Ray Tracing

\label{ray}

Additional understanding of magneto-gravity waves can be gained using a ray tracing technique. This process allows us to explicitly follow the time evolution of a wave as it propagates into a region of increasing magnetic field. We follow the basic technique outlined in \cite{Ligni_res_2009}. In the case of magneto-gravity waves in the WKB limit, the Hamiltonian describing their equations of motion is

\[\label{eqn:hamiltonian} H = \omega = \sqrt{ \frac{{\bf k_\perp}^2 N^2}{{\bf k}^2} + ({\bf k} \cdot {\bf v_A})^2 } \, .\]

In reality, the Hamiltonian contains additional terms that allow for the existence of pure Alfvén waves, although we neglect this subtlety here.

The equations of motion corresponding to the Hamiltonian of Eqn. \ref{eqn:hamiltonian} are

\[\label{eqn:dxdt} \frac{ d{\bf x}}{dt} = \frac{ \partial H}{\partial {\bf k}} = \frac{N^2}{\omega k} \bigg[ \bigg( 1 - \frac{k_\perp^2}{k^2} \bigg) \frac{{\bf k_\perp}}{k} - \frac{k_\perp^2}{k^2} \frac{k_r {\bf \hat{r}}}{k} \bigg] + \frac{\omega_A}{\omega} {\bf v_A} \, ,\]

where \(\omega_A = ({\bf k} \cdot {\bf v_A})\), and

\[\label{eqn:dkdt} \frac{ d{\bf k}}{dt} = - \frac{ \partial H}{\partial {\bf x}} = - \frac{N}{\omega} \frac{k_\perp^2}{k^2} \nabla N - \frac{\omega_A}{\omega} \nabla \big({\bf k} \cdot {\bf v_A} \big) \, .\]

Eqn. \ref{eqn:dxdt} describes the group velocity of the wave, while Eqn. \ref{eqn:dkdt} describes the evolution of its wave vector, which is related to the momentum of the wave. Note that in the absence of a magnetic field in a spherical star, only the radial component of the wave vector changes, and the horizontal component is conserved. This is not surprising because the Hamiltonian is spherically symmetric and thus angular momentum (and hence angular wave vector) is conserved.

However, in the presence of a magnetic field, the last term of Eqn. \ref{eqn:dkdt} breaks the spherical symmetry. Except in the unphysical case of a purely radial field or a constant field, this term is non-zero, and therefore the angular component of the wave vector must change. At the radius \(r_{\rm MG}\) where \(v_A \! \sim \! \omega^2/(N k_\perp)\), each term in Eqn. \ref{eqn:dkdt} is the same order of magnitude, assuming \(|\nabla B|/B \! \sim \! 1/r\). Therefore, the rate of change in horizontal wavenumber is comparable to the rate of change in radial wavenumber at field strengths near \(B_c\). Upon wave reflection or conversion into Alfvén waves, the radial wavenumber will generally change by order unity, i.e., the change in radial wavenumber is \(|\Delta k_r| \! \sim \! |k_r|\). We therefore expect a correspondingly large change in \(k_\perp\), such that \(|\Delta k_\perp| \! \sim \! |k_r|\). Hence, dipole waves will generally obtain high multipole moments when they propagate through strongly magnetized regions of the star.

Joule Damping

A gravity wave propagating through a magnetized fluid induces currents which dissipate in a non-perfectly conducting fluid, causing the wave to damp. For gravity waves in the WKB limit which are not strongly altered by magnetic tension forces, the perturbed radial magnetic field is \(\delta B \! \approx \! \xi_\perp k_r B\), where \(\xi_\perp\) is the horizontal wave displacement. The perturbed current density is \(\delta J \! \approx \! c k_r \delta B/(4 \pi)\), where \(c\) is the speed of light. The volumetric energy dissipation rate is \(\dot{\varepsilon} \! \approx \! (\delta J)^2/\sigma\), where \(\sigma\) is the electrical conductivity. The gravity wave energy density is \(\varepsilon \! \approx \! \rho \omega^2 \xi_\perp^2\), so the local damping rate is

\[\label{eqn:joule} \Gamma_B = \frac{\dot{\varepsilon}}{\varepsilon} \approx \frac{\eta B^2 k_r^4}{(4 \pi)^2 \rho \omega^2} \, ,\]

where \(\eta \! = \! c^2/\sigma\) is the magnetic diffusivity.

The Joule damping rate of Eqn. \ref{eqn:joule} can be compared with the damping rate from radiative diffusion (in the absence of composition gradients), \(\Gamma_r \! = \! k_r^2 \kappa\), where \(\kappa\) is the thermal diffusivity. The ratio of Joule damping to thermal damping is

\[\label{eqn:jouleratio} \frac{\Gamma_B}{\Gamma_r} = \frac{\eta}{\kappa} \frac{B^2 k_r^2}{(4 \pi)^2 \rho \omega^2} = \frac{\eta}{\kappa} \frac{l(l+1) B^2 N^2}{(4 \pi)^2 \rho r^2 \omega^4} \, ,\]

and the second equality follows from using the gravity wave dispersion relation. The maximum magnetic field possible before Lorentz forces strongly alter gravity waves is \(B_c\) (Eqn. \ref{eqn:Bc}), and putting this value into Eqn. \ref{eqn:jouleratio} we find

\[\label{eqn:jouleratio2} \frac{\Gamma_B}{\Gamma_r} = \frac{1}{16 \pi} \frac{\eta}{\kappa} \, .\]

Therefore, for gravity waves, Joule damping cannot exceed thermal damping unless the magnetic diffusivity is significantly larger than the thermal diffusivity. In stellar interiors (and our RGB models), the magnetic diffusivity is typically orders of magnitude smaller than the thermal diffusivity. Therefore Joule damping can safely be ignored. We note that the same result occurs if we use the Alfvén wave dispersion relation in Eqn. \ref{eqn:jouleratio}, so Joule damping is also unimportant for Alfvén waves.

Measurements and Uncertainties

Most of the observational data shown in Fig. \ref{fig:moneyplot} were obtained from \cite{Mosser_2012}. The additional stars KIC 8561221 and KIC 9073950 were analyzed using the same methods as \cite{Mosser_2012}. This analysis provided measured values of dipole mode visibility \(V^2\), \(\nu_{\rm max}\), \(\Delta \nu\), and their associated uncertainties. For KIC9073950, we used the updated KIC \(T_{\rm eff}\) \cite{2014ApJS..211....2H} to calculate mass and its uncertainty from scaling relations. For KIC8561221, mass and uncertainties were obtained from \cite{Garcia_2014}. To calculate values of \(B_c\) for KIC8561221 and KIC9073950, we interpolated in \(\log B_c\) between the tracks shown in Fig. \ref{fig:Bc}, using the measured stellar masses. The uncertainty in \(B_c\) was obtained by performing the same interpolation on the upper and lower bounds of the stellar mass.

\(\varepsilon\) Ophiuchi

The red giant \(\varepsilon\) Ophiuchi, extensively observed with ground-based instruments \cite{De_Ridder_2006} and with the MOST satellite \cite{Barban_2007}, may also exhibit depressed dipole modes. Its temperature of \(\approx 4900 \, {\rm K}\), inferred mass of \(1.85 \pm 0.05 \, M_\odot\) and interferometricly measured radius of \(10.39 \pm 0.07 \, R_\odot\) \cite{Mazumdar_2009} yield \(\nu_{\rm max} \approx 57 \, \mu{\rm Hz}\) and \(\Delta \nu \approx 5.5 \, \mu{\rm Hz}\). This is consistent with the interpretation \cite{De_Ridder_2006,Barban_2007,Mazumdar_2009} that many of the peaks in its MOST power spectrum belong to a series of radial oscillation modes. However, we agree with \cite{Kallinger_2008} that the most likely explanation for the power spectrum is that it is created by a combination of both radial and non-radial modes.

We speculate that the low amplitude and missing dipole modes can be explained if \(\varepsilon\) Ophiuchi is a depressed dipole mode star. At this stage of evolution, we expect the normalized depressed dipole mode power \(V^2\) and lifetime \(\tau\) to be roughly half their normal values. The measured lifetimes of \(\tau \sim 12 \, {\rm days}\) \cite{Kallinger_2008} are dominated by radial and envelope-dominated quadrupole modes, and are consistent with the usual lifetimes of these modes in red giants at this stage of evolution. A more robust conclusion would require a comparison of measured radial mode line widths to dipole mode line widths, and our scenario would predict that the dipole modes should have lifetimes of \(\tau \sim 6 \, {\rm days}\). We suspect that overlapping radial and quadrupole modes may help explain the large line widths found by \cite{Barban_2007}, who considered the peaks to be produced solely by radial modes.