AC26 · Waves inside a star

Stellar Oscillation Modes

How can a flickering point of light reveal a star’s interior? Waves cross a star and return at preferred rhythms. Their frequency spacing reveals the sound travel time; waves that also feel buoyancy can probe the core. Interpreting those rhythms requires a model of the star and of the observations.

Drag an internal boundary and change its sound speed. Watch radial waves combine, then read their rhythms from the saved noisy signal.

Selected model frequency $\nu_3R/c_{\rm e}=1.26$

Center-to-boundary sound travel time $\tau c_{\rm e}/R=1.2$

Selected mode’s internal pressure nodes $N_{\rm node}=2$

Saved observation’s variance $V_y=0.101$

Change a layer and compare modes

Core sound speed

c_{\rm core}/c_{\rm e}

0.650

Internal boundary radius

a/R

0.370

Show this radial mode’s nodes

n

3

Amplitude damping time

\tau_{\rm d}c_{\rm e}/R

24.0

Drag the internal ring or core handle. Orange marks model frequencies; blue comes from the saved noisy signal.

Observation and model assumptions

Observation length

128

Measurement noise

0.0800

Repeated observing gap

12\%

p(R)=0,\qquad\xi_r(R)=\frac{p_r(R)}{\rho\omega^2}

The acoustic boundary has zero pressure and nonzero displacement. This synthetic displacement proxy includes stationary damped modes and exposure averaging; actual stellar brightness requires additional physics.

Analyze the actual TESS and Kepler files

Model assumptions

The controllable model is a constant-density spherical radial acoustic cavity with a pressure-release boundary. It includes no buoyancy, self-gravity or nonadiabatic stellar photometry. Its nonzero boundary-displacement proxy is driven by independent stationary damped modes with white acceleration and exact joint finite-exposure noise. The spectrum uses the saved noisy, gapped record; model frequencies are a separate reference. Spectral peaks and spacing alternatives are diagnostics without precision posteriors. Real TESS/Kepler observations and licensed MESA structures are separate fixed data. Solar-scaled, acoustic-integral, published GYRE and measured spacings retain distinct labels. The single HD128279 track gives conditional model properties, not a reproduced age posterior. Apparent displacement, radial scale and playback speed are schematic.

One saved signal, two ways to see its rhythms

Blue comes from the saved displacement proxy in thousandths of the radius; orange marks separate model modes. Folding uses a measured candidate when available, otherwise a labeled exploratory spacing.

Noisy displacement proxy and actual gaps

Power spectrum of the saved observation

Fold the measured candidate into an échelle diagram

The default noisy observation is already saved and analyzed. Spectral peaks are finite-window diagnostics.

Real TESS / Kepler observations and stellar structures

These fixed observations keep their own time stamps and source versions. Cavity controls change the synthetic experiment above.

Real lightcurve periodogram; published mode markers

Licensed HD128279 model sound speed

TESS sectors38/65 are short observations separated by about two years. Their average periodogram is a discovery view; published frequencies remain external references.

Open the reproducible real-data tutorial

Start with the essentials

Focus question: Which part of the frequency ladder changes when the core sound speed changes?
One-sentence intuition: Different modes sample a layer differently. A frequency pattern contains more structural information than one pitch.

Core mathematical model

Uniform radial cavity

\nu_n=\frac{nc}{2R}

A regular center and pressure-release outer boundary select a discrete ladder. The boundary pressure is zero, while its displacement can be nonzero.

Acoustic travel time

\tau=\int_0^R\frac{dr}{c(r)},\qquad\Delta\nu_{\rm asym}\simeq\frac{1}{2\tau}

This high-order estimate uses the travel time. Low-order frequencies and a real star’s atmosphere can shift the actual spacings.

Saved measurement

y_i=\frac{1}{\Delta t}\int_{t_i-\Delta t/2}^{t_i+\Delta t/2}q(t)\,dt+\epsilon_i

Exposure averages the stochastic boundary-displacement proxy; measurement noise is added afterward. Missing samples keep their actual positions.

Common difficulties

A spectral maximum is noisy

Typical misconceptionThe tallest frequency bin is the exact natural frequency.

Better mental modelRandom excitation, finite duration and gaps move the local maximum. The orange reference and blue saved-data peaks answer different questions.

A spacing does not give a unique age

Typical misconceptionOne measured spacing determines a unique stellar age.

Better mental modelMass, composition and outer layers also change frequencies. The real-model exercise compares one fixed track conditionally.

Run the experiment

01
Start with an even ladder
Choose An even frequency ladder, then inspect the separate spectrum.
What to observe: Uniform sound speed gives evenly spaced model modes, with noisy observed peaks nearby.
02
Move the internal boundary
Drag the highlighted internal ring, then drag the core sound-speed handle vertically.
What to observe: The node pattern and frequency ladder respond continuously to the same structure.
03
Shorten the mode lifetime
Choose Short-lived rhythms blur, keeping the saved record in view.
What to observe: Rapidly changing mode phases broaden peaks. A finer frequency grid would not restore a longer coherent observation.

Stellar Oscillation Modes

One saved signal, two ways to see its rhythms

Read the rhythms of a layered sphere

Core mathematical model

Uniform radial cavity

Acoustic travel time

Saved measurement

Common difficulties

A spectral maximum is noisy

A spacing does not give a unique age

Run the experiment

Start with an even ladder

Move the internal boundary

Shorten the mode lifetime

Related experiments and articles

Read a star through its vibrations