11institutetext:  Dublin Institute for Advanced Studies, Astronomy &
Astrophysics Section, 31 Fitzwilliam Place, Dublin 2, Ireland
11email<EMAIL_ADDRESS>22institutetext: Tartu Observatory, University of
Tartu, Observatooriumi 1, Tõravere, 61602 Tartumaa, Estonia 33institutetext:
University of Western Ontario, Department of Physics & Astronomy, London,
Ontario, N6A 3K7, Canada 44institutetext: Leiden Observatory, Leiden
University, PO Box 9513, 2300RA, Leiden, The Netherlands 55institutetext:
Univ. Grenoble Alpes, CNRS, IPAG, 38000 Grenoble, France 66institutetext:
LUPM, Université de Montpellier & CNRS, Montpellier, Cedex 05, France
77institutetext: Univ. de Toulouse, CNRS, IRAP, 14 avenue Belin, 31400
Toulouse, France

# Misalignment of the outer disk of DK Tau
and a first look at its magnetic field using spectropolarimetry

M. Nelissen 11 P. McGinnis 11 C. P. Folsom 2233 T. Ray 11 A. A. Vidotto 44 E.
Alecian 55 J. Bouvier 55 J. Morin 66 J.-F. Donati 77 R. Devaraj 11

(Accepted 22 December 2022)

Context. Misalignments between a forming star’s rotation axis and its outer
disk axis, although not predicted by standard theories of stellar formation,
have been observed in several classical T Tauri stars (cTTs). The low-mass
cTTs DK Tau is suspected of being among them. In addition, it is an excellent
subject to investigate the interaction between stellar magnetic fields and
material accreting from the circumstellar disk, as it presents clear
signatures of accretion.

Aims. The goal of this paper is to study DK Tau’s average line-of-sight
magnetic field in both photospheric absorption lines and emission lines linked
to accretion, using spectropolarimetric observations, as well as to examine
inconsistencies regarding its rotation axis.

Methods. We used data collected with the ESPaDOnS spectropolarimeter, at the
Canada-France-Hawaii Telescope, and the NARVAL spectropolarimeter, at the
Télescope Bernard Lyot, probing two distinct epochs (December 2010 to January
2011 and November to December 2012), each set spanning a few stellar rotation
cycles. We first determined the stellar parameters of DK Tau, such as
effective temperature and $v\sin i$. Next, we removed the effect of veiling
from the spectra, then obtained least-squares deconvolution (LSD) profiles of
the photospheric absorption lines for each observation, before determining the
average line-of-sight magnetic field from them. We also investigated
accretion-powered emission lines, namely the 587.6 nm Hei line and the Caii
infrared triplet (at 849.8 nm, 854.2 nm and 866.2 nm), as tracers of the
magnetic fields present in the accretion shocks.

Results. We find that DK Tau experiences accretion onto a magnetic pole at an
angle of $\sim 30$\mathrm{\SIUnitSymbolDegree}$$ from the pole of its rotation
axis, with a positive field at the base of the accretion funnels. In 2010 we
find a magnetic field of up to 0.95kG (from the Caii infrared triplet) and
1.77kG (from the Hei line) and in 2012 we find up to 1.15kG (from the Caii
infrared triplet) and 1.99kG (from the Hei line). Additionally, using our
derived values of period, $v\sin i$ and stellar radius, we find a value of
58$\mathrm{\SIUnitSymbolDegree}$ (+18)(-11) for the inclination of the stellar
rotation axis, which is significantly different from the outer disk axis
inclination of 21$\mathrm{\SIUnitSymbolDegree}$ given in the literature.

Conclusion. We find that DK Tau’s outer disk axis is likely misaligned
compared to its rotation axis by 37$\mathrm{\SIUnitSymbolDegree}$.

###### Key Words.:

Stars: individual: DK Tau - Stars: variables: T Tauri - Stars: magnetic field
- Accretion, accretion disks - Techniques: polarimetric - Techniques:
spectroscopic

##  1 Introduction

Stellar magnetic fields are omnipresent and play an essential role in the
formation of stars and planets. Understanding their impact is therefore
crucial to the study of stellar birth. We do not, however, yet possess a
complete picture of how stellar magnetic fields originate, how they evolve
over time, or the extent of their impact on circumstellar disks and the
accretion process in the early stages of a star’s life (see e.g., Bouvier et
al., 2007; Gregory et al., 2012; Folsom et al., 2016; Hartmann et al., 2016;
Villebrun et al., 2019). Stellar magnetic fields of accreting T Tauri stars
play an essential role in driving accretion and strongly impact the geometry
of the accretion flow (see Hartmann et al., 2016). By analyzing the magnetic
field along the line-of-sight and integrated over the visible stellar
hemisphere measured in the acc retion-powered emission lines, one can recreate
a picture of the component of the stellar magnetic field that dominates the
accretion process. This is an integral quantity that relates Stokes I and
Stokes V, making use of spectropolarimetric data. The interested reader is
referred to, for example, Rees & Semel (1979); Donati & Landstreet (2009);
Morin (2012). The Stokes parameters and the magnetic field are connected
through the Zeeman effect. This effect describes the impact of a magnetic
field on a spectrum: its atomic (and molecular) lines are broadened or split,
depending on the strength of the field and the sensitivity of the line in
question (see e.g., Tennyson, 2011; Hussain & Alecian, 2014).

In this work, we analyze the spectropolarimetry of the classical T Tauri star
(cTTs) DK Tau. DK Tau is a young low-mass star surrounded by a circumstellar
disk which is actively accreting from its inner regions. It is a wide binary
(separation 2$\aas@@fstack{\prime\prime}$38, equivalent to 307 au - see e.g.,
Manara et al., 2019), which allows DK Tau A (hereafter ”DK Tau”) to be
spatially resolved with spectropolarimetry and studied on its own. It is
located in the Taurus Molecular Cloud at a distance of 132.6 pc Gaia
Collaboration et al. (2016, 2018, 2022). Its spectral type is K7 (see e.g.,
Johns-Krull, 2007; Fischer et al., 2011) with a heliocentric radial velocity
of +16.2 km s-1 (Kounkel et al., 2019). Rota et al. (2022) measured the
inclination, with respect to the line of sight, of its outer (¿20 au) gaseous
disk axis via projection to be 21$\mathrm{\SIUnitSymbolDegree}$, using the
12CO (J = 2–1) line with ALMA. The star’s rotation period $P$, as determined
using photometry, has been measured as 8.4 days Bouvier et al. (1993) and 8.18
days Percy et al. (2010); Artemenko et al. (2012).

DK Tau presents with significant and variable veiling (a strong signature of
accretion - see e.g., Hartigan et al., 1995; Fischer et al., 2011). In
addition to accretion, it also shows evidence of ejection (see e.g., Hartigan
et al., 1995), in particular inner disk winds and a jet as revealed by the
detection of both low and high velocity forbidden [Oi] emission by Hartigan et
al. (1995) and Banzatti et al. (2019).

The effect of veiling on spectra complicates magnetic field measurements, yet
the study of active accretors is very valuable for understanding the
interaction between stellar magnetic fields and accreting material from the
circumstellar disk. Indeed, our current understanding of magnetospheric
accretion (see e.g., Shu et al., 1994; Romanova et al., 2002; Bessolaz et al.,
2008; Hartmann et al., 2016) involves the truncation of the inner
circumstellar disk at a few stellar radii and the channeling of accretion in
funnel flows by the stellar magnetic field. When the accreted matter falls
onto the star at near free-fall velocities, it produces an accretion shock
close to the stellar surface, which gives rise to continuum and line veiling
Calvet & Gullbring (1998) and generates a localized bright/hot spot at the
level of the chromosphere.

Simple models of stellar formation do not account for a misalignment between
the star’s rotation axis and its outer disk axis. However, misalignments may
be common, as several dippers display a low inclination of their outer disk
axis (see e.g., Ansdell et al., 2020; Sicilia-Aguilar et al., 2020). Dippers
are stars that show flux dips in their light curves. The standard explanation
involves circumstellar material from the inner disk passing in front of the
star, occulting it periodically or aperiodically (see e.g., McGinnis et al.,
2015; Roggero et al., 2021). Dippers therefore require a relatively high
inclination for the inner disk axis (which is here assumed to point in the
same direction as the stellar rotation axis), that is inconsistent with their
measured outer disk axis inclination. Such a misalignment has been directly
measured in few cTTs (see e.g., Alencar et al., 2018; Bouvier et al., 2020).
These observations suggest a more complex formation mechanism than normally
considered, though it is not clear what gives rise to such a misalignment. As
examples of potential causes, Sicilia-Aguilar et al. (2020) suggest the
possibility of two different protostellar collapses, whereas Alencar et al.
(2018) invoke the presence of a massive planet inside the disk gap, and
Benisty et al. (2018) the effects of a low-mass stellar companion. DK Tau
shows signs of being one of these systems with a misaligned outer disk.

In this paper, we describe our observations in Sect. 2. We detail our analysis
and results regarding stellar parameters, veiling, and magnetic
characterizations in Sect. 3. In Section 4 we discuss the implications of the
inclination we measure for DK Tau and its magnetic field. Finally, Sect. 5
contains our conclusions.

##  2 Observations

Our data set is comprised of two sets of circularly polarized spectra of DK
Tau, collected from December 2010 to January 2011, and from the end of
November to the end of December 2012, with the spectropolarimeters ESPaDOnS
(Echelle SpectroPolarimetric Device for the Observation of Stars), mounted at
the CFHT (Canada-France-Hawaii Telescope) 3.6 meter telescope in Hawaii
(Donati et al., 2006), and NARVAL, mounted at the 2 meter TBL (Télescope
Bernard Lyot) on the Pic du Midi in France (Aurière, 2003). These echelle
spectropolarimeters cover the visible domain, from 370 to 1 050 nm, in a
single exposure, and have a resolving power of 65 000. ESPaDOnS has a fiber
aperture of 1$\aas@@fstack{\prime\prime}$66, while NARVAL has one of
2$\aas@@fstack{\prime\prime}$80.

Table 1 lists the dates of the middle of the observations for the two ESPaDOnS
and NARVAL data sets. The total exposure time was 4 996.0 s for each ESPaDOnS
observation and 4 800.0 s for each NARVAL observation. In 2010 a total of 15
observations were taken over 39 days, and in 2012 a total of 12 observations
were taken over 35 days, with the intention of capturing a few rotation cycles
of the target in each set.

The data are public and were downloaded from the archive of the PolarBase
website111http://polarbase.irap.omp.eu (see e.g., Petit et al., 2014). These
observations were made as a result of proposals 10BP12 and 12BP12, with J.-F.
Donati as P.I in both cases and obtained as part of the MaPP (Magnetic
Protostars and Planets) large program at the CFHT. We also downloaded the
corresponding image files for the ESPaDOnS data from the Canadian Astronomy
Data Centre (CADC) website222http://www.cadc-ccda.hia-iha.nrc-cnrc.gc.ca/en.
The data had been previously reduced at the CFHT and TBL. The reduction was
carried out with the LibreESpRIT (for ”Echelle Spectra Reduction: an
Interactive Tool”) reduction package specifically built for extracting
polarization echelle spectra from raw data. This includes subtracting the bias
and the dark frames, and correcting for the variations in sensitivity using
flat field frames (Donati et al., 1997). The spectra were continuum normalized
in addition to the LibreESpRIT automatic continuum normalization, as the
automatic procedure is not tailored for stars presenting with emission lines
and it did not manage to properly adjust the continuum.

Table 1: Dates for the 2010 and 2012 ESPaDOnS and NARVAL data sets.  Date | Heliocentric Julian | Rotation cycle | S/N of the continuum | Airmass | Instrument
---|---|---|---|---|---
(yyyy-mm-dd) | date (UTC) | (8.2 day period) | at the central wavelength |  |
2010-11-26 | 2 455 527.436 03 | 0.00 | 70 | 1.1 | NARVAL
2010-12-09 | 2 455 540.393 30 | 1.58 | 77 | 1.2 | NARVAL
2010-12-10 | 2 455 541.397 11 | 1.70 | 53 | 1.1 | NARVAL
2010-12-13 | 2 455 544.418 52 | 2.07 | 42 | 1.1 | NARVAL
2010-12-14 | 2 455 544.979 74 | 2.14 | 72 | 1.1 | ESPaDOnS
2010-12-15 | 2 455 545.853 82 | 2.25 | 97 | 1.0 | ESPaDOnS
2010-12-16 | 2 455 546.854 61 | 2.37 | 100 | 1.0 | ESPaDOnS
2010-12-17 | 2 455 547.826 04 | 2.49 | 108 | 1.1 | ESPaDOnS
2010-12-18 | 2 455 548.819 03 | 2.61 | 113 | 1.1 | ESPaDOnS
2010-12-19 | 2 455 549.786 41 | 2.73 | 81 | 1.2 | ESPaDOnS
2010-12-19 | 2 455 550.390 86 | 2.80 | 54 | 1.1 | NARVAL
2010-12-24 | 2 455 554.883 63 | 3.35 | 83 | 1.0 | ESPaDOnS
2010-12-26 | 2 455 557.034 94 | 3.61 | 101 | 1.8 | ESPaDOnS
2010-12-30 | 2 455 560.977 48 | 4.09 | 96 | 1.3 | ESPaDOnS
2011-01-03 | 2 455 565.451 64 | 4.64 | 68 | 1.1 | NARVAL
2012-11-19 | 2 456 250.509 43 | 0.00 | 68 | 1.1 | NARVAL
2012-11-25 | 2 456 256.919 80 | 0.78 | 91 | 1.0 | ESPaDOnS
2012-11-28 | 2 456 259.895 31 | 1.15 | 121 | 1.1 | ESPaDOnS
2012-11-29 | 2 456 260.991 38 | 1.28 | 127 | 1.1 | ESPaDOnS
2012-12-01 | 2 456 262.947 48 | 1.52 | 105 | 1.0 | ESPaDOnS
2012-12-02 | 2 456 263.865 70 | 1.63 | 84 | 1.1 | ESPaDOnS
2012-12-04 | 2 456 265.965 15 | 1.89 | 120 | 1.0 | ESPaDOnS
2012-12-07 | 2 456 268.846 50 | 2.24 | 94 | 1.1 | ESPaDOnS
2012-12-09 | 2 456 271.387 97 | 2.55 | 70 | 1.2 | NARVAL
2012-12-10 | 2 456 271.825 04 | 2.60 | 107 | 1.2 | ESPaDOnS
2012-12-12 | 2 456 273.557 58 | 2.81 | 63 | 1.2 | NARVAL
2012-12-23 | 2 456 284.762 49 | 4.18 | 95 | 1.3 | ESPaDOnS

##  3 Analysis & results

###  3.1 Veiling

Accretion shocks are at a higher temperature than the photosphere. This adds
an extra continuum to the stellar continuum, artificially decreasing the depth
of the photospheric absorption lines. This is known as veiling and it varies
with the wavelength, and can also vary from night to night. Its effect needs
to be removed from the spectra in order to analyze the stellar magnetic field
from the photospheric absorption lines.

Veiling ($R$) is defined as the ratio between the flux of the accretion shock
and the photospheric flux. For a normalized spectrum, it can be expressed
using the following equation:

$I_{\textrm{v}}(\lambda)=[I_{\textrm{ph}}(\lambda)+R(\lambda)]N(\lambda)$ (1)

where $I_{\textrm{v}}(\lambda)$ refers to the veiled intensity at wavelength
$\lambda$, $I_{\textrm{ph}}(\lambda)$ to the intensity of the photosphere at
wavelength $\lambda$, $R(\lambda)$ to the veiling at wavelength $\lambda$ and
$N(\lambda)$ is a normalization constant at wavelength $\lambda$.

In order to measure the veiling in our spectra, we used a technique based on
the fitting of a rotationally broadened and artificially veiled weak-lined T
Tauri star (wTTs) spectrum to the spectrum of DK Tau. By choosing a wTTs with
the same spectral type as our cTTs and coming from the same star forming
region333This implies that both T Tauri stars would have the same chemical
composition and very similar age and $\log g$. We also assume that the
microturbulence and macroturbulence velocities should be very similar., this
wTTs can be seen as the purely photospheric version of our star because it
experiences no accretion. We tested several wTTs with a K7 spectral type and a
line-of-sight-projected equatorial rotational velocity $v\sin i$ lower than
the $v\sin i$ of DK Tau (see Sect. 3.2), in order to find the one that
provided the best fit. We ultimately used the spectrum of TAP45 (which has a
$v\sin i$ of 11.5 km s-1 \- see Feigelson et al., 1987; Bouvier et al., 1993)
as a template to estimate the veiling across DK Tau’s spectra as the extra
continuum that has to be added to the wTTs spectrum in order to reproduce the
veiled cTTs spectrum (at the lowest $\chi^{2}$ level).

We obtained values of the veiling as a function of the wavelength (in bins of
$\sim$20 nm), for each spectrum. When the value is less than $\sim$0.4
throughout the spectrum, we see that it is approximately constant in
wavelength, therefore we took the mean value as $R$ for the whole spectrum.
When it is larger than this, we fit a linear relation through the points and
considered this function as our $R(\lambda)$. We then inverted Eq. 1 to
recover the normalized photospheric spectrum of each observation. Figure 1
shows the night with the least veiling and the one with the most veiling on
the top and bottom panel respectively.

Figure 1: Veiling (gray dots), the best linear fit (blue line) and the
standard deviation (light blue shaded region) as a function of wavelength for
the fourth night of the 2010 ESPaDOnS observations (top panel) and for the
seventh night of the 2012 ESPaDOnS observations (bottom panel).

In 2010, the peak values of veiling (at $\sim$550 nm) for each observation
range from 0.2 to 1.8. Three observations out of fifteen have nearly constant
veiling values across their spectrum. In 2012, the peak values of veiling for
each observation range from 0.2 to 1.3 (lower than the value from two years
before). Only one observation out of twelve has a nearly constant veiling
value across its spectrum.

For the two sets of observations, we see that the slope of veiling as a
function of wavelength steepens when the veiling is higher as well. We also
find that when the veiling is high, there is more scatter. We believe this
scatter is due to the correlation of stronger line emission with stronger mass
accretion rate (see e.g., Dodin & Lamzin, 2012; Rei et al., 2018). In that
case, individual line emission would contribute more to the veiling by filling
in absorption lines; whereas for nights when the veiling is low, continuum
veiling would remain the dominant form of veiling. The interpretation of these
trends goes beyond the scope of this work, and will be investigated in a
subsequent paper.

###  3.2 Stellar parameters

Given the importance of accurate stellar parameters, we took advantage of
ESPaDOnS and NARVAL’s continuum normalized high resolution spectra to derive
precise values for DK Tau, in particular of the line-of-sight-projected
equatorial rotational velocity $v\sin i$, needed to investigate the potential
misalignment of DK Tau’s outer disk, and for the effective temperature
$T_{\textrm{eff}}$, which is needed for our analysis. For this, we first
obtained the mean spectrum of the four nights with the least amount of veiling
(i.e., $R$ = 0.2 for all four nights, see Sect. 3.1) in order to get a
spectrum with a higher signal to noise ratio of 136, whereas the individual
spectra had a signal to noise ratio of 103 on average. We then used the ZEEMAN
spectrum synthesis program, developed by Landstreet (1988) and Wade et al.
(2001) and modified by Folsom et al. (2016) to derive the stellar parameters.
Under the assumption of local thermodynamic equilibrium, this code solves
radiative transfer equations. It iteratively compares a synthetic spectrum
(calculated using a grid of model atmospheres and a list of atomic data) with
the observation, by $\chi^{2}$ minimization, to determine free model
parameters. Veiling was included in the model.

We used the Vienna Atomic Line Database (VALD)
website444http://vald.astro.uu.se (see e.g., Ryabchikova et al., 2015) to
acquire a continuous atomic line list ranging from 400 nm to 1 000 nm for a
star of $T_{\textrm{eff}}$ = 4 000 K, logarithmic surface gravity $\log g$ = 4
(in cgs units) and a solar metallicity. We used the MARCS model atmosphere
grid of Gustafsson et al. (2008). In the ZEEMAN code, we specified the
following initial values for the stellar parameters: for $T_{\textrm{eff}}$
and $v\sin i$, we used the values provided in the literature, of
$T_{\textrm{eff}}$ = 4 000 K Herczeg & Hillenbrand (2014), and $v\sin i$ =
12.7 km s-1 McGinnis et al. (2020). We used $\log g$ = 4 (in cgs units), as
this is typical of T Tauri stars (TTs), microturbulence velocity
$v_{\textrm{mic}}$ = 1 km s-1, macroturbulence velocity $v_{\textrm{mac}}$ = 0
km s-1, solar metallicity and a veiling of 0.1. We started with a model with
initial values of $T_{\textrm{eff}}$, $v\sin i$, $\log g$ and veiling for a
fixed value of $v_{\textrm{mic}}$, $v_{\textrm{mac}}$ and metallicity. We then
ran fits with only $T_{\textrm{eff}}$ and $v\sin i$ as free parameters. When
fitting the observation, we used a wavelength range from 400 nm to 1 000 nm.
We ran the code on several windows throughout the spectrum. We calculated the
average of the values obtained for the different spectral windows and took the
standard deviation of the spread as the error bars. We derive a
$T_{\textrm{eff}}$ of 4 150 $\pm$ 110 K and a $v\sin i$ of 13.0 $\pm$ 1.3 km
s-1. We note that our values are in good agreement with the ones found in the
literature (see Herczeg & Hillenbrand, 2014; McGinnis et al., 2020).

We also determined DK Tau’s radius. We first calculated the stellar luminosity
$L_{\star}$ using the J-band magnitudes of Eisner et al. (2007). As DK Tau A
and B are not spatially resolved in their observations, we corrected the
J-band magnitude to remove the contribution of DK Tau B. This was done by
extrapolating the brightness ratio given by Eisner et al. (2007) for the
K-band (of 3.3) to the J-band, taking into consideration the shape of the
continua of the two stars based on their respective spectral types, as well as
the individual extinction each star suffers (i.e., A${}_{\textrm{V}}$ = 0.7
mag for DK Tau A, and A${}_{\textrm{V}}$ = 1.80 mag for DK Tau B - Herczeg &
Hillenbrand, 2014). We find a flux ratio of 4.4 for the J-band. We then
corrected the magnitude for the extinction in DK Tau A (i.e., 0.7 mag) Herczeg
& Hillenbrand (2014)555We chose to use the value quoted by Herczeg &
Hillenbrand (2014) as they give a value for both components of the binary.. It
can be noted that the value for the extinction quoted by Fischer et al. (2011)
is a factor 2 higher. The variability of the extinction is probably connected
to the dipper behavior of the star (Roggero et al., 2021). We also corrected
for a veiling of $R_{J}=0.3\pm 0.1$, based on an interpolation of our
measurements of veiling as a function of wavelength (see Sect. 3.1). This is
very similar to the values observed by Fischer et al. (2011). We then used the
bolometric correction in the J band from Pecaut & Mamajek (2013). We find a
value of $L_{\star}$ = (1.65 $\pm$ 0.25) $L_{\odot}$. Using the relation
$R^{2}=L_{\star}/(4\pi\sigma T_{\textrm{eff}}^{4})$ and our measured value of
$T_{\textrm{eff}}$, we derive a stellar radius of $R_{\star}$ = (2.48 $\pm$
0.25) $R_{\odot}$.

Table 2 summarizes the measured properties of DK Tau. The inclination of the
outer disk axis was measured by Rota et al. (2022) using ALMA observations. We
derive a different value for the inclination $i$ of the stellar rotation axis
(see Sect. 4.1). The stellar rotation period $P$ mentioned in the literature
is based on photometry (see Bouvier et al., 1993; Percy et al., 2010;
Artemenko et al., 2012). The mass $M_{\star}$ was derived by Johns-Krull
(2007) from pre-main sequence evolutionary tracks. Using the Siess et al.
(2000)
models666http://www.astro.ulb.ac.be/~siess/pmwiki/pmwiki.php?n=WWWTools.PMS,
we find that the values of $T_{\textrm{eff}}$ and $L_{\star}$ that we obtain
give a slightly higher $M_{\star}$ than the one of 0.7 $M_{\odot}$ derived by
Johns-Krull (2007), but it agrees with 0.7 $M_{\odot}$ within 2$\sigma$ (see
Appendix A).

Table 2: Summary of the measured properties of DK Tau.

Stellar parameter | Value | Reference
---|---|---
$i$ ($\mathrm{\SIUnitSymbolDegree}$) | 58 (+18)(-11) | This work
Outer disk axis ($\mathrm{\SIUnitSymbolDegree}$) | 21 $\pm$ 3 | Rota et al. (2022)
$P$ (days) | 8.4 | Bouvier et al. (1993)
| 8.18 | Percy et al. (2010)
| 8.18 | Artemenko et al. (2012)
| 8.20 $\pm$ 0.13 | This work
$T_{\textrm{eff}}$ (K) | 4 150 $\pm$ 110 | This work
$v\sin i$ (km s-1) | 13.0 $\pm$ 1.3 | This work
$R_{\star}$ ($R_{\odot}$) | 2.48 $\pm$ 0.25 | This work
$M_{\star}$ ($M_{\odot}$) | 0.68 | Johns-Krull (2007)
$L_{\star}$ ($L_{\odot}$) | 1.65 $\pm$ 0.25 | This work

  * •

The inclination is derived from $v\sin i=2\pi R_{\star}P^{-1}\sin i$ (see
Sect. 4.1).

###  3.3 Least-squares deconvolution

Least-squares deconvolution (LSD) is a cross-correlation technique used to add
the signatures from hundreds of photospheric absorption lines and obtain an
average line profile of very high signal to noise ratio (see e.g., Donati et
al., 1997; Kochukhov et al., 2010). It makes use of a line mask, a list
describing the position and depth of the chosen absorption lines. We first
created a line mask using the VALD line list (with a detection threshold of
0.01, $T_{\textrm{eff}}$ = 4 000 K, $\log g$ = 4 and $v_{\textrm{mic}}$ = 1 km
s-1) and assuming an ATLAS9 stellar atmosphere model (Kurucz, 1993). Next,
using our line mask, we applied LSD to all observations in the range from 500
nm to 1000 nm (i.e., excluding the blue edge of the spectra because of excess
noise), excluding regions with telluric and emission lines, as well as the
lines most affected by accretion777We identified them by comparing with the
spectrum of RU Lup, a cTTs of the same spectral type as DK Tau but presenting
with more accretion, and determining the lines in emission., in order to
obtain a mean line profile for each veiling-corrected spectrum. We set the
effective Landé factor $\overline{g}_{0}$ = 1.4, the central intensity $d_{0}$
= 0.47 and the equivalent wavelength $\lambda_{0}$ = 650.0 nm for all the
nights (see Kochukhov et al., 2010). The LSD profiles were then normalized to
be at the same equivalent width. The latter was done in order to correct for
the residual effects of veiling. For all nights, we obtained a definite Zeeman
signal detection in the LSD profiles, with a false alarm probability smaller
than 0.001 % (see the LSD profiles in Appendix B).

Figure 2: LSD profile (in the heliocentric velocity frame) of Stokes V (top)
and Stokes I (bottom) parameters normalized to the continuum for the sixth
night of the 2010 ESPaDOnS observations. Note on this night scattered light
from the Moon caused the small blue-ward absorption feature. For comparison,
we also show the seventh night of the 2010 ESPaDOnS observations (dashed
line), which did not suffer from moonlight contamination.

The Stokes I LSD profiles of the ESPaDOnS and NARVAL observations made on 19
December 2010 presented a small absorption feature blueward of the main
absorption line (see e.g., Fig. 2). The other nights all showed a single
photospheric absorption line. This small absorption feature is due to
scattered moonlight, as the full Moon was close to DK Tau on that night (with
an angular separation of about 8$\mathrm{\SIUnitSymbolDegree}$), and as the
contamination is at the expected lunar radial velocity in the heliocentric
rest frame888The online applet at
https://astroutils.astronomy.osu.edu/exofast/barycorr.html, based on Wright &
Eastman (2014), allows to calculate the correction applied to geocentric
observations in order to transpose them in the heliocentric rest frame. In our
case, this correction is -8.8 km s-1. Applied to the Moon’s radial velocity of
0 km s-1 in the geocentric rest frame, this translates into a radial velocity
of -8.8 km s-1 in the heliocentric rest frame.. This type of contamination has
been seen before (see e.g., Donati et al., 2011).

We fit the wing of the main absorption feature with that of a Voigt profile
and manually removed the contamination.

###  3.4 Average line-of-sight magnetic field

We measured the magnetic field along the line-of-sight and integrated over the
visible stellar hemisphere, $B_{\text{los}}$999$B_{\text{los}}$ is also
referred to as the longitudinal field $B{{}_{\ell}}$. We chose not to use this
term to avoid confusion with its homonym $B_{\phi}$, the field along the east-
west direction or azimuthal field (see e.g., Vidotto, 2016)., by using
equation 3.3 from Morin (2012):

$B_{\text{los}}(G)=-2.14\times
10^{11}\frac{\int{v\,V(v)\,\text{d}v}}{\lambda_{0}\,g_{\text{eff}}\,c\,\int{[I_{c}-I_{v}]\,\text{d}v}}$
(2)

on the LSD profiles of the photospheric absorption lines (see also Rees &
Semel, 1979; Donati et al., 1997; Wade et al., 2000). In this equation, $v$ is
the radial velocity in the rest frame of the star, $V$ refers to Stokes V,
$\lambda_{0}$ is the wavelength of the line center in nm, $g_{\text{eff}}$ is
the effective Landé factor of the line, $I$ refers to Stokes I and $I_{c}$ to
the unpolarized continuum. We find values ranging from -0.19 $\pm$ 0.05 kG to
0.20 $\pm$ 0.03 kG in 2010 and from -0.13 $\pm$ 0.02 kG to 0.08 $\pm$ 0.02 kG
in 2012 (see Fig. 3, and see the list of values in Appendix B). It should be
noted that, since $B_{\text{los}}$ represents a signed average over the
visible stellar hemisphere, regions of opposite polarities partly cancel out.

We applied a phase dispersion minimization (PDM; Stellingwerf, 1978) technique
on the 2010 $B_{\text{los}}$ values and found a period of 8.20 $\pm$ 0.13
days. Since this is the period of the modulation of the stellar magnetic
field, it accurately represents the stellar rotation period. This period is
consistent with values found in the literature from photometry (8.4 days from
Bouvier et al., 1993, and 8.18 days from Percy et al., 2010 and Artemenko et
al., 2012). Small discrepancies could be explained by differential rotation:
different measurements tracking features at various latitudes.

Figure 3: Average line-of-sight magnetic field $B_{\text{los}}$ as derived
from the photospheric absorption lines and veiling over time, shown folded in
phase with the derived 8.2 day period, for the 2010 (left panel) and 2012
(right panel) datasets. Different colors and symbols represent different
rotation cycles.

###  3.5 Emission lines

After analyzing the LSD profiles of photospheric absorption lines to derive
the associated $B_{\text{los}}$ linked to non-accreting regions, we also
investigated emission lines associated with accretion shocks, in particular
the narrow component of the 587.6 nm Hei line and of the Caii infrared triplet
(IRT - at 849.8 nm, 854.2 nm and 866.2 nm). They can be used to get more
information on DK Tau’s magnetic fields, as they are tracers of the fields
present at the footpoints of accretion funnels (see e.g., Donati et al.,
2007). These lines are known to have multiple components (see e.g., Beristain
et al., 2001; McGinnis et al., 2020), often containing a narrow component
(NC), originating from the accretion shock, and a much broader component (or
components), which likely forms farther out in the accretion columns or in hot
winds. For this reason, we decomposed the profiles using a fit of 2 or more
Gaussians in order to isolate the NC (examples of these fits can be seen in
Appendix C). For the Caii lines, we then averaged the three lines into a
single LSD-like profile in order to increase the signal to noise ratio, as it
has been done in other studies (see e.g., Donati et al., 2007, 2008, 2012).
Since this is a triplet, the shape of all three lines should be the same (see
e.g., Azevedo et al., 2006). In addition, in our data, we see an intensity
ratio close to 1:1:1 and the NCs are not contaminated by the nearby Paschen
emission lines at 850.2 nm, 854.5 nm and 866.5 nm. The NC of the Hei line is
believed to be generated in the post-shock region at the base of the magnetic
accretion funnels that connect the surface of the star to its inner disk. The
NC of the Caii IRT is thought to probe the accretion regions and the
chromosphere.

The Stokes V and Stokes I profiles of the Hei emission line and the Caii IRT,
for both epochs, can be seen in Fig. 4. The Stokes V profiles of the emission
lines show similar signatures with phase. This indicates that the accretion
spot is mostly likely located at a high enough latitude to be visible with the
same polarity at all times. They also indicate that the field is positive
(i.e., pointing toward the observer) at the base of the accretion funnels that
connect the star to its circumstellar disk.

Figure 4: Stokes V and Stokes I profiles (in gray) and average (in red) of the
Hei emission line (top panels) and the Caii IRT (bottom panels), for the 2010
(left panels) and 2012 (right panels) observations.

We measured $B_{\text{los}}$ in the emission lines using the same method as
described in Sect. 3.4. We find values ranging from 0.20 $\pm$ 0.51 kG to 1.77
$\pm$ 0.08 kG for Hei in 2010, from 0.48 $\pm$ 0.09 kG to 1.99 $\pm$ 0.09 kG
for Hei in 2012, from 0.34 $\pm$ 0.04 kG to 0.95 $\pm$ 0.02 kG for Caii in
2010, from 0.26 $\pm$ 0.02 kG to 1.41 $\pm$ 0.05 kG for Caii in 2012. Figure 5
shows the obtained values folded in phase (and the list of values can be found
in Appendix C). The values of $B_{\text{los}}$ are higher in 2012. We find
that the values measured through the Caii IRT are lower than the ones measured
through the Hei line. It has been hypothesized that the lower intensity of
$B_{\text{los}}$ found using the Caii IRT stems from the dilution of the
emission from the accretion shock by chromospheric emission (see e.g., Donati
et al., 2019), which results in the Stokes V/I profile being shallower for the
Caii IRT than for the Hei line.

Figure 5: Average line-of-sight magnetic field $B_{\text{los}}$ over time,
shown folded in phase with an 8.2 day period, for the Hei emission line (top
panels) and the Caii IRT (bottom panels), for the 2010 (left panel) and 2012
(right panel) datasets. Different colors and symbols represent different
rotation cycles.

We find extreme values of $B_{\text{los}}$ in the emission lines that are one
order of magnitude larger than the extreme values of $B_{\text{los}}$ derived
from the LSD profiles of the photospheric absorption lines (see Fig. 3),
showing strong fields in the accretion shocks. This is consistent with the
current understanding of accretion shocks as compact regions with some of the
strongest magnetic field concentrating in dark polar regions at the surface of
the star and magnetic field lines reaching to the circumstellar disk. For both
epochs, we only see the positive pole and never the negative one.

The plots of the $B_{\text{los}}$ in the emission lines in 2012 do not fold
well in phase. We believe this may stem from the location of the accretion
shocks being more dynamic and/or the accretion being more complex, with more
than one accretion shock. This is also observed in the variability of veiling
in 2012 (see Fig. 3), which does not fold well with the rotation phase,
indicating that there is considerable intrinsic variability in the mass
accretion rate.

We derived the equivalent width (EW) of the Hei emission line. Figure 6 shows
the obtained values folded in phase (and the list of values can be found in
Appendix C). When the EW is larger, we are seeing more of the accretion shock
in our line-of-sight. This is also when we measure stronger magnetic fields
using emission lines tracing the accretion shock (see Fig. 5), as expected
following the paradigm of magnetospheric accretion.

Figure 6: Equivalent width (in $\mathring{A}$) of the Hei emission line over
time, shown folded in phase with an 8.2 day period, for the 2010 (left panel)
and 2012 (right panel) datasets. Different colors and symbols represent
different rotation cycles.

###  3.6 Magnetic Obliquity

We used the third equation of Preston (1967), which assumes a pure dipole, a
simplification of the magnetic field present in the accretion shocks, to
calculate an estimate of the magnetic obliquity (i.e., the angle between the
stellar rotation axis and the magnetic field axis) derived from the emission
lines. We used the extreme values found for $B_{\text{los}}$ in the emission
lines and an inclination $i$ of 58$\mathrm{\SIUnitSymbolDegree}$. For the Hei
emission line, we find a magnetic obliquity101010This is not the magnetic
obliquity of the entirety of DK Tau’s magnetic field. It is based solely on
the average line-of-sight magnetic field present in the accretion shocks and
probed through emission lines. Furthermore, the calculation uses the extreme
values for $B_{\text{los}}$ and does not account for variability between
nights. of 26$\mathrm{\SIUnitSymbolDegree}$ for the 2010 epoch, and of
21$\mathrm{\SIUnitSymbolDegree}$ for the 2012 epoch. For the Caii IRT, we find
a magnetic obliquity of 16$\mathrm{\SIUnitSymbolDegree}$ for the 2010 epoch,
and of 23$\mathrm{\SIUnitSymbolDegree}$ for the 2012 epoch. These estimates
are consistent with the Stokes V signatures of the emission lines. They are
also consistent with the magnetic obliquity of
18$\mathrm{\SIUnitSymbolDegree}$ (+8)(-7) in 2011 derived by McGinnis et al.
(2020), using the radial velocity variability of the Hei emission line and
assuming one accretion spot. We therefore have an agreement between the values
derived from the magnetic field that drives the accretion and the value
derived from a result of accretion.

The estimates of the magnetic obliquity are consistent with only seeing the
positive pole in the plots of the $B_{\text{los}}$ in the emission lines,
confirming that DK Tau experiences nearly poleward accretion, with a positive
field at the base of the accretion funnels.

The Hei emission lines show a radial velocity variability with a small
amplitude, which is another indication that the accretion spot is most likely
close to the pole. As the star rotates, if the accretion spot were located at
the equator, the velocity variation would be large as the spot gets red and
blueshifted.

###  3.7 Truncation & co-rotation radii

In order to calculate the truncation radius, we need to know the star’s mass
accretion rate. For this, we measured the equivalent width (EW) of several
emission lines (i.e., H$\alpha$, H$\beta$, H$\gamma$, the Hei lines at 447.1
nm, 667.8 nm and 706.5 nm, as well as the Caii IRT at 849.8 nm, 854.2 nm and
866.2 nm). Since ESPaDOnS and NARVAL’s spectra are not flux calibrated, we
created a template for DK Tau based on SO879, a weak-lined T Tauri star with a
K7 spectral type (described in Stelzer et al., 2013). The template was
corrected for extinction, then scaled to have the same luminosity (i.e., 1.65
$L_{\odot}$) and be at the same distance (i.e., 132.6 pc) as DK Tau. After
correcting the EW for veiling, we used this template to flux calibrate them
through the following formula:

$F_{\text{line}}=EW_{\text{line}}\cdot F_{\text{cont}}$ (3)

where $F_{\text{line}}$ is the flux of the line, $EW_{\text{line}}$ is the
veiling corrected EW of the line and $F_{\text{cont}}$ is the flux of the
continuum of the template at the wavelength of the line in question. Then we
obtained the luminosity in each line. Next, we used the relations in Table
B.1. from Alcalá et al. (2017) to calculate the accretion luminosity from each
line and averaged these values for each night. We then took the average over
all nights in each epoch as the accretion luminosity, and the standard
deviation of the spread in the values found from different nights as the error
bars. We find $L_{\text{acc}}$ = 0.26 $\pm$ 0.18 $L_{\odot}$ in 2010 and
$L_{\text{acc}}$ = 0.49 $\pm$ 0.42 $L_{\odot}$ in 2012. These values are
similar to the ones found e.g., by Fischer et al. (2011) (i.e., 0.17
$L_{\odot}$) or by Fang et al. (2018) (i.e., 0.16 $L_{\odot}$).

We then converted the accretion luminosity into mass accretion rate using Eq.
8 from Gullbring et al. (1998) with the values of $R_{\star}$ and $M_{\star}$
from Table 2 and $R_{\text{in}}$ = 5 $R_{\star}$ (as is typically used). We
find log ($\dot{M}_{\text{acc}}$[$M_{\odot}$ yr-1]) = -7.43 in 2010, and log
($\dot{M}_{\text{acc}}$[$M_{\odot}$ yr-1]) = -7.15 in 2012. These values are
consistent with the one of -7.42 quoted by Gullbring et al. (1998). Finally,
we used Eq. 6 from Bessolaz et al. (2008) to estimate the truncation radius.
This equation assumes an axisymmetric dipole, which is a simplification of DK
Tau’s magnetic topology. It also uses the dipolar field calculated at the
equator as $B_{\star}$. Considering the equatorial value is half of the value
at the pole, we estimated the latter (see Preston, 1967) using the values of
$B_{\text{los}}$ in the emission lines 111111We used the magnetic field
derived from the accretion-powered emission lines, considering it will
dominate over the magnetic field derived from the photospheric absorption
lines at the distance of the truncation radius. . This is an approximation, as
part of the $B_{\text{los}}$ could come from higher order multipoles, in
particular from the octupole, rather than the dipole. We find
$r_{\text{trunc}}$ $\sim$ (5.2 $\pm$ 1.1) $R_{\star}$ for the Hei emission
line in 2010, $r_{\text{trunc}}$ $\sim$ (3.9 $\pm$ 0.8) $R_{\star}$ for the
Caii IRT121212We find lower values for the truncation radius estimates when
using the Caii IRT. This stems from the lower values found for
$B_{\text{los}}$ in those lines (see Sect. 3.5). in 2010, $r_{\text{trunc}}$
$\sim$ (4.7 $\pm$ 1.2) $R_{\star}$ for the Hei emission line in 2012, and
$r_{\text{trunc}}$ $\sim$ (3.9 $\pm$ 1.0) $R_{\star}$ for the Caii IRT in
2012. We calculated the co-rotation radius as well, using Kepler’s third law:
$r_{\text{co-rot}}$ = 6.1 $R_{\star}$. We find that the truncation radius
values are consistent with the co-rotation radius within the error bars. This
implies that DK Tau is unlikely to be in the propeller regime, an unstable
accretion regime, as the truncation radius is not farther than the co-rotation
radius Romanova et al. (2018). For the 2010 epoch, DK Tau may be in the stable
accretion accretion regime, since the $B_{\text{los}}$ and accretion tracers
seem fairly periodic. The truncation radius being slightly smaller than the
co-rotation radius is consistent with this as well Blinova et al. (2016).

##  4 Discussion

###  4.1 Inconsistencies regarding the inclination

Naively one might assume that the inclination angle $i$ of the stellar
rotation axis with respect to the line-of-sight is
21$\mathrm{\SIUnitSymbolDegree}$ based on the inclination of the outer gaseous
disk axis Rota et al. (2022). DK Tau’s lightcurve however classifies the star
as a dipper (Roggero et al., 2021). The traditional explanation invokes
circumstellar material passing in front of the star and occulting it. If the
disk is seen close to edge on, matter lifted above the disk plane could cause
these occultations. However, DK Tau’s outer disk is seen nearly pole on (Rota
et al., 2022), which is inconsistent with this scenario, unless the stellar
rotation axis is at a very different angle than that of the outer disk axis.

Furthermore, based on the star’s rotational properties (see Table 2) and using
the following relation

$v\sin i=\frac{2\pi R_{\star}}{P}\sin i$ (4)

we derive a much higher inclination of 58$\mathrm{\SIUnitSymbolDegree}$
(+18)(-11). Therefore, if DK Tau is in fact seen nearly pole on, then $P$,
$v\sin i$ and $R_{\star}$ are not consistent with each other. As the stellar
radius is the most uncertain of these parameters131313This is because it
depends on evolutionary models as well as an accurate determination of the
effective temperature and stellar luminosity. Both can be subject to fairly
large uncertainties, particularly the luminosity for a star with a dipper
light curve which likely suffers from variable extinction., it is possible
that it may have been underestimated. However, if we consider that $P$, $v\sin
i$ and $i$ are accurately determined, then we would need $R_{\star}$ = (6
$\pm$1) $R_{\odot}$ for this formula to agree, which is unrealistically large
for a TTs.

Another possibility is that the period or $v\sin i$ may be inaccurate.
Regarding $v\sin i$, the value we derive agrees within error bars with the one
measured by McGinnis et al. (2020), despite using two different assessment
methods. It is therefore a value that can be trusted. This leaves the stellar
rotation period.

In the literature, the stellar rotation period of DK Tau has been measured
using photometry with values ranging from 8.18 days (Percy et al., 2010;
Artemenko et al., 2012) to 8.4 days (Bouvier et al., 1993). However, since the
photometry is dominated by flux dips that might be due to extinction events
(Roggero et al., 2021), it is possible that these dips are caused by
circumstellar material that is not located at the co-rotation radius. In that
case, the measured period would not be the same as the stellar rotation
period.

In Sect. 3.4 we derived a period from the rotational modulation of the line-
of-sight magnetic field $B_{\text{los}}$, which should accurately represent
the stellar rotation period. In the context of exoplanet search programs,
$B_{\text{los}}$ is indeed often considered as the most reliable indicator of
stellar rotation period (see e.g., Hébrard et al., 2016). Additionally, the
value we find of 8.20 $\pm$ 0.13 days is consistent with those found in the
literature from photometry. We therefore find that the value for the period
can be trusted.

Moreover, this rotation period can be seen in a number of datasets at our
disposal. For example, we computed bidimensional periodograms of the intensity
of the Hei (at 587.6 nm) emission line, and the Stokes I and Stokes V LSD
profiles of the photospheric absorption lines (see Appendix D). The Hei line
comes from the accretion shock and should therefore vary with the stellar
rotation period. In 2010 we find a period around 8 days for the entire red-
shifted part of the Hei line (from 0 to 50 km s-1), however this period is
very uncertain. The Stokes V profile also shows a period near 8.5 days between
$\sim$-20 and -7 km s-1 and between $\sim$0 and 8 km s-1, but again the
uncertainty is large. The Stokes I profile does not show a clear period. In
2012 there is no clear period found from the Hei line, likely because
accretion is more intrinsically variable in this epoch than 2 years prior.
Again no clear period is observed from the Stokes I profile, but the Stokes V
profile shows a possible periodicity at around 8 days (albeit with a large
uncertainty, same as in 2010).

Furthermore, the variation of the 2010 veiling as a function of time is also
consistent with an 8.2 day period (see Fig. 3). The variation of the 2012
veiling as a function of time, however, does not seem to follow any trend with
the period. This is consistent with the intensity of the HeI line not showing
a clear correlation with period in this epoch, since both are tracing
accretion. This is another indication that there must be some intrinsic
variability in the mass accretion rate in 2012, which masks any rotational
modulation of the accretion spot(s).

We looked at the possibility of the rotation period or $v\sin i$ being
inaccurate and found evidence to the contrary. Because their values appear to
be accurate, we deduce that it is the value for the inclination that is
problematic. We conclude that the inclination measured for the outer
circumstellar disk axis must not represent the inclination of the rotation
axis of the star. This suggests that there is a considerable misalignment
between the rotation axis of DK Tau and its outer disk. When we calculate DK
Tau’s inclination based on its rotational properties using Eq. 4, we find $i$
= 58$\mathrm{\SIUnitSymbolDegree}$ (+18)(-11). This value is based on $v\sin
i$ = (13.0 $\pm$ 1.3) km s-1, $P$ = (8.2 $\pm$ 0.2) days and $R_{\star}$ =
(2.48 $\pm$ 0.25) $R_{\odot}$. It follows that the outer disk axis of DK Tau
is likely misaligned by 37$\mathrm{\SIUnitSymbolDegree}$ with its rotation
axis (see Fig. 7).

Figure 7: Sketch (not to scale) showing DK Tau in the center, surrounded first
by its inner disk, then by its outer disk which is considerably misaligned.
The rotation axis at 58$\mathrm{\SIUnitSymbolDegree}$ is in red, the outer
disk axis at 21$\mathrm{\SIUnitSymbolDegree}$ is in blue, and the line-of-
sight axis is in gray (by C. Delvaux).

Misalignments between the inner and outer circumstellar disk axes of T Tauri
stars are starting to be observed, when combining near infrared
interferometric VLTI/GRAVITY data and millimeter interferometric ALMA data
(see e.g., Ansdell et al., 2020; Bouvier et al., 2020), or with shadows
observed with VLT/SPHERE (see e.g., Benisty et al., 2018; Sicilia-Aguilar et
al., 2020), or with VLTI/GRAVITY (see e.g., Bohn et al., 2022). We find a
misalignment between the outer disk axis and the rotation axis. What of the
inner disk axis? In young dippers like DK Tau, the material that causes the
dips is believed to be located in the inner accretion disk Bouvier et al.
(2007); McGinnis et al. (2015), which need to be observed at high inclinations
in order for material to cross our line-of-sight to produce the dips.
Therefore an inclination of 21$\mathrm{\SIUnitSymbolDegree}$ of the inner disk
of DK Tau is difficult to reconcile with its dipper light curve. In addition,
the inner disk axis is normally expected to be aligned with the stellar
rotation axis. We thus find it very likely that the inner disk axis of DK Tau
has the same inclination of $i=58$\mathrm{\SIUnitSymbolDegree}$$ as was
calculated for its rotation axis. This inclination is sufficiently high to
support the dipper behavior and there are other cases of dippers with similar
inclinations (see e.g., McGinnis et al., 2015; Roggero et al., 2021). DK Tau
is one more example of a TTS with a misalignment between its inner and outer
circumstellar disk axes.

It is also a wide binary system, and the misalignment could stem from the
binary formation mechanism: turbulent fragmentation might generate disk axis
that are more randomly oriented. It is however interesting to note as well
that the outer disk axes of both components of the binary are misaligned by
43$\mathrm{\SIUnitSymbolDegree}$ (see Rota et al., 2022), which is close to
the value (of 37$\mathrm{\SIUnitSymbolDegree}$) of the misalignment between
the inner and outer disk axes of DK Tau A. This could suggest a quasi-
alignment of the inner disk axis of DK Tau A with the outer disk axis of DK
Tau B, assuming that they are not only aligned compared to our line-of-sight,
but that the orientation of their nodes are aligned as well, which is unknown.

###  4.2 Magnetic field in the accretion-powered emission lines

The $B_{\text{los}}$ derived from the photospheric absorption lines (see Sect.
3.4), gives a partial view of DK Tau’s magnetic fields that exclude the
accreting regions, as photospheric absorption lines and accretion-powered
emission lines form in different regions of the stellar surface. It is the
field present in these accreting regions that is understood to best probe the
global stellar magnetic field that reaches to the circumstellar disk. The
magnetic obliquity derived from the accretion-powered emission lines is
therefore likely to be close to the actual value. We find that the low
magnetic obliquity that we derive (see Sect. 3.6), the positive polarity of
the $B_{\text{los}}$ in the emission lines, as well as their Stokes V
signatures and the range of radial velocity of the Hei line are consistent
with the presence of an accretion spot always visible and close to the pole.
This is where the accretion funnels connecting DK Tau to its disk would be
anchored. This is similar to what has been found for several other cTTs (see
e.g., Johnstone et al., 2014; McGinnis et al., 2020).

For the 2010 epoch, we find that the magnetic field in the Caii IRT (and in
the Hei line - see Fig. 5) is at a maximum close to the same phase (i.e.,
around phase 0.3) as the maximum in the veiling (see Fig. 3), which is when
the accretion shock is in our line-of-sight. Around phase 0.5, we see a small
redshifted absorption in the H$\alpha$ line (see Appendix E), indicating that
the accretion column is in our line-of-sight, which is directly after the
maximum of the magnetic field in the emission lines and the increase in
veiling, therefore probably directly after the accretion shock was in our
line-of-sight. Because this small redshifted absorption is not perfectly
simultaneous with the increase in veiling and in the emission lines, it might
be an indication of differential rotation.

##  5 Conclusions

In this paper, we have studied DK Tau, a low-mass classical T Tauri star
(cTTs) with significant veiling (defined as the ratio between the accretion
shock flux and the photospheric flux), using dual-epoch spectropolarimetric
observations (collected in 2010 and 2012). We derive an effective temperature
$T_{\textrm{eff}}$ of 4 150 $\pm$ 110 K and a line-of-sight-projected
equatorial rotational velocity $v\sin i$ of 13.0 $\pm$ 1.3 km s-1, in
agreement with the literature. We find peak values of veiling in the optical
($\sim$550 nm) ranging from 0.2 to 1.8 in 2010, and from 0.2 to 1.3 in 2012.

We derive the line-of-sight magnetic field integrated over the visible
hemisphere $B_{\text{los}}$ from the photospheric absorption lines (linked to
non-accreting regions). We find values ranging from -0.19 $\pm$ 0.05 kG to
0.20 $\pm$ 0.03 kG in 2010 and from -0.13 $\pm$ 0.02 kG to 0.08 $\pm$ 0.02 kG
in 2012.

We recover a rotation period of 8.2 days using the values of $B_{\text{los}}$
for the 2010 dataset. We confirmed the period by analyzing the intensity of
the Hei line in 2010, the intensity of the Stokes V profiles in both 2010 and
2012, and the variation of veiling as a function of time in 2010. They are all
consistent with an 8.2 day period. This also agrees with the values of period
given in the literature from photometry.

We find several inconsistencies related to the inclination of the stellar
rotation axis with respect to the line-of-sight $i$. The measurement of the
inclination of the outer circumstellar disk axis gives a value of
21$\mathrm{\SIUnitSymbolDegree}$ Rota et al. (2022). DK Tau’s lightcurve,
however, classifies it as a dipper (Roggero et al., 2021), for which the
simplest explanation involves a star seen close to edge on. Furthermore, using
Eq. 4, we find that the inclination of 21$\mathrm{\SIUnitSymbolDegree}$, the
period, $v\sin i$ and the radius are not consistent with each other. When
using the values of period, $v\sin i$ and stellar radius that we derive to
estimate the inclination of the stellar rotation axis, we find a value of $i$
= 58$\mathrm{\SIUnitSymbolDegree}$ (+18)(-11). We thus find a substantial
misalignment between DK Tau’s rotation axis (at
58$\mathrm{\SIUnitSymbolDegree}$) and its outer disk axis (at
21$\mathrm{\SIUnitSymbolDegree}$) to be likely.

To complement the partial picture of the $B_{\text{los}}$ derived from the
photospheric absorption lines, we analyzed emission lines that are tracers of
the magnetic fields present in the accretion shocks. We examined the narrow
component of the 587.67 nm Hei emission line and of the Caii infrared triplet
(IRT - at at 849.8 nm, 854.2 nm and 866.2 nm). We found that their Stokes V
profiles show similar signatures with phase, indicating that DK Tau
experiences poleward accretion, with a positive field at the base of the
accretion funnels connecting the star to its circumstellar disk.

We measured $B_{\text{los}}$ within the accretion shocks. We find values
ranging from 0.92 $\pm$ 0.09 kG to 1.77 $\pm$ 0.08 kG for Hei in 2010, from
0.48 $\pm$ 0.09 kG to 1.99 $\pm$ 0.09 kG for Hei in 2012, from 0.42 $\pm$ 0.02
kG to 0.95 $\pm$ 0.02 kG for Caii in 2010, from 0.30 $\pm$ 0.01 kG to 1.15
$\pm$ 0.02 kG for Caii in 2012. The positive polarity of the $B_{\text{los}}$
in the emission lines is again consistent with the presence of an accretion
spot always visible and close to the pole. This geometry is similar to what
has been found for other cTTs (see e.g., Johnstone et al., 2014; McGinnis et
al., 2020).

We derived an estimate of the magnetic obliquity from the emission lines using
the third equation of Preston (1967). This equation assumes a pure dipole,
which is a simplification of the magnetic field present in the accretion
shocks. It is the field present in these accretion shocks that is understood
to best probe the global stellar magnetic field that reaches the circumstellar
disk. The magnetic obliquities derived from the accretion-powered emission
lines are therefore a reasonable reflection of the real dipole present above
the surface of the star. For the Hei emission line, we find a magnetic
obliquity of 26$\mathrm{\SIUnitSymbolDegree}$ for the 2010 epoch, and of
21$\mathrm{\SIUnitSymbolDegree}$ for the 2012 epoch. For the Caii IRT, we find
a magnetic obliquity of 16$\mathrm{\SIUnitSymbolDegree}$ for the 2010 epoch,
and of 23$\mathrm{\SIUnitSymbolDegree}$ for the 2012 epoch. These estimates
are consistent with the magnetic obliquity of 18$\mathrm{\SIUnitSymbolDegree}$
(+8)(-7) in 2011 derived by McGinnis et al. (2020), using the Hei emission
line and assuming one accretion spot.

We also estimated the truncation radius using the values of $B_{\text{los}}$
in the emission lines, and find $r_{\text{trunc}}$ $\sim$ (5.2 $\pm$ 1.1)
$R_{\star}$ for the Hei emission line in 2010, $r_{\text{trunc}}$ $\sim$ (3.9
$\pm$ 0.8) $R_{\star}$ for the Caii IRT in 2010, $r_{\text{trunc}}$ $\sim$
(4.7 $\pm$ 1.2) $R_{\star}$ for the Hei emission line in 2012, and
$r_{\text{trunc}}$ $\sim$ (3.9 $\pm$ 1.0) $R_{\star}$ for the Caii IRT in
2012. We calculated the co-rotation radius as well, and find $r_{\text{co-
rot}}$ = 6.1 $R_{\star}$. We find that the truncation radius values are
consistent with the co-rotation radius within the error bars.

In conclusion, we find that DK Tau, presenting with significant veiling, has
similar magnetic properties to the more moderately accreting cTTs studied so
far. In addition, we find that DK Tau’s outer disk axis is likely to be
misaligned compared to its rotation axis by 38$\mathrm{\SIUnitSymbolDegree}$.
This poses questions with regards to standard models of circumstellar disk
formation. More observations of cTTs are needed to better understand the
prevalence of such misalignments, while the geometry of DK Tau’s system
requires additional studies to characterize it further.

## Acknowledgements

The authors thank A. Natta for helpful discussions and C. Delvaux for the
sketch of DK Tau. We also thank the referee for valuable comments that
improved the paper.

Based on observations obtained at the Canada-France-Hawaii Telescope (CFHT)
which is operated by the National Research Council of Canada, the Institut
National des Sciences de l’Univers of the Centre National de la Recherche
Scientifique of France, and the University of Hawaii. Based on observations
obtained at the Télescope Bernard Lyot (TBL) which is operated by the Institut
National des Sciences de l’Univers of the Centre National de la Recherche
Scientifique of France.

This work has made use of the VALD database, operated at Uppsala University,
the Institute of Astronomy RAS in Moscow, and the University of Vienna.

This project has received funding from the European Research Council (ERC)
under the European Union’s Horizon 2020 research and innovation programme
under grant agreement No 743029 (EASY: Ejection Accretion Structures in YSOs),
as well as grant agreement No 817540 (ASTROFLOW), grant agreement No 742095
(SPIDI) and grant agreement No 740651 (NewWorlds).

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##  Appendix A Stellar parameters

Using the Siess et al. (2000)
models141414http://www.astro.ulb.ac.be/~siess/pmwiki/pmwiki.php?n=WWWTools.PMS,
we checked the compatibility of the values of $M_{\star}$, $T_{\textrm{eff}}$
and $L_{\star}$ and obtained Fig. 8. The range of $T_{\textrm{eff}}$ and
$L_{\star}$ (accounting for their error bars) that we derive correspond to
masses that are close within 2$\sigma$ to the value of $M_{\star}$ = 0.7
$M_{\odot}$ quoted by Johns-Krull (2007).

Figure 8: Hertzsprung-Russell diagram with PMS evolutionary tracks from Siess
et al. (2000). The colored lines correspond to different masses (in
$M_{\odot}$). The gray rectangle highlights our values of $T_{\textrm{eff}}$
and $L_{\star}$ with their error bars.

##  Appendix B Photospheric absorption lines

Figure 9 shows the Stokes V and Stokes I profiles of the photospheric
absorption lines for both epochs. Table 3 lists the values of the
$B_{\text{los}}$, the line-of-sight magnetic field integrated over the visible
hemisphere, for the photospheric absorption lines.

Figure 9: Stokes V and Stokes I profiles (in gray) and average (in red) of the absorption lines, for the 2010 (left panels) and 2012 (right panels) observations.  Table 3: $B_{\text{los}}$ for the photospheric absorption lines.  Date | Rotation cycle | $B_{\text{los}}$
---|---|---
(yyyy-mm-dd) | (8.2 day period) | (G)
2010-11-26 | 0.00 | -148.49
2010-12-09 | 1.58 | 131.92
2010-12-10 | 1.70 | 16.08
2010-12-13 | 2.07 | -185.01
2010-12-14 | 2.14 | -186.52
2010-12-15 | 2.25 | -90.83
2010-12-16 | 2.37 | 30.36
2010-12-17 | 2.49 | 205.23
2010-12-18 | 2.61 | 87.10
2010-12-19 | 2.73 | 31.08
2010-12-19 | 2.80 | 6.72
2010-12-24 | 3.35 | 10.11
2010-12-26 | 3.61 | 100.33
2010-12-30 | 4.09 | -148.73
2011-01-03 | 4.64 | 93.22
2012-11-19 | 0.00 | -29.55
2012-11-25 | 0.78 | -124.88
2012-11-28 | 1.15 | -17.76
2012-11-29 | 1.28 | 81.99
2012-12-01 | 1.52 | -15.75
2012-12-02 | 1.63 | -78.84
2012-12-04 | 1.89 | -74.67
2012-12-07 | 2.24 | -5.17
2012-12-09 | 2.55 | -87.57
2012-12-10 | 2.60 | -102.34
2012-12-12 | 2.81 | -35.25
2012-12-23 | 4.18 | 26.34

##  Appendix C Emission lines

The emission lines associated with accretion shocks have multiple components
(usually a broad and a narrow component). The narrow component is believed to
come from the accretion shock, and that is the component we wish to isolate to
probe the magnetic field in the shock region. We fit several components of
each emission line that we studied and then subtracted all the components
except the narrow one, to get a residual profile. Figure 10 shows two
examples, for the Hei line (at 587.6 nm) and for the average of the Caii IRT
(at 849.8 nm, 854.2 nm and 866.2 nm).

Figure 10: Fit of the Hei line for the first night of the 2010 ESPaDOnS
observations (left panel) and of the average of the Caii IRT for the second
night of the 2012 ESPaDOnS observations (right panel). The observed line is in
black. The different components are in red and their sum is in blue.

Table 4 lists the values of the $B_{\text{los}}$, the line-of-sight magnetic
field integrated over the visible hemisphere, for the Hei line and the Caii
IRT.

Table 5 lists the values of the equivalent width (EW) of the Hei line.

Table 4: $B_{\text{los}}$ for the emission lines.  Date | Rotation cycle | $B_{\text{los}}$ | $B_{\text{los}}$
---|---|---|---
(yyyy-mm-dd) | (8.2 day period) | for Hei (G) | for Caii (G)
2010-11-26 | 0.00 | 688.77 | 341.07
2010-12-09 | 1.58 | 912.45 | 778.92
2010-12-10 | 1.70 | 1601.15 | 781.50
2010-12-13 | 2.07 | -201.64 | 450.74
2010-12-14 | 2.14 | 970.06 | 610.67
2010-12-15 | 2.25 | 1392.66 | 621.40
2010-12-16 | 2.37 | 1473.35 | 949.06
2010-12-17 | 2.49 | 1579.56 | 826.53
2010-12-18 | 2.61 | 1321.08 | 699.57
2010-12-19 | 2.73 | 1759.99 | 663.43
2010-12-19 | 2.80 | 1023.12 | 760.05
2010-12-24 | 3.35 | 1767.61 | 944.87
2010-12-26 | 3.61 | 923.56 | 671.05
2010-12-30 | 4.09 | 1198.59 | 426.48
2011-01-03 | 4.64 | 803.45 | 771.46
2012-11-19 | 0.00 | 1957.62 | 1411.28
2012-11-25 | 0.78 | 1988.06 | 876.93
2012-11-28 | 1.15 | 1130.63 | 838.51
2012-11-29 | 1.28 | 939.35 | 782.20
2012-12-01 | 1.52 | 479.01 | 300.70
2012-12-02 | 1.63 | 606.14 | 397.38
2012-12-04 | 1.89 | 1670.70 | 1052.93
2012-12-07 | 2.24 | 1872.10 | 1149.95
2012-12-09 | 2.55 | 1178.70 | 828.18
2012-12-10 | 2.60 | 1349.05 | 561.12
2012-12-12 | 2.81 | 1137.52 | 263.62
2012-12-23 | 4.18 | 1435.56 | 850.50
Table 5: EW of the Hei line.  Date | Rotation cycle | EW for
---|---|---
(yyyy-mm-dd) | (8.2 day period) | Hei ($\mathring{A}$)
2010-11-26 | 0.00 | 1.37
2010-12-09 | 1.58 | 1.49
2010-12-10 | 1.70 | 1.54
2010-12-13 | 2.07 | 1.32
2010-12-14 | 2.14 | 2.44
2010-12-15 | 2.25 | 2.04
2010-12-16 | 2.37 | 2.05
2010-12-17 | 2.49 | 1.46
2010-12-18 | 2.61 | 1.74
2010-12-19 | 2.73 | 1.50
2010-12-19 | 2.80 | 0.86
2010-12-24 | 3.35 | 2.20
2010-12-26 | 3.61 | 1.50
2010-12-30 | 4.09 | 1.95
2011-01-03 | 4.64 | 1.43
2012-11-19 | 0.00 | 2.61
2012-11-25 | 0.78 | 1.74
2012-11-28 | 1.15 | 1.61
2012-11-29 | 1.28 | 1.44
2012-12-01 | 1.52 | 1.43
2012-12-02 | 1.63 | 1.78
2012-12-04 | 1.89 | 2.57
2012-12-07 | 2.24 | 2.12
2012-12-09 | 2.55 | 1.91
2012-12-10 | 2.60 | 1.67
2012-12-12 | 2.81 | 1.19
2012-12-23 | 4.18 | 1.91

##  Appendix D Bidimensional periodograms

Fig. 11 shows the bidimensional periodograms of the intensity of the Hei (at
587.6 nm) emission line, the Stokes I and Stokes V LSD profiles of the
photospheric absorption lines. Bidimensional periodograms analyze the
intensity of the line in several bins over the velocity range. This allows us
to see if different parts of the lines have different periods and therefore
distinct origins. A dark color on the plots indicates a peak in the
periodogram, meaning that a period was found, but a large spot indicates a
large uncertainty.We would expect to find the same period in all the bins if
the entire line has a single origin. This is seen for instance in the plot of
the Hei line in 2010, where the same period is found for the whole redshifted
part of the line. However, the width of the peak of the periodogram is very
broad, indicating that there is a large uncertainty in this period.

Figure 11: Bidimensional periodograms of the intensity of the Hei (at 587.6
nm) emission line (top panel), the Stokes I (middle panel) and Stokes V LSD
profiles (bottom panel) of the photospheric absorption lines, for the 2010
(left panels) and 2012 epoch (right panels). The power of the periodogram is
showed using the color code. A light color represents a zero power intensity,
while a dark color represents the maximum power intensity.

##  Appendix E H$\alpha$ lines

Fig. 12 shows the H$\alpha$ line for the fourth and fifth night of the 2010
ESPaDOnS observations, corresponding to phase $\sim$0.5. For both nights, we
see a small redshifted absorption, indicating that the accretion column is in
our line-of-sight.

Figure 12: H$\alpha$ line for the fourth night (right panel) and the fifth
night (left panel) of the 2010 ESPaDOnS observations. The dotted line
highlights the continuum.


210mm 297mm

# Design of Induction Machines using
Reinforcement Learning

Yasmin SarcheshmehPour1,2, Tommi Ryyppö1, Victor Mukherjee1, and Alex Jung2  1
Technology Center, ABB Motors and Generators, 00380 Helsinki, Finland  2
Department of Computer Science, Aalto University, 02150 Espoo, Finland

###### Abstract

The design of induction machine is a challenging task due to different
electromagnetic and thermal constraints. Quick estimation of machine’s
dimensions is important in the sales tool to provide quick quotations to
customers based on specific requirements. The key part of this process is to
select different design parameters like length, diameter, tooth tip height and
winding turns to achieve certain torque, current and temperature of the
machine. Electrical machine designers, with their experience know how to alter
different machine design parameters to achieve a customer specific operation
requirements. We propose a reinforcement learning algorithm to design a
customised induction motor. The neural network model is trained off-line by
simulating different instances of electrical machine design game with a reward
or penalty function when a good or bad design choice is made. The results
demonstrate that the suggested method automates electrical machine design
without applying any human engineering knowledge.

###### Index Terms:

Design automation, Induction machines, Neural networks, Reinforcement
learning.

##  I Introduction

Electric motors design and dimension estimation need to be fast, especially in
sales tool in big corporation where thousands of motors are manufactured daily
based on different customer requirements. This is specifically very important
in large scale industry where customer ask for quick dimension of machines
based on their application, efficiency, load curve and operational conditions.
In this case, an electric motor designer select a machine which is closely
matching with the customer requirements from an existing database, often
referred as base machine, and then manually change few machine parameters to
match the customer requirements. Normally, this is done with a deterministic
approach either manually or with an algorithm where machine designer increase
or decrease certain machine design parameters to check if the customer
requirements are met. However, this is still a time consuming process, and can
be an impediment to give customer an immediate feedback of machine design,
price etc during a sales negotiation.

Using meta-heuristic optimisation algorithm to resolve the above mentioned
issue can be challenging as in electric machine design optimisation problem
some fundamental variables have more importance than other geometrical shape
variables. In [1], a reinforcement learning algorithm and evolutionary
optimisation method is proposed to tackle this issue for a linear induction
motor. However, finding fundamental machine variables like pole pair number,
slot number with stator and rotor slot shape dimensions seems a fairly complex
and time consuming process. Also, in [1], the proposed method design machine
from scratch, whereas in our current context, the optimisation space is
comparatively simpler as the base induction machine already have most of the
correct fundamental machine design variables like pole numbers, stator and
rotor slot numbers, machine diameters etc. Due to the manufacturing limit of
our case, only few fixed diameters of stator and rotor laminations and rotor
slots dimensions can be selected. Upon selecting a base machine, 3 machine
design variables (length of the machine, rotor tooth tip height and number of
coil turns) need to be selected automatically to meet the customer
specification. If the goal is not met, then next base machine is selected, for
example, with a bigger diameter and other geometrical parameters, and then
again the 3 machine design variables are searched to met the customer
requirements.

Reinforcement Learning (RL) methods have been applied to optimization problems
arising in important application domains such as arithmetic circuits, chip
placement, aerodynamic design and so on. In all these applications a deep RL
algorithm is chosen to solve the corresponding optimization problem [2]. In
our context, an agent plays different games of electrical machine design and
at each step it checks if the design choices has brought it closer to the
objective. The usefulness of an action is quantified by some reward signal.
After playing several games, the agent learns what kind of design choices like
altering length, coil turns and tooth tip height need to be made to achieve
certain torque speed curve, temperature rise, efficiency, weight of the
machine etc. We propose a novel designing method for the induction machines
based on RL [3] which use proximal policy optimisation (PPO) [4] to update the
policy neural network (NN) based on cumulative reward for every game. The
developed neural network model decreases the machine designing computational
time to less than a minute. To the best knowledge of the authors, this is the
first time when a design automation method of rotating electrical machine is
investigated using RL, Proximal Policy Optimization (PPO) and NN. This model
is developed for sales tool, however, engineers can also achieve an excellent
initial machine design.

##  II Problem Formulation

Designing an electrical machine can be formulated as a parameter optimization
problem that consists of an objective function $F(x)$, design variables $x$ as
well as equality and inequality constraints. In our simplified electrical
machine design problem, the feasibility of the machine design is checked with
certain performance values which are rotor tooth tip height, airgap flux
density, current, torque, and stator temperature rise. These performance
values are directly or indirectly related to customer specific requirements
and manufacturing constraints. In practice, there is a performance flag
$\in\\{-1,0,1\\}$ for each of these performance values that checks one
characteristic of the electrical machine. For every flag, if its relevant
performance value meets the requirements, then the flag value is $0$,
otherwise it is either $+1$ or $-1$. A machine design can be considered as
feasible if and only if all its flags are equal to zero. we model the above
mentioned electrical machine design problem as a RL problem. We solve this RL
problem using PPO.

##  III Method

RL problems can be formulated as Markov Decision Processes (MDPs), consisting
of three key elements, states, actions, and reward [2]. In this setting, each
state is the concatenations of the performance flags’ values, and the previous
action. In this paper, the action space consists of $6$ different actions,
decreasing/increasing length, number of coil turns, and rotor tooth tip height
of the machine. The agent (policy network) starts from an initial state
$s_{0}$, which is a base machine being calculated based on the user specified
requirements and the manufacturing constraints, and the final state $s_{T}$
corresponds to a feasible electrical machine that fits the given set of
requirements and constraints. At each time step $t$, the agent begins in state
$s_{t}$, takes an action $a_{t}$ that changes the design variables, arrives at
a new state ($s_{t+1}$), and receives a reward ($r_{t}$) that is calculated
based on the performance values.

Through repeated episodes (sequences of states, actions, and rewards), the
policy network learns to take actions that maximizes the cumulative reward. We
use a model-free on-policy RL method called PPO [4] to update the parameters
of the policy network. PPO algorithms are policy gradient methods, which
implies that they search the space of policies rather than assigning values to
state-action pairs. As the optimization direction is guided by cumulative
rewards, the reward function is the key element of RL methods.

In this paper, for designing the reward function we only consider the flags’
values. Each flag’s value ($+1$ or $-1$) indicates that the action should
decrease or increase the corresponding performance value. Therefore, for the
flag $f_{i}$, the related reward, $w_{f_{i}}$, is $w_{p}>0$ (positive reward)
as long as the corresponding performance value is modified towards its correct
direction, otherwise it is $w_{n}<0$ (negative reward). As an example, if
$torque$ flag is $+1$, the action that increases the $torque$ performance
value gives a positive reward ($w_{torque}=w_{p}$) and the action that
decreases it gives a negative reward ($w_{torque}=w_{n}$). For combining
multiple objectives into one reward function, we take the sum of the rewards
for all the flags. We continue giving these rewards and penalties until all
the flags’ values reach to $0$.

Sometimes agent can not find the correct machine design after getting stuck in
the different loops. In order to prevent these loops, some priorities are
defined among these flags. In addition to this, a negative penalty is given to
the actions that leads to a machine state which has been already found during
the training. The agent also loses the game if it reaches a fixed maximum
number of total steps. When the agent wins the game by getting all zero flags,
then it receives a huge reward.

##  IV Results

Three different induction machines are selected as a case study for this work.
The PPO model is trained on 75 different variations of these selected
induction machines. The game is designed in a way that for each machine an
initial design $s_{0}$ is calculated based on the power, voltage, and type of
the machine. As mentioned in the section II, the observation state consists of
the values of the design flags and the initial action. Then at each step $t$,
one of the $6$ available actions is chosen. The maximum number of total steps
is 300. Table I indicates the average number of steps for PPO for the selected
machines. Figure 1 also represent PPO steps for a single induction machine. In
the full paper, more details on the RL implementation, hyperparameter tuning,
reward functions and its influence on the game’s success and failure will be
discussed. Also, more detail analysis on the design outcomes of different
machines (different power, size etc) and scaling of the methods for different
base machines will be presented.

TABLE I: The average number of steps for PPO. Machine |  power |  voltage |  PPO steps
---|---|---|---
1 |  2500 kW |  10000 V |  11
2 |  600 kW |  6000 V |  12
3 |  2100 kW |  6000 V |  5
Figure 1: PPO steps for a single induction machine.

## References

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$\displaystyle n_{\gamma}=$ $\displaystyle-
C_{ac|ac}C_{ab|cd}^{2}-C_{ac|ac}C_{ab|cd}Cacbd+2C_{ac|ac}Cacbd^{2}-3C_{ab|ac}C_{ab|cd}C_{ac|bc}$
$\displaystyle-3C_{ab|cd}C_{ac|ad}C_{ac|bc}-6C_{ab|ac}CacbdC_{ac|bc}-6C_{ac|ad}CacbdC_{ac|bc}-3C_{ab|ad}C_{bc|cd}$
$\displaystyle-
C_{ac|ac}C_{ab|cd}+3C_{bc|cd}C_{ac|ad}+3C_{ab|ac}C_{ac|cd}+3C_{ac|ad}C_{ac|cd}+C_{ac|ac}Cacbd$
$\displaystyle+3C_{ab|ad}C_{ac|bc}-3C_{ac|ad}C_{ac|bc},$ $\displaystyle
d_{\gamma}=$
$\displaystyle-3C_{ab|ac}C_{bc|cd}-4C_{ac|ac}C_{ab|cd}+3C_{bc|cd}C_{ac|ad}+6C_{ab|ac}C_{ac|cd}-3C_{ab|ad}C_{ac|cd}$
$\displaystyle+3C_{ac|ad}C_{ac|cd}-2C_{ac|ac}Cacbd-3C_{ab|ac}C_{ac|bc}+3C_{ab|ad}C_{ac|bc}-6C_{ac|ad}C_{ac|bc}+2C_{ac|ac},$
$\displaystyle n_{h_{1}}=$
$\displaystyle(-3C_{ac|ad}C_{ac|cd}+C_{ac|ac})x_{1}+6C_{ab|ac}C_{bc|cd}-3C_{ab|ad}C_{bc|cd}+3C_{ac|ac}C_{ab|cd}$
$\displaystyle-6C_{ab|ac}C_{ac|cd}+3C_{ab|ad}C_{ac|cd}+3C_{ac|ac}Cacbd+3C_{ac|ad}C_{ac|bc}-3C_{ac|ac},$
$\displaystyle d_{h_{1}}=$
$\displaystyle+3C_{ab|ac}C_{bc|cd}-3C_{ab|ad}C_{bc|cd}-C_{ac|ac}C_{ab|cd}+3C_{bc|cd}C_{ac|ad}+C_{ac|ac}Cacbd$
$\displaystyle-3C_{ab|ac}C_{ac|bc}+3C_{ab|ad}C_{ac|bc}-3C_{ac|ad}C_{ac|bc},$
$\displaystyle n_{h_{2}}=$
$\displaystyle(3C_{ab|ac}C_{ac|bc}-C_{ac|ac})x_{2}+3C_{ab|ad}C_{bc|cd}+3C_{ac|ac}C_{ab|cd}-6C_{bc|cd}C_{ac|ad}$
$\displaystyle-3C_{ab|ac}C_{ac|cd}-3C_{ab|ad}C_{ac|bc}+6C_{ac|ad}C_{ac|bc},$
$\displaystyle d_{h_{2}}=$
$\displaystyle-3C_{ab|ac}C_{bc|cd}+3C_{ab|ad}C_{bc|cd}-C_{ac|ac}C_{ab|cd}-3C_{bc|cd}C_{ac|ad}+3C_{ab|ac}C_{ac|cd}$
$\displaystyle-3C_{ab|ad}C_{ac|cd}+3C_{ac|ad}C_{ac|cd}-2C_{ac|ac}Cacbd+C_{ac|ac},$
$\displaystyle n_{x_{1}}=$
$\displaystyle+3C_{ab|ac}C_{bc|cd}-3C_{ab|ad}C_{bc|cd}+3C_{ac|ac}C_{ab|cd}-3C_{ab|ac}C_{ac|cd}$
$\displaystyle d_{x_{1}}=$
$\displaystyle-3C_{ab|ad}C_{bc|cd}-C_{ac|ac}C_{ab|cd}+3C_{bc|cd}C_{ac|ad}+3C_{ab|ac}C_{ac|cd}$
$\displaystyle+3C_{ac|ad}C_{ac|cd}-2C_{ac|ac}Cacbd-C_{ac|ac},$ $\displaystyle
n_{x_{2}}=$
$\displaystyle+3C_{ab|ad}C_{bc|cd}+3C_{ac|ac}C_{ab|cd}-3C_{bc|cd}C_{ac|ad}-6C_{ab|ac}C_{ac|cd}$
$\displaystyle+3C_{ab|ad}C_{ac|cd}+3C_{ab|ac}C_{ac|bc}-3C_{ab|ad}C_{ac|bc}+6C_{ac|ad}C_{ac|bc}-3C_{ac|ac},$
$\displaystyle+3C_{ab|ad}C_{ac|cd}-3C_{ac|ad}C_{ac|cd}+3C_{ac|ac}Cacbd-3C_{ab|ad}C_{ac|bc}+3C_{ac|ad}C_{ac|bc},$
$\displaystyle d_{x_{2}}=$
$\displaystyle-3C_{ab|ac}C_{bc|cd}+3C_{ab|ad}C_{bc|cd}-C_{ac|ac}C_{ab|cd}+C_{ac|ac}Cacbd$
$\displaystyle-3C_{ab|ac}C_{ac|bc}-3C_{ac|ad}C_{ac|bc}+2C_{ac|ac}.$

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