Echelle Spectroscopy ETCs 


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.
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.
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).
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.
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.
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 
In this case the source is a single line of characteristic wavelength l, FWHM, and emitting a selectable Flux (in 10^{16} ergs/s/cm^{2} units).
Seeing limited sources are pointlike sources.
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).
Here you define the sky brightness in mag/arcsec^{2}. 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 XSHOOTER, 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.
The airmass at which the object is observed.
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 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:
\( { \small \begin{equation} \begin{aligned} \mathit{FWHM}_{\text{IQ}} & = \sqrt{\mathit{FWHM}_{\text{atm}}^2(\mathit{s},x,\lambda)+\mathit{FWHM}_{\text{tel}}^2(\mathit{D},\lambda)+\mathit{FWHM}_{\text{ins}}^2} \\ \end{aligned} \end{equation} } \) .
For fibrefed instruments, the instrument transfer function is not applied. For nonAO 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:
\(
\begin{equation}
\begin{aligned}
& {\small \mathit{FWHM}}_{\text{atm}}(\mathit{s},\mathit{x},\lambda) = \mathit{s} \cdot x^{0.6} \cdot (\lambda/500\text{ nm})^{0.2} \cdot \sqrt{1+F_{\text{Kolb}} \cdot 2.183 \cdot ({r_0}/L_{0})^{0.356}} \text{,} \\
& \small{\text{where}}
\end{aligned}
\end{equation}
\) \( \begin{equation} \begin{aligned} {r_0} & = 0.976 \cdot 500.0 \cdot 10^{9}\text{ nm } / \mathit{s} \cdot (180/\pi \cdot 3600) \cdot (\lambda/500.0\text{ nm})^{1.2} \cdot x^{0.6} \\ F_{\text{Kolb}} & = 1/(1+300 \cdot D/L_{0})1 \end{aligned} \end{equation} \) \({ L_{0} }\) is the wavefront outerscale. 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 \({\small 0.0 }\). 
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
The ETC allows the user to set the following:
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 arcsec^{2}. ( = 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 pulldown 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 pulldown 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.
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 

BBS6HER5  CD#1 or CD#2 
BBS2BG24  CD#1 
RBS1BG40  CD#3 
RBS2SHP700  CD#3 
RBS9BK7_5  CD#3 or CD#4 
RBS3OG590  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.
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/cm^{2}/A for point sources and ergs/s/cm^{2}/arcsec^{2}/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 preslit 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_RON^{2})
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.
