# Input Parameters

The model includes an input spectrum (e.g. a template star spectrum), atmospheric parameters , optical instrument path and an observation criterium. The model generates as default the spectral format in a table format reporting for each order number the wavelength of the central column, its y position in pixel units and arcseconds units, the Free Spectral Range (FSR) size, minimum and maximum wavelength, the order starting and ending wavelength and size (Template Spectra range). If required the ETC also calculates for each significative order (in the single line case only for the order where is detected the line) the total efficiency (% units), the Object, Sky and maximum expected counts (in electron units) and the Signal to Noise ratio for three points, the FSR minimum, central column and FSR maximum wavelength. For the wavelength of the central column it is reported also the wavelength value and the spectral bin size (over which the Object, Sky, Imax counts are integrated). The information contained in the spectral format table, relative to the central column wavelength value, can be displayed also in form of graph selecting the appropriate check button in the input page. In this case the ETC generate applet graphs and links to the corresponding data in ASCII format and in gif images format.

• ### Input Flux Distribution

• Blackbody

The target model is a blackbody defined by its temperature and monochromatic apparent magnitude at a given wavelength. Temperature is expected in Kelvin and wavelength in one of the band filters U, B, V, R or I.

• Power Law

The target model is a power law spectrum defined, at the wavelength w, from a continuum flux level Fo, a reference wavelength wc and the power low index p, a floating number, as:

F(w)=Fo*(w/wc)p

where

Fo=10-(0.4*M + Z)

where M is the Object Magnitude and Z is the zero order point in the selected observing band (see table below).

• Continuum

The target model is a spectral distribution constant with the wavelength. This case is a subcase of the Power Law one selecting index p=0.

• Template spectrum

The target model can be defined by a spectral type. As with the blackbody it will be scaled to the provided magnitude and band filter U, B, V, R or I.

• Object Magnitude

Indicates the object magnitude (in the Vega or AB magnitude system) in the broad band filter associated to the filter defined in the Instrument Setup. The spectrum is scaled after integration in the corresponding photometric filter.

Photometric Band

U

B

V

R

I

J

H K

wc

360 440 550 640 790 1250 1650 2160

Z

7.3788 7.1804 7.4425 7.6408 7.9115 8.5058 8.9431 9.4045
• Single Line

In this case the source is a single line of characteristic wavelength l, FWHM, and emitting a selectable Flux (in 10-16 ergs/s/cm2 units).

• ### Spatial Distribution

• Seeing Limited

Seeing limited sources are point-like sources.

• Extended Source

The signal to noise for extended sources is given per wavelength resolution bin on the detector (as indicated in the output table). The magnitude is given per square arcsecond. The detected counts reported on the output table integrates over the solid angle omega determined by the product of PSF (in arcsec) and the slit width (in arcsec).

• ### Sky Conditions

• Sky Brightness

Here you define the sky brightness in mag/arcsec2. The following table (Walker 1987, NOAO Newsletter) gives to the observed sky brightnesses versus moon phase.

Days from
new moon
Sky Brightness

U

B

V

R

I

Z

0

22.0 22.7 21.8 20.9 19.9 18.8

3

21.5 22.4 21.7 20.8 19.9 18.8

7

19.9 21.6 21.4 20.6 19.7 18.6

10

18.5 20.7 20.7 20.3 19.5 18.3

14

17.0 19.5 20.0 19.9 19.2 18.1

For X-SHOOTER, the sky background is reduced to refer to the continuum between emission lines.
The offsets applied to the normal table of night sky brightness are: U:0 mag, B:0 mag, V:0.242 mag, R:0.139 mag, I: 0.633 mag, Z: 1.237 mag.

• Airmass

The airmass at which the object is observed.

• 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 point sources in the ETC.

With the seeing now consistently defined as the atmospheric PSF FWHM outside the telescope at zenith at 500 nm, the IQ FWHM is modeled by the ETC considering the transfer functions of the atmosphere, telescope and instrument, with s=seeing, λ=wavelength, x=airmass and D=telescope diameter:

{ \small \begin{aligned} \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} \\ \end{aligned} } .

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

For point sources and non-AO instrument modes, the atmospheric PSF FWHM with the given seeing $$s$$ (arcsec), airmass $$x$$ and wavelength $${\small \lambda }$$ (nm) is modeled as a gaussian profile with:

 \begin{aligned} & {\small \mathit{FWHM}}_{\text{atm}}(\mathit{s},\mathit{x},\lambda) = \mathit{s} \cdot x^{0.6} \cdot (\lambda/500\text{ nm})^{-0.2} \cdot \sqrt{1+F_{\text{Kolb}} \cdot 2.183 \cdot ({r_0}/L_{0})^{0.356}} \text{,} \\ & \small{\text{where}} \end{aligned} \begin{aligned} {r_0} & = 0.976 \cdot 500.0 \cdot 10^{-9}\text{ nm } / \mathit{s} \cdot (180/\pi \cdot 3600) \cdot (\lambda/500.0\text{ nm})^{1.2} \cdot x^{-0.6} \\ F_{\text{Kolb}} & = 1/(1+300 \cdot D/L_{0})-1 \end{aligned} $${ L_{0} }$$ is the wave-front outer-scale. We have adopted a value of $${ L_{0} }$$=23 m, which is the generally accepted value for Paranal. D is the telecscope diameter in meters. $${r_0}$$ is the Fried parameter at the requested wavelength and airmass. $$F_{\text{Kolb}}$$ is the Kolb factor (ESO Technical Report #12). If the argument of the square root $${\small (1+F_{\text{Kolb}} \cdot 2.183 \cdot ({r_0}/L_{0})^{0.356}) < 0 }$$, which happens when the Fried parameter $${\small {r_0} }$$ reaches its threshold of $${\small r_{\text{t}} = L_{0} \cdot (1.0/2.183)^{1/0.356} }$$, the value of $${\small \mathit{FWHM}}_{\text{atm}}$$ is set to $${\small 0.0 }$$.

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

• ### Instrument Setup

• Mirrors (M1 to M2/M3 Depending on location of Instrument on Telescope). UVES (M1,M2,M3).

The ETC allows the user to set the following:

• Pre Slit Optics: The pre slit filter is selected from a list with an option menu. The ETC allow the following settings: None (no pre-slit filter inserted), ADC (Atmospheric Dispersion Compensator), Depolarizer, Iodine cell.
• Image Slicers:

The UVES instrument can observe with or without inserting image slicers along the optical path. The use of image slicers is important for bad seeing conditions or the highest spectral resolution. The Image Slicers mode is selected from a list with an option menu. The ETC allow the following settings: None (no image slicer inserted), IS#1, IS#2, IS#3. The following table contains the characteristic image slicers data.

 IS IS width (arcsec) IS heigh (arcsec) slit width (arcsec) slit heigh (arcsec) #1 2.0 2.6 0.68 8 #2 1.8 1.9 0.44 8 #3 1.5 1.5 0.30 10

• Slit.

The ETC allow the user to select the slit width. The slit heigh is kept fixed at 10 arcsec. See below to know how the Obj, Sky, Imax and S/N is calculated.

• FLAMES Fiber Feed

Instead of a slit, the FLAMES Medusa fibers can be used to feed light to UVES. The fiber diameter is 1 arcsec, which then serves as the slit. The total sky aperture is 0.785398 arcsec2. ( = pi*(0.5 arcsec)2)

• HARPS

The echelle spectrograph HARPS (La Silla 3.6m) is basically very similar to UVES, but has a simpler set of configurations. The efficiency of Fiber A is factor ~ 1.6 higher than fiber B. For details see this web page. Note that the polarimeters are splitting the light in 2 channels (equally if the star is unpolarized), i.e. for a "lossless" polarimeter circ_A/no_pol_A = lin_A/no_pol_A = 0.5. You can also see page 18 of the user manual. The ETC applies measured polarimeter efficiency factors.

• Observation Mode.

The instrument works in 4 instrument modes: Red Arm, Blue Arm, Dichroic1, and Dichroic2 where the dichroic modes allow the simultaneous exposure of the Red and the Blue Arm. The definition of an 'Observing Mode' requires the selection of the instrument mode, the crossdisperser to be used (Blue Arm: CD1 or CD2, Red Arm: CD3 or CD4), and the central wavelength.
If 'Standard Template' is selected, the predefined central wavelength as given in parentheses (lam_c) in the pull-down menu is used. This wavelength setting corresponds to the wavelength in the Standard Template as provided by the 'Phase 2 Proposal Preparation (P2PP)' tool.
If 'Free Template' is selected, the central wavelength can be set in the wavelength range as given in brackets [lam_0 < lam_c > lam_1] in the pull-down menu for the given instrument mode and crossdisperser. Note, that the suggested instrument mode, crossdisperser, and central wavelength combinations are predefined to allow senseful instrument setups only.

• Below Slit Filters.

The user can insert in the optical path no filter (option None) or set one of the the filters listed in the following table:

Filter Cross Disperser
BBS6-HER5 CD#1 or CD#2
BBS2-BG24 CD#1
RBS1-BG40 CD#3
RBS2-SHP700 CD#3
RBS9-BK7_5 CD#3 or CD#4
RBS3-OG590 CD#4
RBS12_HALPHA CD#3 or CD#4
RBS13_HBETA CD#3
RBS14_OIII5007 CD#3 or CD#4
RBS15_OIII4363 CD#3
RBS16_NII5755 CD#3
RBS17_OI6300 CD#3 or CD#4
RBS18_SII6724 CD#3 or CD#4
RBS19_HeII4686 CD#3

Yellow color highlighted combinations are recommended.

Detailed information on filters, optical components and detectors is available in the relevant instrument user manuals.

# Results

The ETC calculates as default the predicted spectral format. This is presented in a table format. The table reports for each order number the wavelength of the central column, its y position in pix units and arcseconds units, the Free Spectral Range (FSR) size, minimum and maximum wavelength, the order starting and ending wavelength and size: Template Spectra range (TS range).

The FSR is the wavelength range over which two adjacent orders are not overlapping, correspondent to the distance between wavelengths at which the Blaze function is 0.5.

The central wavelength of the FSR is

wc=2*sin(alpha_blaze)/Kech/m

where alpha_blaze is the echelle incidence angle at blaze wavelength, Kech is the echelle constant (grooves/mm) and m is the order number.

The size of the FSR it is approximatively given by

FSR_size=lambda_blaze/m

where lambda_blaze is the blaze wavelength and m the order number.

If required the ETC also calculates for each significative order (in the single line case only for the order where is detected the line) the total efficiency (% units), the Object, Sky and maximum expected counts (in electron units) and the Signal to Noise ratio for three points, the FSR minimum, and maximum and the central column wavelength. For the FSR central wavelength it is reported also the wavelength value and the spectral bin size (over which the Object, Sky, Imax counts are integrated).

To evaluate the total number of counts expected, the ETC use the following "zero order" formula:

For point sources:

N_point=F*D*T*E*S/P For extended sources:

N_extended=F*D*T*E*S*Omega/P

Where

F=Incident Flux (in ergs/s/cm2/A for point sources and ergs/s/cm2/arcsec2/A for extended sources).

D=wavelength resolution bin

T=Exposure time

E=Total efficiency (atmosphere, telescope, optical components, filters, detector, slit losses in case of point sources)

S=Telescope Surface

P=Energy of one photon

Omega=Solid Angle subtended by a rectangle of size equal to the product of the slit width (in arcsecs) and the pre-slit PSF FWHM projected on the sky.

To evaluate the signal to noise ratio the ETC use the following expression:

S/N=N_Obj/sqrt(N_Obj+ S_Sky+ nPixY*n_dark*T/3600+ nBinY*n_RON2)

Where N_Obj and N_Sky are the number of predicted detected counts predicted for the object (using the appropriate expression if point source or extended one) and the sky (extended source).

nPixY is the number of pixels along the Y detector direction equivalent to twice the PSF FWHM (or to the appropriate size if an Image Slicer is inserted).

n_dark is the dark current (1e/pix/h).

T the exposure time in seconds.

nBinY is the number of bins equivalent to nPixY.

n_RON is the read out noise (for the particular chip used in the arm red or blue at the specified read out speed and gain).

The information contained in the spectral format table, relative to the central column wavelength value, can be displayed also in form of graphs selecting the appropriate check button in the input page. In this case the ETC generate applet graphs and links to the corresponding data in ASCII format and in gif images format.

### Version Information

• Version 6.0.1 (July 22, 2015) A bug fix eliminating unexpected behaviour when emission lines were used in instrument modes with multiple detectors. Graphs are disabled in emission line mode. Minor cosmetic changes, results unchanged.

• Version 6.0.0 (February 26, 2015) A unified convention for seeing and image quality applied in all ETCs offered from CfP P96.

• Version 5.0.1 (Aug 27, 2013)
Corrected a bug affecting single lines in the NIR X-SHOOTER arm.
Reformatted the text/layout of target spectrum distribution section in all echelle ETCs.

• 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. Please report significant discrepancies to the ESO user support group usd-help@eso.org

• Version 3.2.13 (December 19, 2011)
X-SHOOTER: Introduced a warning appearing if the NIR detector has entered the extrapolated regime of the non destructive mode.

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