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\newcommand{\msun}{$\mathrm{M}_\odot$}
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\begin{document}

\title{A prevalence of dynamo-generated magnetic fields in the cores of intermediate-mass stars
%Dynamo-driven magnetic fields common in the cores of intermediate- and low-mass stars
%Dynamo-driven magnetic fields common in the cores of intermediate-mass stars
%Dynamo-generated magnetic fields common in the convective cores of intermediate-mass stars
%Common dynamo-generated magnetic fields in the convective cores of intermediate-mass stars %Alternative w/o '(are) common' active verb}


 \author{\textbf{Dennis Stello}, \footnote{\textit{dennis.stello@sydney.edu.au}} \href{https://www.authorea.com/inst/5746}{University of Sydney}\\
\textbf{Matteo Cantiello}, \footnote{\textit{matteo@kitp.ucsb.edu}} \href{https://www.authorea.com/inst/7192}{Kavli Institute for Theoretical Physics} \& \href{https://www.authorea.com/inst/7191}{UCSB} \\
\textbf{Jim Fuller}, \footnote{\textit{jfuller@caltech.edu}} \href{https://www.authorea.com/inst/943}{California Institute of Technology} \& \href{https://www.authorea.com/inst/7192}{Kavli Institute for Theoretical Physics}\\
\textbf{Daniel Huber} \href{https://www.authorea.com/inst/5746}{University of Sydney}\\
\textbf{Rafael A. Garc{\'i}a}, \href{https://www.authorea.com/inst/7321}{Service d'Astrophysique, IRFU/DSM/CEA Saclay} \\
\textbf{Timothy R. Bedding}, \href{https://www.authorea.com/inst/5746}{University of Sydney}\\
\textbf{Lars Bildsten} \href{https://www.authorea.com/inst/7192}{Kavli Institute for Theoretical Physics} \& \href{https://www.authorea.com/inst/7191}{UCSB}\\
\textbf{Victor Silva Aguirre} Department of Physics and Astronomy, Aarhus University \\}


\date{\today}

\maketitle

\selectlanguage{english}
\begin{abstract}
\textit{This is the author's version of the work. It is posted here for personal
use, not for redistribution. The definitive version was published in \href{http://www.nature.com/nature/journal/vaop/ncurrent/full/nature16171.html}{Nature} on
04 January 2016, DOI:10.1038/nature16171}

Magnetic fields play a role in almost all stages of stellar evolution
\citep{Landstreet_1992}. Most low-mass stars, including the Sun, show
surface fields that are generated by dynamo processes in their
convective envelopes \citep{Parker_1955,Donati_2009}. Intermediate-mass
stars do not have deep convective envelopes \citep{Kippenhahn_1990}, although
10\% exhibit strong surface fields that are presumed to be residuals
from the stellar formation process \citep{2008CoSka..38..443P}. These
stars do have convective cores that might produce internal magnetic
fields \citep{Brun_2005}, and these might even survive into later stages
of stellar evolution, but information has been limited by our inability to
measure the fields below the stellar surface \citep{Auri_re_2015}. Here
we use asteroseismology to study the occurrence of strong magnetic
fields in the cores of low- and intermediate-mass stars. We have
measured the strength of dipolar oscillation modes, which can be
suppressed by a strong magnetic field in the core \citep{Fuller15},
in over 3,600 red giant stars observed by \kepler. About 20\% of our sample show
mode suppression but this fraction is a strong function of mass. Strong
core fields only occur in red giants above 1.1 solar masses (1.1\msol), and
the occurrence rate is at least 60\% for intermediate-mass stars
(1.6--2.0\msol), indicating that powerful dynamos were very common in the
convective cores of these stars.

  %
\end{abstract}%



\section{Main}
Red giants are formed when a low- or intermediate-mass star has finished
burning the hydrogen in its core. This leaves an inert helium core
surrounded by a thin hydrogen-burning shell and a very thick outer
convective envelope.
Like the Sun, red giants oscillate in a broad comb-like frequency spectrum
of radial and non-radial acoustic modes that are excited by the turbulent
surface convection \citep{De_Ridder_2009}. 
The observed power spectrum has a roughly Gaussian envelope whose central
frequency, \numax, decreases as a star expands during the 
red giant phase \citep{Stello_2008}. %\citep{Brown_1991,Stello_2009}. 
The comb structure of the spectrum arises from a series of 
overtone modes separated by the so-called large frequency separation,
\dnu. One of these overtone sequences is seen for each spherical degree,
$\ell$.  For observations of unresolved distant stars, 
geometric cancellation prevents detection of modes with $\ell>3$.  
Their spectra are characterised by a pattern of radial ($\ell=0$) and
quadrupolar ($\ell=2$) modes that form close pairs, interspersed with dipolar
($\ell=1$) modes located roughly halfway between successive
radial-quadrupolar pairs. 
The octupolar modes ($\ell=3$) are weak or undetectable.
The dipolar modes have turned out to be particularly useful probes of
internal structure \citep{GarciaStello15}. 
They have been used to distinguish between hydrogen-shell and helium-core
burning stars \citep{Bedding_2011,Stello_2013,Mosser_2014}
%space saver: only keep Bedding
and to measure radial differential rotation \citep{Beck_2011,Mosser_2012}. 
%space saver: only keep Beck
This usefulness arises because
each acoustic non-radial mode in the envelope couples to multiple gravity
modes in the core, forming several observable mixed modes
with frequencies in the vicinity of the acoustic mode \citep{Beck_2011}.
This coupling is strongest for dipole modes, making them the most useful
probes of the core \citep{Dupret_2009}.\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=1.00\columnwidth]{figures/example-spectra3/example-spectra}
\caption{{Oscillation spectra of six red giants observed with \kepler. The
  stars are ordered in three pairs, each representing a different evolution
  stage ranging from the most evolved (lowest oscillation frequencies) on
  the left to the least evolved (highest frequencies) to the right. The
  coloured regions mark the power dominated by modes of different degree
  $\ell=0$--3. For clarity the spectra are smoothed by 3\% of the frequency
  separation between overtone modes, which for the most evolved stars tend
  to create a single peak at each acoustic resonance, even if it comprises
  multiple closely-spaced mixed modes (red peaks in the left and centre
  panels). The slightly downward sloping horizontal dashed line indicates
  the noise level. Observations of each star were made during the first 37
  months of the \kepler\ mission (observing quarters Q0--Q14).%
}}
\end{center}
\end{figure}


Figure 1 shows the oscillation power spectra of red giants at three
different evolutionary stages observed by NASA's \kepler\ mission. 
For ``normal'' stars (upper panels), the dipolar modes (red peaks) have
similar power to the radial modes (black peaks).  However, at each stage 
of evolution we also find stars with greatly suppressed dipolar modes
(lower panels). 
Suppressed dipolar modes have been reported in a few dozen red giant stars,
with an occurrence rate of about 20\%  \citep{Mosser_2011,Garc_a_2014}. The
cause of this phenomenon has been puzzling until recent theoretical work \citep{Fuller15},
which showed that the suppression can be explained if waves entering the
stellar core are prevented from returning to the envelope. 
This occurs for dipolar modes if there are strong magnetic fields in the
core, giving rise to a ``magnetic greenhouse effect'' \citep{Fuller15}.
  

  
  


  We measured the amount of suppression by comparing the integrated power of
the dipolar and radial modes (the dipole mode visibility, $V^2$), averaged
over the four orders centred on \numax\ (see Methods for details).  
While the normal stars show dipole mode visibilities of $V^2 \approx 1.5$,
independent of \numax, \citep{Ballot_2011,Mosser_2011}, the stars with
suppressed modes have $V^2\approx 0.5$ for \numax\ $\simeq 70\,$\muhz\ and  
down to almost zero for the least evolved red giants oscillating above
$200\,$\muhz\ (Fig. 1).   

In Fig. 2 we show the dipole mode visibility for about 3,600 red giants
observed over the first 37 months of the \kepler\ mission (see Methods). 
%space saver: Put next bit in Methods
Our analysis is
restricted to a sample of stars with \numax\ larger than 50\muhz\ and
masses below 2.1\msol\ which, assuming no observational uncertainties, is
expected to include only red giants that have not started burning helium in
their cores \citep{Stello_2013}.   
%space saver: Put next bit in Methods
We cross-matched our sample with those of known helium-burning stars
\citep{Stello_2013,Mosser_2014}, which allowed us to identify and remove a
small fraction of evolved stars burning helium that, due to measurement
uncertainty, had entered our sample ($2\%$ of our sample, almost all with
\numax\ $< 70$\muhz). 

  
  

The stars in Fig. 2 form two distinct branches that gradually merge as the
stars evolve leftwards towards lower \numax.  Most stars fall on the
``normal'' upper branch of $V^2\approx 1.5$, in agreement with previous
results \citep{Mosser_2011}.     
The lower branch, with suppressed dipole modes, agrees remarkably well with
theoretical predictions (black curve). This prediction
assumes that all the wave energy leaking into the stellar core is trapped by 
a magnetic greenhouse effect caused by strong internal magnetic fields
\citep{Fuller15}. 
The decrease of the suppression towards lower \numax\ is a consequence of
the weaker coupling between acoustic waves in the envelope and gravity
waves in the core \citep{Fuller15}.   
With this large sample we have been able to separate the stars in Fig.2
into five different mass intervals, from 0.9 to 2.1\msol. It is striking
how strongly the relative population on the lower branch (stars with
suppressed dipole modes) depends on mass.   
  


We quantify the mass dependence in Fig. 3 by showing the relative number of
dipole-suppressed stars (those below the dashed line in Fig. 2) in narrow
mass intervals.   We see no suppression in red giants below 1.1\msol, which 
coincides with the mass below which they did not have convective cores
during the core-hydrogen-burning phase \citep{Kippenhahn_1990}. The onset
of magnetic suppression above this threshold suggests that at least some of
those stars had convectively driven magnetic dynamos in their cores during
the core-hydrogen-burning (main-sequence) phase. This is supported by 3D
hydrodynamical modeling of these stars \citep{Brun_2005}. Red giants no
longer contain convective cores, leading us to conclude that the strong
magnetic fields in suppressed oscillators are the remnants of the fields
produced by core dynamos during the main sequence. 


Figure 3 shows that the incidence of magnetic suppression increases with
mass, with red giants above 1.6\msol\ showing a remarkable suppression rate
of 50-60\%. These have evolved from main-sequence A-type stars, among which
only up to $\approx 10$\% are observed to have strong fields at their
surfaces \citep{2008CoSka..38..443P}. %{\bf (see Methods for details.)}
%DISCUSSION OF WHY NOT 100\% AND/OR LINKS TO ROTATION AS REFEREE ALLUDES
%TO...COULD ALSO BE REFERED TO AFTER 2nd LAST PARA)
We conclude that these magnetic A stars represent only the tip of the
iceberg, and that a much larger fraction of A stars have strong magnetic
fields hidden in their cores.\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/vis-vs-numax2/vis-vs-numax2}
\caption{{Visibility of dipolar modes for red giants observed with
  \kepler. The abscissa is the central frequency of the oscillations, which
  correlate closely with surface gravity shown at the top axis. Stars
  evolve from right to left in the diagram, corresponding roughly to the
  beginning of the red giant phase to the red giant luminosity bump
  \protect\citep{Salaris_2005}. 
  %space saver: next setence in Methods
  The upper limit on \numax\ is set by the sampling
  of the \kepler\ data. 
  Each panel shows stars in a different mass bracket
  increasing from top to bottom (indicated by the colour bar annotation on
  the right). 
  %space saver: next setence in Methods
  Mass is calculated from asteroseismic scaling relations
  \protect\citep{Stello_2013}, and has a formal 1$\sigma$ uncertainty of 10\%
  \protect\citep{Miglio_2011}. The solid black line shows the theoretical predicted
  visibility of suppressed dipole modes \protect\citep{Fuller15} assuming a stellar
  mass of 1.1, 1.3, 1.5, 1.7, and 1.9\msol\ (top to bottom panel) and a mode lifetime for radial modes of 20 days
  \protect\citep{E_Corsaro_2015}. The fiducial dashed line separates the two
  branches of normal and dipole-suppressed stars. %{\bf Data for the theoretical predictions of suppressed $\ell =1$ modes is also available here for M= \href{https://www.authorea.com/users/2/articles/37186/master/file/figures/vis_vs_numax2/Visibility_1.1M.dat}{1.1}, \href{https://www.authorea.com/users/2/articles/37186/master/file/figures/vis_vs_numax2/Visibility_1.3M.dat}{1.3}, \href{https://www.authorea.com/users/2/articles/37186/master/file/figures/vis_vs_numax2/Visibility_1.5M.dat}{1.5}, \href{https://www.authorea.com/users/2/articles/37186/master/file/figures/vis_vs_numax2/Visibility_1.7M.dat}{1.7} and \href{https://www.authorea.com/users/2/articles/37186/master/file/figures/vis_vs_numax2/Visibility_1.9M.dat}{1.9} \msol\.}%
}}
\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=1.00\columnwidth]{figures/histogram/tmp}
\caption{{Observed fraction of stars with suppressed dipolar modes.  The
  abscissa is the stellar mass (in solar units).  
  %space saver: next two sentences in Methods
  For each mass bin we
  calculated the dipole-suppressed star 
  fraction as the number of stars that fall below the dashed line in
  Fig. 2, relative to all stars in that same mass bin.  
  To make the distinction unambiguously between normal stars and stars with
  suppressed dipoles, we only counted stars with \numax\ $>70$\muhz. 
  The 1$\sigma$ uncertainty in the fractions (grey vertical errorbars) are based on
  Poisson statistics of the total star counts (inset: blue plus green) and of
  the number of dipole-suppressed stars (inset: green).  The vertical
  dotted line separates stars  
  for which hydrogen-core burning took place in either a radiative or
  convective environment for solar metallicity \protect\citep{Kippenhahn_1990}.%
}}
\end{center}
\end{figure}


In Fig. 4 we show the observed \numax\ and inferred mass of all the stars
superimposed on a contour plot of minimum magnetic field strengths required
for mode suppression \citep{Fuller15}. For stars with suppressed modes
(filled red circles), the underlying colour provides a lower bound to the
field strength at the hydrogen-burning shell. For stars without suppressed
modes (open black circles), the underlying colour represents an upper limit
to the field at the hydrogen-burning shell; above or below the shell the
field could potentially be larger. Hence, normal and dipole-suppressed
stars that fall in the same regions of Fig. 4 may have core field strengths
that are only slightly different. However, we expect that the
dipole-suppressed stars on average exhibit stronger core fields than their
normal counter parts. 



  Considering again the low-mass stars ($< 1.1$\msol), of which none show
suppression, we see from Figure 4 that  radial magnetic fields above $\approx 100 \,
{\rm kG}$ are not present at the hydrogen-burning shell when the stars are
just below the red giant luminosity bump (\numax\ $\approx 70-100 \,
\mu{\rm Hz}$). 
Assuming magnetic flux conservation from the main-sequence phase, this
suggests that  radial fields above $\approx 5 \, {\rm kG}$ do not exist within the
cores of Sun-like stars \citep{Fuller15}. 
Large scale fields in the solar interiors have been discussed in order to
explain the properties of the tachocline \citep{GoughMcIntyre98}. However,
our results do not rule out strong horizontal fields near the
radiative-convective boundary because those fields would be outside the
core and could not cause mode suppression when the star evolves into a red
giant. 

  
  
  


  Turning to higher masses we see that, for a given \numax, stars above
1.4\msol\ require increasingly strong magnetic fields to suppress their
dipolar modes. From Figure 4, there is no clear upper limit to the field
strengths present in red giant cores, given that dipole-suppressed stars are
common even when field strengths $ B> 1 \, {\rm MG}$ are required for
suppression.  However, the hint of a decline in the occurrence of
dipole-suppressed stars above 2\msol\ seen in Fig. 3 suggests there may be
a mass above which dynamo-generated magnetic fields can no longer cause
oscillation mode suppression in intermediate-mass stars. 
%Such a cutoff might be caused by the different core structure of stars more massive than 2\msol, which evolve through the red giant phase much faster than stars of lower mass. 



The high occurrence rate of dipole mode suppression demonstrates that 
internal magnetic fields, generated by a convective core dynamo during the main sequence,  can  persist through the red giant phase. This indicates that these dynamo-generated fields are
frequently able to settle into long-lived stable configurations
%,a result
%that was not certain from magnetohydrodynamical simulations
\citep{Braithwaite_2004,Duez_2010}. The occurrence rate of
suppressed dipole modes in intermediate-mass red giants is much higher
than the occurrence rate of
strong fields at the surfaces of the main-sequence A stars from which they
evolved.  The latter fields are thought to 
be a relic of the star's formation process \citep{Moss_1987,Braithwaite_2004}.
%MC: We do not know if it's a dynamo process or simply some kind of magnetic flux conservation or a combination of the two (hybrid hypothesis)
%have been generated by a pre-hydrogen-core burning dynamo during star formation
%\citep{Moss_2004}. 
We conclude that fields generated during
 convective core-hydrogen-burning are able to settle into stable equilibrium
configurations much more commonly (more than $60\%$ of the time) than
fields  generated or inherited during star formation (less than $10\%$ of the time).  


  
  
  
  
  
  
  
  


  Our results show that main-sequence stars with no observable magnetic field
at the surface can still harbour strong fields in the core that survive
into the red giant phase. 
The presence of internal magnetic fields might play an important role for
angular momentum transport and  potentially influence internal mixing processes.
Fields too weak to suppress dipolar oscillation modes may exist in normal
red giants, and these fields may nevertheless transport enough angular
momentum to help explain the measured rotation rates of red giant cores
\citep{Mosser_2012,Cantiello_2014}.  
The presence of strong internal magnetic fields might help explain extra internal mixing observed in 
red giant branch and asymptotic giant branch stars \citep{Busso_2007}. Interestingly the magnetic field strength necessary for magnetic buoyancy mixing \citep{Nucci_2014} is similar to the field strength required for dipole mode suppression \citep{Fuller15}.

  
  
  
  
  
  
  
  
  
  


  
After some time, intermediate-mass red giants also start burning helium in
their cores. Suppressed dipolar modes in those so-called red clump stars
will reveal whether the fields survive until helium-core burning, and
whether they can account for magnetic fields observed in stellar remnants
such as white dwarfs.  Like intermediate-mass stars, more massive stars
($M>10$\msun) also undergo convective hydrogen-core burning that generates
a magnetic dynamo, and which may produce the magnetic fields observed in
many neutron stars.\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=1.00\columnwidth]{figures/b-crit-nu-mass1/b-crit-nu-mass-6-6}
\caption{{Critical magnetic field strength required to suppress dipole mode
    oscillations. The abscissa is the 
    observed central frequency of the oscillations.  The ordinate is the
    inferred asteroseismic mass.  The coloured contours indicate the minimum
    magnetic field at the hydrogen shell required for
    mode suppression (the critical field, $B_c$).  Filled red
    circles mark stars with observed suppressed modes, and open circles
    mark normal (not suppressed) stars. The cross shows a typical 1$\sigma$ errorbar
    for the data points.    
    The uncertainty in $B_c$ due to uncertainty in mass is negligible for
    stars below 1.4\msol\ and is no more than 25\% for the more massive
    stars. %{\bf  The data for the \textit{Kepler} targets (including their critical field $B_c$) can be downloaded from \href{https://www.authorea.com/users/2/articles/37186/master/file/figures/b_crit_nu_mass1/bcrit_data.dat}{this link}.}%%
}}
\end{center}
\end{figure}


  \section*{Methods}

\subsection{Dipole-mode visibilities} 

For each star in our sample we derived the frequency power spectrum as the
Fourier transform of the \kepler\ light curve (up to observing quarter 14)
and adopted the values of \dnu, \numax, and mass from previous work \citep{Stello_2013}. To correct the
spectrum for the background noise, we measured and subtracted a linear
slope anchored on the median power on either side of the central
power excess, defined by the frequency ranges 0.75(\numax\ $-$ 2\dnu) to
(\numax\ $-$ 2\dnu) and (\numax\ $+$ 2\dnu) to 1.25(\numax\ $+$ 2\dnu).
We then selected a 4\dnu-wide range of the spectrum centred on \numax.
The location of each mode was found by first folding this central part of
the spectrum using \dnu\ as the folding frequency such that modes of the
same spherical degree each formed a single peak.  The folded spectrum was
smoothed by a Gaussian filter with a width of 0.1\dnu. We finally  
correlated the folded spectrum with a model spectrum comprising of three
Lorenzian profiles, one for each degree $\ell=0$, 1, and 2 with relative
heights 1.0, 0.5, and 0.8, and widths of 5\%, 10\%, and 5\% of
\dnu, respectively.
The centres of each Lorenzian profile was fixed relative to one
another such that the one representing the $\ell=2$ modes was 0.12\dnu to the
left (lower frequency) of the $\ell=0$ profile, and the $\ell=1$ profile was
0.52\dnu to the right (higher frequency) \cite{Huber_2010}. 
The shift between the model and the observed folded spectrum that gave the
largest correlations, provided the location of each mode. 
%With this method the pattern of the multiple mixed 
%modes around the acoustic resonance is smoothed away, creating a single
%broad peak centred at the resonant frequency.
%Separations between mixed 
%modes could therefore not be mistaken for the small separation betweeen
%radial and quadrupole modes, which ensured a unique mode identification.
The region of the power spectrum associated with each spherical degree were
set to be 0.16, 0.53, and 0.16 times \dnu\ wide for $\ell=0$--2,
respectively, with the regions located according to the mode location found
by the correlation with the model spectrum. The remaining region was
associated with $\ell=3$ modes (Fig. 1).
The dipole-mode visibility was derived as the integrated power of the
dipole modes relative to that of the radial modes following the approach by 
previous studies \citep{Mosser_2011}.
%For normal stars the integrated power in the dipole modes is typically
%about 50\% larger than for the radial modes ($V^2\simeq 1.5$) 
%[This 'frequency
%of maximum power' shows tight correlation with stellar surface gravity[REF
%Brown]]. 

%\subsection{DISUCSSION OF WHY NOT 100\% SUPPRESSED AND OR LINKS TO ROTATION AS REFEREE ALLUDES TO} 
%The resulting strong magnetic fields must have survived long after the
%dynamo terminated at the end of hydrogen-core burning in order to be
%detected during the red giant phase. Therefore, dynamo-generated fields are
%frequently able to settle into long-lived stable configurations
%\citep{Braithwaite_2004,Braithwaite_2006,Duez_2010}, a result that was not
%certain from the 3D hydrodynamical simulations. 
%The gradual increase in suppression may result from the increasing size of
%the convective core in more massive stars, which generates stronger
%magnetic fields that survive after hydrogen burning ceases in the core. 
%[Jim/Matteo elaborate?]

  
 \subsection{Code availability}
We used the commercially available IDL routine lnp\_test to derive the Fourier spectra of the Kepler light curves.
To derive the mode visibilities we used build-in IDL routines and subroutines written by others, which are not made publicly available for practical reasons.
For the stellar evolution calculations we used MESA (publicly available at http://mesa.sourceforge.net/)

%For the stellar evolution calculations we used ATON (not publicly available). For the N-body simulations we used Starlab (publicly available at http://www.sns.ias.edu/~starlab/). For the synthetic populations, we used a program (HBSINT) written explicitly for this work (not publicly available).
 
%equest from mazzali@mpa-garching.mpg.de.
  
%We have opted not to make the code available for practical reasons. However, the methods we adopt are published in the literature and are commonly used in the community. Amongst others, the publications mentioned in the above Methods section contain tests of our code against problems with known solutions and also with respect to solutions obtained with similar codes from independent authors.
 
%The calculations presented in this paper were performed using LIPAD, a proprietary software product funded by the Southwest Research Institute that is not publicly available. It is based upon the N-body integrator SyMBA, which is publicly available at http://www.boulder.swri.edu/swifter/.
 
%We choose not to make the pipeline code available at this time because it is not fully documented for public use but plan to do so in early 2016.
  

\section*{Acknowledgments} 

This paper was written collaboratively, on the web, using \href{https://www.authorea.com}{Authorea}. JF acknowledges partial support from NSF under grant no. AST-1205732 and through a Lee DuBridge Fellowship at Caltech. RAG acknowledge the support of the European Community's Seventh Framework Programme (FP7/2007-2013) under grant agreement No. 269194 (IRSES/ASK), and from the CNES.
This project was supported by NASA under TCAN grant number NNX14AB53G and the NSF under grants PHY 11-25915 and AST 11-09174.

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