Stars are not perfectly spherically symmetric. They are deformed by rotation and magnetic fields. Until now, the study of stellar shapes has only been possible with optical interferometry for a few of the fastest-rotating nearby stars. We report an asteroseismic measurement, with much better precision than interferometry, of the asphericity of an A-type star with a rotation period of 100 days. Using the fact that different modes of oscillation probe different stellar latitudes, we infer a tiny but significant flattening of the star’s shape of ΔR/R = (1.8 ± 0.6) × 10 −6 . For a stellar radius R that is 2.24 times the solar radius, the difference in radius between the equator and the poles is ΔR = 3 ± 1 km. Because the observed ΔR/R is only one-third of the expected rotational oblateness, we conjecture the presence of a weak magnetic field on a star that does not have an extended convective envelope. This calls to question the origin of the magnetic field.

Keywords

According to Clairaut’s theorem, slowly rotating stars are oblate spheroids ( 1 , 2 ). Other factors may affect the shapes of stars, such as magnetic fields, thermal asphericities, large-scale flows, or strong stellar winds. Global poloidal magnetic fields tend to make stars oblate, whereas toroidal magnetic fields tend to make them prolate ( 3 , 4 ). The tidal interaction of a star with a close stellar companion or a giant planet is yet another cause of stellar deformation ( 5 ). Thus, measuring the asphericity of stars can place constraints on a wide range of phenomena beyond the standard model of stellar structure and evolution. Direct imaging of the deformed shapes of nearby stars requires a resolution better than a milli–arc second. The elongated projected shape of the rapidly rotating A star Altair has been observed with infrared interferometry to have a flattening ΔR/R = 0.14 ± 0.03 ( 6 , 7 ). Vega, another rapidly rotating A star, has an apparent deformation that is too small to be measured because it is seen almost pole-on ( 8 ). Here, we present with unprecedented precision the first measurement of stellar asphericity by means of asteroseismology ( 9 ), for the star KIC 11145123, which has an equatorial velocity two orders of magnitude smaller than either Altair or Vega. This work is motivated by helioseismology’s ability to probe the Sun’s asphericities and their temporal variations with the 11-year solar magnetic cycle ( 10 , 11 ).

RESULTS

The star KIC 11145123 belongs to the class of hybrid pulsators (12). It oscillates both in a high-frequency band (15 to 25 day−1) and in a low-frequency band (below 5 day−1). The observed modes of oscillation are acoustic (p), gravity (g), and mixed (p and g) modes. Modes with dominant p-mode character are seen in the high-frequency band. These modes oscillate throughout most of the star, with larger oscillation amplitudes near the surface. They are labeled with the radial order n, which counts the number of nodes in the radial direction with a positive sign for nodes in the p-mode cavity and a negative sign for nodes in the g-mode cavity. Most known hybrid pulsators, including KIC 11145123, belong to the γ Doradus–δ Scuti class (13). Oscillations in these stars are likely to be excited by the opacity (p and mixed modes) and the convective-blocking (g modes) mechanisms.

Oscillations of KIC 11145123 were observed in intensity over a period of T = 3.94 years by Kepler (14). Because the oscillations are purely harmonic, random errors in the inferred mode frequencies scale as T−3/2 times the noise-to-signal ratio of the periodic oscillations (15), and thus, the mode frequencies can be determined with astounding precision. In the p-mode frequency band, Kurtz et al. (12) report frequency errors between 5 × 10−7 day−1 and 10−4 day−1. The stellar model that best explains the observed mode frequencies implies that KIC 11145123 is a terminal-age main sequence A star. It has a small convective core (r < 0.04 R), in which the fraction of hydrogen content is less than 5%. Outside this convective core, energy is transported by radiation up to the surface layers of the star. In the top few thousand kilometers, there are very thin convective layers associated with the ionization of helium and hydrogen. See Table 1 for a summary of the basic stellar parameters.

Table 1 Parameters of the star KIC 11145123 and the best-fit seismic model. View this table:

In spherically symmetric stars, the eigenfunctions of stellar oscillations are proportional to spherical harmonics (θ, ϕ) of degree l and azimuthal order m = −l, −l + 1, … l, where θ is the colatitude and ϕ is the longitude. Internal rotation and departures from spherical symmetry lift the (2l + 1)–fold degeneracy in m of the mode frequencies, ν nlm . The antisymmetric component of the frequency splittings in a multiplet, δν nlm = (ν nlm − ν nl,−m )/2, is a weighted average over the stellar volume of the stellar angular velocity (16). KIC 11145123 is one of a very few stars in which these rotational splittings have unambiguously been detected in both the p-mode and g-mode bands. The observed frequency splittings imply an internal rotation period of more than 105 days and a surface rotation period of less than 99 days, showing that the star rotates a little more quickly at the surface than in the core (12). Internal angular momentum transfer or external accretion mechanisms have spun up the atmosphere, a result of theoretical interest (17).

Stellar asphericity is measured through the symmetric component of the splittings (9) (1)

This differential measurement exploits the different sensitivities in latitude of modes with different values of |m| at fixed l and n. As shown in Fig. 1, modes with larger values of |m| are confined to lower latitudes. For p modes, the mean frequencies = (ν nlm + ν nl,−m )/2 are not sensitive to rotation at first order and inform us about the inverse acoustic stellar radius at specific latitudes, with increased sensitivity at lower latitudes for larger |m|. For a spherical star, = ν nl0 and s nlm = 0 for all m. Latitudinal variations in stellar shape or wave speed will cause a nonzero s nlm . The s nlm values are positive for prolate spheroids and negative for oblate spheroids. Latitudinal variations in the wave speed may result from variations in a magnetic field or chemical composition.

Fig. 1 Latitudinal dependence of mode kinetic energy density. Dipole (l = 1, left panel) and quadrupole (l = 2, right panel) modes of oscillation. The arrow points along the stellar rotation axis. Scalar eigenfunctions of stellar oscillation are proportional to (cosθ) eimϕ, where are associated Legendre functions. Polar plots of the kinetic energy density E lm (θ) = c lm [ (cosθ)]2 sinθ, where θ is the colatitude, for the modes with azimuthal orders m = 0 (black), m = 1 (red), and m = 2 (blue) are shown. The constants of normalization, c lm , are such that E lm (θ) = 1 for each (l, m). For dipole modes, we see that E 10 is maximum at latitude λ = π – θ = ±63° and E 11 is maximum at the equator. For quadrupole modes, E 20 peaks at λ = 0° and ±59°, E 21 is maximum at λ = ±39°, and E 22 is maximum at the equator. For reference, the dashed curves show a highly distorted (oblate) stellar shape of the form r(θ) = 1 – 0.15 P 2 (cosθ), where P 2 (x) = (3x2 − 1)/2 is the second-order Legendre polynomial.

In the p-mode frequency band of KIC 11145123, five multiplets have been identified (12) and assigned values of (n, l, m) by comparison with the best-fit seismic model. These are two dipole (l = 1) and three quadrupole (l = 2) multiplets, for which all 2l + 1 azimuthal modes are identified. The measured values of s nlm are tabulated in Table 2. Among these, the quadrupole multiplet near 23.5 day−1 does not provide frequencies with sufficient precision to affect the results of this study. The s nlm values are plotted in Fig. 2 for the four multiplets of interest. The average of all values, <s nlm > = (−1.4 ± 0.5) × 10−5 day−1, is negative (3 standard deviations away from zero); therefore, the star is oblate.

Table 2 Mode frequencies and frequency shifts. Values of s nlm , as defined by Eq. 1, are given for the dipole and quadrupole multiplets in the p-mode range of KIC 11145123. The frequency shifts expected from rotational oblateness, , are computed using Eq. 4. Mode identification is according to the best-fit stellar model, with R = 2.24 R ⊙ and M = 1.46 M ⊙ (12). Mode amplitudes are measured to a precision of 0.01 mmag. View this table:

Fig. 2 Symmetric component of observed frequency splittings s nlm . Observations are plotted as red circles with error bars. Each value is associated with the mode index given in the last column of Table 2. The data points are labeled by the values of (l, m). The theoretical values for rotational oblateness alone, , are plotted as open squares. Note that the last two values of s nlm from Table 2 are not plotted here because they are associated with errors that are too large to provide additional constraints.

As mentioned in the Introduction, several physical mechanisms can make a star aspherical to stellar oscillations. One mechanism that must be present is rotational oblateness, which is relatively easy to compute when rotation is slow. The centrifugal force distorts the equilibrium structure of a rotating star. The corresponding perturbation to the mode frequencies scales as the ratio of centrifugal to gravitational forces (2)where Ω is the star’s surface angular velocity; R and M are the radius and mass of the star, respectively; and G is the universal constant of gravity. Using Ω/2π = 0.01 day−1 for KIC 11145123, we have ε = 1.34 × 10−6 (R/R ⊙ )3 (M/M ⊙ )−1, where R ⊙ and M ⊙ are reference solar values (19). The mass and radius of the star are not known to the same level of confidence as the rotation. Our best-fit seismic model (12) has a metallicity of Z = 0.01, a mass of 1.46 M ⊙ , and a radius of 2.24 R ⊙ . For this stellar model, the ratio of the centrifugal to gravitational forces becomes (3)

This is a very small number, but it is not small compared to the relative errors of the most precise frequencies in the p-mode range from Table 2. Note that ε is roughly half the solar value (ε ⊙ = 1.8 × 10−5).

For slow rotators, rotational oblateness is described by a quadrupole distortion of the stellar structure. To leading order, the contribution of rotational oblateness to s nlm can be written as (16, 20) (4)where the dependence on m and l is due to the latitudinal sensitivity of the modes of oscillation (Fig. 1). The amplitude of the effect is proportional to the degenerate mode frequency ν nl of the nonrotating reference model and to the numbers Δ nl , which are mode-weighted radial averages of the stellar distortion (see Table 2 and Materials and Methods). The numerical values of are listed in Table 2 and are overplotted in Fig. 2 for the available modes. We find that the theoretical values are of the same sign and same order of magnitude as the measured s nlm . As illustrated in Fig. 3, a good representation of the measurements is (5)

Fig. 3 nlm to theoretical prediction for rotational oblateness Ratios of observed sto theoretical prediction for rotational oblateness The horizontal solid line and the gray area indicate the average and the 1-σ bounds, <s nlm / > = 0.35 ± 0.12. Each value is associated with the mode index given in the last column of Table 2. The distributions of the data points and their errors are consistent with a single value for the ratio of s nlm / .

Hence, the star is more round than rotational oblateness would imply. Equation 5 also implies that the modes of oscillation see a quadrupole distortion of the shape of the star. The amplitude of the distortion is smaller than would be expected from rotation alone; therefore, an additional physical ingredient is needed.