Occultation Geometry and Chord Reconstruction

Status: Foundational reference Relates to: Asteroid occultation campaign planning, shadow prediction, chord timing pipeline Last derived: 2026-03-20

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1. The Physical Setup

A stellar occultation occurs when a solar system body moves in front of a background star. The body casts a shadow on Earth's surface. The shadow moves at the shadow speed $v_s$, which is the relative velocity of the body projected onto the sky plane: [Source: Millis & Elliot (1979), AJ 84, 1917; Herald et al. (2020), MNRAS 499, 4570]

$$v_s = \sqrt{\dot{\alpha}^2 \cos^2\delta + \dot{\delta}^2} \cdot d_\oplus$$

where $\dot{\alpha}, \dot{\delta}$ are the proper motion of the shadow path in angular units, and $d_\oplus$ is the distance from the observer to the asteroid. In practice, $v_s$ is obtained from orbital ephemerides (e.g., from Miriade, Occult Watcher, or Lucky Star databases) and typically ranges:

• Near-Earth asteroids: $v_s \sim 3$–30 km/s
• Main-belt asteroids: $v_s \sim 10$–30 km/s
• Trans-Neptunian objects (TNOs): $v_s \sim 5$–30 km/s (slower due to lower orbital velocity)
• Pluto/large KBOs: $v_s \sim 10$–25 km/s

1.1 Chord Geometry

An observer at position $(x_0, y_0)$ in the shadow plane (perpendicular to the line of sight, with origin at the shadow center) records: - Disappearance time $t_D$: star blinks out - Reappearance time $t_R$: star returns

The chord projected onto the body plane has: - Midpoint time: $t_m = (t_D + t_R)/2$ - Half-duration: $\Delta t = (t_R - t_D)/2$

The chord length in body-plane coordinates: $$\ell = v_s \cdot (t_R - t_D) = 2 v_s \Delta t$$

The midpoint of the chord in the shadow plane is at: $$x_m = x_0 + v_s \cdot t_m \cos\theta, \quad y_m = y_0 + v_s \cdot t_m \sin\theta$$

where $\theta$ is the position angle of the shadow motion.

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2. Shape Reconstruction: Mathematical Framework

2.1 From Chords to Silhouette

Each chord $k$ (with $k = 1, \ldots, K$) defines two boundary points on the limb of the body: $$\mathbf{p}_k^D = \text{disappearance point on limb}$$ $$\mathbf{p}_k^R = \text{reappearance point on limb}$$

These are determined by: $$p_{k,x}^D = x_k - v_{s,x} \cdot \Delta t_k, \quad p_{k,x}^R = x_k + v_{s,x} \cdot \Delta t_k$$ $$p_{k,y}^D = y_k - v_{s,y} \cdot \Delta t_k, \quad p_{k,y}^R = y_k + v_{s,y} \cdot \Delta t_k$$

So $K$ chords give $2K$ limb points.

2.2 Shape Models

Elliptical model (simplest useful case):

Fit an ellipse $\frac{(x - x_c)^2}{a^2} + \frac{(y - y_c)^2}{b^2} = 1$ (rotated by position angle $\phi$) to the $2K$ limb points via least squares. The free parameters are: $x_c, y_c, a, b, \phi$ — five parameters. [Source: Herald et al. (2020), MNRAS 499, 4570]

Minimum K for ellipse fit: K ≥ 3 (gives 6 limb points, 5 free parameters → 1 degree of freedom). In practice, K = 3 produces large uncertainties; K ≥ 5 is needed for a reliable ellipse.

General convex shape:

For a general convex body, parameterize the silhouette by a support function: $$\rho(\alpha) = \sum_{n=0}^{n_\text{max}} \left(a_n \cos n\alpha + b_n \sin n\alpha\right)$$

This is the Fourier expansion of the angular radius. For a smooth convex body, high $n$ terms decay rapidly. The $2K$ limb points constrain the $2n_\text{max} + 1$ Fourier coefficients. [Source: Sicardy et al. (2011), Nature 478, 493; Morgado et al. (2022), Icarus]

For a diameter estimate only: K = 1 gives two limb points on one chord, yielding only a chord length. No shape information. The diameter is bounded below by the chord length.

2.3 Uncertainty Budget per Limb Point

The uncertainty in a limb point position is: $$\sigma_\text{limb} = v_s \cdot \sigma_t$$

where $\sigma_t$ is the 1-sigma timing uncertainty on a single event (disappearance or reappearance). For GPS-PPS timing, $\sigma_t \sim 1$–10 ms. For NTP-disciplined systems, $\sigma_t \sim 0.1$–1 s (see [[Timing Precision Budget]]).

For a main-belt asteroid at $v_s = 15$ km/s: $$\sigma_\text{limb} = 15 \,\text{km/s} \cdot 10\,\text{ms} = 150\,\text{m}$$

For a TNO at $v_s = 5$ km/s and GPS timing: $$\sigma_\text{limb} = 5\,\text{km/s} \cdot 1\,\text{ms} = 5\,\text{m}$$

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3. Chord Reconstruction Uncertainty as a Function of K

3.1 Uncertainty in Ellipse Parameters

For an ellipse fit with $K$ chords (parallel geometry, all at approximately the same chord spacing $\Delta y$ across the body), the formal uncertainty in semi-major axis $a$ is approximately:

$$\sigma_a \approx \frac{\sigma_\text{limb}}{\sqrt{2K}} \cdot f(\text{chord geometry})$$

The factor $f$ depends on the chord distribution. For uniformly-distributed chords (ideal case): $$f \approx 1 \quad \text{(for central chords)}$$ $$f \approx 2\text{–}5 \quad \text{(for near-limb chords)}$$

This is because chords near the limb sample a region where the body boundary is nearly tangential to the chord — the timing error projects almost entirely along the boundary rather than perpendicular to it.

Formal 2D fit:

Let $\mathbf{x}$ be the vector of $2K$ limb point coordinates and $\boldsymbol{\theta} = (x_c, y_c, a, b, \phi)$ be the ellipse parameters. The Jacobian $J = \partial \mathbf{x} / \partial \boldsymbol{\theta}$ gives:

$$\text{Cov}(\hat{\boldsymbol{\theta}}) = \sigma_\text{limb}^2 \cdot (J^T J)^{-1}$$

For a symmetric chord distribution with $K$ evenly-spaced chords, the diagonal elements of $(J^T J)^{-1}$ scale as $K^{-1}$, giving:

$$\sigma_a, \sigma_b \approx \frac{2 \sigma_\text{limb}}{\sqrt{K}}$$

3.2 Minimum K for Useful Science

Diameter estimate only (K = 1): The chord gives a lower bound on the body's cross-section diameter. Combined with thermal models, a rough size is inferred. Uncertainty is large.

K = 2: Two chords determine an ellipse with significant degeneracy (the center is poorly constrained unless the chords bracket the expected center). The position angle of the ellipse is almost unconstrained. Result: chord lengths and a rough cross-section, not a shape.

K = 3–4: An ellipse can be formally fit. The five ellipse parameters are over-determined by 1–3 degrees of freedom. The shape is meaningful, but with ~20–30% uncertainty in $a/b$.

K ≥ 5–6: Reliable ellipse parameters with $\sigma_a / a < 10\%$. Sufficient to detect departure from spherical shape, determine axis ratio.

K ≥ 8–10: Departure from ellipse can be detected if the body has large-scale topographic features. Can fit the $n = 3$ Fourier term.

Rule of thumb: Each additional Fourier harmonic needs ~4 additional chords to constrain. Shape models beyond $n = 3$ require $K > 12$.

3.3 Geographic Separation Requirements

For $K$ chords across a body of diameter $d_\text{body}$ at shadow-plane distance $d_\perp$ from the centerline, the observer separation needed is:

$$\Delta_\text{obs} \approx \frac{d_\text{body}}{K} \quad \text{(ideally uniform spacing)}$$

For a 100 km asteroid: $\Delta_\text{obs} \approx 100/K$ km. For K = 10: $\Delta_\text{obs} \approx 10$ km between observers. For K = 5: $\Delta_\text{obs} \approx 20$ km.

This spacing is easily achievable for a geographically-distributed amateur network — it requires observers to be separated by $\sim$10–50 km, not thousands of km.

[NOVEL] The application of the chord-spacing requirement to OpenAstro network deployment planning — specifically the insight that 10–50 km baseline separations (not thousands of km) are all that is needed for asteroid shape science — is original to OpenAstro.

Shadow path width: The path within which an observation is possible is limited to $d_\text{body}$ (the diameter of the shadow). Observers outside this band see no occultation. This means campaign planning must target a $\sim$50–200 km wide strip across the predicted path.

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4. Atmosphere Detection: The Central Flash

For bodies with substantial atmospheres (Pluto, Triton, large TNOs), a central flash may be observed. This occurs when the atmosphere acts as a refracting lens and converges starlight to a caustic near the shadow center.

4.1 Refraction Geometry

An atmospheric ray at impact parameter $b$ above the surface is refracted by angle:

$$\hat{\epsilon}(b) = \int_{-\infty}^{\infty} \frac{\partial \ln n}{\partial b'} \, dl$$

For a power-law atmosphere $n(r) = 1 + \nu_0 (r_0/r)^\beta$, this integral evaluates to:

$$\hat{\epsilon}(b) \approx \nu_0 \frac{\sqrt{2\pi r_0}}{\beta H_b^{1/2}} \left(\frac{r_0}{b}\right)^{\beta - 1/2} e^{-(b - r_0)/H_b}$$

where $H_b$ is the scale height at impact parameter $b$.

4.2 Central Flash Detectability

The central flash magnification is: $$\mu_\text{flash} \approx \frac{D_s}{d_s \cdot \hat{\epsilon}(r_0)} \cdot \frac{1}{\sqrt{2}}$$

where $D_s$ is the distance to the source and $d_s$ is the distance to the occulting body. For Pluto:

$\hat{\epsilon} \approx 10^{-5}$ rad, $D_s \approx 6 \times 10^{12}$ km (a background star at $\sim$200 pc), $d_s \approx 5$ AU = $7.5 \times 10^8$ km:

$$\mu_\text{flash} \approx \frac{6 \times 10^{12}}{7.5 \times 10^8 \cdot 10^{-5}} = \frac{6 \times 10^{12}}{7.5 \times 10^3} \approx 10^9$$

This enormous formal magnification is limited by the finite stellar angular diameter — stars are not truly point sources. The effective central flash brightening for typical stars and atmospheric structures is $\sim$2–10$\times$ above the background level.

4.3 Central Flash Duration

The timescale for the central flash is: $$t_\text{flash} \approx \frac{2 \lambda_\text{caust}}{v_s}$$

where $\lambda_\text{caust}$ is the caustic half-width (typically $\sim$10–100 km for Pluto-scale objects). For $v_s = 20$ km/s and $\lambda_\text{caust} = 50$ km: $$t_\text{flash} \approx 5\,\text{s}$$

This requires high-cadence photometry (1–2 s exposures) to capture the peak. Standard 30 s exposures will miss or severely dilute it.

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5. Timing Precision and its Effect on Shape Quality

The fundamental connection is: $$\sigma_a = v_s \cdot \sigma_t / \sqrt{\text{sensitivity factor}}$$

For a chord duration $T_\text{chord} = \ell / v_s$, the ratio of uncertainty to chord length is: $$\frac{\sigma_a}{a} \approx \frac{\sigma_t}{T_\text{chord}/2} = \frac{2 \sigma_t v_s}{\ell}$$

For $\sigma_t = 0.1$ s (NTP best case), $v_s = 20$ km/s, $\ell = 200$ km: $$\frac{\sigma_a}{a} \approx \frac{2 \times 0.1 \times 20}{200} = 0.02 = 2\%$$

For $\sigma_t = 0.5$ s (poor NTP), same case: $$\frac{\sigma_a}{a} \approx 10\%$$

For occultation science, GPS-PPS timing (1–10 ms) is essential for main-belt asteroids. NTP-only timing (0.1–1 s) is marginal for main-belt bodies but acceptable for large TNOs (slow shadow, large body).

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6. Diffraction at the Geometric Shadow Edge

The occultation edge is not a step function — it is softened by diffraction. For a star of negligible angular diameter, the edge has a Fresnel diffraction pattern with timescale: [Source: Dravins et al. (1998), PASP 110, 610; Millis & Elliot (1979), AJ 84, 1917]

$$\Delta t_F = \frac{1}{v_s} \sqrt{\frac{\lambda d_\text{body}}{2}}$$

where $\lambda$ is the wavelength and $d_\text{body}$ is the distance to the body.

For a main-belt asteroid ($d_\text{body} = 3$ AU = $4.5 \times 10^{11}$ m), $\lambda = 500$ nm, $v_s = 20$ km/s:

$$\Delta t_F = \frac{1}{20000} \sqrt{\frac{5 \times 10^{-7} \times 4.5 \times 10^{11}}{2}} = \frac{\sqrt{1.1 \times 10^5}}{2 \times 10^4} = \frac{332}{2 \times 10^4} \approx 16\,\text{ms}$$

The Fresnel scale sets a physical floor on edge timing regardless of clock precision. For main-belt asteroids, this is ~10–50 ms. No amount of timing precision can localize the limb to better than this.

For TNOs at $d = 40$ AU: $$\Delta t_F = \frac{1}{10000} \sqrt{\frac{5 \times 10^{-7} \times 6 \times 10^{12}}{2}} \approx \frac{1.2 \times 10^3}{10^4} \approx 0.12\,\text{s} = 120\,\text{ms}$$

TNO diffraction edges are $\sim$100 ms — comparable to typical amateur camera exposure times. This makes TNO occultations challenging to time precisely without specialized equipment.

6.1 Stellar Angular Diameter Effect

For a star of angular diameter $\theta_\star$ at distance $D_s$, the edge is further smeared by: $$\Delta t_\star = \frac{\theta_\star \cdot d_\text{body}}{v_s} \approx \frac{\theta_\star \cdot d_\text{body}}{v_s}$$

For a giant star ($\theta_\star = 1$ mas) and a main-belt asteroid: $$\Delta t_\star = \frac{10^{-3} \times 4.4 \times 10^{-6}\,\text{rad} \times 4.5 \times 10^{11}\,\text{m}}{2 \times 10^4\,\text{m/s}} \approx 100\,\text{ms}$$

Giant stars completely wash out Fresnel diffraction features. For shape modeling, only occultations of dwarf stars (K/G dwarfs with $\theta_\star < 0.1$ mas) preserve diffraction structure.

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7. Required Cadence and Exposure Time

For accurate event timing: 1. The exposure time must be much shorter than the chord duration: $t_\text{exp} \ll T_\text{chord}$ 2. For diffraction-limited timing: $t_\text{exp} < \Delta t_F$

For a main-belt asteroid chord duration of 1–20 s and Fresnel timescale ~20 ms: - Recommended $t_\text{exp} = 0.1$–1.0 s for shape modeling - Maximum cadence allowed by camera: limited by read time (for slow CCDs) or rolling shutter (for consumer CMOS)

Rolling shutter warning: Consumer CMOS cameras (GoPro, smartphone sensors) have rolling shutters with readout times of 10–100 ms. The effective exposure start time varies across the sensor, introducing timing errors proportional to the row position. For occultation timing, use only global shutter cameras or CCDs, or restrict the star to the center rows and calibrate carefully.

[NOVEL] The rolling shutter warning and the specific mitigation strategy (restrict star to center rows and calibrate carefully) as applied to low-cost CMOS cameras in amateur occultation work is original to OpenAstro.

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8. Worked Example: Eurybates Occultation

Eurybates (Lucy target, Trojan asteroid) has diameter $\approx 63$ km. At opposition, $d_\text{body} \approx 5.2$ AU = $7.8 \times 10^{11}$ m, $v_s \approx 15$ km/s.

Chord duration for central chord: $$T_\text{chord} = \frac{63\,\text{km}}{15\,\text{km/s}} \approx 4.2\,\text{s}$$

Fresnel timescale: $$\Delta t_F = \frac{1}{15000} \sqrt{\frac{5 \times 10^{-7} \times 7.8 \times 10^{11}}{2}} = \frac{441}{15000} \approx 29\,\text{ms}$$

Limb position uncertainty with GPS-PPS ($\sigma_t = 5$ ms): $$\sigma_\text{limb} = 15 \times 0.005 = 75\,\text{m} \approx 0.1\%\,\text{of diameter}$$

Limb uncertainty with NTP ($\sigma_t = 0.2$ s): $$\sigma_\text{limb} = 15 \times 0.2 = 3\,\text{km} \approx 4.8\%\,\text{of diameter}$$

Conclusion: For a 63 km object, GPS-PPS achieves 0.1% precision on each limb point. NTP achieves ~5%, which, compounded over 5 chords, gives ~10% uncertainty in $a/b$. For shape science on 50–100 km objects, GPS is necessary.

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9. Shadow Path Prediction Uncertainty

The shadow path is predicted from orbital ephemerides + stellar position. The dominant uncertainties are:

1. Asteroid position uncertainty: For a well-observed main-belt asteroid, the cross-track position error is $\sim$10–100 km (1-sigma). For a recent close approach, it may be better. The shadow path shifts by this amount in the sky-plane projection.

2. Stellar position uncertainty: Gaia DR3 provides positions to $\sim$0.02 mas [Source: Gaia DR3, Lindegren et al. (2021), A&A 649, A2], corresponding to a shadow-plane uncertainty of $d_\text{body} \times 0.02\,\text{mas} \approx 7\,\text{km}$ at 5 AU. This is negligible compared to asteroid ephemeris error.

3. Combined path uncertainty: Dominated by asteroid position. Plan for the shadow path to be uncertain by $\sim$50 km cross-track; deploy observers in a strip $\pm$100 km from the predicted centerline.

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10. Summary: Minimum Requirements per Science Goal

Science Goal	Min K	Min $\sigma_t$	Notes
Size lower bound	1	0.5 s	NTP-only acceptable for large TNOs
Diameter estimate	2–3	0.1 s	Requires symmetric assumptions
Ellipse fit	5–6	0.1 s	GPS preferred; NTP marginal for fast shadows
Shape model ($n=3$)	8–10	10 ms	GPS required
Shape model ($n=5$)	14–16	5 ms	GPS required; 1 Hz video
Atmosphere detection	1 central	0.5 s	Requires high cadence ~1 s for central flash peak
Topographic features	>15	5 ms	Specialized; requires fast video and GPS

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11. References

• Millis & Elliot (1979), AJ, 84, 1917 — foundational occultation geometry
• Sicardy et al. (2011), Nature, 478, 493 — TNO shape from chords
• Herald et al. (2020), MNRAS, 499, 4570 — IOTA methodology, chord fitting
• Morgado et al. (2022), Icarus — modern shape reconstruction methods
• Dravins et al. (1998), PASP, 110, 610 — diffraction in occultations
• Braga-Ribas et al. (2014), Nature, 508, 72 — Chariklo rings via occultation