The Direct Effect of Toroidal Magnetic Fields on Stellar Oscillations: An Analytical Expression for the General Matrix Element

Where is the solar dynamo located and what is its modus operandi? These are still open questions in solar physics. Helio- and asteroseismology can help answer them by enabling us to study solar and stellar internal structures through global oscillations. The properties of solar and stellar acoustic modes are changing with the level of magnetic activity. However, until now, the inference on subsurface magnetic fields with seismic measures has been very limited. The aim of this paper is to develop a formalism to calculate the effect of large-scale toroidal magnetic fields on solar and stellar global oscillation eigenfunctions and eigenfrequencies. If the Lorentz force is added to the equilibrium equation of motion, stellar eigenmodes can couple. In quasi-degenerate perturbation theory, this coupling, also known as the direct effect, can be quantified by the general matrix element. We present the analytical expression of the matrix element for a superposition of subsurface zonal toroidal magnetic field configurations. The matrix element is important for forward calculations of perturbed solar and stellar eigenfunctions and frequency perturbations. The results presented here will help to ascertain solar and stellar large-scale subsurface magnetic fields, and their geometric configuration, strength, and their change over the course of activity cycles.Comment: 20 pages, accepted for publication in The Astrophysical Journa

Determination of fundamental asteroseismic parameters using the Hilbert transform

Context. Solar-like oscillations exhibit a regular pattern of frequencies. This pattern is dominated by the small and large frequency separations between modes. The accurate determination of these parameters is of great interest, because they give information about e.g. the evolutionary state and the mass of a star. Aims. We want to develop a robust method to determine the large and small frequency separations for time series with low signal-tonoise ratio. For this purpose, we analyse a time series of the Sun from the GOLF instrument aboard SOHO and a time series of the star KIC 5184732 from the NASA Kepler satellite by employing a combination of Fourier and Hilbert transform. Methods. We use the analytic signal of filtered stellar oscillation time series to compute the signal envelope. Spectral analysis of the signal envelope then reveals frequency differences of dominant modes in the periodogram of the stellar time series. Results. With the described method the large frequency separation $\Delta\nu$ can be extracted from the envelope spectrum even for data of poor signal-to-noise ratio. A modification of the method allows for an overview of the regularities in the periodogram of the time series.Comment: 7 pages, 7 figures, 2 tables, submitted to A&

Stellar magnetic activity and variability of oscillation parameters - An investigation of 24 solar-like stars observed by Kepler

Context. The Sun and solar-like stars undergo activity cycles for which the underlying mechanisms are not well understood. The oscillations of the Sun are known to vary with its activity cycle and these changes provide diagnostics on the conditions below the photosphere. Kepler has detected oscillations in hundreds of solar-like stars but as of yet, no widespread detection of signatures of magnetic activity cycles in the oscillation parameters of these stars have been reported. Aims. We analyse the photometric short cadence Kepler time series of a set of 24 solar-like stars, which were observed for at least 960 days each, with the aim to find signatures of stellar magnetic activity in the oscillation parameters. Methods. We analyse the temporal evolution of oscillation parameters by measuring mode frequency shifts, changes in the height of the p-mode envelope, as well as granulation time scales. Results. For 23 of the 24 investigated stars, we find significant frequency shifts in time. We present evidence for magnetic activity in six of them. We find that the amplitude of the frequency shifts decreases with stellar age and rotation period. For the most prominent example, KIC 8006161, we find that, similar to the solar case, frequency shifts are smallest for the lowest and largest for the highest p-mode frequencies. Conclusions. These findings show that magnetic activity can be routinely observed in the oscillation parameters for solar-like stars, which opens up the possibility to place the solar activity cycle in the context of other stars by asteroseismology.Comment: 25 pages, 34 figures. Accepted for publication in Astronomy & Astrophysic

The Effect of Toroidal Magnetic Fields on Solar Oscillation Frequencies

Solar oscillation frequencies change with the level of magnetic activity. Localizing subsurface magnetic field concentrations in the Sun with helioseismology will help us to understand the solar dynamo. Because the magnetic fields are not considered in standard solar models, adding them to the basic equations of stellar structure changes the eigenfunctions and eigenfrequencies. We use quasi-degenerate perturbation theory to calculate the effect of toroidal magnetic fields on solar oscillation mean multiplet frequencies for six field configurations. In our calculations, we consider both the direct effect of the magnetic field, which describes the coupling of modes, and the indirect effect, which accounts for changes in stellar structure due to the magnetic field. We limit our calculations to self-coupling of modes. We find that the magnetic field affects the multiplet frequencies in a way that depends on the location and the geometry of the field inside the Sun. Comparing our theoretical results with observed shifts, we find that strong tachocline fields cannot be responsible for the observed frequency shifts of p modes over the solar cycle. We also find that part of the surface effect in helioseismic oscillation frequencies might be attributed to magnetic fields in the outer layers of the Sun. The theory presented here is also applicable to models of solar-like stars and their oscillation frequencies

GONG p-mode parameters through two solar cycles

We investigate the parameters of global solar p-mode oscillations, namely damping width $\Gamma$, amplitude $A$, mean squared velocity $\langle v^2\rangle$, energy $E$, and energy supply rate $\mathrm{d}E/\mathrm{d} t$, derived from two solar cycles' worth (1996 - 2018) of Global Oscillation Network Group (GONG) time series for harmonic degrees $l=0 - 150$. We correct for the effect of fill factor, apparent solar radius, and spurious jumps in the mode amplitudes. We find that the amplitude of the activity related changes of $\Gamma$ and $A$ depends on both frequency and harmonic degree of the modes, with the largest variations of $\Gamma$ for modes with $2400\,\mu\text{Hz}\le \nu \le 3300\,\mu\text{Hz}$ and $31\le l \le 60$ with a min-to-max variation of $26.6\pm0.3\%$ and of $A$ for modes with $2400\,\mu\text{Hz}\le\nu\le 3300\,\mu\text{Hz}$ and $61\le l \le 100$ with a min-to-max variation of $27.4\pm0.4\%$. The level of correlation between the solar radio flux $F_{10.7}$ and mode parameters also depends on mode frequency and harmonic degree. As a function of mode frequency, the mode amplitudes are found to follow an asymmetric Voigt profile with $\nu_{\text{max}}=3073.59\pm0.18\,\mu\text{Hz}$. From the mode parameters, we calculate physical mode quantities and average them over specific mode frequency ranges. This way, we find that the mean squared velocities $\langle v^2\rangle$ and energies $E$ of p modes are anti-correlated with the level of activity, varying by $14.7\pm0.3\%$ and $18.4\pm0.3\%$, respectively, and that the mode energy supply rates show no significant correlation with activity. With this study we expand previously published results on the temporal variation of solar p-mode parameters. Our results will be helpful to future studies of the excitation and damping of p modes, i.e., the interplay between convection, magnetic field, and resonant acoustic oscillations.Comment: Accepted for publication in Solar Physics. 33 pages, 16 figures, 5 table

Seismic signatures of stellar magnetic activity — what can we expect from TESS?

Asteroseismic methods offer a means to investigate stellar activity and activity cycles as well as to identify those properties of stars which are crucial for the operation of stellar dynamos. With data from CoRoT and Kepler, signatures of magnetic activity have been found in the seismic properties of a few dozen stars. Now, NASA's Transiting Exoplanet Survey Satellite (TESS) mission offers the possibility to expand this, so far, rather exclusive group of stars. This promises to deliver new insight into the parameters that govern stellar magnetic activity as a function of stellar mass, age, and rotation rate. We derive a new scaling relation for the amplitude of the activity-related acoustic (p-mode) frequency shifts that can be expected over a full stellar cycle. Building on a catalog of synthetic TESS time series, we use the shifts obtained from this relation and simulate the yield of detectable frequency shifts in an extended TESS mission. We find that, according to our scaling relation, we can expect to find significant p-mode frequency shifts for a couple hundred main-sequence and early subgiant stars and for a few thousand late subgiant and low-luminosity red giant stars

Empirical relations for the sensitivities of solar-like oscillations to magnetic perturbations

Oscillation mode frequencies of stars are typically treated as static for a given stellar model. However, in reality they can be perturbed by time varying sources such as magnetic fields and flows. We calculate the sensitivities of radial p-mode oscillations of a set of models for masses between 0.7–3.0M⊙ from the main sequence to the early asymptotic giant branch. We fit these mode sensitivities with polynomials in fundamental stellar parameters for six stages of stellar evolution. We find that the bestfitting relations differ from thos proposed in the literature and change between stages of stellar evolution. Together with a measure of the strength of the perturbation, e.g., of the level of magnetic activity, the presented relations can be used for assessing whether a star’s observed oscillation frequencies are likely to be close to the unperturbed ground state or whether they should be adjusted

They do change after all : 25 years of GONG Data reveal variation of p-mode energy supply rates

It has been shown over and over again that the parameters of solar p modes vary through the solar activity cycle: frequencies, amplitudes, lifetimes, energies. However, so far, the rates at which energy is supplied to the p modes have not been detected to be sensitive to the level of magnetic activity. We set out to re-inspect their temporal behaviour over the course of the last two Schwabe cycles. For this, we use Global Oscillation Network Group (GONG) p-mode parameter tables. We analyse the energy supply rates for modes of harmonic degrees l = 0-150 and average over the azimuthal orders and, subsequently, over modes in different parameter ranges. This averaging greatly helps in reducing the noise in the data. We find that energy supply rates are anticorrelated with the level of solar activity, for which we use the F10.7 index as a proxy. Modes of different mode frequency and harmonic degrees show varying strengths of anticorrelation with the F10.7 index, reaching as low as r = -0.82 for low frequency modes with l = 101-150. In this first dedicated study of solar p-mode energy supply rates in GONG data, we find that they do indeed vary through the solar cycle. Earlier investigations with data from other instruments were hindered by being limited to low harmonic degrees or by the data sets being too short. We provide tables of time-averaged energy supply rates for individual modes as well as for averages over disjunct frequency bins

Cycle dependence of a quasi-biennial variability in the solar interior

We investigated the solar cycle dependence on the presence and periodicity of the Quasi-Biennial Oscillation (QBO). Using helioseismic techniques, we used solar oscillation frequencies from the Global Oscillations Network Group (GONG), Michelson Doppler Imager (MDI), and Helioseismic and Magnetic Imager (HMI) in the intermediate-degree range to investigate the frequency shifts over Cycles 23 and 24. We also examined two solar activity proxies, the F10.7 index and the Mg ii index, for the last four solar cycles to study the associated QBO. The analyses were performed using Empirical Mode Decomposition (EMD) and the Fast Fourier Transform (FFT). We found that the EMD analysis method is susceptible to detecting statistically significant Intrinsic Mode Functions (IMFs) with periodicities that are overtones of the length of the data set under examination. Statistically significant periodicities, which were not due to overtones, were detected in the QBO range. We see a reduced presence of the QBO in Cycle 24 compared to Cycle 23. The presence of the QBO was not sensitive to the depth to which the p-mode travelled, nor the average frequency of the p-mode. The analysis further suggested that the magnetic field responsible for producing the QBO in frequency shifts of p-modes is anchored above approximately 0.95 R⊙

The PAC2MAN mission: a new tool to understand and predict solar energetic events

An accurate forecast of flare and CME initiation requires precise measurements of the magnetic energy build up and release in the active regions of the solar atmosphere. We designed a new space weather mission that performs such measurements using new optical instruments based on the Hanle and Zeeman effects. The mission consists of two satellites, one orbiting the L1 Lagrangian point (Spacecraft Earth, SCE) and the second in heliocentric orbit at 1AU trailing the Earth by 80$^\circ$ (Spacecraft 80, SC80). Optical instruments measure the vector magnetic field in multiple layers of the solar atmosphere. The orbits of the spacecraft allow for a continuous imaging of nearly 73\% of the total solar surface. In-situ plasma instruments detect solar wind conditions at 1AU and ahead of our planet. Earth directed CMEs can be tracked using the stereoscopic view of the spacecraft and the strategic placement of the SC80 satellite. Forecasting of geoeffective space weather events is possible thanks to an accurate surveillance of the magnetic energy build up in the Sun, an optical tracking through the interplanetary space, and in-situ measurements of the near-Earth environment.Comment: Accepted for publication in the Journal of Space Weather and Space Climate (SWSC