# Magnetic topologies of young suns:
The weak-line T Tauri stars TWA 6 and TWA 8A

C. A. Hill1, C. P. Folsom1, J.-F. Donati1, G. J. Herczeg2, G. A. J.
Hussain3,1, S. H. P. Alencar4, S. G. Gregory5 and the MaTYSSE collaboration.
1IRAP, Université de Toulouse, CNRS, UPS, CNES, 14 Avenue Edouard Belin,
Toulouse, F-31400, France
2Kavli Institute for Astronomy and Astrophysics, Peking University, Yi He Yuan
Lu 5, Haidian Qu, Beijing 100871, China
3ESO, Karl-Schwarzschild-Str. 2, D-85748 Garching, Germany
4Departamento de Fìsica - ICEx-UFMG, Av. Antônio Carlos, 6627, 30270–901 Belo
Horizonte, MG, Brazil
5University of St Andrews, St Andrews, KY16 9SS, UK
E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

We present a spectropolarimetric study of two weak-line T Tauri stars (wTTSs),
TWA 6 and TWA 8A, as part of the MaTYSSE (Magnetic Topologies of Young Stars
and the Survival of close-in giant Exoplanets) program. Both stars display
significant Zeeman signatures that we have modelled using Zeeman Doppler
Imaging (ZDI). The magnetic field of TWA 6 is split equally between poloidal
and toroidal components, with the largest fraction of energy in higher-order
modes, with a total unsigned flux of 840 G, and a poloidal component tilted
$35\degr$ from the rotation axis. TWA 8A has a 70 per cent poloidal field,
with most of the energy in higher-order modes, with an unsigned flux of 1.4 kG
(with a magnetic filling factor of 0.2), and a poloidal field tilted $20\degr$
from the rotation axis. Spectral fitting of the very strong field in TWA 8A
(in individual lines, simultaneously for Stokes $I$ and $V$) yielded a mean
magnetic field strength of $6.0\pm 0.5$ kG. The higher field strengths
recovered from spectral fitting suggests that a significant proportion of
magnetic energy lies in small-scale fields that are unresolved by ZDI. So far,
wTTSs in MaTYSSE appear to show that the poloidal-field axisymmetry correlates
with the magnetic field strength. Moreover, it appears that classical T Tauri
stars (cTTSs) and wTTSs are mostly poloidal and axisymmetric when mostly
convective and cooler than $\sim 4300$ K, with hotter stars being less
axisymmetric and poloidal, regardless of internal structure.

###### keywords:

stars: magnetic fields – techniques: polarimetric – stars: formation – stars:
imaging – stars: individual: TWA 6 – stars: individual: TWA 8A

††pubyear: 2018††pagerange: Magnetic topologies of young suns:  The weak-line
T Tauri stars TWA 6 and TWA 8A–C

##  1 Introduction

During the first few hundred thousand years of low-mass star formation,
class-I pre-main sequence (PMS) stars accrete significant amounts of material
from their surrounding dusty envelopes. After around 0.5 Myr, these protostars
emerge from their dusty cocoons and are termed classical T Tauri stars (cTTSs
/ class-II PMS stars) if they are still accreting from their surrounding
discs, or weak-line T Tauri stars (wTTSs / class-III PMS stars) if they have
exhausted the gas from the inner disc cavity. During the PMS phase, stellar
magnetic fields have their largest impact on the evolution of the star. These
fields control accretion processes and trigger outflows/jets (Bouvier et al.,
2007), dictate the star’s angular momentum evolution by enforced spin-down
through star-disc coupling (e.g., Davies et al. 2014), and alter disc dynamics
and planet formation (Baruteau et al., 2014). Moreover, as PMS stars are
gravitationally contracting towards the MS, the change in stellar structure
from fully to partly convective is expected to alter the stellar dynamo
mechanism and the resulting magnetic field topology.

Previous work through the MaPP (Magnetic Protostars and Planets) survey
revealed that the large-scale topologies of 11 cTTSs remained relatively
simple and mainly poloidal when the host star is still fully or largely
convective, but become much more complex when the host star turns mostly
radiative (Gregory et al., 2012; Donati et al., 2013). This survey concluded
that these fields likely originated from a dynamo, varying over time-scales of
a few years (Donati et al., 2011, 2012, 2013), and resembling those of mature
stars with comparable internal structure (Morin et al., 2008).

The nature of the magnetic fields of wTTSs and how they depend on fundamental
parameters is less well known. These evolutionary phases are the initial
conditions in which disc-less PMS stars initiate their unleashed spin up
towards the zero-age main sequence (ZAMS). Hence, it is crucial to
characterize their magnetic fields and how they depend on mass, temperature,
age and rotation. To this end, we are performing a spectropolarimetric study
of around 30 wTTSs through the MaTYSSE (Magnetic Topologies of Young Stars and
the Survival of close-in giant Exoplanets) programme, mainly allocated on
ESPaDoNS at the Canada-France-Hawaii Telescope (CFHT), complemented by
observations with NARVAL on the Telescope Bernard Lyot, and with HARPS on the
ESO 3.6-m Telescope. By using Zeeman Doppler Imaging (ZDI) to characterize the
magnetic fields of wTTSs, we are able to test stellar dynamo theories and
models of low-mass star formation. Moreover, by filtering out the activity-
related jitter from the radial velocity (RV) curves, we are able to
potentially detect hot Jupiters (hJs; see Donati et al. 2016), and thus verify
whether core accretion and migration is the most likely mechanism for forming
close-in giant planets (e.g., Alibert et al., 2005).

Here, we present our detailed analysis of the wTTSs TWA 6 and TWA 8A as part
of the MaTYSSE survey. Both targets are members of the TW Hydrae association,
which, at an age of $10\pm 3$ Myr (Bell et al., 2015), is in transition
between the T Tauri and the post T Tauri phase, and thus provides a very
interesting period in which to study the properties of the member stars as
they spin-up towards the ZAMS. Our phase-resolved spectropolarimetric
observations are documented in Section 2, with the stellar and disc properties
presented in Section 3. We discuss the spectral energy distributions, several
emission lines, and the accretion status of both stars in Section 3.2. In
Section 4 we present our results after applying our tomographic modelling
technique to the data. In Section 5 we present our results of our spectral
fitting to the Stokes  _I_ and Stokes  _V_ spectra, and in Section 6 we
discuss our analysis of the filtered RV curves. Finally, we discuss and
summarize our results and their implications for low-mass star and planet
formation in Section 7.

##  2 Observations

Spectropolarimetric observations of TWA 6 were taken in February 2014, with
observations of TWA 8A taken in March and April 2015, both using ESPaDOnS at
the 3.6-m CFHT. Spectra from ESPaDOnS span the entire optical domain (from
370–1000 nm) at a resolution of 65,000 (i.e., a resolved velocity element of
4.6 kms-1) over the full wavelength range, in both circular or linear
polarization (Donati, 2003).

A total of 22 circularly-polarized (Stokes  _V_) and unpolarized (Stokes  _I_)
spectra were collected for TWA 6 over a timespan of 16 nights, corresponding
to around 29.6 rotation cycles (where $P_{\text{rot}}=0.5409$ d, Kiraga 2012).
Time sampling was fairly regular, with the longest gap of 6 nights occurring
towards the end of the run. For TWA 8A, 15 spectra were collected with regular
time sampling over a 15 night timespan, corresponding to around 3.2 rotation
cycles (where $P_{\text{rot}}$ = 4.638 d, Kiraga 2012).

All polarization spectra consist of four individual sub-exposures (each
lasting 406 s for TWA 6, and 1115 s for TWA 8A), taken in different
polarimeter configurations to allow the removal of all spurious polarization
signatures at first order. All raw frames were processed using the Libre
ESpRIT software package, which performs bias subtraction, flat fielding,
wavelength calibration, and optimal extraction of (un)polarized échelle
spectra, as described in the previous papers of the series (Donati et al.
1997, also see Donati et al. 2010, 2011, 2014), to which the reader is
referred for more information. The peak signal-to-noise ratios (S/N, per 2.6
kms-1 velocity bin) achieved on the collected spectra range between 111–197
(median 164) for TWA 6, and 209–369 (median 340) for TWA 8A, depending on
weather/seeing conditions. All spectra are automatically corrected for
spectral shifts resulting from instrumental effects (e.g., mechanical
flexures, temperature or pressure variations) using atmospheric telluric lines
as a reference. This procedure provides spectra with a relative RV precision
of better than 0.030 kms-1 (e.g. Moutou et al., 2007; Donati et al., 2008). A
journal of all observations is presented in Table 1 for both stars.

Table 1: Journal of ESPaDOnS observations of TWA 6 (first 22 rows) and TWA 8A (last 15 rows), each consisting of a sequence of 4 subexposures lasting 406 s and 1115 s for TWA 6 and TWA 8A, respectively. Columns 1–4 list (i) the UT date of the observation, (ii) the corresponding UT time at mid exposure, (iii) the Barycentric Julian Date (BJD), and (iv) the peak signal-to-noise ratio (per 2.6 kms-1 velocity bin) of each observation. Columns 5 and 6 respectively list the S/N in Stokes  _I_ LSD profiles (per 1.8 kms-1 velocity bin), and the rms noise level (relative to the unpolarized continuum level $I_{\text{c}}$ and per 1.8 kms-1 velocity bin) in the Stokes  _V_ LSD profiles. Column 6 indicates the rotational cycle associated with each exposure, using the ephemerides given in Equation 1. Date | UT | BJD | S/N | S/N${}_{\textsc{lsd}}$ | $\sigma_{\textsc{lsd}}$ | Cycle
---|---|---|---|---|---|---
(2014) | (hh:mm:ss) | (2456693.9+) |  |  | (0.01%) |
Feb 04 | 11:15:00 | 0.07239 | 160 | 1796 | 5.6 | 0.151
Feb 04 | 12:17:52 | 0.11605 | 184 | 2260 | 4.5 | 0.232
Feb 07 | 10:28:34 | 3.04027 | 134 | 1619 | 6.2 | 5.638
Feb 07 | 11:30:04 | 3.08299 | 131 | 1559 | 6.4 | 5.717
Feb 07 | 13:04:25 | 3.14851 | 158 | 1849 | 5.4 | 5.838
Feb 09 | 09:27:53 | 4.99822 | 168 | 1990 | 5.1 | 9.258
Feb 09 | 10:46:58 | 5.05313 | 169 | 1973 | 5.1 | 9.359
Feb 09 | 11:48:45 | 5.09605 | 132 | 1639 | 6.1 | 9.439
Feb 10 | 11:08:25 | 6.06807 | 163 | 1914 | 5.3 | 11.236
Feb 10 | 12:10:38 | 6.11128 | 178 | 2126 | 4.7 | 11.316
Feb 11 | 09:37:38 | 7.00507 | 183 | 2273 | 4.4 | 12.968
Feb 11 | 11:05:43 | 7.06624 | 197 | 2461 | 4.1 | 13.081
Feb 11 | 11:54:04 | 7.09981 | 169 | 1922 | 5.2 | 13.143
Feb 12 | 09:21:45 | 7.99407 | 164 | 1933 | 5.2 | 14.797
Feb 12 | 11:22:19 | 8.07780 | 159 | 1869 | 5.4 | 14.951
Feb 12 | 12:49:30 | 8.13834 | 160 | 1855 | 5.4 | 15.063
Feb 13 | 10:35:31 | 9.04533 | 111 | 1514 | 6.6 | 16.740
Feb 13 | 13:06:11 | 9.14997 | 180 | 2094 | 4.8 | 16.934
Feb 19 | 09:23:25 | 14.99545 | 160 | 1847 | 5.4 | 27.741
Feb 19 | 10:51:16 | 15.05645 | 181 | 2223 | 4.5 | 27.853
Feb 19 | 12:29:16 | 15.12451 | 192 | 2387 | 4.2 | 27.979
Feb 20 | 12:06:25 | 16.10867 | 137 | 1345 | 7.5 | 29.799
(2015) |  | (2457107.9+) |  |  |  |
Mar 25 | 11:43:05 | 0.06756 | 338 | 3847 | 2.6 | 0.020
Mar 26 | 11:04:43 | 1.04090 | 343 | 3812 | 2.6 | 0.230
Mar 27 | 11:40:03 | 2.06545 | 340 | 3863 | 2.6 | 0.451
Mar 28 | 11:30:18 | 3.05868 | 341 | 3841 | 2.6 | 0.665
Mar 29 | 12:16:00 | 4.09040 | 302 | 3348 | 3.0 | 0.887
Mar 30 | 11:49:48 | 5.07220 | 369 | 4244 | 2.4 | 1.099
Mar 31 | 08:28:33 | 5.93245 | 357 | 4071 | 2.5 | 1.285
Apr 01 | 08:26:55 | 6.93130 | 349 | 3960 | 2.5 | 1.500
Apr 03 | 11:25:50 | 9.05552 | 253 | 2670 | 3.8 | 1.958
Apr 04 | 11:34:15 | 10.06136 | 355 | 4091 | 2.5 | 2.175
Apr 05 | 08:42:10 | 10.94184 | 332 | 3764 | 2.7 | 2.365
Apr 06 | 08:30:36 | 11.93379 | 353 | 4013 | 2.5 | 2.579
Apr 08 | 09:13:10 | 13.97061 | 202 | 1754 | 5.7 | 3.018
Apr 09 | 07:18:42 | 14.90036 | 253 | 2819 | 3.6 | 3.218
Apr 09 | 08:45:35 | 14.95143 | 309 | 3621 | 2.8 | 3.229

##  3 Stellar and disc properties

Both stars are part of the TW Hya association (TWA, e.g., Jayawardhana et al.,
1999; Webb et al., 1999; Donaldson et al., 2016), one of the closest young
star associations at a distance of $\simeq 50$ pc (e.g., Zuckerman & Song,
2004). Furthermore, at an age of $10\pm 3$ Myr (Bell et al., 2015), TWA is at
a crucial evolutionary phase where star-disk interactions have ceased, and
where the T Tauri stars are rapidly spinning up as they continue their
gravitational contraction towards the main sequence (e.g., Rebull et al.,
2004).

Both stars are classed as T Tauri due to strong Li i 6708 Å absorption (e.g.,
Webb et al., 1999), with mean equivalent widths (EW) of around 0.45 Å (20
kms-1) and 0.38 Å (17 kms-1) for TWA 6 and TWA 8A, respectively (slightly
lower than the 0.56 Å and 0.53 Å found by Torres et al. 2003). Furthermore,
our spectra show that the strength of Li i 6708 Å absorption does not vary
significantly for either star, indicating a lack of veiling (in agreement with
Herczeg & Hillenbrand 2014), and confirming their status as wTTSs (see Section
3.2 and B for further discussion). Moreover, both stars show very regular
periodic light-curves that do not appear like those of cTTS, further
supporting their non-accreting status.

For TWA 6, we adopt the photometric rotation period of 0.5409 d found by
Kiraga (2012) for the remainder of this work, as this is in excellent
agreement with the $0.54\pm 0.01$ d period of Lawson & Crause (2005), and the
$0.54090\pm 0.00005$ d period of Skelly et al. (2008). For TWA 8A, we adopt
the photometric period of 4.638 d (Kiraga, 2012), in excellent agreement with
the $4.65\pm 0.01$ d period found by Lawson & Crause (2005), the $4.66\pm
0.06$ d period of Messina et al. (2010), and the 4.639 d period found by
applying a Lomb-Scargle periodogram analysis to SuperWASP photometric data
(Butters et al., 2010). The rotational cycles of TWA 6 and TWA 8A (denoted
$E_{1}$ and $E_{2}$ in Equation 1) are computed from Barycentric Julian Dates
(BJDs) according to the (arbitrary) ephemerides:

BJD (d) $\displaystyle=2456693.9+0.5409E_{1}$ (for TWA 6) BJD (d)
$\displaystyle=2457107.9+4.638E_{2}$ (for TWA 8A) (1)

###  3.1 Stellar properties

To determine the $T_{\text{eff}}$ and $\log{g}$ of our target stars, we
applied our automatic spectral classification tool (discussed in Donati et al.
2012) to several of the highest S/N spectra for both stars. We fit the
observed spectrum using multiple windows in the wavelength ranges 515–520 nm
and 600–620 nm (using Kurucz model atmospheres, Kurucz 1993), in a similar way
to the method of Valenti & Fischer (2005). This process yields estimates of
$T_{\text{eff}}$ and $\log{g}$, where the optimum parameters are those that
minimize $\chi^{2}$, with errors bars determined from the curvature of the
$\chi^{2}$ landscape at the derived minimum.

For TWA 6, we find that ${T_{\text{eff}}=4425\pm 50}$ K and ${\log{g}=4.5\pm
0.2}$ (with $g$ in cgs units). While two-temperature modelling such as that
carried out by Gully-Santiago et al. (2017) would provide a better estimate of
$T_{\text{eff}}$ and the fractional spot coverage, for our purposes, a
homogeneous model is sufficient. For TWA 6, we adopt the $V$ and $B$
magnitudes of $10.88\pm 0.05$ and $12.19\pm 0.05$ from (Messina et al., 2010),
and assuming a spot coverage of the visible stellar hemisphere of $\sim 30$
per cent (typical for such active stars, see Section 4), we derive an
unspotted $V$ magnitude of $10.6\pm 0.2$. We note that assuming a different
spot coverage (such as 0 or 50 per cent) places our derived parameters within
our quoted error bars. Using the relation from Pecaut & Mamajek (2013), the
expected visual bolometric correction for TWA 6 is $BC_{\text{v}}=-0.70\pm
0.04$, and as there is no evidence of extinction to TWA members (e.g. Stelzer
et al., 2013), we adopt $A_{\textsc{v}}=0$. Combining $V$, $BC_{\text{v}}$,
$A_{\textsc{v}}$ and the trigonometric parallax distance found by Gaia of
$63.9\pm 1.4$ pc (corresponding to a distance modulus of $4.03\pm 0.05$, Gaia
Collaboration et al. 2016; Gaia Collaboration et al. 2018, in excellent
agreement with the $59.59\pm 3.6$ pc of Donaldson et al. 2016), we obtain an
absolute bolometric magnitude of $5.85\pm 0.29$, or equivalently, a
logarithmic luminosity relative to the Sun of $-0.44\pm 0.12$. When combined
with the photospheric temperature obtained previously, we obtain a radius of
$1.0\pm 0.2$ R☉.

Coupling $P_{\text{rot}}$ (see Equation 1) with the measured $v\sin{i}$ of
$72.6\pm 0.5$ kms-1 (see Section 4), we can infer that $R_{\star}\sin{i}$ is
equal to $0.78\pm 0.01$ R☉, where $R_{\star}$ and $i$ denote the stellar
radius and the inclination of its rotation axis to the line of sight. By
comparing the luminosity-derived radius to that from the stellar rotation, we
derive that $i$ is equal to $49\degr^{+15}_{-8}$, in excellent agreement with
that found using our tomographic modelling (see Section 4). Using the
evolutionary models of Siess et al. (2000) (assuming solar metallicity and
including convective overshooting), we find that TWA 6 has a mass of $0.95\pm
0.10$ M☉, with an age of $21\pm 9$ Myr (see the H-R diagram in Figure 1, with
evolutionary tracks and corresponding isochrones). Similarly, using the
evolutionary models of Baraffe et al. (2015), we obtain a mass of $0.95\pm
0.10$ M☉ and an age of $17\pm 7$ Myr.

For TWA 8A, our spectral fitting code yields a best-fit at
${T_{\text{eff}}=3800\pm 150}$ K and ${\log{g}=4.7\pm 0.2}$, however, this
$T_{\text{eff}}$ is in the regime where the Kurucz synthetic spectra are
considered unreliable in terms of temperature. To address this issue, we are
currently working on a more advanced spectral classification tool based on
PHOENIX model atmospheres and synthetic spectra (see Allard, 2014). In the
mean time for the work presented here, we determined $T_{\text{eff}}$ for TWA
8A from the observed $B-V$ value and the relation between $T_{\text{eff}}$ and
$B-V$ for young stars from Pecaut & Mamajek (2013) (and by assuming
$A_{\textsc{v}}=0$). We adopt $V=12.265\pm 0.023$ and $B=13.70\pm 0.03$ from
Henden et al. (2016), with $B-V=1.434\pm 0.038$. Using this $B-V$ with the
relation between intrinsic colour and $T_{\text{eff}}$ for young stars found
by Pecaut & Mamajek (2013), and assuming $A_{\textsc{v}}=0$, we derive
$T_{\text{eff}}=3690\pm 130$ K. Combining the observed $V$ magnitude with the
expected $BC_{\text{v}}$ for TWA 8A of $-1.50\pm 0.19$ (Pecaut & Mamajek,
2013) with the trigonometric parallax distance of $46.27\pm 0.19$ pc as found
by Gaia (Gaia Collaboration et al. 2016; Gaia Collaboration et al. 2018,
corresponding to a distance modulus of $3.326\pm 0.009$, in excellent
agreement with the $47.2\pm 2.8$ pc of Donaldson et al. 2016 and
$46.9_{-2.9}^{+3.3}$ pc of Riedel et al. 2014), we obtain an absolute
bolometric magnitude of $7.1\pm 0.3$, or equivalently, a logarithmic
luminosity relative to the Sun of $-0.93\pm 0.11$. When combined with the
photospheric temperature obtained previously, we obtain a radius of $0.8\pm
0.2$ R☉. Combining this radius with the mass derived below (from Baraffe et
al. 2015 evolutionary models), we estimate $\log{g}=4.3\pm 0.3$.

Combining $P_{\text{rot}}$ (see Equation 1) with the $v\sin{i}$ of $4.82\pm
0.16$ kms-1 (see Section 5), we find $R_{\star}\sin{i}=0.44\pm 0.03$ R☉,
yielding $i=32\degr^{+13}_{-8}$, in good agreement with our tomographic
modelling (see Section 4). Using Siess et al. (2000) models we find M$=0.45\pm
0.10$ M☉, with an age of $11\pm 5$ Myr. Using the evolutionary models of
Baraffe et al. (2015), we find M=$0.55\pm 0.1$ M☉, with an age of $13\pm 6$
Myr.

We note that we do not consider the formal error bars on the derived masses
and ages to be representative of the true uncertainties, given the inherent
limitations of these evolutionary models. Furthermore, we note that for
internal consistency with previous MaPP and MaTYSSE results, the values from
the Siess et al. (2000) models should be referenced. We note that the ages
derived here are consistent with the age of the young TWA moving group (of
$10\pm 3$ Myr, Bell et al. 2015), and that both evolutionary models suggest
that TWA 6 has a mostly radiative interior, where as TWA 8A is mostly (or
fully) convective.

The temperatures measured here are hotter than expected from spectral types
estimated from red-optical spectra that cover TiO and other molecular bands
(White & Hillenbrand, 2004; Stelzer et al., 2013; Herczeg & Hillenbrand,
2014). This discrepancy is consistent with past wavelength-dependent
differences in photospheric temperatures from young stars, which may be
introduced by spots (e.g. Bouvier & Appenzeller, 1992; Debes et al., 2013;
Gully-Santiago et al., 2017). The interpretation of these differences is not
yet understood. Use of the lower temperatures that are measured at longer
wavelengths from molecular bands would lead to lower masses and younger ages.
Our temperatures are accurate measurements of the photospheric emission from
5000–6000 Å and are consistent with all temperature measurements for stars in
the MaTYSSE program.

Figure 1: H-R diagram showing the stellar evolutionary tracks provided by
Siess et al. (2000, blue solid lines) and Baraffe et al. (2015, black solid
lines) for masses of 0.3, 0.4, 0.5, 0.9 and 1.0 M☉. Blue dashed lines show the
corresponding isochrones for ages 5, 10 & 20 Myr, and blue dotted lines mark
the 0 and 50 per cent fractional radius for the bottom of the convective
envelope, both for for Siess et al. (2000) models.

###  3.2 Spectral energy distributions

Spectral Energy Distributions (SEDs) of TWA 6 and TWA 8A were constructed
using photometry sourced from the DENIS survey (DENIS Consortium, 2005), the
AAVSO Photometric All Sky Survey (APASS, Henden et al., 2015), the GALEX all-
sky imaging survey (Bianchi et al., 2011), the TYCHO-2 catalogue (Høg et al.,
2000), the WISE, Spitzer and Gaia catalogues (Wright et al., 2010; Werner et
al., 2004; Gaia Collaboration et al., 2016; Gaia Collaboration et al., 2018),
and Torres et al. (2006). We note that deep, sensitive sub-mm and mm
photometry are not currently available for our targets. Comparing the SEDs
(shown in Fig. 2) to PHOENIX-BT-Settl synthetic spectra (Allard, 2014), we
find that neither TWA 6 nor TWA 8A have an infrared excess up to 23.675
$\mu$m, indicating that both objects have dissipated their circumstellar
discs. Given that the SEDs of TWA 6 and TWA 8A show no evidence of an infra-
red excess, both stars are likely disc-less and are not accreting (also see
e.g., Weinberger et al. 2004; Low et al. 2005). However, for completeness, in
Appendix B we present several metrics that determine the accretion rates from
emission lines (if accretion were present), with our analysis showing that
chromospheric emission likely dominates the line formation for both targets,
confirming their classification as wTTSs.

Figure 2: Spectral energy distributions (SEDs) of TWA 6 (top) and TWA 8A
(bottom), where the photometric data (see text) are shown as black dots, and
where PHOENIX-BT-Settl model spectra (Allard, 2014) are shown as a red line.
For the model spectra we adopt $T_{\text{eff}}=4400$ K and 3700 K for TWA 6
and TWA 8A respectively, and $\log{g}=4.5$ for both stars, as well as the
other parameters given in Table 2, adopting the extinction relation of
Cardelli et al. (1989). Furthermore, we assume that both stars have a 30 per
cent surface coverage of cool starspots (see Section 4.1), and so the
displayed spectra have a 30 per cent contribution from a spectrum that is 1000
K cooler.

###  3.3 Emission line analysis

We find that TWA 6 shows core Ca ii infrared triplet (IRT) emission (see Fig.
15) with a mean equivalent width (EW) of around 0.3 Å (10.7 kms-1), similar to
what is expected from chromospheric emission for such PMS stars (e.g. Ingleby
et al., 2011), and lower than that for accreting cTTSs (e.g. Donati et al.,
2007). The core Ca ii IRT emission is somewhat variable, with both red and
blue-shifted peaks (where the red-shifted emission is generally larger), and
where the emission is significantly higher at cycles 9.258, 9.359, 14.951 and
15.063. We note that there are some differences in the Stokes  _V_ line
profiles of the Ca ii IRT, that are likely due to their different atmospheric
formation heights. We note that no significant Zeeman signatures are detected
in Ca ii H&K, Ca ii IRT or He i 5875.62 Å and so the emission is likely
chromospheric rather than from the magnetic footpoints of an accretion funnel.
TWA 6 also shows single-peaked H$\alpha$ and H$\beta$ emission that displays
relatively little variability over the $\sim 30$ rotation cycles (see Fig.
15). For H$\alpha$, significantly higher flux is seen in cycles 9.258, 9.359
and 9.439, with the extra emission arising in a predominantly red-shifted
component. Moreover, cycle 14.797 displays a significantly higher flux that is
symmetric about zero velocity. This higher flux is also seen in H$\beta$, with
larger emission for cycles 9.258 and 9.359 (both asymmetric, red-shifted),
14.797 (symmetric) and 14.951 (asymmetric, red-shifted). Given that these
emission features occur at similar phases in Ca ii IRT, H$\alpha$ and
H$\beta$, and are also short lived, they likely stem from the same formation
mechanism in the form of stellar prominences that are rotating away from the
observer. This conclusion is also supported by the mapped magnetic topology,
as we see closed magnetic loops off the stellar limb, along which prominence
material may flow. To better determine the nature of the emission and its
variability, one can calculate variance profiles and autocorrelation matrices,
as described in Johns & Basri (1995) and given by:

$V_{\lambda}=\left[\frac{\sum^{n}_{i=1}\left(I_{\lambda,i}-\overline{I_{\lambda}}\right)^{2}}{n-1}\right]^{\frac{1}{2}}$
(2)

Fig. 18 shows that the H$\alpha$ emission varies from around -200 kms-1 to
+300 kms-1 (similar to that found previously for TWA 6 by Skelly et al. 2008),
well beyond the $v\sin{i}$ of 72.6 kms-1, and with most of the variability in
a red-shifted component. Furthermore, the autocorrelation matrix shows strong
correlation of the low-velocity components, indicating a common origin. We
find that H$\beta$ and He i D3 show negligible variability, with a relatively
low spectral S/N limiting the analysis.

In the case of TWA 8A, core Ca ii IRT emission is present with a mean EW of
around 0.37 Å (11.9 kms-1, see Fig. 16). This emission is mostly non-variable,
with only cycle 1.958 showing significantly higher (symmetric) emission.
Furthermore, the Zeeman signatures in the Stokes  _V_ line profiles (see Fig.
17) have the same sign as those of the absorption lines (see Fig. 3), and so
are of photospheric origin. TWA 8A also displays double-peaked H$\alpha$ and
H$\beta$ emission, with a peak separation of around 40 kms-1. This separation
lies well within the co-rotation radius, and is only a few times larger than
the $v\sin{i}$ of 4.82 kms-1, indicating that the source of the emission is
chromospheric. The lines are somewhat variable, with a significant increase in
emission (for both H$\alpha$ and H$\beta$) at cycles 1.958, 2.579 and 3.018.
Fig. 19 shows the variance profiles and autocorrelation matrices of H$\alpha$,
H$\beta$ and He i D3, Here we see that for H$\alpha$, the variability
concentrates in two peaks centred around -50 kms-1 and +75 kms-1 (ranging $\pm
150$ kms-1), with variability in H$\beta$ likewise occurring in two peaks
centred around -75 kms-1 and +65 kms-1 (ranging $\pm 150$ kms-1), with both
autocorrelation matrices showing the low-velocity components to be highly
correlated. For He i D3 we find that the variability is single peaked, centred
around zero velocity, with only low-velocity components showing significant
correlation. We also note that the H$\alpha$ emission of TWA 8A shows strong
Zeeman signatures (see Fig. 17) that are opposite in sign to those of the
absorption lines (see Fig. 5), as expected for chromospheric emission.

##  4 Tomographic modelling

In order to map both the surface brightness and magnetic field topology of TWA
6 and TWA 8A, we have applied our dedicated stellar-surface tomographic-
imaging package to the data sets described in Section 2. In doing this, we
assumed that the observed variability is dominated by rotational modulation
(and optionally differential rotation). Our imaging code simultaneously
inverts the time series of Stokes  _I_ and Stokes  _V_ profiles into
brightness maps (featuring both cool spots and warm plages) and magnetic maps
(with poloidal and toroidal components, using a spherical harmonic
decomposition). For brightness imaging, a copy of a local line profile is
assigned to each pixel on a spherical grid, and the total line profile is
found by summing over all visible pixels (at a given phase), where the pixel
intensities are scaled iteratively to fit the observed data. For magnetic
imaging, the Zeeman signatures are fit using a spherical-harmonic
decomposition of potential and toroidal field components, where the weighting
of the harmonics are scaled iteratively (Donati, 2001). The data are fit to an
aim $\chi^{2}$, with the optimal fit determined using the maximum-entropy
routine of Skilling & Bryan (1984), and where the chosen map is that which
contains least information (where entropy is maximized) required to fit the
data. For further details about the specific application of our code to wTTSs,
we refer the reader to previous papers in the series (e.g., Donati et al.,
2010, 2014, 2015).

As with previous studies of wTTSs, we applied the technique of Least-Squares
Deconvolution (LSD, Donati et al., 1997) to all of our spectra. Given that
relative noise levels are around $10^{-3}$ in a typical spectrum (for a single
line), with Zeeman signatures exhibiting relative amplitudes of $\sim 0.1$ per
cent, the use of LSD allows us to create a single ‘mean’ line profile with a
dramatically enhanced S/N, with accurate error bars for the Zeeman signatures.
LSD involves cross-correlating the observed spectrum with a stellar line-list,
and for this work, stellar line lists were sourced from the Vienna Atomic Line
Database (VALD, Ryabchikova et al., 2015), computed for $T_{\text{eff}}=4500$
K and $\log{g}=4.5$ (in cgs units) for TWA 6, and $T_{\text{eff}}=3750$ K and
$\log{g}=4.5$ for TWA 8A (the closest available to our derived spectral-types,
see Section 3.1). Only moderate to strong atomic spectral lines were included
(with line-to-continuum core depressions larger than 40 per cent prior to all
non-thermal broadening). Furthermore, spectral regions containing strong lines
mostly formed outside the photosphere (e.g. Balmer, He, Ca ii H&K and Ca ii
IRT lines) and regions heavily crowded with telluric lines were discarded (see
e.g. Donati et al. 2010 for more details), leaving 6088 and 5953 spectral
lines for use in LSD, for TWA 6 and TWA 8A, respectively. Expressed in units
of the unpolarized continuum level $I_{\text{c}}$ (and per 1.8 kms-1 velocity
bin), the average noise level of the resulting Stokes  _V_ signatures range
from 4.1–7.5$\times 10^{-4}$ (median of $5.3\times 10^{-4}$) for TWA 6, and
2.4–5.7$\times 10^{-4}$ (median of $2.6\times 10^{-4}$) for TWA 8A.

Figure 3: LSD circularly-polarized (Stokes  _V_ , top/red curve) and
unpolarized (Stokes  _I_ , bottom/blue curve) profiles of TWA 6 (top,
collected on 19-02-2014, cycle 27.979) and TWA 8A (bottom, collected on
26-03-2015, cycle 0.223). Clear Zeeman signatures are detected in both LSD
Stokes  _V_ profiles in conjunction with the unpolarized line profiles. The
mean polarization profiles are expanded by a factor of 10 shifted upwards by
0.04 for display purposes.

The disc-integrated average photospheric LSD profiles are computed by first
synthesizing the local Stokes $I$ and $V$ profiles using the Unno-Rachkovsky
analytical solution to the polarized radiative transfer equations in a Milne-
Eddington model atmosphere, taking into account the local brightness and
magnetic field. Then, these local line profiles are integrated over the
visible hemisphere (including linear limb darkening, with a coefficient of
0.75, as observed young stars, e.g. Donati & Collier Cameron 1997) to produce
synthetic profiles for comparison with observations. This method provides a
reliable description of how line profiles are distorted due to magnetic fields
(including magneto-optical effects, e.g., Landi Degl’Innocenti & Landolfi
2004). The main parameters of the local line profiles are similar to those
used in our previous studies; the wavelength, Doppler width, equivalent width
and Landé factor being set to 670 nm, 1.8 kms-1, 3.9 kms-1 and 1.2,
respectively.

We note that while Zeeman signatures are detected at all times in Stokes  _V_
LSD profiles for both stars (see Figure 3 for an example), TWA 8A exhibits
much larger longitudinal field strengths ($\textbf{B}_{l}$), similar to those
of e.g. mid M dwarfs (see Morin et al., 2008), with values shown in Fig. 4, as
calculated from the LSD profiles. Here we clearly see the periodicity in field
strength, with the maximum $\textbf{B}_{l}$ around phase 0.37, coincident with
the phase of the aligned dipole of the magnetic field (see Fig. 7) being
viewed along the line of sight, with the minimum $\textbf{B}_{l}$ seen around
half a rotation later. TWA 8A also exhibits significant Zeeman broadening in
the Stokes  _I_ profiles that we model in Section 5, with almost no
distortions due to brightness inhomogeneities on the surface.

As part of the imaging process we obtain accurate estimates for
$v_{\text{rad}}$ (the RV the star would have if unspotted), equal to $17.5\pm
0.1$ kms-1 and $8.34\pm 0.10$ kms-1, the inclination $i$ of the rotation axis
to the line of sight, equal to $46\degr\pm 10\degr$ and $42\degr\pm 10\degr$,
for TWA 6 and TWA 8A, respectively, and for TWA 6 the $v\sin{i}$ equal to
$72.6\pm 0.1$ kms-1 (see Table 2, in excellent agreement with the values
derived in Section 3.1). For TWA 8A, we fixed the $v\sin{i}$ to 4.82 kms-1, as
this was determined by direct spectral fitting in Section 5 and is more
accurate than that derived from ZDI.

Figure 4: The longitudional field strengths $\langle\textbf{B}_{l}\rangle$ for TWA 8A, as measured from the LSD profiles. Table 2: Main parameters of TWA 6 and TWA 8A as derived from our study, with $v_{\text{rad}}$ noting the RV that the star would have if unspotted, the equatorial rotation rate $\Omega_{\text{eq}}$ and the difference between equatorial and polar rotation rates $d\Omega$ (as inferred from the modelling of Section 4). Note, the stellar masses and ages are those determined from Siess et al. (2000) models, with values from Baraffe et al. (2015) given in parenthesis. The $\log{g}$ for TWA 8A is estimated from its mass (using Baraffe et al. (2015) models) and $R_{\star}$. | TWA 6 | TWA 8A
---|---|---
$M_{\star}$ (M☉) |  $0.95\pm 0.10$ ($0.95\pm 0.10$) |  $0.45\pm 0.10$ ($0.55\pm 0.10$)
$R_{\star}$ (R☉) | $1.1\pm 0.2$ | $0.8\pm 0.2$
Age (Myr) |  $21\pm 9$ ($17\pm 7$) |  $11\pm 5$ ($13\pm 6$)
$\log{g}$ (cgs units) | $4.5\pm 0.2$ | $4.3\pm 0.3$
$T_{\text{eff}}$ (K) | $4425\pm 50$ | $3690\pm 130$
log($L_{\star}$/L☉) | $-0.40\pm 0.12$ | $-0.93\pm 0.11$
$P_{\text{rot}}$ (d) | $0.54095\pm 0.00003$ | $4.578\pm 0.006$
$v\sin{i}$ (kms-1) | $72.6\pm 0.1$ | $4.82\pm 0.16$
$v_{\text{rad}}$ (kms-1) | $17.5\pm 0.1$ | $8.34\pm 0.1$
$i$ (°) | $46\pm 10$ | $42\pm 10$
Distance (pc) | $63.9\pm 1.4^{\text{a}}$ | $46.27\pm 0.19^{\text{a}}$
$\Omega_{\text{eq}}$ (rad d-1) | $11.6199\pm 0.0005$ | -
$d\Omega$ (rad d-1) | $0.0098\pm 0.0014$ | -
References: (a) Gaia Collaboration et al. 2016; Gaia Collaboration et al.
2018.

###  4.1 Brightness and magnetic imaging

The observed LSD profiles for TWA 6 and TWA 8A, as well as our fits to data,
are shown in Fig. 5. For TWA 6, we obtain a reduced chi-squared
$\chi_{\text{r}}^{2}$ fit equal to 1 (where the number of fitted data points
is equal to 4312, with simultaneous fitting of both Stokes  _I_ and Stokes
_V_ line profiles). For TWA 8A, the low $v\sin{i}$ means that there is little
modulation of the Stokes  _I_ line profiles, with the strong magnetic fields
causing significant Zeeman broadening of the lines. Indeed, we are able to
model the Stokes  _I_ line profiles sufficiently well using a stellar model
with a homogeneous surface brightness, with our fits to the Stokes  _V_ line
profiles yielding $\chi_{\text{r}}^{2}=1.04$ (for 930 fitted data points). We
note that, given the substantially larger $v\sin{i}$ of TWA 6 as compared to
TWA 8A, combined with more complete phase coverage, the reconstructed maps of
TWA 6 have an effective resolution around 10 times higher.

Figure 5: Maximum-entropy fit (thin red line) to the observed (thick black
line) Stokes  _I_ (first and third panels) and Stokes  _V_ (second and fourth
panels) LSD photospheric profiles of TWA 6 (first two panels) and TWA 8A (last
two panels). Note that for TWA 8A the velocity scales are different.
Rotational cycles are shown next to each profile. This figure is best viewed
in colour.

The brightness map of TWA 6 includes both cool spots and warm plages (see Fig.
6), with no true polar spot, but rather a large spotted region centred around
$60\degr$ latitude (centred around phase 0.6), with the majority of plages at
a similar latitude on the opposing hemisphere. These features introduce
significant distortions to the Stokes  _I_ profiles (see Fig. 5), introducing
large RV variations (with maximum amplitude 6.0 kms-1, see Section 6).
Overall, we find a spot and plage coverage of $\simeq 17$ per cent (10 and 7
per cent for spots and plages, respectively), similar to that found for V819
Tau, V830 Tau (Donati et al., 2015), and Par 2244 (Hill et al., 2017).

Note that the estimates of spot and plage coverage should be considered as
lower limits only, as Doppler imaging is mostly insensitive to small-scale
structures that are evenly distributed over the stellar surface (hence the
larger minimal spot coverage assumed in Section 3.1 to derive the location of
the stars in the H-R diagram).

Figure 6: Map of the logarithmic brightness (relative to the quiet
photosphere) at the surface of TWA 6. The star is shown in flattened polar
projection down to latitudes of $-30\degr$, with the equator depicted as a
bold circle, and 30°and 60°parallels as dashed circles. Radial ticks indicate
the phases of observation. This figure is best viewed in colour.

###  4.2 Magnetic field imaging

Using our imaging code, we have reconstructed the magnetic fields of our
target stars using both poloidal and toroidal fields, each expressed using a
spherical-harmonic (SH) expansion, with $\ell$ and $m$ denoting the mode and
order of the SH (Donati et al., 2006). For a given set of complex coefficients
$\alpha_{\ell,m},~{}\beta_{\ell,m}$ and $\gamma_{\ell,m}$ (where
$\alpha_{\ell,m}$ characterizes the radial field component, $\beta_{\ell,m}$
the azimuthal and meridional components of the poloidal field term, and
$\gamma_{\ell,m}$ the azimuthal and meridional components of the toroidal
field term), one can construct an associated magnetic image at the surface of
the star, and thus derive the corresponding Stokes  _V_ data set. Here, we
carry out the inverse, where we reconstruct the set of coefficients that fit
the observed data.

For TWA 6, our reconstructed fields presented in Fig. 7 are limited to SH
expansions with terms $\ell\leq 20$. Given the high $v\sin{i}$ of TWA 6
(combined with good phase coverage), we are able to resolve smaller-scale
magnetic fields, and indeed such a large number of modes are required to fit
the observed Stokes  _V_ signatures. We note, however, that including higher-
order terms ($>20$) only marginally improves our fit. Such high-degree modes
indicate that the magnetic fields in TWA 6 concentrate on smaller, more
compact spatial-scales. In contrast, our fits to the Stokes  _V_ observations
of TWA 8A only require terms up to $\ell\leq 10$, with higher order terms
providing only a marginal improvement. Hence, the magnetic field of TWA 8A is
concentrated at larger spatial scales.

The reconstructed magnetic field for TWA 6 is split almost evenly between
poloidal and toroidal components (53 and 47 per cent, respectively), with a
total magnetic energy $\langle B\rangle=840$ G, where $\langle B\rangle$ is
given by

$\langle
B\rangle=\oiint_{\theta,\phi}\left(\mathbf{B}_{\alpha}^{2}+\mathbf{B}_{\beta}^{2}+\mathbf{B}_{\gamma}^{2}\right)^{1/2}d\theta
d\phi$ (3)

The poloidal field is mostly axisymmetric (49 per cent), with the largest
fraction of energy (58 per cent) in modes with $\ell>3$, and with 30 per cent
of energy in the dipole mode ($\ell=1$, with a field strength of 550 G). On
large scales, the poloidal component is tilted at $35\degr$ from the rotation
axis (towards phase 0.34). The toroidal component is also mostly axisymmetric,
with the largest fraction of energy (68 per cent) in modes with $\ell>3$, and
with 17 per cent of energy in the octupole ($\ell=3$) mode. These components
combine to generate an intense field of $\geq 2$ kG at $45\degr$ latitude
around phase 0.50–0.75 and 0.20–0.35, as well as an off-pole 2 kG spot at
phase 0.75. We note that the large spotted region reconstructed in the
brightness map (around $60\degr$ latitude at phase 0.6, see Fig. 6) aligns
well with these intense fields, suggesting that they are related.

In the case of TWA 8A, the reconstructed field is 71 per cent poloidal and 29
per cent toroidal, with a total unsigned flux of 1.4 kG, and with a magnetic
filling factor of $f_{v}=0.2$ (where $f_{v}$ is equal to the fraction of the
stellar surface that is covered by the mapped magnetic field using Stokes  _V_
data). The poloidal field is mostly axisymmetric (70 per cent), with 16 per
cent of the energy in the dipole ($\ell=1$, with a field strength of 0.72 kG),
21 per cent in the quadrupole ($\ell=2$), 18 per cent in the octupole
($\ell=3$), and with the remaining 44 per cent of energy in modes with
$\ell>3$. On large scales (several radii from the star), the poloidal
component may be approximated by an $B=0.69$ kG aligned-dipole tilted at
$20\degr$ from the rotation axis (towards phase 0.37). The toroidal component
is mostly non-axisymmetric, with the majority of energy (55 per cent) in modes
with $\ell>3$, and with 21, 6 and 18 per cent in modes with $\ell=3,2$ & $1$.
These components combine to generate intense fields in excess of 2 kG in
around phases 0.08, 0.42 and 0.75 on the stellar surface, centred around
$20\degr$ latitude in the radial field component and around $35\degr$ in the
meridional field component. Given the filling factor of $f_{v}=0.2$, this
suggests that surface magnetic fields can locally reach over 10 kG. Moreover,
the high fraction of energy in high-order modes suggests that there are a
large number of small-scale magnetic features, a conclusion also supported by
the direct spectral fitting in Section 5.5.

Figure 7: Map of the radial (left), azimuthal (middle) and meridional (right)
components of the magnetic field B at the surface of TWA 6 (top) and TWA 8A
(bottom). Magnetic fluxes in the colourbar are expressed in G. Note that the
magnetic filling factor for TWA 8A is $f_{v}=0.2$. The star is shown in
flattened polar projection as in Figure 6. This figure is best viewed in
colour.

In Figure 8 we use the a potential field approximation (e.g., Jardine et al.
2002) to extrapolate the large-scale field topologies of TWA 6 and TWA 8A.
These topologies are derived solely from the reconstructed radial field
components, and represent the lowest possible states of magnetic energy,
providing a reliable description of the magnetic field well within the Alfvén
radius (Jardine et al., 2013).

Figure 8: Potential field extrapolations of the radial magnetic field
reconstructed for TWA 6 (left) and TWA 8A (right), viewed at phases 0.95 and
0.70, with inclinations of $45.6\degr$ and $31\degr$, respectively. Open and
closed field lines are shown in blue and white, respectively, whereas colours
at the stellar surface depict the local values of the radial field (as shown
in the left panels of Figure 7). The source surfaces at which the field
becomes radial are set at distances of 2.6 $R_{\star}$ for TWA 6 and 10.7
$R_{\star}$ for TWA 8A, as these are close to the co-rotation radii (where the
Keplerian orbital period equals the stellar rotation period, and beyond which
the field lines tend to open under the effect of centrifugal forces, Jardine
2004), and are smaller than or similar to the Alfvén radii of $>10$
$R_{\star}$ (Réville et al., 2016). This figure is best viewed in colour. Full
animations may be found for both TWA 6 and TWA 8A at https://imgur.com/hSkhYLT
and https://imgur.com/AdKptUx.

###  4.3 Surface differential rotation

The level of surface differential rotation of TWA 6 was determined in a
similar manner as that carried out for other wTTSs (e.g., Skelly et al., 2008;
Skelly et al., 2010; Donati et al., 2014, 2015). Assuming that the rotation
rate at the surface of the star varies with latitude $\theta$ as
$\Omega_{\text{eq}}-d\Omega\sin^{2}{\theta}$ (where $\Omega_{\text{eq}}$ is
the rotation rate at the equator and $d\Omega$ is the difference in rotation
rate between the equator and the pole), we reconstruct brightness and magnetic
maps at a fixed information content for many pairs of $\Omega_{\text{eq}}$ and
$d\Omega$ and determine the corresponding reduced chi-squared
$\chi_{\text{r}}^{2}$ of our fit to the observations. The resulting
$\chi_{\text{r}}^{2}$ surface usually has a well defined minimum to which we
fit a parabola, allowing an estimate of both $\Omega_{\text{eq}}$ and
$d\Omega$ (and their corresponding error bars).

Fig. 9 shows the $\chi_{\text{r}}^{2}$ surface we obtain (as a function of
$\Omega_{\text{eq}}$ and $d\Omega$) for both Stokes $I$ and $V$, for TWA 6. We
find a clear minimum at $\Omega_{\text{eq}}=11.6199\pm 0.0005$
$\text{rad~{}d}^{-1}$ and $d\Omega=0.0098\pm 0.0014$ $\text{rad~{}d}^{-1}$ for
Stokes  _I_ data (corresponding to rotation periods of $0.54073\pm 0.00002$ d
at the equator and $0.54118\pm 0.00002$ d at the poles; see left panel of
Figure 9), with the fits to the Stokes  _V_ data of
$\Omega_{\text{eq}}=11.622\pm 0.004$ $\text{rad~{}d}^{-1}$ and
$d\Omega=0.018\pm 0.011$ $\text{rad~{}d}^{-1}$ showing consistent estimates,
though with larger error bars (right panel of Figure 9). We note that both
these periods are in excellent agreement with those found previously by Skelly
et al. (2008) and Kiraga (2012).

For TWA 8A, we were able to constrain the rotational period to $4.578\pm
0.006$ d (corresponding to $\Omega_{\text{eq}}=1.3724\pm 0.0019$
$\text{rad~{}d}^{-1}$), in good agreement with the photometric period of 4.638
d found by Kiraga (2012). However, given that the observations span only $\sim
3$ rotation cycles, the recurrence of profile distortions across different
latitudes is severely limited, and so we were unable to constrain surface
shear. Hence, for our fits with ZDI we have assumed solid body rotation.

Figure 9: Variations of $\chi_{\text{r}}^{2}$ as a function of
$\Omega_{\text{eq}}$ and $d\Omega$ for TWA 6, derived from modelling of our
Stokes  _I_ (red) and Stokes  _V_ (blue) LSD profiles at a constant
information content. For both Stokes  _I_ and Stokes  _V_ , a clear and well
defined parabola is observed, shown by the 1, 2 and $3\sigma$ ellipses
(depicting 68.3, 95.5 and 99.7 per cent confidence levels, respectively), with
the $3\sigma$ contour tracing the 5.5 per cent increase in
$\chi^{2}_{\text{r}}$ (or equivalently a $\chi^{2}$ increase of 11.8 for 2156
fitted data points). This figure is best viewed in colour.

##  5 Magnetic field strength from individual lines

TWA 8A has a very strong photospheric magnetic field that can be detected in
some individual lines, allowing direct spectral fitting to derive the strength
of the magnetic field. As this is not the case for TWA 6, it is not included
in the following analysis. For TWA 8A, Stokes  _V_ signatures are visible in
over 20 lines, mostly redwards of 8000 Å where the S/N is largest. Of
particular interest are a set of eleven strong Ti i lines between 9674 and
9834 Å ten of which are detected in Stokes  _V_ , and one which has a Landé
factor of zero (9743.6 Å, see Fig. 10). These atomic lines have minimal
blending from molecular lines, and while there is a some blending from
telluric lines, it can be corrected. These lines have the added advantage that
all but two of them are from the same multiplet, which mitigates the impact of
some systematic errors (e.g., errors in $T_{\text{eff}}$) on our measurements
of the magnetic field. A detailed description of these lines is given in Table
4.

Figure 10: Detections of Zeeman broadening in the observation of TWA 8A on
March 27. The panels show the Stokes  _I_ spectrum at the top and the
corresponding Stokes  _V_ spectrum below for the full set of lines used in our
fits (see Table 4). Dashed lines show the observation before telluric
correction and solid lines show the spectrum after telluric correction. Over-
plotted in a red solid line is our best fit using our third model to fit both
Stokes  _I_ and Stokes  _V_ simultaneously.

Figure 11: The distribution of surface magnetic field strengths for TWA 8A, as
determined from ZDI and direct spectral fitting of Ti i lines. Blue bars show
the fraction of the total mapped magnetic field strength from ZDI, for fields
of a given bin. Top panel: Comparison between the magnetic field strengths
determined from fitting Stokes  _I_ data. Red circles show the mean filling
factors for each field strength using our third model to simultaneously fit
Stokes  _I_ and $V$ spectra (see Section 5.5). Black circles show the combined
filling factors for both the positive and negative fields. Thus, one can
directly compare the recovered field strengths for Stokes  _I_ data from ZDI
and direct spectral fitting by comparing the blue bars and the black circles,
respectively. One can see that a significantly larger fraction of higher-
strength fields are recovered by direct spectral fitting, as compared to that
from ZDI (see discussion in Section 5.5). Bottom panel: Comparison between the
magnetic field strengths determined from fitting Stokes  _V_ data. Black
circles show the resulting filling factors after subtracting the contributions
of the negative fields from those of the positive fields. As Stokes  _V_
profiles are sensitive to the sign of the line-of-sight component of
$\mathbf{B}$, significant cancellation of fields may occur, and so we must
compare our fits with ZDI to Stokes  _V_ profiles, to these black circles. In
this case, we see that ZDI recovers a similar fraction of field strengths for
the 5 and 10 kG bins, but significantly more for the 2 kG bin, and
significantly less for the 15 and 20 kG bins (see discussion in Section 5.5).

###  5.1 Telluric correction

Before a detailed analysis of the Stokes  _I_ spectra may be carried out, we
must first correct for the large number of telluric water lines present
between 9670–9840 Å. Telluric lines are not expected to produce circular
polarisation, and we see no indication of them in Stokes  _V_ , hence we
conclude that their impact on the Stokes  _V_ spectrum is negligible.

As we did not expect to detect magnetic fields in individual telluric blended
lines, we did not observe a hot star for telluric calibration. Fortuitously,
on some nights, other programs with ESPaDOnS at the CFHT observed the hot
stars HD 63401 (PI J.D. Landstreet) and HD 121743 (PI G.A. Wade). HD 63401 is
a 13500 K Bp star (e.g., Bailey, 2014) and HD 121743 is a 21000 K B star
(e.g., Alecian et al., 2014), with both stars having virtually no photospheric
lines in the wavelength range of interest, apart from Paschen lines. Our
observations of TWA 8A on the nights of March 25 to April 1, as well as April
5 and 6, had suitable telluric reference observations that were sufficiently
close in time and obtained under sufficiently similar conditions.

The telluric reference spectra were first continuum normalised by fitting low
order polynomials through carefully selected continuum regions, then dividing
by those polynomials, independently for each spectral order. The telluric
reference spectra were then scaled in the form $I^{a}$, where $I$ is the
continuum normalised spectrum and $a$ the scaling factor. The scaling factor
$a$ and the radial velocity shift for the telluric lines were determined by
fitting the modified reference spectrum to telluric lines of the science
spectrum through $\chi^{2}$ minimisation. Telluric lines around the
photospheric lines of interest ($\sim$9650–9850 Å) were included, as well as
some telluric lines in the range 9300–9500 Å where there are fewer strong
photospheric lines. The science spectrum was then divided by the scaled
shifted telluric spectrum. A example spectrum before and after telluric
correction is shown in Fig. 10.

###  5.2 Spectrum synthesis

To constrain the strength of the photospheric magnetic field, we have modelled
individual lines in the Stokes  _I_ and Stokes  _V_ spectra of TWA 8A.
Furthermore, as one of the Ti i lines has a Landé factor of zero, and is
narrower in Stokes  _I_ as compared to the other Ti i lines, the magnetic
field can also be strongly constrained by the Stokes  _I_ spectrum.

To generate synthetic spectra, we used the Zeeman spectrum synthesis program
(Landstreet, 1998; Wade et al., 2001; Folsom et al., 2012). This program
includes the Zeeman effect and performs polarized radiative transfer in Stokes
$IQUV$. The code uses plane-parallel model atmospheres and assumes LTE, and
produces disk-integrated spectra. Zeeman includes quadratic Stark, radiative,
and van der Waals broadening, as well as optional microturbulence
($v_{\text{mic}}$) and radial-tangential macroturbulence. A limitation of the
code for use in very cool stars is that it does not include molecular lines,
or calculations of molecular reactions in the abundances for atomic species.
The Ti i lines in the 9674–9834 Å region are blended with a few very weak
molecular lines, and so Zeeman can produce accurate spectra for this region,
however most of the spectral region bluewards of this is problematic.

For input to the code we used MARCS model atmospheres (Gustafsson et al.,
2008) and atomic data taken from VALD (Ryabchikova et al., 2015) (see Table 4
for the properties of the atomic lines). VALD data for these particular Ti i
lines were also used by Kochukhov & Lavail (2017) for a similar analysis, and
were deemed reliable. Additionally, we can reproduce these Ti lines with near-
solar abundances, implying that the oscillator strengths are likely close to
correct.

To model the magnetic field of TWA 8A, we adopted a uniform radial magnetic
field. While this is an unrealistically simple magnetic geometry, the ZDI
analysis found the magnetic geometry to be more complex than a simple dipole.
Therefore we leave the geometric analysis to ZDI and adopt the simplest
possible geometry here to avoid additional weakly constrained geometric
parameters. Furthermore, since this analysis is applied to individual
observations, a full magnetic geometry cannot be reliably derived. The model
we implement here includes a combination of magnetic field strengths $B$, each
with their own filling factor $f$, with the sum of the filling factors
(including a region of zero field) equal to unity.

We fit synthetic spectra using a Levenberg-Marquardt $\chi^{2}$ minimisation
routine (similar to Folsom et al., 2012; Folsom et al., 2016), with the radial
magnetic field strengths and filling factors as optional additional free
parameters. The code was updated to allow fitting observed Stokes $I$ spectra,
$V$ spectra, or $I$ and $V$ simultaneously, with wavelength ranges carefully
set around the lines of interest. In order to place uncertainties on the
fitting parameters, we use the square root of the diagonal of the covariance
matrix, as is commonly done. This is then scaled by the square root of the
reduced $\chi^{2}$, to very approximately account for systematic errors. These
formal uncertainties may still be underestimates, and a further consideration
of uncertainties is discussed in Sect. 5.5.

###  5.3 Fitting the Stokes $I$ spectrum

Our initial fits were carried out with the observation on March 27 since the
Stokes  _V_ LSD profile for this night has one of the simplest shapes,
indicating a more uniform magnetic field in the visible hemisphere.

Measurements of magnetic fields in Stokes  _I_ spectra are constrained by both
the width and the desaturation of lines with different Landé factors. Fitting
the Stokes  _I_ spectrum to determine magnetic field strengths requires
constraints on several other stellar parameters which influence line width and
depth. Here, we adopt the $T_{\text{eff}}$ and $\log{g}$ values derived in
Section 3 (see Table 2). Since our choice of lines is dominated by one
multiplet, adopting these values is a small source of uncertainty. We note
that these lines are not well adapted to constraining $T_{\text{eff}}$ and
$\log{g}$ spectroscopically. We include $v\sin{i}$ and $v_{\text{mic}}$ as
free parameters in the fit, since they can play an important role in line
shape and strength, and can only be determined spectroscopically.
$v_{\text{mic}}$ is constrained by desaturation of strong (on the curve of
growth) lines and, given the lack of weak lines in our spectral range, is
determined with only a modest accuracy by different degrees of desaturation of
different strong lines. Macroturbulence is assumed to be zero, since it is
likely much smaller than the $v\sin{i}$ of $\sim$5 kms-1. Ti abundance is
included as a free parameter, however, we caution the reader that this may not
provide reliable results, as the code neglects the fraction of Ti bound in
molecules. Nevertheless, this free parameter is necessary to avoid the code
fitting line strength entirely by varying magnetic field and $v_{\text{mic}}$.

When fitting the spectra of TWA 8A we adopted three main models, each of
increasing complexity, to better constrain the nature of the magnetic field.
These three models (described below) are used to fit Stokes  _I_ spectra only,
Stokes  _V_ only, and both Stokes  _I_ and $V$ simultaneously.

Our first model consists of fitting the Stokes  _I_ spectrum using just one
magnetic region with a corresponding filling factor, yielding a best-fit
magnetic field strength of $B_{1}=5.65\pm 0.10$ kG with a filling factor
$f_{1}=0.597\pm 0.016$, but at a reduced $\chi^{2}$ of 19.6. Fits with $f$
fixed to 1 consistently fail to reproduce the line shape, with a core that is
far too wide and with wings that are too narrow, implying that only a fraction
of the star is covered by very strong magnetic fields.

Our second model increases the number of free parameters by including two
magnetic regions and filling factors, achieving a visibly much better fit with
a reduced $\chi^{2}$ of 12.9, and with field strengths of $B_{1}=4.71\pm 0.08$
kG with $f_{1}=0.648\pm 0.015$, and $B_{2}=15.61\pm 0.25$ kG with
$f_{2}=0.133\pm 0.007$. This second model does a better job of simultaneously
reproducing the narrow core and broad wings of the magnetically sensitive
lines, although the high field strength region produces a sharper change in
the shape of the wings than seen in the observation, implying that the star
has a more continuous distribution of magnetic field strengths than our model.

Our third model again increases the number of free parameters to improve the
fit. However, rather than add additional sets of magnetic field strengths and
filling factors, which may become more poorly conditioned or not converge
well, we instead adopt a grid of fixed magnetic field strengths with filling
factors as free parameters (in a similar way to e.g. Johns-Krull et al. 1999,
2004). This provides an approximate distribution of magnetic field strengths
on the visible hemisphere of the star. Using our third model for fitting
Stokes  _I_ only, we use bins of 0, 2, 5, 10, 15 and 20 kG. Bins of $\sim 5$
kG allow for smooth model line profiles, and so smaller bins (that would be
less well constrained) are not necessary. Adding bins above 20 kG improves the
$\chi^{2}$ fit by a small but formally significant amount. However, the impact
on the synthetic line is small and only affects the far wings of the line in
Stokes  _I_. Small changes in the far wings of the line are most vulnerable to
systematic errors, such as weak lines that are not accounted for, errors in
the telluric correction, errors in continuum normalisation, or very weak
fringing, all of which could approach the strength of the line this far into
the wing. Thus we limit the magnetic field to 20 kG, and caution that even for
this bin the filling factor may be overestimated. The resulting best fit
parameters for Stokes  _I_ only for March 27 using our third model is
presented in Table 3, with a reduced $\chi^{2}$ of 10.6.

Yang et al. (2008) studied TWA 8A and derived some magnetic quantities based
on Stokes  _I_ observations in the IR. They adopted literature values for the
stellar parameters of $T_{\text{eff}}=3400$ K, $\log{g}=4.0$ and
$v\sin{i}=4.0$kms-1. Their “Model 1” corresponds to our first model with one
filling factor and magnetic field strength. They report only the product of
their filling factor and magnetic field strength as 2.3 kG, which is close to
our value for March 27 of $3.37\pm 0.11$ kG, although not within uncertainty.
Their “Model 2” corresponds to our second model with two filling factors and
magnetic field strengths. They report the quantity $\langle|B_{f}|\rangle=2.7$
kG, which is comparable but again not consistent with our value of $5.13\pm
0.14$ kG. The “Model 3” of Yang et al. (2008) is closest to our third model
with a grid of filling factors, although they only fit filling factors for
field strengths of 2, 4 and 6 kG. They report
$\langle|B_{f}|\rangle=\sum_{i}B_{i}f_{i}$ of 3.3 kG. The equivalent value
from our fit is $\langle|B_{f}|\rangle=5.90\pm 0.44$ kG, which is again
inconsistent. We note that, if we perform our fit using the three bins of 2,
4, and 6 kG used by Yang et al. (2008), we find $\langle|B_{f}|\rangle=3.96\pm
0.38$ kG. While this is much closer to their “Model 3” results, we find that
the fit to our data is much worse in the wings of the lines, so we consider
this model to be less accurate for our spectra. The IR spectra of Yang et al.
(2008) had a much lower S/N than our observations, and so the wings of the
lines may not have been detected as clearly as in our spectra. Indeed, the
very strong magnetic field with a very small filling factor necessary to fit
the wings of our magnetically sensitive lines is likely the cause of the
difference between our results, as well as intrinsic variability of the field.

###  5.4 Fitting the Stokes $V$ spectrum

In order to fit the Stokes  _V_ spectrum we adopt the best fit $v\sin{i}$,
$v_{\text{mic}}$ and Ti abundance from fitting Stokes  _I_ with our third
model, since these parameters cannot be well constrained from $V$ spectra (see
Table 3).

When directly fitting the Stokes  _V_ spectrum, it becomes immediately
apparent that a filling factor (much less than unity) is necessary. To produce
Stokes  _V_ profiles with the widths of the observed lines, a very strong
magnetic field is necessary. However, to reproduce the amplitudes of the
Stokes  _V_ profiles, a weaker field is necessary, or a very strong field
covering a small portion of the star. This can be easily seen by comparing the
widths of the observed Stokes  _I_ and $V$ profiles (see Fig. 5) and noting
that the $V$ profiles remain stronger in the far wings compared to the $I$
profiles.

Fitting the Stokes  _V_ profiles with our first model yields a best fit of
$B_{1}=7.09\pm 0.19$ kG and $f_{1}=0.081\pm 0.004$, with a reduced $\chi^{2}$
of 2.27. However, this provides a poor fit to the line profiles, in particular
the outer and inner parts of the line cannot be well fit simultaneously. We
find a much better fit when using our second model, with a reduced $\chi^{2}$
of 1.58, and field strengths and filling factors of $B_{1}=4.70\pm 0.19$ kG
with $f_{1}=0.078\pm 0.004$, and $B_{2}=14.94\pm 0.26$ kG with $f_{2}=0.051\pm
0.003$, implying $\langle|B_{f}|\rangle=1.13\pm 0.05$ kG. The filling factors
and $\langle|B_{f}|\rangle$ derived here are much smaller than those derived
from Stokes  _I_. Stokes  _V_ is sensitive to the sign of the line-of-sight
component of $\mathbf{B}$, while Stokes  _I_ is sensitive to the magnitude of
$\mathbf{B}$. The difference in filling factors is likely due to cancelation
in $V$ of nearby regions with opposite sign.

We also fit the Stokes  _V_ spectra with our third model, where our use of
positive fields is still appropriate as the disc integrated field is positive
for March 27, and indeed at all other phases. Our fit yields a reduced
$\chi^{2}$ of 1.56, where the parameters are summarised in Table 3. The
improvement in the fit using our third model is modest compared to the first
and second models, but it is clearly better visually, with a formally
significant improvement of nearly $3\sigma$. We note that the distribution of
filling factors is quite different from that of the Stokes  _I_ fit, with most
of the surface having no magnetic field detected in Stokes  _V_ , and the
remaining field lying more in the 5 and 15 kG bins.

Using our fits to approximate the longitudinal magnetic field ($B_{l}$), we
have taken the line of sight component of the model magnetic field, averaged
over the stellar disc and weighted by the brightness of the continuum, i.e.

$B_{l,\rm{syn}}=\sum_{i}\frac{\int I_{c}f_{i}B_{i}\cos(\theta){\rm
d}\Omega}{\int I_{c}{\rm d}\Omega}$ (4)

where $f_{i}$ is the filling factor for component $i$, $B_{i}$ is the purely
radial magnetic field for that component, $\theta$ is the angle between the
line of sight and the radial field. $I_{c}$ is the continuum brightness at for
a point on the disc (accounting for limb darkening), and the integral of
d$\Omega$ is over the visible disk.

From Eqn. 4 we derive $B_{l,\rm{syn}}=0.78\pm 0.04$ kG and $0.82\pm 0.08$ kG
for our second and third models, respectively. These values agree to within
their uncertainties, and are comparable to (but roughly 1.7 times larger than)
the actual observed $B_{l}$ values for this phase, as calculated from the LSD
profiles (see Fig. 4). Indeed, if we calculate an observed $B_{l}$ from just
the Ti i 9705.66 Å line (using the telluric corrected spectrum), rather than
an LSD profile, we find $0.88\pm 0.13$ kG for March 27. Moreover, the
behaviour of this Ti i line with rotational phase is consistent with the LSD
profile, except that it shows a higher field strength. This implies that the
signal in the Stokes  _V_ LSD profiles may not be adding perfectly coherently,
producing a lower amplitude $V$ profile. This is not surprising as, due to the
very large field strength, Zeeman splitting patterns of individual lines begin
to matter for the line profile shapes. Thus, simply scaling amplitudes by
effective Landé factors is a less effective approximation for such strong
fields.

###  5.5 Simultaneous fitting of Stokes $I$ and $V$

As we detect magnetic fields in both Stokes  _I_ and $V$ observations, our
model should be able to reproduce these signatures simultaneously. This
requires us to allow a combination of positive and negative magnetic fields,
resulting in a cancellation of much of the signal in Stokes  _V_ while
allowing for a large unsigned magnetic flux in Stokes  _I_. This is evident
from the much smaller filling factor in our fits of Stokes  _V_ compared to
our fits to Stokes  _I_.

Firstly, we performed simultaneous fits to Stokes  _I_ and $V$ using a simple
model with three magnetic regions - two with positive fields and one with a
negative field. A model with one positive field and one negative field is
insufficient to reproduce the shapes of the Stokes  _I_ or $V$ line profiles.
For this simple model, the best fit magnetic parameters are $B_{1}=+4.76\pm
0.07$ kG with $f_{1}=0.360\pm 0.007$, $B_{2}=-5.05\pm 0.09$ kG with
$f_{2}=0.282\pm 0.007$, and $B_{3}=+15.92\pm 0.20$ kG with $f_{3}=0.098\pm
0.004$ (with $v\sin{i}=5.26\pm 0.17$ kms-1, $v_{\text{mic}}=1.00\pm 0.06$
kms-1 and [Ti/H]$=-6.947\pm 0.013$). This fit gives a reduced $\chi^{2}$ of
7.94, and fits the $I$ spectrum similarly well to our best model from fitting
Stokes  _I_ only (see above), although it is too strong in the wings of $V$,
implying that there should be additional cancellation. This model implies a
total $\langle|B_{f}|\rangle$ of 4.70 kG, and a synthetic $B_{l,\rm{syn}}$
(allowing for cancellation) of 1.28 kG, although (as noted) this is likely too
large.

Using our third model (with a grid of magnetic field strengths and filling
factors, see above), we again require both negative and positive magnetic
fields. As with fitting only Stokes  _I_ or Stokes  _V_ , we use bins of 0 G,
$\pm 2$ kG, $\pm 5$ kG, $\pm 10$ kG, $\pm 15$ kG, and $\pm 20$ kG, for a total
of 11 bins. The results of our fit with this model, with 11 filling factors as
well as $v\sin{i}$, $v_{\text{mic}}$ and [Ti/H], are presented in Table 5,
with a reduced $\chi^{2}$ of 6.33 - clearly an improvement over the simple
three magnetic-region model. Our fit to the observation taken on March 27 is
shown in Fig. 10, showing a good fit to both Stokes  _I_ and $V$ spectra,
including matching the width of the magnetically-insensitive line with a Landé
factor of zero.

A summation of the filling factors for bins with the same $|B|$ yields a very
similar distribution to that for the fit to Stokes  _I_ only, with differences
much smaller than the formal uncertainties. This can be understood as Stokes
_I_ is sensitive to the total magnetic field strength but not the orientation
of the magnetic field. Similarly, the difference between filling factors for
bins with the same $|B|$ but opposite sign produces a distribution very
similar to that of the fit to Stokes  _V_ only. This can also be understood
since Stokes  _V_ is sensitive to the line-of-sight component of the magnetic
field only, with the spatially unresolved (within the same model pixel)
components of opposite orientation cancelling out. For our observation on
March 27, we find a total $\langle|B_{f}|\rangle=5.71\pm 0.22$ kG and
$B_{l,\rm{syn}}=0.78\pm 0.15$ kG. This $\langle|B_{f}|\rangle$ is consistent
with our fit of only Stokes  _I_ with our third model, and $B_{l,\rm{syn}}$ is
consistent with our fit of only Stokes  _V_.

Over the rotation of TWA 8A, this set of results shows $B_{l,\rm{syn}}$ to
range from $640\pm 150$ to $840\pm 140$ G, with $\langle|B_{f}|\rangle$
ranging from $5.71\pm 0.22$ to $6.36\pm 0.22$ kG, and varying coherently with
rotation phase.

Given the high S/N of our observations, the results we present here may be
limited by systematic errors, and our uncertainties may be underestimated. To
investigate the impact of uncertainties in $T_{\text{eff}}$ and $\log{g}$, we
re-fit the observation on March 27 with these two parameters changed by $\pm
1\sigma$. The change in $T_{\text{eff}}$ produces at most a change of
$0.5\sigma$ in the other parameters, and often smaller changes than that, and
so we conclude that the uncertainty on $T_{\text{eff}}$ has a minor
contribution to the total uncertainty. Changing $\log{g}$ by $1\sigma$ has a
large impact on $v\sin{i}$ and [Ti/H] (4–5$\sigma$) and on $v_{\text{mic}}$
($2\sigma$), although it has a much smaller impact on the magnetic filling
factors of only $\sim 1\sigma$, rising to $3\sigma$ for the 2 kG and 5 kG bins
when $\log{g}$ is decreased by $1\sigma$. In that case, the filling factor
shifts from the 2 kG bin into the 0 and 5 kG bins, underscoring the
uncertainty of the 2 kG bin. The relatively large uncertainty in $\log{g}$
changes the line broadening, but does so independently of Landé factor, and so
$v\sin{i}$ and $v_{\text{mic}}$ are more sensitive to $\log{g}$ than filling
factors. It is possible that our $v_{\text{mic}}$ is an over-estimate, since
typical $v_{\text{mic}}$ values for PMS M-dwarfs are not well known. To
estimate an upper limit on this uncertainty, we re-ran the fit with
$v_{\text{mic}}=0$, finding that the best fitting $v\sin{i}$ decreases by 1
kms-1, that [Ti/H] increases by 0.1 dex, and that filling factors generally
change by less than $1\sigma$ (except for the 10 kG bin which decreases by
$2\sigma$). From these tests we conclude that our formal uncertainties may be
underestimated by a factor $\lesssim 2$, mostly due to the large uncertainty
in $\log{g}$ and the (potentially) larger systematic errors on the filling
factors for the 2 and 20 kG bins.

Having established an analysis method for the observation of March 27 using
our third model to fit both Stokes  _I_ and $V$, we performed this analysis on
all observations for which we could perform reliable telluric correction,
providing us with ten sets of results, shown in Table 5. Taking an average
over all 10 observations, we find a mean magnetic field strength of
$\langle|B_{f}|\rangle=6.0\pm 0.5$ kG, where the amount of magnetic energy in
each bin is shown in Table 3. The standard deviation of these results is close
to the mean uncertainty for all parameters, suggesting that our formal
uncertainties account well for random errors, with the larger standard
deviation likely due to the rotational modulation.

In Fig. 11 we compare the magnetic field strength distribution on TWA 8A as
determined by our ZDI maps in Section 4.2, to our direct spectral fitting
here. As our ZDI map has a continuous distribution of field strengths, we have
created histograms using the same bins as that for the direct spectral
fitting, allowing for a direct comparison of recovered field strengths. For
Stokes  _I_ , we find that 75 per cent of the field strength recovered by ZDI
is in the 2 kG bin, with a 15 per cent in the 5 kG bin, and 9 per cent at
higher field strengths. In comparison, direct spectral fitting yields 32 per
cent of the magnetic field to be 2 kG, with almost 46 per cent in the 5 kG
bin, and with 22 per cent of fields in the 10, 15 and 20 kG bins. For Stokes
_V_ , the line profiles are sensitive to the sign of the line-of-sight
component of $\mathbf{B}$, and so there is likely significant cancellation of
fields of opposite polarity. Hence, our fits to Stokes  _V_ LSD profiles with
ZDI recover only the uncanceled magnetic fields. Therefore, for comparison to
direct spectra fitting, we must subtract the filling factors determined for
the negative fields from the positive fields, yielding the fraction of
uncanceled fields that could be fit with ZDI. For ZDI we find that 80 per cent
of the surface has a 0 G field, with 15 per cent of the field in the 2 kG bin,
3 per cent in the 5 kG bin, 1 per cent in the 10 kG bin, and with 1 per cent
at higher field strengths. In comparison, for direct spectral fitting we find
that less than 1 per cent of the field is 2 kG, with 3.6 per cent of the field
at 5 kG, 1.4 per cent at 10 kG, and with 4.5 per cent at higher field
strengths. Thus, with ZDI we recover most of the magnetic flux up to 10 kG,
but are not as sensitive to fields higher than this. Moreover, our results
demonstrate that we underestimate the fraction of high field strengths using
the ZDI technique with LSD profiles Stokes  _V_ spectra. As mentioned
previously, this may be due to the signal in the Stokes  _V_ LSD profiles not
adding perfectly coherently, as variations in line splitting patterns cause
variations in line shapes, and so scaling amplitudes by effective Landé
factors is less accurate. Moreover, there may be significant cancellation in
Stokes  _V_ profiles as it is sensitive to the sign of the line-of-sight
component of $\mathbf{B}$. The recovery of small-scale, high-field-strength
features would likely be improved if linear polarization spectra (Stokes $Q$
and $U$) were included in the ZDI modelling, and would likely increase the
recovered total magnetic field energy (see Rosén et al., 2015).

Table 3: Best fit parameters from direct spectral fitting of TWA 8A. The first and second columns respectively give the results of fitting Stokes $I$ and $V$ separately (using our third model) for the spectrum taken on 27 Mar 2015. Parameters with no error bars for the $V$ fit were held fixed. The third column shows the results of fitting Stokes $I$ and $V$ simultaneously, where we present the mean over the 10 nights that could be reliably telluric-corrected, with error bars given as the standard deviations. Values for fits to individual nights are presented in Table 5. | Stokes $I$ only | Stokes $V$ only | Stokes $I$ and $V$
---|---|---|---
| 27 Mar 2015 | 27 Mar 2015 | mean
$v\sin{i}$ (kms-1) | $4.77\pm 0.23$ | 4.77 | $4.82\pm 0.16$
$v_{\text{mic}}$ (kms-1) | $1.15\pm 0.08$ | 1.15 | $1.08\pm 0.05$
$[$Ti/H$]$ | $-7.006\pm 0.017$ | -7.01 | $-6.976\pm 0.022$
0 G | $0.002\pm 0.084$ | $0.859\pm 0.014$ | $0.017\pm 0.016$
+2 kG | $0.312\pm 0.067$ | $0.023\pm 0.010$ | $0.161\pm 0.016$
+5 kG | $0.483\pm 0.044$ | $0.057\pm 0.007$ | $0.245\pm 0.019$
+10 kG | $0.090\pm 0.020$ | $0.016\pm 0.005$ | $0.055\pm 0.004$
+15 kG | $0.060\pm 0.013$ | $0.041\pm 0.005$ | $0.047\pm 0.003$
+20 kG | $0.053\pm 0.011$ | $0.004\pm 0.004$ | $0.035\pm 0.004$
-2 kG | - | - | $0.155\pm 0.010$
-5 kG | - | - | $0.209\pm 0.007$
-10 kG | - | - | $0.041\pm 0.005$
-15 kG | - | - | $0.010\pm 0.003$
-20 kG | - | - | $0.027\pm 0.004$
$\langle|B_{f}|\rangle$ (kG) | $5.9\pm 1.0$ | $1.2\pm 0.3$ | $6.0\pm 0.5$

##  6 Filtering the activity jitter

As well as characterizing magnetic fields of wTTSs, the MaTYSSE program also
aims to detect close-in giant planets (called hot Jupiters, hJs) to test
planetary formation and migration mechanisms. In particular, characterizing
the number and position of hJs will allow us to quantiatively assess the
likelihood of the disc migration scenario, where giant planets form in the
outer accretion disc and then migrate inward until they reach the central
magnetospheric gaps of cTTSs (see e.g., Lin et al. 1996; Romanova & Lovelace
2006). Given that we map the surface brightness of the host star, we are able
to use our fits to the observed data to filter out the activity-related jitter
from the RV curves (where the RV is measured as the first-order moment of the
LSD profile; see Donati et al. 2014, 2015). After subtraction of the RV
jitter, we may look for periodic signals in the RV residuals to reveal the
presence of hJs. Indeed, this method has so far yielded two detections of hJs
in the MaTYSSE sample, around both V830 Tau (Donati et al., 2015, 2016, 2017)
and TAP 26 (Yu et al., 2017).

For TWA 6, the unfiltered RVs have an rms dispersion of 3.8 kms-1. The
predicted RV due to stellar activity and the filtered RVs are shown in Fig.
12. We find that RV residuals exhibit an rms dispersion of $\sim 0.20$ kms-1,
with a maximum amplitude of 0.51 kms-1. This is well above the intrinsic RV
precision of ESPaDOnS (around 0.03 kms-1, e.g. Moutou et al. 2007; Donati et
al. 2008), however, given the high $v\sin{i}$, the accuracy of the filtering
process is somewhat reduced, with an intrinsic uncertainty of around 0.1
kms-1. Indeed, we find no significant peaks in a periodogram analysis, and so
we find that TWA 6 is unlikely to host a hJ with an orbital period in the
range of what we can detect (i.e. not too close to the stellar rotation period
or its first harmonics; see Donati et al. 2014). We find a $3\sigma$ error bar
on the semi-amplitude of the RV residuals equal to 0.19 kms-1, translating
into a planet mass of $\simeq 3.1$ MJup orbiting at $\simeq 0.1$ au (assuming
a circular orbit in the equatorial plane of the star; see Figure 13).

For TWA 8A, the unfiltered RVs have an rms dispersion of 0.13 kms-1. Given
that the surface brightness of TWA 8A is compatible with that of a homogeneous
star, we were unable to filter the RVs in the same manner. However, the
measured RVs (shown in Fig. 12) do display a clear periodic signal that is
equal to the stellar rotation period, implying that there are starspots on the
surface, even though the modulation of the line profiles is minimal.

Figure 12: Top left panel: RV variations (in the stellar rest frame) of TWA 6
a function of rotation phase, as measured from our observations (open blue
circles) and predicted by the tomographic brightness map of Figure 6 (green
line). RV residuals are also shown (red crosses, with values and error bars
scaled by a factor of 4 for clarity), and exhibit a rms dispersion equal to
0.20 kms-1. RVs are estimated as the first order moment of the Stokes  _I_ LSD
profiles rather than through Gaussian fits, due to their asymmetric and often
irregular shape. Top right panel: The same as the top left panel after phase-
folding the data and model. Note that the model shows little variation over
the $\sim 30$ rotation cycles, showing the very low level of differential
rotation. Bottom panel: The measured RVs of TWA 8A as a function of rotation
phase. Note that the filtered RVs are not shown for TWA 8A as the line
profiles are compatible with a star of uniform brightness. The unfiltered RVs
show a period signal that is equal to the stellar rotation period. This figure
is best viewed in colour.
Figure 13: The $1\sigma$ and $3\sigma$ upper limits (solid and dashed lines,
respectively) on the recovered planet mass as a function of orbital distance,
using the RVs shown in Fig. 12 for TWA 6. This figure is best viewed in
colour.

##  7 Summary and discussion

Figure 14: H-R diagram showing the MaTYSSE wTTSs (black line border and
labelled) and the MaPP cTTSs (no border). The size of the symbols represents
the surface-averaged magnetic field strength (with a larger symbol meaning a
stronger field), the colour of the symbol represents the fraction of the field
that is poloidal (with red being completely poloidal), and the shape of the
symbols represents the axisymmetry of the poloidal field component (with
higher axisymmetry shown as a more circular symbol). Also shown are
evolutionary tracks from Siess et al. (2000) (black dashed lines, ranging from
0.3–1.9M☉), with corresponding isochrones (black dotted lines, for ages of
0.5, 1, 3, 5 & 10 Myr), and lines showing 100% and 50% convective interior by
radius (blue dashed).

We report the results of our spectropolarimetric observations collected with
ESPaDOnS at CFHT of two wTTSs, namely TWA 6 and TWA 8A, in the framework of
the international MaTYSSE Large Program. Our spectral analysis reveals that
the two stars have quite different atmospheric properties, with photospheric
temperatures of $4425\pm 50$ K and $3690\pm 130$ K and logarithmic gravities
(in cgs units) of $4.5\pm 0.2$ and $4.3\pm 0.3$. The stars are significantly
different in mass, with TWA 6 being $1.0\pm 0.10$ M☉ and TWA 8A being around
half that at $0.45\pm 0.10$ M☉. Likewise, the radii are also different with
$1.0\pm 0.2$ R☉ for TWA 6 and $0.8\pm 0.2$ R☉ for TWA 8A, viewed at
inclinations of $46\degr\pm 10\degr$ and $31\degr\pm 10\degr$. Using the Siess
et al. (2000) evolutionary models (for direct comparison to other MaTYSSE and
MaPP results), we estimate their ages to be $21\pm 9$ Myr and $11\pm 5$ Myr,
with TWA 6 being mostly radiative, and TWA 8A being fully convective. We note
that these masses, ages and internal structures depend strongly on the adopted
temperatures.

With a rotation period of $0.54095\pm 0.00003$ d, TWA 6 is the most rapidly
rotating wTTS yet mapped with ZDI, and one of the fastest rotators in TWA (see
de la Reza & Pinzón 2004). By contrast, TWA 8A has a much slower period of
$4.578\pm 0.006$ d, which is very similar to the median period of 4.7 d of the
TWA 1–13 group (Lawson & Crause, 2005), and also more similar to that of other
wTTSs such as V819 Tau ($P_{\text{rot}}=5.53113$ d, Donati et al. 2015), as
well as Par 1379 ($P_{\text{rot}}=5.585$ d, Hill et al. 2017).

We find that neither TWA 6 nor TWA 8A have an infrared excess up to 23.675
$\mu$m. Hence, both stars have likely dissipated their circumstellar accretion
discs, with either no accretion taking place, or with accretion occurring at
an undetectable level, given that standard accretion-rate metrics based on the
equivalent widths of H$\alpha$, H$\beta$ and He i $D_{3}$ are strongly
affected by chromospheric emission.

The H$\alpha$, H$\beta$ and Ca ii IRT emission for both stars is mostly non-
variable, with only a few spectra showing excess emission that is attributable
to flaring events or prominences. In particular, TWA 6 shows excess red-
shifted emission in the H$\alpha$, H$\beta$ and Ca ii IRT lines in three
spectra, however, these features are not long lasting and are not periodic.
Indeed, the magnetic topology at these phases is such that excess emission
could be due to off-limb prominence material that is rotating away from the
observer in closed magnetic loops.

Using Zeeman Doppler Imaging, we have derived a surface brightness map of TWA
6, and the magnetic topologies of both stars. We find that TWA 6 has many cool
spots and warm plages on its surface, with a total coverage of around 17 per
cent. We detect no significant modulation of the Stokes  _I_ lines profiles
for TWA 8A, and so find its surface to be compatible with a uniformly bright
star. The reconstructed magnetic fields for TWA 6 and TWA 8A are somewhat
different in strength, and dramatically different in topology. TWA 6 has a
field that is split equally between poloidal and toroidal components, with the
largest fraction of energy in higher order modes (with $\ell>3$), with a total
unsigned flux of $\langle B\rangle=840$ G and where the large-scale
magnetosphere is tilted at $35\degr$ from the rotation axis. On the other
hand, TWA 8A has a highly poloidal field, with most of the energy in the high
order modes with $\ell>3$. The field strength is sufficiently large that the
Stokes  _I_ lines profiles are significantly Zeeman broadened, with Zeeman
signatures clearly detected in individual Stokes  _V_ spectral lines. We
derive a total unsigned flux of $\langle B\rangle=1.4$ kG, using a magnetic
filling factor $f$ equal to 0.2 (meaning that 20 per cent of the surface was
covered with the mapped magnetic features), where on large scales the
magnetosphere is tilted at $20\degr$ from the rotation axis.

For TWA 8A, our simultaneous fits to both Stokes  _I_ and $V$ spectra yields a
mean magnetic field strength of $\langle|B_{f}|\rangle=6.0\pm 0.5$ kG, with a
significant fraction of energy in high-strength fields ($>5$ kG). Given that
we recover a larger fraction of high magnetic field strengths from our direct
modelling of Stokes  _I_ profiles, with those fields having small filling
factors, a significant proportion of magnetic energy likely lies in small-
scale fields that are unresolved by ZDI. The difference between direct
spectral fitting and ZDI is likely due to several factors; Firstly, by the
cancellation of near-by regions of different sign in Stokes  _V_ (providing
most of the difference between Stokes  _I_ and $V$ in single lines); Secondly,
by the signal in Stokes  _V_ LSD profiles not adding perfectly coherently due
to the non-self similarity of different lines in Stokes  _V_ , with scaling
amplitudes by effective Landé factors yielding a less accurate line profile
(most of the difference between single lines and LSD profiles). Hence, small-
scale high-strength magnetic fields are not recovered with LSD, and are thus
not reconstructed with ZDI.

Compared to Tap 26, another wTTS that has a similar mass, age and rotation
rate (Yu et al., 2017), TWA 6 has a larger toroidal field component (50 per
cent for TWA 6 versus 30 per cent for Tap 26), with a total field strength
that is around twice as large. Likewise, the field of TWA 6 is also around
twice as strong as those of the slower rotating (but similarly massive) wTTSs,
V819 Tau and V830 Tau (Donati et al., 2015). In the case of TWA 8A, we find
that is has a weaker (poloidal) dipole field (of $B=0.72$ kG) compared to LkCa
4 (with $B=1.6$ kG), a wTTSs with a similar rotation rate and a slightly
higher mass ($P_{\text{rot}}=3.374$ d, 0.8 M☉). Moreover, compared to main-
sequence M dwarfs with a similar mass and period, namely EV Lac ($\langle
B\rangle=0.57$ kG) and GJ 182 ($\langle B\rangle=172$ G), we see that TWA 8A
has a slightly stronger magnetic field.

In Fig. 14 we compare the magnetic field topologies of all cTTSs and wTTSs so
far mapped with ZDI in an H-R diagram. Fig. 14 also indicates the fraction of
the field that is poloidal, the axisymmetry of the poloidal component, and
shows PMS evolutionary tracks from Siess et al. (2000). In contrast to cTTSs
of the MaPP project, the wTTSs that have been analysed (so far) in the MaTYSSE
sample do not appear to show many obvious trends with internal structure. The
magnetic field strength does not appear to change significantly after the star
becomes mostly radiative, with the largely convective V830 Tau, V819 Tau and
V410 Tau hosting a similarly strong dipole field to the mostly radiative TAP
26, and with the largely convective Par 2244 hosting a similarly strong field
mostly radiative TWA 6. Moreover, the percentage of poloidal field does not
appear to change from when the star is fully convective to when it is mostly
radiative (e.g., V410 Tau and TWA 6 are both around 50 per cent poloidal).
However, the degree of axisymmetry of the poloidal field appears to correlate
with the strength of the magnetic field, given that LkCa 4 and TWA 8A (two
stars with significantly stronger fields of 1.2 kG and 1.4 kG, respectively)
are mostly axisymmetric ($\gtrsim 70$ per cent). Considering both cTTSs and
wTTSs as a whole, it appears that stars are mostly poloidal and axisymmetric
when they are mostly convective and cooler than $\sim 4300$ K. Moreover, stars
hotter than $\sim 4300$ K appear to be less axisymmetric and less poloidal,
regardless of their internal structure. We note that the wTTSs studied thus
far clearly show a wider range of field topologies compared to those of cTTSs,
with large scale fields that can be more toroidal and non-axisymmetric,
consistent with the fact that most of them are largely radiative or are higher
mass. We also note that a more complete analysis will be possible once the
remainder of the MaTYSSE sample has been analysed.

Through our tomographic modelling, we were able to determine that TWA 6 has a
non-zero surface latitudinal-shear at a confidence level of over 99.99 per
cent for the brightness map, and 90 per cent for the magnetic map, as measured
over the 16 nights of observation. Its shear rate is around 56 times smaller
than the Sun, with an equator-pole lap time of $640^{+110}_{-80}$ d. Given the
lack of variability in the lines profiles and the small number of observed
rotations ($\sim 3$ cycles), we were unable to measure the shear rate for TWA
8A. Out measured shear rate for TWA 6 is similar to that found for V410 Tau,
V819 Tau, V830 Tau and LkCa 4 (Skelly et al., 2010; Donati et al., 2014,
2015), which are all of similar mass.

Finally, the brightness map of TWA 6 was used to predict the activity related
RV jitter due to stellar activity, allowing us to filter the measured RVs in
the search for potential hJs (in the same manner as Donati et al. 2014, 2015).
Here, the activity jitter was filtered down to a rms RV precision of $\sim
0.20$ kms-1 (from an initial unfiltered rms of 3.8 kms-1). While this is well
above the RV precision of ESPaDOnS, the high $v\sin{i}$ decreases the accuracy
of the filtering process, with an intrinsic uncertainty of around 0.1 kms-1.
We find no significant peaks in a periodogram analysis, and find that TWA 6 is
unlikely to host a hJ with an orbital period in the range of what we can
detect, with a $3\sigma$ error bar on the semi-amplitude of the RV residuals
equal to 0.19 kms-1, translating into a planet mass of $\simeq 3.1$ MJup
orbiting at $\simeq 0.1$ au.

## Acknowledgements

This paper is based on observations obtained at the CFHT, operated by the
National Research Council of Canada, the Institut National des Sciences de
l’Univers of the Centre National de la Recherche Scientifique (INSU/CNRS) of
France and the University of Hawaii. We thank the CFHT QSO team for its great
work and effort at collecting the high-quality MaTYSSE data presented in this
paper. MaTYSSE is an international collaborative research programme involving
experts from more than 10 different countries (France, Canada, Brazil, Taiwan,
UK, Russia, Chile, USA, Switzerland, Portugal, China and Italy). Observations
of TWA 8A are supported by the contribution to the MaTYSSE Large Project on
CFHT obtained through the Telescope Access Program (TAP), which has been
funded by the“the Strategic Priority Research Program - The Emergence of
Cosmological Structures" of the Chinese Academy of Sciences (Grant No.11
XDB09000000) and the Special Fund for Astronomy from the Ministry of Finance.
GJH is supported by general grants 11473005 and 11773002 awarded by the
National Science Foundation of China. We also thank the IDEX initiative at
Université Fédérale Toulouse Midi-Pyrénées (UFTMiP) for funding the STEPS
collaboration program between IRAP/OMP and ESO and for allocating a ‘Chaire
d’Attractivité’ to GAJH. JFD acknowledges funding from the European Research
Council (ERC) under the H2020 research & innovation programme (grant agreement
#740651 NewWorlds). SHPA acknowledges financial support from CNPq, CAPES and
Fapemig. 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.

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##  Appendix A Line profiles of Ca ii infrared triplet, H$\alpha$, and
H$\beta$, for TWA 6 and TWA 8A

Line profiles of the Ca ii infrared triplet, H$\alpha$ and H$\beta$ are shown
in Fig. 15 and Fig. 16 for TWA 6 and TWA 8A, respectively.

Figure 15: For TWA 6. Left panel: The Ca ii infrared triplet, with line
profiles of the 8498.0.2 Å 8542.09 Å and 8662.14 Å components shown (left to
right) as black solid lines, where the mean line profile is shown in red, with
the cycle number displayed on the right of the profiles. Right panel:
H$\alpha$ and H$\beta$ line profiles, shown in the same manner, additionally
showing the co-rotation radius as a dashed blue line.

Figure 16: Same as Fig. 15 but for TWA 8A, with the left panel showing the Ca
ii infrared triplet, and the right panel showing H$\alpha$ and H$\beta$ line
profiles.

Figure 17: Stokes  _V_ line profiles of TWA 8A with the Ca ii IRT shown in the
left panel, and H$\alpha$ shown in the right panel. $3\sigma$ errorbars are
shown in red on the left side of the line profiles.

Figure 18: Left panel: The normalized variance profile of H$\alpha$ for TWA 6.
There is variance from around -200 kms-1 up to around +300 kms-1. Right panel:
The autocorrelation matrix for H$\alpha$, where black means perfect
correlation and white means perfect anticorrelation.

Figure 19: Same as Fig. 18 but for TWA 8A. The top row shows normalized
variance profiles for H$\alpha$, H$\beta$ and He i D3 (left to right), with
the bottom row showing the corresponding autocorrelation matrices.

##  Appendix B Accretion status of TWA 6 and TWA 8A

The SEDs of TWA 6 and TWA 8A show no evidence of an infra-red excess (see Fig.
2), suggesting that both stars are disc-less. Nevertheless, we may use our
high-quality spectra of both targets to determine their accretion status using
several metrics.

Following our previous studies (e.g., Hill et al., 2017), one may estimate the
level of surface accretion in TTSs by adopting the relations between line
luminosity $L_{\text{line}}$ and the accretion luminosity $L_{\mathrm{acc}}$
of Alcalá et al. (2017). For this purpose we determined $L_{\text{line}}$ by
assuming blackbody scaling using the stellar radius $R_{\star}$ and
$T_{\text{eff}}$ given in Table 2. Then, the mass accretion rate
${\dot{M}_{\mathrm{acc}}}$ was calculated using the relationship

$\dot{M}_{\mathrm{acc}}=\frac{L_{\mathrm{acc}}R_{\star}}{GM_{\star}(1-\frac{R_{\star}}{R_{\text{in}}})}$
(5)

where $R_{\text{in}}$ denotes the truncation radius of the disc, and is taken
to be $5R_{\star}$ (Gullbring et al., 1998).

For TWA 6, we detect weak He i $D_{3}$ emission with an EW of around 0.03 Å
(1.6 kms-1), corresponding to ${\log{\dot{M}_{\mathrm{acc}}}\simeq-10.6}$
M☉yr-1. We find the H$\beta$ emission (see Fig. 15) to have an EW ranging
between 0.69–1.44 Å (average of 0.96 Å, equivalent to 59.5 kms-1,
corresponding to ${\log{\dot{M}_{\mathrm{acc}}}\simeq-10.1}$ M☉yr-1), and the
H$\alpha$ emission to have an EW ranging between 2.26–4.03 Å (average of 2.85
Å, equivalent to 130.2 kms-1, corresponding to
${\log{\dot{M}_{\mathrm{acc}}}\simeq-10.2}$ M☉yr-1).

For TWA 8A, we detect moderate He i $D_{3}$ emission with an EW of around 0.3
Å (17.4 kms-1), corresponding to ${\log{\dot{M}_{\mathrm{acc}}}\simeq-9.7}$
M☉yr-1. We find the H$\beta$ emission (see Fig. 16) to have an EW ranging
between 3.1–6.8 Å (average of 3.9 Å, equivalent to 238.0 kms-1, corresponding
to ${\log{\dot{M}_{\mathrm{acc}}}\simeq-9.7}$ M☉yr-1), and the H$\alpha$
emission to have an EW ranging between 5.9–10.9 Å (average of 7.2 Å,
equivalent to 326.9 kms-1, corresponding to
${\log{\dot{M}_{\mathrm{acc}}}\simeq-9.8}$ M☉yr-1).

These accretion rates would suggest that both stars are weakly accreting,
however, as discussed in Hill et al. (2017), chromospheric activity in TTS
becomes a significant influence on the strength and width of emission lines in
the low accretion regime. Pertinently, the large convective turnover times of
TTSs (Gilliland, 1986) combined with their rapid rotation means they possess a
low Rossby number, placing them well within the saturated activity regime
(e.g., Reiners et al. 2014). Indeed, the H$\alpha$ line luminosity is observed
to saturate in young stars at around $\log[L(H\alpha)/L_{\text{bol}}]=-3.3$ or
lower (Barrado y Navascués & Martín, 2003), and as both our target stars show
line luminosities that are similar to (or below) this level, with
$\log[L(H\alpha)/L_{\text{bol}}]$ equal to $-3.19$ for TWA 6 and $-3.78$ for
TWA 8A, any estimate of accretion rates based on line luminosities (especially
H$\alpha$, and to a lesser extent H$\beta$) must be considered to be
significantly influenced or even dominated by chromospheric activity.

The distinction between emission due to accretion and that due to
chromospheric activity has been characterized by several authors, yielding a
distinct threshold between these regimes. Using the empirical spectral-type-
dependant relationship between the EW(H$\alpha$) and the accretion rate of
Barrado y Navascués & Martín (2003), the defining threshold of an accreting
TTS is EW(H$\alpha$) equal to 5.1 Å and 12.2 Å for K5 and M3 spectral types,
respectively (appropriate for TWA 6 and TWA 8A, see Section 3.1). Given that
the maximum EW(H$\alpha$) of TWA 6 and TWA 8A are equal to 4.0 Å and 10.9 Å ,
both stars lie below these limits and fall into the non-accreting regime
(where line broadening is dominated by chromospheric activity).

Elsewhere, Manara et al. (2017) derived an empirical relationship between a
star’s spectral-type and the point at which line emission may be dominated by
chromospheric activity (termed chromospheric accretion ‘noise’). In the case
of TWA 6, this threshold is at
$\log{(L_{\text{acc,noise}}/L_{\text{star}})}=-2.3\pm 0.1$. Given that the
average line luminosities $\log{(L_{\text{acc}}/L_{\text{star}}})$ for
H$\alpha$, H$\beta$ and He i $D_{3}$ are respectively equal to $-3.19\pm
0.01$, $-3.04\pm 0.02$ and $-3.63\pm 0.03$, the luminosity of all three
emission lines are significantly below the threshold of chromospheric noise.
Likewise for TWA 8A, this threshold is estimated as
$\log{(L_{\text{acc,noise}}/L_{\text{star}})}=-2.59\pm 0.13$. Here, the
average line luminosities for H$\alpha$, H$\beta$ and He i $D_{3}$ are
respectively equal to $-3.78\pm 0.02$, $-3.65\pm 0.03$ and $-3.62\pm 0.02$,
where again, all emission is well below the threshold where one can
distinguish between accretion and chromospheric emission.

Thus, the accretion rates determined above for TWA 6 and TWA 8A must be taken
to be upper limits, given that chromospheric emission is likely the dominant
broadening mechanism. Hence, our target stars are likely not accreting (or are
doing so at an undetectable level), thus confirming their classification as
wTTSs - a result consistent with past work by White & Hillenbrand (e.g. 2004);
Kastner et al. (e.g. 2016).

##  Appendix C Magnetic fields from direct spectral fitting

Table 4: Atomic data used in the direct spectrum fitting, from VALD, for the major lines. Additional much weaker lines were included in the spectrum synthesis for completeness, but are omitted here for brevity. The quantities low and high refer to the lower and upper level of the transition, respectively. Term symbols are provided to identify lines of the same multiplet.  Species | Wavelength (Å) | $\log gf$ |  $E_{\rm low}$ (Ev) | $J_{\rm low}$ | $J_{\rm high}$ | Landé $g_{\rm low}$ | Landé $g_{\rm high}$ | Multiplet terms
---|---|---|---|---|---|---|---|---
Ti i | 9675.54 | -0.804 | 0.8360 | 4 | 4 | 1.34 | 1.35 | a 5F – z 5F∘
Ti i | 9688.87 | -1.610 | 0.8129 | 1 | 2 | 1.00 | 1.50 | a 5F – z 5F∘
Ti i | 9705.66 | -1.009 | 0.8259 | 3 | 3 | 1.26 | 1.26 | a 5F – z 5F∘
Ti i | 9718.96 | -1.181 | 1.5025 | 4 | 3 | 1.00 | 0.95 | a 1G – z 1F∘
Ti i | 9728.41 | -1.206 | 0.8181 | 2 | 2 | 1.00 | 1.00 | a 5F – z 5F∘
Ti i | 9743.61 | -1.306 | 0.8129 | 1 | 1 | 0.00 | 0.00 | a 5F – z 5F∘
Ti i | 9770.30 | -1.581 | 0.8484 | 5 | 4 | 1.34 | 1.55 | a 5F – z 5F∘
Ti i | 9783.31 | -1.428 | 0.8360 | 4 | 3 | 1.26 | 1.48 | a 5F – z 5F∘
Ti i | 9783.59 | -1.617 | 0.8181 | 2 | 1 | 0.00 | 1.49 | a 5F – z 5F∘
Ti i | 9787.69 | -1.444 | 0.8259 | 3 | 2 | 1.00 | 1.50 | a 5F – z 5F∘
Ti i | 9832.14 | -1.130 | 1.8871 | 5 | 4 | 1.21 | 1.21 | a 3G – y 3F∘
Table 5: Best fit parameters from direct spectral fitting of TWA 8A, using our third model that fits Stokes $I$ and $V$ simutaneously, for observations that could be adequately telluric-corrected. Each column gives the fitted parameters for the spectrum obtained on the date given at the top. Mean values are presented in Table 3. | 2015-03-25 | 2015-03-26 | 2015-03-27 | 2015-03-28 | 2015-03-29 | 2015-03-30 | 2015-03-31 | 2015-04-01 | 2015-04-05 | 2015-04-06
---|---|---|---|---|---|---|---|---|---|---
$v\sin{i}$ (kms-1) | $4.75\pm 0.19$ | $4.78\pm 0.17$ | $4.72\pm 0.18$ | $5.08\pm 0.17$ | $4.61\pm 0.18$ | $4.66\pm 0.16$ | $4.66\pm 0.16$ | $4.95\pm 0.17$ | $4.92\pm 0.17$ | $5.02\pm 0.17$
$v_{\text{mic}}$ (kms-1) | $1.18\pm 0.07$ | $1.07\pm 0.06$ | $1.14\pm 0.06$ | $1.13\pm 0.06$ | $1.08\pm 0.06$ | $1.06\pm 0.05$ | $1.06\pm 0.06$ | $1.08\pm 0.06$ | $1.01\pm 0.06$ | $1.02\pm 0.06$
$[$Ti/H$]$ | $-7.007\pm 0.014$ | $-6.985\pm 0.012$ | $-7.005\pm 0.013$ | $-6.974\pm 0.012$ | $-6.981\pm 0.012$ | $6.988\pm 0.011$ | $-6.966\pm 0.011$ | $-6.963\pm 0.012$ | $-6.934\pm 0.011$ | $-6.952\pm 0.012$
+2 kG | $0.168\pm 0.023$ | $0.182\pm 0.017$ | $0.167\pm 0.018$ | $0.154\pm 0.018$ | $0.150\pm 0.018$ | $0.176\pm 0.016$ | $0.175\pm 0.016$ | $0.165\pm 0.018$ | $0.147\pm 0.019$ | $0.125\pm 0.019$
+5 kG | $0.229\pm 0.010$ | $0.227\pm 0.009$ | $0.259\pm 0.010$ | $0.239\pm 0.010$ | $0.221\pm 0.010$ | $0.228\pm 0.009$ | $0.239\pm 0.009$ | $0.268\pm 0.010$ | $0.258\pm 0.010$ | $0.282\pm 0.010$
+10 kG | $0.053\pm 0.008$ | $0.061\pm 0.006$ | $0.051\pm 0.007$ | $0.056\pm 0.007$ | $0.052\pm 0.006$ | $0.056\pm 0.006$ | $0.061\pm 0.006$ | $0.055\pm 0.006$ | $0.058\pm 0.006$ | $0.048\pm 0.007$
+15 kG | $0.052\pm 0.007$ | $0.044\pm 0.006$ | $0.049\pm 0.006$ | $0.051\pm 0.006$ | $0.045\pm 0.006$ | $0.047\pm 0.005$ | $0.043\pm 0.005$ | $0.043\pm 0.006$ | $0.045\pm 0.006$ | $0.047\pm 0.006$
+20 kG | $0.034\pm 0.005$ | $0.030\pm 0.004$ | $0.027\pm 0.005$ | $0.042\pm 0.005$ | $0.038\pm 0.005$ | $0.034\pm 0.004$ | $0.037\pm 0.004$ | $0.037\pm 0.005$ | $0.038\pm 0.004$ | $0.038\pm 0.005$
-2 kG | $0.163\pm 0.023$ | $0.166\pm 0.017$ | $0.145\pm 0.018$ | $0.159\pm 0.018$ | $0.148\pm 0.018$ | $0.173\pm 0.016$ | $0.161\pm 0.016$ | $0.144\pm 0.018$ | $0.146\pm 0.019$ | $0.141\pm 0.019$
-5 kG | $0.221\pm 0.010$ | $0.202\pm 0.009$ | $0.205\pm 0.010$ | $0.202\pm 0.010$ | $0.217\pm 0.010$ | $0.213\pm 0.009$ | $0.199\pm 0.009$ | $0.209\pm 0.010$ | $0.205\pm 0.010$ | $0.213\pm 0.010$
-10 kG | $0.044\pm 0.008$ | $0.045\pm 0.006$ | $0.036\pm 0.007$ | $0.053\pm 0.007$ | $0.041\pm 0.006$ | $0.041\pm 0.006$ | $0.036\pm 0.006$ | $0.039\pm 0.006$ | $0.035\pm 0.006$ | $0.040\pm 0.007$
-15 kG | $0.010\pm 0.006$ | $0.000\pm 0.006$ | $0.010\pm 0.006$ | $0.010\pm 0.006$ | $0.012\pm 0.006$ | $0.010\pm 0.005$ | $0.012\pm 0.005$ | $0.011\pm 0.006$ | $0.009\pm 0.006$ | $0.013\pm 0.006$
-20 kG | $0.022\pm 0.005$ | $0.028\pm 0.004$ | $0.024\pm 0.005$ | $0.035\pm 0.005$ | $0.023\pm 0.005$ | $0.023\pm 0.004$ | $0.024\pm 0.004$ | $0.028\pm 0.005$ | $0.031\pm 0.004$ | $0.029\pm 0.005$
0 kG | $0.004\pm 0.039$ | $0.017\pm 0.031$ | $0.028\pm 0.033$ | $0.000\pm 0.033$ | $0.051\pm 0.032$ | $0.000\pm 0.029$ | $0.014\pm 0.029$ | $0.000\pm 0.032$ | $0.028\pm 0.033$ | $0.023\pm 0.033$