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Read the interior of a star

AC26 · L4

Read core rotation from mixed modes

Sandbox Physics ·

Mixed modes respond differently to the core and envelope, allowing their frequency splittings to separate two average rotation rates. Read Li, Deheuvels and Ballot 2024, arXiv:2405.12116v1, sections 2.2–2.5, Figure 6 and Appendix E. Figure 6 tests a crucial complication: rotation can change the coupling between nearby modes, making a simple symmetric-splitting model unreliable.

Begin with the peaks the paper actually fits

Open section 2.2 before looking at rotation rates. The authors first fit the radial and quadrupole pressure-dominated modes with Lorentzian profiles. Their Equation 1 calls eta the half width at half maximum. A full-width value is twice that number; silently exchanging the conventions changes damping interpretations.

The fitted radial and quadrupole frequencies establish the acoustic pattern. Curvature and small frequency offsets matter: a single perfectly even radial ladder is too restrictive for a real evolved star. The dipole pressure-mode reference also has an additional shift parameter because its position between radial modes varies with evolution.

The Zenodo files preserve distinct stages. Fitting-detail tables give fitted quantities and uncertainties. Dipole bin_points are approximate starting positions in the power spectrum. These high-precision printed values guide a search; their frequency uncertainties still require a fit.

Reading: Li, Deheuvels & Ballot 2024, Core and envelope rotation rates for 2006 red giants · Li, Deheuvels & Ballot, Zenodo13207485 v2

Read the two cavity phases with units attached

In section 2.3, a mixed mode joins an acoustic cavity in the envelope and a buoyancy cavity in the core. The phase in each cavity describes how the wave fits through it. The coupling factor tells how strongly the cavities communicate. Solving their phase relation produces many mixed modes around the pressure-mode pattern.

Use ordinary frequency in hertz when combining inverse frequency with the gravity-mode period spacing in seconds. For a mode near 190 microhertz, inverse frequency is about 5263 seconds. A period spacing near 86 seconds corresponds to dozens of gravity-wave orders, not a tiny fraction of one.

The phase convention needs particular care. The paragraph after Equation 9 explicitly warns that its catalog gravity-phase offset follows a convention displaced by one half from the proper pure-gravity weak-coupling definition. Keep the printed equations and catalog convention together. Add the stated half-cycle conversion when comparing to the corrected convention; do not mix the two definitions in a zero-coupling test.

A numerical solver also needs to distinguish roots from tangent poles. A continuous sine/cosine form or segmented phase calculation avoids accepting a pole as a mode. Checking the uncoupled limits and the number of expected branches is more informative than a plot that merely looks like a dense comb.

tan⁡θp=qtan⁡θg\tan\theta_{\rm p}=q\tan\theta_{\rm g}
qq is the coupling factor. This asymptotic relation describes real buoyancy/acoustic mixed-mode physics, separate from the controllable radial acoustic cavity.
θp=πν−νpΔν,θg=πΔΠ1(1ν−1νg)\theta_{\rm p}=\pi\frac{\nu-\nu_{\rm p}}{\Delta\nu},\qquad\theta_{\rm g}=\frac{\pi}{\Delta\Pi_1}\left(\frac1\nu-\frac1{\nu_{\rm g}}\right)
The p reference is a pure acoustic-mode frequency; the g reference uses the paper’s stated period/phase convention. The phases are dimensionless only when frequency and period units agree.

Reading: Li, Deheuvels & Ballot 2024, Core and envelope rotation rates for 2006 red giants

Separate core and envelope with different mode weights

Rotation separates modes with different azimuthal orders, labeled by mm. A dipole multiplet can contain three components. The paper’s first approach expresses the splitting as a weighted combination of core and envelope rotation.

The weight ζ\zeta is the fraction of the mode inertia in the gravity cavity. Inertia measures the amount of moving material for a chosen surface displacement normalization. A gravity-dominated mode has weight near one and responds strongly to the core; a pressure-dominated mode has more envelope sensitivity.

Read the factors of two and pi before interpreting a table. The catalog Omega_g and Omega_p columns are angular rates in millionths of a radian per second; the source instructions require division by 2π2\pi to obtain cyclic microhertz. The gravity-mode splitting in the formula has another factor of one half.

For KIC3860179, the catalog gives angular values 3.717347 and 0.510525. They correspond to cyclic rotation frequencies about 0.592 and 0.0813 microhertz. A pure gravity-dominated splitting in this approximation is about 0.296 microhertz, not 3.717 microhertz. These are conversions of external catalog results, not a rotation fit to our 275-day subset.

δνrot=ζΩcore4π+(1−ζ)Ωenv2π\delta\nu_{\rm rot}=\zeta\frac{\Omega_{\rm core}}{4\pi}+(1-\zeta)\frac{\Omega_{\rm env}}{2\pi}
This first-order relation assumes identified modes with the stated coupling approximation. Ω\Omega is angular frequency in radians per second, while the splitting is cyclic frequency.

Reading: Li, Deheuvels & Ballot 2024, Core and envelope rotation rates for 2006 red giants · Li, Deheuvels & Ballot, Zenodo13207485 v2

Use Figure 6 to see where symmetric splitting fails

Near degeneracy means that neighboring unperturbed frequencies are close enough for the rotational perturbation to mix their eigenfunctions. The rotation rate can still be small compared with the oscillation frequency, while being important compared with the separation between adjacent modes.

The paper’s second approach shifts the pure p and g frequencies before solving the coupling relation for each azimuthal order. This lets the coupling pattern respond to rotation. Adding a symmetric shift only after the mixed modes have been calculated cannot represent the same behavior in general.

In Figure 6, the reference case is KIC11245496. Red points show synthetic multiplet asymmetries from a separate near-degenerate calculation. Gray results show the second approach’s fit. The background indicates the inertia weight; the stronger asymmetries gather near pressure-dominated modes. Follow those axes and symbols before drawing a conclusion about a negative envelope-rotation estimate.

Appendix E explains the independent matrix viewpoint. Expand displacement in an orthonormal set of nearby unperturbed modes. The diagonal matrix contains their squared angular frequencies, and the rotation operator supplies both diagonal and off-diagonal perturbations. Diagonalizing the whole matrix includes mixing between modes. Its eigenvalues are squared angular frequencies and must be converted accordingly.

νg,m=νg−mΩcore4π,νp,m=νp−mΩenv2π\nu_{{\rm g},m}=\nu_{\rm g}-m\frac{\Omega_{\rm core}}{4\pi},\qquad\nu_{{\rm p},m}=\nu_{\rm p}-m\frac{\Omega_{\rm env}}{2\pi}
These use the paper’s azimuthal sign convention. Each shifted reference enters the coupling calculation for its own azimuthal component.
M=A+R,Aij=ω0,i2δij,Rij=⟨δLrotξi,ξj⟩M=A+R,\qquad A_{ij}=\omega_{0,i}^{2}\delta_{ij},\qquad R_{ij}=\langle\delta\mathcal L_{\rm rot}\boldsymbol\xi_i,\boldsymbol\xi_j\rangle
The inner product is mass weighted. Actual stellar eigenfunctions and rotation kernels are needed to reproduce the paper’s specific reference star.

Reading: Li, Deheuvels & Ballot 2024, Core and envelope rotation rates for 2006 red giants

Calculate a small independent near-degeneracy example

The supplied mixed-mode-limits script uses two orthonormal p/g basis modes and a positive squared-frequency coupling matrix. It compares a Hermitian eigensolver with the independent quadratic characteristic roots, and verifies that zero coupling returns the pure modes.

With an illustrative pure-mode separation of 0.03 microhertz, its rotational multiplet asymmetry reaches about 0.170 microhertz. Moving the pure frequencies 3 microhertz apart reduces the largest asymmetry to about 0.000448 microhertz. The mode mixing becomes much less important when the separation is large.

This two-mode example tests the mechanism and numerical limits. It has no calibrated conversion to the paper’s coupling factor, no full stellar eigenfunction kernels and no claim to reproduce Figure 6. It helps identify why retaining off-diagonal mixing changes the answer before undertaking the larger stellar calculation.

python3 science/mixed-mode-limits.py --input input/helmi__HD128279__HD_128279_observed_modes.csv --out rerun/mixed-mode-limits.json

Reading: Li, Deheuvels & Ballot 2024, Core and envelope rotation rates for 2006 red giants

Read population results with their selection still attached

The paper begins with 2495 stars showing clear mixed-mode patterns. It identifies 800 doublets and 1206 triplets, yielding 2006 core-rotation measurements. The remaining stars do not show the same resolved splitting information. Missing splitting is not a measured zero rotation.

For 243 stars whose envelope rates are significantly above zero, the paper reports core-to-envelope ratios around 20 with a substantial spread. That statement belongs to the detected-envelope subset. It should not become a universal ratio assigned to every red giant. Mode visibility, inclination, observing resolution and the fitting model affect which stars enter the subset.

The raw catalog also contains sentinel values. Some auxiliary uncertainty entries and the luminosity description require care. The supplied exercise preserves them unchanged and excludes unresolved auxiliary fields instead of silently repairing their signs or interpreting luminosity as a different quantity. Read the source notes before using the table as a new population prior.

Reading: Li, Deheuvels & Ballot 2024, Core and envelope rotation rates for 2006 red giants · Li, Deheuvels & Ballot, Zenodo13207485 v2

Follow surface effects through a change of mode identity

Now return to Lindsay 2025, Appendix A and Figure 5. The near-surface layers of a stellar model shift predicted acoustic frequencies. The Ball–Gizon family represents that shift using inverse and cubic frequency terms divided by the declared mode inertia.

For mixed modes, changing an acoustic reference can also change its coupling to the gravity cavity. A smooth shift added blindly to every final mixed-mode frequency may not preserve identities. The HD128279 table gives a concrete warning: its uncorrected dipole column is not in the same row order as the corrected dipole column. Row subtraction does not identify a physical surface shift.

Keep the two stars and studies separate. The Kepler rotation catalog measures a population of red giants; the HD128279 TESS example compares metal-poor stellar models and age estimates. Their connection is mode physics. A result about one star’s conditional model age does not supply another star’s rotation uncertainty.

For the next paper, write down the mode-identification step, angular/cyclic units, phase convention, inertia normalization, resolution and near-degenerate treatment before reading the headline rate. Those choices explain what the number can mean.

δνsurf=a−1(ν/νac)−1+a3(ν/νac)3I\delta\nu_{\rm surf}=\frac{a_{-1}(\nu/\nu_{\rm ac})^{-1}+a_3(\nu/\nu_{\rm ac})^3}{\mathcal I}
The coefficients have frequency units for this normalization. I\mathcal I and its surface normalization must be declared. The source paper has a corrigendum. Our comparison uses the published model table as an external reference; reproducing the full correction would require the stellar eigenfunctions and their normalization.

Reading: Lindsay et al. 2025, Precise Asteroseismic Ages for the Helmi Streams · Ball & Gizon 2014, A new correction of stellar oscillation frequencies for near-surface effects · Lindsay et al., Zenodo15102877 v1 model/observed-mode archive

Exercises and explanations

  1. A catalog angular rate is 4×10−6 rad s−14\times10^{-6}\,\mathrm{rad\,s^{-1}}. What is its cyclic frequency and the pure-gravity splitting under the first-order dipole formula?

    Read the explanation

    Divide by 2π2\pi to obtain about 0.637 microhertz. The pure-gravity splitting is half that, about 0.318 microhertz. Keep the angular rate, cyclic rotation rate and mode splitting as three distinct quantities.

  2. Why can rotation be perturbatively small relative to a mode frequency and still cause strong near-degenerate mixing?

    Read the explanation

    The relevant comparison for mixing is the small separation between nearby modes. A shift tiny relative to 190 microhertz can exceed a separation of 0.03 microhertz, so off-diagonal coupling can strongly change the eigenfunctions.

  3. Why is a source dipole seed insufficient for a core-rotation measurement?

    Read the explanation

    It has no fitted frequency error or secure multiplet identity. Rotation inference also needs azimuthal assignments, mode inertia/coupling information and an appropriate treatment of near degeneracy and the observing window.

Try the prediction

Change a layer and read its modes

  1. Inspect the real Kepler spectrum and identify how many candidate peaks lie between neighboring radial references.
  2. Run the independent two-mode calculation and compare near and far frequency separations.
  3. Return to Figure 6 and compare its asymmetric multiplets with a symmetric first-order splitting. Identify the stellar mode information needed for a rotation measurement.

Sources and calculation scope

  1. Lindsay et al. 2025, Precise Asteroseismic Ages for the Helmi Streams

    Pinned arXiv2507.01091v1; sections II.2/III/V.3/V.4, Figures 3/5, Tables 3/4/6 and Appendix A. Published as ApJ 989, 189; full optimization and its age errors are external results.

  2. Lindsay et al., Zenodo15102877 v1 model/observed-mode archive

    CC-BY4 dataset metadata; complete selected HD128279 observations, history/index, eight GYRE structures and published mode tables. One fixed-physics track, not the 840-track optimizer ensemble.

  3. Li, Deheuvels & Ballot 2024, Core and envelope rotation rates for 2006 red giants

    Pinned 2405.12116v1; sections 2.2–2.5, near-degeneracy mock Figure 6 and Appendix E. Published as A&A 688, A184. Keep angular/cyclic units and g-phase convention explicit.

  4. Li, Deheuvels & Ballot, Zenodo13207485 v2

    CC-BY4 rotation catalog and selected KIC3860179 mode files. Dipole bin_points are approximate seeds; fitted l0/l2 tables have separate uncertainties. Ambiguous auxiliary fields remain excluded.

  5. Ball & Gizon 2014, A new correction of stellar oscillation frequencies for near-surface effects

    Inverse/cubic inertia-weighted surface terms; Lindsay2025 Appendix A supplies the concrete application. The original paper has a corrigendum DOI10.1051/0004-6361/201424325e; this course does not claim a full independent reproduction.

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