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Sandbox Physics
Read the interior of a star

AC26 · L3

From a light curve to stellar modes

Sandbox Physics ·

A spectral peak contains both a stellar rhythm and the effects of observing it. To estimate a frequency, start with the saved samples, their times and exposures, then fit a model that includes stochastic excitation and the observing window. A fine frequency grid helps evaluate that model; it does not create extra observing time.

Save what was observed

A lightcurve is a sequence of times, measured fluxes and uncertainties. Quality flags identify samples that may need excluding. Keep those fields and the source version together. If a sample is missing, it has not become a measurement of zero brightness.

For the synthetic cavity, the observable is a declared dimensionless boundary-displacement proxy. A detector averages it during an exposure and then adds measurement noise. This differs from the instantaneous pressure at the acoustic boundary, which is constrained to zero. Real stellar photometry is a separate observation with its own calibration.

Take the measurement time as the exposure midpoint. The observation equation below defines the exposure duration and noise explicitly. The numerical analysis receives these saved values; it must not read the generating mode frequencies as if they had been measured.

yi=1Δi∫ti−Δi/2ti+Δi/2q(t) dt+ϵi,ϵi∼N(0,σi2)y_i=\frac{1}{\Delta_i}\int_{t_i-\Delta_i/2}^{t_i+\Delta_i/2}q(t)\,dt+\epsilon_i,\qquad\epsilon_i\sim\mathcal N(0,\sigma_i^2)
qq is the underlying displacement proxy, tit_i the midpoint, Δi\Delta_i the exposure duration and σi\sigma_i the measurement-noise standard deviation.

Reading: MAST public data use and acknowledgment · Lindsay et al. 2025, Precise Asteroseismic Ages for the Helmi Streams

A mode is continually excited and damped

A coherent sine wave keeps a single phase forever. Solar-like stellar modes do not. Convective forcing continually supplies new motion while damping removes energy. Their amplitudes and phases wander, broadening the spectral peaks.

A useful linear model is a second-order oscillator with natural angular frequency ω0\omega_0 and amplitude damping rate γ\gamma. White acceleration represents forcing with a correlation time much shorter than the mode evolution being modeled. Its strength is chosen so the stationary displacement variance is VqV_q.

The natural frequency and the tallest point of the continuous power spectrum are not exactly the same when damping is appreciable. In the weak-damping limit the peak is approximately Lorentzian near the natural frequency. Its cyclic-frequency full width at half maximum is approximately damping rate divided by pi. Always state whether a quoted width is full width or half width.

The browser records four independently forced modes with the same declared damping time and chosen displacement-proxy variances. Those weights are part of the synthetic measurement definition, not a prediction of stellar radiative brightness.

q¨+2γq˙+ω02q=η(t),τamp=1/γ\ddot q+2\gamma\dot q+\omega_0^2q=\eta(t),\qquad\tau_{\rm amp}=1/\gamma
The forcing η\eta is white acceleration. The amplitude damping time is distinct from an energy-decay time.
Sq(ω)∝1(ω02−ω2)2+4γ2ω2,Γν≃γπS_q(\omega)\propto\frac{1}{(\omega_0^2-\omega^2)^2+4\gamma^2\omega^2},\qquad\Gamma_{\nu}\simeq\frac{\gamma}{\pi}
The width approximation applies for weak damping. The peak occurs at ω02−2γ2\sqrt{\omega_0^2-2\gamma^2} when that expression is real.

Reading: MESA School 2026, Lab 3: Asteroseismology · Lindsay et al. 2025, Precise Asteroseismic Ages for the Helmi Streams

Exposure and gaps reshape the spectrum

An exposure averages changing displacement. For a coherent sinusoid and a rectangular exposure, the amplitude is reduced by the sinc factor below. Stochastic motion also changes during the exposure, so an exact simulation includes the covariance of the integrated forcing; multiplying a sampled noisy sine by a sinc factor is not the same stochastic process.

The observing window records which times were sampled. Multiplying a complete time series by that window mixes neighboring frequencies. Repeated gaps create side peaks, and a strong mode can leak power into a neighboring search band. A peak that survives changing the plotting grid may still be a window feature.

Consider two short TESS sectors separated by about two years. The calendar span is long, but the modes are stochastically excited and may have lost phase memory between sectors. Treating them as one perfectly coherent sine wave would give misleadingly narrow errors. Independent-sector analysis and a physically justified stochastic transition answer different, explicitly stated questions.

Aexp(ν)=sin⁡(πνΔ)πνΔA_{\rm exp}(\nu)=\frac{\sin(\pi\nu\Delta)}{\pi\nu\Delta}
This is the coherent-sinusoid amplitude response of a rectangular exposure. Recorded power receives its square. A real frame comb has its own response.

Reading: MAST public data use and acknowledgment · Lindsay et al. 2025, Precise Asteroseismic Ages for the Helmi Streams

Normalize power before interpreting it

For equally spaced saved samples, subtract the mean of the observed values and set only the missing positions to zero for the transform. Keep the original mask separately. A one-sided periodogram can then be normalized so its frequency integral equals the variance of the observed, mean-subtracted values.

The expression below uses the cadence, the number of retained samples and the actual mask. The zero and Nyquist bins have a factor of one instead of two. This normalization provides a useful independent Parseval check, although the gap window still changes the spectral shape.

Zero padding gives a finer plotted frequency grid. The intrinsic time information is unchanged. Similarly, oversampling a Lomb–Scargle periodogram does not make all plotted bins independent. A smooth curve with thousands of points may contain far fewer independent spectral measurements.

An échelle diagram takes a candidate spacing, divides the spectrum into consecutive intervals of that width and stacks them. A nearly regular frequency ladder forms a ridge. Half-spacing aliases or the wrong candidate can also make appealing patterns, so retain alternatives and decide between them using identified modes.

Pk=2 δtNobs∣∑j=0N−1wj(yj−yˉ)exp⁡(−2πijk/N)∣2P_k=\frac{2\,\delta t}{N_{\rm obs}}\left|\sum_{j=0}^{N-1}w_j(y_j-\bar y)\exp(-2\pi i jk/N)\right|^2
wjw_j is one for an observed sample and zero otherwise. Away from endpoint bins, the density units are signal squared times time. The frequency step is 1/(Nδt)1/(N\delta t).

Reading: MAST public data use and acknowledgment

Use the same stochastic process in the likelihood

A likelihood evaluates how probable the saved samples would be under a proposed mode frequency and its nuisance assumptions. Here a linear Gaussian state-space calculation provides an exact reference for the declared oscillator. Its state contains displacement and velocity. A transition propagates them through an exposure; an observation is the average displacement during that exposure.

The noise that changes the final state also contributes to the exposure average. Those two noises are correlated. Ignoring that correlation changes the likelihood, even if the instantaneous oscillator simulation looks plausible. The update below conditions the state at the end of the exposure, then propagates across the actual unobserved gap.

Before using intervals, compare this sequential calculation with a separate full-covariance Gaussian calculation. The latter evaluates the analytic displacement covariance at all pairs of exposure times and uses a Cholesky factorization. In the supplied test, the two log likelihoods agree within about 2.3×10−132.3\times10^{-13}. This is numerical agreement for the declared model, not proof that a one-mode model describes a real multi-mode star.

xend=Fxstart+wi,yi=Hxstart+vi+ϵi,Cov⁡(wi,vi)=S\mathbf x_{\rm end}=F\mathbf x_{\rm start}+\mathbf w_i,\quad y_i=H\mathbf x_{\rm start}+v_i+\epsilon_i,\quad\operatorname{Cov}(\mathbf w_i,v_i)=S
x\mathbf x contains displacement and velocity. FF propagates through this exposure, HH averages it, and SS retains the process/observation cross-covariance before the next gap.
U=HPHT+R+σi2,C=FPHT+SU=HPH^\mathsf T+R+\sigma_i^2,\qquad C=FPH^\mathsf T+S
PP is the prior state covariance, RR the exposure-integral noise variance, and UU the innovation variance.
mend=Fm+Cyi−HmU,Pend=FPFT+Q−CCTU\mathbf m_{\rm end}=F\mathbf m+C\frac{y_i-H\mathbf m}{U},\qquad P_{\rm end}=FPF^\mathsf T+Q-\frac{CC^\mathsf T}{U}
QQ is the process covariance. This update includes exposure noise before the next real gap is propagated.

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

Work an interval example and test what it promises

The standalone known-mode exercise uses natural frequency 0.8 in inverse time units, damping time 12, stationary displacement variance 0.5 and measurement noise 0.3. Exposures last 0.1 time units, the underlying cadence is 0.125, and 428 samples survive the gaps over a 64-unit observing interval. These nuisance values are declared known for this exercise.

The inference receives only the saved measurements and that model family. It evaluates frequency on a grid under a uniform prior from 0.65 to 0.95. Multiply the likelihood at each frequency by that prior, then normalize the area to one to obtain the posterior density. A central 95 percent interval leaves 2.5 percent posterior probability below its lower endpoint and 2.5 percent above its upper endpoint. For the first saved realization, the posterior median is about 0.79797 and the central 95 percent interval is about 0.78395 to 0.81211. Doubling the numerical grid checks integration accuracy; it does not double the experiment’s resolving power.

Next repeat the experiment with new noise. At a fixed true frequency, the three supplied lifetime/noise cases give 99, 89 and 95 coverages out of 100 for nominal 95 percent intervals. The 89 case is retained as a failed universal-coverage interpretation. A proper Bayesian credible interval need not contain every fixed parameter with the same repeated-experiment frequency.

A separate check draws the true frequency from the declared prior before each independently generated observation, using a disjoint random-seed namespace. Its counts are 97, 94 and 98 out of 100, with binomial uncertainty reported. This checks prior-averaged conditional coverage. It does not calibrate unknown stellar backgrounds, blended modes, uncertain lifetimes or a unique sound-speed inversion. Those require their own data model and validation.

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

Fit neighbouring components in a real stellar spectrum

The single-mode exercise isolates the origin of frequency uncertainty. A red giant adds overlapping components: several mixed modes can occur within one radial interval, and their broad profiles and window side peaks share power. A separate maximum search around each published frequency cannot allocate that shared power.

The companion calculation jointly fits sixteen components to the two saved HD128279 lightcurves. Four radial, eight dipole and four quadrupole labels are supplied as conditions, together with a search half-width of 0.65 microhertz around each source frequency. Frequencies and positive mode strengths are estimated with a local background for each sector.

The model passes every component through the actual quality-mask window before adding the expected powers. The covariance sum below explains that operation: each pair of observing times contributes according to its correlation and Fourier phase. Finite exposure is included. An independent matrix calculation quantifies the mean-spectrum correction from linear detrending.

Assuming amplitude-damping times of 0.5, 2 and 8 days changes the fitted components. At 8 days, one dipole pair lies near 184.186 and 185.936 microhertz; at 2 days it lies near 184.379 and 185.969. Several components become weak or hit search edges at 0.5 days. These differences describe what the data can separate under each assumption.

The averaged-spectrum contrast selects point estimates. Correlated oversampled frequency bins are not assigned an independent-bin probability. Three sets of 96 independent stochastic injections test the actual masks, exposure, lifetime mismatch and a crowded pair. Error quantiles from 48 training records produce intervals whose heldout coverage falls as low as 38 of 48 for one component. The failed coverage is retained; the real results remain conditional frequency estimates.

E[P(ν)]=1N∑i,jwiwjCΔ(ti−tj)cos⁡ ⁣[2πν(ti−tj)]\mathbb E[P(\nu)]=\frac{1}{N}\sum_{i,j}w_iw_jC_{\Delta}(t_i-t_j)\cos\!\left[2\pi\nu(t_i-t_j)\right]
wiw_i records the sampling mask and NN counts retained samples. CΔC_{\Delta} is the finite-exposure covariance. This mean omits the detrend projector; its correction is quantified separately.

Reading: Lindsay et al. 2025, Precise Asteroseismic Ages for the Helmi Streams · Lindsay et al., Zenodo15102877 v1 model/observed-mode archive

Exercises and explanations

  1. Why can a contrast on a dense spectral grid select a fit, yet fail to justify error bars based on independent bins?

    Read the explanation

    A defined contrast can select parameters closest to the data. Error bars additionally describe how the data fluctuate under repetition. A finite, gapped lightcurve correlates neighbouring bins, so independent-bin errors would count shared information repeatedly. Intervals need that correlation model or independent injections through the observing window.

  2. A spectrum is recomputed on a frequency grid ten times finer. What has improved, and what has not?

    Read the explanation

    The numerical sampling of the spectrum or likelihood has improved. The observation length, excitation history, exposure and gaps remain the same, so the data’s intrinsic information and stochastic uncertainty have not gained a factor of ten.

  3. Why must exposure-integral noise be correlated with the oscillator state at the end of that exposure?

    Read the explanation

    The same forcing acts while the detector averages the motion and while the final state develops. It therefore contributes to both quantities. The cross-covariance term accounts for that shared random history.

  4. Can the known-mode interval width be used as an uncertainty for a TESS dipole peak?

    Read the explanation

    No. The exercise fixes amplitude, damping and noise and contains one identified oscillator. A TESS dipole peak may blend with other modes and a background, and its identity or linewidth can be uncertain. Its intervals need a model and calibration including those effects.

Try the prediction

Change a layer and read its modes

  1. Keep the same structural preset and shorten damping time. Compare peak width with the orange natural-frequency references.
  2. Increase observing gaps in the advanced group. Locate side peaks without calling every new maximum a new physical mode.
  3. Export the saved signal and replay its analysis. Explain which fields are observations, which are source assumptions, and which are separately labeled truth.

Sources and calculation scope

  1. MESA School 2026, Lab 3: Asteroseismology

    Official classroom treatment of oscillation frequencies, sound-travel integrals and model evolution. The controllable constant-density cavity here is a separate declared teaching model.

  2. 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.

  3. 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.

  4. MAST public data use and acknowledgment

    Original mission PUBLIC products downloaded using the supplied MAST product URI; preserve NASA/Kepler/TESS/SPOC/STScI attributions and frozen data releases.

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