.
Here
the user can choose between:
 Given the S/N (intended as the one per pixel at grism central
wavelength for the spectroscopic case), compute the requested Exposure Time.
 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 flatfield 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 flatfield
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 zeropoint 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 photoelectrons/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.
LensSurface
= π * [ (Mirror_Radius)
^{
2
}
: (Centr_Obstr_Radius)
^{
2
}
]

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 photoelectrons/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 photoelectrons/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:
A_{fib}
= (spatial_sampling)
^{2}

Number of fibers covering the source:
N_{fib}
= (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 twodimensional 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 4482 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
twodimensional 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 userselected 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
 Plate
Scale of the CCD.
 Sky
Background per pixel = signal from the sky in the given (or evaluated) exposure
time, integrated over 1 pixel.
 Detector
parameters: readout noise, dark current, saturation level.
 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
 Number of pixels in the PSF area ImageQualityFWHM
^{2}
arcsecs: Source signal in the PSF area.
 Peak
pixel value (source + sky).
 S/N
at PSF central pixel.
Extended sources
 Source
signal per pixel.
 Peak
pixel value (source + sky).
Extended sources  Integral Photometry
 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.
 Source
counts in the aperture area (object only).
6.1.2 SPECTROSCOPY
General
 Central
wavelength and dispersion of the selected grism, as computed by the ETC.
 Plate
scale of the instrument CCD.
 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).
 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.
 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***".
 Detector
parameters: readout noise, dark current, saturation level.
 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:
 Slit
losses = fraction of light coming from the source that is lost due to the
selected ImageQualityFWHM and slit width.
 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.
 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:
 Total
object signal at central wavelength = signal from the source, summed over (2*
radius) / (Plate scale) pixels and 1 dispersing element.
 Object
signal at central wavelength = source signal over 1 pixel at the center of the
spatial profile and 1 dispersing element.
 S/N
at central wavelength = signal to noise over 1 pixel (i.e. not over 2*radius in
the spatial direction)
 Source:
intensity factor (spatial summing) ): see Section
4.3.5.4.
 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:
 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).
 Number
of fibers fully covering the source: see Section
4.3.6.1.
 Fiber
diameter in pixels.
Extended
sources:
 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.

Fiber diameter projection onto the detector in pixels.
6.2 GRAPHICAL
OUTPUT
Possible
graphical outputs are:
 Resultant source spectrum as a function of lambda (1dimensional).
 Sky spectrum: same as source (spectroscopy only).
 Resultant spectrum: sum of source + sky spectrum (spectroscopy only).
 Total system efficiency (for source, i.e. including atmospheric extinction).
 S/N as a function of lambda, i.e. S/N along the spectrum (spectroscopy).
 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.
 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.