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\title{Observing Convectively Excited Gravity Modes in Main Sequence Stars}


\author{Tyler Lux}
\affil{Michigan State University}



\date{\today}

\maketitle




\textbf{Abstract}

This paper will primarily focus on a study by \cite{Shiode_2013} on how gravity modes could be excited by convection in massive main sequence stars.  The first portion of this paper will explain the more commonly understood method of how gravity modes are driven by adiabatic expansion at the core and why gravity modes produced this way are so difficult to observe. The second part of this paper will briefly go over the methods for detecting gravity modes and the observational challenges faced.  The third portion will look at models \cite{Shiode_2013} constructed using the MESA stellar evolution code that were used to estimate mode frequencies, excitation amplitudes and where in the stellar interior of various sized main sequence stars, gravity modes would propagate.  The final portion of this paper will look at future advancements in detecting gravity modes and promising observations taken from the Kepler space satellite. 

\section{Introduction}

The detection of gravity modes (g-modes) is crucial for the understanding of the solar interior.  Although some teams over the past couple decades have claimed to have detected g-modes, to date there has been no concrete evidence of g-modes being detected in main sequence stars.  Future concrete g-mode discoveries would provide researchers with a powerful tool for obtaining more detailed data on the stellar interior. P-modes do not provide enough quality information about the deep interior of stars because P-modes are typically excited by outer convective regions and typically do not penetrate deep depths in the star.  Old style adiabatic expansion driven g-modes are excited by operations near the core, but are often too dampened to for modern instruments to make meaningful observations. Gravity modes from other stellar types such as white dwarfs have been observed. \cite{Shiode_2013} showed that g-modes, which produce variations in magnitude, could potentially be detected for main sequence stars with $M > 5M[?]$ and may even be detectable in the 2-3 solar mass range \cite{Shiode_2013}.  These modeled g-modes are excited in the outer convective region of the stellar interior which is different from the g-modes previously sought after that are excited near the core driven by adiabatic expansion. G-modes propagating in the outer regions could make their instrumental detection more likely.  These g-modes may penetrate deep into the star and be able to provide large amounts of information about the interior of stars that p-modes have not been successful at. Gravity modes are difficult to detect in smaller main sequence stars (MS) around \texttildelow$1{M}_{[?]}$ due to smaller MS stars tending to have smaller convective regions and larger radiative sections in the core.  

\section{Old School G-Modes}

\subsection{Conditions for Gravity Mode Oscillations}
Unlike the more easily detectable p-modes that use pressure as the restoring force, g-modes are waves of dense material that use gravity as its restoring force.  The power these g-mode waves require to oscillate comes from adiabatic expansion in the stellar interior.  A star that could produce a gravity mode would have a core with a very poor temperature gradient.  When a hot gas packet begins to move upwards away from the hot center it starts to cool and when a gas cools it becomes denser.  This cooling gas becomes dense enough that the buoyant force that caused it to rise is now too small to overcome the gravitational force of the stellar core and the gas begins falling back towards the denser center. This makes gravity the restoring force that restores the density wave back to its starting place.  These oscillations happen deep in the stellar interior and are extremely difficult to detect.  This initial adiabatic expansion and the restoring gravitational pull from the core result in this oscillating movement of large amounts of gas in the interior, and this is what drives g-modes.  Because this process happens deep in the stars interior, the oscillations weaken dramatically as they pass through the outer convective region which acts as a dampening cushion in a similar way to how grabbing a tunning fork quickly dampens out the fork's vibrating oscillations.  Unlike p-modes, which often have much larger amplitudes at much higher frequencies making them more easily detectable, these types of g-modes will only produce small amplitudes given they are strong enough to reach the surface. 

\subsection{Observational Challenges for Detecting G-Modes}

One of the big challenges faced with detecting g-modes is lowering the upper limit for g-mode amplitudes. The upper limit defines the current possible range of a potential g-mode amplitude for a given a mode frequency.  The current predicted theoretical amplitude for a low order high frequency solar g-mode is estimated to be $~{10}^{-2}$cm/s and $~{10}^{-3}$cm/s for moderate to high order modes \cite{Kumar_1996}.  Current upper limits for g-mode amplitude are under 10 mm/s and vary based on the instrument and length of the time series.  The GOLF instrument on board the SOHO spacecraft is roughly 6.5 mm/s because its data set was taken over six years.  BiSON has calculated an even lower upper limit of 3.5 mm/s from a ground based instrument using a nine year time series.   The upper limit comes from a probability limit calculated in \cite{Appourchaux_2000}.  The upper limit is important because instrumental observations return large amounts of noise and to hastily lower the the probability limit and thus the upper limit would mean accepting more peak values that could just be solar noise and potentially missing real g-mode peaks \cite{Appourchaux_2003}. Lowering the upper limit is extremely advantageous to helping find an actual g-mode as long as it is done carefully.  A lower upper limit means a smaller range of data that must be processed and a higher likelihood of finding a g-mode.  However, lowering the upper limit has been a very slow process.  Alternatives to drastically lowering the upper limit will be discussed in the Future Observational Prospects section. 



\section{G-Mode Detection Techniques}

There are a number of methods for detecting g-modes from instrument data.  Spectrum estimators are a common technique with the Fourier spectrum being one of the most widely used estimators \cite{Appourchaux_2003}.  The Fourier transform applied to large oscillation redshift data can identify potential g-modes. Mode masking is another technique used for detecting g-modes. Masking is often used to detect modes produced by the sun.  Masking helps boost the signal-to-noise ratio and assist in masking other mode peaks that would otherwise interfere with potential g-modes \cite{Wachter_2003}.  Patterns is also a technique used on low frequency stars that takes advantage of the asymptotic predictable behavior of low frequency stars.  However low frequency g-modes have a very poor signal-to-noise ratio that makes this method difficult. One other promising method for mode detection is the use of multiple instruments \cite{Appourchaux_2003}.  This would help clean up the signal and make and modes more prominent and make noise less of an obstacle. 


\section{Convective Excitation of Gravity Modes}
\subsection{Stochastic Excitation Physics}
The main sequence stars \cite{Shiode_2013} discusses are massive main sequence stars ranging from 2M[?] to 30M[?] .  These main sequence stars have dramatically different interiors.  Unlike the sun, massive main sequence stars have a convective region surrounding their cores as a apposed to radiative region. And have a thin outer radiative layer. G-modes are predicted to propagate through this radiative layer.  Although this radiative envelope still dampens the oscillations these G-modes propagate much closer to the outer radius and are estimated to not nearly as dampened as much as the g-modes in a 1M[?] star would. This makes them prime candidates for detection.

Convection excites gravity modes in the stellar interior.  The energy supplied to these g-modes influences the variating magnitudes the g-modes release so it is important to understand how power is transferred from the convective region to the g-modes.  \cite{Shiode_2013}, using the findings of various studies, defines the power spectrum of energy supplied to g-modes as


\begin{equation}
\frac{d \dot{E_{g}} }{\textit{d} \textrm{ ln }\omega \textrm{ } d\ell} \sim {\cal M}_{{\rm c}}L_{{\rm conv}}\left(\frac{\omega
                           }{\omega _{{\rm c}}}\right)^{-a} \ell ^{b}(\frac{H}{r})^{b+1} (1 + \ell \frac{H}{r}) \exp [-(\frac{\ell
                           }{\ell _{\rm max}})^{2}]
\end{equation}

\textbf{\cite{Shiode_2013}; \cite{Press_1981}; \cite{Garcia_Lopez_1991}; \cite{Goldreich_1990}; \cite{Kumar_1999}}

Where the frequency range is $ \omega \leq \omega_{{c}} ( [?] \upsilon_{{conv}}/H )$, and the mode $  >  1$.  The convective mock number is ${\cal M}_{{\rm c}} ([?] \upsilon_{{conv}}/c_{{sound}})$ where $\upsilon_{{conv}}$ is the convective velocity.  The maximum mode $_{{max}} = (r/H)( \selectlanguage{greek}ω/ω\selectlanguage{english}_{{c}})^{3/2}$ and r is the radius from the convective zone and H is the pressure scale height.  The radius from the convective zone and the pressure scale height are roughly proportional to each other.  Their ratio is $\sim 1$ for a convective shell \cite{Fuller_2015}. 

\cite{Shiode_2013} integrates over frequency $\omega$ and mode  to arrive at $\dot{E_{g}} \sim {\cal M}_{{\rm c}}L_{{\rm conv}} $ Both sides are energy over time.  This makes sense as increasing $\upsilon_{{conv}}$ would increase the kinetic energy of the convective fluid, thus adding more energy to the gravity modes. \cite{Shiode_2013} chose to use an uncertain dimensionless constant in order to make the gravity wave power exactly $\dot{E_{g}} = {\cal M}_{{\rm c}}L_{{\rm conv}} $. This simplified exact wave power function will allow for modeling using the MESA stellar evolution code.



\section{Stellar Modeling Using MESA}
\subsection{Convective Power Spectrum}

\cite{Shiode_2013} created main sequences models ranging from ${1M}_{[?]}$through ${30M}_{[?]}$.  All the stars range from zero age to terminal age main sequence. (ZAMS - TAMS)
These spectrum graphs in Figure 1 from \cite{Shiode_2013} show the convection region for a 2M star, but with varying $\eta$.  The $\eta$ value corresponds to an unknown uncertainty of whether excitation mainly takes place in the convective region or in a thinner overshoot region that huddles near the boundary between the radiative and convective regions. A $\eta$ value of 1.0 such as on the left graph means it's assumed the convection is excited within an entire pressure scale height of the convection region.  A value $\eta$ less than 1 such as 0.1 for the right graph corresponds to a fraction of a pressure scale height of the convective region. This is a thinner more compact region where the excitation would occur called the overshoot region.  These modeled graphs show that the bulk of the wave energy that ultimately creates the gravity modes is in the lowest frequency modes as seen by the dark red coloring.  As the oscillating frequency increases. The wave energy lessens.  Similarly as the frequency increases the highest wave energy seems to be in higher and higher modes of .  The difference in $\eta $ shows that excitation occurring in a thinner overshoot region$(\eta = 0.1)$ also increases the energy input to higher modes of .  A thinner envelope has smaller convective eddy currents that can only excite with higher frequencies and higher modes of .\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=1.00\columnwidth]{figures/convective-excitation-spectrum/convective-excitation-spectrum}
\caption{{\textbf{Figure 1.} Figure 1 from \protect\cite{Shiode_2013} Convective Excitation Spectrum for a model 2M[?] MS.  Where \eta = 1.0(Left) and  \eta = 0.01(Right) models.%
}}
\end{center}
\end{figure}

\subsection{Modeled Brightness and Velocity Perturbations}
\cite{Shiode_2013} found that for models that had excitation that were not hidden and dampened by an overshoot region that was 10 percent of the pressure scale height thick and that had peak excitation frequency were able to generate brightness fluctuations of roughly 10 {\mu}mags or less.

In figure 2, taken from \cite{Shiode_2013}, we can see that the magnitude decreases with higher frequencies modes making the perturbations brighter. We can also see that star models with excited convective regions that fill the entire pressure scale height produce lower magnitudes than compared to the $<1 \eta $ which produce dimmer higher magnitude perturbations. Althought this does not apply to the younger ZAMS, the older midMS and TAMS have higher and higher mode cut off frequencies. This tells us that g-modes may be rarer with older MS stars. Combined with this, a smaller excited convective region increases the cut off frequency even further. For a TAMS with a 5 percent excited convective region, modes can not be produced unless it has a frequency of almost 10\selectlanguage{greek}μ\selectlanguage{english}Hz.

Figure 3 shows velocity perturbations per modes of different frequencies.  We can see that for convective regions that have excitation convection fill the entire pressure scale height, there is an increase in perturbation velocity.  However, for the convective regions with smaller excitation there is actually a decrease in velocity as mode frequency increases. \cite{Shiode_2013} found excitation velocity amplitudes of $\delta{\nu}_{disk} \leq  0.3mm/s$.  This velocity is consistent with the \cite{Kumar_1996}velocities discussed in second 2.2 and within the limits of observational instruments.\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=1.00\columnwidth]{figures/log-brightness-perturbations/log-brightness-perturbations}
\caption{{\textbf{Figure 2.}  Figure 5 from \protect\cite{Shiode_2013}.  The top chart's open circles correspond to mode frequencies and the bottom charts correspond to calculated ${\delta}{m}_{rms} $%
}}
\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/VelocityPerturbations/VelocityPerturbations}
\caption{{\textbf{Figure 3}. Figure 8 from \protect\cite{Shiode_2013} Velocity perturbations per g-modes of different frequencies.%
}}
\end{center}
\end{figure}

\section{Future Observational Prospects}
\subsection{Lowering The Upper Limit Through Gravity Waves}
A hopeful prospect for the detection of g-modes is through gravitational wave detection.  The ESA/NASA mission LISA could potentially have lowered the upper limit by a few mm/s in only a year of observing time\cite{Appourchaux_2003}. LISA would have been a gravitational wave telescope and would have been able to detect gravity waves coming from the oscillations deep within the sun's interior.  However, the mission was canceled due to lack of funding, but a similar mission by the ESA called ASTROD may still be underway.  It would operate with two spacecrafts located on the far side of the sun in Earth's orbit and use similar equipment to that of mission LISA to detect gravitational waves.  This would dramatically lower the upper amplitude detection limit.

\subsection{Kepler Observations}
    Recent analysis of Kepler satellite data by \cite{Balona_2011} have shown low frequency modes in slowly pulsating B-type(SPB) stars. \cite{Balona_2011} claims that seven stars in their study show evidence of g-modes.  Figure 4 shows one of those seven, KIC 10960750. The figure shows the majority of frequencies below 3 ${d}^{-1}$ which are in the accepted range of for a g mode. Even though the majority of the detected amplitudes do not exceed 700 ppm, the Kepler data used was taken over 320 days, which allowed much of the noise to be factored out.  The current noise level is only several ppm \cite{Balona_2011}.  These distinct modes being primarily low frequency fit well with the convective excitation spectrum discussed in section 4.2.  This demonstrates strong evidence for an observationally detected g mode.  \cite{Balona_2011} also claims that some of the higher frequency modes, like the one at 10.449 ${d}^{-1}$ for KIC 10960750 in Figure 4, may actually be g modes themselves and that the star's rotational frequency may be leading to scattering low frequency modes into frequencies that appear higher to the observer \cite{Balona_2011}.\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=1.00\columnwidth]{figures/KIC-10960750-g-modes/KIC-10960750-g-modes}
\caption{{\textbf{Figure 4.} Part of Figure 6 from \protect\cite{Balona_2011} shows pulsations of different frequencies in parts per million of the B type star KIC 10960750.  The left side show the original data and the right side shows a zoomed look of the data with some terms subtracted.%
}}
\end{center}
\end{figure}

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