apero_pol_spirou

Contents

1. Description

SHORTNAME: POLAR

Polarimetry

The polarimetry module for APERO was adapted from the spirou-polarimetry. SPIRou as a polarimeter can measure either circular (Stokes V) or linear (Stokes Q or U) polarization in the line profiles. Each polarimetric measurement is performed by 4 exposures obtained with the Fresnel rhombs set at different orientations (see Section 3.1 of Donati et al. 2020).

Index positions of the Fresnel rhombs (RHB1 and RHB2) for exposures taken in each observing mode of SPIRou.

Observing

Exp1

Exp1

Exp2

Exp2

Exp3

Exp3

Exp4

Exp4

mode

RHB1

RHB2

RHB1

RHB2

RHB1

RHB2

RHB1

RHB2

Stokes IU

P16

P2

P16

P14

P4

P2

P4

P14

Stokes IQ

P2

P14

P2

P2

P14

P14

P14

P2

Stokes IV

P14

P16

P2

P16

P2

P4

P14

P4

In the Table above we provide the index position of each Fresnel rhomb, as they appear in the FITS header, for each exposure in the corresponding polarimetric mode.

These indices are used by APERO to recognize exposures within a polarimetric sequence, and then correctly apply the method introduced by Donati et al. 1997 to calculate polarimetric spectra.

The polarization spectra of SPIrou are calculated using the technique introduced by Donati et al. 1997, which is summarized as follows. Let \(f_{i\parallel}\) and \(f_{i\perp}\) be the extracted flux in a given spectral element of fiber A and B channels, where \(i=\{1,2,3,4\}\) gives the exposure number in the polarimetric sequence. Note that the extracted flux can be either the extracted spectrum or the extracted telluric corrected spectrum; by default in APERO, we use the telluric corrected spectrum. The total flux of unpolarized light (Stokes I) is calculated by the sum of fluxes in the two channels and in all exposures, i.e.,

\[F_{I} = \sum_{i=1}^{4}{(f_{i\parallel} + f_{i\perp})}\]

Let us define the ratio of polarized fluxes as

\[r_{i} = \frac{f_{i\parallel}}{f_{i\perp}}\]

which gives a relative measurement of the flux between the two orthogonal polarization states. In an ideal system, :math`r=1` means completely unpolarized light, and other values provide the amount (or the degree) of polarization that can be calculated as in Equation 1 of Donati et al. 1997, i.e.,

\[P = \frac{f_{\parallel} - f_{\perp}}{f_{\parallel} + f_{\perp}} = \frac{r - 1}{r + 1}\]

Therefore, in principle, one could obtain the amount of polarization with a single exposure. However, this measurement is not optimal, since it only records the two states of polarization at the same time but not at the same pixel. To obtain a measurement that records the same state of polarization at the same pixel, it suffices to take a second exposure with one of the quarter-wave analyzers rotated by \(90^{\circ}\) with respect to the first exposure, consisting of the 2-exposure mode. One can also use the 4-exposure (2 pairs) mode, where the polarization state in the two channels is swapped between pairs, which better corrects for slight deviations of retarders from nominal characteristics (retardance and orientation) and also corrects for the differences in transmission between the two channels caused, for example, by different throughput of the two fibers, or by a small optical misalignment. For this reason, SPIRou only operates in the 4-exposure mode, which is accomplished by rotating the analyzers accordingly in each exposure, as detailed in the table above. The equation to calculate the degree of polarization for the 4-exposure mode can be obtained in two different ways, by using the Difference method or by the Ratio method, as defined in sections 3.3 and 3.4 of Bagnulo et al. 2009 and also in Equation 3 of Donati et al. 1997. The degree of polarization for a given Stokes parameter \(X=\{U, Q, V\}\) in the Difference method is calculated by

\[P_{X} = \frac{1}{4}\sum_{k=1}^{2}{\left(\frac{r_{2k-1}-1}{r_{2k-1}+1} - \frac{r_{2k}-1}{r_{2k}+1}\right)}\]

and for the Ratio method the degree of polarization is given by

\[P_{X} = \frac{(\prod_{k=1}^{2}{r_{2k-1}/r_{2k}})^{1/4} - 1}{(\prod_{k=1}^{2}{r_{2k-1}/r_{2k}})^{1/4} + 1}\]

Another advantage of using two pairs of exposures is that one can calculate the null polarization (NULL1 and NULL2) as in equations 20 and 26 of Bagnulo et al. 2009, which provides a way to quantify the amount of spurious polarization. The null polarization for the Difference method is given by

\[NULL_{X} = \frac{1}{4}\sum_{k=1}^{2}{\left[(-1)^{k-1}\left(\frac{r_{2k-1}-1}{r_{2k-1}+1} - \frac{r_{2k}-1}{r_{2k}+1}\right)\right]}\]

and for the Ratio method the null polarization is given by

\[NULL_{X} = \frac{\left(\prod_{k=1}^{2}{r_{2k-1}/r_{2k}}\right)^{\frac{(-1)^{k-1}}{4}} - 1}{\left(\prod_{k=1}^{2}{r_{2k-1}/r_{2k}}\right)^{\frac{(-1)^{k-1}}{4}} + 1}\]

Finally, the uncertainties of polarimetric measurements can be calculated from the extracted fluxes and their uncertainties (denoted here by \(\sigma\)) by equations A3 and A10 of Bagnulo et al. 2009. In the Difference method, the variance for each spectral element is given by

\[\sigma_{X}^{2} = \frac{1}{16} \sum_{i=1}^{4}{ \left\{ \left[ \frac{2 f_{i\parallel} f_{i\perp}}{(f_{i\parallel} + f_{i\perp})^{2}} \right]^{2} \left[ \frac{\sigma_{i\parallel}^{2}}{f_{i\parallel}^{2}} + \frac{\sigma_{i\perp}^{2}}{f_{i\perp}^{2}} \right] \right\} }\]

and in the Ratio method the variance is given in terms of the flux ratio $r_{i}$, i.e.,

\[\sigma_{X}^{2} = \frac{\left( \frac{r_{1}}{r_{2}} \frac{r_{4}}{r_{3}} \right)^{1/2}} { 4 \left[ \left( \frac{r_{1}}{r_{2}} \frac{r_{4}}{r_{3}} \right)^{1/4} + 1\right]^{4}} \sum_{i=1}^{4}{\left[ \frac{\sigma_{i\parallel}^{2}}{f_{i\parallel}^{2}} + \frac{\sigma_{i\perp}^{2}}{f_{i\perp}^{2}} \right]}\]

Applying this formalism to SPIRou spectra, we obtain values that vary continuously throughout the spectrum and are systematically above or below zero for each spectrum, which we refer to here as the `continuum polarization’. For general scientific applications with SPIRou, this continuum polarization is actually spurious as it reflects small differences in the injection between beams, and must therefore be fitted and removed. This step is mandatory before performing measurements in spectral lines. APERO applies an iterative sigma-clip algorithm to fit either a polynomial or a spline to model the continuum polarization.

Least-Squares Deconvolution

The least-squares deconvolution method (LSD) is an efficient technique that combines the signal from thousands of spectral lines retaining the same line profile information to obtain a mean velocity profile for the intensity, polarization, and null spectra. A common application of this technique concerns the measurement of the Zeeman split into Stokes V (circularly polarized) profiles. The Zeeman split is a physical process where electronic transitions occurring in the presence of a magnetic field have their main energy transition level split into two additional levels, forming a double line in the intensity spectrum. An interesting feature of these lines is that they are circularly polarized and their polarizations have opposite signs. Therefore, by observing the circularly polarized spectrum one can obtain a characteristic Stokes V profile that provides a way to detect and characterize the magnetism in stellar photospheres with great sensitivity.

APERO implements the LSD calculations using the formalism introduced by Donati et al. 1997, summarized as follows. Let us first consider the weight of a given spectral line i, \(w_{i} = g_{i} \lambda_{i} d_{i}\), where g is the Landé factor (magnetic sensitivity), \(\lambda\) is the central wavelength, and d is the line depth. Then one can construct the line pattern function

\[M(v)= \sum_{i=1}^{N_{l}}{w\delta(v - v_{i})}\]

where \(N_{l}\) is the number of spectral lines considered in the analysis, \(\delta\) is the Dirac function, and v is the velocity. The transformation from wavelength (\(\lambda\)) to velocity space is performed by the relation \(dv/d\lambda = c / \lambda\), where c is the speed of light.

The LSD profile is calculated by the following matrix equation:

\[\rm{\bf Z} = \left( \rm{\bf M}^{t}.\rm{\bf S}^{2}.\rm{\bf M} \right)^{-1} \rm{\bf M}^{t} . \rm{\bf S}^{2} . \rm{\bf P}\]

where \(\rm{\bf P}\) is the polarimetric spectrum, and \(\rm{\bf S}\) is the covariance matrix, a diagonal matrix where each element in the diagonal is given by \(S_{jj}=1/\sigma_{j}\), with \(\sigma_{j}\) being the uncertainty in the polarimetric spectrum.

Note that one can also calculate the null polarization LSD profile by substituting the polarimetric spectrum \(\rm{\bf P}\) by the null spectrum \(\rm{\bf N}\). The intensity LSD is also possible, by using the flux spectrum \(\rm{\bf F}\), but in this case the line weight is simply given by the line depth, i.e, \(w_{i} = d_{i}\).

In practice, LSD requires a few important steps to be executed by APERO. First, each individual spectrum is cleaned using a sigma-clip rejection algorithm to minimize the impact of outliers in the LSD profile. Then we set a grid of velocities to calculate the LSD profile, where the grid is defined by the following parameters: an initial velocity, \(v_{0}\), a final velocity, \(v_{f}\), and the total number of points in the grid, \(N_{v}\).

Next, a fast and accurate method is necessary to project the spectral values onto the velocity grid. Finally, an appropriate catalog of spectral lines (line mask) needs to be adopted for the LSD calculations. APERO selects the line mask from a repository of masks, where the selection is based on the proximity to the effective temperature of the star observed. The APERO masks are computed using the VALD catalog (Piskunov et al. 1995) and a MARCS model atmosphere (Gustafsson et al. 2008) with an effective temperature ranging from 2500 to 5000 K in steps of 500 K, and the same surface gravity of \(\log g=5.0\) dex. The lines that are effectively used in the LSD analysis are selected with line depths above a given threshold, which is set to 3% by default and with a Lande factor of \(g_{\rm eff}>0\), resulting in a total of approximately 2500 atomic lines that cover the full spectral range of SPIRou.

The LSD analysis is not computed in a standard automated run of APERO but the module is supplied and can be activated with the use of a single keyword in the APERO profiles or run after processing.

2. Schematic

No schematic set

3. Usage

apero_pol_spirou.py {obs_dir}[STRING] {options}

{obs_dir}[STRING] // [STRING] The directory to find the data files in. Most of the time this is organised by nightly observation directory

4. Optional Arguments

--exposures[FILE:EXT_E2DS_FF,TELLU_OBJ] // List of exposures to add (order determined by recipe)
--exp1[FILE:EXT_E2DS_FF,TELLU_OBJ] // Override input exposure 1
--exp2[FILE:EXT_E2DS_FF,TELLU_OBJ] // Override input exposure 2
--exp3[FILE:EXT_E2DS_FF,TELLU_OBJ] // Override input exposure 3
--exp4[FILE:EXT_E2DS_FF,TELLU_OBJ] // Override input exposure 4
--objrv[FLOAT] // Object radial velocity [km/s]
--lsdmask[STRING] // LSD mask
--output[STRING] // Output file
--output_lsd[STRING] // Output LSD file
--lsd // Run LSD analysis
--noqccheck // Do not check quality control of inputs
--blazefile[FILE:FF_BLAZE] // [STRING] Define a custom file to use for blaze correction. If unset uses closest file from calibDB. Checks for an absolute path and then checks 'directory' (CALIBDB=BADPIX)
--plot[0>INT>4] // [INTEGER] Plot level. 0 = off, 1 = interactively, 2 = save to file
--wavefile[FILE:WAVESOL_REF,WAVE_NIGHT,WAVESOL_DEFAULT] // [STRING] Define a custom file to use for the wave solution. If unset uses closest file from header or calibDB (depending on setup). Checks for an absolute path and then checks 'directory'
--no_in_qc // Disable checking the quality control of input files

5. Special Arguments

--xhelp[STRING] // Extended help menu (with all advanced arguments)
--debug[STRING] // Activates debug mode (Advanced mode [INTEGER] value must be an integer greater than 0, setting the debug level)
--listing[STRING] // Lists the night name directories in the input directory if used without a 'directory' argument or lists the files in the given 'directory' (if defined). Only lists up to 15 files/directories
--listingall[STRING] // Lists ALL the night name directories in the input directory if used without a 'directory' argument or lists the files in the given 'directory' (if defined)
--version[STRING] // Displays the current version of this recipe.
--info[STRING] // Displays the short version of the help menu
--program[STRING] // [STRING] The name of the program to display and use (mostly for logging purpose) log becomes date | {THIS STRING} | Message
--recipe_kind[STRING] // [STRING] The recipe kind for this recipe run (normally only used in apero_processing.py)
--parallel[STRING] // [BOOL] If True this is a run in parellel - disable some features (normally only used in apero_processing.py)
--shortname[STRING] // [STRING] Set a shortname for a recipe to distinguish it from other runs - this is mainly for use with apero processing but will appear in the log database
--idebug[STRING] // [BOOLEAN] If True always returns to ipython (or python) at end (via ipdb or pdb)
--ref[STRING] // If set then recipe is a reference recipe (e.g. reference recipes write to calibration database as reference calibrations)
--crunfile[STRING] // Set a run file to override default arguments
--quiet[STRING] // Run recipe without start up text
--nosave // Do not save any outputs (debug/information run). Note some recipes require other recipesto be run. Only use --nosave after previous recipe runs have been run successfully at least once.
--force_indir[STRING] // [STRING] Force the default input directory (Normally set by recipe)
--force_outdir[STRING] // [STRING] Force the default output directory (Normally set by recipe)

6. Output directory

DRS_DATA_REDUC // Default: "red" directory

7. Output files

Outputs

name

description

HDR[DRSOUTID]

file type

suffix

input file

POL_DEG

Polarimetry 2D degree of polarisation file

POL_DEG

.fits

_pol

EXT_E2DS_FF, TELLU_OBJ

NULL_POL1

2D Null polarisation 1 file

NULL_POL1

.fits

_null1_pol

EXT_E2DS_FF, TELLU_OBJ

NULL_POL2

2D Null polarisation 2 file

NULL_POL2

.fits

_null2_pol

EXT_E2DS_FF, TELLU_OBJ

STOKESI_POL

Polarimetry 2D stokes I polarisation file

STOKESI_POL

.fits

_StokesI

EXT_E2DS_FF, TELLU_OBJ

LSD_POL

Least squares deconvolution file

LSD_POL

.fits

_lsd_pol

EXT_E2DS_FF, TELLU_OBJ

POL_CALIB

Polarimetry 2D shifted wavelength solution and blaze calibration file

POL_CALIB

.fits

_pol_calib

EXT_E2DS_FF, TELLU_OBJ

S1DW_POL

Polarimetry 2D degree of polarisation file (constant wavelength binning)

S1DW_POL

.fits

_s1d_w_pol

EXT_E2DS_FF, TELLU_OBJ

S1DV_POL

Polarimetry 2D degree of polarisation file (constant velocity binning)

S1DV_POL

.fits

_s1d_v_pol

EXT_E2DS_FF, TELLU_OBJ

S1DW_NULL1

1D Null polarisation 1 file (constant wavelength binning)

S1DW_NULL1

.fits

_s1d_w_null1

EXT_E2DS_FF, TELLU_OBJ

S1DV_NULL1

1D Null polarisation 1 file (constant velocity binning)

S1DV_NULL1

.fits

_s1d_v_null1

EXT_E2DS_FF, TELLU_OBJ

S1DW_NULL2

1D Null polarisation 2 file (constant wavelength binning)

S1DW_NULL1

.fits

_s1d_w_null2

EXT_E2DS_FF, TELLU_OBJ

S1DV_NULL2

1D Null polarisation 2 file (constant velocity binning)

S1DV_NULL2

.fits

_s1d_v_null2

EXT_E2DS_FF, TELLU_OBJ

S1DW_STOKESI

Polarimetry 1D stokes I polarisation file (constant wavelength binning)

S1DW_STOKESI

.fits

_s1d_w_stokesi

EXT_E2DS_FF, TELLU_OBJ

S1DV_STOKESI

Polarimetry 1D stokes I polarisation file (constant velocity binning)

S1DV_STOKESI

.fits

_s1d_v_stokesi

EXT_E2DS_FF, TELLU_OBJ

8. Debug plots

POLAR_FIT_CONT
POLAR_CONTINUUM
POLAR_RESULTS
POLAR_STOKES_I
POLAR_LSD
EXTRACT_S1D_WEIGHT
EXTRACT_S1D

9. Summary plots

SUM_EXTRACT_S1D