# From moving stars to a testable inference

Six short modules lead from the original galaxy views to the current methods result. Each uses the project's actual evidence. The [research article](https://raw.githubusercontent.com/djova/bar-halo-benchmark/publication-2026-09-24.2/publication/web/paper.md) gives the complete argument; the [evidence guide — historical website-only](https://djova.ca/galaxy-bar/guide.html) is a directory, not a prerequisite.

## 1. Stars and patterns {#learn-1}

**Prerequisite:** positions, velocities and the idea of an orbit.

A bar is a density pattern. Its stars need not remain together like points painted on a solid rod. In the [galaxy view — historical website-only](https://djova.ca/galaxy-bar/index.html#experiment), choose **Follow stars**, then switch between the inertial view and a qualified pattern frame. You are following recorded particle identities, not an artist's animation. A coherent orientation is evidence for a bar; visible elongation alone does not prove resonant trapping.

The galaxy clock is in **Gyr** (billions of years). Density brightness is logarithmic and normalized per frame; a brighter frame is not necessarily a stronger bar. The measured cosine amplitude is a separate quantity.

**Worked example — the factor of two.** If \(\Sigma(\phi)\propto1+a\cos(2\phi)\), the normalized complex coefficient is \(C_2=\frac{\int\Sigma e^{2i\phi}\,\mathrm d\phi}{\int\Sigma\,\mathrm d\phi}\). Expanding the numerator gives \(e^{2i\phi}+(a/2)e^{4i\phi}+a/2\). The oscillating terms integrate to zero over a full azimuth; the constant \(a/2\) survives. Dividing by the denominator gives \(C_2=a/2\). Thus \(A_2=2|C_2|=|a|\) for this profile. General positive distributions can have higher harmonics and A₂ above one; A₂ is not a fraction of stars.

**Try:** Does a local A₂ of 1.2 mean 120% of the stars are in the bar?

<details><summary>Reason it through</summary><p>No. It is a cosine-amplitude convention. Its aperture, radial coherence and higher harmonics matter. The observed S⁴G comparison uses a related amplitude convention but not an identical measurement pipeline.</p></details>

[Expert definitions and observational selection — historical website-only](https://djova.ca/galaxy-bar/methodology.html) · [Next: orbital populations](#learn-2)

## 2. Orbital populations {#learn-2}

**Prerequisite:** module 1; a probability distribution describes how much material occupies a range of states.

Two galaxies can have similar density profiles but different orbital motions. A distribution function specifies population in phase space, including motion. A selected Gaussian tracer population is a physical modeling choice unless its sampling weights correct it back to the intended DF.

An **orbital action** characterizes the motion around an orbit, rather than the particle's position at one instant. In an integrable, time-independent potential, actions stay constant while their conjugate angles advance; angular momentum is one familiar example. We use actions as coordinates for organizing an orbital population, within the reference potential's assumptions.

For the twofold corotation resonance, the slow action is \(J_s=L_z/2\) and the slow angle is \(\psi=2(\theta_\phi-\phi_{\rm bar})\). Here \(L_z\) is specific angular momentum about the symmetry axis, \(\phi_{\rm bar}\) is the bar's orientation and \(\theta_\phi\) is an orbital angle variable. This angle variable is generally **not** the star's instantaneous geometric azimuth. The local coordinate \(x\) labels the initial slow action relative to the initial resonance, in the experiment's scaled units; the other, fast actions are held fixed.

In the local experiment, the **kernel tells us what the dynamics do** at each starting action; the **weight tells us how much material is there**. Reweighting a fixed external response is legitimate because the particles do not change its prescribed field. That separation cannot simply be assumed in a live halo.

**Worked example.** An exponential \(w_{\mathrm E}=e^{gx}\) and Gaussian \(w_\sigma=e^{gx-x^2/(2\sigma^2)}\) both have \(w(0)=1\) and \(w'(0)=g\). Their logarithmic slopes are \(g\) and \(g-x/\sigma^2\). A moving resonance encounters the mismatch away from zero. Same initial slope does not mean same sampled gradient.

[Explore the recorded populations — historical website-only](https://djova.ca/galaxy-bar/population-response.html#population-explorer) · [Expert model and normalization](https://raw.githubusercontent.com/djova/bar-halo-benchmark/publication-2026-09-24.2/publication/web/paper.md#model) · [Next: transfer](#learn-3)

## 3. Angular-momentum exchange {#learn-3}

**Prerequisite:** a force can change momentum; module 2.

Torque is the rate of angular-momentum change. Integrating torque over time gives a transfer. In a live isolated galaxy, stars and halo can exchange it. In an imposed-field calculation the source is prescribed, so the test particles' transfer is not a self-consistent history of the bar.

The reduced experiment records the bar impulse \(B=-\int\sin\psi\,\mathrm dt\), separately from the imposed noise and the moving coordinate. The reported response is **\(B_{\mathrm{noise}}-B_{\mathrm{smooth}}\)**, averaged with a specified population weight.

**Worked example — contrast versus total.** If the smooth bar transfers +10 units to a population and the noisy bar transfers +8, the contrast is −2. Both transfers remain positive; the negative contrast means suppression, not transfer back to the bar. These numbers are an arithmetic illustration, not a simulated result.

[Follow the same recorded orbit and torque — historical website-only](https://djova.ca/galaxy-bar/response-coupling.html) · [Expert observable](https://raw.githubusercontent.com/djova/bar-halo-benchmark/publication-2026-09-24.2/publication/web/paper.md#model) · [Next: resonances](#learn-4)

## 4. Resonances and a moving pattern {#learn-4}

**Prerequisite:** periodic motion and module 3.

Weak repeated pushes can accumulate if their phases remain correlated with an orbit. Near a resonance, a particular combination of orbital and bar angles changes slowly. Some trajectories librate around a phase; others circulate. As the imposed bar slows, the resonant action moves. Trapping, crossing and population gradients can then influence transfer.

The local slow-angle equations are \(\mathrm d\psi=-j\,\mathrm dt\) and \(\mathrm dj=(-\sin\psi-s)\,\mathrm dt+\sqrt{2D}\,\mathrm dW_t\). The symbol \(W_t\) denotes a standard Wiener process: over a time step \(\Delta t\), its independent random increment has mean zero and variance \(\Delta t\). Thus the noise impulse has variance \(2D\Delta t\). The term −s shifts the coordinate origin; it is not a physical bar torque.

First turn off the noise, \(D=0\). A phase-locked equilibrium requires \(j=0\) and \(\sin\psi=-s\). For \(|s|<1\), the deterministic constant-coefficient system has a stable libration region; at \(|s|=1\), it reaches a degenerate boundary. With noise present, this structure helps organize temporary trapping, but does not imply permanent phase locking. Escape and residence times are separate stochastic questions.

**Worked example.** With D = 0, at s = 1.2 no constant phase solves sinψ = −s; at s = 0.25 such phases exist. Neither calculation alone determines the sign of the noise-induced total transfer.

[Recorded resonance trajectories — historical website-only](https://djova.ca/galaxy-bar/noise-sweep.html#paths) · [Expert local model](https://raw.githubusercontent.com/djova/bar-halo-benchmark/publication-2026-09-24.2/publication/web/paper.md#model) · [Next: noise and inference](#learn-5)

## 5. Noise and numerical inference {#learn-5}

**Prerequisite:** averages and variance; modules 3–4.

Particle graininess, a chosen stochastic operator and physical collisions are different sources of fluctuations. Changing numerical softening or particle count tests the calculation. Imposing additive Brownian action noise defines a model; it does not calibrate an SIDM cross-section. Conservation checks, convergence checks and a test of model adequacy answer different questions.

**Worked example — a non-flat screen with constant diffusion.** For a stationary Ornstein–Uhlenbeck process \(\mathrm dJ=-\gamma J\,\mathrm dt+\sqrt{2D}\,\mathrm dW\), constant D coexists with restoring drift. Its increment variance gives:

```math
\frac{\operatorname{Var}[J(t+\tau)-J(t)]}{2\tau}
=D\,\frac{1-e^{-\gamma\tau}}{\gamma\tau}.
```

At \(\gamma=0.25\) and lags 1, 2, 4, this is about 0.8848D, 0.7869D and 0.6321D. It is not flat, despite constant diffusion. Pooled state-dependent drift adds another ambiguity. Short softened gravitational trajectories can instead be ballistic; taking lag to zero is not automatically a diffusion measurement.

**Try:** Are two endpoint times, two windows and two antithetic partners eight independent experiments?

<details><summary>Reason it through</summary><p>No. They share dynamics and random histories. The independent sampling units in the kernel uncertainty are the eight separately seeded batches. Retaining pairing helps estimate contrasts but does not create independence.</p></details>

[Retained correction and synthetic controls — historical website-only](https://djova.ca/galaxy-bar/response.html) · [Expert uncertainty](https://raw.githubusercontent.com/djova/bar-halo-benchmark/publication-2026-09-24.2/publication/web/paper.md#accuracy) · [Next: the contribution](#learn-6)

## 6. When does an approximation preserve the answer? {#learn-6}

**Prerequisite:** gradients, signed sums and modules 2–5.

The primitive kernel \(Q_{\mathrm p}\) weights the initial population gradient: \(\mathcal R[w]=-\int w'Q_{\mathrm p}\,\mathrm dx\) when the boundary term vanishes. An absolute mismatch integral can qualify an approximation. It can also decline an accurate one because taking absolute values destroys useful cancellation.

**Worked example — reconstruct the sign.** The original early Gaussian gives approximately \(-0.011058+0.012565=+0.001508\). The halo gives \(-0.004835+0.000934=-0.003901\). Rounding explains the final displayed digit. These are **gradient-coordinate contributions**, not two literal orbital cohorts. Matching central density and slope did not preserve their balance.

[Inspect Gaussian 128's conservative rejection — interactive — requires the canonical site](https://djova.ca/galaxy-bar/paper.html?population=gaussian128&time=10&window=40#interactive-decision). [Static values](https://raw.githubusercontent.com/djova/bar-halo-benchmark/publication-2026-09-24.2/publication/web/results/accuracy.json) are available without the interactive controls. Its independent intrinsic error is within 5%, but its absolute allowance declines to certify it. That is an informative limitation of the rule, not evidence that the recorded response was wrong.

**What has been learned?** Population approximations can change a finite-time answer; some inexpensive approximations work closely here; the kernel can expose the mismatch and assess its error with stated numerical assumptions. The eight supported qualifications share one new dynamical condition. No full-halo or SIDM prediction follows.

[Read the complete article](https://raw.githubusercontent.com/djova/bar-halo-benchmark/publication-2026-09-24.2/publication/web/paper.md) · [Check the evidence yourself](https://raw.githubusercontent.com/djova/bar-halo-benchmark/publication-2026-09-24.2/publication/web/reproduce.md)

## Keep symbols and clocks separate {#notation}

| Model | Symbol or clock | Meaning |
|---|---|---|
| Original galaxy | t in Gyr; positions in kpc | Physical simulation time and distance |
| Original galaxy | Toomre Q | Disc stability parameter; realized baseline minimum ≈ 1.31, target 1.5 |
| Prescribed Cartesian field | Model time and model action | Analytical-potential units; no automatic Gyr conversion |
| Reduced resonance | T, x, j, ψ, s, η | Dimensionless time, initial action, moving slow action, resonant angle, sweep and noise |
| Reduced resonance | \(Q_{\mathrm p}\) (Q in earlier records) | Primitive response kernel, unrelated to Toomre Q |
| Reduced resonance | R[w] | Noise-minus-smooth accumulated bar transfer with specified weight |
| All models | An unavailable estimate | Missing or unqualified, never silently zero |

The original particles were sampled from an approximately self-consistent equilibrium model. The softened evolution force differs from the construction potential, so initial adjustment is a documented limitation. A successful local resonance test does not remove that galaxy-model limitation.