Asteroseismic detection of latitudinal differential rotation in 13 Sun-like stars

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Science  21 Sep 2018:
Vol. 361, Issue 6408, pp. 1231-1234
DOI: 10.1126/science.aao6571

Stellar oscillations show differential rotation

The Sun rotates faster at its equator than at its poles. This process is known as differential rotation and is seen in the motion of sunspots. Helioseismology has shown that the effect extends into the Sun's interior. It has not been possible to measure whether other stars also experience equivalent differential rotation. Benomar et al. used the Kepler spacecraft to monitor stellar oscillations of a group of Sun-like stars. By decomposing the oscillations into separate frequencies, they searched for signs of differential rotation. Several stars do indeed seem to have equators that spin faster than their poles, and none indicated the opposite pattern.

Science, this issue p. 1231


The differentially rotating outer layers of stars are thought to play a role in driving their magnetic activity, but the underlying mechanisms that generate and sustain differential rotation are poorly understood. We report the measurement using asteroseismology of latitudinal differential rotation in the convection zones of 40 Sun-like stars. For the most significant detections, the stars’ equators rotate approximately twice as fast as their midlatitudes. The latitudinal shear inferred from asteroseismology is much larger than predictions from numerical simulations.

Analysis of acoustic oscillations visible on the Sun’s surface by using helioseismology has been critical for constraining its rotation profile. Helioseismology has revealed that the rotation rate of the Sun’s convection zone decreases with latitude (1, 2). This latitudinal differential rotation has a magnitude of 11% of the average rate from equator to midlatitudes (≈45° latitude) and 30% between equator and the poles. At the base of the convection zone, the Sun transitions to solid-body rotation. How such a rotation profile is established and maintained is still poorly understood. However, it is likely that differential rotation plays a role in sustaining the solar magnetic field through a dynamo mechanism (35).

So far, little is known about the latitudinal differential rotation in other stars, and classical methods for investigating it are primarily sensitive to the near-surface layers. Most studies rely on photometric variability from starspots at different latitudes (6) or use Doppler imaging to track magnetic features at the surface and their migration in latitude (7, 8). Another approach involves studying the Fourier transform of the spectroscopic line profiles (9, 10).

Asteroseismology provides an opportunity to probe rotation inside stars, including Sun-like pulsators (1113), because it studies the resonant frequencies of waves within the body of the star. Among these are acoustic waves that travel at various depths and have a varying sensitivity to rotation with latitude. These acoustic waves can be used to infer the star’s internal rotation both in radius and latitude (1417).

The NASA Kepler spacecraft has provided high-precision, long-duration photometric time series for many stars, which is necessary for the study of the differential rotation of Sun-like stars with asteroseismology.

Oscillations of Sun-like stars are driven by the stochastic convective motion of material in the outer envelope of the stars. Each mode of oscillation is identified by the overtone number n and spherical harmonic functions of angular degree l and azimuthal order m. Modes with the same n and l but different m appear as multiplets of 2l + 1 components. The splitting of modes in a multiplet provides information about the rotation profile and on the physical processes acting within the star (such as flows, internal structure, and tidal forces).

An efficient means for quantifying the impact of rotation on the split frequencies of acoustic waves is through the use of Clebsch-Gordan a-coefficients a1, a2, a3, ... (18). This decomposition has been used extensively in helioseismology but not in the analysis of other stars. For stars other than the Sun, only low-degree modes (l ≤ 3) can be observed, and so only the coefficients a1, a2, and a3 can be determined. The coefficient a1 is an average of the rotation rate. The a2 coefficient is related to the asphericity (19), and a3 is a measure of the latitudinal differential rotation (16). The coefficient a3 is positive when the pole rotates slower than the equator—that is, if the star exhibits a solarlike rotation profile (fig. S1). Conversely, a3 is negative for antisolar rotation profiles. Simulations tend to show that fast rotation causes solarlike rotation, whereas slower rotation rates lead to antisolar rotation (2022), although simulations found an antisolar rotation at solar rotation rates (21, 22). This is evidently incompatible with solar observations, and constraints from other stars are necessary to resolve this issue.

Pulsations appear as Lorentzian profiles in a power spectrum of the star’s photometric time series (23). The analysis of the pulsations is performed by fitting a model to the power spectrum by using a Markov chain Monte Carlo (MCMC) sampling technique (11). The contribution of a1 to mode frequencies is easily measured, but the absolute value of a3 is approximately two orders of magnitude smaller than a1, so it is only possible to measure it by using multiyear observations from space instruments such as Kepler.

We performed a combined measurement of the a1 and a3 coefficients for 40 stars of mass between 0.9 and 1.5 M (where M is the mass of the Sun), observed as part of the Kepler LEGACY sample (24, 25). By integrating the probability density functions of a3 determined by fitting the power spectra, we computed the detection significance for either solar or antisolar latitudinal differential rotation (Fig. 1 and table S1). Applying a detection threshold at a probability of 84% (excluding a3 = 0 with a detection significance >1σ for a Gaussian), we found that none of the stars unambiguously show antisolar rotation, whereas 32% (13 stars) show significant solarlike rotation. Five stars have a detection probability of more than 97.5% (or a significance >2σ). The excess of solarlike rotators may be due to an observational limitation. Antisolar differential rotation is theoretically expected (20, 26) for slow rotators, in which a1 and a3 are difficult to measure. Our sample consists predominantly of fast rotators, so we may only be sensitive to stars with solarlike rotation profiles. Our most statistically significant detections have relatively high rotation rates compared with those below the detection threshold.

Fig. 1 Detection of solar rotation and antisolar rotation.

(A) Histogram of the detection significance for solarlike versus antisolar rotation by using Kepler asteroseismic light curves (colored bars). The cumulative distribution is also shown (black dots). All stars with conclusive detections of a3 ≠ 0 (>84% significance) have solarlike rotation (a3 > 0). (B) Average internal rotation as measured with a1. Stars with high detection significance rotate faster than those with low significance. (C) The latitudinal differential rotation coefficient a3.

The Rossby number is an indicator of the influence of rotation on a fluid compared with convective motion. Current theory (20) predicts that solarlike rotators should have a Rossby number less than unity. For the majority of our significant detections, we found a Rossby number less than 0.8 (fig. S2).

We next sought to estimate the rotational shear between the equator and higher latitudes. For this, we selected two representative stars (HD 173701 and HD 187160) in terms of a1, a3, and internal structure (thickness of convection zone). Their power spectra are shown in figs. S3 and S4 with the best-fitting model. HD 173701 is a cool dwarf of spectral type G8 (the Sun is of spectral type G2), with a mass 0.974 ± 0.029 M and age 4.69 ± 0.44 billion years (24). It is known to be very active (27), with irregular variability on time scales that range from 8 to 40 days, likely caused by starspots. The probability density function deduced from the fit indicates that a1 = 574.6 ± 82.0 nHz and a3 = 28.4 ± 12.5 nHz (fig. S5). HD 187160 is of spectral type F9 and has a mass and age of 1.09 ± 0.03 M and 3.28 ± 0.16 billion years. Thus, HD 187160 is similar to a younger version of the Sun. It has a visible magnitude of 7.4 and a faint, cool spectroscopic binary companion of magnitude 8.7 (28). The average internal rotation period is ~9 days (a1 = 1185.0 ± 51.7 nHz). The light curve of HD 187160 shows variability at two distinct time scales, one at ≈9.4 days and another at ≈17.5 days. We attribute this variability to starspots on the surfaces of HD 187160 and its fainter noninteracting companion, respectively. We measured a3 = 53.7 ± 25.0 nHz (fig. S6).

We have evaluated the potential systematic errors associated with the mode-fitting methodology using simulated power spectra for HD 173701 and HD 187160 (23). These systematic errors are small, meaning that a3 in both cases is positive, with a probability higher than 97.5%. This confirms that these stars have poles that rotate slower than the equator.

To obtain a latitudinal rotation profile, we used an inversion of the a-coefficients (16). Because we only measured a1 and a3, we were restricted to a two-parameter rotation model for this. We assumed a solarlike rotation profile Ω(θ) = Ω0 + 3Ω1(5cos2θ – 1)/2 in the convection zone, where θ is the colatitude (complementary angle of the latitude so that θ = 0 is the pole and θ = 90° is the equator). The average internal rotation rate is given by Ω0, and the term in Ω1 ∝ Ωpole – Ωeq measures the contrast in rotation (latitudinal shear) from the equator to the pole in the convection zone. In the equation, Ωpole and Ωeq are the pole and equator rotation rates, respectively. The convection zone extends from the surface down to 0.687 Embedded Image for HD 173701 and to 0.785 Embedded Image for HD 187160, where Embedded Image is the stellar radius. Because of its relatively fast rotation, HD 187160 was investigated for possible variations of a1 with the overtone n (fig. S7). We did not find evidence of significant variation. Studies of radial differential rotation of low-mass main-sequence stars indicate that variation in a1 should not exceed 30% (29), and we therefore do not expect this to have an effect on the inversion.

The fundamental properties of the stellar models assumed for the inversion and the results of the inversion are provided in tables S2 and S3, respectively. The two-dimensional rotation profiles and the probability density of the rotation rate are shown in Fig. 2 at different latitudes for both stars. By construction, the profile is symmetric around the equator and around the rotation axis, so only one quadrant is shown. The rotation rate in the interior matches that of the envelope at a latitude of 26.5° (θ = 63.5°) because of the structure of the two-zone rotation profile. As is shown by the probability density of the rotation profile, the uncertainty on the rotation rate increases substantially toward the poles, beyond a value of ~45° latitude. At high latitudes, the modes become much less sensitive to rotation, thus yielding less information. This limit is imposed by the lack of visible modes with an angular degree higher than l = 2 in the spectrum. We found that both HD 173701 and HD 187160 exhibit a latitudinal shear from equator to 45° latitude ΔΩ45eq, which is approximately five times greater than that of the Sun.

Fig. 2 Rotation profiles from inversion for HD 173701 and HD 187160.

(A and B) HD 173701. (C and D) HD 187160. (A) and (C) display the most likely rotation profile (colors) and the interface between radiative and convective zone (dashed lines). (B) and (D) show the probability density, π(θ,Ω), at each latitude of the rotation profile in the convection zone (red-shaded region). The 1σ confidence interval is highlighted with dashed lines. The latitudinal differential rotation is well constrained for colatitudes θ > 45°.

As suggested by numerical simulations, the fast rotation of HD 187160 means that the functional form of its rotation profile may not be solarlike (23). We have considered this possibility for HD 187160 (fig. S8) and found that alternative rotation profiles have larger latitudinal shear than the solarlike rotation profile. However, computation of the Bayesian evidence (necessary to determine the goodness of the fit) shows that the solarlike profile is the most likely model.

Because of loss of angular momentum, such as from magnetic braking, a star’s rotation is expected to slow with its age (1). This age-rotation relation is apparent in our data: Fig. 3, A and B, shows a1 as a function of the age given by stellar models (25).

Fig. 3 Internal rotation rate and differential rotation.

The solar a-coefficients are for the solar rotation profile (gray dot). (A) Measured values of a1, as a function of age and mass for 40 stars. The well-established decrease in rotation rate with stellar age (gyrochronology) is apparent. (B) Same as (A) but for most significant detections. (C) The distribution of ΔΩ45eq. The median is 64%. The gray region shows the 1σ dispersion.

As shown in Fig. 3C, the latitudinal shear between the equator and midlatitude for the subset of 13 stars with significant detections is ≈60%, albeit with a large scatter. HD 173701 and HD 187160 are representative of this ensemble, with a differential rotation of ≈50%. The Sun has a significantly lower shear factor than that of the considered ensemble. The difference is of more than 1σ from the dispersion of the ensemble.

This unexpectedly large shear poses a challenge to theoretical models. The balance between angular momentum transport (owing to anisotropy in the turbulent flow) and small-scale flows, which act as enhanced turbulent viscosity, plays a dominant role in regulating latitudinal shear (30). The large shear we found indicates a correspondingly large anisotropy in the turbulence, leading to efficient angular momentum transport and suppression of turbulent viscosity. In a typical stellar convection zone, turbulent anisotropy, driven by rotation, substantially affects large-scale flows. Thus, enhanced angular momentum transport and diminished turbulent viscosity amplify large-scale flows and suppress small-scale flows. This could be caused by a very efficient small-scale dynamo, in which small-scale flows are suppressed (31). In addition, large-scale magnetic fields tend to reduce shear through the Lorentz force (32). However, our results indicate that Lorentz-force feedback is ineffective in the stars we investigated. Thus, the large-scale magnetic field is likely transported efficiently into the deeper regions of the star in which rigid rotation is expected. For this, magnetic pumping is a candidate mechanism (33).

Supplementary Materials

Materials and Methods

Figs. S1 to S8

Tables S1 to S4

References (3475)

References and Notes

  1. Materials and methods are available as supplementary materials.
Acknowledgments: The authors thank T. R. White and S. Kamiaka for their helpful discussions. Funding: This work is supported by NYUAD institute grant G1502. Research funding from the German Aerospace Center (Grant 50OO1501) and the Max Planck Society (PLATO Science) is acknowledged. Funding for the Stellar Astrophysics Centre is provided by The Danish National Research Foundation (grant DNRF106). The research was supported by the ASTERISK project (ASTERoseismic Investigations with SONG and Kepler) funded by the European Research Council (grant agreement 267864). Funding for the Kepler mission is provided by the NASA Science Mission directorate. Author contributions: O.B., M.B., and L.G. designed the research. O.B. performed seismic analyses and lead in writing the manuscript. M.B. performed seismic inversion. M.B.N. analyzed the light curves variability. All authors discussed the results and contributed to sections of the manuscript. Competing interests: None declared. Data and materials availability: This paper includes data collected by the Kepler mission, provided by the Kepler Asteroseismic Science Operations Center (KASOC) and available at Further details on the data used are listed in table S4. The TAMCMC analysis code is available at
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