Research ArticleASTRONOMY

Shape of a slowly rotating star measured by asteroseismology

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Science Advances  16 Nov 2016:
Vol. 2, no. 11, e1601777
DOI: 10.1126/sciadv.1601777
  • 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 Embedded Image (cosθ) eimϕ, where Embedded Image are associated Legendre functions. Polar plots of the kinetic energy density Elm(θ) = clm [Embedded Image(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, clm, are such that Embedded Image Elm(θ) = 1 for each (l, m). For dipole modes, we see that E10 is maximum at latitude λ = π – θ = ±63° and E11 is maximum at the equator. For quadrupole modes, E20 peaks at λ = 0° and ±59°, E21 is maximum at λ = ±39°, and E22 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 P2(cosθ), where P2(x) = (3x2 − 1)/2 is the second-order Legendre polynomial.

  • Fig. 2 Symmetric component of observed frequency splittings snlm.

    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, Embedded Image, are plotted as open squares. Note that the last two values of snlm from Table 2 are not plotted here because they are associated with errors that are too large to provide additional constraints.

  • Fig. 3 Ratios of observed snlm to theoretical prediction for rotational oblateness Embedded Image.

    The horizontal solid line and the gray area indicate the average and the 1-σ bounds, <snlm/Embedded Image> = 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 snlm/Embedded Image.

  • Table 1 Parameters of the star KIC 11145123 and the best-fit seismic model.
    Photometry (12, 18)Seismology (12)
    Spectral typeALate main sequence
    Kepler visual magnitude13
    Effective temperature, Teff8050 ± 200 K7032 K
    Surface gravity, log g (cgs)4.0 ± 0.23.904
    Surface rotation rateΩ/2π = 0.01 day−1
    MassM = 1.46 M
    RadiusR = 2.24 R
    Initial abundances(X, Z) = (0.65, 0.010)
    Radius of convective core0.04 R
    Hydrogen core abundanceXc = 0.033
  • Table 2 Mode frequencies and frequency shifts.

    Values of snlm, 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, Embedded Image, 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.

    nlmνnlm ± σnlm (day−1)Amplitude (mmag)snlm (10−5 day–1)ΔnlEmbedded Image (10−5 day−1)Index
    −12−216.7258824 ± 0.0000017 2.33
    −116.7339455 ± 0.00001860.21
    016.7420104 ± 0.00000750.53
    116.7500755 ± 0.00001100.360.1 ± 1.30.40−1.01
    216.7580083 ± 0.00005040.08−6.4 ± 2.60.40−3.92
    21−118.3558305 ± 0.00000291.36
    018.3660001 ± 0.00001350.29
    118.3761210 ± 0.00000341.13−2.3 ± 1.40.68−7.73
    02−218.9869603 ± 0.00000780.50
    −118.9967001 ± 0.00000600.65
    019.0064482 ± 0.00000560.69
    119.0161736 ± 0.00001010.39−1.1 ± 0.80.39−1.14
    219.0259102 ± 0.00000860.45−1.4 ± 0.80.39−4.35
    31−121.9933315 ± 0.00000640.61
    022.0018915 ± 0.00002860.14
    122.0104220 ± 0.00000640.61−1.5 ± 2.90.68−9.26
    22−223.5455350 ± 0.00003030.13
    −123.5553428 ± 0.00005820.07
    023.5651835 ± 0.00004800.08
    123.5749885 ± 0.00006070.06−1.7 ± 6.40.20−0.67
    223.5847898 ± 0.00001630.24−2.2 ± 5.10.20−2.28

Supplementary Materials

  • Supplementary Materials

    This PDF file includes:

    • fig. S1. Mode kinetic energy density.
    • fig. S2. Radial dependence of centrifugal distortion.

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