Probing the interior physics of stars through asteroseismology

Yearslong time series of high-precision brightness measurements have been assembled for thousands of stars with telescopes operating in space. Such data have allowed astronomers to measure the physics of stellar interiors via nonradial oscillations, opening a new avenue to study the stars in the Universe. Asteroseismology, the interpretation of the characteristics of oscillation modes in terms of the physical properties of the stellar interior, brought entirely new insights in how stars rotate and how they build up their chemistry throughout their evolution. Data-driven space asteroseismology has delivered a drastic increase in the reliability of computer models mimicking the evolution of stars born with a variety of masses and metallicities. Such models are critical ingredients for modern physics as a whole because they are used throughout various contemporary and multidisciplinary research fields in space science, including the search for life outside the Solar System, archaeological studies of the Milky Way, and the study of single and binary supernova progenitors, among which are future gravitational wave sources. The specific role and potential of asteroseismology for those modern research fields are illustrated. The review concludes with current limitations of asteroseismology and highlights how they can be overcome with ongoing and future large infrastructures for survey astronomy combined with new theoretical research in the era of high-performance computing. This review presents results obtained through major community efforts over the past decade. These breakthroughs were achieved in a collaborative and inclusive spirit that is characteristic of the asteroseismology community. The review’s aim is to make this research field accessible to graduate students and readers coming from other fields of physics, with incentives to enjoy and join future applications in this domain of astrophysics.

Figure 1

Hertzsprung-Russel diagram (HRD) showing the position of different classes of pulsating stars. The abbreviation of the classes follows the nomenclature used by [2] in Chap. 2, to which we refer for extensive discussions of all indicated classes in terms of the excitation mechanisms, along with the typical periods and amplitudes of the oscillations. The hatching line style used in each of the ellipses marks the dominant type of oscillation mode in each class: $//$ for gravity modes and $\backslash \backslash$ for pressure modes. The recently discovered stochastic low-frequency (SLF) variability in O-type stars and blue supergiants is discussed in the text and has been added as a compararison with previous versions of this plot. The solid black lines and the black dotted line represent standard evolutionary model tracks, with birth masses and evolutionary timescales as indicated. The borders of the classical instability strip are plotted with gray lines, while the double line represents the zero-age main sequence (ZAMS). Early versions of this figure were made by Jørgen Christensen-Dalsgaard (Aarhus University) and Pieter Degroote (KU Leuven). Adapted from [423].

Figure 2

Snapshot of the angular dependence of the radial component of the displacement vector ${\xi }_r$ at one point in the oscillation cycle for various nonradial modes, seen under an inclination angle of 60°. White bands indicate the positions where ${\xi }_r=0$; red and blue represent areas at the stellar surface moving in (out) at the chosen time. Shown from left to right are first row, axisymmetric ($m=0$) modes with $l=1,2,3$; second row, sectoral ($l=|m|$) modes with $l=1,2,3$; third row, tesseral ($l\ne |m|$) modes with $(l,|m|)=(3,1),(6,4),(15,5)$. High-degree modes such as the two in the third row are usually not detected in space photometry due to cancellation effects when integrating the flux variations across the visible stellar disk.

Figure 3

Excerpts of 110 d duration (of the total $\sim 1500\mathrm{d}$) extracted from the Kepler long-cadence ($\sim 30\hspace{0.17em}\hspace{0.17em}\min $ per point) light curves (in black dots) of seven slowly pulsating B stars (cf. Fig. 1) indicated with their Kepler input catalog identification (KIC) ([96]). The amplitude spectrum obtained from a Fourier transform of the full Kepler light curves is overplotted in red. These stars exhibit nonradial gravity modes with individual mode periods of the order of 1 d. The light curves reveal a gallery of diverse beating patterns among the modes and a gradual shift in maximum amplitude from low to high frequency as their rotation frequency changes from low in the top panel to high in the bottom panel. Despite their large amplitudes of several to $\sim 12\hspace{0.17em}\mathrm{ppt}$, none of these stars were known to have nonradial oscillations prior to the Kepler mission; it is notoriously difficult to detect modes with such periodicities from ground-based data given their similarity with the rotation period of Earth. Adapted from [427].

Figure 4

Part of the Kepler light curve (top left panel) and the power density spectrum (bottom left panel, black) of the red-giant star KIC 007949599. Overplotted in the bottom left panel are the model fits from Table 1 given by [298], based on their Eq. (2), representing two Lorentzian components due to granulation variability at low frequencies (dashed red and dot-dashed purple lines) and a Gaussian component due to stochastic $p$-mode oscillations (dotted yellow line). K2 light curve (top right panel) and its power density spectrum (bottom right panel, black) of the blue supergiant star $\rho$ Leo. Overplotted in the bottom right is a Lorentzian model fit (dashed red line) using the formalism given by [84] to represent the power excess due to the low-frequency stochastic variability. In both bottom panels, the blue dashed horizontal lines indicate the photon noise level and the solid green lines represent the superposition of the individual model components representing the variability. Adapted from [298], [8], and [85].

Figure 5

Shifts in the periods of dipole ($l=1$) triplet g modes of two equilibrium models [blue (orange), without (with) atomic diffusion, including radiative levitation] with the same input physics for parameters $M=1.7{\mathit{M}}_{\odot }$, $X_{\mathrm{ini}}=0.7154$, $Z_{\mathrm{ini}}=0.022$, and at two different evolutionary stages expressed in terms of the central hydrogen fraction $X_c$. The g modes were computed assuming rigid rotation with a period of 14.4 d. The radial orders $n$ are labeled. The vertical line indicated on the left side in the upper panel denotes the measurement uncertainty for such mode periods from a 4-yr nominal Kepler light curve. Adapted from [383].

Figure 6

Left panel: core (near-core) rotation rates derived from mixed (gravity) modes for stars in core-hydrogen burning (purple and gray), hydrogen-shell burning (pink and red), and core-helium burning (after the helium flash in orange and avoiding the helium flash in cyan). The circles indicate stars with asteroseismic estimates of ${\mathrm{\Omega }}_{\mathrm{core}}$ and $\log g$ taken from [4]; their errors are smaller than the symbol size. The squares are additional stars with asteroseismic determinations for ${\mathrm{\Omega }}_{\mathrm{core}}$ but with less reliable values for $\log g$ from spectroscopy and/or stellar models from [338] (622 $\gamma$ Dor stars, indicated in gray) and from [528] (72 core-helium-burning red giants in blue). Their uncertainties for $\log g$ range from 0.2 (blue squares) to 0.5 dex (gray squares) and are omitted for clarity. Right panel: all stars with an additional measurement of the envelope (from $p$ modes) or surface (from rotational modulation) rotation frequency. Adapted from [4].

Figure 7

Schematic representation of mixing profiles due to various transport processes in stars with a convective core (indicated in gray) and a radiative envelope for exponentially decaying diffusive core overshooting (upper panels) and convective penetration (lower panels) as CBM (purple). Four types of envelope mixing based on different theoretical frameworks are considered (pink), as labeled. Based on envelope mixing profiles for $5M_{\odot }$ and $3M_{\odot }$ ZAMS models computed by [224] and [465], respectively. Adapted from [434].

Figure 8

Propagation diagrams showing the mode cavities of axisymmetric p and g modes in four stellar models halfway through the core-hydrogen-burning stage of evolution. The models have masses of $1{\mathit{M}}_{\odot }$, $1.7{\mathit{M}}_{\odot }$, $5{\mathit{M}}_{\odot }$, and $15{\mathit{M}}_{\odot }$ from top left to bottom right. The thick solid black line indicates $N(r)$, while the dotted black and gray lines represent $S_1(r)$ and $S_2(r)$, respectively. The values of the dipole (quadrupole) mode frequencies are indicated as black (gray) horizontal lines. The position of the nodes of ${\xi }_r$ are indicated as thick black and gray dots for $l=1$ and 2, respectively. The red region is the g-mode cavity for dipole modes; it is extended by the orange part for quadrupole modes. The dark blue region is the mode cavity of quadrupole ($l=2$) p modes. It is extended by the light blue region for dipole ($l=1$) p modes. The modes correspond to evanescent waves in the white regions in the stellar envelope. The g modes cannot propagate in the convective core of the three most massive stellar models, nor in the outer $\sim 26\%$ of the convective envelope of the $1{\mathit{M}}_{\odot }$ model, where $N^2(r)<0$. Partially based on models from [384].

Figure 9

Radial (left panels) and horizontal components (middle panels) of the Lagrangian displacement of four indicated axisymmetric ($m=0$) p and g modes for $l=1$ (top panels) and $l=2$ (bottom panels) of a stellar model with $M_{\star }=1.7{\mathit{M}}_{\odot }$ halfway through its core-hydrogen-burning stage of evolution. The right panels show the rotation kernel defined by Eq. (45), which represents the probing power of an oscillation mode. Partially based on pulsation computations from [383].

Figure 10

Enlargement (red) of the observed envelope of oscillation signal revealed by the power density spectrum (black) of the solar analog 16 Cyg A as deduced from data assembled with the Kepler satellite. The $p$ modes are labeled by their degree $l$. The large frequency separation based on the detected radial mode frequencies is indicated. Adapted from [116].

Figure 11

Top panel: observed amplitude spectrum (black) in terms of the period for the $\gamma$ Dor star KIC 11721304 from its light curve observed with the Kepler satellite. The mode periods with dominant amplitude are indicated with red dashed vertical lines as a guide for the eye. Bottom panel: the period-spacing pattern deduced from the dipole sectoral prograde modes of consecutive radial order $n$ indicated in the top panel. Adapted from [559].

Figure 12

Same as Fig. 11, but for the $\gamma$ Dor star KIC 12066947 exhibiting both prograde dipole gravitoinertial modes with $(k,m)=(0,+1)$ and retrograde Rossby modes with $(k,m)=(-2,-1)$. In contrast to the case of KIC 11721304 shown in Fig. 11, the errors in the period-spacing pattern are smaller than the symbol size. Adapted from [559].

Figure 13

Same as Fig. 11, but for the $\gamma$ Dor star KIC 6425437 exhibiting retrograde Yanai modes with $(k,m)=(-1,-1)$. In contrast to the case of KIC 11721304 shown in Fig. 11, the errors in the period-spacing pattern are smaller than the symbol size. Adapted from [559].

Figure 14

Snapshot of 3D hydrodynamical simulations representing the temperature fluctuations induced by IGWs excited by stochastic forcing at the transition layer between the convective core and the bottom of the radiative envelope of a $3{\mathit{M}}_{\odot }$ ZAMS star. The color coding represents fluctuations up to ${10}^5\mathrm{K}$ with respect to an equilibrium model. Adapted from [197].

Figure 15

Excerpt of the Kepler light curve (top panel), phase folded according to the orbital frequency (middle panel), and LS amplitude spectrum (bottom panel) of the eccentric eclipsing binary KIC 8112039. All but two frequencies (not indicated by a thin red vertical line) in the bottom panel are caused by tidal excitation. Adapted from [577].

Figure 16

Schematic representation of the procedure of asteroseismic modeling for an ensemble of stars that summarizes the steps discussed in Sec. 3c. See the text for the meaning of the notation. The green boxes involve statistical methods in the topics of maximum likelihood estimation, pattern recognition, and model selection.

Figure 17

Planetary radii as a function of orbital period, where the properties of exoplanet host stars with and without asteroseismic estimation are compared. Asteroseismology of the host star not only provides the age of the exoplanetary system but also improves the planetary radii by a factor of $\sim 2$ compared to the case where such data are not available. Adapted from [127].

Figure 18

Frequency deviations with respect to a fourth-order polynomial fit to the measured frequencies for the modes of degree $l=0$ (blue triangles), $l=1$ (gray squares), $l=2$ (red circles), and $l=3$ (orange diamonds) of 16 Cyg A, based on the PD shown in Fig. 10. The full line represents a fit to the oscillatory signal caused by sharp features in the star’s sound speed in the second ionization zone of helium. Adapted from [562].

Figure 19

Excerpt of the Kepler light curve (top panel), phase folded according to an orbital period of 14 d (middle panel), and the corresponding LS amplitude spectrum (bottom panel) of KIC 4142768. This eclipsing binary reveals self-excited $\kappa$-driven p modes and tidally excited g modes occurring at exact multiples of the orbital frequency (indicated in red). Adapted from [245].