A first asteroseismological analysis on WD J1916$+$3938, the ZZ Ceti star discovered in the Kepler mission field
A. D. Romero$^{1,2}$
L. G. Althaus$^{1,2}$
and M. M. Miller Bertolami$^{1,2}$
$^{1}$Facultad de Ciencias Astron\'omicas y Geof\'isicas
Universidad Nacional de La Plata
Argentina
$^{2}$Consejo Nacional de Investigaciones Cient\'ificas y T\'ecnicas (CONICET)
DAV stars, also called ZZ Ceti variables, are pulsating white dwarfs with atmospheres rich in H. Asteroseismology of DAV stars can provide valuable clues about the origin, structure and evolution of DA white dwarfs. Recently, a new DAV star, WD J191643.83$+$393849.7, has been discovered in the field of the Kepler spacecraft. It is expected that further monitoring of this star in the next years will enable astronomers to obtain the best lightcurve of a pulsating DA white dwarf ever recorded, and thus to know with unprecedented precision the hidden details of the internal structure of this star. In this paper, we perform a first asteroseismological analysis of WD J191643.83$+$393849.7 on the basis of fully evolutionary DA white-dwarf models. Specifically, we employ a complete set of evolutionary DA white-dwarf structures covering a wide range of effective temperatures, stellar masses, and H envelope thicknesses. These models have been obtained on the basis of a complete treatment of the evolutionary history of progenitors stars. We compute $g$-mode adiabatic pulsation periods for this set of models and compare them with the pulsation periods exhibited by WD J191643.83$+$393849.7. Based on a tentative estimation of the mean period spacing of the star, we find that the stellar mass should be substantially large ($\gtrsim 0.80 M_{\odot}$), in agreement with the spectroscopically derived stellar mass. Also, from period-to-period fits we find an asteroseismological model characterised by a low effective temperature, rather high stellar mass and a thin H envelope. The possibility that this rather massive pulsating white dwarf can be further monitored with Kepler with a high degree of detail turns the star WD J191643.83$+$393849.7 into a promising and unique object to study the physics of crystallization and carbon/oxygen phase diagrams at high densities.
paper.tex
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\title[Asteroseismology of WD J1916$+$3938]
{A first asteroseismological analysis on WD J1916$+$3938, the ZZ Ceti star 
discovered in the \emph{Kepler} mission field}

\author[A.  H.  C\'orsico, A. D. Romero, L.  G.  Althaus,  \& 
M. M. Miller Bertolami]
       {A.             H.             C\'orsico$^{1,2}$\thanks{E-mail:
           acorsico@fcaglp.unlp.edu.ar  (AHC)}, A.  D. Romero$^{1,2}$,
         L.     G.    Althaus$^{1,2}$,     and     M.    M.     Miller
         Bertolami$^{1,2}$\\ $^{1}$Facultad de Ciencias Astron\'omicas
         y   Geof\'isicas,   Universidad   Nacional   de   La   Plata,
         Argentina\\   $^{2}$Consejo  Nacional   de  Investigaciones
         Cient\'ificas y T\'ecnicas (CONICET), Argentina\\ }
\begin{document}

\date{}

\maketitle

\label{firstpage}

\begin{abstract}
DAV stars, also  called ZZ Ceti variables, are  pulsating white dwarfs
with atmospheres rich in H.  Asteroseismology of DAV stars can provide
valuable clues about  the origin, structure and evolution  of DA white
dwarfs. Recently,  a new DAV star, WD  J191643.83$+$393849.7, has been
discovered  in  the field  of  the  \emph{Kepler}  spacecraft.  It  is
expected that further  monitoring of this star in  the next years will
enable astronomers  to obtain  the best lightcurve  of a  pulsating DA
white dwarf ever recorded,  and thus to know with unprecedented
precision the hidden  details of the internal structure  of this star.

In this paper,  we perform a first asteroseismological  analysis of WD
J191643.83$+$393849.7   on  the   basis  of   fully   evolutionary  DA
white-dwarf  models.   Specifically,  we  employ  a  complete  set  of
evolutionary  DA  white-dwarf  structures  covering a  wide  range  of
effective  temperatures, stellar masses,  and H  envelope thicknesses.
These models have  been obtained on the basis  of a complete treatment
of the evolutionary history of progenitors stars.  We compute $g$-mode
adiabatic pulsation  periods for this  set of models and  compare them
with  the  pulsation periods  exhibited  by WD  J191643.83$+$393849.7.
Based  on a tentative  estimation of  the mean  period spacing  of the
star, we  find that  the stellar mass  should be  substantially large
($\gtrsim  0.80  M_{\odot}$), in  agreement with  the  spectroscopically
derived stellar  mass.  Also, from  period-to-period fits we  find an
asteroseismological   model   characterised   by   a   low   
effective temperature, rather high stellar mass and a thin H 
envelope. The possibility that this rather massive
pulsating white dwarf can be further monitored with \emph{Kepler} with
a high degree of detail turns the star WD  J191643.83$+$393849.7 into 
a promising and unique object to study the physics of crystallization 
and carbon/oxygen phase diagrams at high densities.
\end{abstract}

\begin{keywords}
asteroseismology -- stars: evolution -- stars: interiors -- stars: oscillations
-- stars: individual: ZZ Ceti stars -- stars: white dwarfs
\end{keywords}

\section{Introduction}

Pulsating DA (H-rich atmospheres) white  dwarfs, also known as ZZ Ceti
or DAV variable stars, comprise the most numerous and studied class of
degenerate pulsators.   Their instability strip, probably  a pure one,
is centered at an effective  temperature of around $12\,000$ K (Winget
\& Kepler  2008; Fontaine  \& Brassard 2008;  Althaus et  al.  2010a).
The  luminosity  variations  (of  up  to $0.30$  mag)  are  caused  by
nonradial  $g$-mode  pulsations of  low  degree  ($\ell  \leq 2$)  and
periods between $\approx  70$ and $\approx 1500$ s,  excited through a
combination of  the $\kappa-\gamma$  mechanism acting in  the hydrogen
partial ionization zone (Dolez \&  Vauclair 1981; Winget et al.  1982)
and the ``convective driving'' mechanism --- once a sufficiently thick
outer convective zone has been developed (Brickhill 1991; Goldreich \&
Wu  1999).  ZZ Ceti  asteroseismology ---  the comparison  between the
observed   periods  and   the  periods   computed  for   a   suite  of
representative stellar models --- has the potential of disentangle the
internal structure  of DA white dwarfs, allowing  to place constraints
on the  stellar mass, the thickness  of the outer  envelopes, the core
chemical  composition, weak  magnetic fields  and slow  rotation rates
from the observed period patterns  (Winget \& Kepler 2008; Fontaine \&
Brassard 2008; Althaus et al. 2010a).

A total of  148 ZZ Ceti stars are currently  known (Castanheira et al.
2010).  To this  list we must add the recently  discovered DAV star WD
J191643.83$+$393849.7   (\emph{Kepler}  ID   4552982   hereinafter  WD
J1916$+$3938), a  ZZ Ceti located  in the \emph{Kepler}  mission field
and identified through ground-based times series photometry (Hermes et
al.  2011; hereinafter HEA11).  This  star ($T_{\rm eff}= 11\, 129 \pm
115$ K  ; $\log g=  8.34 \pm 0.06$) exhibits  low-amplitude luminosity
variations  with periods between  $\approx 800$  and $\approx  1450$ s
(see column 2 of Table \ref{table1}).  It is expected that an extended
monitoring  of this  object with  \emph{Kepler} could  bring  the best
lightcurve of a pulsating white dwarf ever recorded (HEA11), even with
a better quality  than the lightcurves of the  brighter stars observed
through  uninterrupted  ground-based  campaigns  of  the  Whole  Earth
Telescope (WET; Nather et al. 1990).

In  order  for  white  dwarf  asteroseismology  to  provide  realistic
constraints on the  internal structure of white dwarfs,  it is crucial
the  use of stellar  models characterised  by consistent  and detailed
chemical  profiles  to   accurately  assess  the  adiabatic  pulsation
periods. This requirement has been tackled successfully in the case of
the hot DOVs or GW Vir stars (see C\'orsico et al. 2007a, 2007b, 2008,
2009). Very  recently, Romero et  al. (2011) have performed  the first
asteroseismological analysis of a set of 44 ZZ Ceti stars based on the
new generation  of fully evolutionary DA white  dwarf models presented
in  Althaus  et  al.   (2010b).   These models  are  characterised  by
realistic  chemical profiles  throughout the  star and  cover  a wide
range of stellar  masses, thicknesses of the H  envelope and effective
temperatures. The analysis of a large  number of ZZ Ceti stars has the
potential  to characterise  the  global properties  of  the class,  in
particular the thicknesses  of the H envelope and  the stellar masses.
The study  of Romero et al.   (2011) revealed that DA  white dwarfs in
the  solar  neighbourhood  could  harbor  a  broad  range  of  H-layer
thickness.   On  the  other  hand,  this analysis  was  successful  in
finding, for the first time, an univocal asteroseismological model for
the archetypal  ZZ Ceti star G117$-$B15A,  one of the  targets of that
analysis.

Motivated by  the exciting discovery  of the first pulsating  DA white
dwarf in the field of the \emph{Kepler} mission, and encouraged by the
availability  of the most  complete set  of detailed  evolutionary and
pulsation models for DAV stars up to date (Romero et al 2011), we   
present in this paper a   first  asteroseismological  analysis   of  WD
J1916$+$3938.   We   are  well  aware   that  the  asteroseismological
potential of this pulsating star will probably be fully exploited with
further  observations  by  \emph{Kepler},  which will  allow  to  have
available a period spectrum substantially  richer than that we have at
hand at present just  from ground-based photometry.  However, we think
that a  first asteroseismological analysis  of this star based  on the
seven  periods ---  a representative  number  of periods  for ZZ  Ceti
standards  ---  currently  available  is worth  being  attempted.   By
considering the  mean period spacing  exhibited by the star,  we found
that  WD J1916$+$3938  should be  substantially massive,  in excellent
agreement  with  the  spectroscopic  inference.  We  also  found  that
period-to-period    fits   favour    an    asteroseismological   model
characterised by a high stellar mass, a low effective temperature, and
a rather thin H envelope.

The paper  is organized  as follows.  In  the next Section  we briefly
describe the  computer codes and  stellar models employed.   The short
Section \ref{smd}  is devoted to  the derivation of  the spectroscopic
mass of  WD J1916$+$3938.  We  make inferences about the  stellar mass
from the  period spacing data in  Section \ref{period-spacing}, taking
into account  its dependence not  only with the  effective temperature
and mass, but  instead also with the thickness of  the H envelope.  In
Section \ref{period-fit}  we extract  information of the  star through
asteroseismological period fits.  Section \ref{rotation} is devoted to
infer  a rotation  period  of  the star.   Finally,  we enumerate  our
results in the Conclusions (Section \ref{conclu}).

\begin{table}  
\centering  
\caption{The observed frequencies, periods, and amplitudes of 
WD J1916$+$3938 (columns 1 to 3), according to HEA11, and the periods of a  
phenomenological model (see Section \ref{period-spacing}) with a fixed 
($\ell= 1$) period spacing of $38.54$ s
(columns 4 to 6).}  
\begin{tabular}{cccccc}  
\hline  
\hline  
$\nu$ & $\Pi_i^{\rm obs}$  &  $A_i$   & $\Delta k$ & $\Pi_{\rm mod}$ &  $\Pi_i^{\rm obs}-\Pi_{\rm mod}$ \\  
$[\mu$Hz$]$ & $[$ s $]$ & $[\%]$    &             &   $[$ s $]$    &    $[$ s $]$            \\
\hline  
1213.7 &      823.9 &  0.38   & $2$  & $819.53$ & $ 4.37$ \\  
1198.9 &      834.1 &  0.32   & $5$  & $935.60$ & $-1.01$ \\
1070.1 &      934.5 &  0.36   & ---   &   ---   &  ---   \\
1032.1 &      968.9 &  0.44  & $6$  & $974.14$ & $-5.24$ \\  
 918.3 &     1089.0 &  0.25  & $9$  & $1089.75$& $-0.75$ \\ 
 854.8 &     1169.9 &  0.23  & $11$ & $1166.82$& $ 3.08$ \\ 
 696.0 &     1436.7 &  0.24 & $18$ & $1436.58$& $ 0.12$ \\
\hline  
\hline  
\end{tabular}  
\label{table1}  
\end{table}  

\section{Numerical tools and models}
\label{numerical}

\begin{figure}
\begin{center}
\includegraphics[clip,width=240pt]{figure1.eps}
\caption{The location of  WD J1916$+$3938 (black dot) and  a sample of
  44  bright  DAV  stars (red  dots)  on  the  $T_{\rm eff}-  \log  g$
  plane. Also included are our DA white dwarf evolutionary tracks with
  thick H  envelopes (solid curves).   For the sequences  with stellar
  masses of $0.877, 0.837, 0.770$  and $0.705 M_{\odot}$, we have also
  included the evolutionary tracks corresponding to models with the 
  thinnest H envelopes we considered (see  Romero et al. 2011). The  
  spectroscopic 
  stellar mass
  of WD  J1916$+$3938 inferred from  thick H envelope tracks  is $M_*=
  (0.805\pm  0.04) M_{\odot}$, while  from thin  H envelope  tracks is
  $M_*=   (0.797   \pm  0.04)   M_{\odot}$.   The   location  of   our
  asteroseismological  model  for this  star  is  marked  with a  blue
  circle; see Sect. \ref{best-fit}.}
\label{gteff}
\end{center}
\end{figure}

The  present   asteroseismological  study   is  based  on   the  fully
evolutionary  DA  white  dwarf  models  described in  Althaus  et  al.
(2010b) and  Romero et  al.  (2011), generated  with the  {\tt LPCODE}
evolutionary code.  To the best of our knowledge, these models are the
first  complete  set  of  DA  white  dwarfs  models  characterised  by
consistent  chemical   profiles  for   both  the  core   and  envelope
\emph{simultaneously}.  This feature renders these models particularly
suitable for asteroseismological studies  of DA white dwarfs.  Details
about  the  input  physics  and  numerical methods  included  in  {\tt
  LPCODE},   and   relevant  issues   related   to  the   evolutionary
computations are  given in Althaus et  al.  (2010b) and  Romero et al.
(2011),  and we  refer  the  interested reader  to  those papers  (and
references therein).   Here, we briefly  note that the DA  white dwarf
models employed in this study  are the result of complete evolutionary
calculations  of  progenitor  stars  for solar-like  metallicity  ($Z=
0.01$). It is worth mentioning that these models are in agreement with
the semi-empirical white dwarf initial-final mass relationships 
(Salaris et al. 2009; Althaus et al. 2010b).
The  evolution of eleven evolutionary  sequences with initial
stellar mass in  the range $1-5 M_{\odot}$ has  been computed from the
ZAMS through the thermally-pulsing and mass-loss phases on the AGB and
finally  to the  domain of  planetary nebulae  and white  dwarfs.  The
values of the stellar mass and  the H envelope thickness of our set of
white dwarf models are shown in Table 1 of Romero et al.  (2011).  
Convection is modeled by using the standard mixing length theory 
with $\alpha = 1.61$. For
our asteroseismological  analysis, we have considered a  large bank of
models with  stellar mass and  mass of the  H envelope varying  in the
ranges  $0.525  \lesssim  M_*  \lesssim 0.877  M_{\odot}$,  and  $-9.4
\lesssim \log(M_{\rm  H}/M_*) \lesssim -3.6$,  respectively, being the
range of the values of $M_{\rm  H}$ dependent on $M_*$ (see Table 1 of
Romero et  al.  2011).  For simplicity,  the mass of He  has been kept
fixed at the value predicted by the evolutionary computations for each
sequence.   A detailed justification  of this  assumption is  given in
Romero et al.   (2011).  The thickest H envelopes  (or ``canonical'' H
envelopes) considered are characterised  by the maximum allowed values
of   $M_{\rm  H}$   as   predicted  by   standard  stellar   evolution
calculations, which  are dependent  on the stellar  mass of  the white
dwarf.  Note that  the largest  value  of $M_{\rm  H}/M_*$ is  usually
arbitrarily   fixed   around    $\sim   10^{-4}$   in   previous   DAV
asteroseismological studies, regardless the value of the stellar mass.
H envelopes  thinner than  the canonical ones  have been  generated in
order to extend  the exploration of the parameter  space of the models
for  asteroseismology.  The properties  of our  DA white  dwarf models
have  been  exhaustively  described  in  Romero et  al.   (2011).   In
particular, we emphasise again  that these models are characterised by
consistent and detailed chemical profiles  {\sl from the centre to the
  surface},  which  are  needed  to  correctly  assess  the  adiabatic
pulsation  periods   of  DAVs,  a   crucial  aspect  of   white  dwarf
asteroseismology.  The pulsation computations have been performed with
the  adiabatic  nonradial pulsation  code  described  in C\'orsico  \&
Althaus (2006)  that uses the so-called  ``Ledoux modified'' treatment
(Tassoul et  al.  1990) to  assess the run of  the Brunt-V\"ais\"al\"a
frequency.

\section{Spectroscopic mass determination}
\label{smd}

The  spectroscopic  mass of  WD  J1916$+$3938  can  be derived  simply
interpolating from the tracks in  the $\log g- T_{\rm eff}$ diagram of
Fig.   \ref{gteff}, given  the value  of  $\log g$  and $T_{\rm  eff}$
inferred from the spectroscopic analysis of HEA11.  If we consider for
the moment only the tracks corresponding to DA white dwarf models with
thick (canonical)  H envelopes, we  found a rather high  stellar mass,
$M_*=  (0.805 \pm  0.040) M_{\odot}$.   If, instead,  we  consider the
evolutionary tracks of models with  thin H envelopes, the stellar mass
of WD J1916$+$3938 turns be slightly lower, of $M_*= (0.797 \pm 0.040)
M_{\odot}$. These values are somewhat smaller ($\sim 2.4 \%$) than the
value derived  by HEA11 using the  older tracks of  Wood (1990), $M_*=
(0.82 \pm 0.04) M_{\odot}$.

\section{Inferences from the period spacing}
\label{period-spacing}

WD J1916$+$3938 exhibits seven periods  in the range $824-1437$ s with
amplitudes  up  to $0.44  \%$.   The  large  aliasing present  in  the
ground-based data  set of HEA11 prevents from  a secure identification
of   true  periodicities.    Here,  in   absence  of   any  additional
observational  constraint,  we  shall  assume  that  all  the  periods
detected correspond to genuine  eigenmodes of the star.  An inspection
of  the  list  of  periods   shown  in  the  second  column  of  Table
\ref{table1}  suggests that, due  to their  proximity, the  periods at
$823.9$  and $834.1$  s could  correspond to  different values  of the
harmonic  degree\footnote{Other possibilities  are that  these periods
  correspond to components of a $\ell= 1$ triplet induced by rotation,
  or that  both periods are associated  to modes with  the same $\ell$
  and consecutive radial order  $k$, being their very short separation
  due to  strong mode  trapping.}.  If we  look for a  nearly constant
period spacing among the periods, we found that a period separation of
about $38.5$ s could exists if we discard for the moment the period at
834.1 s.  Thus,  somewhat arbitrarily, we can assume  that this period
is  associated  to  a $\ell=  2$  mode  and  that the  remainder  ones
constitute a series of $\ell= 1$ modes.  If we take this assumption as
a valid  one, we can  compute an observed  mean period spacing  for WD
J1916$+$3938 by means of  a nonlinear least-squares fit by considering
these  six remainder  periods. We  obtain  $\overline{\Delta \Pi}_{\rm
  obs}= 38.54 \pm  0.29$ s. In column 1 of  Table \ref{table1} we list
the observed  periods and  compare six of  them (those supposed  to be
$\ell=  1$) with the  periods associated  to a  phenomenological model
characterised by a  fixed period spacing of $38.54$  s (column 5). The
periods  of  this  model  are  obtained by  means  of  the  expression
$\Pi_{\rm mod}= 38.54 \times \Delta k + 742.92$ $[$s$]$, where $\Delta
k$ (the relative radial order of the modes) is shown in column 4.  The
residuals between  $\Pi_i^{\rm obs}$ and $\Pi_{\rm mod}$  are shown in
the sixth column, being the average of these residuals 2.44 s.

For  $g$-modes  with  high   radial  order  $k$  (long  periods),  the
separation  of consecutive  periods ($|\Delta  k|= 1$)  becomes nearly
constant  at a  value  given  by the  asymptotic  theory of  nonradial
stellar  pulsations.   Specifically,  the  asymptotic  period  spacing
(Tassoul et al.  1990) is given by:

\begin{equation} 
\Delta \Pi_{\ell}^{\rm a}= \Delta \Pi_{\ell}^{\rm a} (M_*, M_{\rm  H}, 
T_{\rm eff}, {\rm  CE})  = \Pi_0 / \sqrt{\ell(\ell+1)},  
\label{aps}
\end{equation}

\noindent where

\begin{equation}
\Pi_0= 2 \pi^2 \left[ \int_{r_1}^{r_2} \frac{N}{r} dr\right]^{-1}.
\end{equation}


\begin{figure}
\begin{center}
\includegraphics[clip,width=240pt]{figure2.eps}
\caption{The   run   of  the   asymptotic   period  spacing,   $\Delta
  \Pi_{\ell}^{\rm a}$,  as computed using Eq. (\ref{aps}),  for our DA
  white  dwarf sequences  with different  values of  the  stellar mass
  (indicated  at the  right of  each  curve) and  thick (canonical)  H
  envelope thicknesses.   The location of WD  J1916$+$3938 is depicted
  with   a  black   dot  ($T_{\rm   eff}=   11\,  129   \pm  115$   K,
  $\overline{\Delta \Pi}_{\rm obs}= 38.54 \pm 0.29$ s). The boundaries
  of  the empirical  ZZ  Ceti instability  strip  are displayed  with
  vertical dotted lines. By simple linear interpolation, we found that
  the mass of the star is $M_*= 0.851\pm 0.030 M_{\odot}$ with a thick
  H envelope.} 
\label{asymp-can}
\end{center}
\end{figure}

\noindent For DAV stars, the asymptotic period spacing is sensitive to
the  stellar  mass, the  effective  temperature,  the  mass of  the  H
envelope, and the  convective efficiency (CE) (see Tassoul et  al. 1990 for
an  excellent  discussion  of  this  issue). For  simplicity,  in  our
analysis we  shall neglect  the dependence of  $\Delta \Pi_{\ell}^{\rm
  a}$ with  the convective efficiency\footnote{$\Delta \Pi_{\ell}^{\rm
  a}$ increases when the convective efficiency is 
increased; see Fig. 36 of Tassoul et al. (1990).}.  In principle,  
one can compare
the  asymptotic period  spacing computed  from a  grid of  models with
different masses,  effective temperatures, and  H envelope thicknesses
with the mean period spacing exhibited by the star, and then infer the
value of the  stellar mass by adopting the  effective temperature from
the spectroscopic analysis.  This  method has been applied in numerous
studies of  pulsating PG 1159  stars (see, for instance,  C\'orsico et
al.  2007a,b, 2008, 2009 and references therein).  For the approach to
be  valid,  the  periods  exhibited  by the  pulsating  star  must  be
associated with  high order $g$-modes ($k  \gg 1$), that  is, the star
must be within the asymptotic regime of pulsations.  Fortunately, this
condition is fulfilled by WD J1916$+$3938, because it pulsate in quite
long  periods  with  (presumably)  $k\gtrsim  16$ for  $\ell=  1$  and
$k\gtrsim  30$  for  $\ell=  2$  if the  stellar  mass  and  effective
temperature   of  this   star   are  assumed   to   be  $M_*   \approx
0.8\ M_{\odot}$ and $T_{\rm eff}  \approx 11\, 000$ K, as suggested by
spectroscopy.

In  Fig. \ref{asymp-can} we  show the  run of  the dipole  ($\ell= 1$)
asymptotic period  spacing in terms  of the effective  temperature for
our complete  set of  DA white  dwarf sequences in  the case  of thick
(canonical)  H  envelopes.   Also  displayed  is the  location  of  WD
J1916$+$3938  with  its  associated  uncertainties.  By  adopting  the
spectroscopic effective temperature  of WD J1916$+$3938 ($T_{\rm eff}=
11\,  129 \pm  115$  K; HEA11)  we found  that  the mass  of the  star
according to our computed mean period spacing is $M_*= 0.851 \pm 0.031
M_{\odot}$,  which is  $\approx  5.5  \%$ larger  than  --- but  still
compatible  with  --- the  spectroscopic  estimate  ($M_* \approx  0.8
M_{\odot}$).  This  comparison makes  sense because both  estimates of
the stellar mass  of WD J1916$+$3938 are based  ultimately on the same
DA white dwarf evolutionary tracks. The  above method to estimate  
the  stellar mass  from the  asymptotic period
spacing strongly depends on the precise value of $T_{\rm eff}$ as derived from
the spectroscopic analysis.   If we ignore  the constraint imposed  by the
effective temperature, and look for the possible \emph{range} of stellar mass
just for the star to be within the ZZ Ceti instability strip, we found
that $M_* \gtrsim 0.77 M_{\odot}$. This lower limit is free of 
the uncertainties inherent to the spectroscopic effective temperature 
determination, and is still indicating a rather high stellar mass
for WD J1916$+$3938. 

\begin{figure}
\begin{center}
\includegraphics[clip,width=240pt]{figure3.eps}
\caption{The run  of the asymptotic  period spacing for  the sequences
  with $M_*= 0.837 M_{\odot}$ (dashed blue) and $M_*= 0.877 M_{\odot}$
  (solid  red)  corresponding  to  the canonical  H  envelopes  (thick
  curves), and  for the  case of thin  H envelopes (thin  curves).  In
  each case,  the value of  $\log(M_{\rm H}/M_*)$ is  indicated.}
\label{asymp-837-877}
\end{center}
\end{figure}

In the  above derivation  of the stellar  mass, we have  neglected the
dependence of the asymptotic period  spacing with the thickness of the
H envelope. As is well known,  for a given stellar mass the asymptotic
period spacing is larger for  models with thinner H envelopes (Tassoul
et al.   1990).  This is  shown in Fig. \ref{asymp-837-877},  where we
depict the run  of $\Delta \Pi_{\ell}^{\rm a}$ for  the sequences with
$M_*= 0.837  M_{\odot}$ and $M_*= 0.877 M_{\odot}$  and several values
of the H  envelope thickness for each sequence.   Note that the curves
for  different  H  envelope   thicknesses  corresponding  to  the  two
different  values  of the  stellar  mass  overlap,  somewhat that  can
complicate the analysis.  For instance, regarding the value of $\Delta
\Pi_{\ell}^{\rm a}$, and  for a fixed $T_{\rm eff}$,  a DA white dwarf
model with $M_*= 0.837 M_{\odot}$ and $\log(M_{\rm H}/M_*)= -7.36$ can
readily  mimic a model  with $M_*=  0.877 M_{\odot}$  and $\log(M_{\rm
  H}/M_*)= -8.37$. Obviously, much more ambiguities can still be found
if we consider the curves of $\Delta \Pi_{\ell}^{\rm a}$ corresponding
to the remainder sequences with lower stellar masses, not shown in the
plot  in the  interest  of clarity.   Now,  going to  the  case of  WD
J1916$+$3938,  we could naively  conclude that  that this  star should
have a stellar  mass $M_* \sim 0. 877 M_{\odot}$ and  a rather thick H
envelope of $\log(M_{\rm H}/M_*)\sim -5.40$.  But we also could easily
accommodate a sequence with a stellar mass $M_* > 0.877 M_{\odot}$ and
a  thinner H  envelope,  if we  had  available a  sequence with  these
characteristics.  Thus,  given  the  intrinsic degeneracy  of  $\Delta
\Pi_{\ell}^{\rm a}$ with  $M_*$ and $M_{\rm H}/M_*$, all  we can do is
just to place  a lower limit for the stellar  mass of WD J1916$+$3938.
That is, for a $\overline{\Delta  \Pi}_{\rm obs}= 38.54$ and a $T_{\rm
  eff}= 11\, 129$ K, the stellar mass of the star must be $M_* \gtrsim
0.837 M_{\odot}$ and we cannot say anything about the thickness of the
H  envelope.   Again,  if  we  relax the  constraint  imposed  by  the
spectroscopic  value of  $T_{\rm eff}$,  and we  look for  the allowed
values  of $M_*$  for  the star  to  be residing  within  the ZZ  Ceti
instability strip, we  again obtain the lower limit  $M_* \gtrsim 0.77
M_{\odot}$, which  is completely independent of the  estimation of the
effective temperature of WD J1916$+$3938.

We close this Section by noting that, if the convective efficiency 
characterising our set of DA white dwarf 
models were larger than assumed, then 
the asymptotic period spacing should be larger for each sequence. As a result, 
the lower limit for the stellar mass of WD J1916$+$3938 derived from the 
period spacing should be larger, thus pointing again to the conclusion 
that this DA white dwarf must be rather massive.

\section{Information from the individual pulsation periods}
\label{period-fit}

Besides using the information implicit  in the period spacing, we also
can place constraints on  the stellar mass, effective temperature, and
details  of the  internal  structure of  WD  J1916$+$3938 through  its
individual pulsation  periods.   In this approach
we  seek  a pulsation  DA  white dwarf  model  that  best matches  the
pulsation  periods of  WD J1916$+$3938.   We  assume that  all of  the
observed  periods  correspond  to  normal  modes, that  is,  they  are
associated with  physical displacements  and carry out  information of
the internal structure of the  star. The goodness of the match between
the theoretical pulsation periods  ($\Pi_k^{\rm th}$) and the observed
individual  periods ($\Pi_i^{\rm  obs}$)  is measured  by  means of  a
quality function defined as:

\begin{equation}
\Phi = \Phi(M_*, M_{\rm H},
T_{\rm eff})= \frac{1}{N}\sum_{i=1}^N |\Pi_k^{\rm th}-\Pi_i^{\rm obs}|
\end{equation}

\noindent where $N$  (= 7) is the number of  observed periods.  The DA
white dwarf model  that shows the lowest value  of $\chi^2$ is adopted
as the ``best-fit model''.  We evaluate the function $\Phi = \Phi(M_*,
M_{\rm  H}, T_{\rm  eff})$  for  stellar masses  in  the range  $0.525
\gtrsim M_*/M_{\odot}  \gtrsim 0.877$.  For  the effective temperature
we adopt the range $14\,000  \gtrsim T_{\rm eff} \gtrsim 9\,000$ K and
employ  a  much more  finer  grid  ($\Delta  T_{\rm eff}=  10-30$  K).
Finally, we  vary the thickness of  the H envelope in  the range $-9.4
\lesssim \log(M_{\rm  H}/M_*) \lesssim -3.6$,  respectively, being the
ranges of the values of $M_{\rm H}$ dependent on $M_*$ (see Table 1 of
Romero  et al.  2011).  In our  analysis,  we employ  a  set of  about
$15\,500$ evolutionary DA white dwarf  models. Each of our models have
about $2\,000$ mesh points,  which guarantee us that eigenfunctions of
$g$-modes with radial orders reasonably high (for $\ell= 1, 2$) can be
spatially resolved.

We  first assumed  that  the  harmonic degree  of  the seven  observed
periods of WD  J1916$+$3938 is $\ell= 1$ from the  outset.  We found a
best-fit  solution for a  model with  $M_*= 0.837  M_{\odot}$, $T_{\rm
  eff}=   11\,391$   K   and   $\log(M_{\rm   H}/M_*)=   -7.36$.    In
Fig. \ref{best-fit} we  show the quality function $\Phi$  in terms the
effective temperature for  the case of $M_*= 0.837  M_{\odot}$ and for
all the values of the thickness  of the H envelope considered for this
stellar  mass.  The best-fit  solution corresponds  to the  minimum of
$\Phi$. Our  asteroseismological solution is a bit  ($\approx 2.4 \%$)
hotter than WD J1916$+$3938. The stellar mass of the best-fit model is
in  good agreement  with the  spectroscopic estimate  ($M_*\approx 0.8
M_{\odot}$)  and   with  the   derivation  from  the   period  spacing
($M_* \gtrsim  0.837 M_{\odot}$). This is quite encouraging, because
the three  estimates have been derived from  very different approaches
and assumptions.   In Table \ref{table2} we show  a comparison between
the  observed periods  and the  periods of  the best-fit  model (model
1). The  value of the quality  function is $\Phi= 3.18$  s. Note that,
not surprisingly, the periods at 823.9 s and 834.1 s are associated to
a  single theoretical  period at  825.25  s.  This  because these  two
periods are  very close, and  probably correspond to  different $\ell$
values.

 \begin{figure}
\begin{center}
\includegraphics[clip,width=240pt]{figure4.eps}
\caption{The quality  function $\Phi(M_*, M_{\rm H},  T_{\rm eff})$ in
  terms of the  effective temperature for the different  values of the
  hydrogen  thickness  (shown  with  different  colors  and  symbols),
  corresponding to  the sequence with  $M_*= 0.837 M_{\odot}$  and for
  the case in which we assume that all the seven periods correspond to
  modes  with $\ell=  1$ (model  1 in  Table \ref{table2}).   The gray
  strip correspond to the effective temperature of WD J1916$+$3938 and
  its  uncertainties  according to  spectroscopic  analysis of  HEA11.
  Note  in  particular the  location  of  the  best-fit solution  (the
  minimum of $\Phi$ at $T_{\rm eff} \sim 11\, 400$ K).}
\label{best-fit}
\end{center}
\end{figure}

\begin{table*}
\centering
\caption{Possible  asteroseismological solutions for  WD J1916$+$3938.
  Model 1 is the best-fit model  when we assume that the seven periods
  exhibited by the star are $\ell= 1$; model 2 corresponds to the case
  in which  we discard the period  at 834.1 and again  assume that all
  the six remainder periods  are $\ell= 1$; model 3 is  the best-fit model
  when we consider the seven periods  and allow the values $\ell= 1$ and
  $\ell= 2$ for  them; and finally, model 4 correspond  to the case in
  which we consider a period at  829 s instead of the periods at 823.9
  s and 834.1 s, and again suppose that all the periods are $\ell= 1$.}
\begin{tabular}{ccccccccccc}
\hline
\hline
Model & $T_{\rm eff}$ & $M_*/M_{\odot}$ & $\log(M_{\rm He}/M_*)$ & $\log(M_{\rm H}/M_*)$ & $\Pi_i^{\rm obs}$   &  $\Pi_k^{\rm th}$ & $\ell$ & $k$ & $|\Delta|$ & $\Phi$ \\
 & $[$K$]$      &                &                       &                     &   $[$s$]$     &   $[$s$]$    &   &     &    $[$s$]$ & $[$s$]$\\  
\hline
1 & $ 11\,391$  &   $0.837$       &    $-2.50$            &   $-7.36$          &   823.9   &    825.25  &   1  &  18  & -1.35   &  3.18   \\
  &             &                 &                       &                    &   834.1   &    825.25  &   1  &  18  &  8.85   &           \\
  &             &                 &                       &                    &   934.5   &    930.08  &   1  &  20  & 4.42    &           \\
  &             &                 &                       &                    &   968.9   &    971.58  &   1  &  21  & -2.68   &           \\
  &             &                 &                       &                    &   1089.0  &   1089.11  &   1  &  24  & -0.11   &           \\
  &             &                 &                       &                    &   1169.9  &   1169.59  &   1  &  26  &  0.31   &           \\
  &             &                 &                       &                    &   1436.7  &   1432.14  &   1  &  32  &  4.56   &           \\
\hline
2 & $ 11\,391$  &   $0.837$       &    $-2.50$            &   $-7.36$          &   823.9   &    825.25  &   1  &  18  & -1.35   &   2.24   \\
  &             &                 &                       &                    &   934.5   &    930.08  &   1  &  20  & 4.42    &           \\
  &             &                 &                       &                    &   968.9   &    971.58  &   1  &  21  & -2.68   &           \\
  &             &                 &                       &                    &   1089.0  &   1089.11  &   1  &  24  & -0.11   &           \\
  &             &                 &                       &                    &   1169.9  &   1169.59  &   1  &  26  &  0.31   &           \\
  &             &                 &                       &                    &   1436.7  &   1432.14  &   1  &  32  &  4.56   &           \\
\hline
3 & $ 11\,391$  &   $0.837$       &    $-2.50$            &   $-7.36$          &   823.9   &    825.25  &   1  &  18  & -1.35 &   1.14  \\
  &             &                 &                       &                    &   834.1   &    832.76  &   2  &  32  &  1.34 &           \\
  &             &                 &                       &                    &   934.5   &    932.69  &   2  &  36  &  1.81 &           \\
  &             &                 &                       &                    &   968.9   &    971.58  &   1  &  21  & -2.68 &           \\
  &             &                 &                       &                    &   1089.0  &   1089.11  &   1  &  24  & -0.11 &           \\
  &             &                 &                       &                    &   1169.9  &   1169.59  &   1  &  26  & 0.31  &           \\
  &             &                 &                       &                    &   1436.7  &   1437.09  &   2  &  56  & -0.39 &           \\
\hline
4 & $ 11\,376$  &   $0.837$       &    $-2.50$            &   $-7.36$          &   829.0   &  825.88    &   1  &  18  & 3.12  &   2.56    \\
  &             &                 &                       &                    &   934.5   &  930.87    &   1  &  20  & 3.63  &           \\
  &             &                 &                       &                    &   968.9   &  972.42    &   1  &  21  & -3.52 &           \\
  &             &                 &                       &                    &   1089.0  &  1090.11   &   1  &  24  & -1.11 &           \\
  &             &                 &                       &                    &   1169.9  &  1170.68   &   1  &  26  & -0.78 &           \\
  &             &                 &                       &                    &   1436.7  &  1433.51   &   1  &  32  &  3.19 &           \\
\hline
\end{tabular}
\label{table2}
\end{table*}

Next, we  tried to find a  best-fit model by discarding  the period at
834.1 s  (so, $N= 6$),  to be consistent  with the assumption  made to
derive   a    mean   period   spacing    of   the   star    in   Sect.
\ref{period-spacing}.  We  found the  same best-fit model  than before
(see Table \ref{table2},  model 2), but the quality of  the fit is, as
expected, substantially improved ($\Phi= 2.24$ s).

In a further step, we relaxed  the constraint of $\ell= 1$ for all the
observed periods of  WD J1916$+$3938, and we redone  our period fit by
allowing the values $\ell= 1$ or $\ell= 2$. We obtain exactly the same
best-fit  model  as  before.    In  Table  \ref{table2}  we  show  the
theoretical periods and the $k$- and $\ell$-identifications (model 3).
For  this period  fit, we  obtain  $\Phi= 1.14$  s. This  is the  best
agreement with the  observed periods we obtain, although  in this case
the $\ell$-identification  is not  the same as  that assumed  in Sect.
\ref{period-spacing} to derive the period spacing value of the star.

Finally,  we have  carried  out an  additional  period-fit, this  time
supposing that  the periods at 823.9 s  and 834.1 are the  $m= +1$ and
$m=  -1$ components  of a  rotational  triplet, in  which the  central
component ($m= 0$) is missing (see next Section). Under  this
assumption,  we estimate the  period of  the $m=  0$ component  as the
average of  the $m= +1$ and  $m= -1$ components.  Thus,  we consider a
period of 829  s in our period-fit procedure,  instead the two periods
at 823.9 s and 834.1.  Here, we  adopt $\ell= 1$ for the set of $N= 6$
periods at  the outset.   In this case,  we obtain virtually  the same
asteroseismological  model than before,  with $M_*=  0.837 M_{\odot}$,
$T_{\rm eff}=  11\, 376$ K,  $\log(M_{\rm H}/M_*)= -7.36$,  and $\Phi=
2.557$ s.  The  theoretical periods and $\ell, k$  values are shown in
Table \ref{table2} (model 4).

We conclude  that, even by adopting several  different assumptions for
the assignation of the $\ell$  and $k$ values to the observed periods,
we found the same  asteroseismological model which is characterised by
an  effective  temperature of  $T_{\rm  eff}=  11\,380$  K, a  surface
gravity of $\log g= 8.39$, a stellar mass of $M_*= 0.837 M_{\odot}$, a
mass  of  H of  $\log(M_{\rm  H}/M_*)= -7.36$,  and  a  He content  of
$\log(M_{\rm He}/M_*)=  -2.50$. The location of the  best-fit model in
the $T_{\rm eff}- \log g$ plane is depicted in Fig. \ref{gteff}.

\section{Stellar rotation}
\label{rotation}

Here, we suppose that the periods at 823.9 s and 834.1 are the $m= +1$
and $m= -1$ components, respectively, of a rotational triplet in which
the central component  ($m= 0$) is absent from  the pulsation spectrum
of  WD J1916$+$3938.   If this  assumption is  valid, we  can  infer a
frequency spacing of $\Delta  \nu= \nu(m=+1)- \nu(m=-1)= 14.8$ $\mu$Hz
between the extreme  components of the triplet. By  assuming rigid and
slow    rotation,    we   have    $\delta    \nu=   -m\    \Omega_{\rm
  rot}\ \left(1-C_{k,\ell}\right)$,  being $\delta \nu=  \Delta \nu/2=
7.4$  $\mu$Hz and  $C_{k,\ell}\sim 0.5$  in the  asymptotic  limit for
$\ell= 1$.  Then, we can infer a mean rotation period of $P_{\rm rot}=
1/\Omega_{\rm rot}=  18.77$ hs for  WD J1916$+$3938.  This is  in line
with the values derived for  other ZZ Ceti stars from asteroseismology
(see, for instance,  Table 4 of Fontaine \&  Brassard 2008).  The fact
that  a DA  white  dwarf  like this,  characterised by  an
effective temperature of $T_{\rm eff} \sim 11\, 400 - 11\, 100$ K, and
pulsating in high order $g$-modes with long periods, which usually
have associated rotational kernels that  are mostly  confined to  the outer
layers of the star, prevent us  to attempt a more profound analysis of
the internal rotation  of WD  J1916$+$3938, like the one performed very recently
by C\'orsico et al. (2011) for the GW Vir star PG 0122+200.

\section{Conclusions}
\label{conclu}

Tempted by  the exciting discovery  of the first pulsating  DA white
dwarf in  the \emph{Kepler} mission  field by HEA11, and
encouraged by  the availability of  the most complete set  of detailed
evolutionary and pulsation models for  DAV stars up to date (Romero et
al.    2011),   in   this    paper   we   have   presented   a   first
asteroseismological analysis of WD  J1916$+$3938.  This, even with the
certainty that the full asteroseismological potential of this DAV star
will be  exploited with further  observations by \emph{Kepler}  in the
next years.

We summarize our main findings below:

\begin{itemize}

\item[-]    We    derive    a    spectroscopic    mass    of    $M_*=
  0.797-0.805\ M_{\odot}$ for WD J1916$+$3938  on the basis of the new
  DA white dwarf  evolutionary tracks of Romero et  al.  (2011).  This
  value is smaller than the estimate of HEA11, of
  $M_*= 0.82\  M_{\odot}$, based on  the older models of  Wood (1990).
  In our analysis,  we have taken into account the dependence of the
  tracks in the $T_{\rm eff}-\log g$ diagram with the thickness of the
  H envelope  (see Fig.  \ref{gteff}).

\item[-] By assuming that all the periods exhibited by WD J1916$+$3938
  are normal  modes of  the star, and  supposing that all  the periods
  except one (at 834.1 s) are associated to $\ell= 1$ modes, we derive
  an average period spacing of $\overline{\Delta \Pi}_{\rm obs}= 38.54
  \pm  0.29$ s.   On the  basis of  the asymptotic  theory  of stellar
  pulsations,  we estimate  a stellar  mass of  $M_*= 0.851  \pm 0.031
  M_{\odot}$,  corresponding  to  a  thick  H  envelope. If  we take  
  into  account the strong dependence of  the
  asymptotic  period spacing with  the thickness of the H envelope, 
  we found an evident degeneracy of $\Delta \Pi_{\ell}^{\rm a}$ with $M_*$ 
  and $M_{\rm H}$. In virtue of this, we can conclude that 
  $M_* \gtrsim  0.837 M_{\odot}$, but we cannot say anything about  
  the thickness of the H envelope.  Finally, if  we ignore  the  
  constraint  imposed  by the  effective
  temperature   as  derived   by  spectroscopy,   we  found   that  WD
  J1916$+$3938 should have a stellar mass $M_* \gtrsim 0.77 M_{\odot}$
  in order for the star to be within the ZZ Ceti instability strip.

\item[-]  By using the  individual pulsation  periods exhibited  by WD
  J1916$+$3938, and  taking into  account several assumptions  for their
  $\ell$- and $k$-identification, we found an
  asteroseismological model  with $T_{\rm  eff}= 11\,380$ K,  $\log g=
  8.39$,  $M_*= 0.837  M_{\odot}$, $\log(M_{\rm  H}/M_*)=  -7.36$, and
  $\log(M_{\rm   He}/M_*)=   -2.50$.    The   stellar  mass   of   the
  asteroseismological   model   is   in   good  agreement   with   the
  spectroscopic  estimate ($M_*\approx  0.8 M_{\odot}$)  and  with the
  derivation   from   the   period   spacing   ($M_*\gtrsim   0.837
  M_{\odot}$). This is quite  encouraging, because the three estimates
  have been derived from very different approaches and assumptions. On
  the other hand, the effective temperature of the asteroseismological
  model is somewhat higher than  the spectroscopic one  ($T_{\rm eff}=
  11\,129 \pm  115$ K). 

\item[-] In passing,  we have estimated a rotation  period of 18.77 hs
  for WD  J1916$+$3938.  The reliability  of this result rests  on the
  assumption that the modes with periods  at 823.9 s and 834.1 are the
   $m=  +1$ and  $m=  -1$ components,  respectively, of  a
  $\ell= 1$ frequency triplet due to rotation.

\end{itemize}

In closing, we  think that the conclusion that  WD J1916$+$3938 should
be a  rather massive ZZ Ceti  star, with $M_*  \gtrsim 0.8 M_{\odot}$,
appears to  be robust. The possibility that this rather massive
pulsating white dwarf can be further monitored with \emph{Kepler} with
a high degree of detail turns this star into a promising and unique target
to study the physics of crystallization and carbon/oxygen phase diagrams 
at high densities. In this regard, we  note that, for  a stellar
mass of  $M_* \sim 0.88  M_{\odot}$, crystallization of  matter should
occur at  the centre of  a white dwarf star  at $T_{\rm
  eff}  \approx  10\,500$  K according to the  phase  diagram  of
Segretain \& Chabrier (1993), and at $T_{\rm eff} \approx 9\,600$ K if
the new phase  diagram recently proposed by   Horowitz et  al.  (2010) 
is considered  (Althaus et
al. 2011). Indeed,  based on direct molecular dynamics simulations for dense 
carbon-oxygen mixtures, Horowitz et al. (2010) find substantially 
lower crystallization temperatures and that the shape of the carbon-oxygen 
phase diagram is of the azeotropic form --- 
and not of the spindle type --- as previously believed. Our analysis  from 
the period  spacing of  WD J1916$+$3938
suggests that  this star  could well have  a stellar mass  much higher
than $0.8  M_{\odot}$ and  necessarily a rather  thin H  envelope.  If
this  were  the case,  this  star  could have  a  substantial
fraction  of its  interior  crystallized depending on the adopted
carbon-oxygen phase  diagram, an issue that could be explored
with
future asteroseismological analysis  that exploit the new high-quality
photometric   data  coming   from  the   ongoing  monitoring   by  the
\emph{Kepler} mission.

\section*{Acknowledgments}

Part of  this work  was supported by  AGENCIA through the  Programa de
Modernizaci\'on   Tecnol\'ogica  BID   1728/OC-AR, and by   the  PIP
112-200801-00940  grant from CONICET.  
This research  has made  use of NASA's Astrophysics Data System.

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