Abstract

A large fraction of nocturnal UAP reports are attributed, correctly or otherwise, to sunlight reflecting from satellites or orbital debris. The attribution is frequently made on plausibility grounds alone, without testing whether sunlight could physically have reached an object along the reported line of sight at the reported time.

That test is available and is purely geometric. Earth casts a shadow into space; an observer on the dark side of the terminator sees only those objects high enough to remain outside it. The minimum altitude required rises through the night, and rises fastest in the direction opposite the Sun. Where the required altitude exceeds the orbital population that actually exists, the flare hypothesis is eliminated — not as a judgment of likelihood, but as a consequence of geometry.

This paper documents the model, presents closed-form solutions for both branches of the shadow boundary, describes an implementation released as a free browser-based field tool, reports validation against independent closed-form solutions, and states the model's limits explicitly. A worked case study demonstrates the exclusion of a Starlink attribution by a factor of twenty to sixty in required altitude.

The method is a falsifier, not a confirmer. A negative result is strong and defensible. A positive result eliminates nothing.

1. Problem statement

1.1 The attribution problem

Satellite flare is among the most commonly invoked prosaic explanations for nocturnal point-source UAP reports, and it is often correct. Tumbling rocket bodies, defunct satellites, and specular glints from solar panels routinely produce brief brightenings of several magnitudes. The explanation is reached for so readily, however, that it is frequently applied without test.

This produces error in both directions. Genuine satellite events are recorded as unexplained when the investigator lacks the tools to confirm them. Non-satellite events are closed as satellites when the explanation merely sounds reasonable. Both outcomes degrade the evidential value of a case file.

The specific failure this paper addresses is the second: a satellite attribution that is geometrically impossible but is accepted because nobody checked whether the object could have been in sunlight.

1.2 What can and cannot be determined

A single ground observer reporting a point source in the night sky provides two angles — elevation and azimuth — and no range. Altitude cannot be recovered from a single-station observation. Any method claiming to determine an object's height from one witness is unsound.

The illumination question is different in kind. It does not ask where the object was. It asks a conditional: if an object had been along this line of sight, at what altitude would sunlight have reached it? That question has a unique answer determined entirely by the observer's position, the time, and the reported direction. All three are known or estimable from a standard field report.

Comparing that answer against the known orbital population then supports one of two conclusions:

• The required altitude falls within the populated orbital shells. The satellite hypothesis survives. Nothing further is established.
• The required altitude exceeds where relevant hardware exists. The satellite hypothesis is eliminated.

The asymmetry is fundamental and is the central epistemic claim of this paper. Sunlight is necessary for a passive reflector to be visible; it is nowhere near sufficient.

2. Geometric model

2.1 Coordinate frame and assumptions

The model uses a local East-North-Up frame centred on the observer, who is placed at radius R = 6371 km from Earth's centre along the local zenith, the volumetric mean radius given in the NASA Earth Fact Sheet [1].

Assumptions:

1. Earth is spherical.
2. Sunlight arrives as parallel rays, so the shadow is a cylinder of radius R aligned with the anti-solar direction.
3. The atmosphere is treated as an opaque annulus of thickness 80 km, giving an effective shadow radius R_eff = 6451 km.
4. Refraction is neglected.

Assumptions 2 and 4 are relaxed and quantified in Section 6.

Assumption 3 requires comment because it is a modeling choice rather than a physical constant. Sunlight grazing the limb below roughly 80–100 km traverses enough atmosphere that scattering and absorption reduce it to a level unable to produce a detectable reflection. The geometric shadow is therefore too small. The 80 km figure is conservative in the direction that matters: it raises the required altitude, making exclusions harder to obtain and correspondingly stronger when they are. It is not derived from first principles, but it sits close to the notional eclipsing layer of approximately 85 km used in lunar-eclipse work [31], corroborating that the value is of the right order.

2.2 The illumination condition

Let ŝ be the unit vector toward the Sun and û the unit vector along the observer's line of sight, both in the local frame. A point at slant range t has position

r(t) = Rẑ + tû

where ẑ is the local zenith. That point is sunlit if either:

• it lies in the sunward hemisphere, r·ŝ > 0; or
• its perpendicular offset from the Earth–Sun axis exceeds R_eff.

The exit range is the smaller positive root of the two conditions. Writing q = ẑ·ŝ, p = ẑ·û, and w = û·ŝ:

Plane crossing:t_plane = −Rq / w, valid when w > 0

Cylinder exit: (1 − w²)t² + 2R(p − qw)t + R²(1 − q²) − R_eff² = 0

The exit altitude follows from

h = √(R² + 2Rpt + t²) − R

This formulation is general. It handles any solar position and any look direction without special cases, and it is what the released tool implements.

2.3 Closed forms

For the special case where the object lies in the plane containing the Sun and the zenith, useful closed forms exist. Let β be the Sun's depression angle below the horizon and E the object's elevation.

Zenith:

h_min = R (sec β − 1)

Anti-solar azimuth, E > β — the sightline exits the near side of the shadow:

d = R (1 − cos β) / sin(E − β)

Anti-solar azimuth, E < β — the sightline crosses the shadow axis and exits the far side:

d = R (1 + cos β) / sin(β − E)

At E = β the sightline is parallel to the cylinder axis and never exits it.

Both anti-solar expressions derive from the same quadratic. Substituting x = d sin(E − β) into the cylinder condition gives

x² + 2R cos β · x − R² sin²β = 0

with roots x = R(−cos β ± 1). For E > β the positive root applies; for E < β the sine is negative and the other root yields a positive range.

2.4 The sub-β floor

The far branch has a minimum. Since (1 + cos β)/sin β = cot(β/2), the horizon case gives slant range R·cot(β/2) and altitude

h_floor = R (csc(β/2) − 1)

This is the least altitude at which sunlight can reach any direction below elevation β. It is attained at the horizon and rises without bound as E approaches β from below.

The operational significance is direct. Geostationary orbit is at 35,786 km. For β greater than about 18°, the entire region of sky below the Sun's depression angle requires altitudes at or beyond the geostationary belt — a region containing only sparse graveyard debris and high-apogee objects, none of them visually detectable. Below elevation β, satellite illumination is not impossible, but it is excluded for every populated orbital regime.

2.5 An error worth documenting

An earlier formulation of this work asserted that nothing below elevation β is ever sunlit in the anti-solar direction. That claim is false, and the error is instructive.

The quadratic in Section 2.3 always has one positive root. The earlier derivation kept the root corresponding to the near-side exit and discarded the other as lying behind the observer — true for E > β, false for E < β. The two-branch structure was thereby collapsed into a single branch plus a spurious impossibility.

The practical consequence was small, because far-branch distances are enormous and the resulting case conclusions were unchanged. The methodological consequence was not: an assertion of impossibility cannot be independently checked, whereas a stated altitude requirement can. The corrected formulation is strictly more useful in a case file for exactly that reason, and the released tool reports which branch applies to each result.

3. Orbital population reference

An illumination requirement is only meaningful against the population it is compared to.

3.1 Shell illumination limits at zenith

The maximum Sun depression at which a circular shell remains sunlit at zenith is

β_max = arccos(R_eff / (R + h))

Two features govern most real cases. First, nearly all trackable hardware sits below 2000 km, so β beyond about 40° leaves only MEO and GEO illuminated anywhere in the sky. Second, the productive window for overhead low-Earth-orbit passes is narrow: below β ≈ 10° the sky is too bright, and above β ≈ 20° the LEO shells begin entering shadow near the zenith, leaving roughly β = 10°–18°.

3.2 The sunward horizon is the exception, and it matters most

The β_max column above is computed at zenith, and generalising it into a blanket detection window is the single most likely way to misuse this method. Sunlight persists far longer at low elevation toward the Sun's azimuth than it does overhead, and that is precisely the geometry in which the most commonly misreported satellite events occur.

Sitrec's Starlink documentation [42] describes this region as the "flare band" — the zone of Earth's surface from which low-elevation Starlink flares are visible at a given moment, bounded by circles centred on the anti-solar point, roughly parallel to the terminator, and covering on the order of 10% of the globe at any time. Near the equinoxes some locations remain within it for five hours or more in the middle of the night. West notes that this is exactly why such events are misreported: they occur at times when an observer has learned not to expect satellites.

The illumination model of §2 reproduces this structure independently. For the case-study geometry of §7 — 34.51° N, 7 May 2026, 22:05 local, Sun at azimuth 323.3° and β = 29.6°, nearly three hours after sunset — the maximum elevation at which a 570 km object remains sunlit varies with bearing as follows:

The operational rule follows directly. A large β does not exclude Starlink; a large β combined with an anti-solar bearing does. Any statement of the form "the Sun was too far below the horizon for satellites" is unsound unless the reported bearing is also stated. The tool is azimuth-dependent throughout and handles this correctly; the risk lies in summarising its output as a depression-angle threshold.

3.3 The Starlink flare mechanism

Starlink flares are not generic tumbling glints. The satellite's phased-array antenna is a flat panel held in a horizontal attitude, so when the satellite is low in the observer's sky it can reflect sunlight specularly toward the ground for a few seconds — typically five to twenty — tracing a short straight path before fading [42]. Several can appear in the same patch of sky at once, on crossing trajectories, at brightness comparable to Venus and penetrating both light pollution and thin cloud. The combination of short duration, multiple objects, and crossing paths produces an impression of a smaller number of maneuvering objects.

Two consequences for the investigator. First, this profile — brief, bright, straight, low, sunward, multiple — is a positive signature to test for rather than merely a hypothesis to exclude. Second, because the mechanism is a predictable specular geometry rather than a random tumble, it can be simulated directly against real orbital elements, which is what Sitrec does and what this tool does not attempt.

3.4 Angular rate as an independent test

Illumination is necessary but not sufficient, and a second geometric constraint is available at no additional cost. A circular orbit at altitude h has speed v = √(μ/(R+h)) with μ = 398,600.4 km³/s² [1], bounding the apparent angular rate at ω_max = v/d for slant range d.

This test is powerful in combination with the illumination test because the two constraints move in opposition. A deep-twilight sighting demands a high altitude for illumination, and high altitude implies slow apparent motion. A report of a bright, fast-moving object late at night is therefore doubly inconsistent with an orbital explanation: it fails illumination at low altitude and fails kinematics at high altitude.

4. Detection model

Where illumination is geometrically permitted, a first-order brightness estimate indicates what instrumentation would have been required. This component of the model is substantially weaker than the geometry and is presented as guidance rather than measurement.

Apparent magnitude is scaled from a reference at 1000 km slant range, the standard-distance convention used in the published satellite photometry literature [17]:

m = m₀ + 5 log₁₀(d / 1000 km)

The Starlink figure is conservative against measurement: Mallama et al. report a mean of 7.87 at 1000 km for Gen2 Mini satellites in brightness-mitigation mode [19].

Specular flare is modeled as a generic 5-magnitude boost. Real glints range from 3 to 8 magnitudes and persist from one to ten seconds [20]. This generic treatment is a deliberate simplification and a known weakness: the dominant real-world case, the Starlink flat-panel horizon flare described in §3.3, has a specific and predictable specular geometry that a single scalar boost does not capture. Where the reported profile matches that signature, a simulation against real orbital elements [42] will substantially outperform this model. Heavier tails occur: original-design Starlink satellites were observed flaring by up to 10 magnitudes in April 2020, with no comparable events reported since [21].

Detector thresholds are capped by sky brightness, which is itself a function of β. The figures in the following table are the author's working estimates. No controlled study is known that measures these detector classes against moving point sources under common conditions, and they should not be treated as measured values:

The sky limiting magnitudes interpolated by the tool as a function of β are likewise practical estimates rather than output from a published twilight sky-brightness model.

Not modeled: moonlight, which can raise sky background by 2–3 magnitudes; atmospheric extinction, roughly 0.3 mag at 30° elevation, 1 mag at 10°, and 3+ mag near the horizon; light pollution; and phase angle, which for a diffuse reflector can vary brightness by more than a magnitude across a pass.

Users should treat the detection table as establishing order of magnitude only. The geometric results are far more robust and should carry the evidential weight in any case file.

5. Implementation

The tool is a single self-contained HTML file with no build step, no dependencies, no backend, and no data collection. It runs from local disk, which matters for field use without connectivity, and is hosted as a static page for general access.

Inputs: date, local time, UTC offset, observer latitude and longitude, object elevation, compass bearing, optional magnetic declination correction, optional angular rate, and six behavioral checkboxes.

Solar position is computed by the NOAA general solar position algorithm [4][5], which derives from Meeus [7] and is accurate to approximately one arcminute over the relevant epoch. NOAA notes that its own calculator implementing these equations is no longer actively maintained; the published equation set is unaffected, and the NOAA calculator [6] remains the natural independent cross-check. This removes the need for the investigator to obtain β from a separate source.

Outputs:

• A graded finding across five bands, from the whole orbital population, was lit through sunlight geometry excludes an orbiting object
• Sun depression, angular offset from the anti-solar point, minimum sunlit altitude, slant range to the shadow boundary, which branch of the shadow boundary applies, penumbral fade band width, sky limiting magnitude, and orbital angular rate ceiling
• A cross-section diagram on a logarithmic radial scale showing Earth's shadow, the orbital shells, and the reported line of sight
• A curve of shadow ceiling against elevation angle along the reported bearing, permitting immediate assessment of sensitivity to elevation error
• A detection table by object class and instrument
• A formatted case note for direct inclusion in a report

Behavioral flags are evaluated independently of the geometry and can override a positive geometric result. Reported sound, strobing or coloured lights, hovering or reversal, station-keeping beyond fifteen minutes, and disappearance followed by return are each inconsistent with an orbiting object regardless of illumination. Reported reddening before fading is the single flag that supports the satellite hypothesis, for the reasons given in §6.3, and it is presented distinctly.

Accessibility. The interface uses a dark background with foreground contrast ratios at or above 7:1 against panel backgrounds, meeting WCAG 2.2 Success Criterion 1.4.6 for normal text [34]. That threshold is not arbitrary for this readership: W3C documents the 7:1 ratio as compensating for contrast sensitivity loss equivalent to approximately 20/80 vision, and the 4.5:1 Level AA ratio as corresponding to 20/40, described as typical acuity at roughly age 80 [35]. The sunlit/shadow color distinction lies on the blue-yellow axis, preserving it under red-green color deficiency, and every color-coded state also carries a text label.

6. Validation and error budget

6.1 Agreement with closed-form solutions

The general vector solver was compared against the independent closed forms of Section 2.3 across both branches. Agreement is at machine precision, with relative error below 10⁻¹⁴ throughout.

The sub-β floor formula of Section 2.4 was likewise verified against the solver at the horizon across β from 6° to 60°, agreeing to the kilometer at every value.

6.2 Cylindrical versus conical umbra

The Sun subtends approximately 0.533° at 1 AU [2], so Earth's true umbra is a cone converging to a point 1,367,149 km from Earth's center rather than a cylinder. The cone is narrower than the cylinder everywhere, so the true exit occurs earlier, and the cylindrical model overstates the required altitude.

The correction is 1–7% and always in the same direction. It is not implemented in the released tool, on the grounds that it is smaller than the uncertainty in a witness's elevation estimate and that its sign makes the model conservative for exclusions. Investigators reporting a marginal result should note that the true requirement is slightly lower than stated.

6.3 Penumbra

The finite solar diameter also means the shadow boundary is not sharp. An object crossing the penumbral band dims progressively over several seconds and commonly reddens as it does so. The band is roughly 20 km of altitude at β = 18° and 40 km at β = 30°, computed in the tool from the two solar limbs.

The reddening arises because sunlight still reaching an object near the shadow boundary has grazed the atmospheric limb, and Rayleigh scattering through the lower atmosphere preferentially removes short wavelengths — the same process that reddens a sunset [31]. Umbral photometry of lunar eclipses quantifies the resulting differential extinction: at the center of the umbra, blue light is attenuated by roughly 22 magnitudes against roughly 11 magnitudes in red [32].

One qualification matters, because the opposite intuition is widespread. Ozone absorbs in the Chappuis band, approximately 430–750 nm, which removes red light rather than blue. Spaceborne multi-wavelength observation of the January 2019 eclipse found the red-to-blue ratio falling below unity near the umbra, producing the bluish band astronomers photograph at the edge of totality [33]. Ozone therefore works against reddening, and dominates only in a narrow zone at the very edge of the umbra where the grazing path lies high in the stratosphere. Reddening is a lower-atmosphere effect; any bluish cast is an ozone effect.

This has diagnostic value. A witness describing a light that turned orange and faded over several seconds has described a shadow entry, and has thereby substantially corroborated an orbital explanation.

6.4 Refraction

Refraction is neglected. Its effect on β is largest immediately after sunset, where it reaches roughly half a degree, and diminishes rapidly thereafter. For sightings within a few minutes of sunset — where illumination rarely excludes anything in any case — the reported β should be treated as approximate.

6.5 Input uncertainty

Input error dominates model error in nearly every real case. Naked-eye elevation estimates are routinely wrong by ten degrees or more; compass bearings taken from memory are worse. The tool's shadow-ceiling curve exists specifically to make this visible: an investigator can read directly how far the conclusion would move under a plausible error in the elevation estimate.

Where a conclusion is robust across the full plausible range of input error — as in the case study below — it should be stated as such. Where it is not, it should not be stated at all.

7. Case study

Report. Witness in eastern Arizona (34.51° N, 110.07° W), 7 May 2026 at 22:05 local (UTC−7). A light low in the southeastern sky, elevation approximately 12°, bearing 120°–140°. Prevailing explanation offered: Starlink flare.

Solar geometry. Sunset occurred at approximately 19:06 local. At 22:05, two hours and fifty-nine minutes later, the Sun was at elevation −29.63°, azimuth 323.3°. The anti-solar point was therefore at azimuth 143.3°, placing the reported bearing range 3.3° to 23.3° from it — nearly down the axis of Earth's shadow.

Illumination requirement at 12° elevation:

Comparison to the population. SpaceX's Gen2 authorization covers altitudes from 340 km to 614 km [11], with subsequent modification lowering the 525, 530, and 535 km shells to 480, 485, and 475 km, respectively [12], and the January 2026 grant adding shells between 340 km and 485 km [13]. SpaceX announced in January 2026 that approximately 4,400 satellites at around 550 km would be lowered to 480 km over the course of that year [15][16]. Insertion and parking orbits are lower still [14]. Taking 620 km as a generous ceiling on any Starlink hardware, the reported direction required 12,400–33,900 km for solar illumination: a factor of twenty to sixty above the constellation's maximum altitude.

The kinematic test compounds this. At the required altitudes, maximum apparent angular rate is 0.005–0.016 °/s, corresponding to ten to thirty minutes to traverse a single fist-width. Starlink flares are fast-moving. Both constraints fail.

Robustness. The conclusion is unchanged if the bearing was magnetic rather than true (local declination approximately 9.5° E, giving true bearings of 129.5°–149.5°, requiring 21,900–31,300 km); if the elevation estimate is wrong by ten degrees in either direction; or if the date is read in day-month rather than month-day convention.

Where Starlink was illuminated. At that moment the lowest sunlit altitude anywhere in that sky was 262 km, but only in a narrow band low toward the northwestern horizon — the Sun's own azimuth, 323.3°. At zenith the requirement was already 1051 km. Had the sighting been reported low in the northwest, the Starlink hypothesis would have been live, and the §3.2 table shows a 570 km object remained sunlit there up to 27.5° elevation. Toward the southeast at 12° elevation, it was not.

This is the point at which the finding is most open to challenge, and it survives the challenge cleanly. An investigator aware that middle-of-the-night Starlink horizon flares occur — and they do, for hours at a time, at this latitude and season — might reasonably ask whether the depression angle alone is doing too much work here. It is not. The exclusion rests on bearing, not on β: the reported direction was roughly 180° from where flares were possible. The same conclusion follows from Sitrec's flare-band construction by an independent route [42], which is worth stating in the case file because it means the finding does not depend on this tool being right.

Alternative candidate. At the reported time and location, Antares — a distinctly red slow irregular variable ranging between apparent magnitude 0.6 and 1.6 [39] — stood at azimuth 131.8°, elevation 9.8°. This falls at the center of the reported bearing range and within approximately 2° of the reported elevation, comfortably inside naked-eye estimation error. At under 10° elevation, the star is viewed through roughly five air masses, where scintillation produces rapid brightening, dimming, and color flashes routinely described by witnesses as flaring. The star's intrinsic variability compounds this. The hypothesis fits the reported position better than any orbital object.

Finding as recorded. The Starlink attribution is excluded on geometric grounds. A bright-star misidentification is under evaluation and can be tested directly by showing the witness the sky as it appeared, using planetarium software set to the sighting time and location.

8. Investigative protocol

The following sequence is recommended for any nocturnal point-source report where an orbital explanation is proposed.

1. Obtain solar geometry for the exact time and place [6][10]. Do not estimate from memory or from a different date.
2. Re-establish the witness's position and sightline by site revisit where possible. Record whether bearings are magnetic or true, and obtain local declination from the current geomagnetic model [37][38]. Record how the elevation was estimated.
3. Run the illumination screen. Record the required altitude, the branch, and the angular rate ceiling.
4. Test robustness across the plausible range of input error before stating any conclusion.
5. If the hypothesis is eliminated, state it as an altitude requirement with the supporting geometry, not as a bare assertion.
6. If the hypothesis survives, it remains untested. Proceed to pass prediction against orbital elements dated within a few days of the sighting [24][28][30]; compare observed and predicted angular rate; compare observed and predicted brightness. Where the profile suggests a Starlink horizon flare, Sitrec [42][43] simulates the specular geometry directly against real elements and is the stronger instrument for that step. Note that it loads current satellites by default; for any event more than a day or two old, the elements must be reloaded for the sighting date, and the in-view accuracy indicator checked.
7. Check bright stars and planets in the reported quadrant [39][40]. Low-elevation scintillation is among the most common sources of reported flaring.
8. Retrieve instrumental records where available — ADS-B history for aircraft [41], and any local all-sky or sky-monitoring capture covering the window. Independent timestamps and measurable angular rates outrank witness estimates in every case. Buettner et al. demonstrate the combined TLE-plus-ADS-B reconstruction on a corroborated five-witness report, and remain the closest published precedent for the method set out here [23].

Step 4 deserves emphasis. A conclusion that survives only under the witness's exact reported numbers is not a conclusion.

9. Limitations

The following are stated plainly because a screening tool that overstates its reach is worse than none.

The tool tests one hypothesis. It screens satellite and debris illumination. It says nothing about aircraft, balloons, drones, meteors, atmospheric optics, or any other class of explanation.

The tool never identifies an object. A surviving hypothesis is not an identification. Identification requires matching a specific cataloged object by pass prediction.

Altitude is not recoverable. The model computes a conditional requirement, not the object's actual position.

Positive results are weak. Where the geometry permits illumination, it has excluded nothing and carries no evidential weight.

Illumination is not visibility. An object outside Earth's shadow may still be undetectable owing to size, albedo, phase angle, extinction, sky brightness, or horizon obstruction. The detection model addresses this only to order of magnitude.

The brightness model is weak. Magnitude estimates depend on parameters not available from a witness report. They should inform judgment, not determine it.

Input error dominates. In most real cases, uncertainty in the witness's angular estimates exceeds every modeling approximation combined.

A depression angle alone excludes nothing. The most likely misuse of this method is to summarise a result as a β threshold and discard the bearing. Low-elevation sunward geometry keeps low Earth orbit illuminated for hours after every other part of the sky has gone dark (§3.2). Exclusions must be stated with the bearing attached.

10. Availability

The tool is released free for use, hosted at https://www.satorufo.com. It is a single HTML file that may be saved and run offline from local storage, including on a phone during fieldwork. It requires no account, collects no data, and transmits nothing.

References

Bracketed numbers throughout refer to the companion reference list, Orbital Illumination Screening for UAP Field Investigation — References, which gives full citations and verified hyperlinks for all forty-four sources, grouped by the section they support.

The three components of this paper that rest on author estimate rather than published source are identified explicitly in §4 and §2.1: the detector limiting magnitudes, the sky limiting magnitude as a function of β, and the 80 km opaque-atmosphere allowance.

Appendix A: Summary of formulae

R = 6371 km; R_eff = 6451 km; μ = 398,600.4 km³/s²

Appendix B: Notation