Earth's Shadow Filtering Overview

• Earth’s Shadow Filtering is a suite of methods that define and model the penumbra and umbra to quantify variations in sunlight and atmospheric refraction.
• It combines physics-based, numerical integration and empirical models to simulate refraction, absorption, and the effects of atmospheric filtering.
• Key applications include satellite orbit tracking, urban remote sensing, and SETI artifact detection, providing enhanced precision in observational data.

Earth’s shadow filtering refers to the ensemble of methods and effects by which Earth’s shadow—whether cast onto artificial satellites, the Moon, or through imaging domains—modulates illumination, radiative forces, and observational backgrounds. Such filtering finds broad application in celestial mechanics, satellite photometry, artifact detection, and eclipse modeling. It is rooted in the physically complex interplay between geometric shadow boundaries and atmospheric effects, and is computationally represented in both empirical and physics-driven models.

1. Celestial Geometry and Definition of Earth’s Shadow

Earth’s shadow is comprised of two principal regions: the penumbra (partial shadow) and the umbra (full shadow), defined by the spatial relationships among the Earth, Sun, and the target object (satellite, Moon, etc.). The umbra is the locus where the Sun is fully occluded by Earth, while the penumbra corresponds to regions where only a portion of the solar disk is blocked (Mallama, 2021).

For objects in near-Earth orbits—particularly geosynchronous debris—the umbral cone is set by the Earth’s and Sun’s angular sizes and the heliocentric separation. The length $L$ of the umbral cone may be approximated by $L \approx R_\oplus / \tan(\theta_\odot)$, with $\theta_\odot \sim 0.5^\circ$ as the solar angular radius. In SETI artifact searches, the “dark full shadow” (DFS) plus refraction fringe typically subtends an angular radius $\varrho + \gamma \sim 8$–$9^\circ$ for GEO altitudes, where $\gamma$ quantifies the fringe extent attributed to refracted sunlight (Villarroel et al., 19 Oct 2025).

2. Atmospheric Effects: Refraction, Absorption, and Filtering

Earth’s atmosphere modulates the geometric boundary of its shadow via refraction, absorption, and focusing. Rays grazing the limb are refracted $\sim 70'$ (twice the sunset/sunrise displacement), supplying even the shadow’s core (umbral region) with diffuse sunlight. The intensity after refraction and absorption is modeled as

$\frac{i(t)}{i(0)} = E^m$

where $E$ is the transmission coefficient per unit air mass and $m$ is the cumulative air mass traversed. Absorption is wavelength-dependent; blue photons are attenuated more strongly than red, leading to reddened appearances of eclipsed objects (Mallama, 2021, Mallama, 2021).

Focusing further concentrates the refracted sunlight into the umbral region, with geometric and physical broadening manifest as umbral enlargement—observed to be $87$–$211~$km larger than naive geometric predictions. This results in a gradual, rather than abrupt, decline in observed brightness at eclipse boundaries (Mallama, 2021).

3. Computational Models and Numerical Integration Schemes

Accurate tracking of objects in high Earth orbit or during eclipse phenomena necessitates robust numerical schemes. One notable approach is symplectic integration, wherein the Hamiltonian governing debris orbits is split into analytically tractable (Keplerian, rotational) and perturbative (geopotential, third-body, SRP) components. For the latter, explicit shadow modeling is included via smooth switching functions:

• Cylindrical shadow model: $$\nu_c(\mathbf{r}) = \frac{1}{2} \left\{ 1 + \tanh\left[\gamma\,s_c(\mathbf{r})\right] \right\}$$ with $s_c(\mathbf{r})$ structurally negative inside the shadow.
• Penumbra/conical model: $$\nu_p(\mathbf{r}) = \frac{1}{2} \left\{ 1 + \tanh\left[ \frac{\delta\,2\pi R_\oplus}{\Delta h(\mathbf{r})} s_c(\mathbf{r}) \right] \right\}$$

Parameters $\gamma$, $\delta$, and $\Delta h$ are chosen for near-on/off filtering, but permit analytic derivatives required for variational calculations (Hubaux et al., 2012). These models are not equivalent; orbital evolution under penumbral transitions, in particular for high area-to-mass ratio objects, diverges substantially from hard-edged (cylindrical) modeling.

Symplectic schemes paired with analytic filtering enable large time steps and bounded energy errors over centuries, outperforming non-symplectic Adams–Bashforth–Moulton approaches in numerical stability (Hubaux et al., 2012).

4. Empirical and Data-Driven Filtering in Satellite and Urban Analysis

Shadow filtering principles extend to empirical contexts such as urban planning and remote sensing (Al-Hilaly et al., 2018). Complex astronomical dependencies (solar declination, zenith, azimuth, local latitude) may be collapsed, in specific regions, to a single parameter Gaussian model:

$$\mathrm{SLP}(dn) = a_1 e^{- \frac{(dn - b_1)^2}{C_1}} + a_2 e^{- \frac{(dn - b_2)^2}{C_2}}$$

with coefficients fitted to observed shadow lengths across the year. For building height inference from satellite imagery, $\mathrm{SLP}$ enables conversion between measured shadow length and real structure height, simplifying design impact quantification and architectural planning. Empirical fits, with $R^2 \sim 0.9913$ and RMSE of $2.337$, deliver typical errors near $1.4\%$ in height estimation for the paper area. However, this method is region-specific, contingent upon non-variable imaging times and environmental factors.

5. Shadow Filtering in Photogrammetry and Neural Rendering

Multi-view satellite photogrammetry is complicated by time-variable shadows and illumination. The Shadow Neural Radiance Field (S-NeRF) architecture augments conventional NeRF by learning both a solar light visibility field $s(\mathbf{x},\omega_s)$ and a sky illumination color field $\mathrm{sky}(\omega_s)$ (Derksen et al., 2021). The composite radiance at each point is computed as

$$\ell(\mathbf{x},\omega_s) = s(\mathbf{x},\omega_s)\,\mathbb{1}_3 + (1-s(\mathbf{x},\omega_s))\,\mathrm{sky}(\omega_s)$$

with final color $$c(\mathbf{x},\omega_s) = a(\mathbf{x})\,\ell(\mathbf{x},\omega_s)$$, where $a(\mathbf{x})$ is albedo.

Training includes “Solar Correction” rays enforcing physical consistency of $s$ with accumulated transparency. The network reconstructs sharper shadow boundaries and more accurate elevation models (altitude MAE $\sim 4$–$7$ m in shadows versus $8$–$12$ m for conventional NeRF). S-NeRF enables novel view synthesis, albedo reconstruction, and transient object filtering even with unlabeled data.

6. Earth’s Shadow as an Observational Filter for Artifact Searches

Earth’s shadow creates a background with greatly diminished reflected sunlight—particularly within the umbra—making it a valuable discriminant for optical SETI investigations (Villarroel et al., 19 Oct 2025). Detection procedures label transients via imaging (e.g., ZTF’s 30 s exposures), applying segmentation and photometric scoring. Candidates are retained if inside the DFS ($\rho < 6^\circ$) and passing a battery of shape/morphology/association cuts.

Of $\sim11,000$ one-off transients found in ZTF data, only $2.4\%$ (262) fall within the shadow, and a handful remain unexplained after screening. The estimated event rate for possible self-luminous artifacts is $3\times10^{-7}$ per hour per square degree. Planned improvements will exploit multi-telescope parallax to distinguish fast-moving transient phenomena, further refining artifact discovery. This suggests Earth’s shadow is a practical filter for reducing false positives in optical technosignature searches.

7. Implications and Applications

Earth’s shadow filtering has significant impacts across dynamical astronomy, satellite cataloging, urban design, photogrammetry, and SETI. In orbital debris tracking, symplectic integration with analytic shadow models underpins precise long-term prediction and minimization of energy drift, supporting collision risk assessment. Satellite photometry and lunar eclipse analyses leverage physical filtering models to interpret color and intensity changes, quantify atmospheric effects, and model extended umbral boundaries. Urban planners and remote sensing engineers can simplify shadow quantification via empirical Gaussian modeling, albeit within specified locales and conditions.

The application of Earth’s shadow as an observational filter opens new avenues in artifact detection, offering a low-background regime for intrinsic emissions and complementing other technosignature search modalities. The plausibility of rare or brief optical events as candidate probes suggests continued development of high time-resolution, multi-baseline survey techniques.

In sum, Earth’s shadow filtering comprises a suite of physical, computational, data-driven, and observational techniques for manipulating the effects of Earth’s occlusion of sunlight. The discipline synthesizes atmospheric physics, geometric modeling, high-precision integration, and modern machine learning to support robust analysis and discovery in the dynamic near-Earth environment.