The Trembling of Stars: Asteroseismology and Measuring the Unreachable

The Hook: An Opaque Object, Precisely Measured

A star is the most opaque object imaginable. What we see is nothing but a thin surface layer. There is no telescope that looks inside a star, and no probe you could send in.

And yet: for more than 16,000 red giants the mean density has been determined seismically. For about 300 red giants we know how fast their core rotates — a quantity for which there is no other route at all. And for more than a dozen, the magnetic field strength in the core region has now been determined, in a region for which there were previously no direct measurements whatsoever.

The trick is the same one geophysicists used to map the Earth's core: you listen. Stars with convective envelopes oscillate in a particular way — the turbulence in their outermost layers excites standing sound waves that travel through the star, are reflected near the surface, and build up into resonances. Each of these resonances has a path through a particular depth. Its frequency is therefore an integral over the physics along that path. Anyone who can measure a star's frequency spectrum does not have a picture of its interior — but does have a measurement record.

This discipline is called asteroseismology. Conny Aerts's review in Reviews of Modern Physics (volume 93, article 015001, 2021) states the foundation plainly: "Yearslong time series of high-precision brightness measurements have been assembled for thousands of stars with telescopes operating in space." Years-long, gap-free, high-precision brightness measurements — only these make the spectrum visible. It cannot be done from the ground; day, night and weather chop up the time series.

This article pursues three questions. First: how do you get from a light curve to a star's mass, radius and age? Second: where exactly do the convenient formulas used for this break down — and how far can the breakage be repaired? And third, to me the most interesting question: what do you learn from a case in which a spectacular asteroseismic discovery was announced and subsequently could not be confirmed?

Part 1: What Makes a Star Ring

Two Restoring Forces, Two Families of Modes

An oscillation needs a restoring force. In stars there are two candidates, and they produce two fundamentally different families of modes.

p modes (pressure modes) are sound waves. The restoring force is the pressure gradient. They travel through the star, are reflected near the surface and form standing waves — acoustic resonances in a sphere. In the Sun their periods are on the order of minutes.

g modes (gravity waves) are something quite different. Their restoring force is buoyancy: a parcel of material displaced upward in a stably stratified zone is heavier than its surroundings there and falls back. g modes can therefore only propagate in stably stratified, that is non-convective, zones; in convection zones they are evanescent, their amplitude falling off exponentially. For the Sun this means they oscillate deep inside, in the radiative zone — precisely where one most urgently wants to look — and reach the surface only as a minute residual amplitude.

In their review of the asteroseismology of solar-type and red-giant stars (2013, DOI 10.1146/annurev-astro-082812-140938), Chaplin and Miglio state the excitation mechanism for the observable modes precisely: these stars "show rich spectra of solar-like oscillations, which are excited and intrinsically damped by turbulence in the outermost layers of the convective envelopes." Convection is therefore both loudspeaker and damper. This is why these oscillations are called stochastically excited: there is no clock, only broadband noise that nudges all modes in a frequency range with irregular amplitudes. (There are also stars that pulsate without a convective envelope, via heat-engine mechanisms; this article does not treat them.)

Mixed Modes: The Door-Opener to the Core

The most interesting thing happens when a star changes character. In red giants the core contracts and becomes extremely dense while the envelope swells up. As a result the frequency ranges of p and g modes slide over each other, and mixed modes arise. The description in Li and colleagues (Nature 610, issue 7930, pages 43–46, 2022) is brief and exact: "In red giant stars, non-radial acoustic waves couple to gravity waves and give rise to mixed modes, which behave as pressure (p) modes in the envelope and gravity (g) modes in the core."

Mixed modes are the real door-opener of this field. Such a mode is a gravity wave in the core and a sound wave in the envelope. It is visible at the surface — because it is acoustic there — and at the same time it carries information from the core, because it oscillated there. The cores of red giants are thus, paradoxical as it sounds, seismically more accessible than the solar core. For the outer layers the opposite still holds: with its thousands of resolved p modes, the Sun is the best-measured star of all.

Part 2: The Two Numbers That Reveal Almost Everything

A measured spectrum has two conspicuous properties even before a single mode has been identified.

First, a regularity. The radial modes are almost equidistant in frequency. Their mean spacing is called the large frequency separation Δν.

Second, an envelope. The amplitudes form a hump. Its peak is called ν_max, the frequency of maximum oscillation power.

From these two numbers — plus the effective temperature — mass and radius can be estimated. Saskia Hekker has compiled the relations used for this in a review (Frontiers in Astronomy and Space Sciences 7:3, 2020, DOI 10.3389/fspas.2020.00003). There they read:

$$\Delta\nu_0 = \left(\frac{M}{M_\odot}\right)^{1/2} \left(\frac{R}{R_\odot}\right)^{-3/2} \Delta\nu_\odot$$

$$\nu_\mathrm{max} = \frac{M/M_\odot}{(R/R_\odot)^2 \sqrt{T_\mathrm{eff}/5777\,\mathrm{K}}} \; \nu_{\mathrm{max},\odot}$$

(The normalization of 5777 K is as printed at the cited place; other equations in the same work use 5772.0 K. The difference is a tenth of a per mille and irrelevant here — it merely shows how many small conventions are baked into such formulas.)

Why the First Formula Is Plausible — and the Second Is Justified Differently

The Δν relation has a transparent meaning. Hekker gives the justification as follows: Δν "is physically justified as Δν is in an asymptotic approximation equal to the inverse of the sound travel time through the star". Kjeldsen and Bedding showed in 1995 that, with estimates for the pressure and temperature inside the star, one obtains "Δν ∝ sqrt(M/R^3), i.e., that the large frequency separation is directly proportional to the square root of the mean density of the star".

The core of the argument is a dimensional estimate: in the asymptotic approximation Δν is the inverse of the sound travel time through the star, and that travel time scales as √(R³/GM). From this follows Δν ∝ √(M/R³) — the square root of the mean density, not the central one.

What matters is what is not being said. The sound speed is c² = Γ₁P/ρ, so it depends on P/ρ and thereby on temperature and composition, not simply on density. That these dependences drop out of the scaling relation holds only approximately — and the rest of this article rests on exactly that.

With ν_max the situation is different, and here precision matters. Hekker writes that the ν_max relation was "defined empirically based on homology arguments with another typical dynamical timescale of the atmosphere, i.e., the acoustic cut-off frequency". According to the same work the cut-off frequency ν_ac scales as ν_ac ∝ g·T_eff^(−1/2) with the surface gravity g. It would be wrong to claim that the ν_max relation has no theoretical basis at all: the review explicitly reports that Belkacem and colleagues in 2011 "aimed to provide a theoretical basis for the scaling between ν_max and ν_ac". The correct finding is: Δν is asymptotically justified, ν_max was introduced empirically via homology arguments, and a theoretical grounding was attempted later.

An Uncomfortable Detail: Which Sun?

Both formulas are calibrated on the Sun. So one needs Δν_⊙ and ν_max,⊙ — and these are not unique. Kjeldsen and Bedding used Δν_⊙ = 134.9 μHz and ν_max,⊙ = 3.05 mHz. Hekker's Table 1 lists the values used over the years; they scatter between 134.88 μHz and 135.5 μHz for Δν_⊙ and between 3050 μHz and 3166 μHz for ν_max,⊙. The review notes: "Over the years, several authors have adopted different solar values based on internal calibrations."

For Δν that is only a few parts per thousand — but for the ν_max reference value it is nearly 4%. Anyone who wants masses accurate to 10% cannot ignore such calibration questions.

Part 3: Where the Rules of Thumb Break — and How Far They Can Be Repaired

Here it gets honest. In practice both relations are applied homologously: they presuppose that the target star is structurally similar to the reference star — including the asymptotically justified Δν relation, as soon as it is carried over to other stars via a single calibration point. Hekker names this weakness directly: the relations "provide decent estimates of the observed oscillations for a large range of stars", but "the inherent shortcomings of such relations, i.e., they rely on a homologous stellar structure between the target star and the reference, have been apparent."

The documented deviations, each in the form in which the review reports them:

Particularly instructive is an aside in Gaulme: the effective temperature is itself an error channel. Lower it by 100 K and the seismic masses and radii fall by 3.1% and 1.0% respectively. An error in the spectroscopy therefore migrates straight into the seismic mass.

On top of this comes the surface effect — defined in the review as "a frequency-dependent offset between observed and modeled frequencies that affects Δν" and attributed elsewhere (after Serenelli et al. 2017) to "the neglect of nonadiabatic effects in the outer most layers". The outermost layers of a star are precisely the ones stellar models describe worst — and they shift the frequencies systematically.

The Part You Must Not Leave Out: Corrections

It would be a distortion to stop at the deviations, because the uncorrected relations have long ceased to be the working standard. Hekker's review reports that Brogaard et al. (2018) and Themeßl et al. (2018) found in binary comparisons "that asteroseismic scaling relations without corrections to the Δν scaling relations would overestimate the masses and radii" — the problem therefore lies explicitly with the missing corrections, not with the method itself. On Brogaard et al. (2016) the same work records that they concluded "that they are accurate to within their uncertainties for giant stars".

Tanda Li and five co-authors take the route of detailed modelling rather than rules of thumb for 3,642 red giants (arXiv:2201.09577, DOI 10.3847/1538-4357/ac4fbf). Their finding: "We find a systematic offset of ~15% in mass and ~7% in radius between the modeling solutions and the scaling relations." They give the typical uncertainties of their model solutions as "4.5% for mass … 1.7% for radius", and on independent validation they write: "We also achieve good accuracy for estimated masses and radii, based on comparing with those determined for eclipsing binaries" — without quoting a number for it.

The correct summary is therefore not "the scaling relations are unreliable", but: their domain of validity has been mapped, the deviations have a known direction, and they are largely tractable through corrections or through detailed modelling.

One more piece of context: the review gives no overall accuracy of its own for masses and radii across all stellar classes. It does, however, quote class-specific numbers — besides the ≤4% for radii of main-sequence stars (Huber et al. 2012), for instance that the reference function of Guggenberger et al. (2016) allows masses and radii to be recovered "with an accuracy of 5 and 2 per cent, respectively", or that Bellinger (2019) quotes "uncertainties of 0.032 M⊙ (3.3 per cent), 0.011 R⊙ (1.1 per cent), and 0.56 Gyr (12 per cent)". Each of these numbers applies to a particular stellar class, method or correction. A single number for "the accuracy of the scaling relations" does not exist.

Part 4: The Harvest — From 66 Stars to 16,094

Two catalogues show the scale that space photometry has reached.

The precision work. Mikkel N. Lund and 19 co-authors analysed 66 main-sequence stars with solar-like oscillations in the "Kepler seismic LEGACY project" (arXiv:1612.00436, DOI 10.3847/1538-4357/835/2/172). For each star individual modes were identified and measured for frequency, amplitude and line width using a Bayesian Markov chain Monte Carlo procedure — "peak-bagging" in the jargon. The data basis: Kepler short-cadence data, "of which each star has at least 12 months". The paper emphasizes what this is good for: "The frequencies and frequency ratios can tightly constrain the fundamental parameters of these solar-type stars."

That is the laborious variant: not two summary numbers, but the full mode spectrum, compared with stellar models.

Mass production. Jie Yu and six co-authors, by contrast, systematically measured 16,094 oscillating red giants from the Kepler data up to the end of the mission (arXiv:1802.04455, DOI 10.3847/1538-4365/aaaf74). The catalogue quotes uncertainties of 1.6% for ν_max, 0.6% for Δν, 7.8% for the seismic mass, 2.9% for the radius, and 0.01 dex for log g. (The paper's abstract attaches the attribute "typical" explicitly to ν_max; the remaining values appear in the same list.)

These numbers have to be read correctly — and this is the point at which many popular accounts stumble. They are the formal uncertainties of the catalogue, not the accuracy relative to independent measurements. Part 3 showed that for red giants in binary systems, systematic overestimates of about 5% in radius and about 15% in mass were found. Precision and correctness are two different things, and with 16,094 stars precision becomes cheap while correctness does not.

Why Mass Fares Worse Than Radius

That mass is usually considerably more uncertain than radius follows from inverting both relations jointly. Solving the two formulas from Part 2 for M and R gives:

$$\frac{M}{M_\odot} \propto \left(\frac{\nu_\mathrm{max}}{\nu_{\mathrm{max},\odot}}\right)^{3} \left(\frac{\Delta\nu}{\Delta\nu_\odot}\right)^{-4} \left(\frac{T_\mathrm{eff}}{T_\odot}\right)^{3/2}$$

$$\frac{R}{R_\odot} \propto \left(\frac{\nu_\mathrm{max}}{\nu_{\mathrm{max},\odot}}\right)^{1} \left(\frac{\Delta\nu}{\Delta\nu_\odot}\right)^{-2} \left(\frac{T_\mathrm{eff}}{T_\odot}\right)^{1/2}$$

All the exponents are two to three times larger for mass than for radius; input errors are amplified correspondingly more strongly. In Yu's catalogue that appears as 7.8% versus 2.9%. And Gaulme's aside is a clean cross-check: the same temperature change shifts the mass by 3.1% and the radius by 1.0% — a ratio of exactly 3:1, as the exponents 3/2 and 1/2 demand.

Better determined than either on its own is only their combination M/R³, that is the mean density: it depends on Δν alone, the quantity measured most precisely at 0.6%. This ranking can, however, be reversed as soon as independent additional information comes in — a radius from a Gaia parallax plus flux, or a dynamical mass from a binary system.

Part 5: Looking Into the Core — Rotation, and a Model That Does Not Fit

Now it gets exciting, because here asteroseismology delivers something no other method can: the rotation profile inside a star.

Rotation lifts the degeneracy of the modes in azimuthal order: a mode splits into a symmetric multiplet. Formally this recalls Zeeman splitting, but physically it works via the Coriolis force rather than a magnetic moment — and the fact that a real magnetic field shifts this multiplet asymmetrically becomes important below. The size of the splitting is a weighted mean of the rotation rate over the region in which the mode has its amplitude. For mixed modes that weight lies predominantly in the core.

2012: The Core Rotates Faster Than the Surface

Paul G. Beck and 23 co-authors reported in Nature (DOI 10.1038/nature10612, arXiv:1112.2825) the detection of non-rigid rotation inside red giants. Their wording is remarkably cautious: "Comparing with theoretical stellar models, we conclude that the core must rotate at least ten times faster than the surface." Not "ten times as fast", but at least ten times — a lower bound. And they placed the result correctly: "This observational result confirms the theoretical prediction of a steep gradient in the rotation profile towards the deep stellar interior."

2012: But Far Slower Than Angular Momentum Conservation Demands

Benoît Mosser and 20 co-authors extended this to about 300 red giants (arXiv:1209.3336, DOI 10.1051/0004-6361/201220106) and found a nuanced picture. Ascending the giant branch, the core rotation increases slightly: "we observe a small increase of the core rotation of stars ascending the red giant branch." At the same time: "an important slow down is observed for red-clump stars compared to the red giant branch", and from this the overall statement "the mean core rotation significantly slows down during the red giant phase", where "the slow-down occurs in the last stages of the red giant branch."

The finding is therefore not that expectation is turned on its head, but that it is quantitatively missed by a wide margin. The core of a star on the giant branch contracts — pure angular momentum conservation would have to accelerate it dramatically. What is observed is only a small increase, followed by a slow-down in the final stages. The core is therefore losing angular momentum outward. (In the comparison with red-clump stars there is the additional point that their core is no longer degenerate after the helium flash and has expanded again; that comparison cannot be read in terms of angular momentum conservation alone.)

How Much Stronger Must the Coupling Be?

Patrick Eggenberger, Josefina Montalbán and Andrea Miglio quantified exactly this (arXiv:1207.1023, DOI 10.1051/0004-6361/201219729). Their finding for the red giant KIC 8366239: "Meridional circulation and shear mixing alone are found to produce an insufficient internal coupling so that an additional mechanism for the internal transport of angular momentum is needed during the post-main sequence evolution." They state the additional viscosity derived from the observed splittings as tightly constrained: ν_add = 3 × 10⁴ cm² s⁻¹.

That is a model-dependent effective parameter for one star, not a measured transport coefficient — and follow-up work has spread it out widely. Moyano and colleagues (A&A 663, A180, 2022, DOI 10.1051/0004-6361/202243389) report: "The additional viscosity is found to vary by approximately two orders of magnitude in the mass range M ∼ 1–2.5 M⊙", and for the red clump "an acceptable value of the additional viscosity for red clump stars is ν_add = (3 ± 2)×10^5 cm^2 s^−1" — an order of magnitude above younger giants of comparable mass.

Li and colleagues put the problem even more directly. The rotation measurements, they write, led to the conclusion "that purely hydrodynamical processes of angular momentum transport from the core are too inefficient. Magnetic fields could produce the additional required transport." Except: "due to the lack of direct measurements of magnetic fields in stellar interiors, very little is currently known about their properties."

2022: The Magnetic Field, Measured Directly

That is precisely the gap Gang Li, Sébastien Deheuvels, Jérôme Ballot and François Lignières closed (Nature 610, issue 7930, pages 43–46, 2022, DOI 10.1038/s41586-022-05176-0). The principle: a magnetic field shifts mode frequencies just as rotation does — but differently. Rotation splits a dipole multiplet symmetrically; a magnetic field breaks that symmetry. "The fields induce shifts that break the symmetry of dipole mode multiplets."

The result, in the published version: "We thus measure field strengths ranging from about 30 kilogauss to about 100 kilogauss in the vicinity of the hydrogen-burning shell and place constraints on the field topology." For comparison: the magnetic field at the solar surface is orders of magnitude weaker.

Two caveats belong with this. It concerned three stars — "the cores of three red giant stars observed with the Kepler satellite" — so a first measurement, not a statistic. And the field strength is determined in the vicinity of the hydrogen-burning shell, not at the geometric centre.

2023 to 2026: Three Stars Become a Field

The sample has grown since. Gang Li, Sébastien Deheuvels, Tanda Li, Jérôme Ballot and François Lignières filtered 13 stars with clearly asymmetric multiplets out of 2,500 Kepler giants with clear mixed-mode patterns (arXiv:2309.13756): "we measured average core magnetic fields ranging from 20 to 150kG, corresponding to 5% to 30% of the critical field strengths." Their evolutionary finding is especially interesting: "We found that the field strengths decrease with stellar evolution, despite the fact that the cores of these stars are contracting."

Sébastien Deheuvels, Jérôme Ballot, François Lignières, Gang Li and M. Villate examined eight further Kepler giants with l = 1 doublets in 2026 (arXiv:2604.09901, DOI 10.1051/0004-6361/202659344) and arrive at considerably stronger fields: "These solutions correspond to fields with intensities ranging from 100 to 700 kG that are confined well below the H-burning shell."

And one caveat worth knowing: asymmetric splitting is not explicable by magnetism alone. J. M. Joel Ong, Lisa Bugnet and Sarbani Basu have shown that rotation on its own also produces asymmetric splittings when mixed modes lie close together (arXiv:2210.01928, related DOI 10.3847/1538-4357/ac97e7): "for evolved stars exhibiting mixed modes, avoided crossings between different multiplet components are known to yield asymmetric rotational splitting, particularly for near-degenerate mixed-mode pairs". They explicitly investigate the observational consequences for measurements of asymmetric splitting, "which has been proposed as a magnetic-field diagnostic". I am of the opinion that this mechanism must be read as a possible confounder of the magnetic interpretation — the paper itself issues no such warning, but merely works through the competing effect.

That magnetic fields are the additional angular-momentum transporter has become more plausible through this series of measurements. It is not proven.

Part 6: Age — The Currency of Galactic Archaeology

A star's mass and radius can also be determined by other means — its age only with far more difficulty. It is not a directly measurable quantity but is inferred by comparison with evolutionary models. Alternatives do exist (isochrones for star clusters, white-dwarf cooling, nucleocosmochronometry, gyrochronology), but they are either restricted to special cases or considerably less precise. And age is the quantity on which the history of the Milky Way hangs.

Two examples show what asteroseismic ages can do.

Kepler-444. T. L. Campante and 40 co-authors studied a compact system of five planets with sizes between Mercury and Venus (arXiv:1501.06227, DOI 10.1088/0004-637X/799/2/170). The host star is "the densest star with detected solar-like oscillations". The result: "We use asteroseismology to directly measure a precise age of 11.2+/-1.0 Gyr for the host star, indicating that Kepler-444 formed when the Universe was less than 20% of its current age and making it the oldest known system of terrestrial-size planets." An uncertainty of 1.0 Gyr on 11.2 Gyr — about 9%.

The early assembly of the Milky Way. Josefina Montalbán and 18 co-authors combined asteroseismic ages with kinematics and chemical abundances (Nature Astronomy 2021, DOI 10.1038/s41550-021-01347-7, arXiv:2006.01783). Precision achieved: "precise stellar ages (~11%)". This made a chronological question decidable: did the Milky Way form its thick disc before or after the in-fall of the satellite galaxy Gaia-Enceladus/Sausage? The finding: "we provide compelling evidence in favour of models in which the Galaxy had already formed a substantial population of its stars (which now reside mainly in its thick disc) before the in-fall of the satellite galaxy Gaia-Enceladus/Sausage … around 10 billions years ago".

That is the real lever. Without ages you have a heap of stars with orbits and abundances, but no ordering. With ages you get a chronology — and thus a galactic history you can test against cosmological models. The authors themselves name why this was not possible before: "it has so far been challenging owing to difficulties in obtaining precise ages for these oldest stars."

And the Same Caveat as for the Masses

The 9% and the 11% are precisions. For correctness, what Part 4 showed for masses applies — and for ages even more sharply, because here models alone are speaking. B. Nsamba and six co-authors worked out, for 34 solar-type stars, how strongly the choice of input physics and surface correction alone shifts the result (arXiv:1804.04935, DOI 10.1093/mnras/sty948): models with and without diffusion differ by "16%" in age, the uncertainty in the solar metallicity mixture contributes "6.7% in age", and even the best surface-correction methods leave "~8%" in age. Their conclusion is unambiguous: "The internal systematics in age are significantly larger than the statistical uncertainties."

A seismic age of "11.2 ± 1.0 Gyr" is therefore a precise statement within one family of models. The scatter across families of models is larger than the quoted error bar.

Part 7: The Case of the Solar g Modes — A Lesson in Evidence

Now to the part that seems to me most valuable for transfer to other fields.

The Discovery

The Sun's g modes are the holy grail of helioseismology: they oscillate in the radiative zone and carry information about the solar core, but reach the surface only as a minute residual amplitude. Writing in Astronomy & Astrophysics (volume 604, article A40, 2017, DOI 10.1051/0004-6361/201730460), Fossat and colleagues say of the rotational characteristics of the g modes that these "have, until now, remained unknown", and note elsewhere that earlier claimed detections in this part of the spectrum "have always been doubted and have never been confirmed, despite numerous efforts."

Then Eric Fossat and 13 co-authors reported the detection. Their approach was indirect and elegant: instead of looking for individual g modes, they looked for a collective modulation that periodic changes in the deep solar structure impose on the directly observable p modes. The data basis: two decades of the GOLF instrument on the SOHO satellite. The results were spectacularly concrete: the asymptotic period spacing P₀ "is measured to be 34 min 01 s, with a 1 s uncertainty", a rotation of 1277 ± 10 nHz averaged with weights over the modes' sensitivity kernels, and from this "a rapid rotation frequency of 1644 ± 23 nHz (period of one week) of the solar core itself, which is a factor 3.8 ± 0.1 faster than the rotation of the radiative envelope."

A solar core turning almost four times faster than the layer above it. That would have been a major finding.

The First Re-examination: Fragile

Hannah Schunker, Jesper Schou, Patrick Gaulme and Laurent Gizon repeated the analysis (Solar Physics volume 293, article 95, 2018, DOI 10.1007/s11207-018-1313-6). Their title is already the verdict: "Fragile Detection of Solar g-Modes by Fossat et al."

They were initially able to reproduce the result qualitatively — "we were able to qualitatively reproduce the results of Figure 10 in that article. However, our results are slightly different…" The significance of the central peak at 210 nHz came out at 4.0 σ for them, against 4.7 σ in the original paper.

Then they began varying the analysis parameters:

• Start time. Shift the beginning of the 16.5-year time series by a few hours and the detection vanishes: "When we removed two hours and ten hours, there were no significant peaks." The decisive sentence: "the parameter is unrelated to the properties of the Sun."
• Cadence. "The signal that is evident in the autocorrelation for a four-hour cadence vanishes for a 3.9 hour cadence."
• Duty cycle. With reduced data coverage the significance falls to 3.7 σ.

Their conclusion: "We conclude that this reported detection of gravity modes is extremely fragile and should be treated with utmost caution." And on interpretation, explicitly in the conditional: "the peaks found by Fossat et al. (2017) could be an artefact of the analysis method and the choice of cadence."

The Second: Not Robust

P. H. Scherrer and D. O. Gough additionally examined data from other instruments — LOI and MDI on SOHO, HMI on SDO, and the ground-based networks BiSON and GONG (arXiv:1904.02820, DOI 10.3847/1538-4357/ab13ad, 2019). They "found the evidence reported by Fossat et al. not to be robust" and conclude: "We are therefore led to doubt the claim." They name two structural problems: the inferred g-mode amplitudes "appear to exceed the upper bound reported by Appourchaux et al. (2010)", and "the theory of the procedure implies that the principal modes claimed to have been measured should be undetectable." They also point out that alternative evaluations of the same method can lead to a core rotation only about twice as fast — or even slower — than the envelope.

The Third: Not Confirmed, and the Follow-Up an Artefact

The hardest test comes from Thierry Appourchaux and Thierry Corbard (Astronomy & Astrophysics volume 624, article A106, 2019, DOI 10.1051/0004-6361/201935196). They report: "Their claims of detection of g modes cannot be confirmed when changing parameters such as sampling interval, length of time series, or photomultipliers", and: "Other instrument such as GONG and BiSON do not confirm their detection."

Decisive is their noise finding. The authors of the original work had followed up in 2018 (Fossat & Schmider, A&A 612, L1, DOI 10.1051/0004-6361/201832626), treating a high correlation between spectrum and g-mode model as confirmation. Appourchaux and Corbard reproduce this correlation — and show that it proves nothing: "the same occurs for pure noise due to the large number of peaks present in the model." On the follow-up paper they state clearly in the abstract: "We conclude that the detection performed in Fossat and Schmider (2018) is an artefact of the methodology." In the conclusion somewhat more cautiously: "We believe that the detection of g modes as made in FS18 is the result of an artefact of the analysis methodology related to the large number of modes used in the analysis." On the original paper they stay with: "We cannot confirm the g-mode detection claims of F17 and FS18."

A review by K. Belkacem, C. Pinçon and G. Buldgen (Amplitudes of Solar Gravity Modes: A Review) summarizes the state of play thus: "Again, the detection has not been confirmed and was questioned by Schunker et al. (2018) and Appourchaux and Corbard (2019)", and: "the quest of solar g-modes, while at present unsuccessful, is still ongoing." Their own theoretical assessment: the low-frequency solar gravity modes are "currently at best at the limit of the detection with the GOLF instrument."

No retraction or correction notice for the original 2017 paper is to be found in the sources checked in the course of this research.

What This Case Says — and What It Does Not

Here one has to stay careful, because it is tempting to claim more than the sources support. None of the re-examinations declares the 2017 finding formally refuted. The degrees of strength are graded: "extremely fragile" (Schunker et al.), "not to be robust" and "led to doubt the claim" (Scherrer and Gough), "cannot be confirmed" (Appourchaux and Corbard, for the 2017 paper) — and only for the follow-up paper of 2018 does the word "artefact" appear.

And here is the formulation that matters: a detection that vanishes when the start time of the time series is shifted by two hours is not a dependable detection. That does not establish that no solar signal is present — a weak genuine signal near the detection threshold can fluctuate with the particular realization of the data, and that is exactly the explanation Appourchaux and Corbard offer. It establishes that the evidence presented is not sufficient to distinguish the signal from noise. The start time of the measurement is a property of the experiment and not of the object under study — which is why a claim resting on it must not count as established.

What, then, does 4.7 σ mean? It means: assuming pure noise and this one analysis path, a peak this high would be very unlikely. Both conditions are essential. The number says nothing about the stability of the result across analysis paths — and it is additionally biased upward as long as the degrees of freedom in the analysis (cadence, start time, assumed P₀) are not folded into it. Appourchaux and Corbard demonstrate this directly: they obtain high correlations even for P₀ values far outside the physically plausible range.

Part 8: What Comes Next — PLATO

Asteroseismology came into being through space telescopes and depends on them. CoRoT and Kepler made the field. The next dedicated mission is ESA's PLATO (PLAnetary Transits and Oscillations of stars).

The mission overview by Heike Rauer and a very large collective of authors (arXiv:2406.05447) describes the payload as "26 cameras with 12cm aperture each" and the duration as "For at least four years". The goal is explicitly both: planet hunting and "asteroseismic monitoring of a large number of stars". On accuracy it says: "planets will be characterised for their radius, mass, and age with high accuracy (5 %, 10 %, 10 % for an Earth-Sun combination respectively)."

That sentence must be read with care, and I deliberately soften it: in both sources checked, the 10% for age appears in the context of planet characterization. ESA's cosmos.esa.int page phrases it in parallel — radii with "5 per cent precision", masses "better than 10 per cent precision", ages "10 per cent precision" — and separately names the means: "determination of accurate stellar masses, radii, and ages from asteroseismology", there without a number. A target accuracy explicitly identified as being for stellar ages I did not find in the official sources checked. Physically the connection is clear — the age of a planetary system is the age of its star — but the number itself is not stated there as a stellar-age requirement. And after Part 6, such a requirement would in any case be a statement about precision, not automatically about correctness.

On the launch date there is a discrepancy worth knowing: the mission overview (arXiv version from the end of 2024) states "a launch date end 2026". ESA's esa.int page, at the time of this research, gives "Launch: planned for January 2027", with Ariane 6 from Europe's Spaceport in French Guiana and a halo orbit around the Lagrange point L2. What is dependable is therefore: early 2027, date still in motion.

Part 9: The Bridge — Inverse Problems and the Treachery of the Pipeline

What does this have to do with software architecture and operations? More than it looks — because asteroseismology is a textbook case of an inverse problem: you have no access to a system's state, only to its output signal, and must infer the state from the signal.

That is exactly the situation with a distributed system in production. You cannot look inside. You have latency distributions, error rates, queue lengths, GC pauses — a spectrum of observables. From this spectrum you reconstruct an internal state, via a model.

Three parallels seem to me to hold — and I mark them explicitly as my interpretation, not as statements of the cited works:

1. Scaling relations are heuristics with a domain of validity. Δν and ν_max deliver mass and radius from two numbers — cheap, fast, feasible for 16,094 stars. They hold as long as the homology assumption carries. For metal-poor stars the uncorrected relation overestimates the mass by about 16%; with corrections to the reference values, or via detailed modelling, that bias is largely tractable. The counterpart in engineering is every capacity formula of the kind "throughput ≈ cores × rate" or "cost ≈ requests × price per request". These formulas are not wrong; they are calibrated on a reference case. The real engineering question is rarely "is the formula right?" but "how far is my case from the reference case, and in which direction does the formula then err?" For metal-poor stars the direction and the magnitude are known — which is why correction is possible. For your own capacity formula, usually neither is known, and therefore correction is not possible.

2. Precision is not correctness. Yu and colleagues' catalogue quotes 7.8% mass uncertainty; the binary test shows about 15% systematic overestimation by the uncorrected relations. Both are true at once, because they are two different quantities: scatter within the method versus deviation from an independent reference. For ages the gap is larger still — a 9% error bar against a 16% shift from a single different assumption about diffusion. Many monitoring dashboards with three decimal places have this problem. The number is precise because the same code always computes it the same way. Whether it is correct is known only to whoever holds an independent measurement alongside it — a load test against the capacity formula, a manual count against the metric, a second tool against the first.

3. Pipeline sensitivity is the cheapest validity criterion there is — and the first one to check. This is, for me, the most valuable lesson from the g-mode case. Schunker and colleagues built no new instrument and developed no new theory. They varied a parameter that has nothing to do with the object under study — the start time of the time series — and the finding vanished.

It is not, however, the hardest criterion. That remains independent replication with other data and other instruments, as Scherrer and Gough carried out with MDI, LOI, HMI, GONG and BiSON, and as Appourchaux and Corbard followed up with the individual GOLF photomultipliers. Pipeline variation can overturn a finding, but it cannot confirm one.

The same check usually costs little in practice — and in my experience is rarely done systematically. A performance gain that appears only at this warm-up time. An A/B result that flips when the evaluation window is shifted by a day. An error rate that depends on the bucketing of the time axis. A cost analysis that only works out at this month boundary. In all these cases the result hangs on a parameter that is a property of the measurement and not of the system measured.

The conceptual distinction behind this is constantly confused, and it is subtler than it looks. Significance quantifies how unlikely the data would be under the null hypothesis — for one fixed analysis path. Robustness is the stability of the result across analysis paths. The two are not merely different, they are connected: anyone who could choose the path after looking at the data has reported a systematically inflated significance. A result can therefore be nominally highly significant and at the same time not robust — and then the significance figure was already too optimistic. That is precisely the state the g-mode announcement is in.

The Central Takeaway

Asteroseismology shows how far a system inaccessible from the outside can be characterized when you can measure its eigenmode spectrum and hold a workable model alongside it — and where the limits of that procedure lie. Three levels:

1. Scientifically it has delivered what was previously unreachable: the rotation profile inside stars; magnetic field strengths in the core region (in 2022 first in three red giants at about 30 to 100 kG in the vicinity of the hydrogen-burning shell, in 2023 in 13 stars at 20 to 150 kG, in 2026 in eight more at 100 to 700 kG further down); and ages of individual stars with formal uncertainties around 10% (Kepler-444: 11.2 ± 1.0 Gyr; the Milky Way data set of Montalbán and colleagues: about 11%).
2. Methodologically it shows the price of shortcuts with unusual clarity — and how that price is paid. The scaling relations are applied homologously and break down where the structure deviates: about 16% mass overestimation for metal-poor stars, about 15% for red giants in the binary test. This is not a weakness of the field but its strength: the deviations are mapped, published, and largely tractable through corrections or detailed modelling. The same caveat applies to ages, where the choice of input physics alone shifts the result by up to 16% — more than the quoted error bar.
3. Epistemically the case of the solar g modes provides the best short case study in robustness I know. A 4.7 σ finding whose detection vanishes when the start time of the time series is shifted by two hours is not established — and that holds entirely independently of whether it turns out to be true in the end.

A concrete prompt for this week: take one numerical result on which a decision is currently being based in your environment — a benchmark, an A/B result, a capacity or cost projection. Find in it the one parameter that is a property of the measurement and not of the system: the evaluation window, the warm-up time, the bucket boundary, the start time, the sampling interval. Then vary it once by a plausibly small amount and recompute. If the statement stands, you have hardened it cheaply. If it flips, you have found something more important than any further decimal-place update.

A question to reflect on: for which of your currently most dependable metrics do you actually not know how it changes when an arbitrary parameter of the measurement — not of the system — is shifted a little? And what would it cost you to try that exactly once?

Fact Check

166 claims were checked: 50 hard facts (numbers, years, titles, author counts, volumes, article numbers, DOIs), 59 verbatim quotations, 28 source attributions and 29 synthesis claims, plus a systematic check for internal contradictions. 29 claims were corrected or weakened — of these 8 were discarded or set right as false, 20 were weakened and 1 was marked as not verifiable; in addition, 10 internal contradictions were resolved. The check was carried out by two independent reviewing instances that received only the draft and the source list and whose brief was to find errors.

The three most important corrections:

1. The draft concluded from the start-time dependence that the g-mode signal "cannot come from the Sun". That is a fallacy and supported by no source: a weak genuine signal near the detection threshold may fluctuate with the realization of the data. What is correct is only that the detection is not dependable. The passage was reworded accordingly, and the same weakening was carried through into Part 9 and the Central Takeaway.
2. The draft justified Δν ∝ √(mean density) on the grounds that "the sound speed is related to density". That is physically wrong (c² = Γ₁P/ρ depends on temperature and composition) and contradicted the finding of Miglio et al. (2012), cited two sections later, that two models of equal mean density differ by 3.3% in Δν. The justification now runs correctly via the sound travel time.
3. The draft presented the uncorrected scaling relations as the state of the field and left out the fact that the deviations are routinely corrected. The findings of Brogaard et al. (2016/2018) and Themeßl et al. (2018) from the same review were added, as was the modelling of T. Li et al. (2022) for 3,642 red giants. Part 9, point 1 and the Central Takeaway were rewritten accordingly.

Further corrections: a quotation from Fossat et al. (2017) had shifted the subject — what was unknown were the rotational characteristics of the g modes, not the g modes themselves. The paper by Li et al. (2022) was wrongly called a "review"; it is a first measurement in three stars. Beck et al. (2012) were credited with a priority ("the first such measurement") that the abstract does not claim. The field-strength quotation from Li et al. (2022) was a hybrid of preprint and published version and is now quoted from the published version. The draft left out Mosser's own counter-finding of a small increase in core rotation while ascending the giant branch, and explained the slow-down of clump stars by a contracting core — their core has expanded after the helium flash. The claim that Hekker's review quotes no hard accuracy figure was too strong; it quotes several class-specific ones. The error-propagation argument via the exponents −3/2 and +1/2 was not sound and was replaced by the correct joint inversion of both relations (M ∝ ν_max³ Δν⁻⁴ T^{3/2}, R ∝ ν_max Δν⁻² T^{1/2}). The statement about 4.7 σ inverted the conditioning direction of a p-value. The claims "g modes exist only in stably stratified zones", "red giants are easier to see into than the Sun", "pipeline sensitivity is the hardest validity criterion" and "asteroseismology is the working proof" were weakened, as were several universal statements about software engineering practice ("always", "never", "every", "is almost never done"). Newly incorporated were Appourchaux & Corbard (2019) as the third and hardest re-examination of the g-mode announcement, the review assessment by Belkacem et al., the magnetic-field follow-up papers of 2023 and 2026, the competing rotational mechanism for asymmetric splittings from Ong, Bugnet & Basu, the spread in viscosity in Moyano et al. (2022), and the age systematics in Nsamba et al. (2018). Not verifiable was the empirical blanket statement about how widespread pipeline-sensitivity checks are in software practice; it is now marked as personal experience. Volume, issue and pages are given only where a source opened during this run prints them.

Automated AI check against primary sources; no human expert review.

Cross-References in the Vault

• The Bottleneck at Mass Eight: The Hoyle State and How Stars Invent Carbon – the nuclear physics running inside the stars measured seismically here; asteroseismology supplies the boundary conditions against which models of these burning zones can be tested.
• Which One of You Was It? The Duhem-Quine Problem and the Underdetermination of Theory – the g-mode case is a textbook instance of exactly that structure: does the finding contradict the Sun, the instrument, or the analysis pipeline?
• The Cosmic Tension: Why the Universe Seems to Have Two Rates of Expansion – there too the decisive question is whether a discrepancy is physical or methodological; the same figure of thought, a different object.
• Messengers from the Hidden: Neutrinos, the Ghost Particles, and the Birth of a New Astronomy – the second channel into the solar core: neutrinos carry information out of the burning zone, oscillations carry it via the structure.
• The Clock That Feels a Centimetre: Optical Atomic Clocks and Measuring Time at the Quantum Limit – as here, everything hangs on a frequency measurement; and as here, distinguishing precision from correctness is the real work.

Sources and Further Reading

Reviews

• C. Aerts – Probing the interior physics of stars through asteroseismology, Reviews of Modern Physics 93, 015001 (2021), DOI 10.1103/RevModPhys.93.015001: https://link.aps.org/doi/10.1103/RevModPhys.93.015001 (preprint: https://arxiv.org/abs/1912.12300)
• W. J. Chaplin, A. Miglio – Asteroseismology of Solar-Type and Red-Giant Stars (2013), DOI 10.1146/annurev-astro-082812-140938: https://arxiv.org/abs/1303.1957
• S. Hekker – Scaling Relations for Solar-Like Oscillations: A Review, Frontiers in Astronomy and Space Sciences 7:3 (2020), DOI 10.3389/fspas.2020.00003: https://www.frontiersin.org/journals/astronomy-and-space-sciences/articles/10.3389/fspas.2020.00003/full
• K. Belkacem, C. Pinçon, G. Buldgen – Amplitudes of Solar Gravity Modes: A Review: https://arxiv.org/html/2210.09229

Catalogues and modelling

• M. N. Lund et al. – Standing on the shoulders of dwarfs: the Kepler asteroseismic LEGACY sample I – oscillation mode parameters, DOI 10.3847/1538-4357/835/2/172: https://arxiv.org/abs/1612.00436
• J. Yu et al. – Asteroseismology of 16000 Kepler Red Giants: Global Oscillation Parameters, Masses, and Radii, DOI 10.3847/1538-4365/aaaf74: https://arxiv.org/abs/1802.04455
• T. Li et al. – Asteroseismology of 3,642 Kepler Red Giants: Correcting the Scaling Relations based on Detailed Modeling, DOI 10.3847/1538-4357/ac4fbf: https://arxiv.org/abs/2201.09577

Rotation, angular momentum transport and magnetic fields inside stars

• P. G. Beck et al. – Fast core rotation in red-giant stars revealed by gravity-dominated mixed modes, Nature (2012), DOI 10.1038/nature10612: https://arxiv.org/abs/1112.2825
• B. Mosser et al. – Spin down of the core rotation in red giants, DOI 10.1051/0004-6361/201220106: https://arxiv.org/abs/1209.3336
• P. Eggenberger, J. Montalbán, A. Miglio – Angular momentum transport in stellar interiors constrained by rotational splittings of mixed modes in red giants, DOI 10.1051/0004-6361/201219729: https://arxiv.org/abs/1207.1023
• F. D. Moyano et al. – Asteroseismology of evolved stars to constrain the internal transport of angular momentum. V., A&A 663, A180 (2022), DOI 10.1051/0004-6361/202243389: https://www.aanda.org/articles/aa/full_html/2022/07/aa43389-22/aa43389-22.html
• G. Li, S. Deheuvels, J. Ballot, F. Lignières – Magnetic fields of 30 to 100 kG in the cores of red giant stars, Nature 610 (7930), 43–46 (2022), DOI 10.1038/s41586-022-05176-0: https://arxiv.org/abs/2208.09487
• G. Li, S. Deheuvels, T. Li, J. Ballot, F. Lignières – Internal magnetic fields in 13 red giants detected by asteroseismology: https://arxiv.org/abs/2309.13756
• S. Deheuvels, J. Ballot, F. Lignières, G. Li, M. Villate – Near-critical magnetic fields in Kepler red giants (2026), DOI 10.1051/0004-6361/202659344: https://arxiv.org/abs/2604.09901
• J. M. J. Ong, L. Bugnet, S. Basu – Mode Mixing and Rotational Splittings: I. Near-Degeneracy Effects Revisited, related DOI 10.3847/1538-4357/ac97e7: https://arxiv.org/abs/2210.01928

Stellar ages

• T. L. Campante et al. – An ancient extrasolar system with five sub-Earth-size planets, DOI 10.1088/0004-637X/799/2/170: https://arxiv.org/abs/1501.06227
• J. Montalbán et al. – Chronologically dating the early assembly of the Milky Way, Nature Astronomy (2021), DOI 10.1038/s41550-021-01347-7: https://arxiv.org/abs/2006.01783
• B. Nsamba et al. – Asteroseismic modelling of solar-type stars: internal systematics from input physics and surface correction methods, DOI 10.1093/mnras/sty948: https://arxiv.org/abs/1804.04935

The case of the solar g modes

• E. Fossat et al. – Asymptotic g modes: Evidence for a rapid rotation of the solar core, A&A 604, A40 (2017), DOI 10.1051/0004-6361/201730460: https://arxiv.org/abs/1708.00259
• H. Schunker, J. Schou, P. Gaulme, L. Gizon – Fragile Detection of Solar g-Modes by Fossat et al., Solar Physics 293, 95 (2018), DOI 10.1007/s11207-018-1313-6: https://doi.org/10.1007/s11207-018-1313-6
• E. Fossat, F. X. Schmider – More about solar g modes, A&A 612, L1 (2018), DOI 10.1051/0004-6361/201832626: https://doi.org/10.1051/0004-6361/201832626
• P. H. Scherrer, D. O. Gough – A Critical Evaluation of Recent Claims Concerning Solar Rotation (2019), DOI 10.3847/1538-4357/ab13ad: https://arxiv.org/abs/1904.02820
• T. Appourchaux, T. Corbard – Searching for g modes. II. Unconfirmed g-mode detection in the power spectrum of the time series of round-trip travel time, A&A 624, A106 (2019), DOI 10.1051/0004-6361/201935196: https://www.aanda.org/articles/aa/full_html/2019/04/aa35196-19/aa35196-19.html

The future

• H. Rauer et al. – The PLATO Mission: https://arxiv.org/abs/2406.05447
• ESA – Plato (mission overview, launch date): https://www.esa.int/Science_Exploration/Space_Science/Plato
• ESA Cosmos – PLATO (accuracy requirements): https://www.cosmos.esa.int/web/plato

Created as part of the daily learning workflow. Area of interest: Astrophysics. Estimated reading time: ~30 minutes.

← All articles