Compact-Star Asteroseismology

Compact-star asteroseismology asks what frequencies, damping times, and mode splittings reveal about mass, radius, composition, rotation, elasticity, and magnetic structure. The inverse problem is intrinsically many-to-one: a peak is not a unique equation-of-state measurement, and a transient need not be a normal mode. A defensible result therefore joins a forward oscillation model to a versioned dataset, calibrated instrument response, explicit selection, and a likelihood that carries nuisance parameters. The observational status and data examples here are current through 2026-08-18.

Helpful background. Stress–Energy and Conservation supplies conserved mode energies; the Einstein equation supplies the background spacetime; Relativistic Matter and Compact Objects orients stellar structure; Black-Hole Perturbation Theory supplies quasinormal-mode conventions; Gravitational Radiation and Waveforms supplies detector observables; and Computational Relativity supplies numerical forward models.

From stellar structure to a spectrum

For a nonrotating equilibrium, linearized fluid and Einstein equations define complex eigenfrequencies

$$\omega_n=2\pi f_n-\frac{i}{\tau_n}$$

when perturbations vary as $e^{-i\omega t}$. A damped free mode has $\operatorname{Im}\omega_n<0$ and $\tau_n>0$. The real frequency identifies an oscillation timescale and $\tau_n$ its damping time. The fluid f-mode roughly tracks the dynamical frequency,

$$f_f\sim \frac{1}{2\pi}\sqrt{\frac{GM}{R^3}},$$

while p-modes are sensitive to compressibility and sound speed, g-modes to stratification and composition gradients, and spacetime w-modes primarily to the relativistic gravitational sector. Kokkotas and Schmidt give the mode classification and the eigenvalue boundary conditions in Kokkotas and Schmidt 1999, §§4.1–4.2 and 5.4.

The scaling is diagnostic, not a calibrated universal relation. Rotation splits azimuthal components and mixes harmonic indices; magnetic tension joins fluid, crustal, and Alfvén motions; superfluid entrainment adds counter-moving degrees of freedom. Redshift converts a local clock to the frequency measured at infinity. Every forward-model prediction must therefore state the mass, equation of state, thermal and compositional profile, rotation law, magnetic configuration, boundary conditions, and frequency convention. In $c=1$ units, $f$ has dimension $L^{-1}$ and $GMf$ is dimensionless.

What an instrument actually measures

Two important data classes have different response models. Gravitational-wave strain is a calibrated time series

$$d_k(t)=h_k(t;\vartheta,\lambda)+n_k(t),$$

where $k$ labels a detector, $\vartheta$ contains stellar parameters, and $\lambda$ contains calibration and waveform nuisance parameters. The official Gravitational Wave Open Science Center (GWOSC) GWOSC GW170817 event page, strain data release and posterior-sample version is an example of the event-level products required for reproduction; it does not constitute a detected post-merger stellar mode. Abbott and collaborators reported the inspiral detection and its analysis in Abbott et al. 2017, Methods and pp. 161101-1–161101-9.

High-energy observations of magnetar flares instead begin with photon event lists or binned counts. Their instrument response includes clock corrections, dead time, energy redistribution, effective area, telemetry gaps, and barycentric timing. Watts and Strohmayer’s analysis of high-frequency oscillatory power in the 2004 SGR 1806–20 flare illustrates energy- and rotation-phase-dependent searches in Watts and Strohmayer 2006, Figures 1–3 and pp. L117–L120. Interpreting such a quasi-periodic oscillation as a crustal, Alfvén, or magnetoelastic mode requires an additional physical model; the periodogram alone does not choose among them.

Likelihood, priors, and selection

For approximately stationary Gaussian strain noise with one-sided power spectral density $S_k(f)$, a common likelihood is

$$\ln {\cal L}(d\mid\vartheta,\lambda) =-\frac{1}{2}\sum_k \left\langle d_k-h_k(\vartheta,\lambda) \mathrel{\big|}d_k-h_k(\vartheta,\lambda)\right\rangle_k+\text{constant},$$ $$\langle a\mid b\rangle_k =4\,\operatorname{Re}\int_{f_{\min}}^{f_{\max}} \frac{\tilde a(f)\tilde b^*(f)}{S_k(f)}\,df .$$

The posterior is proportional to this likelihood times priors on masses, radii or equation-of-state parameters, spin, mode amplitude, excitation time, damping time, sky position, and calibration. A photon-count analysis generally needs a Poisson likelihood or a validated non-Poisson noise model rather than this Gaussian expression. Searching many start times, frequency bins, energy bands, or rotational phases incurs a trials factor; red noise and burst envelopes can imitate narrow excesses.

Selection belongs inside the inference. A detection threshold favors louder, longer-lived, and favorably oriented signals. A catalog-level population model must include the selection function or completeness as a function of source and instrument parameters. Even a single-event upper limit depends on the searched band, vetoes, duty cycle, background construction, waveform family, and prior volume. Publishing only the most significant peak suppresses the information needed to calibrate false-alarm probability.

Degeneracies and systematic uncertainty

A measured f-mode frequency mainly constrains a combination close to mean density, not $M$ and $R$ separately. Frequency and damping time together can reduce that degeneracy, but only through a chosen family of stellar models. Mode identification is another discrete uncertainty: an f-mode, a rotationally split component, and a magnetoelastic QPO can occupy overlapping frequency regions in different model classes. Unknown excitation physics further decouples absence of a peak from absence of the mode.

Important systematics include detector calibration, background nonstationarity, finite-duration spectral leakage, waveform truncation, interpolation of mode tables, equation-of-state parameterization, magnetic geometry, crust and superfluid microphysics, and an incorrect mode label. These should be varied or marginalized, not hidden inside a narrow credible interval. Cross-detector coherence, off-source background trials, injected-signal recovery, posterior predictive checks, and recovery of synthetic signals at known parameters are independent diagnostics.

No data were downloaded or reanalyzed for this page. A reproducible study would pin the machine-readable event-list or strain data product, calibration and pipeline version, quality flags, preprocessing, software environment, random seeds, priors, likelihood implementation, selection calculation, and posterior samples. The GWOSC event products provide one public starting point; equivalent magnetar work requires the mission archive’s exact observation identifiers and response files. GWOSC documents access to its versioned public products on the GWOSC catalog and data-release interface.

Check a frequency-only inference

Suppose a calibrated relation uses only $f_f\propto\sqrt{GM/R^3}$. Model A has mass $1.4\,M_\odot$ and radius $R_A$. Model B has mass $2.0\,M_\odot$. What radius $R_B$ gives the same predicted frequency, and what does the result show?

Solution

Equal frequencies require equal mean-density combinations:

$$\frac{M_A}{R_A^3}=\frac{M_B}{R_B^3}, \qquad \frac{R_B}{R_A}= \left(\frac{M_B}{M_A}\right)^{1/3} =\left(\frac{2.0}{1.4}\right)^{1/3}\simeq1.126 .$$

Thus a star about $12.6\%$ larger in radius can produce the same leading-order frequency if it is also more massive. The datum identifies a curve in the $(M,R)$ plane, not a unique point. A mass measurement, damping time, or another securely identified mode is needed to break the degeneracy, and the calibrated relation itself carries model uncertainty.

References

• Abbott, B. P., et al. “GW170817: Observation of Gravitational Waves from a Binary Neutron Star Inspiral.” Physical Review Letters 119 (2017): 161101. DOI.
• Gravitational Wave Open Science Center. “GW170817 Event Page.” LIGO Laboratory, 2017, accessed 2026-08-18. Data products.
• Kokkotas, Kostas D., and Bernd G. Schmidt. “Quasi-Normal Modes of Stars and Black Holes.” Living Reviews in Relativity 2 (1999): 2. DOI.
• Watts, Anna L., and Tod E. Strohmayer. “Detection with RHESSI of High-Frequency X-Ray Oscillations in the Tail of the 2004 Hyperflare from SGR 1806–20.” Astrophysical Journal Letters 637 (2006): L117–L120. DOI.