The Stellar templates for stars of different spectral type/class
SIPGI provides for LUCI flux calibration are listed in the following
Table (credits to ESO
- Pickles, 1998).
a0i.dat
b3v.dat
f6v.dat
k0iv.dat
m1v.dat
rg0v.dat
a0iii.dat
b57v.dat
8i.dat
k0v.dat
m2.5v.dat
rg5iii.dat
a0iv.dat
b5i.dat
f8iv.dat
k1iii.dat
m2i.dat
rg5v.dat
a0v.dat
b5ii.dat
f8v.dat
k1iv.dat
m2iii.dat
rk0iii.dat
a2i.dat
b5iii.dat
g0i.dat
k2i.dat
m2v.dat
rk0v.dat
a2v.dat
b6iv.dat
g0iii.dat
k2iii.dat
m3ii.dat
rk1iii.dat
a3iiisp.dat
b8i.dat
g0iv.dat
k2v.dat
m3iii.dat
rk2iii.dat
a3v.dat
b8v.dat
g0v.dat
k34ii.dat
m3v.dat
rk3iii.dat
a47iv.dat
b9iii.dat
g2i.dat
k3i.dat
m4iii.dat
rk4iii.dat
a5iii.dat
b9v.dat
g2iv.dat
k3iii.dat
m4v.dat
rk5iii.dat
a5v.dat
f02iv.dat
g2v.dat
k3iv.dat
m5iii.dat
wf5v.dat
a7iii.dat
f0i.dat
g5i.dat
k3v.dat
m5v.dat
wf8v.dat
a7v.dat
f0ii.dat
g5ii.dat
k4i.dat
m6iii.dat
wg0v.dat
b0i.dat
f0iii.dat
g5iii.dat
k4iii.dat
m6v.dat
wg5iii.dat
b0v.dat
f0v.dat
g5iv.dat
k4v.dat
m7iii.dat
wg5v.dat
b12iii.dat
f2ii.dat
g5v.dat
k5iii.dat
m8iii.dat
wg8iii.dat
b1v.dat
f2v.dat
g8iii.dat
k7v.dat
o5v.dat
wk1iii.dat
b1i.dat
f2iii.dat
g8i.dat
k5v.dat
m9iii.dat
wk0iii.dat
b2ii.dat
f5i.dat
g8iv.dat
m0iii.dat
o8iii.dat
wk2iii.dat
b2iv.dat
f5iii.dat
g8v.dat
m0v.dat
o9v.dat
wk3iii.dat
b3i.dat
f5iv.dat
k01ii.dat
m10iii.dat
rf6v.dat
wk4iii.dat
b3iii.dat
f5v.dat
k0iii.dat
m1iii.dat
rf8v.dat
The sampling is at 5 Angstrom. The templates are in vacuum. The
conversion from air to vacuum wavelength has been performed by the
PyAstronomy
library using the Edlen (1953) and Ciddor (1996) prescriptions. The
syntax for the filenames is as follows: xxy.dat where xx is the
spectral type in lower case letter (e.g. a0, g5), and y the luminosity
class in roman lower case letters (i,ii, iii, iv, v) (credits to ESO
- Pickles, 1998).
The typical Stellar templatemust contain two columns and have a
constant sampling: wavelength (Angstrom), fluxes per wavelenght unit
(arbitrary units), plus an arbitrary header on top of the file.
The Spectro-photometric standards SIPGI provides for MODS flux
calibration (credits LBTO).
G191-B2B
GD 71
Feige 34
Feige 66
Feige 67
GD 153
Hz 43
Hz 44
BD+332642
BD+284211
Feige 110
The sampling is at 10 Angstrom. The spectra are in vacuum.
Note
G191-B2B is an HST primary white dwarf star; GD71 is an HST
primary white dwarf standard stars. The NIR extension is based on
HST model spectrum; Feige 66 has only relatively low-quality near-IR
extension data, and should only be used in the blue or out to about
9200Å GD153 is an HST primary white dwarf star. The NIR extension is
based on the HST fluxes shifted by -0.02mag. Hz 43 is an HST primary
white dwarf star. The NIR extension is based on the HST model
spectrum. A faint binary M-dwarf companion, Hz43B (V=14.3), is
located 3-arcsec away, so this star is not recommended during poor
seeing. BD+28 4211 has a faint red companion 2.8-arcsec away, and is
only recommended for use in the blue channel in good seeing (credits
LBTO).
The typical Spectro-photometric standard must be composed by three
columns: wavelength (Angstrom), magnitude or fluxes, sampling, plus an
arbitrary header on top of the file.
9.2. Appendix B: keywords necessary to SIPGI to import and categorize raw data
The list of keywords that must be correctly filled in LUCI data for a
successful import of raw frames. Old version indicates the name of
the keyword in old files.
and \(D_{i,j}\) is the values of the (i,j)th pixel of the Master
Dark, (N,M) the Master Dark dimension in [px] and \(\bar{D} =
\frac{1}{NM}\sum_i^N\sum_j^M D_{i,j}\). In this case, the final error of
the pre-reduced frame dark subtracted is:
The error on the reduced frames largely depends on the type of slit,
i.e. distorted or un-distorted, and on the sky subtraction methods.
9.3.2.1. Error for un-distorted slits in reduced frames.
For un-distorted slits, the first operation performed by the Reduce
Observation recipe is the sky subtraction. Starting from the
prereduced frames, the error on each pixel (i,j) of the sky-subtracted
image is:
\[E_{SKYSUB,i,j} = \sqrt{E_{i,j}^2 + E_{SKY}^2}\]
where, here and after, \(E_{i,j}\) is replaced by
\(E_{i,j}^{'}\) if the pre-reduced frames are dark subtracted, and
\(E_{SKY}\) depends on the sky methods. In particular:
sky method median
\[E_{SKY} = \frac{\sqrt{\sum_j E_{i,j}^2}}{N}\]
sky method fit/wfit: in case of weighted fit (wfit), the recipe
estimates the weights as:
\[w_{j}=\frac{1}{E_{i,j}^2},\]
then estimates the weighted fit and computes the chi^2:
where \(\bar{d}_{i,j}\) is the pixel value of the pre-reduced image, and
\(\hat{d}_j\) is the value of the weighted fit at the jth pixel.
The final error is:
\[E_{SKY} = \sqrt{\frac{\chi^2}{N-M-1}},\]
In case of a simple fit \(w_{j}\) = 1.
sky method ABBA:
\[E_{SKY} = E_{i,j}\]
where \(E_{i,j}\) is the pixel error of the frame that is
subtracted.
sky method ABBA + median/fit/wfit: if different sky mehods are
combined, the relative errors are added in quadrature.
After the sky subtraction, the recipe rectifies the spectra with the
wavelength calibration and extract the 2D spectra. The error on the
rectified EXR2D is:
\[E_{EXR2D,i,j} = E_{RESAMPLE,i,j},\]
where \(E_{RESAMPLE,i,j}\) is obtained resampling the error of the
input pixels (for more details see \(r_e(\lambda)\) in Appendix D).
The final step is the extraction of 1D spectra. SIPGI offers two
possibilities for the spectra extraction: the sum and
the Horne extraction. The error associated to the 1D spectra is:
where \(d_{i,j}^{*}\) are the pixel values of the EXR2D image, and
\(\hat{d}_j\) is the value of the weighted fit at the jth pixel.
The final error is:
\[E_{SKY} = \sqrt{\frac{\chi^2}{N-M-1}},\]
In case of a simple fit \(w_{j}\) = 1.
sky method ABBA:
\[E_{SKY} = E_{EXR2D,i,j}\]
where \(E_{EXR2D,i,j}\) is the pixel error of the EXR2D
subtracted image.
sky method ABBA + median/fit/wfit: if different sky methods are
combined, the relative errors are added in quadrature.
For the distorted slits the DAVIES method is also provided for the sky
subtraction but in this version of SIPGI no error propagation is
implemented in this case.
The final step is the extraction of the 1D spectra that follow the
same scheme of un-distorted slits, i.e.:
\[E_{EXR1D,i} = \sum_j E_{SKYSUB,i,j},\]
where the sum is extended to all the jth pixels of the detected
object. In case of a Horne extraction, the error is a weighted sum.
where \(e_{i,j,k}\) is the error of the (i,j)th pixel in the (k)th frame.
9.4. Appendix D: 2D spectra extraction and resampling
The 2D extraction procedure is basically an image warping process and it involves a re-sampling function: the process of transforming the raw spectra from one coordinate system described by the models (OPT, CRV and IDS), to another where the spectra are perfectly rectified and wavelength calibrated.
SIPGI goes along the slices used during the Master Lamp creation (see Fig. 7.5), moving in the coordinates system defined by the 3 models.
For each slice and for each wavelength value of the extraction range, it computes the re-sampled flux value \(r_f(\lambda)\) and error \(r_e(\lambda)\)
of the 2D spectrum.
The recipe computes the re-sampling using a filter derived from exponential functions; the library defines a low-pass filter in the Fourier space, using the \(C^{\infty}\) function defined by:
The parameter \(s\) defines the sharpness of the filter and SIPGI uses \(s=5\). The value 0.5 is the cut-off frequency to conform to the Nyquist rate. The actual 1D re-sampling kernel in the image domain, is obtained computing the inverse Fourier transform of \(H_s(f)\). This gives us the \(h_k(x)\), as shown in Fig. 9.1.
Fig. 9.1 On the left side the \(H_5(f)\) spectrum in the Fourier space. On the right side the \(h_5(x)\) kernel profile in the image domain, obtained by the inverse fast Fourier transform of \(H_5(f)\).
Since the extraction procedure works with 2D images, the 2D re-sampling kernel \(w\) is obtained by combining 2 separate 1D kernels along the 2 orthogonal axes \(x\) and \(y\)
\[w_s(x,y) = h_s(x)\cdot h_s(y).\]
The shape of the 2D re-sampling kernel \(w_5\) is shown in the figure Fig. 9.2.
Fig. 9.2 In this figure is shown the shape of the \(w_5\) 2D kernel with a radius 2 pixels large.
The 2D extraction process, for each slice and each \(\lambda\), follows these steps:
the 3 models are used to find the point where the re-sampling kernel must be applied, i.e. the models get the center of the \(w_5\) kernel (red dot in the Fig. 9.3);
SIPGI collects 16 pixels covered by the the 4x4 pixels box (green pixels in the Fig. 9.3);
Fig. 9.3 The yellow pixels represent the image pixels; the red dot is the kernel center computed by the models; the blue pixels are the 4x4 kernel box.
the re-sampled flux value \(r_f(\lambda)\) is computed by the formula
These steps will be repeated for each re-sampled values and packed into a final spectrum to obtain a wavelength calibrated spectrum corrected by distortions.
9.5. Appendix E: bad pixels correction and cosmic rays cleaning
The bad pixels correction is performed interpolating the good pixels values around the bad pixel.
The same algorithm is used to clean the cosmic rays.
Here the recipe step by step, using as example a cluster of 4 bad pixels (A, B, C and D) illustrated in the Fig. 9.4:
The recipe starts from the bad pixel to correct (B in the example) and it moves along the 4 cardinal directions (North-South, East-West, NorthEast-SouthWest and NorthWest-SouthEast) to find the first good pixels available. The good pixels found in the example are the orange pixels.
For each direction the recipe should found 2 good pixels: one before (\(d_b\)) and one after (\(d_a\)) the bad one.
If both pixels are found the algorithm computes the weighted mean of the two values, using as weights (\(w_b\) and \(w_a\)) the distances of the \(d_b\) and \(d_a\) pixels from the bad pixel.
On the edges of the frame, the weighted mean along some direction could not be computed because some good pixel candidate could be out of frame.
For this reason, we can have less then 4 weighted means. For the example of the pixels B the recipe will found only 3 good \(m_k\) values (along the E-W, NE-SW and NW-SE directions) since the pixel after the bad one in the N-S direction falls out of frame.
The final value of the bad pixel is median of all available \(m_k\) values
Fig. 9.4 An example of the bad pixel treatment. The yellow pixels represent the image pixels. The gray pixels are a cluster of 4 bad pixels (A, B, C, and D). The orange pixels are the good pixels used by the recipe to correct the central bad pixel B.
