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hyperrefSuppressing link with empty target on input line 178.

11institutetext: Astronomical Institute, Slovak Academy of Sciences, Tatranská
Lomnica, Slovakia
11email<EMAIL_ADDRESS><EMAIL_ADDRESS>22institutetext: Leibniz-Institute for
Astrophysics Potsdam, Germany
33institutetext: Instituto de Astrofísica de Canarias (IAC), Vía Láctea s/n,
38205 La Laguna, Tenerife, Spain
44institutetext: Departamento de Astrofísica, Universidad de La Laguna, 38205,
La Laguna, Tenerife, Spain
55institutetext: Max-Planck-Institut für Sonnensystemforschung, Justus-von-
Liebig-Weg 3, 37077 Göttingen, Germany
66institutetext: Institut für Sonnenphysik (KIS), Schöneckstr. 6, 79104
Freiburg, Germany

# The dependence of the magnetism of a near-limb sunspot on height

M. Benko 11 H. Balthasar 22 P. Gömöry 11 C. Kuckein 334455 and S.J. González
Manrique 11 6 6

(Received November 13, 2023; accepted November 14, 2023)

###### Abstract

Context. The physical parameters of the sunspot are not fully understood,
especially the height dependence of the magnetic field. So far, it is also an
open question as to which heights the He i 10830 Å spectral line is formed at.

Aims. Our aim is to investigate the magnetic and dynamical properties in the
atmosphere above a sunspot, from the photosphere to the chromosphere. We
analyzed the photospheric and chromospheric magnetic field properties of a
stable sunspot in AR 12553 on June 20, 2016 using spectropolarimetric
observations obtained with the GREGOR Infrared Spectrograph (GRIS) at the
1.5-meter GREGOR telescope.

Methods. A spectral-line inversion technique was used to infer the magnetic
field vector and Doppler velocities from the full Stokes profiles. In total,
three spectral lines were inverted and the variation of the magnetic
properties was qualified using the average values of the radial circles. The
sunspot is located close to the solar limb, and thus this allows us to make a
geometrical determination of the height of the spectral line He i 10830 Å.

Results. We find the height of helium spectral line to be 970 km above the
photospheric spectral lines directly from observation at a stable sunspot. The
total magnetic field strength decreases with height over the sunspot; the
rates are $-0.34\,\rm{G}\,\rm{km}^{-1}$ for the umbra and
$-0.28\,\rm{G}\,\rm{km}^{-1}$ for the penumbra. The inclination increases with
increasing height in the umbra, but decreases in the penumbra. In the umbra,
the vertical component ($B_{z}$) decreases with height, while the horizontal
component ($B_{hor}$) remains almost constant. In the penumbra this is
reversed, as $B_{z}$ remains nearly constant over height, while $B_{hor}$
decreases. We also observe fast velocities with 30 $\rm{km\,s}^{-1}$ in small
chromospheric patches on the central side of the spot.

Conclusions.  The key parameters depending on height in the sunspot are the
$B_{z}$ component of the magnetic field for the umbra and the $B_{hor}$
component of the magnetic field for the penumbra. The observation revealed
supersonic downward velocities in and near the outer penumbra.

###### Key Words.:

Sun: sunspot – Sun: photosphere – Sun: chromosphere – Sun: activity – Methods:
observational – Methods: data analysis – Techniques: high angular resolution

##  1 Introduction

The magnetic field is the principal driver of physical processes in the solar
atmosphere, with its strength and vector being the main aspect that
characterizes active regions (ARs). The most dominant magnetic structures in
ARs are sunspots (Hale 1908).
The photospheric appearance of a sunspot is characterized by a dark umbral
core surrounded by a filamentary penumbra. Typically, the magnetic field
strength and inclination of a sunspot are studied via spectral lines that have
a high sensitivity (a large Landé factor, $g_{eff}$) to the presence of a
magnetic field. The magnetic field of sunspots is typically analyzed by means
of inversion codes. A large number of studies have investigated the global
properties of the magnetic field in sunspots (for an overview see the reviews
by Solanki (2003) and Borrero & Ichimoto (2011), and for detailed results see
Westendorp Plaza et al. (1998); Moran et al. (2000); Bellot Rubio et al.
(2004); Sánchez Cuberes et al. (2005); Orozco Suarez et al. (2005); Balthasar
& Gömöry (2008); Jurčák (2011); Balthasar et al. (2013); Bommier (2013);
Socas-Navarro et al. (2015); Tiwari et al. (2015); Joshi et al. (2017); Benko
et al. (2018, 2021), most of which focus on photospheric lines). The magnetic
field of the umbra is stronger and more vertical than the penumbra. Sunspots
are visible structures in both layers, the photosphere and the chromosphere
(Carlsson et al. 2019). Investigations of the chromospheric magnetic field
have also been performed and often focused on the He I infra-red triplet
around 10830 Å (Merenda et al. 2011; Schad et al. 2015; Xu et al. 2012; Joshi
et al. 2016).

Figure 1: Overview of the sunspot NOAA 12553 in the intensity of the spectral
continuum, re-assembled from the scan. The x and y axes are given in
arcseconds. The green arrow indicates the direction to the solar disk center.
The blue arrow is pointing north. The highlighted red rectangle denotes the
specific area chosen to calculate the height of spectral lines, as is
described in Section 3.2. The purple ring limits the area near the
Stokes-$V$-zero line chosen to calculate average magnetic field parameters, as
is described in Section 4.1.

The magnetic field in sunspots is weaker in the chromosphere than in the
photosphere. Despite the thorough investigation of sunspots, several unsolved
questions remain about these phenomena. One of them is the height dependence
of their magnetic field (see the review of Balthasar (2018)). Sunspots are in
balance with the atmosphere around them due to the pressure of the stratified
gas. The magnetic field in sunspots has its own pressure and reduces the inner
gas pressure. To maintain the hydrostatic equilibrium, the magnetic field
strength has to decrease with height. For this gradient, a controversy arises
between studies using various spectral lines and those employing the divB = 0
condition. Multiple spectral lines are formed at different heights and deliver
the height dependence of the magnetic field.
On the other hand, the divergence of $B$ has to be zero, and the vertical
gradient has to balance the horizontal ones, which can be obtained from two-
dimensional maps. Deriving the difference of the total magnetic field
strengths from two or more spectral lines, values vary between 1.0 to 4.0
$\rm{G\,km}^{-1}$. Following $\rm{div}B=0$, the values vary between 0.2 and
1.0 $\rm{G\,km}^{-1}$ for different sunspots and spectral lines.
To determine these gradients, it is important to get the formation heights of
the spectral lines. For the photosphere, typically absorption contribution
functions (ACFs) (Grossmann-Doerth 1994; Balthasar & Gömöry 2008) or response
functions (Ruiz Cobo & del Toro Iniesta 1992) are calculated, based either on
a model atmosphere or the inversion of observations. For chromospheric lines
such as He i 10830 Å, formation heights are not well known, especially when a
magnetic field affects them. For a horizontally extended atmosphere, Avrett et
al. (1994) determined heights up to 2000 km for the helium line. However, the
formation height can change a lot if magnetic fields come into the game
(Libbrecht et al. 2021; Felipe et al. 2023). So far, it remains an open
question as to which heights above the sun the helium line is formed at.
In the chromosphere above ARs, fast downflows occur, which exceed the local
sound speed (Penn & Kuhn 1995; Schmidt et al. 2000; Lagg et al. 2004).
Supersonic downflows in ARs remain a subject of incomplete comprehension,
although they were observed many times in solar arch filament systems (AFS,
Bruzek 1967, 1969; Lagg et al. 2007; González Manrique et al. 2018, 2020). A
sunspot is likely accompanied by an arch filament system (Sowmya et al. 2022)
and these supersonic velocities are consistently observed close to the
footpoints (e.g., Schmidt et al. 2000; Lagg et al. 2007; Schad et al. 2011;
González Manrique et al. 2018; Yadav et al. 2019; González Manrique et al.
2020). Various interpretations exist for these supersonic downflows. One of
the explanations is that they are a consequence of magnetic flux emergence
(González Manrique et al. 2018); another suggestion involves siphon flow
attributed to differences in the magnetic field (e.g., Rueedi et al. 1992)
between the footprints.
This study investigates the line formation of a He i 10830 Å by using dipole
approximation and the height dependence of the magnetic field in a sunspot
situated in NOAA 12553. Additionally, our focus in this paper is to notice
observed supersonic downflows at the sunspot’s edge in He i 10830 Å. The paper
is organized as follows: Observations and data analysis are outlined in
Section 2 and Section 3 respectively. The results are presented in Section 4
and discussed in Section 5.

Figure 2:  Example of synthesized and observed Stokes profiles $I$, $Q$, $U$
and $V$, along with the corresponding response functions to the magnetic field
for a) calcium and b) silicon lines. The calcium intensities
($I/I_{\mathrm{c}}$) are clipped between 0.4 to 0.9. The log $\tau$ scale is
valid for the response functions. The colored crosses mark the locations in
the continuum map.

##  2 Observations

The sunspot, which was part of NOAA 12553, was observed on June 20, 2016 at
$x=$ 831$\aas@@fstack{\prime\prime}$1 and $y=$
-232$\aas@@fstack{\prime\prime}$3, corresponding to a heliocentric angle of
$54^{\circ}$ ($\mu=0.58$). The observation was acquired with the GREGOR
Infrared Spectrograph (GRIS; Collados et al. 2012) which is mounted at the
1.5-meter GREGOR solar telescope (Denker et al. 2012; Schmidt et al. 2012).
The AR was observed in spectropolarimetric mode, and the GREGOR polarimetric
calibration unit (Hofmann et al. 2012) was used to carry out the polarimetric
calibration of the data.
We recorded the spectral lines Si i 10827 Å (upper photosphere), He i 10830 Å
(upper chromosphere), and Ca i 10839 Å (deeper photosphere). The telescope
resolution element is 0$\aas@@fstack{\prime\prime}$18 at the used wavelength
range under ideal conditions. The slit size is
$70\text{\,}\mathrm{\SIUnitSymbolMicro m}$, corresponding to
0$\aas@@fstack{\prime\prime}$25\. We recorded the Stokes parameters of the
spectral lines above, between 10:57 UT and 11:23 UT. The observed sunspot was
sequentially scanned. The slit orientation (horizontal in Fig 1) was
perpendicular to the nearby solar limb. The scan consists of 340 slit steps
perpendicular to the slit. The step size of the scan was
0$\aas@@fstack{\prime\prime}$134, with ten accumulations with single exposure
times of $100\text{\,}\mathrm{ms}$. The slit length was 477 pixels of a size
of 0$\aas@@fstack{\prime\prime}$136\. The field of view (FOV) of the acquired
data (shown in Fig. 1) has a size of 64$\aas@@fstack{\prime\prime}$82 along
the slit and 45$\aas@@fstack{\prime\prime}$56 in the stepping direction. The
spectral sampling is 18 mÅ per pixel. According to the open data policy, the
observation can be freely downloaded from the GRIS data
archive.111https://sdc.leibniz-kis.de
The silicon line and the calcium line are both sensitive to the magnetic field
(Zeeman triplets), with an effective Landé factor of $g_{eff}=1.5$. The He i
10830 Å line consists of three spectral lines. The first component appears at
10 829.09 Å with $g_{eff}=2.0$, and is called the blue component. The second
component of He i appears at 10 830.25 Å with $g_{eff}=1.75$, and the third
component at 10 830.34 Å with $g_{eff}=1.25$ is overlapping with the second
one, causing them to produce the partial Paschen-Back effect. The latter two
components together are called the red component of the helium line.

Figure 3: Schematic sketch of the dipole field in the sunspot magnetic field.
The geometry allows one to estimate the formation height of the measured
Stokes-$V$ parameter of the spectral lines in the sunspot regions. The line-
of-sight vector is characterized by angle $\Theta$. $\Delta x_{1}$ and $\Delta
x_{2}$ are the distances of the Stokes-$V$ zero line between neighboring
spectral lines. $\Delta h_{1}$ and $\Delta h_{2}$ are the real height of the
spectral lines. Figure 4: Morphology of the magnetic Stokes-$V$ zero line
displayed on the intensity continuum. The individual colors represent values
of Stokes-$V$ for Ca i 10839 Å, Si i 10827 Å and He i 10830 Å. The graph shows
the calculated height difference for silicon and calcium (orange), and for
helium and silicon (blue).

##  3 Data analysis

In this section, we summarize the data processing performed on observations
from the GREGOR telescope. The process includes photometric correction, dark
current subtraction, flat-fielding, polarimetric calibration, and cross-talk
removal, and was performed with the standard GRIS data reduction software, as
has been described by Collados (1999, 2003). The spectrograph still exhibits a
continuum intensity variation that has to be corrected. For this purpose as
well as for the wavelength calibration, we used the IAG spectral atlas
obtained with the Fourier Transform Spectrograph (Reiners et al. 2016). The
continuum correction curve was obtained by fitting a polynomial to the ratio
of the disk center slope from our observations and the slope of the atlas
spectrum. The telluric lines in the observed spectral region were used for the
wavelength calibration. The calculated spectral dispersion was 18 mÅ per
pixel. Finally, the observed spectrum was normalized to the continuum level of
the quiet Sun at the disk center, which was necessary for the inversions.

###  3.1 Inversions

The Stokes parameters were inverted using the Stokes inversions based on
response functions (SIR; Ruiz Cobo & del Toro Iniesta 1992). The inversion
code provides physical parameters such as the components of the magnetic field
vector, the temperature, and the Doppler velocity. We limited the inversions
to one node for the Doppler velocity, magnetic field strength, inclination and
azimuth, keeping these values height-independent. Only the temperature is
height-dependent, with one, three, and four nodes in the different cycles of
SIR. We used a value of 750 m s-1 for the macroturbulence and do not change it
during the whole inversion process. Independent inversions were performed for
both the silicon and calcium lines, see plotted Stokes profiles in Fig 2.
Using SIR, the nodes must be selected following certain rules, and often they
do not fit the formation height of the individual lines. Therefore, we prefer
separate inversions with one node for the magnetic field vector, which then
addresses the formation height of the used spectral lines (see the calculated
response functions in Fig 2). The response functions quantify the sensitivity
of an observable (in this case the Stokes parameters) to a specific
atmospheric property (in this case the total magnetic field strength,
$B_{\mathrm{tot}}$) in the specific optical depth.
The GRIS spectropolarimeter enables the study of the chromosphere through the
He i 10830 Å line, which is formed in the chromosphere. The Stokes parameters
of the helium triplet were inverted using the HAnle and ZEeman Light (HAZEL)
code. We followed the description of the original paper by Asensio Ramos et
al. (2008) and the HAZEL manual.222https://aasensio.github.io/hazel The
inversion conditions were defined in a configuration file and include the
determination of Doppler velocity, magnetic field vector, thermal width and
optical depth. The inversion process was conducted in two cycles with a single
atmospheric component.
To solve the 180∘-ambiguity, we assumed a single azimuth center for the
sunspot and a radial structure inside the spot as an ideal configuration. The
ideal azimuth vector has a radial structure and the real azimuth values must
differ by less than ± $90^{\circ}$ from the ideal case. If abrupt
discontinuities in the Cartesian components of the magnetic field remained
after this procedure, we did a manual correction of the azimuth by 180∘ on one
side of the discontinuity until we got a continuous magnetic field. A detailed
description of this procedure is given by Balthasar & Gömöry (2008). For
helium at the northern tip of the spot, after the described corrections we
still encountered discontinuities in the Cartesian components of the magnetic
field. We could remove them by changing the azimuth by 90∘ in some locations.
The magnetic field strength is low at these locations, and we cannot exclude
that we are dealing here with the ambiguities of the Hanle effect. Therefore,
we eventually applied this correction to remove the discontinuities. The final
step was to transform the coordinate system of the magnetic field to the local
reference frame and to correct the geometrical foreshortening of the data maps
by applying the method described by Verma et al. (2012).

###  3.2 Heights of the He i 10830 Å

In this subsection, we describe a method of obtaining the formation heights of
Stokes-$V$ parameters in the sunspot. Assuming a dipole configuration of the
magnetic field as a reasonable first approximation (see Lites & Skumanich
1990), the zero line of the Stokes-$V$ parameter allows us to estimate the
relative height of the spectral lines. For this, we made use of the dipole
property that all field lines cross a selected radial vector from the center
of the dipole under the same angle $\alpha$, for which the following equation
is valid

$\tan\alpha=0.5\tan\beta,$ (1)

where $\beta$ is the angle between the radial vector and the axis of the
dipole (see Kertz 1969). The measured Stokes-$V$ zero-line positions of the
three spectral lines were shifted due to their formation height. Along this
radial vector, we can determine the distance ratios, and because of the
intersection theorem, we have the same ratios along the normal to the solar
surface. Here we know the heliocentric angle that corresponds to the zero line
in the Stokes-$V$ maps for the different lines. The measured Stokes-$V$ zero-
lines were shifted with respect to each other in each spectral line,
corresponding to different heights in the solar atmosphere. The valid values
are only along the line from the disk center via the sunspot center to the
limb (azimuth $\psi=0^{\circ}$), including an offset of $\pm 5^{\circ}$ to
obtain a few more data points, allowing averaging, and $\sin\Delta\psi$ is
smaller than 0.1 so that the transversal component is negligible. A simple
schematic sketch of the geometry sunspot’s magnetic field is displayed in Fig.
3 for the three spectral lines. The blue lines represent magnetic field lines
for three different spectral lines and the red line represents the radial
dipole line that crossed the magnetic field lines. The height distance between
the spectral lines was calculated like

$\Delta h_{1}=\Delta x_{1}\cos{(90^{\circ}-\theta)};\quad\Delta h_{2}=\Delta
x_{2}\cos{(90^{\circ}-\theta)}$ (2)

where $\Delta h_{1}$ is the height distance between Ca i 10839 Å and Si i
10827 Å and $\Delta h_{2}$ is height distance between Si i 10827 Å and He i
10830 Å. The $\Delta x$ values are the corresponding distances of the
Stokes-$V$ zero lines.

Figure 5: Three layers covered stratification of the magnetic field vector in
the solar atmosphere. The displayed heights are not in scale. The left image
displays the vector magnetic field in the spherical coordination system (total
magnetic field strength, $B_{\mathrm{tot}}$, inclination, $\gamma$, and
azimuth, $\psi$). Inclination values were clipped at 90∘; their maximum is
99∘. The right image displays the vector magnetic field in the Cartesian
coordination system ($B_{\mathrm{x}}$, $B_{\mathrm{y}}$, $B_{\mathrm{z}}$).
The vector stratification comprises of the Ca i 10839 Å and Si i 10827 Å for
the photosphere and He i 10830 Å for the chromosphere. The black contours
represent the umbra-penumbral and the penumbra-quiet Sun boundaries.

Figure 6: Azimuthal averages of the magnetic field components (strength,
$B_{tot}$, inclination, $\gamma$, vertical component, $B_{\mathrm{z}}$, and
horizontal component, $B_{\mathrm{hor}}$) at different distances from the
sunspot center are shown. The data are presented for each spectral line in the
umbra and penumbra. The x axis represents the distance from the center of the
sunspot. The vertical black lines indicate the boundaries between the umbra
and the penumbra and between the penumbra and the quiet Sun.

##  4 Results

Figure 7: Averages of magnetic field parameters (strength, $B_{\mathrm{tot}}$,
inclination, $\gamma$, $B_{\mathrm{z}}$ and the $B_{\mathrm{hor}}$ component)
and standard deviations of each spectral line in the umbra and penumbra. The x
axis displays the height in the solar atmosphere. The height $x=0$ corresponds
to $\tau=1$. Blue bullets represent the umbra, red ones the penumbra, and
violet ones the Stokes-$V$ zero line.

###  4.1 Magnetic field of the studied sunspot from the photosphere to
chromosphere

In the first step, we determined the formation height by applying the
procedure described in Section 3.2. Fig 4 shows Stokes-$V$ zero lines and the
calculated heights of the selected spectral lines. The position of Si i 10827
Å should be located above that of Ca i 10839 Å, but the distance is below the
resolution element of the GREGOR telescope at this wavelength; thus, we cannot
distinguish between these two lines. We obtain a height difference of 971
$\pm$ 64 km between the helium and the photospheric lines. The heights of the
two photospheric lines were calculated from ACFs for the models M4 by
Kollatschny et al. (1980) and T93_27 by Schleicher (1976) and interpolated
between them according to the local conditions (see Balthasar & Gömöry 2008),
with 100 km being obtained for calcium and 220 km for silicon at the
Stokes-$V$ zero line. We prefer ACFs because they provide a more general mean
than response functions which depend on the selected physical parameter.
As a result of independently inverted spectral lines (Ca i 10839 Å, Si i 10827
Å, and He i 10830 Å), we obtain the morphology of the magnetic field vector in
spherical and Cartesian coordinate systems, covering the magnetic field
strength, $B_{\mathrm{tot}}$, inclination, $\gamma$, azimuth, $\psi$, and the
Bx, By, and Bz components of the sunspot. The retrieved results are displayed
in Fig. 5. Dark contours indicate the umbral boundary at an intensity level of
0.5 $I_{0}$, where $I_{0}$ is the quiet Sun intensity of the local continuum,
and the outer penumbral boundary at $B_{\text{hor}}=490\,\text{G}$, as defined
by Benko et al. (2018). This arrangement of results indicates the three-
dimensional structure of the magnetic field in the sunspot from the
photosphere to the chromosphere.

Units | Umbra 1 | Penumbra | near V zero line | Umbra 2
---|---|---|---|---
$B_{\mathrm{tot}}$ [ $\rm{G\,km}^{-1}$] | $-0.3400$ | $-0.2800$ | $-0.4400$ | $-0.5200$
$\gamma$ ${[^{\circ}\,\rm{km}^{-1}]}$ | 0.0043 | $-0.0087$ | $-0.0022$ | 0.0064
$B_{\mathrm{hor}}$ [ $\rm{G\,km}^{-1}$] | 0.0230 | $-0.2800$ | $-0.3500$ | 0.0350
$B_{z}$ [ $\rm{G\,km}^{-1}$] | $-0.3900$ | $-0.0160$ | $-0.2800$ | $-0.5900$
Table 1: Overview table of the inferred gradients of the magnetic field
parameters at the individual parts of the sunspot shown in Fig.7 The umbra and
penumbra values are average values for the whole area. The calculated
gradients for umbra 2 were determined with a deeper formation height of the
helium line. The values near the Stokes-$V$ zero line are average values from
a narrow ring around the whole umbra.

The spot is a stable one with a positive polarity; thus, the field lines are
directed upward. The total magnetic field strength values amount to 3100 G
located in the umbra for Ca i 10839 Å. The values decrease from the umbra to
the outer penumbra. The inclination angle of the vector magnetic field varies
between $0^{\circ}$ to the $100^{\circ}$. The value of $0^{\circ}$ occurs in
the center of the umbra, and the inclination increases to $100^{\circ}$ at the
outer penumbra. Values above 90∘ indicate a downward-directed magnetic field
(see Westendorp Plaza et al. (2001)). The azimuth clearly shows a radial
structure across from the umbra to the outer penumbra. The azimuth angle
increases continuously in a counterclockwise direction from the zero line.
The distribution of the magnetic field vector in the chromosphere is similar
to that in the photosphere. The transformed results into the Cartesian
coordinate system are displayed in the second column of Fig. 5. The vertical
component, $B_{z}$, corresponds to the direction perpendicular to the solar
surface, and the horizontal components, Bx and Bz, correspond to the plane
tangential to the solar surface. Bx is in the direction from the disk center
to the solar limb, and By is perpendicular to the Bx direction. While Bz
dominates in the umbra, Bx and By dominate in the penumbra. Bz reaches a
maximum value of above 3000 G at some points in the umbra and decreases toward
the outer edge of the penumbra.
To obtain the radial dependence of the magnetic parameters, we determined mean
values along ellipses, omitting those pixels outside the scanned range (see
Fig. 1) for examples. These ellipses have an increasing distance from the
center of the spot. The results are displayed in Fig. 6. The average values of
the vector magnetic field as a function of the radial distance from the
sunspot center to the quiet Sun are plotted in Fig.6. The values of
$B_{\mathrm{tot}}$, $\gamma$, $B_{\mathrm{z}}$, and the $B_{\mathrm{hor}}$ are
displayed in different line colors. The umbra-penumbra boundary is indicated
at 0.32 $R/R_{p}$, where $I$ is 0.5, and the outer penumbra boundary is at 1
$R/R_{p}$. The inclination of the vector magnetic field is almost vertical in
the umbra, but a deviation from the vertical direction appears in the
chromosphere because the magnetic field widens.
The inclination increases in the radial direction across the height. The
inclination in deep layers (calcium) approaches 90∘ faster than that of other
lines between 0.6 R/Rp and 0.73 R/Rp, and from this point the inclination is
almost constant to 1.0 R/Rp. The maximum values are around 80 ∘. For helium,
the maximum inclination is 71 ∘ at 0.93 R/Rp and then it decreases again. Such
a decrease might be caused by the fact that the shape of the spot is not
perfect, so the ellipses close to the outer penumbral boundary partly cross
areas outside it.
The vertical, $B_{\mathrm{z}}$, component of the magnetic field vector reaches
its maximum values from 2500 G to 3100 G in the umbra, and it decreases
outward. In the inner penumbra, it reaches values between 1600 G and 2100 G
and continues to decrease to a value of approximately 200 G at the outer
penumbra. The horizontal, $B_{\mathrm{hor}}$, component of the magnetic field
vector has values between 100 G and 350 G, increasing toward the penumbra.
Values between 650 G and 1000 G appear at the inner penumbra. The maximum
values are reached at the 0.5 $R/R_{p}$ and from this point, the
$B_{\mathrm{hor}}$ values decrease to 400 G at the outer penumbra.
Figure 7 shows the height dependence of a) $B_{\mathrm{tot}}$, b) $\gamma$, c)
$B_{\mathrm{z}}$, and d) $B_{\mathrm{hor}}$ for the observed spectral lines.
The figures present the average values of the selected components for the
entire umbra and penumbra. In addition, a narrow ring near the Stokes-$V$-zero
line is depicted in Fig. 1. Notably, the helium Stokes-$V$-zero line
represents a specific region where we encounter challenges in achieving
reliable inversions and proper fits for the spectral lines. We suspect that we
may be dealing with a situation similar to recent sunspot inversions using
HAZEL that were published by Lindner et al. (2023). It is plausible that the
Stokes-$V$ signal is formed at a different atmospheric height than Stokes-$Q$
and Stokes-$U$, as is suggested by Díaz Baso et al. (2019). We applied the
height differences determined in Subsection 3.2 and, in addition, those
determined by (Felipe et al. 2023), for an umbral area where strong shocks due
to propagating slow magnetoacoustic waves happened. Such shocks can reduce of
the response of the helium line to the magnetic field.
In the case of umbra 1, we assumed that the heights obtained in Section 3.2
are valid all over the spot. The heights derived from ACFs are 110 km for
calcium and 180 km for silicon in the umbra. For umbra 2, we took a helium
height at 750 km in the umbral atmosphere, corresponding to the special case
investigated by (Felipe et al. 2023) mentioned above. The calculated heights
of the photospheric lines remain the same as in the case of umbra 1. For the
penumbra, we obtain from ACFs photospheric heights of 125 km for calcium and
275 km for silicon. As was said above, the formation heights of the
photospheric near the Stokes-$V$ zero lines (see Fig.3) are 100 km and 220 km,
respectively. The vertical bars represent the standard deviations from the
average value. We applied a linear fit to the data. Table1 shows the gradient
values determined by linear fits used in Figure 7.
The umbral average values of magnetic field strength decrease linearly with
increasing height, from 2600 G to 2270 G. The penumbral values of the magnetic
field strength also decrease with increasing height. The penumbral value
decreases from 1460 G to 1180 G. The measured gradient is -0.34
$\rm{G\,km}^{-1}$ for umbra 1 and -0.52 $\rm{G\,km}^{-1}$ for umbra 2. A value
of -0.28 $\rm{G\,km}^{-1}$ is obtained for the penumbra. The gradient for
umbra 1 is low compared to the values in Table 2 of Balthasar (2018), while
the result for umbra 2 and the penumbral gradient are comparable to the values
there.
Figure 7 b) shows the height dependence of inclination, where the penumbral
value decreases in the penumbra, while the umbral value has a tendency to
increase with height. The umbral value increases from $17^{\circ}$ to
$21^{\circ}$. The penumbral inclination decreases from $62^{\circ}$ to
$53^{\circ}$. The measured gradient is 0.0042 ${}^{\circ}\,\rm{km}^{-1}$ for
the umbra and -0.0087 ${}^{\circ}\,\rm{km}^{-1}$ for the penumbra.
Fig. 7 c) shows the $B_{z}$ component of the magnetic field vector. The umbral
value shows a clear decrease with increasing height from 2500 G for 110 km to
2100 G in 1070 km over the solar surface. The penumbral value is almost
constant. The interval values are between 754 G and 739 G. The measured
gradient is -0.39 $\rm{G\,km}^{-1}$ for the umbra and 0.016 $\rm{G\,km}^{-1}$
for the penumbra. The last investigated parameter is $B_{\mathrm{hor}}$,
displayed in Fig. 7 d). The umbral values are almost constant with increasing
height. The umbral values vary from 750 G to 770 G. The penumbral value
clearly decreases depending on the increasing height. The penumbral value is
1130 G at a height of 110 km and the value decreases to 860 G for 1071 km
above the solar surface. The measured gradient is 0.023 $\rm{G\,km}^{-1}$ for
the umbra and -0.28 $\rm{G\,km}^{-1}$ for the penumbra.

###  4.2 Downflows in the He i 10830 Å

Figure 8: a) Group of Stokes parameters $I$, $Q$, $U$ and $V$ with a higher
Doppler shift located in the outer penumbra. b) Location of the group of
Stokes parameters with a higher Doppler shift in the continuum. c) Location of
the group of Stokes parameters with a higher Doppler shift in the silicon
core.  Figure 9: Overview of the helium doppler maps in the sunspot. The left
helium Doppler map displays velocities inverted using a one-component
atmosphere. The green contours show sunspot intensity boundaries. The purple
contour shows areas with high Doppler shifts. The right images show results
from inversions with two chromospheric models and one photospheric model. The
green line shows the boundary between the penumbra, and the quiet Sun.

The considered area consists of smaller areas where the Stokes-profile shapes
are similar due to the fact that the Stokes profiles were formed under similar
thermodynamic parameters. At this stage of data processing, we used a special
algorithm to classify the observed data before the inversion. While various
methods exist for data categorization, choosing this approach proves
advantageous in terms of time efficiency. The categorization of the data is
helpful for an inversion technique with the goal of determining the physical
parameters of the observed atmosphere. Additionally, it serves as a valuable
method of quickly scanning and obtaining a comprehensive overview of the
entire observed map. One such technique, which we make use of in this paper,
is called k-means clustering (MacQueen 1967). This technique is often used to
group common spectral profiles in large datasets (see, e.g., Pietarila et al.
2007; Viticchié & Sánchez Almeida 2011; Panos et al. 2018; Robustini et al.
2019; Sainz Dalda et al. 2019; Kuckein et al. 2020; González Manrique et al.
2024). We also used the MiniBatchKMeans algorithm to reduce the computation
time. Both of these algorithms are part of the scikit-learn library of Python.
The first step of the algorithm is to assign the initial location of the
centroids. The number of clusters is defined by the user. In our case, we
opted for 63 groups to properly cover the diversity of spectra with, all with
different characteristic Doppler shifts. It is the minimum number that
includes all of the supersonic velocities observed in helium. The creators of
the method suggest setting an odd number as it helps one to obtain a clear
majority.
The selected profiles are shown in Fig. 8 a) and correspond to clustered
Stokes profiles $I$, $Q$, $U$ and $V$ with a higher Doppler shift. Figure 8 b)
shows two small patches indicating the location of these profiles in the
continuum intensity. The small patches represent areas where the one-component
helium inversions do not deliver good results. The right patch is definitely
part of the penumbra and it has effects on the behavior of the penumbra and
its shape at this place. The left one is just outside the penumbra according
to the photospheric boundary. Figure 8 c) complements the continuum image by
showing the observed map in the silicon core, where we see penumbra-like
structures overlapping with the left patch. These structures are not visible
in the continuum map.
In some special locations, where the red helium component becomes rather
asymmetric or even split, an inversion with one atmospheric component is not
sufficient. Here we need two atmospheric models (e.g., Kuckein et al. 2020).
The corresponding results are shown in the small images in Fig. 9. In the
second chromospheric atmosphere, the velocities are supersonic (exceeding the
standard sound speed of 10 $\rm{km\,s}^{-1}$, which is typical for the
formation temperature of the He i 10830 Å triplet around 8000–10,000 K). They
have a tendency to be smaller than 30 $\rm{km\,s}^{-1}$ inside the penumbra
(right patch) and somewhat larger outside that boundary (left patch). The
average value of the second helium component is about 30 $\rm{km\,s}^{-1}$,
referred to as the fast velocity component. The average value is 3
$\rm{km\,s}^{-1}$ $\pm$ 2 $\rm{km\,s}^{-1}$ for silicon and 2
$\rm{km\,s}^{-1}$ $\pm$ 0.7 $\rm{km\,s}^{-1}$ for the first helium component,
which is a slow velocity component.
The separated regions evolve under different atmospheric conditions. While the
right region is part of the penumbra, the left region is only part of the
superpenumbra, where the photospheric Evershed flow does not occur. This area
is located at a sharp end of the penumbra structure due to the presence of
fast velocities that occur in higher parts of the solar atmosphere and are
related to the fast temporal evolution of the sunspot.

##  5 Discussion and conclusions

In this analysis, we investigated a sunspot near the limb, observed with
GREGOR/GRIS. Not many sunspots that are close to the limb have been studied in
detail before. We analyzed the magnetic topology and (thermo)dynamic structure
of the sunspot NOAA 12553 at different heights using the spectral lines Ca i
10839 Å, Si i 10827 Å, and He i 10830 Å, which allowed us to determine the
stratification from the deep photosphere to the chromosphere. The
chromospheric height of He i 10830 Å was determined using the properties of a
dipole field that we took as a first approximation of the magnetic field of
the sunspot. The distance of the Stokes-$V$ zero lines for the different
spectral lines was then transferred into different heights. Since the
positions of the photospheric lines fall close together within the resolution
element of the GREGOR telescope, we cannot use this method to determine the
height difference of the photospheric lines. However, we find that the He i
line is formed about 970 km higher than the photospheric lines at this
location. Felipe et al. (2023) find for certain locations in the umbra a
formation height of the helium line only 450 km higher than the silicon line.
These values are lower than those calculated by Avrett et al. (1994) for
plages and the quiet Sun, in which the helium line originates 2000 km higher
than the photosphere. For a quiet Sun simulation, de la Cruz Rodríguez et al.
(2019) displays the formation height of helium between 1000 and 3000 km, and
at special locations even higher. Nevertheless, the situation in sunspots
would be different. In our investigation, the mean value of the magnetic field
in the umbra amounts to 2600 G at the height of the calcium line and decreases
to a value of 2270 G at the formation height of the helium line. These values
are typical of a large sunspot. The decrease rate is $-0.34$
$\rm{G\,km}^{-1}$, if we assume a helium formatio n height of 1071 km. This
value is a bit lower than the values listed by Balthasar (2018), but if we
assume a helium formation height of 750 km, the value is -0.52
$\rm{G\,km}^{-1}$, a value within the range of Balthasar (2018). The vertical
component of the magnetic field strength, $B_{\mathrm{z}}$, decreases with
height in the umbra. The $B_{\mathrm{z}}$ component amounts to values of 2500
G in the lowest layer of the photosphere, 2300 G in the upper part of the
photosphere, and only 2100 G in the lower layer of the chromosphere. The
values of the magnetic inclination indicate an almost vertical magnetic field
in the umbra; mean values vary between $17^{\circ}$ and $21^{\circ}$. As Figs.
6 and 7 show, the inclination slightly increases with height, which means that
the magnetic field is more inclined in higher layers. This result is in
agreement with previous publications. The penumbra of the observed sunspot is
usually dominated by a more horizontal magnetic field. The magnetic field
strength of the penumbra in the photosphere shows a value of 1450 $\rm{G}$ for
the calcium line and decreases with height to a value of 1180 $\rm{G}$ for the
helium line. Our calculated average value of the penumbral inclination is
$62^{\circ}$ for the calcium line, and in the chromosphere the mean
inclination is 53∘. The value of the inclination angle doesnt not change with
increasing height in the inner penumbra, which is in contrast to the results
reported by Borrero & Ichimoto (2011), Balthasar & Gömöry (2008), and van
Noort et al. (2013), who found that the magnetic field inclination decreases
with height in the penumbra. However, our results are consistent with the
findings of Sánchez Cuberes et al. (2005), who also found that the inclination
in the penumbra is independent of height. In the outer penumbra, we find
values around 80∘ for the calcium line indicating an almost horizontal field
in deep layers.
The Cartesian components, $B_{\mathrm{x}}$ and $B_{\mathrm{y}}$, dominate in
the penumbra, in contrast to the umbra, where the $B_{\mathrm{z}}$ component
dominates. This finding corresponds to the magnetic field distribution found
in many previous publications. The vertical component of the magnetic
induction vector, $B_{\mathrm{z}}$, decreases with height in the umbra. The
$B_{\mathrm{z}}$ component exceeds values of 2500 G in the lowest layer of the
photosphere, 2340 G in the upper part of the photosphere, and only 2100 G in
the lower layer of the chromosphere. The vertical component of the magnetic
field vector in the penumbra is almost constant with the height, which is in
agreement with the variation of the magnetic inclination. The inferred
magnetic field component, $B_{\mathrm{z}}$, falls into the interval from
$-1500$ G to $-1900$ G for the umbra (Benko et al. 2018) and into the interval
from $-600$ G to $-820$ G in the penumbra. Balthasar & Gömöry (2008) show that
the $B_{\mathrm{z}}$ component reaches smaller values with increasing height
in the umbra and that in the outer penumbra the $B_{\mathrm{z}}$ value
increases with height.
The azimuth angle of the magnetic field vector is radially oriented all over
the sunspot and changes continuously from $0^{\circ}$ and $360^{\circ}$, as
was to be expected. The azimuth does not change with height.
At the sunspot’s edge, we observe regions with supersonic velocities at the
chromosphere. The chromosphere above ARs is known for hosting fast downflows
surpassing the local sound speed (e.g., Penn & Kuhn 1995; Schmidt et al. 2000;
Solanki et al. 2003; Lagg et al. 2004; González Manrique et al. 2016, 2018,
2020). A statistical study conducted by Sowmya et al. (2022) revealed
supersonic downflows in all ARs, comprising 0.2–6.4$\%$ of the observed FOV,
suggesting their relatively common occurrence in the upper chromospheres of
ARs. Line-of-sight velocities of these supersonic downflows can reach up to 50
$\rm{km\,s}^{-1}$, with a majority (approximately 92$\%$) coexisting with
subsonic flow components (Sowmya et al. 2022). An alternative might be a
footpoint of the inverse (superpenumbral) Evershed flow. However, supersonic
velocities have only been observed in the transition region or above, but not
in chromospheric layers (see the review of Solanki (2003)). A recent
investigation by Romano et al. (2023) shows that the superpenumbral filaments
end at the outer boundary of the penumbra, but yields velocities clearly below
10 $\rm{km\,s}^{-1}$ from chromospheric spectral lines.
Future investigations with a new generation of solar telescopes with a higher
resolution, for instance the Daniel K. Inouye Solar Telescope (Rimmele et al.
2020; Rast et al. 2021) or the European Solar Telescope (Quintero Noda et al.
2022), will contribute to solving the remaining open questions.

###### Acknowledgements.

We express our thanks to Dr. A. Asensio Ramos for helpful discussions and his
comments to improve the results. We are indebted to Dr. A. Pietrow for
carefully reading the manuscript and giving many valuable comments to improve
the paper. The 1.5-meter GREGOR solar telescope was built by a German
consortium under the leadership of the Leibniz-Institut für Sonnenphysik in
Freiburg (KIS) with the Leibniz-Institut für Astrophysik Potsdam (AIP), the
Institut für Astrophysik Göttingen (IAG), the Max-Planck-Institut für
Sonnensystemforschung in Göttingen (MPS), and the Instituto de Astrofísica de
Canarias (IAC), and with contributions by the Astronomical Institute of the
Academy of Sciences of the Czech Republic (ASCR). This work was supported by
the project VEGA 2/0043/24. MB is grateful for the support of the Stefan
Schwarz grant of the Slovak Academy of Sciences. SJGM is grateful for the
support of the European Research Council through the grant
ERC-2017-CoG771310-PI2FA, and by the Spanish Ministry of Science and
Innovation through the grant PID2021-127487NB-I00. CK acknowledges funding
from the European Union’s Horizon 2020 research and innovation programme under
the Marie Skłodowska-Curie grant agreement No 895955.

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Let $f:\Sigma^+\rightarrow \Sigma^*$ be a $\var$-transduction, and let $\bim=\tuple{\laut,\raut,\bout,\lfinal,\rfinal}$ be a $\var$-bimachine realizing $f$.
We denote by $\sim_\epsilon$ the empty word congruence over $\Sigma$, and by $\laut_\epsilon$ and $\raut_\epsilon$ the left and right automata associated with $\sim_\epsilon$, respectively.
Since ${\sim_\epsilon}\in \var$ by hypothesis, $\laut_\epsilon$ and $\raut_\epsilon$ are $\var$-automata.
We define $\bim'=\tuple{\laut\times\laut_\epsilon,\raut\times\raut_\epsilon,\bout',\overline \epsilon,\overline \epsilon}$ with
$\bout'((\cla u _\laut,\cla u_\epsilon),\sigma,(\cla v _\raut,\cla v_\epsilon))=\bout(\cla u _\laut,\sigma,\cla v _\raut)$ if $u,v\neq \epsilon$,  $\lfinal(\cla v _\raut)\bout(\cla u _\laut,\sigma,\cla v _\raut)$ if $u=\epsilon$ and $v\neq \epsilon$,  $\bout(\cla u _\laut,\sigma,\cla v _\raut)\rfinal(\cla u _\laut)$ if $u\neq\epsilon$ and $v= \epsilon$, and $\lfinal(\cla v _\raut)\bout(\cla u _\laut,\sigma,\cla v _\raut)\rfinal(\cla u _\laut)$ if $u,v= \epsilon$.
By construction $\bim'$ realizes $f$, and since ${({\approx_\bim}\sqcap{\sim_\epsilon})} \finer {\approx_{\bim'}}$, $\bim'$ is indeed a $\var$-bimachine.

Let $\var$ be an congruence class equivalent to a fragment .
Then a transduction is a $\var$-transduction if and only if it is definable by a 2--transducer.

According to Proposition <ref>, we have that $\var$-bimachines without final outputs exactly characterize $\var$-transductions.
Thus, from Lemma <ref> we have the equivalence between $\var$-transductions and 2--transductions.

From the previous corollary we can immediately deduce:

Let $\var$ be an congruence class equivalent to a fragment  such that  and 2- define the same pointed languages.
Then a transduction is a $\var$-transduction if and only if it is an -transduction.

§.§.§ From  to 2-and back

When  is a fragment equivalent to some congruence class , logical transductions defined with 2- formulas trivially have the same expressive power as -bimachines.
In this section we shall give some sufficient properties on a logical fragment under which -formulas and 2--formulas are equivalent, define the same pointed languages.
For this we introduce a third logical formalism for pointed words:

A pointed word over an alphabet $\Sigma$ can alternatively be seen a logical structure over $\struct_{\Sigma\uplus \dot \Sigma}=\set{(\sigma(x))_{\sigma\in\Sigma},(\dot\sigma(x))_{\sigma\in\Sigma},x<y}$, a word over an extended alphabet which contains a pointed copy of each letter.
Note that the number of pointed positions is not necessarily one.
However we will see in the next example that restricting to one pointed position can be enforced by a formula $\phi_\text{pointed}$.

The language defined by the formula in Example <ref> is:

\[\set{u\alpha u'\dot\gamma v|\ u,u',v\in \Sigma^*}\cup\set{u \dot\gamma v\beta v'|\ u,v,v'\in \Sigma^*}\]

We can define this language by an -sentence over the extended alphabet $\Sigma\uplus \dot \Sigma$.
First we define the formula:

$$\phi_{\text{pointed}}:=\tuple{\exists x\ \dot\Sigma(x)} \wedge \tuple{\forall x,y\ \dot\Sigma(x)\wedge \dot\Sigma(y) \rightarrow x=y}$$

The formula $\phi_{\text{pointed}}$ specifies that exactly one position holds a pointed letter, with predicate $\dot\Sigma(x)$ being a shortcut for $\bigvee_{\sigma\in\Sigma}\dot\sigma(x)$.
Finally we obtain the formula:

$$\phi_{\text{pointed}}\wedge \exists c\ \dot\gamma(c)\wedge \tuple{\tuple{\exists x\ x<c \wedge \alpha(x)}\vee \tuple{\exists x\ x>c \wedge \beta(x)}}$$

This is just the formula of Example <ref> where $\con$ is replaced by the variable $c$ which is existentially quantified, and holds a pointed letter.

Our goal here is to show that -formulas and 2--formulas define the same pointed languages under the following assumption (1)-(3). Assumption (4) is used to show that a corresponding congruence class would be .

* -formulas over an alphabet $\Sigma$ and -formulas over the extended alphabet $\Sigma\uplus \dot \Sigma$ define the same pointed languages.

* A language over the alphabet $\Sigma$ is definable by an -formula over $\Sigma$ if and only if it is definable by an -formula over a larger alphabet $\Sigma\cup\Gamma$.

* -languages are closed under pointed concatenation, meaning that for any two -languages $L_1,L_2$ over an alphabet $\Sigma$ and a fresh symbol $\sharp$, $L_1\cdot\sharp\cdot L_2$ is an -language over $\Sigma\uplus\set{\sharp}$.

* $\set{\epsilon}$ is an -language.

These assumptions may seem quite strong and not necessary, and they are, however they cover several well studied logical fragments of , as
shown in Section <ref>.

It seems that most of the known logical fragments that have access to the linear order and for which there is an equivalent congruence class satisfy properties (1)-(4).
A non-example is the fragment of first-order logic with successor, , which is not closed under pointed concatenation.

The following results establish that under assumptions (1)-(3),
, and 2-coincide.
Figure <ref> summarizes how these assumptions are used to prove it.

[baseline=0, inner sep=0, outer sep=0, minimum size=0pt]

(0,1) node (Fc) ;
(2,1) node (F) ;
[white] (Fc) – (F) node[black,midway]$=$ node[black,midway,above=1mm] (1);

(4,1) node (2F) 2-;
[white] (F) – (2F)
node[black,midway,above=1mm] (nsub) $\subseteq$
node[black,midway,below=1mm] (nsup) $\supseteq$;

(3,2.5) node (two) (2);
[white] (two) – (nsub)
node[black,midway] 270$\implies$
node[black,midway,right=2mm] Lemma <ref>;

(3,-0.5) node (three) (2)-(3);
[white] (three) – (nsup)
node[black,midway] 90$\implies$
node[black,midway,right=2mm] Lemma <ref>;

Logics , 2-and coincide under assumptions (1)-(3).

Let  be a fragment equivalent to some congruence class $\var$.
Under assumption (2), any -definable pointed language is 2--definable.

Let $\phi$ be an -formula recognizing a pointed language over the extended alphabet.
Since  is equivalent to the congruence class , let $\aut$ be a $\var(\Sigma\uplus \dot\Sigma)$-automaton, with transition relation $\Delta$, recognizing $L(\phi)$.
Then we define

$$\psi= \bigvee_{(p,\dot \sigma,q)\in \Delta} (\phi_I^p,\phi_q^F)_{\set{\sigma}}$$

where $\phi_I^p,\phi_q^F$ are defined respectively as -formulas recognizing the language of words going from the initial states to $p$ and from $q$ to the final states. These languages are indeed definable in , since they are recognized by an automaton obtained from $\aut$ by changing the initial and final states (which does not modify the transition congruence).
Since the words recognized by $\aut$ only have one pointed letter, the formulas defined are -formulas over the regular alphabet, under assumption (2).

Let  be a fragment equivalent to some congruence class $\var$.
Under assumption (2)-(3), any 2--definable pointed language is -definable.

Let us show that under assumption (3), a 2- formula can be transformed into a disjunction of -formulas recognizing pointed concatenations of -languages.
Let $F=\bigvee_{1\leq i\leq n}(\phi_i,\phi_i')_{\Gamma_i}$ be a 2- formula.
Up to decomposing $(\phi_i,\phi_i')_{\Gamma_i}$ into a disjunction, we can assume that $\Gamma_i=\set{\sigma_i}$.
According to assumption (3), there exists an -formula $\chi_i$ such that $\sem {\chi_i}=\sem{\phi_i}\dot{\sigma}_i\sem{\phi_i'}$.
Under assumption (2), these can be seen as -languages over $\Sigma\uplus \dot\Sigma$.
Since -languages are $\var$-languages, they are closed under Boolean operations.
Hence, there exists an -formula $\chi$ recognizing the pointed language $\bigcup_{1\leq i\leq n} \sem{\chi_i}$.

Let  be a fragment equivalent to some congruence class $\var$.
Under assumptions (1)-(3), the logics  and 2- are equivalent.

According to Lemmas <ref> and <ref>, under assumptions (2) and (3), a pointed language is -definable if and only if it is 2--definable.
Then, by assumption (1) we obtain the equivalence.

We give sufficient conditions under which a congruence class corresponding to a logical fragment is .

Let be a congruence class equivalent to some fragment satisfying properties (1) and (4).
Then is .

According to (4), $\set{\epsilon}$ is an -language, and according to (1), $\set{\epsilon}$ is an -language over any alphabet $\Sigma$.
Thus for any alphabet $\Sigma$ there exists a congruence in $\var(\Sigma)$ recognizing $\set{\epsilon}$. Since $\var$ is closed under taking coarser congruences, it contains all the empty word congruences.

Let $\var$ be a congruence class equivalent to some fragment  satisfying properties (1)-(4).
Then a transduction is a $\var$-transduction if and only if it is an -transduction.

From Proposition <ref>, we know that is . Hence from Corollary <ref> and Proposition <ref>, we obtain the result.

§.§ Decidable fragments

The fragments , and 1 all satisfy properties (1)-(4).

The fragments trivially satisfy (4).
Let us show that satisfies properties (1)-(3). The proof for can easily be obtained with the same reasoning.
Let us re-state the assumptions:

* -formulas over an alphabet $\Sigma$ and -formulas over the extended alphabet $\Sigma\uplus \dot \Sigma$ define the same pointed languages.

* A language over the alphabet $\Sigma$ is definable by an -formula over $\Sigma$ if and only if it is definable by an -formula over a larger alphabet $\Sigma\cup\Gamma$.

* -languages are closed under pointed concatenation, meaning that for any two -languages $L_1,L_2$ over an alphabet $\Sigma$ and a fresh symbol $\sharp$, $L_1\cdot\sharp\cdot L_2$ is an -language over $\Sigma\uplus\set{\sharp}$.

Let us show (1):
Let $\phi$ be an -formula over the extended alphabet $\Sigma\uplus \dot \Sigma$ recognizing a pointed language.
We syntactically replace any atomic formula $\dot\sigma(x)$ by $x=c\wedge \sigma(x)$.
Thus we obtain an -formula recognizing the same pointed language.
Conversely let $\phi$ be an -formula. We define as in Example <ref> the formula  $\phi_{\text{pointed}}:=\tuple{\exists x\ \dot\Sigma(x)} \wedge \tuple{\forall x,y\ \dot\Sigma(x)\wedge \dot\Sigma(y) \rightarrow x=y}$ which is in .
Then we define $\phi'$ which is obtained from $\phi$ by syntactically replacing any atomic formula $\sigma(\con)$ by $\exists x\ \dot\sigma(x)$, $\sigma(x)$ by $\sigma(x)\vee \dot\sigma(x)$, $x<\con$ by $\exists y\ \dot\Sigma(y) \wedge (x<y)$ and $\con<x$ by $\exists y\ \dot\Sigma(y) \wedge (y<x)$.
Finally we obtain the -formula $\phi_{\text{pointed}}\wedge \phi'$, over the extended alphabet $\Sigma\uplus \dot \Sigma$, recognizing the same pointed language.

Let us show (2):
Let $\phi$ be an -formula over $\Sigma$ recognizing some language $L$.
Then in particular it is an -formula over $\Sigma\cup\Gamma$.
Let $\alpha=\neg \exists x\ \bigvee_{\gamma\in \Gamma\setminus\Sigma} \gamma(x)$ be a formula which specifies that no new letter appears in a word.
Then the formula $\phi\wedge \alpha$ is an -formula over $\Sigma\cup\Gamma$ which recognizes $L$.
Conversely let $\phi$ be an -formula over $\Sigma\cup\Gamma$ recognizing some language $L \subseteq \Sigma^*$.
Then one only has to syntactically change $\gamma(x)$ by $\bot$ for every predicate $\gamma \in \Gamma\setminus\Sigma$ in $\phi$ to obtain a formula $\phi'$ over $\Sigma$.
One can see that any model of $\phi$ is a model of $\phi'$ and vice versa.

Let us show (3):
Given a formula $\phi$, we define $\phi^{<\con}$ which is the formula $\phi$ where all the quantifications are guarded and restrict variables to positions before the position $\con$. Formally, if $\phi=\exists x\ \chi(x)$ , we define $\phi^{<\con} =\exists x\ (x<\con)\wedge\chi^{<\con} (x)$.
Let $\phi_1,\phi_2$ be two  -languages $L_1,L_2$ over an alphabet $\Sigma$ and let $\sharp\notin\Sigma$ be a fresh alphabet symbol.
We define $\Phi=\phi_1^{<\con}\wedge\phi_2^{>\con}\wedge \sharp(\con)\wedge \forall x\ \sharp(x)\rightarrow (x=\con)$, which recognizes the pointed language $L_1\dot\sharp L_2$.
Finally, from (1) there exists an -formula recognizing the same language.

Now we show that the fragment 1 satisfies properties (1)-(3).
Let us show (1):
Let $\phi$ be a 1-formula over the extended alphabet $\Sigma\uplus \dot \Sigma$ recognizing a pointed language.
We syntactically replace any atomic formula $\dot\sigma(x)$ by $x=c\wedge \sigma(x)$.
Thus we obtain a 1-formula recognizing the same pointed language.
Conversely let $\phi$ be a 1-formula. We define the formula  $\phi_{\text{pointed}}:=\tuple{\exists x\ \dot\Sigma(x)} \wedge \tuple{\forall x,y\ \dot\Sigma(x)\wedge \dot\Sigma(y) \rightarrow x=y}$ which is in 1.
Then we define $\phi'$ which is obtained from $\phi$ by syntactically replacing any atomic formula $\sigma(\con)$ by $\dot\sigma(c)$, $\sigma(x)$ by $\sigma(x)\vee \dot\sigma(x)$, $x<\con$ by $(x<c)$ and $\con<x$ by $ c<x$ where $c$ is a variable which does not appear in $\phi$.
Finally we obtain the 1-formula $\phi_{\text{pointed}}\wedge \exists c\ \dot\Sigma(c) \wedge \phi'$, over the extended alphabet $\Sigma\uplus \dot \Sigma$, recognizing the same pointed language.

Let us show (2):
Let $\phi$ be a 1-formula over $\Sigma$ recognizing some language $L$.
Then in particular it is a 1-formula over $\Sigma\cup\Gamma$.
Let $\alpha=\neg \exists x\ \bigvee_{\gamma\in \Gamma\setminus\Sigma} \gamma(x)$ be a formula which specifies that no new letter appears in a word.
Then the formula $\phi\wedge \alpha$ is a 1-formula over $\Sigma\cup\Gamma$ which recognizes $L$.
Conversely let $\phi$ be a 1-formula over $\Sigma\cup\Gamma$ recognizing some language $L \subseteq \Sigma^*$.
Then one only has to syntactically change $\gamma(x)$ by $\bot$ for every predicate $\gamma \in \Gamma\setminus\Sigma$ in $\phi$ to obtain a formula $\phi'$ over $\Sigma$.
One can see that any model of $\phi$ is a model of $\phi'$ and vice versa.

Let us show (3):
Given a formula $\phi$, we define $\phi^{<\con}$ which is the formula $\phi$ where all the quantifications are guarded and restrict variables to positions before the position $\con$.
Formally, if $\phi=\exists x_1,\ldots,x_n\ \chi$ where $\chi$ is quantifier-free , we define $\phi^{<\con} =\exists x_1,\ldots,x_n\ (x_1<\con)\wedge\ldots\wedge (x_n<\con)\wedge \chi$.
If $\phi=Bool(\phi_1,\ldots,\phi_n)$ where the $\phi_i$s are 1-formulas and $Bool(v_1,\ldots,v_n)$ is some propositional formula, then we define $\phi^{<\con} =Bool(\phi_1^{<\con},\ldots,\phi_n^{<\con})$.
Let $\phi_1,\phi_2$ be two  1-languages $L_1,L_2$ over an alphabet $\Sigma$ and let $\sharp\notin\Sigma$ be a fresh alphabet symbol.
We define $\Phi=\phi_1^{<\con}\wedge\phi_2^{>\con}\wedge \sharp(\con)\wedge \forall x\ \sharp(x)\rightarrow (x=\con)$, which recognizes the pointed language $L_1\dot\sharp L_2$.
Finally, from (1) there exists a 1-formula recognizing the same language.

Given a transducer realizing a transduction $f$, one can decide if:

* $f$ is -definable.

* $f$ is -definable.

* $f$ is 1-definable.

It was shown by [32, 25] that  and $\ap$ are equivalent.
It was also shown in [36] that  and $\da$ are equivalent.
The equivalence between 1 and the class $\jtrivial$ of $\mathcal J$-trivial congruences is due partly to [34] and can be found in [15].
From Theorem <ref> and Lemma <ref> we have the equivalences between -transductions and $\ap$-transductions, -transductions and $\da$-transductions, and 1-transductions and $\jtrivial$-transductions respectively.
In the articles mentioned above it is also shown that the equivalences are effective, meaning that the congruence classes $\jtrivial$, $\da$ and $\ap$ are decidable, thus from Theorem <ref> we obtain the result.

Extensions of the above fragments with additional predicates can easily be shown to satisfy assumptions (1)-(4).
A table from [16] sums up many results of decidability concerning fragments of with additional predicates.
For all these fragments, our Theorem <ref> should carry over. Similarly, the proof for 1 should easily transfer to i, for any positive integer $i$.

§ CONCLUSION

We have shown that the problem of deciding whether a bimachine realizes an aperiodic transduction is -complete. A question which remains open is whether the problem is still in if the input is given as a transducer instead of a bimachine.
Our main result is that any rational transduction can be completely characterized by a finite set of minimal bimachines, and we have introduced a logical formalism which allows to lift some known equivalences from rational languages to rational transductions.

One possible direction of investigation would be to formalize in algebraic terms sufficient, and possibly necessary, conditions under which a logic-algebra equivalence for languages can be lifted to transductions.
It would also be interesting to investigate the cases which are precisely not covered by our approach such as the logics or .
Another direction would be to try to lift the present results further to regular functions, functions realized by two-way transducers.
Indeed a characterization of a regular transduction in terms of minimal congruences would for instance yield a way to decide if a regular function is first-order definable. In [4] such a characterization is given, but with the stronger origin semantics, where every output position originates from an input position, which means that two transducers which realize the same function but produce outputs at different positions will not be considered equivalent under this semantics. Obtaining a similar characterization in origin-free semantics is a very ambitious objective and for now there is no known canonical way to define a regular function.

§ ACKNOWLEDGEMENT

The authors thank Anca Muscholl for her contribution in
early stages of this work.

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