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# Lorentz gauge-invariant variables in torsion-based theories of gravity

Daniel Blixt  [  Laboratory of Theoretical Physics, Institute of Physics,
University of Tartu, W. Ostwaldi 1, Tartu 50411, Estonia  Rafael Ferraro  [
Instituto de Astronomía y Física del Espacio (IAFE, CONICET-UBA), Casilla de
Correo 67, Sucursal 28, 1428 Buenos Aires, Argentina.  Departamento de Física,
Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Ciudad
Universitaria, Pabellón I, 1428 Buenos Aires, Argentina.  Alexey Golovnev  [
Centre for Theoretical Physics, The British University in Egypt, 11837 El
Sherouk City, Egypt  María-José Guzmán  [  Laboratory of Theoretical Physics,
Institute of Physics, University of Tartu, W. Ostwaldi 1, Tartu 50411, Estonia

###### Abstract

General relativity dynamics can be derived from different actions –which
depart from the Einstein-Hilbert action in boundary terms– and for different
choices of the dynamical variables. Among them, the teleparallel equivalent of
general relativity is a torsion-based theory for the tetrad field. More
general torsion-based theories have been built in the last years, intending to
supersede general relativity. There are two current ways to formulate such
theories; one includes a spin-connection and the other does not. We discuss
the notion of Lorentz gauge invariance in such theories, and give a simple but
important proof that both formulations are physically equivalent.

##  I Introduction

Fernparallelismus or teleparallelism is firstly known as an attempt by
Einstein to base a unified theory of the electromagnetic and gravitational
fields on the mathematical structure of absolute teleparallelism. In this
framework, spacetime is characterized by a curvatureless connection together
with a metric tensor field, both defined in terms of a dynamical tetrad or
vierbein field (a field of orthonormal tangent-space bases). This was an
episode in Einstein’s research that lasted for three years from 1928 to 1931.
Although the pertinent mathematical structures had been developed before by
Cartan and Weitzenböck, they were introduced just as mathematical concepts,
and the search for field equations for the tetrads was the novelty, efforts
which eventually were abandoned [2].

The concepts of teleparallelism were not further explored until three decades
later, when Møller [3] revived Einstein’s original idea, but in order to find
a tensorial complex for the gravitational energy-momentum density. After this,
the Lagrangian formulation for teleparallel gravity was written by Pellegrini
and Plebianski [4]. Later, Hayashi and Shirafuji [5] proposed the New General
Relativity (NGR), a teleparallel theory based on the curvatureless Weitzenböck
connection (which is linear in first derivatives of the tetrad). The NGR
Lagrangian combines the three quadratic invariants emerging from the
decomposition of the Weitzenböck torsion in its irreducible parts. Three
parameters enter the NGR Lagrangian, which can be fixed to render NGR the
teleparallel equivalent of general relativity (TEGR). The equivalence between
TEGR and GR lies in the fact that TEGR dynamics for the tetrad yields the GR
dynamics for the metric whenever the tetrad is thought as an orthonormal basis
in the tangent space, so linking the tetrad to the metric. The orthonormality
property is clearly invariant under local Lorentz transformations of the
tetrad. The equivalence between TEGR and GR would suggest that the TEGR
Lagrangian should exhibit such a gauge invariance. Actually, TEGR Lagrangian
is not invariant but pseudo-invariant [6]: a local Lorentz transformation of
the tetrad produces a boundary term in the TEGR action, which does not affect
the dynamics.

In the middle of the 2000’s, Ferraro and Fiorini proposed a non-linear
extension of TEGR theory that consisted in replacing the torsion scalar
$\mathbb{T}$ in the Lagrangian with a convenient function of it. For a Born-
Infeld-like Lagrangian, they have shown that the inflationary early-universe
is a solution for ordinary sources such as radiation or dust [7]. This opened
the study of the so called $f(\mathbb{T})$ theories of modified teleparallel
gravity [8, 9, 10]. We should notice that, although pseudo-invariance works
like a full invariance at the level of the TEGR equations of motion, this
feature is lost when passing to $f(\mathbb{T})$ gravity. The divergence
associated with the boundary term now remains encapsulated within the argument
of the function $f(\mathbb{T})$ in the Lagrangian. Thus the equations of
motion are no longer invariant under local Lorentz transformations of the
tetrad (however, they remain invariant under global Lorentz transformations).
The loss of this gauge freedom implies that $f(\mathbb{T})$ theories contain
some degrees of freedom (d.o.f.) associated with the local orientation of the
tetrad field (i.e., with the parallelization of the spacetime). How many extra
d.o.f. are involved in $f(\mathbb{T})$ and other modified teleparallel
gravities is a subject under study; but given the non-linear background-
dependent nature of its constraints, it can be expected that the extra d.o.f.
will depend on the point in the phase space [11, 12, 13, 14, 15, 16]. As long
as the matter couples just to the metric, or even the Levi-Civita connection,
the sole possibility of observing these elusive extra d.o.f. would be via
changes of gravitational dynamics, but not at the level of the motion of
matter in a given spacetime.

The loss of the Lorentz gauge freedom can be avoided from the beginning, by
providing the TEGR Lagrangian with a divergence term to guarantee its full
local Lorentz invariance. However, this would lead to a sort of $f(R)$ theory
(non-linear extension of the GR Lagrangian), which would depart from the
teleparallel foundations. A different strategy consists of improving the
Weitzenböck connection, by replacing it with a more general one that is able
to provide a better behavior under local Lorentz transformations. In this way,
we endow the theory with a full Lorentz gauge invariance. For reasons that
will be clearer in the next Section, this strategy is called covariantization.

In recent years we have been witnessing intense discussions about the local
Lorentz invariance in modified teleparallel theories of gravity, such as
$f(\mathbb{T})$ [17]. Some authors object the covariantization procedure as
inconsistent since the components of the more general curvatureless connection
cannot be fixed by varying the covariantized action [18, 19]. Therefore the
connection must be chosen by resorting to some criterion, which is not very
different to directly adopting the Weitzenböck connection. Of course, there
might be some preferences based on symmetry considerations, but those would
often be not unique, neither available in cases of no symmetry.

Some others view the local-Lorentz-covariant representations as the only
consistent approach [20, 21], by assuming that the fundamental tetrad variable
of teleparallel gravity must be regarded as the frame an observer is free to
choose, and with the spin connection somehow related to inertial effects. In
such case, the Lorentz gauge freedom is frozen by parallelizing according to
some physical criterion. But if one thinks of the tetrad as a geometric object
to describe gravity, just a set of orthonormal vectors without caring about
observers, then such a “physical” criterion becomes meaningless. In any case,
the extra d.o.f. will enter into play and they turn out to be rather
unhealthy, at least for the standard $f(\mathbb{T})$ theory [22, 23] and NGR
[24, 25].

All these disagreements seem odd, as there is evidence that the covariant and
the (original) pure tetrad approaches would be equivalent [17, 26]. Moreover,
the general curvatureless connection introduces trivial primary constraints in
the Hamiltonian analysis [27, 28] which are evidently first-class [29]. For
instance, the so called preferred frame effects or “frame-dependent artifacts”
are resolved only very superficially [30, 20, 21]. In a covariantized model
one is allowed to choose absolutely any available tetrad for a given metric,
but at the price of taking an appropriate spin connection. On the other hand,
two different solutions for the same metric ansatz of the pure tetrad approach
would also give two different solutions in the covariantized version, in
disguise of two different possible choices of the spin connection for a given
frame [26].

It is not to say that the covariant approach has no physical meaning, since
for example, it naturally comes from taking teleparallel gravity as a gauge
theory of translations [31]. Of course, this gives good hope for formulating
conserved quantities such as energy, for authors who believe that these
quantities should be well defined for gravity despite that there is no
fundamental symmetry which would make it justified, so that in a really
canonical way one gets rather a holographic notion of conserved quantities,
necessarily paying attention to boundary terms [32].

Leaving aside the controversies, the covariant methods can be used for more
convenience of doing calculations [26]. However, what we will show in this
article is that the pure-tetrad formulation is precisely the same as simply
taking the gauge invariant variables in the covariant version, for any
teleparallel model which is globally Lorentz-invariant. This is yet another,
very simple and direct proof that the two versions have no physical
difference.

##  II Teleparallel and modified teleparallel gravity

In its standard formulation, teleparallel gravity is built from the
Weitzenböck connection, which is the connection that makes the tetrad
$e^{b}_{\nu}$, and its inverse $e_{b}^{\mu}$, parallel transported as
$0=\nabla_{\mu}e^{b}_{\nu}=\partial_{\mu}e^{b}_{\nu}-\Gamma^{\alpha}_{\mu\nu}e^{b}_{\alpha}$.
Then, its components are

$\Gamma^{\alpha}_{\mu\nu}\equiv e_{a}^{\alpha}\partial_{\mu}e^{a}_{\nu}\ .$
(1)

The Weitzenböck connection is compatible with the metric
$g_{\mu\nu}=\eta_{ab}e^{a}_{\mu}e^{b}_{\nu}$, or
$g_{\mu\nu}e_{a}^{\mu}e_{b}^{\nu}=\eta_{ab}$ (the basis $\\{\mathbf{e}_{a}\\}$
is orthonormal). $\Gamma^{\alpha}_{\mu\nu}$ results to be a curvatureless
connection whose torsion is

$T^{\alpha}_{\hphantom{\alpha}\mu\nu}=\Gamma^{\alpha}_{\mu\nu}-\Gamma^{\alpha}_{\nu\mu}=e_{a}^{\alpha}\,(\partial_{\mu}e^{a}_{\nu}-\partial_{\nu}e^{a}_{\mu})\
.$ (2)

The torsion scalar

${\mathbb{T}}=\frac{1}{4}T_{\alpha\beta\mu}T^{\alpha\beta\mu}+\frac{1}{2}T_{\alpha\beta\mu}T^{\beta\alpha\mu}-T_{\mu}T^{\mu}\
,$ (3)

where $T_{\mu}=T^{\alpha}_{\hphantom{\alpha}\alpha\mu}$ is the torsion vector,
is directly related to the Levi-Civita curvature scalar $\overset{\circ}{R}$
of the metric $g_{\mu\nu}$ (that is the GR Lagrangian),

$e\overset{\circ}{R}=-e{\mathbb{T}}+\partial_{\mu}(2eT^{\mu}),$ (4)

where $e=\det e^{a}_{\mu}=(-\det g_{\mu\nu})^{1/2}$. On this basis, the TEGR
Lagrangian density is taken equal to $\pm e\mathbb{T}$, with the sign
depending on the chosen signature. Thus we obtain a TEGR Lagrangian quadratic
in first derivatives of the tetrad. The second derivatives contained in the GR
Lagrangian have been cornered in the boundary term of Eq. (4), which is not
relevant for the dynamics.

In Eq. (4), the l.h.s. depends just on the metric, which is invariant under
local Lorentz transformations of the tetrad,

$e^{a}_{\mu}\longrightarrow L^{a}_{c}(x)~{}e^{c}_{\mu}\ \ \Longrightarrow\ \
g_{\mu\nu}\longrightarrow g_{\mu\nu}\ ,$ (5)

where $L^{a}_{c}$ is a matrix belonging to the Lorentz group (so it is
$\eta_{ab}L^{a}_{c}L^{b}_{d}=\eta_{cd}$). We remark that we are not talking
about the behavior under diffeomorphisms, since each term in the r.h.s. of Eq.
(4) is separately a scalar density. Indeed the expression (1) transforms as an
affine connection under coordinate changes, which implies that
$T^{\alpha}_{\hphantom{\alpha}\mu\nu}$ in Eq. (2) really transforms as the
components of a tensor. We are focusing our analysis on the behavior of the
r.h.s. of Eq. (4) under local changes of the orientation of the tetrad. Of
course, if the l.h.s. of Eq. (4) possesses local Lorentz invariance, then the
r.h.s. will possess it too. However, separately each term in the r.h.s. is
only globally Lorentz invariant. This is because the components of the torsion
in Eq. (2) are made of the components
$\partial_{\mu}e^{a}_{\nu}-\partial_{\nu}e^{a}_{\mu}$ of the exterior
derivative of the tetrad $d{\mathbf{e}}^{a}$, which changes as

$d{\mathbf{e}}^{a}\longrightarrow
L^{a}_{c}~{}d{\mathbf{e}}^{c}+d(L^{a}_{c})\wedge{\mathbf{e}}^{c}\ .$ (6)

Only if the Lorentz transformation is global, then the 1-forms $d(L^{a}_{c})$
will be zero; thus $d{\mathbf{e}}^{a}$ will behave as a 2-form valued in the
tangent space. In a more prosaic language, only for global transformations,
the label “$a$” in $d{\mathbf{e}}^{a}$ will behave as a “contravariant” index
under Lorentz transformations of the tetrad, in the sense that
$d{\mathbf{e}}^{a}$ will transform like ${\mathbf{e}}^{a}$.

The violation of the local Lorentz invariance does not show up in the
equations of motion of TEGR since its Lagrangian density differs only by a
boundary term $\mathbb{B}=\partial_{\mu}(2eT^{\mu})$ from the GR one. This
makes TEGR pseudo-invariant under local Lorentz transformations of the tetrad
[33]. Pseudo-invariance is a rather low price to be paid for substituting the
GR Lagrangian, which contains second derivatives of the metric, with the
simpler TEGR Lagrangian that is built from first derivatives of the tetrad.
However, there are many generalizations, such as NGR, $f(\mathbb{T})$ gravity,
or even higher derivative ones, which do violate the local Lorentz invariance
at the level of equations of motion. The most popular higher derivative models
are $f(\mathbb{T},\mathbb{B})$ ones, even though they do not go much beyond
the more usual modified gravities and $f(\mathbb{T})$ gravity. Due to the
basic relation (4), they can obviously be represented as simply
$f(\overset{\circ}{R},\mathbb{T})$, apparently inheriting all the potential
problems that $f(\mathbb{T})$ could have [22, 23] .

##  III Lorentz-covariantization

The components of the torsion in Eq. (2) will become invariant under local
Lorentz transformations of the tetrad if the ordinary exterior derivative
$d{\mathbf{e}}^{a}$ is covariantized by endowing it with a spin connection
term:

${\mathcal{D}}{\mathbf{e}}^{a}\equiv
d{\mathbf{e}}^{a}+\omega^{a}_{\hphantom{a}b}\wedge{\mathbf{e}}^{b}\ .$ (7)

The 1-forms $\omega^{a}_{\hphantom{a}b}$ must accompany the change of the
tetrad by transforming as components of a spin connection to absorb the
undesirable term in Eq. (6):

${\mathbf{e}}^{a}\rightarrow
L^{a}_{c}\,{\mathbf{e}}^{c},\quad\omega^{a}_{\hphantom{a}b}\rightarrow
L^{a}_{c}\,\omega^{c}_{\hphantom{c}d}\,(L^{-1})^{d}_{b}-(L^{-1})^{c}_{b}\
d(L^{a}_{c})\ .$ (8)

Thus it turns out to be ${\mathcal{D}}{\mathbf{e}}^{a}\rightarrow
L^{a}_{c}(x)\,{\mathcal{D}}{\mathbf{e}}^{c}$.

When talking about invariance or covariance, it is very important to precisely
specify what kind of transformations are being considered, since the invoked
property is not only about the algebraic structure of the involved group but
also about how it acts. Let us introduce the following two definitions:

Definition 1: If a teleparallel theory is Lorentz invariant under the
simultaneous transformations of the tetrad field and the spin connection of
Eq. (8), then it will be called Lorentz invariant of type I.

Definition 2: If a teleparallel theory is Lorentz invariant under the
transformation (5) of the tetrad field (alone), then it will be called Lorentz
invariant of type II.

Remark: Naturally, if a theory violates the Lorentz invariance only at the
boundary, the word “pseudo” is added to the above definitions.

Any theory with an explicitly introduced spin connection is usually Lorentz-
invariant of type I, in the full meaning of that, without the prefix “pseudo”.
However, in case of teleparallel models, the type II is at best a “pseudo-
invariance” which gets broken by almost every modification away from TEGR. On
the other hand, given the fulfillment of type I invariance, the
transformations of type II can equivalently be viewed as transformations of
the spin connection alone.

In its pure tetrad formulation, TEGR uses the Weitzenböck spin connection
$\omega^{a}_{\hphantom{a}b}=0$. But a teleparallel theory covariantly
formulated should be built from the (Lorentz invariant) torsion
$T^{\alpha}_{\hphantom{\alpha}\mu\nu}=e_{a}^{\alpha}({\mathcal{D}}{\mathbf{e}}^{a})_{\mu\nu}$
111${\mathbf{T}}^{a}={\mathcal{D}}{\mathbf{e}}^{a}$ is the 1st Cartan’s
structure equation. It defines the relation between torsion and connection..
Since teleparallelism uses curvatureless connections, so implying that gravity
comes exclusively from the torsion field, the spin connection should be chosen
within the family of curvatureless spin connections to which the Weitzenböck
connection belongs. The more general connection of this type can be obtained
by local-Lorentz transforming the Weitzenböck connection (modulo possible
global issues of cohomology type)

$\omega^{a}_{\hphantom{a}b}=-(\Lambda^{-1})^{c}_{b}\ d\Lambda^{a}_{c}\ ,$ (9)

where the $\Lambda$’s are matrices belonging to the Lorentz group 222The basis
$\\{{\mathbf{e}}_{a}\\}$ will no longer be parallel transported if the
Weitzenböck connection is replaced with (9), but the connection will be still
metric. Anyway, the “parallelization” can be always retrieved by passing to
the Weitzenböck connection through a local Lorentz transformation.. We use
“$\Lambda$” instead of “$L$” because they are new variables of the theory,
characterizing the spin connection, while “$L$” indicates the local Lorentz
transformations of the variables.

Now the (type I) simultaneous Lorentz transformation (8) can be displayed as

$e^{a}_{\mu}\longrightarrow
L^{a}_{c}(x)\,e^{c}_{\mu}\,,\quad\Lambda^{a}_{b}\longrightarrow
L^{a}_{c}(x)\,\Lambda^{c}_{b}.$ (10)

As explained above, any (modified) teleparallel model which is globally
Lorentz invariant in its pure tetrad formulation (every model discussed in the
literature we know) becomes locally Lorentz invariant upon this
covariantization procedure, though with respect to the (type I) simultaneous
transformation (8).

For many popular models, it has been shown that the equation of motion for the
spin connection, which results from varying the action with respect to the
$\Lambda$’s variables, just reproduces the antisymmetric part of the tetrad
equations. Why this happens is rather evident [17]: in the Lorentz-covariant
action, the antisymmetric variation of the tetrad gets precisely compensated
by variation of the spin connection (keeping it inside the flat metric-
compatible class).

Therefore, in the covariant version we can always choose the
$\omega^{a}_{\hphantom{a}b}=0$ gauge, even right inside the action which
otherwise is not always a harmless choice to do, which brings us back to the
pure tetrad formalism.

Moreover, any gauge choice does not influence the value of the torsion tensor,
therefore it is not only that this choice does not influence the physical
contents of equations of motion, but it does not change the global quantities,
like the full value of the action, either. If we found some solution in the
covariant version, we can choose a gauge and make it a pure-tetrad solution,
with the same metric and the same torsion tensor.

##  IV Lorentz gauge-invariant variables

Now, when we know that any globally Lorentz-invariant modified teleparallel
model can be covariantized by replacing the partial derivatives of the tetrad
for the Lorentz-covariant ones, let us make a simple rewriting of the
covariant derivative (7) with the connection (9):

$\displaystyle({\mathcal{D}}{\mathbf{e}}^{a})_{\mu\nu}$ $\displaystyle=$
$\displaystyle
2\;\partial_{[\mu}e^{a}_{\nu]}-2\;(\Lambda^{-1})^{c}_{b}\,(\partial_{[\mu}\Lambda^{a}_{c})\,e^{b}_{\nu]}$
(11) $\displaystyle=$ $\displaystyle
2\;\partial_{[\mu}e^{a}_{\nu]}+2\;\Lambda^{a}_{c}\,(\partial_{[\mu}(\Lambda^{-1})^{c}_{b})\,e^{b}_{\nu]}$
(13) $\displaystyle=$ $\displaystyle\Lambda^{a}_{c}\
2\left[(\Lambda^{-1})^{c}_{b}\,\partial_{[\mu}e^{b}_{\nu]}+(\partial_{[\mu}(\Lambda^{-1})^{c}_{b})\,e^{b}_{\nu]}\right]\
\ $ (15) $\displaystyle=$ $\displaystyle\Lambda^{a}_{c}\
2\,\partial_{[\mu}{\tilde{e}}^{\;c}_{\;\nu]}\ ,$ (17)

which means that ${\mathcal{D}}{\mathbf{e}}^{a}=\Lambda^{a}_{c}\
d{\tilde{\mathbf{e}}}^{c}$, where

${\tilde{\mathbf{e}}}^{c}\equiv(\Lambda^{-1})^{c}_{b}\ {\mathbf{e}}^{b}$ (18)

is a Lorentz-invariant quantity which does not change at all under the
simultaneous local Lorentz transformations (10). Thus the Lorentz-covariant
torsion tensor is

$T^{\alpha}_{\hphantom{\alpha}\mu\nu}=e_{a}^{\alpha}\,({\mathcal{D}}{\mathbf{e}}^{a})_{\mu\nu}={\tilde{e}}_{\;a}^{\;\alpha}\;(\partial_{\mu}{\tilde{e}}^{\;a}_{\;\nu}-\partial_{\nu}{\tilde{e}}^{\;a}_{\;\mu})\
,$ (19)

which is the torsion for the Weitzenböck connection associated with
${\tilde{\mathbf{e}}}^{a}$. Then, the covariant formulation is dynamically
equivalent to a pure tetrad formulation. The only dynamical object of the
covariant formulation is ${\tilde{\mathbf{e}}}^{a}$:

$\mathcal{L}^{\mathrm{covariant}}(e,\Lambda)=\mathcal{L}^{\mathrm{pure\
tetrad}}(\tilde{e})\ ,$ (20)

and therefore the locally Lorentz-invariant variables (18) simply satisfy the
equations of motion of the pure tetrad version of the model. Thus, conclusions
made in the pure tetrad formulation also apply to the covariant formulation,
and it is evident that the spin connection contains purely gauge degrees of
freedom.

The formula (20) is valid for any modified teleparallel model which can be
written in terms of the metric (and its Levi-Civita connection) and the
spacetime components of the torsion tensor. Indeed, the metric is obviously
invariant under any kind of the Lorentz transformations, be it type I or type
II, while for the torsion tensor we have

$T^{\alpha}_{\hphantom{\alpha}\mu\nu}(e,\omega(\Lambda))=T^{\alpha}_{\hphantom{\alpha}\mu\nu}(\tilde{e},0).$

Therefore, it includes $f(\mathbb{T})$ models as well as NGR and its non-
linear generalizations, and even models of $f(\mathbb{T},\mathbb{B})$ and many
other types.

What we have shown here is that taking the $\omega^{a}_{\hphantom{a}b}=0$
gauge is equivalent to a simple change of the variables. Very similarly, a
Lagrangian of the form

$\mathcal{L}(\phi,\psi)=\frac{1}{2}\partial_{\mu}(\phi-\psi)\partial^{\mu}(\phi-\psi)$

can be transformed to
$\mathcal{L}(\chi,\psi)=\frac{1}{2}(\partial_{\mu}\chi)(\partial^{\mu}\chi)$
by a simple change of variable $\phi\to\chi=\phi-\psi$ which removes its
dependence on $\psi$. It is nothing but rewriting the Lagrangian in gauge-
invariant variables.

A source of concern might be eligibility of restricting to gauge-invariant
variables only, inside the action. For example, using the vector potential is
important for the action principle of electrodynamics. In fact, its gauge-
invariant quantities $F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$
depend on derivatives of the fundamental variables. The variation of
$\mathcal{L}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}$ with respect to $F_{\mu\nu}$
would give the trivial equation $F_{\mu\nu}=0$ while its variation with
respect to $A_{\nu}$ gives the correct result of
$\overset{\circ}{\triangledown}_{\mu}F^{\mu\nu}=0$.

The reason is that the condition of $\delta A$ vanishing at infinity is
stronger than the same condition for $\delta F$. This is not about a precise
type of asymptotic behavior, the same is true for vanishing everywhere outside
a big enough ball. For example, vanishing of an otherwise arbitrary function
$x(t)$ both in the past and in the future means also that $\dot{x}(t)$
integrates to zero, what gives a non-trivial condition on admissible functions
$\dot{x}(t)$ even if the boundary contributions are totally neglected. In
other words, the usual variational principle in terms of $A_{\mu}$ imposes
extra restrictions on the allowed variations of $F_{\mu\nu}$.

The stronger condition on admissible variations means that the action must be
stationary with respect to a smaller class of variations. It makes the
corresponding equations admit more solutions. Note also that the vanishing
field strength is of course a particular case of the divergenceless one. This
is a common effect which plays, for example, an important role in
understanding mimetic gravity models [34].

However, this is not the case with the choice of variables presented in this
work; covariant and pure-tetrad formulations are equivalent at the level of
the action too. This is because the gauge-invariant variables (18) do not
depend on any derivatives of the fundamental quantities; we have presented a
purely algebraic relation which imposes no restriction on the class of
variations of gauge-invariant variables.

There are two different sides of the traditional understanding of the role of
spin connection [21]. One assumes a particular spin connection for a given
tetrad, which removes inertial effects. This assumption that one could
objectively separate gravity from inertia goes against all experimental
evidence. And it is related to the picture of (fictional) global translations
being gauged, and requires some, necessarily voluntary, choice of the
reference tetrad. By now, it is known that the standard recipes [21] of
determining the spin connection can give non-unique results [35], and beyond
the simplest cases they might even easily get wrong, with equations of motion
not being satisfied if not in TEGR [36].

Secondly, there are opinions in the literature concerned about making the
action and other global quantities finite, and it has been sometimes claimed
that the spin connection plays a regularizing role. However, our change of
variables is performed directly at the level of all the fundamental
geometrical quantities, and does not involve neglecting any boundary term. It
means that the action, or some conserved quantities, can be regularized in the
pure tetrad approach with no less success than in the covariant one. And in
fact, if there is a finite-action covariant solution, we can always make a
type I (simultaneous) Lorentz transformation which brings this particular
solution, preserving all its physical properties, to the zero spin connection
case.

To state it once more, there are different types of the local Lorentz
(pseudo-)invariance in the case of TEGR. One way is to write the action fully
in terms of the Lorentz-covariant derivatives. Then we have the type I
invariance, in practically any model we might think of, in these terms. On the
other hand, if we think of the tetrad as four vectors composing the
fundamental degrees of freedom, then there is no reason to want this full-
fledged local Lorentz invariance.

However, even the pure-tetrad action of TEGR appears to possess some
invariance which goes beyond the explicitly maintained diffeomorphism
invariance. This is the type II Lorentz pseudo-invariance. And this is the
invariance that gets lost in most generalizations. Even though it is different
from what most experts in general metric-affine gravity are used to call local
Lorentz invariance, it is algebraically related with the same group, the
Lorentz group. Note also that the influence of all these symmetries of the
action can be seen in the Hamiltonian analysis at the level of the primary
constraints [29].

Of course, what we get in models of $f(T)$ gravity is non-trivial dependence
on which tetrad to choose. But this is not an obstacle, since it grants a new
model with more degrees of freedom. These are represented by the tetrad, a
fundamental variable which carries more information than what the metric does.
The covariantization procedure is equivalent to formally rewriting the very
same model in locally Lorentz invariant terms. This is done by simply sharing
the extra modes with the newly introduced variables, the components of the
spin connection. However it does not get rid of the very fact that the model
does have some new dynamical content which a metric alone cannot provide.
Getting something more on top of the metric is the essence of the local
Lorentz symmetry (type II) violation, and the covariant version simply
rewrites it in different terms.

##  V Conclusions

We have proven that the covariant and pure-tetrad formulations of teleparallel
theories are fully equivalent, as long as matter is not coupled to the
Lorentzian extra degrees of freedom. Due to the nature of the (curvatureless
and metric) Weitzenböck connection (1), its torsion can be covariantized by
replacing the tetrad ${\mathbf{e}}^{a}$ with the “gauge invariant” tetrad
${\tilde{\mathbf{e}}}^{a}$ (see Eq. (19)), which is equivalent to splitting
the original tetrad into two factors (see Eq. (18)). This splitting introduces
six new variables – the Lorentz matrices $\Lambda^{a}_{b}$ – deprived of
independent dynamics; they are clearly spurious variables that will have an
impact on the constraint algebra. Since the covariantized torsion is the
Weitzenböck torsion of the tetrad ${\tilde{\mathbf{e}}}^{a}$, then
${\tilde{\mathbf{e}}}^{a}$ satisfies the same dynamics as the original
variables ${\mathbf{e}}^{a}$. Therefore, although the covariant formulation is
Lorentz invariant of type I, it is not dynamically different from the pure
tetrad formulation.

When the Lorentz invariance of type II is broken in the pure tetrad
formulation, it gives rise to new dynamics. This dynamics also appears in the
covariant formulation, because the Lorentz type II breaking is also present.
Certainly, a pure-tetrad Lorentz rotation is no longer an obvious symmetry
once non-zero spin connection terms are added. Therefore, what is broken by
generalizations of the covariant TEGR would rather be a “type
$\tilde{\mathrm{II}}$” invariance, which represents local Lorentz rotations of
our Lorentz-invariant variables (18).

The results here exposed are rooted in the fact that Weitzenböck connection
(1) has already the most general form for a curvatureless connection. From
this perspective, there is a one-to-one relation between tetrads and
curvatureless tetrad-compatible connections (up to global linear
transformations and possible topological obstructions). This is analogous to
the (metric based) GR formulation, where there exists a one-to-one relation
between metrics and torsionless metric-compatible (Levi-Civita) connections.
However, since the relation tetrad-metric is not one-to-one but is subject to
a local Lorentz invariance, the teleparallel formulations are expected to be
endowed with such invariance. This expectation seems to come from considering
the metric as more fundamental than the tetrad. However, such invariance is
not mandatory for building a dynamical theory of tetrads, since both the
gravity Lagrangian and the coupling matter-gravity can be written only in
terms of tetrads (without prejudice to those matter Lagrangians that in fact
exhibit a local Lorentz invariance).

Our findings are applicable to the entire class of teleparallel theories built
from the torsion tensor, which include TEGR, $f(\mathbb{T})$, NGR,
$f(T_{ax},T_{vec},T_{ten})$, $f(\mathbb{T},\mathbb{B})$, teleparallel
Horndeski gravity, among others.

Acknowledgments: The authors thank Laur Järv for helpful feedback on a
previous version of this manuscript. DB and MJG have been supported by the
European Regional Development Fund CoE program TK133 “The Dark Side of the
Universe”. MJG was funded by the Estonian Research Council grant MOBJD622. RF
was supported by Consejo Nacional de Investigaciones Científicas y Técnicas
(CONICET) and Universidad de Buenos Aires.

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11institutetext: INAF-Osservatorio Astronomico di Palermo, Piazza del
Parlamento 1, 90134 Palermo, Italy
11email<EMAIL_ADDRESS>22institutetext: Università degli Studi di
Palermo, Piazza Marina 61, 90133 Palermo, Italy 33institutetext: Harvard-
Smithsonian Center for Astrophysics, 02138 Garden St, Cambridge, MA

# X-ray flares of the young planet host DS Tuc A.

I. Pillitteri 11 C. Argiroffi 2211 A. Maggio 11 G. Micela 11 S. Benatti 11 F.
Reale 22 S. Colombo 11 S.J. Wolk 33

(Received ; accepted )

The discovery of planets around young stars has increased the study of the
early phases of planetary formation and evolution. Stars are strong emitters
at X-ray and UV wavelengths in the first billion of years and this strongly
affects the evaporation, thermodynamics and chemistry of the atmospheres of
young planets around them. In order to investigate these effects in young
exoplanets we observed the 40 Myr old star DS Tuc A with XMM-Newton and
recorded two X-ray bright flares, with the second event occurring about 12 ks
after the first one. Their duration from the rise to the end of the decay was
of about $8-10$ ks in soft X-rays (0.3-10 keV). The flares were also recorded
in the band 200–300 nm with the UVM2 filter of the Optical Monitor. The
duration of the flares in UV was about 3 ks. The observed delay between the
peak in the UV band and in X-rays is a probe of the heating phase followed by
the evaporation and increase of density and emission measure of the flaring
loop. The coronal plasma temperature at the two flare peaks reached 54–55 MK.
The diagnostics based on temperatures and time scales of the flares applied to
these two events allow us to infer a loop length of $5-7\times 10^{10}$ cm,
which is about the size of the stellar radius. We also infer values of
electron density at the flare peaks of $2.3-6.5\times 10^{11}$ cm-3, and a
minimum magnetic field strength of order of 300–500 G needed to confine the
plasma. The energy released during the flares was of order of $5-8\times
10^{34}$ erg in the band $0.3-10$ keV and $0.9-2.7\times 10^{33}$ erg in the
UV band (200-300 nm). We speculate that the flares were associated with
Coronal Mass Ejections (CMEs) that hit the planet about 3.3 hr after the
flares and dramatically increasing the rate of evaporation of the planet. From
the RGS spectra we retrieved the emission measure distribution and the
abundances of coronal metals during the quiescent and the flaring states. In
agreement with what inferred from time resolved spectroscopy and EPIC spectra,
also from the analysis of RGS spectra during the flares we infer a high
electron density.

###### Key Words.:

Stars: flare – Stars: coronae – Stars: activity – X-rays: stars

##  1 Introduction

X-rays are emitted by solar type stars during their early evolution (Vaiana et
al., 1981; Pallavicini et al., 1981; Feigelson & Montmerle, 1999) and are a
powerful proxy for the structure of the stellar coronae and the magnetic
fields that confine the hot coronal plasma. During the first hundred million
years, solar type stars exhibit very high magnetic and coronal activity, as
well as high stellar rotation rates (Pizzolato et al., 2003). The loss of
angular momentum during the Main Sequence is accompanied by a decrease of the
X-ray luminosity by a factor of $10^{3}$ during the first billion of years
(Pallavicini et al., 1981; Vaiana et al., 1981; Favata et al., 2003).

The interest for the study of the activity of stars at high energies has been
renewed with the discovery of extrasolar planets. High energy photons and
particles can affect planetary atmospheres and their habitability. Fluxes at
high energies (X-rays and EUV, XUV) have a strong role in shaping and
transforming the primary atmospheres of the planets and their evaporation.
Because of the decrease of stellar activity, most of the effects due to XUV
irradiation are expected to happen in the first Gyr (Penz & Micela, 2008;
Cecchi-Pestellini et al., 2009; Kubyshkina et al., 2018). Collectively, the
action of the magnetized stellar winds and the coronal mass ejections (CMEs)
can make the planets lose a significant fraction of their atmospheres and
change their compositions through photo-chemical processes (see Drake et al.,
2013 and Kay et al., 2019 and references therein). However, the effects of the
sudden and violent release of energy during flares and CMEs onto the planets
are still poorly understood due to the lack of proper monitoring of the
stellar coronae required in order to build a reliable distribution of flaring
rates vs. age, and an adequate time-resolved modeling of the dynamics and
chemistry of the planetary gaseous envelopes. In this context, the study of
young stars hosting planets, like DS Tuc, in X-ray and UV bands is of high
relevance.

DS Tuc is composed by a pair of two young stars at a distance of 44.1 pc from
the Sun, with an age of $\sim 40$ Myr and spectral types G6 (component A and
host of the planet) and K3 (component B). The planet around DS Tuc A was
independently discovered in NASA TESS observations (TOI-200) by Benatti et al.
(2019) and Newton et al. (2019). DS Tuc Ab has a radius of 0.5 RJ, a mass
upper limit of 14.4 M${\oplus}$ (Benatti et al., 2021) and orbits its host
star in 8.14 days. These characteristics suggest a very low density for the
planet and that it is prone to strong atmospheric evaporation. This is one of
few very young planets discovered so far. Its properties can serve to test
models of dynamical and atmospheric evolution. For the purpose of
investigating its coronal emission, DS Tuc has been observed with XMM-Newton
in two different observations. In Benatti et al. (2021) we have presented the
analysis and the global properties of the coronae of the system of DS Tuc and
related the X-ray emission to the model of atmospheric evaporation during the
Main Sequence.

In the optical band TESS has observed DS Tuc A in six sectors allowing Colombo
et al. (2022) to perform a reconstruction of the distribution of the optical
flares and their main properties. They also inferred the energies of the X-ray
counterparts of TESS flares through the scaling laws of Flaccomio et al.
(2018) before X-rays observations were available. Here, we study two bright
flares of DS Tuc A and discuss their impact on the atmosphere of DS Tuc Ab.

The paper is structured as follows: Sect. 2 describes the observations and the
data analysis, Sect. 3 presents our results and Sect. 4 contains our
discussion and conclusions.

##  2 XMM-Newton observations and data analysis

DS Tuc has been observed two times in X-rays with XMM-Newton during
observations ObsId=0863400901 (P.I. S. Wolk) and 0864340101 (P.I. A. Maggio)
with exposure times of about 40 ks and 30 ks, respectively. From the first
observation (Benatti et al., 2021) we measured X-ray fluxes and luminosities
of the pair of stars of DS Tuc A and B, separated by $\sim 5.3\arcsec$ and
partially resolved in MOS detectors. The A component was 1.3 times more
luminous in X-rays than DS Tuc B. We also inferred a mean coronal temperature
and emission measure of both stars. Some degree of variability in form of
flares was observed in DS Tuc B while DS Tuc A was quieter. From these we
estimated the time scale of the evaporation of DS Tuc Ab and the evolution of
its mass and radius in the next 5 Gyr.

The second observation of DS Tuc with XMM-Newton was performed on April 11th
2021 (obsId 0864340101) 111A detailed log of the observation is available at:
https://xmmweb.esac.esa.int/cgi-
bin/xmmobs/public/obs_view.tcl?search_obs_id=0864340101 , again with EPIC as
main instrument. We adopted the Medium filter and the SmallWindow mode for
both MOS and pn. In addition, we acquired simultaneous high resolution spectra
with RGS and a light curve in the band $200-300$ nm with the Optical Monitor
(OM) using the UVM2 filter in Fast mode. For the OM we obtained five
continuous exposures of 4400 s and one exposure of 3880 s, for each exposure a
light curve with a time binning of 10 s was obtained.

The ODFs files downloaded from the XMM-Newton
archive222https://nxsa.esac.esa.int/nxsa-web/#home were reduced with SAS ver
18.0.0, to obtain event tables for EPIC instruments, light curves for OM and
spectra for the two RGS high-resolution spectrometers. We selected the EPIC
events for having energies in the band $0.3-10.0$ keV and FLAG == 0 and
PATTERN ¡= 12 as prescribed by the SAS reduction guide. We checked that the
overall background was low during the observation, thus the full exposure was
retained for further analysis. A visual inspection of the pn and MOS images
revealed that DS Tuc A underwent strong flaring activity and its flux
dominated the emission of the nearby DS Tuc B. Fig. 1 shows two MOS1 images
relative to two different time intervals and referring to the quiescent and
flare peak intervals of DS Tuc A.

Spectra and light curves of DS Tuc A were obtained from events in a circular
region of radius 75″centered on the X-ray image centroid (see Fig. 1). Spectra
of the background were accumulated from an offset circular region of radius
45″. Given the brightness of DS Tuc A in this observation we checked for any
pile up that could have affected the spectra of MOS and pn. We generated the
pattern of single, double, triple, and quadruple events as a function of the
energy as prescribed by the SAS guide. The diagnostic modeling of the curve
did not show evidence of significant pile up so we retained the entire list of
events in the region.

The high count statistics recorded in the EPIC instruments allowed us to
perform time resolved spectroscopy of the two flares. To this purpose we
accumulated the spectra in 13 time intervals that define the initial quiescent
state, the flare rises, peaks and decays (see Fig. 2). For each time interval
the spectra of the source and the background were created with SAS, along with
the corresponding response matrices and effective area ARF files.

Figure 1:  Top panel: MOS 2 image of DS Tuc with the regions used for
accumulating source and background events. The scale is logarithmic to enhance
the background level otherwise not visible on a linear scale. The background
region was offset to one of the corner of the chip to avoid the diffraction
spikes from the central source. Bottom left: an image of MOS 1 during the
first 6 ks of observation and corresponding to interval 1 in Fig. 2. We marked
the SIMBAD positions (J2000) of DS Tuc A and B and the vectors of proper
motions. Bottom right: image during interval 3 corresponding to the peak of
the first flare. In both images the scale limits and color maps are identical,
the binning is 8 pixels corresponding to $\sim 0.4\arcsec$ and a smoothing
with a Gaussian with $\sigma=1.5$ pixels has been applied to images. We added
contour levels with the same ranges for marking the source centroid.

For the OM we obtained five exposures of 4400 s and one of 3880 s. In each
exposure, a light curve with time binning of 10 s was accumulated with the
standard SAS task omfchain, and the result is displayed in Fig. 2.

For extracting the RGS spectra we used the SAS task rgsproc and rgscombine
with custom good time intervals defined as interval nr. 1 for quiescent and 2
to 13 for flares. We summed the RGS spectrum of the first XMM-Newton
observation (ObsId 0863400901) to the spectrum of the quiescent interval nr.
1, since both are representative of a quiescent phase or at least with low
level of variability. Furthermore, we added RGS 1 and RGS 2 spectra of the
quiescent and flaring phases for achieving a better count statistics.

Figure 2:  Light curves of MOS 1 (red curve, bin size 150 s) and OM in UVM2
filter (blue curve, bin size 10 s). OM rate is scaled down by a factor 30 to
allow a direct comparison of the rise and decay times of both instruments
during the flares. The time intervals used for time resolved spectroscopy (see
Sect. 3.2) are indicated by the alternating light/dark gray areas and numbered
on top of each panel.

Figure 3:  Spectra of MOS 1 relative to the time intervals 1 (black line and
crosses) and 3 (red crosses). The best fit model to the spectrum of the plasma
during interval 1 with its 3 component is shown along with the $\chi$
residuals in the bottom panel.  Table 1:  Best fit parameters of the 3T model
for the quiescent spectrum observed during interval 1. We list the parameters
obtained from the best fit to each EPIC spectrum and the joint fit of pn and
MOS. Unabsorbed flux and luminosity are calculated in the band 0.3-10.0 keV.

Parameter | MOS1 | MOS2 | pn | Joint
---|---|---|---|---
NH (1020 | 1.5 (0–4.4) | 1.55 (0–4.25) | 2.08 (0.44–3.68) | 2.3 (1.0–3.4)
log T1 (K) | 6.54 (6.48–6.61) | 6.67 (6.6–6.76) | 6.57 (6.52–6.62) | 6.59 (6.56–6.63)
log EM1 (cm-3) | 52.89 (52.69–53.16) | 52.95 (52.8–53.08) | 52.99 (52.83–53.12) | 53.02 (52.91–53.10)
log T2 (K) | 7.03 (6.99–7.05) | 7.06 (7.04–7.07) | 7.01 (7–7.03) | 7.03 (7.02–7.04)
log EM2 (cm-3) | 52.95 (52.73–53.17) | 53.12 (53.08–53.19) | 53.13 (52.98–53.24) | 53.14 (53.02 53.20)
log T3 (K) | 7.42 ($>$7.25) |  $>$7.64 | 7.58 ($>$7.34) | 7.87 ($>7.53$)
log EM3 (cm-3) | 52.51 (51.97–52.73) | 51.97 (51.67–52.15) | 52.32 (51.97–52.6) | 52.08 ($>51.91$)
Z/Z$\odot$ | 0.22 (0.13–0.34) | 0.14 (0.11–0.17) | 0.13 (0.0984–0.1804) | 0.13 (0.11–0.16)
$\chi^{2}$ | 52.05 | 55.74 | 60.37 | 190.82
D.o.F. | 47 | 48 | 48 | 159
P($\chi^{2}>\chi^{2}_{0}$) | 0.28 | 0.21 | 0.11 | 0.04
log fX (erg s-1 cm-2) | -11.11 | -11.13 | -11.1 | -11.09
log LX (erg s-1) | 30.26 | 30.23 | 30.27 | 30.27

Table 2:  Best fit parameters from time resolved spectroscopy of EPIC spectra.
For each time interval (column 1) are reported the column equivalent
absorption (NH), the temperature (kT), the emission measure (E.M.) the global
abundance scaling factor (Z/Z$\odot$), the unabsorbed flux and luminosity in
the 0.3-10 keV band, the $\chi^{2}$, the degrees of freedom and the
probability related to the $\chi^{2}$ statistics. The parameter ranges refer
to the 90% confidence intervals.

Interval | NH | log (T/K) | log E.M. | Z/Z$\odot$ | log fX | log LX | $\chi^{2}$ | d.o.f. | Prob
---|---|---|---|---|---|---|---|---|---
| 1020 cm-2 |  | cm-3 |  | erg s-1 cm-2 | erg s-1 |  |  |
| MOS1
2 | 1.31 (0.17–2.55) | 7.75 (7.67–7.83) | 53.6 (53.57–53.63) | 0.29 (0.25–0.34) | -10.45 | 30.92 | 52.05 | 47 | 0.28
3 | 0.71 (0.06–1.39) | 7.62 (7.59–7.65) | 53.96 (53.94–53.98) | 0.56 (0.5–0.62) | -10.13 | 31.24 | 55.74 | 48 | 0.21
4 | 0 (0–0.58) | 7.51 (7.47–7.54) | 53.84 (53.81–53.87) | 0.47 (0.41–0.53) | -10.29 | 31.08 | 60.37 | 48 | 0.109
5 | 0.45 (0–1.13) | 7.51 (7.48–7.55) | 53.58 (53.56–53.61) | 0.49 (0.45–0.54) | -10.45 | 30.91 | 52.05 | 47 | 0.28
6 | 0.61 (0–1.32) | 7.41 (7.37–7.46) | 53.34 (53.31–53.37) | 0.35 (0.31–0.38) | -10.68 | 30.69 | 55.74 | 48 | 0.21
7 | 0 (0–0.44) | 7.36 (7.32–7.42) | 53.17 (53.13–53.22) | 0.3 (0.27–0.33) | -10.79 | 30.57 | 60.37 | 48 | 0.109
8 | 2.22 (1.02–3.52) | 7.73 (7.67–7.81) | 53.65 (53.62–53.68) | 0.45 (0.39–0.52) | -10.37 | 31 | 52.05 | 47 | 0.28
9 | 1.67 (0.68–2.72) | 7.66 (7.61–7.71) | 53.93 (53.9–53.96) | 0.49 (0.41–0.58) | -10.16 | 31.21 | 55.74 | 48 | 0.21
10 | 0.27 (0–1.49) | 7.61 (7.56–7.67) | 53.77 (53.74–53.8) | 0.51 (0.43–0.59) | -10.29 | 31.07 | 60.37 | 48 | 0.109
11 | 0.83 (0.05–1.66) | 7.53 (7.48–7.57) | 53.6 (53.57–53.63) | 0.44 (0.4–0.49) | -10.46 | 30.91 | 52.05 | 47 | 0.28
12 | 0.15 (0–0.92) | 7.37 (7.32–7.43) | 53.15 (53.1–53.19) | 0.31 (0.28–0.34) | -10.79 | 30.57 | 55.74 | 48 | 0.21
13 | 1.28 (0.11–2.52) | 7.2 (7.09–7.41) | 52.22 (51.82–52.46) | 0.25 (0.22–0.29) | -11.04 | 30.32 | 60.37 | 48 | 0.109
| MOS2
2 | 0.65 (0–1.96) | 7.65 (7.58–7.73) | 53.64 (53.59–53.66) | 0.21 (0.17–0.26) | -10.48 | 30.88 | 52.05 | 47 | 0.28
3 | 2.04 (1.33–2.78) | 7.61 (7.58–7.64) | 53.96 (53.94–53.98) | 0.54 (0.48–0.6) | -10.14 | 31.23 | 55.74 | 48 | 0.21
4 | 0.49 (0–1.41) | 7.56 (7.51–7.6) | 53.81 (53.78–53.84) | 0.51 (0.44–0.58) | -10.28 | 31.09 | 60.37 | 48 | 0.109
5 | 1.06 (0.32–1.84) | 7.5 (7.46–7.54) | 53.57 (53.54–53.59) | 0.46 (0.42–0.5) | -10.48 | 30.89 | 52.05 | 47 | 0.28
6 | 1.28 (0.54–2.06) | 7.38 (7.34–7.42) | 53.35 (53.31–53.38) | 0.33 (0.3–0.37) | -10.69 | 30.68 | 55.74 | 48 | 0.21
7 | 0 (0–0.78) | 7.38 (7.34–7.44) | 53.16 (53.11–53.2) | 0.29 (9.27–0.32) | -10.8 | 30.57 | 60.37 | 48 | 0.109
8 | 0.34 (0–1.57) | 7.66 (7.59–7.73) | 53.69 (53.65–53.72) | 0.36 (0.3–0.42) | -10.39 | 30.98 | 52.05 | 47 | 0.28
9 | 1.23 (0.14–2.4) | 7.73 (7.67–7.79) | 53.88 (53.85–53.91) | 0.51 (0.43–0.59) | -10.18 | 31.19 | 55.74 | 48 | 0.21
10 | 1.19 (0–2.48) | 7.57 (7.52–7.63) | 53.8 (53.77–53.84) | 0.4 (0.32–0.49) | -10.31 | 31.06 | 60.37 | 48 | 0.109
11 | 0.5 (0–1.33) | 7.53 (7.49–7.57) | 53.59 (53.56–53.62) | 0.4 (0.35–0.44) | -10.47 | 30.9 | 52.05 | 47 | 0.28
12 | 1.14 (0.27–2.08) | 7.35 (7.29–7.41) | 53.13 (53.07–53.18) | 0.3 (0.27–0.34) | -10.81 | 30.55 | 55.74 | 48 | 0.21
13 | 2.58 (1.34–4.02) | 7.09 (7.02–7.24) | 52.45 (52.08–52.73) | 0.25 (0.19–0.3) | -11.03 | 30.34 | 60.37 | 48 | 0.109
| pn
2 | 0.24 (0–0.81) | 7.7 (7.66–7.75) | 53.72 (53.7–53.74) | 0.18 (0.16–0.2) | -10.37 | 30.99 | 52.05 | 47 | 0.28
3 | 0.16 (0–0.52) | 7.64 (7.62–7.67) | 54.05 (54.04–54.07) | 0.38 (0.36–0.41) | -10.06 | 31.3 | 55.74 | 48 | 0.21
4 | 0.62 (0.15–1.08) | 7.49 (7.47–7.52) | 53.92 (53.9–53.94) | 0.35 (0.32–0.37) | -10.23 | 31.14 | 60.37 | 48 | 0.109
5 | 0.56 (0.22–0.93) | 7.45 (7.43–7.48) | 53.66 (53.64–53.68) | 0.3 (0.29–0.32) | -10.45 | 30.92 | 52.05 | 47 | 0.28
6 | 1.65 (1.28–2.05) | 7.38 (7.35–7.41) | 53.42 (53.4–53.44) | 0.22 (0.21–0.23) | -10.65 | 30.71 | 55.74 | 48 | 0.21
7 | 1.54 (1.12–1.98) | 7.34 (7.3–7.38) | 53.28 (53.25–53.31) | 0.19 (9.18–0.2) | -10.75 | 30.61 | 60.37 | 48 | 0.109
8 | 0.98 (0.43–1.56) | 7.71 (7.66–7.75) | 53.79 (53.77–53.81) | 0.25 (0.23–0.27) | -10.3 | 31.07 | 52.05 | 47 | 0.28
9 | 0.08 (0–0.6) | 7.67 (7.64–7.7) | 54.01 (53.99–54.02) | 0.31 (0.28–0.35) | -10.11 | 31.25 | 55.74 | 48 | 0.21
10 | 0.54 (0–1.14) | 7.56 (7.52–7.6) | 53.86 (53.84–53.89) | 0.31 (0.28–0.34) | -10.26 | 31.1 | 60.37 | 48 | 0.109
11 | 0.25 (0–0.66) | 7.52 (7.49–7.55) | 53.67 (53.65–53.69) | 0.26 (0.24–0.28) | -10.44 | 30.93 | 52.05 | 47 | 0.28
12 | 1.13 (0.72–1.56) | 7.31 (7.28–7.36) | 53.24 (53.21–53.28) | 0.19 (0.17–0.2) | -10.78 | 30.59 | 55.74 | 48 | 0.21
13 | 1.28 (0.72–1.86) | 7.14 (7.05–7.23) | 52.58 (52.43–52.76) | 0.13 (0.11–0.15) | -11.02 | 30.34 | 60.37 | 48 | 0.109

Table 3:  Slope in the plane log T vs log EM/2 ($\xi$), loop length, electron
density and minimum magnetic field derived for the first and second flare for
each EPIC instrument. Errors in $\xi$ are quoted at 1 $\sigma$ level and
propagated to the other quantities. Uncertainties in temperatures and decay
time $\tau_{d}$ amount to another 3%–4%.

Parameter — Instrument | MOS1 | MOS2 | pn
---|---|---|---
$\xi_{1}$ | 0.58 ( 0.47 - 0.68 ) | 0.61 ( 0.52 - 0.69 ) | 0.69 ( 0.56 - 0.82 )
$\xi_{2}$ | 0.54 ( 0.48 - 0.6 ) | 0.81 ( 0.72 - 0.89 ) | 0.73 ( 0.67 - 0.79 )
L1 (1010 cm) | 5.83 ( 3.8 - 7.25 ) | 5.55 ( 4.27 - 6.52 ) | 6.95 ( 5.28 - 8.11 )
L2 (1010 cm) | 3.33 ( 2.56 - 3.96 ) | 4.91 ( 4.43 - 8.15 ) | 4.82 ( 4.4 - 5.18 )
ne,1 (1011 cm) | 2.6 ( 1.9 - 5 ) | 3 ( 2.4 - 4.4 ) | 2.3 ( 1.9 - 3.5 )
ne,2 (1011 cm) | 6.5 ( 5 - 9.7 ) | 3.8 ( 1.8 - 4.4 ) | 4.4 ( 3.9 - 5 )
B1 (G) | 320 ( 270 - 440 ) | 310 ( 270 - 370 ) | 290 ( 260 - 350 )
B2 (G) | 500 ( 440 - 600 ) | 350 ( 240 - 370 ) | 390 ( 370 - 420 )

##  3 Results

We discuss separately the quiescent emission and the flaring emission observed
in the present observation.

###  3.1 Quiescent phase

The quiescent interval before the first flare was about 6.5 ks with an average
rate of 0.9 ct/s in MOS1, this interval served to estimate the coronal plasma
properties before the ignition. We used a sum of 3 APEC thermal components
plus a global absorption to model the EPIC spectra and obtain a statistically
satisfactory description of the spectrum. Table 1 lists the best fit
parameters and Fig. 3 shows the spectrum during time interval 1 and 3 for
comparison. We performed the best fit procedure to each EPIC spectrum and to
them jointly obtaining very similar results.

The main thermal component of the quiescent spectrum is
$\log\mathrm{(T/K)}\sim 7.02$, with the addition of a soft component
$\log\mathrm{T/K}\sim 0.54$ and a hotter component, much less constrained,
$\log\mathrm{T/K}\geq 7.4$. By setting a fixed very low value to the
equivalent column of gas absorption (N${}_{\mathrm{H}}=8\times 10^{18}$ cm-2)
we can constrain the hot component at around $\log T\sim 7.4$ (90% confidence
range: 7.31–7.58) while the other parameters remain unchanged. However the
very low gas absorption is not consistent with the values of visual extinction
and reddening.

The temperatures are consistent with the values determined from the analysis
of spectra acquired during the first XMM-Newton observation (Benatti et al.,
2021). The average flux of DS Tuc A in interval 1 is about 1.9 times the flux
observed during the first XMM-Newton observation. However, some degree of
contamination from DS Tuc B in this estimate has to be accounted for in the
analysis of this first time interval. Visual inspection of the image relative
to the events of the quiescent interval shows that DS Tuc A is still the most
prominent source and DS Tuc B has a lower count statistics. The contribution
to flux of DS Tuc A from its stellar companion DS Tuc B is of order of 40%
during interval 1. After correcting for this contamination fraction, we infer
that the quiescent flux of DS Tuc A in interval 1 was about 15% higher than
the average X-ray flux measured during the first XMM-Newton observation.

###  3.2 Flares

We observed two prominent flares after the initial phase of quiescence that
lasted for about 6.5 ks (Fig. 2) from the start of the XMM-Newton observation.
The second flare occurred about 12 ks after the beginning of the first flare.
In Fig. 2 we show a light curve of the OM with scaled count rate to allow a
straightforward comparison of the timing of the flares. Each flare starts
simultaneously in X-rays and UV, however in the UV band the flares peak
earlier and decays quicker than in X-ray. When the flux reaches each flare
peak in UV the rise of the flux in X-rays does flatten while continuing to
increase. The peak in UV corresponds to the end of the heating phase during
the flares while the loop continues to be filled with plasma evaporating from
its feet and thus increasing the emission measure and the flux in X-rays
(Namekata et al., 2017). This timing and shape of the flares in the near UV
band and in X-rays is analog to the Neupert effect seen in the Sun (Neupert,
1968). Neupert noticed that time-integrated microwave fluxes closely match the
rising portions of soft X-ray (SXR) emission. The effect has been demonstrated
since by others (e.g., Kahler et al., 1988), and was later dubbed as the
Neupert effect by Hudson (1991). Estimating the time-integrated microwave
fluxes from the non-thermal hard X-ray (HXR) emission, the Neupert effect has
been generalized, where the time derivative of the SXR, i.e.,
$dF_{SHXR}(t)/dt$, can be considered as a proxy for the HXR emission
$F_{HXR}(t)$ (Dennis & Zarro, 1993, Veronig et al., 2002). In our case, near
UV flux measured with OM is a proxy of the $dF_{HXR}/dt$, with HXR being the
emission in 0.3-10 keV probed by EPIC.

In the UV we measured rise times of $180\pm 5$ s and $70\pm 5$ s for the first
and second flare respectively. In X-rays we measured rise times of $900\pm 75$
s and $750\pm 75$ s, respectively. The delay between the peak of each flare in
UV and X-rays amount to $\sim 680$ s and $\sim 930$ s, respectively. In a
linear-log plane the decay phase of the flares appear first linear meaning
that the first part of the decay can be modeled with an exponential
relationship between count rate and time, then it does flatten deviating from
a pure exponential decay. With a linear best fit modeling to the first part of
the decay of both flares in the linear-log plane we estimated e-folding times
of 3.55 ks and 2.31 ks for flares 1 and 2, respectively. With similar
procedure we determined that in UV the flare decays are shorter than in X-rays
and of order of 390 s and 375 s for flares 1 and 2, respectively.

The evolution of the plasma temperature and its emission measure can provide
diagnostics about the size of the magnetic loop that hosted the flare, the
presence of prolonged heating and the minimum magnetic field required to
confine the plasma (Reale, 2007, 2014). In order to follow the evolution of
the flares we divided the light curve during the flares in 12 intervals that
identify the rise, the peak and the decay (resolved with 3 different segments
each) of each flare as indicated in Fig. 2. For each interval we modeled the
extracted spectra with a sum of a flaring component described by one APEC
optically thin plasma thermal emission model added to the 3-T model that
describes the quiescent emission (cf. Sect. 3.1). We left free to vary also
the global absorption and the global abundances. For each time segment the
best fit parameters of spectra of MOS 1, 2 and pn are listed in Table 2. We
did not perform a joint best fit to the 3 EPIC spectra of each time interval
because the diagnostics of the flares are calibrated for each instrument
separately. The absorption is of order of $10^{20}$ cm-2, in a few cases it is
not well constrained but we found the 90% confidence level value always below
$4\times 10^{20}$ cm-2. These values are in agreement with the estimates of
optical extinction towards DS Tuc (cf. Benatti et al., 2021).

Figure 4:  Log T (MK) vs. E.M. during the evolution of the two flares. Left
panel: log T and E.M. derived from MOS1 spectra; central panel: values derived
from the MOS2 spectra; right panel: parameters derived from pn spectra. Solid
symbols refer to the first flare, open symbols refer to the second flare. The
numbers refer to the time intervals. The error bars mark the 90% confidence
interval.

Fig. 4 shows the scatter plots of the flare temperatures vs. the emission
measures (E.M.) for each EPIC instrument. Points 2 to 7 refer to the first
flare, points 8 to 13 refer to the second flare (cf. Table 2). In the spectra
of MOS 1 and pn the peak temperatures in both flares are detected in segments
2 and 8 for flare 1 and 2, respectively, and of order of $\log\mathrm{T/K}\sim
7.7-7.75$. In MOS 2 for the second flare the peak temperature is detected in
segment 9, however its value is consistent with that measured in segment 8
with the other two instruments. The peaks of E.M. are reached in segments 3
and 9 for MOS 1 and pn with values of $\log EM(\mathrm{cm}^{-3})\approx 54$.
Owing to our choice of the time intervals, the peak of E.M. in MOS 2 is seen
in segment 8 instead of 9, but still consistent with the values determined
from MOS 1 and pn. We inferred a metallicity value of $Z$ between 0.2 during
the decay and 0.5 at the flare peaks.

The slopes between points 3 and 7, and 9 to 13 in the log T / EM plane, the
temperatures at the peaks of the flares and the e-folding time of the decay of
the flares can be used to infer the size of the loops that hosted the flares
using the diagnostics of Reale (2007) and Reale (2014). Table 3 shows the
slopes derived for each flare and EPIC instrument, the total loop lengths, the
estimates of the electron densities and the minimum strength of the magnetic
field required to constrain the plasma in the loop at the peak of both flares.
We inferred a full loop length of order of the stellar radius ($5-7\times
10^{10}$ cm). To estimate the electron density we first calculated the volume
of the flaring loops assumed as tubes with a ratio between base radius and
length of 0.1. Then we derived the electron densities from E.M., as
$n_{e}=\sqrt{EM/V}$ (Maggio et al., 2000). Electron densities at the flare
peaks are found of order of $2-6.5\times 10^{11}$ cm-3. We can also infer the
minimum strength of the magnetic field needed to confine the plasma as (e.g,
Maggio et al., 2000)

$B=\sqrt{16\pi k_{B}n_{e,\mathrm{peak}}T_{\mathrm{peak}}}$

where $k_{B}$ is the Boltzmann’s constant, $n_{e,\mathrm{peak}}$ and
$T_{\mathrm{peak}}$ are the density and the temperature at the flare peak. We
estimated a minimum magnetic field between 300 G and 500 G.

###  3.3 RGS lines

Figure 5: RGS spectra corresponding to quiescent and flaring phases. The
identifications of some of the strongest lines are reported on top. For
clarity reasons the two spectra are slightly smoothed.

We analyzed the first-order RGS spectra of both XMM-Newton observations to
infer the emission measure distribution (EMD), abundances, and electron
density of the X-ray emitting plasma during the quiescent and flaring phases.
First order RGS spectra cover the range $\approx 4-38$ Å with a resolution
$0.06$ Å FWHM. The exposures of the quiescent and flaring intervals are 45.53
ks and 22.15 ks long, respectively. The quiescent spectrum was accumulated
summing up the events collected during the whole first XMM-Newton observation
and the first 6.2 ks of the present observation. For the flaring phase we
considered the remaining 22.2 ks encompassing the two flares altogether. The
RGS spectra of the first order are made of 11200 and 20700 net counts,
respectively. For the analysis of the RGS data we used the IDL package
PINTofALE, (Kashyap & Drake, 2000) and the XSPEC v12.11b software (Arnaud,
1996). For the line emissivities we adopted the CHIANTI atomic database
(v7.13, Dere et al., 1997).

We identified the strongest emission lines and measured their fluxes by
fitting the RGS spectra in small wavelength intervals of $\sim 1.0$ Å width.
The width of each wavelength interval is set in order to include, and hence
fit simultaneously, blended lines. Each observed line was fitted assuming that
its shape is fully described by the RGS line spread function (the intrinsic
width of coronal emission lines is expected to be negligible compared to the
RGS line spread function). For each wavelength window encompassing the lines
to fit we took into account the continuum contribution in the best fit
procedure, leaving temperature and normalization as free parameters. Given the
small width of each wavelength window, the continuum component from the best
fit acts as an additive constant value. This additive constant contribution,
mimicking the apparent continuum level, ensures that the contributions of
unresolved small lines and wings of other strong lines in the vicinity are
taken into account. The measured line fluxes are listed in Table 4. In the
case of spectral features suspected to be due to the blend of more than one
line, we listed the different ions contributing to the observed flux.

To determine absolute abundances and EMD values, and to better constrain the
hottest components of the EMD, continuum flux measurements are also needed.
The continuum flux level obtained within the line fitting procedures is often
an overestimate of the real continuum level, because of the significant
contribution of small unresolved lines, that is known to affect large portions
of the RGS wavelength range. Therefore, to obtain reliable measurements of the
continuum, we selected small wavelength intervals where line contamination is
known to be small (on average less than 10%). The continuum flux measurements
in these intervals were obtained by integrating the observed net counts. The
total emissivity function associated to these continuum measurements was
anyhow computed taking into account also the small fraction of emission line
contribution in addition to the continuum one. The selected intervals and the
corresponding measured fluxes are listed in Table 4.

Table 4: Measured fluxes of emission lines and continuum intervals in the
X-ray spectrum of DS Tuc.

|  |  | quiescent | flaring |
---|---|---|---|---|---
$\lambda^{a}$ | Ion | $\log T_{\rm max}^{b}$ | $F_{obs}^{c}$ | $(F_{obs}-F_{pred}^{d})/\sigma$ | $F_{obs}^{c}$ | $(F_{obs}-F_{pred}^{d})/\sigma$ | $EMD^{e}$
Lines
8.42 | MgXII MgXII | 7.00 |  5.7$\;\pm\;$ | 3.5 | -1.3 |  32.4$\;\pm\;$ | 10.1 | -0.3 | $\ast$
9.17 | MgXI | 6.80 |  20.0$\;\pm\;$ | 3.9 | 2.6 |  32.0$\;\pm\;$ | 9.1 | 1.7 | $\ast$
9.23 | MgXI | 6.80 |  3.2$\;\pm\;$ | 3.2 | 0.6 |  27.1$\;\pm\;$ | 8.8 | 2.8 |
9.31 | MgXI | 6.80 |  14.2$\;\pm\;$ | 3.4 | 2.9 |  7.7$\;\pm\;$ | 7.8 | 0.1 |
10.62 | FeXIX FeXIX FeXVII FeXXIV FeXXIV | 7.30 |  1.6$\;\pm\;$ | 2.6 | -1.4 |  51.9$\;\pm\;$ | 12.8 | -0.0 | $\ast$
10.98 | FeXXIII NeIX NaX FeXXIII FeXVII FeXXIV | 7.20 |  22.8$\;\pm\;$ | 4.5 | 3.0 |  62.1$\;\pm\;$ | 13.8 | 0.4 | $\ast$
11.43 | FeXXII FeXXIV FeXVIII | 7.20 |  15.3$\;\pm\;$ | 4.8 | 2.2 |  41.7$\;\pm\;$ | 13.1 | 1.1 | $\ast$
11.74 | FeXXIII | 7.20 |  0.8$\;\pm\;$ | 2.7 | -2.0 |  27.1$\;\pm\;$ | 12.6 | -0.7 | $\ast$
11.77 | FeXXII | 7.10 |  13.3$\;\pm\;$ | 4.5 | 1.1 |  20.8$\;\pm\;$ | 12.5 | -0.1 | $\ast$
12.13 | FeXVII NeX NeX | 6.75 |  106.4$\;\pm\;$ | 8.4 | 1.5 |  201.8$\;\pm\;$ | 18.6 | -0.2 | $\ast$
12.28 | FeXXI FeXVII | 7.05 |  20.9$\;\pm\;$ | 6.0 | -1.3 |  54.0$\;\pm\;$ | 14.8 | -0.0 | $\ast$
12.83 | FeXX FeXX FeXX | 7.05 |  27.8$\;\pm\;$ | 5.5 | 1.2 |  41.6$\;\pm\;$ | 12.8 | 0.2 | $\ast$
13.45 | FeXIX NeIX FeXIX FeXXI | 6.60 |  57.1$\;\pm\;$ | 7.2 | 0.2 |  127.2$\;\pm\;$ | 15.6 | 0.5 | $\ast$
13.52 | NeIX FeXIX | 6.95 |  30.3$\;\pm\;$ | 6.5 | 1.5 |  34.9$\;\pm\;$ | 13.4 | -0.4 |
13.70 | NeIX | 6.55 |  30.5$\;\pm\;$ | 6.2 | 1.7 |  48.0$\;\pm\;$ | 12.6 | -0.0 |
13.74 | FeXIX | 6.95 |  13.1$\;\pm\;$ | 4.7 | 2.3 |  5.9$\;\pm\;$ | 7.6 | 0.2 |
13.82 | FeXIX FeXVII | 6.90 |  12.3$\;\pm\;$ | 4.0 | -0.3 |  35.8$\;\pm\;$ | 9.2 | 1.5 |
14.20 | FeXVIII FeXVIII FeXVIII | 6.90 |  42.4$\;\pm\;$ | 4.8 | -0.4 |  65.0$\;\pm\;$ | 9.7 | -0.3 | $\ast$
14.37 | FeXVIII FeXVIII FeXVIII | 6.90 |  15.0$\;\pm\;$ | 3.8 | 0.0 |  28.0$\;\pm\;$ | 8.7 | 0.6 | $\ast$
14.54 | FeXVIII FeXVIII FeXVIII | 6.90 |  5.6$\;\pm\;$ | 3.5 | -2.1 |  23.8$\;\pm\;$ | 8.6 | 0.5 | $\ast$
15.01 | FeXVII | 6.75 |  97.9$\;\pm\;$ | 5.9 | 0.9 |  152.9$\;\pm\;$ | 11.8 | 0.2 | $\ast$
15.18 | FeXIX OVIII OVIII | 6.50 |  19.7$\;\pm\;$ | 4.9 | 1.4 |  42.5$\;\pm\;$ | 10.6 | 1.2 | $\ast$
15.26 | FeXVII | 6.75 |  26.2$\;\pm\;$ | 4.7 | -0.1 |  8.2$\;\pm\;$ | 8.7 | -4.0 | $\ast$
16.01 | FeXVIII OVIII OVIII | 6.50 |  34.1$\;\pm\;$ | 4.5 | -1.6 |  100.0$\;\pm\;$ | 10.4 | 0.9 | $\ast$
16.07 | FeXVIII | 6.85 |  35.7$\;\pm\;$ | 4.5 | 3.8 |  40.0$\;\pm\;$ | 9.0 | 1.4 | $\ast$
16.78 | FeXVII | 6.75 |  49.5$\;\pm\;$ | 4.6 | -1.4 |  109.1$\;\pm\;$ | 10.3 | 1.4 | $\ast$
17.05 | FeXVII | 6.75 |  76.3$\;\pm\;$ | 6.6 | 0.7 |  215.8$\;\pm\;$ | 15.7 | 6.0 |
17.10 | FeXVII | 6.70 |  59.9$\;\pm\;$ | 7.7 | 0.3 |  34.5$\;\pm\;$ | 14.0 | -4.6 |
17.62 | FeXVIII | 6.90 |  7.9$\;\pm\;$ | 3.0 | -1.5 |  8.7$\;\pm\;$ | 6.5 | -1.6 | $\ast$
18.63 | OVII | 6.35 |  9.8$\;\pm\;$ | 3.1 | 1.4 |  17.2$\;\pm\;$ | 7.4 | 1.0 | $\ast$
18.97 | OVIII OVIII | 6.50 |  213.9$\;\pm\;$ | 8.8 | 0.2 |  516.6$\;\pm\;$ | 18.8 | -0.1 | $\ast$
21.60 | OVII | 6.30 |  46.4$\;\pm\;$ | 7.1 | 0.5 |  74.1$\;\pm\;$ | 14.0 | 0.2 | $\ast$
21.81 | OVII | 6.30 |  0.7$\;\pm\;$ | 3.4 | -1.8 |  14.2$\;\pm\;$ | 11.5 | 0.4 |
22.10 | OVII | 6.30 |  37.8$\;\pm\;$ | 6.8 | 2.5 |  28.8$\;\pm\;$ | 11.5 | -0.1 |
24.78 | NVII NVII | 6.30 |  25.4$\;\pm\;$ | 5.3 | -0.8 |  72.1$\;\pm\;$ | 12.0 | 0.5 | $\ast$
33.73 | CVI CVI | 6.15 |  33.1$\;\pm\;$ | 8.9 | -0.8 |  98.4$\;\pm\;$ | 19.2 | 0.5 | $\ast$
Continuum
$[8.49$$\div$ | $8.90]$ |  | 7.75 |  25.5$\;\pm\;$ | 4.2 | 0.7 |  192.1$\;\pm\;$ | 11.6 | -0.0 | $\ast$
$[29.70$$\div$ | $30.36]$ |  | 6.55 |  27.9$\;\pm\;$ | 11.4 | -0.3 |  165.3$\;\pm\;$ | 21.3 | 0.1 | $\ast$

a Wavelengths (Å). b Temperature (K) of maximum emissivity. c Observed fluxes
(${\rm 10^{-6}\,ph\,s^{-1}\,cm^{-2}}$) with uncertainties at the 68%
confidence level. d Predicted fluxes considered here those computed in the low
density limit. e Flux measurements selected for the $EMD$ reconstruction.

We derived the EMD and the abundances corresponding to the quiescent and
flaring phases by applying the Markov chain Monte Carlo method of Kashyap &
Drake (1998). Flux measurements used for the EMD reconstruction are listed in
Table 4. We discarded density sensitive lines and lines whose fluxes are
highly uncertain because of strong blending. Measured fluxes were first
converted into unabsorbed fluxes assuming a hydrogen column density
$N_{\mathrm{H}}=2\times 10^{20}\mathrm{cm^{-2}}$, this is an average value
inferred from the analysis of EPIC spectra (cf. Sect.3.1), and from that
reported by Benatti et al. (2021). Given the range of formation temperatures
of the measured fluxes, we reconstructed the EMD over a regular logarithmic
temperature grid, with bins of width of 0.10 dex, and encompassing the range
of $\log\mathrm{T}(\mathrm{K})$ from 6.0 to 8.0 dex. We show in Fig. 6 the
emissivity functions associated with the line and continuum flux measurements
used for the EMD derivation. The hot tails that characterize the shapes of the
emissivity functions of several lines, as well as that of the continuum
intervals, make their associated flux measurements highly sensitive to the
amount of hot plasma components. Moreover, since we use both line and
continuum flux measurements to reconstruct the EMD, the inversion procedure
provides simultaneously the absolute abundances for those elements for which
at least one line has been measured. For the measured spectral features that
include lines of different elements, the total emissivity, that is needed for
the EMD reconstruction, depends on the starting abundance values of the
elements involved. For this reason, the EMD and abundance reconstruction have
been performed recursively. The first attempts of EMD reconstruction indicated
that the abundances of the quiescent and flaring phases do not differ
significantly from each other. Because of that, in the final EMD derivation of
the quiescent phase we assumed the same set of abundances obtained from the
analysis of the flaring phase, where, due to the higher line fluxes, they are
determined with better precision. The EMD and the abundances derived for the
X-ray emitting plasma of DS Tuc A are reported in Table 5. The two EMDs are
also plotted in Fig. 7.

Figure 6: Emissivity functions of the ion lines (colored curves) and continuum
(black curves) measured in RGS spectra.

Figure 7: EMDs derived from the RGS line fluxes for the quiescent (blue) and
flaring phases (red). Asterisks and solid points mark the EM derived from EPIC
detectors for the quiescent (blue symbol, 3T APEC) and flaring intervals, (red
symbols and interval numbers, 1T APEC). Table 5: EMD and abundances of DS Tuc.

| quiescent | flaring
---|---|---
log T (K) |  log EMD (cm-3) |  log EMD (cm-3)
6.00 |  48.36${}^{+1.50}_{-0.59}$ |  49.08${}^{+0.89}_{-1.01}$
6.10 |  49.43${}^{+1.05}_{-0.45}$ |  49.36${}^{+0.96}_{-0.59}$
6.20 |  50.11${}^{+0.77}_{-0.46}$ |  50.48${}^{+0.45}_{-0.61}$
6.30 |  50.20${}^{+0.59}_{-0.37}$ |  50.42${}^{+0.58}_{-0.45}$
6.40 |  51.66${}^{+0.09}_{-0.99}$ |  51.00${}^{+0.38}_{-0.41}$
6.50 |  51.05${}^{+0.47}_{-0.18}$ |  51.42${}^{+0.57}_{-0.37}$
6.60 |  51.51${}^{+0.45}_{-0.14}$ |  52.60${}^{+0.06}_{-0.38}$
6.70 |  52.17${}^{+0.10}_{-0.32}$ |  52.00${}^{+0.48}_{-0.40}$
6.80 |  52.28${}^{+0.11}_{-0.34}$ |  52.28${}^{+0.15}_{-0.42}$
6.90 |  51.68${}^{+0.50}_{-0.18}$ |  52.02${}^{+0.37}_{-0.17}$
7.00 |  52.30${}^{+0.11}_{-0.30}$ |  52.61${}^{+0.09}_{-0.43}$
7.10 |  52.27${}^{+0.15}_{-0.26}$ |  52.39${}^{+0.33}_{-0.23}$
7.20 |  51.85${}^{+0.22}_{-0.50}$ |  52.54${}^{+0.26}_{-0.52}$
7.30 |  50.95${}^{+0.57}_{-0.50}$ |  52.53${}^{+0.27}_{-0.25}$
7.40 |  50.60${}^{+0.71}_{-0.61}$ |  52.67${}^{+0.31}_{-0.66}$
7.50 |  49.35${}^{+1.28}_{-0.45}$ |  53.25${}^{+0.05}_{-0.67}$
7.60 |  49.73${}^{+0.83}_{-0.81}$ |  52.47${}^{+0.40}_{-0.24}$
7.70 |  48.42${}^{+1.80}_{-0.43}$ |  52.28${}^{+0.34}_{-0.21}$
7.80 |  47.43${}^{+2.57}_{-0.52}$ |  51.98${}^{+0.69}_{-0.29}$
7.90 |  47.33${}^{+2.80}_{-0.54}$ |  52.42${}^{+0.45}_{-0.98}$
8.00 |  47.47${}^{+2.47}_{-0.79}$ |  51.19${}^{+0.89}_{-0.68}$
Elem | $A_{X}/A_{X\odot}$
C |  0.73${}^{+0.28}_{-0.16}$
N |  0.85${}^{+0.35}_{-0.15}$
O |  0.54${}^{+0.16}_{-0.08}$
Ne |  0.91${}^{+0.30}_{-0.16}$
Mg |  0.41${}^{+0.19}_{-0.10}$
Fe |  0.26${}^{+0.07}_{-0.05}$

Abundances are in solar units (Anders & Grevesse 1989, GeCoA, 53, 97).

It is possible to constrain the X-ray emitting plasma density $n_{\mathrm{e}}$
by inspecting the diagnostics provided by He-like line triplets and other
density-sensitive line ratios. Among the emission lines detected in the
flaring and quiescent RGS spectra, density-sensitive line ratios are provided
by the He-like triplets of Mg XI (resonance $r$ at 9.17 Å, intercombination
$i$ at 9.23 Å, forbidden $f$ at 9.31 Å), Ne IX ($\lambda_{r}=13.45$ Å,
$\lambda_{i}=13.55$ Å, $\lambda_{f}=13.70$ Å), and O VII ($\lambda_{r}=21.60$
Å, $\lambda_{i}=21.81$ Å, and $\lambda_{f}=22.10$ Å), and by the pair of Fe
XVII lines at 17.05Å and 17.10 Å (Mauche et al., 2001).

However, we did not analyzed the He-like Ne IX triplet at $\sim 13.5$ Å,
because of its severe blending with strong Fe lines makes its measured fluxes
highly uncertain. The O VII triplet, that has a maximum formation temperature
of $\sim 2$ MK, allows us to measure the density of relatively cold coronal
plasma. The Mg XI and Fe XVII density diagnostics instead probe hotter plasma
components, since their line maximum formation temperatures are at $\sim 6$
MK.

The $f/i$ line ratio of the O VII triplet is high and compatible with the low
density limits for both the quiescent and flaring phases. Density sensitive
lines of Mg XI and Fe XVII, of the quiescent and flaring phases, are displayed
in Fig. 8. Both the quiescent and flaring spectra in the Mg XI region are
quite noisy due to the low S/N ratio of the lines, and to the significant
continuum emission in this region, especially during flares. The Fe XVII lines
are well exposed, with a good $S/N$ ratio in both the quiescent and flare
spectra. However, the Fe XVII lines are separated by 0.05 Å which is
comparable with the RGS spectral resolution (0.06 Å) making the two lines only
marginally resolved. Because of all these sources of uncertainty, the Mg XI
and Fe XVII line fluxes were obtained by freezing line positions, and leaving
the line fluxes as free parameters for the best fit of the line profiles.

The observed $f/i$ ratio of Mg XI and the $F(17.10)/F(17.05)$ ratio of Fe XVII
change significantly during the quiescent and flaring phases. During quiescent
emission both Mg XI and Fe XVII have both ratios compatible with the low
density limit: $f/i>2$ and $F(17.10)/F(17.05)=0.87^{+0.13}_{-0.12}$, both
indicating $n_{\mathrm{e}}<10^{13}$ cm-3. Conversely, during the flaring
phases, both ratios decrease significantly ($f/i=0.28^{+0.36}_{-0.28}$ and
$F(17.10)/F(17.05)=0.16\pm 0.07$) indicating a high plasma density
($n_{\mathrm{e}}\sim 10^{14}$ cm-3), as shown in Fig. 9. This value is derived
from relatively cold lines and they probe the cooler plasma in the loop which
is toward its feet (Argiroffi et al., 2019). Conversely, the plasma density
inferred from the emission measure and the loop volume described in Sect. 3.2
is valid for hotter plasma towards the apex of the loop during the flare peak.
Also, in deriving the electron density from the flare emission measure, the
choice of the volume has some degree of arbitrarity.

The abundances of elements inferred from the line fluxes, scaled to the values
of the solar photosphere as a function of First Ionization Potential (FIP),
are reported in Fig. 10. Apart from the high uncertainty of the abundance of
C, a trend with FIP is visible with Fe, Mg and perhaps O being under abundant
owing to their low FIP. The trend is in agreement with the inverse low FIP
effect observed in active stars.

Figure 8: Observed (histogram with error bars) and best fit (solid black
curve) spectra, of the quiescent and flaring phases, in the regions of the Mg
XI triplet and Fe XVII density sensitive lines. Best fit functions
corresponding to individual line are also shown with solid grey lines. Left
panels; $r$, $f$, and $i$ labels indicate the resonance, inter-combination,
and forbidden lines of the Mg XI triplet.

Figure 9: Predicted line ratios ($f/i$ for O VII and Mg XI and
$F(17.10)/F(17.05)$ for Fe XVII) computed at different plasma densities,
together with the observed line ratios observed during the flaring phase.

##  4 Discussion and conclusions

In this paper we have presented the analysis of an XMM-Newton observation of
DS Tuc A which is one of the youngest known planet hosts. DS Tuc Ab is an
inflated planet with a radius of about 0.5 RJ and with mass less than 14.4 M⊕
that will likely shrink to a size of about 2 $R_{\oplus}$ in the next hundreds
of Myr (Benatti et al., 2021). Observations in X-rays are important to
understand the processes that can modify both dynamics and chemistry of the
primary atmospheres of newly born planets. In this context, the present work
allows a deep characterization of the coronal emission and time variability of
the star and its effects on its close-in planet.

We detected two bright flares in DS Tuc A that permitted a time-resolved
spectroscopy of the two events. The energy released during the first and
second flare are about $8\times 10^{34}$ erg and $5\times 10^{34}$ erg.

Colombo et al. (2022) used TESS light curves of DS Tuc to study the rate of
flares of DS Tuc in the optical band with an analysis of all available sectors
through an iterative method based on Gaussian processes. They found that the
frequency of flares with energy (in optical band) larger than $2\times
10^{32}$ erg amounts to about 2 per day. By using the relationship between
flare energy released in the optical band and in X-rays inferred by Flaccomio
et al. (2018) from the flares observed in the NGC 2264 young star forming
region, this limit corresponds to X-ray energies of order of 1031 erg. The
flares we detected in the present XMM-Newton observation seem very rare events
that likely released energies in the optical band in excess of $5\times
10^{35}$ erg which are on the very high energy tail of the flare energies
observed with TESS by Colombo et al (2022). Furthermore, in the optical band
there is a significant fraction of pairs of flares separated by a few ks,
similar to the separation of the two flares we observed in X-rays. The cause
could be a triggering mechanism of the same active region after the first
flare has been ignited in a scenario where the corona is densely packed with
magnetic structures somehow interacting with each other. The UV energy of the
flares was calculated from the count rates and the conversion factor for the
UVM2 filter (assuming a band width of 100 Å) in the case of a WD at 10,000 K
333See https://www.cosmos.esa.int/web/xmm-newton/sas-watchout-uvflux.. These
energies amount to 2.7 and $0.9\times 10^{33}$ erg for flares 1 and 2,
respectively, and result a factor 30 to 50 lower than the energies released in
X-rays.

In pre Main Sequence stars of Orion, X-ray flares that release energies $\geq
5.5\times 10^{35}$ erg constitute the upper 50% of the sample of the bright
flares observed in COUP (cf. Fig. 9 in Wolk et al., 2005). It is plausible
that the frequency of such energetic events decreases between the age of young
pre-Main Sequence stars in Orion ($3-5$ Myr) and the age of DS Tuc, about 40
Myr.

With an age of 40 Myr the X-ray activity of DS Tuc A has just started its
decline in the Main Sequence phase and probed by a value of the ratio $\log
L_{\mathrm{X}}/L_{\mathrm{bol}}\sim-3.55$ (Benatti et al., 2021). The average
X-ray luminosity of DS Tuc A in this observation was $9.1\times 10^{30}$ erg
s-1which is about a factor 10 higher than recorded in the first XMM-Newton
observation (Benatti et al., 2021). During the flares the ratio $\log
L_{\mathrm{X}}/L_{\mathrm{bol}}$ reached the value of $\approx-2.15$ or
$\approx 0.7\%$ of its bolometric luminosity.

From the ephemeris of Benatti et al. (2019) we estimated that the planet was
at the orbital phases $\phi$ between $\sim-0.06$ and $\sim-0.11$ during this
observation (being $\phi=0$ the phase of the transit and $\sim-0.008$ the
phase of the first contact) and thus the planet was observed between 19 and 12
hours before the transit. The stellar rotation period is about 2.9 days which
amounts to $\approx 250.5$ ks, thus the present XMM-Newton exposure covered
about 12% of the rotation period. The flares did not show any dimming due to
self-eclipse, we infer that either they occurred in active regions near the
equator and in the central portion of the stellar disk or the active regions
were at very high stellar latitudes and coronal loops with sizes similar to
the stellar radius anchored at high stellar latitudes could be visible even
when the feet of the loops are behind the visible disk.

The values of peak luminosity are about $1.8\times 10^{31}$ erg/s, these
values correspond to a flux of about $100$ erg s-1 cm-2at the surface of the
planet, which is of order of 0.1 W m-2. Conversely, the X-ray flux received by
the planet from the quiescent star is about 0.01 W m-2. An hypothetical Earth
around DS Tuc A at 1 AU would have received $6.3\times 10^{-4}$ W m-2 in the
band $0.3-10$ keV, which corresponds to a GOES X6 flare in the band $1-8$ Å.
We speculate that Coronal Mass Ejections (CMEs) were associated with those
flares. Drake et al. (2013) (see also Osten & Wolk, 2015; Moschou et al.,
2019) give an estimate of the mass and kinetic energy associated with powerful
flares observed in the Sun. From their equation 1, which applies to young
stellar coronae in the saturated regime like in the case of DS Tuc A, we
estimated that the mass associated with CMEs was around $5\times 10^{-15}$
$M_{\odot}$. We also infer that the kinetic energy associated with the CMEs is
$\approx 10^{34}$ erg From these numbers we also infer a velocity of the CME
material of order of 1000 km/s. With these velocities the CME would have
reached DS Tuc Ab in about $3.3$ hr after the ignition of the flares.
Simulations of CMEs hitting close-in planets at a few stellar radii from their
stars (see Alvarado-Gómez et al., 2022; Hazra et al., 2022) demonstrate that
the CMEs associated with the flares could be even more disruptive for the
outer atmosphere resulting in dramatic changes of the rate of evaporation with
respect to the effect of a steady stellar wind or from X-ray irradiation alone
due to a flare.

The RGS spectra enable us to reconstruct the distribution of Emission Measure
(EMD) for the quiescent and flaring states. The EMDs are very similar in the
range $6<\log T<7.1$ while they differ significantly above $\log T>7.1$ with
the maximum difference occurring at $\log T\approx 7.5$. The difference is due
to the dense plasma heated during the flares. The integral of the difference
between the flare and quiescent phase EMDs in the range of temperature $7<\log
T<8$ amounts to $\log EM\approx 53.5$ which is consistent, on average, with
the values inferred from the time resolved spectroscopy of EPIC spectra.
Qualitatively, hard X-rays produced during the flares can penetrate more
deeply in the planet’s atmosphere along with a cascade of electrons that
induce further dissociation and photo-chemistry. Such a mechanism can be non
linear if the atmosphere is unable to restore its state before the next flare.
Louca et al. (2022) modeled the response of several planetary atmospheres to
flares determining the conditions under which the initial conditions are
restored and the consequent enhancement of several species. In this respect, a
more systematic X-ray monitoring of stars with close in planets would be
beneficial for the comprehension of the formation and evolution of planetary
atmospheres.

Figure 10: Ratio $A_{X}/A_{X\odot}$ vs. first ionization potential derived
from the abundances analysis of RGS spectra

###### Acknowledgements.

We thank the anonymous referee for their useful comments and suggestions that
improved the quality of this manuscript. IP, AM, SC and GM acknowledge
financial support from the ASI-INAF agreement n.2018-16-HH.0, and from the
ARIEL ASI-INAF agreement n.2021-5-HH.0. SJW was supported by the Chandra X-ray
Observatory Center, which is operated by the Smithsonian Astrophysical
Observatory for and on behalf of the National Aeronautics Space Administration
under contract NAS8-03060. Based on observations obtained with XMM-Newton, an
ESA science mission with instruments and contributions directly funded by ESA
Member States and NASA.

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