From a ring’s moving particles to its missing starlight

Turn positions into a column with a stated weight

Give each numerical tracer a normalized two-dimensional Gaussian kernel. Its integral is one, so summing the kernels measures number per area without changing the number of tracers. Multiply by an effective optical area per tracer to obtain a dimensionless normal optical depth. The independent integral check recovers the total assigned optical area, including the periodic azimuth boundary.

The browser sets that area so the uniform interior of the initial strip approaches normal depth 0.5. Its smoothing length is 28 m; the strip is 680 m wide and 1600 m long. These are teaching choices. The same saved positions enter the scene, kernel, ideal transmission and saved detector record. A separate attractive stripe is never substituted for a calculation.

Finite resolution matters. For the frozen convergence case with inter-tracer collisions disabled and moon contacts retained, the default 1024-tracer mean column differs from an 8192-tracer reference by about eleven percent. The change from 4096 to 8192 is below one percent. Step-size refinement gives a much smaller change in that observable. These comparisons describe this softened local model; the collisional tracer discretization is not a validated physical viscosity calculation.

W_h(\boldsymbol r)=\frac{e^{-|\boldsymbol r|^2/(2h^2)}}{2\pi h^2},\quad\int W_h\,d^2r=1,\quad\tau_n(\boldsymbol r)=A_{\rm opt}\sum_jW_h(\boldsymbol r-\boldsymbol r_j)

The kernel normalizes the spatial interpolation; the optical area gives the column its physical meaning.

Reading: Rein & Tremaine, Symplectic integrators in the shearing sheet, arXiv v2 · Cassini UVIS alpha Arae2008-261 egress, original PDS3 product label · Salo & Mondino-Llermanos, Photometric modelling of self-gravity wakes and overstable oscillations

Project the same column through the sightline

For a locally uniform thin slab, a sightline at an elevation above the plane crosses a path longer than the normal path by the reciprocal sine of that elevation. Integrating direct-light attenuation through the longer path gives the transmission below. The model neglects multiple scattering into the aperture, finite stellar size, diffraction and unresolved nonuniformity along a grazing path.

For normal depth 0.5 at 35 degrees, transmission is about 0.418. At 10 degrees, the same column transmits about 0.0562. No particle was added. This example explains why the normal depth, viewing geometry and detector brightness must remain distinct quantities.

\frac{dI}{ds}=-\alpha I,\quad\frac{I}{I_0}=e^{-\int\alpha ds},\quad T=e^{-\tau_n/|\sin B|}

Separate the local extinction law, its integral and the thin-slab projection.

T_{35^\circ}=e^{-0.5/\sin35^\circ}\simeq0.418,\qquad T_{10^\circ}\simeq0.0562

Lower elevation lengthens the path through the same normal optical area.

Reading: Cassini UVIS alpha Arae2008-261 egress, original PDS3 product label · Colwell et al., Cassini UVIS stellar occultation observations of Saturn’s rings

Let the detector count, then invert its saved counts

Each synthetic sample has 400 expected unocculted source photons and 4 known background photons. Draw one Poisson count with the expected transmitted source plus background. Subtract the background once, then divide by the unocculted source expectation. The ideal gray curve is the forward expectation; mint points and error bars come from the saved count.

If a sample contains 170 photons, the estimated transmission is 0.415. At 35 degrees its estimated normal depth is about 0.504, with a counting-only propagated uncertainty near 0.045. The square-root count approximation is useful away from zero; near background it is asymmetric and the logarithm can fail.

A zero or negative net source count has no real optical-depth logarithm. Keep its count and transmission estimate, omit the logarithm, and report the censoring. Clipping it to a tiny positive value would manufacture a very large measured depth. Background and stellar-model uncertainties are held fixed here and in the tutorial’s conditional counting calculation.

N\sim\operatorname{Poisson}(S_0T+b),\qquad\widehat T=\frac{N-b}{S_0},\qquad\widehat\tau_n=-|\sin B|\ln\widehat T

The inverse uses the acquired count, with the same source and background model as the forward calculation.

\sigma_T\simeq\frac{\sqrt N}{S_0},\qquad\sigma_{\tau_n}\simeq|\sin B|\frac{\sqrt N}{N-b}

Linear propagation is conditional on fixed source/background and a positive, adequately measured net signal.

Reading: Cassini UVIS alpha Arae2008-261 egress, original PDS3 product label · Colwell et al., Cassini UVIS stellar occultation observations of Saturn’s rings

Work the default snapshot before touching a control

The table below gives the actual warmed-up seed 1010 synthetic defaults and contrasting presets. Each run uses 1024 tracers, five inverse-frequency time units, step 0.02 and the stated Gaussian kernel. The values were generated by the frozen acquisition script, with units and parameters preserved in its public output.

Compare the default and grazing rows: their ideal particle column is the same, but their photon transmission differs because the elevation changed. Compare the edge-moon row with the default: the spatial distribution itself changed. Use both comparisons when deciding whether a dark trough is geometric or dynamical.

Actual frozen synthetic snapshot results; seed1010,1024tracers, elapsed time five inverse angular frequencies, step0.02. Gaussian width28m; source400and background4expected photons per sample. Generated by freeze-synthetic.ts, independently remeasured by the public Python replay.
Preset	Elevation / degree	Mean transmission	Peak ideal normal depth
Moon-forced	$35$	$0.495$	$0.574$
No moon	$35$	$0.458$	$0.617$
Edge moon	$35$	$0.427$	$3.57$
Elastic	$35$	$0.498$	$0.672$
Grazing	$10$	$0.109$	$0.574$

Reading: Rein & Tremaine, Symplectic integrators in the shearing sheet, arXiv v2 · Cassini UVIS alpha Arae2008-261 egress, original PDS3 product label

A global density wave adds a different equation

Now leave the browser patch. In a thin, tightly wound, linear self-gravitating disk, the collective dispersion relation balances epicyclic restoring motion, self-gravity and pressure. The pattern frequency combines the moon’s pattern speed and an integer azimuthal order. At an inner Lindblad resonance, the forcing frequency in a particle’s frame matches the radial epicyclic frequency.

Near resonance, take the Kepler limit, ignore the pressure term on the chosen long-wave branch, and treat background surface density as slowly varying. Expand the difference of squared frequencies to first order in distance from resonance. Solving for radial wavenumber gives the inverse-distance wavelength law below. It predicts shorter spatial wavelengths farther from the resonance; it does not follow from Beer attenuation.

For an illustrative pattern with four spiral arms, reference radius 125000 km, angular frequency 0.00014 persecond and surface density 400 kg per square meter, the leading formula gives wavelength about 14.9 km at 50 km from resonance and 7.47 km at 100 km. This worked example uses an ideal linear wave, not a fit to the strong Cassini structure.

(\omega-m\Omega)^2=\kappa^2-2\pi G\Sigma|k|+c^2k^2,\qquad\omega=m\Omega_p

Here surface mass density is distinct from optical depth; the epicyclic frequency equals the orbital frequency only in the Kepler limit.

\kappa_L=m(\Omega_L-\Omega_p),\quad\kappa^2-m^2(\Omega-\Omega_p)^2\simeq\frac{3(m-1)\Omega_L^2}{r_L}(r-r_L)

Differentiate both squared frequencies at the inner resonance before keeping the first-order radial term.

|k|\simeq\frac{3(m-1)\Omega_L^2(r-r_L)}{2\pi G\Sigma r_L},\qquad\lambda_r=\frac{2\pi}{|k|}\simeq\frac{4\pi^2G\Sigma r_L}{3(m-1)\Omega_L^2(r-r_L)}

This expression applies on the stated branch and inside the linear, slowly varying, tightly wound approximation.

Reading: Goldreich & Tremaine, The excitation and evolution of density waves · Colwell et al., Cassini UVIS stellar occultation observations of Saturn’s rings · Salo & Mondino-Llermanos, Photometric modelling of self-gravity wakes and overstable oscillations

Diagnose the inference before reporting mass

To infer surface density from that law, identify the resonance and azimuthal order, account for Saturn’s nonspherical gravity, isolate a suitable wave branch and test nonlinear and overlapping structures. Optical depth also needs an extinction-to-mass relation before it becomes surface density. A numerical tracer weight supplies none of those missing measurements.

The real tutorial therefore reports reconstructed optical depth and local spatial scales, plus failure tests. Its local single-harmonic fits have very large reduced chi-squared, and coarsening changes the shortest recovered scale. Those are useful findings about what the simple analysis cannot resolve. A mass inference would first need a wave model that explains those residuals and remains stable under the resolution test.

Reading: Cassini UVIS alpha Arae2008-261 egress, original PDS3 product label · Colwell et al., Cassini UVIS stellar occultation observations of Saturn’s rings · Goldreich & Tremaine, The excitation and evolution of density waves

Exercises and explanations

A fixed column is observed at lower elevation and looks darker. Which quantity changed in the slab model?
Read the explanation
The slant optical depth increased because the path grew longer. The normal optical depth and assigned particle column did not change. Correct the geometry before comparing normal depths.
With 400 expected source photons and 4 background photons, a sample has 3 photons. What should be reported?
Read the explanation
Its net transmission estimate is negative 0.0025. Keep the count and signed transmission, but do not take its real logarithm or replace it with an invented huge depth. A likelihood or upper-limit analysis is needed near background.
A fitted spatial wavelength is halved while all ideal linear-wave parameters except distance from resonance are fixed. What distance would the formula predict, and why might that inference fail in the real cutout?
Read the explanation
The inverse-distance law predicts twice the distance. Nonlinear crests, multiple interfering waves, a wrong resonance identification or unresolved sampling can break that interpretation. The real tutorial’s poor residuals and rebinning instability must be addressed first.

Try the prediction

Check the forward and inverse connection

Pause a snapshot and export it. Recompute one transmitted fraction from its saved photon count and background.
Lower the opening angle while keeping particles fixed, then distinguish the ideal column from the noisy logarithmic estimate.

Sources and calculation scope

Rein & Tremaine, Symplectic integrators in the shearing sheet, arXiv v2
19April2011 revision; Sections2–3, Hill equations and exact epicycle splitting; Section4 and Figures1–4, free/perturbed trajectory tests. We implement a planar exact-free-motion/symmetric-kick method with an explicitly softened prescribed moon, not their full many-body calculation.
Cassini UVIS alpha Arae2008-261 egress, original PDS3 product label
Product creation27November2020; CO-SR-UVIS-HSP-2/4-OCC-V3.0. Read all12column definitions, elevation54.4degrees,110–190nm band,121-byte records, background-unsubtracted mean counts, star/background models, samples-per-bin, sensitivity threshold and NOTE_FLAG definitions. Exact archived label and1001source rows are preserved for offline replay.
Colwell et al., Cassini UVIS stellar occultation observations of Saturn’s rings
AJ140,1569–1578,2010. Observing geometry, reduction and occultation table including alpha Arae Rev85/2008-261. Consulted primary indexed article text and original PDS product definitions; the tutorial repeats the archived binned-count conversion, not the paper’s full unbinned calibration or global wave analysis.
Goldreich & Tremaine, The excitation and evolution of density waves
Published ApJ222,850–858,15June1978; full author-archive PDF read, SectionII equations4–17 (especially9,11–15): thin self-gravitating sheet, tightly wound dispersion, resonances and wave winding. Course gives its own Kepler-limit expansion; finite pressure, oblateness and nonlinear waves remain separate limits.
Salo & Mondino-Llermanos, Photometric modelling of self-gravity wakes and overstable oscillations
Published3March2025, A&A695,A37. Selected original indexed sections4.2/Figure5 (weak-gravity asymmetry and correlation checks), Figure4 (photometric regimes),8.3/AppendixB (transport contributions), and hybrid-model limitations. Primary full-text HTML was available through indexed excerpts; the whole29-page numerical study was not reproduced.