Calculation rules and references

1. Time and positions

• Ephemerides of the Sun, Moon and planets — astronomy-engine (Don Cross, MIT licence): truncated VSOP87 series for the planets; the Moon from the Improved Lunar Ephemeris (1954, after E. W. Brown), as adapted by Montenbruck & Pfleger, Astronomy on the Personal Computer; Pluto by numerical integration with the DE405 masses; precession and nutation IAU 2000B.
• ΔT (TT − UT1) — a table of values observed by the US Naval Observatory from 1973, then its predictions for about eight years (maia.usno.navy.mil, deltat.data and deltat.preds), in place of the Espenak & Meeus (2006) polynomials that the engine extrapolates.
• Atmospheric refraction — Sæmundsson's formula R = 1.02′ / tan(h + 10.3/(h + 5.11)), as given by Meeus, Astronomical Algorithms (1991, pp. 101-102), clamped one degree below the horizon (astronomy-engine, “normal” mode).
• Parallactic angle — q = atan2(sin H, tan φ cos δ − sin δ cos H), Meeus, Astronomical Algorithms, 2nd ed., ch. 14.
• Orientation of planets and the Moon (central meridian, axis tilt, libration) — the IAU rotation models (pole and prime meridian angle W), as provided by astronomy-engine.
• Projection of the All-sky (Zenith) view — stereographic projection centred on the zenith: a point at zenith distance z falls at a radius proportional to tan(z/2) from the centre, so the horizon (z = 90°) is a circle and angles are preserved, unlike in an equidistant planisphere. Zooming toward the pointer keeps the aimed point on its pixel exactly (the radius scales with tan(F/4), F being the field), and the view centre may stray from the zenith by at most (F₀ − F)/2 degrees, capped at 90°, F₀ being the field at which the whole horizon circle fits: at F₀ the centre is pinned on the zenith, and zooming out pulls it back along the great circle that joins them. A geometric convention of the app, not a published formula; checked by check-ciel-entier.mjs.
• Compass of the All-sky view (phone and tablet) — like a pocket compass: the map turns about the zenith so that the device's heading is at the top and every cardinal point of the map points to its true direction. The heading is the horizontal part of the device's top edge weighted by the vertical part of its back, plus that of the back weighted by the rest: the top edge when flat (it alone follows a pivot of the device), the back when upright. The All-sky map is the sky seen from below (east left of north), which no rotation can align with the ground when the screen faces up; so, screen up (the back's vertical component below −0.25, about 15° below the horizontal), the map is mirrored, and screen down (overhead) it is not, with a dead band between to avoid flicker. App rule: a geometric convention, checked by check-ciel-entier.mjs.

2. Atmosphere and sky background

• Extinction coefficient — k = 1.0857 (τRayleigh + τozone + τaerosols) at 550 nm, with τ_Rayleigh = 0.0973 at sea level scaled by exp(−altitude / 8000 m), τ_ozone = 0.032, and the aerosol optical depth of the day from the Copernicus Atmosphere Monitoring Service (served by Open-Meteo). When it is unknown, τ_aerosols = 0.05, which gives k ≈ 0.195 at sea level, consistent with the usual ~0.2 mag per airmass (Hardie 1962; Green, Spherical Astronomy, 1985, p. 300).
• Airmass of a star — Young & Irvine (1967, AJ 72, 945), finite (~40) at the horizon.
• Airmass of the scattering layer — X(Z) = (1 − 0.96 sin²Z)−1/2, Krisciunas & Schaefer (1991, PASP 103, 1033), after Schaefer (1990, PASP 102, 212).
• Sky brightness is added in nanolamberts, never in magnitudes, with the conversion used by Garstang (1989, PASP 101, 306) and Krisciunas & Schaefer (1991). It depends on the direction viewed, not only on the zenith value.
  • Natural sky (airglow, zodiacal light, integrated starlight): 21.9 mag/arcsec² at the zenith (Benn & Ellison 1998, La Palma Technical Note 115), brightened by the layer airmass and dimmed by extinction, B(z) = Bzenith · X · 10−0.4 k (X − 1) (Garstang 1989; Schaefer 1990).
  • Light pollution: the Bortle class you logged at the place, otherwise the World Atlas of Artificial Night Sky Brightness (Falchi et al. 2016, Science Advances 2, e1600377), resampled and embedded, otherwise an estimate from nearby towns (an app heuristic).
  • Moonlight: the four formulas of Krisciunas & Schaefer (1991) as published — lunar illuminance by phase angle, Rayleigh and Mie scattering function, extinction to the Moon and along the line of sight. The same function draws the lunar halo.
  • Twilight: from −5° to −15°, the V-band zenith polynomial of Patat et al. (2006, A&A 455, 385, Table 1, Paranal, fitted over 95° ≤ ζ ≤ 105°): 11.84 + 1.518 x − 0.057 x² mag/arcsec² with x = ζ − 95° — 11.84 at −5°, 13.30 at −6°, 18.00 at −10°, 19.67 at −12°. Above −4°, where Patat has no data, an app model: 10.67, 8.85 and 7.71 at −4°, −2° and 0°, from the scattering model of the sky colour below, anchored on Patat at −5°; about 4 with the Sun 10° up (order of magnitude). Joined by straight lines in magnitudes; the twilight term vanishes at −18°. The sky is drawn 0.6 mag brighter towards the Sun and 0.6 mag darker opposite it (app estimate, of the order of the gradient seen in the same measurements). Until 3.5 the anchors were 10.0, 16.0 and 19.6 at 0°, −6° and −12°: civil and nautical twilight were up to 2.7 mag too dark.
  • Total solar eclipse: a floor of 12.88 mag/arcsec² at the zenith during totality (measured on 9 March 2016, J. Phys. Conf. Ser. 771, 012012).
• Sky colour (terrestrial view, atmosphere on; display only — no visibility rule reads it) — single scattering in a spherical atmosphere (Nishita et al. 1993, SIGGRAPH), multiple scattering approximated after Hillaire (2020, Comput. Graph. Forum 39, EGSR), refraction followed on the rays towards the Sun (refractivity 2.78 × 10⁻⁴ at sea level, Ciddor 1996, Appl. Opt. 35, 1566, decreasing with density), 12 wavelengths from 390 to 720 nm. The air is the one of the extinction above: Rayleigh optical depth 0.0973 at 550 nm, scale height 8 km; ozone 0.032, cross-sections of Serdyuchenko et al. (2014, AMT 7, 625) at 223 K, layer centred at 25 km; aerosols from the day's AOD above the site (fallback 0.05), scale height 1.2 km, Ångström exponent 1.3, single-scattering albedo 0.9, asymmetry 0.8 (Bruneton & Neyret 2008). Solar spectrum ASTM G173-03 (extraterrestrial); CIE 1931 matching functions approximated by Wyman, Sloan & Shirley (2013, JCGT 2(2), 1); white balance on the Sun. Checked against Hulburt (1953, J. Opt. Soc. Am. 43, 113: the blue twilight zenith comes from ozone), the three photographed twilights of Saito & Iwabuchi (2015, AMT 8, 4295), and the zenith V brightness and B − V colour of Patat et al. (2006, Table 1) at Paranal: brightness within 0.1 mag at −6° and −7°, then too dark (0.4 mag at −8°, 1.25 at −10°); B − V within 0.25 from −5° to −7° (the model is bluer). The display brightness does not come from this model. App choices: only the hue comes from the model; the brightness stays the app's sky gradient, modulated ×0.6 to ×1.4 so that each altitude circle keeps its mean; saturation ×2.4 relative to white; fade to the night palette between −12° and −18°.
• Bortle scale to sky brightness — the middle of each SQM range commonly published with the scale (J. E. Bortle, “Introducing the Bortle Dark-Sky Scale”, Sky & Telescope, February 2001; unihedron.com): class 1 to 9 = 21.85, 21.6, 21.4, 20.85, 19.75, 18.8, 18.25, 17.75, 17.2 mag/arcsec².
• Naked-eye limiting magnitude — the threshold model of Garstang (1989, PASP 101, 306), as used by Schaefer (1990, PASP 102, 212): threshold illuminance I = c (1 + √(k·B))², B in nanolamberts, m = −16.57 − 2.5 log₁₀ I, with the rod constants (c = 10^−9.8, k = 10^−1.9) and the cone constants (c = 10^−8.35, k = 10^−5.9), the more sensitive of the two winning. The branches cross near 18.6 mag/arcsec²: 21.9 → 6.58, 20.85 → 6.03, 18.0 → 4.19, 13.3 (end of civil twilight) → 3.49. Under bright skies the cones keep the limit near 4.2 whatever the background; glare from the Moon's disc itself is not modelled. Neither Schaefer's observer factors nor extinction are included: the value is a ceiling. Until 3.5 only the rod branch was used (18.0 → 3.97).
• Glare from the Moon — light from the lunar disc scattered inside the eye, added to the background against which the naked eye judges a star; neither to the drawn sky nor to the sky an instrument sees. It enters every naked-eye judgement — stars on the map, the object card's naked-eye verdict and its “the naked eye is enough” floor, the evening lists in naked-eye mode, the Conditions limiting magnitude, the hourly rate of meteor showers, the Journal's logged limiting magnitude and its verdicts' naked-eye floor — and no instrument verdict nor the Telescope view. CIE general disability glare equation (CIE 146:2002, Vos & van den Berg): Lveil / E = 10/θ³ + (5/θ² + 0.1 p/θ)(1 + (A/62.5)⁴) + 0.0025 p cd/m² per lux, θ in degrees from 0.5° (the disc) to 100°; the Moon's illuminance at the eye is Krisciunas & Schaefer's I* (0.29 lux at full Moon outside the atmosphere) with point-source extinction. App choice: a reference observer aged 40 with medium pigmentation (p = 0.5). Example, full Moon 50° high: limit 1.75 at 1°, 3.35 at 3°, 3.72 at 5°, 4.05 at 15°.
• Limiting magnitude in an instrument — the same model with Schaefer's two instrument factors: a star gains the full aperture, (D / 7 mm)², capped when the exit pupil exceeds the eye's; the sky background is darkened by 5 log₁₀(7 / exit pupil) below 7 mm. Optics transmission and seeing are neglected, so the value is a ceiling.

3. Telescope and eyepiece

• Magnification = telescope focal length / eyepiece focal length (times the Barlow factor); true field = apparent field / magnification; exit pupil = aperture / magnification. Binoculars use their stated magnification and true field.
• Resolving power — Dawes' criterion, 116″ / D(mm).
• Limiting magnitude shown for the instrument — 2 + 5 log₁₀ D(mm), calibrated on the naked eye (D = 7 mm gives 6.2), under a dark sky. Any verdict for tonight uses the sky-dependent limit of section 2 instead.
• Stars drawn in the Telescope view — each point is judged with the telescopic limiting magnitude under the sky background of that point (section 2, Moon glare excluded since the Moon is outside the eyepiece field): the naked-eye limit for that sky dimmed by 5 log₁₀(7 / p) mag/arcsec² (exit pupil p < 7 mm), raised by 5 log₁₀(D_eff / 7), with D_eff = min(D, 7·D/p) — the same rule the object panel applies to a star. It is capped by the dark-sky limit above. Until 3.5.2 the Telescope view used the naked-eye limit.
• Useful magnification — from D/7 (7 mm exit pupil) to 2×D (0.5 mm exit pupil).
• Surface brightness in the eyepiece — conservation of luminance: never brighter than to the naked eye, and fainter by 5 log₁₀(7 / exit pupil) mag/arcsec² below a 7 mm exit pupil.
• Recommended magnification — the one that brings the object to about 100′ apparent diameter, Clark's “Optimum Magnified Visual Angle” (R. N. Clark, Visual Astronomy of the Deep Sky, 1990), bounded to the useful range.

4. Visibility of deep-sky objects

• Eye contrast threshold — Crumey (2014, “Human contrast threshold and astronomical visibility”, MNRAS 442, 2600), fitted to Blackwell's measurements (1946, J. Opt. Soc. Am. 36, 624). For a uniform target of area A (steradians) on a background of luminance B (cd/m²):
  C = [((r₁ B−1/4 + r₂)² / A)q + (k₁ B−1/4 + k₂)q]1/q,
  r₁ = 6.505×10⁻⁴, r₂ = −8.461×10⁻⁴, k₁ = 7.633×10⁻³, k₂ = −7.174×10⁻³, q = 0.6 (eqs. 44, 47-49). Valid from 10⁻⁵ to 3.4×10⁻² cd/m²; below, the threshold is held at its cut-off value (eqs. 50-52) — held as a contrast without a filter, as an absolute threshold for the part of the background a filter removes (see the filter item below). The first term is Riccò's law for small targets, the second the large-target limit.
• Units — B(cd/m²) = 1.08×10⁵ × 10−0.4 μ for a sky of μ mag/arcsec² (Crumey's correspondence, 21.83 mag/arcsec² ↔ 2×10⁻⁴ cd/m²). The area is the apparent one in the eyepiece, magnification squared.
• Background seen through the instrument — darkened by the exit pupil, (min(d, 7 mm) / 7 mm)², and by a 75 % transmission (F_t = 1.33); the object's contrast is unchanged (Crumey §3.2).
• Field factors (Crumey §3.1-3.2) — F = 2 for an average observer with the naked eye (limiting magnitude 6.2 under 21.83 mag/arcsec²), and a further F_T = 2 at the eyepiece. A region is easy when its contrast over the sky exceeds F × C; marginal when it only exceeds the threshold of a trained observer, F = 1 (limiting magnitude about 7 according to Crumey); otherwise out of reach.
• The brightest region, not the average — the verdict scans regions from the core outwards and keeps the best detected one:
  • Galaxies: Sérsic profile (Sérsic 1963, Bol. Asoc. Argentina Astron. 6, 41; n = 4 is de Vaucouleurs' law, 1948, Ann. d'Astrophys. 11, 247), with b_n ≈ 2n − 1/3 + 4/(405 n) (Ciotti & Bertin 1999, A&A 352, 447). n from the Hubble type: 4 for E, S0, Sa, Sab; 2 for Sb, Sbc; 1 for later types and irregulars (Graham & Worley 2008, MNRAS 388, 1708). Effective radius R_e = 0.30, 0.40 or 0.50 R₂₅, derived from Freeman's law (1970, ApJ 160, 811) for discs and a typical de Vaucouleurs profile for bulges — orders of magnitude, not measured per galaxy.
  • Globular clusters: Plummer profile (1911, MNRAS 71, 460), projected light fraction R² / (R² + a²), with a = 0.40 × catalogue radius (the radius where the cluster is 4.5 mag below its centre).
  • Planetary nebulae and unknown types: the mean over the catalogued extent.
  • Diffuse nebulae: their catalogued extent is photographic, so the mean is a floor. Never “out of reach” while the most concentrated profile the model knows (Sérsic n = 4, R_e = 0.3 R) would still reach the trained-observer threshold over at least 1′; “easy” as soon as the least concentrated (exponential disc) already passes. A nebula whose catalogue magnitude is that of its embedded cluster is judged as that cluster (section 5).
  • Open clusters: resolved into stars, not judged by surface brightness; out of reach only when their total magnitude is fainter than tonight's limiting magnitude.
• Necessary condition — an extended object is never easier than a star of the same total magnitude: fainter than the instrument's limiting magnitude means out of reach.
• Naked-eye floor — an instrument never does worse than the naked eye under the same sky: when the naked-eye verdict is better, it is the one shown. This floor is always judged without a filter: nobody holds a filter in front of the naked eye.
• Visual nebula filters (UHC, OIII, H-beta, CLS) — a filter changes two numbers: the sky background, fainter by
  Δsky = −2.5 log₁₀ [∫ S T V′ dλ / ∫ S V′ dλ], and the object, fainter by Δobj, the same ratio over its own light. V′(λ) is the CIE 1951 scotopic efficiency (the dark-adapted eye at threshold; Hα at 656 nm weighs 4×10⁻⁴ of its peak, so it does not count). S for the sky is the smoothed continuum of the UVES night-sky atlas (Hanuschik 2003, A&A 407, 1157), read between the lines and used for every Bortle class: airglow and lamp lines are ignored, which makes Δsky an underestimate — the cautious side.
  • Passbands — one generic shape per type, from at least two manufacturers' published bands (Astronomik, Lumicon): 90 % in band, 3 nm linear edges, half-maximum at UHC 481–506 nm (+ beyond 645 nm), OIII 493–505 nm, H-beta 481–491 nm, CLS 445–535 nm (+ beyond 650 nm). Result: Δsky ≈ 1.7 (UHC), 2.5 (OIII), 2.7 (H-beta), 0.4 (CLS) mag. Duo-band and UV/IR-cut filters stay descriptive.
  • Emission objects — lines only: Hβ 486.1, [O III] 495.9 and 500.7, Hα 656.3 nm; 5007/4959 = 2.98 (Storey & Zeippen 2000, MNRAS 312, 813), Hα/Hβ = 2.86 when not measured (case B at 10⁴ K, Osterbrock & Ferland 2006). Planetary nebulae use their measured I(5007)/I(Hβ) (Acker et al. 1992, VizieR V/84), or 10.6 when absent (app value: the median of the 144 planetaries of the catalogue with a measured ratio, saturated 5007 lines scaled by 2.98).
  • Class fallback — an app heuristic, not a published formula: [O III]/Hβ = 1 for an H II region (0.3 to 3 with the exciting star, Osterbrock & Ferland 2006) and 4 for a supernova remnant (radiative-shock filaments of the Veil, 3 to 5, Fesen, Blair & Kirshner 1982, ApJ 262, 171).
  • Continuum objects — galaxies, clusters, stars and reflection nebulae: Δobj = Δsky, no gain. Dark nebulae, nebulae of unknown subclass and unknown types get no filter figure at all.
  • In the verdict — the filter in the optical path of the active visual setup enters the detection verdict of the object panel, the eyepiece table, the night programme and the Observer profile alike: the sky becomes μ + Δsky, the object m + Δobj, and every other step above runs unchanged on those two numbers. The necessary condition uses the filtered sky and a star dimmed by Δsky, the same continuum as the sky. Descriptive filter types change nothing.
  • Below 10⁻⁵ cd/m² — there Crumey treats the background as effectively zero (eqs. 45-52; for stars, eqs. 70-71, “threshold assumed constant”): the absolute threshold ΔB stays at its cut-off value, so the contrast threshold rises as 1/B. A filter therefore only gains on the part of the background above the cut-off: the threshold is multiplied by max(1, min(10⁻⁵, B) / Bf), B and B_f the backgrounds at the eye without and with the filter. Under a background already below the cut-off, the filter only dims the object. Holding the contrast instead would have counted as a gain the very background that no longer hindered the eye — under the darkest skies and at high power, the case of a small bright planetary nebula.
  • Recommended filter (object panel, “Your filters”) — for an emission nebula, the verdict of the recommended eyepiece is computed for each computed filter in the case (UHC, OIII, H-beta, CLS) and without a filter; the recommended one has the best score (the margin above the threshold, in magnitudes), except that any candidate within 0.2 mag of the best yields to a broader one (no filter, then CLS, UHC, OIII, H-beta, by main passband width), since it dims the field stars less. The 0.2 mag tie is an app choice, about the uncertainty of the line ratios (±20 %), not a published value. Field stars are dimmed by Δsky (a continuum). With no computed filter in the case, the same choice among UHC, OIII and H-beta names the type that would help. Each filter's line also shows its gain over no filter, the difference of the two scores in magnitudes; within the same 0.2 mag it reads “barely any effect”, since such a gap is inside the line-ratio uncertainty.
• Nebula outlines — OpenNGC outlines (Mattia Verga, CC BY-SA 4.0; outlines/outlines_catgen.dat, commit 08288992), hand-traced in Aladin (CDS) on the DSS2 colour survey at three histogram settings: level 1 the faint extent, level 2 standard, level 3 the bright core; simplified to 0.002°. Each outline is attached to its catalogue object by designation, and rejected if its centre lies more than 3° away; region outlines (Orion, Cygnus, Vela…) are not used. Drawn for nebulae, nebulous clusters, supernova remnants and planetary nebulae once the object spans 8 px on screen, at opacities 0.45, 0.75 and 1 by level (app choices); open clusters keep their circle and add their nebulosity outline (the Pleiades). The same projected outline is used for clicking.

5. Deep-sky magnitudes

• Catalogue — OpenNGC (Mattia Verga, CC BY-SA 4.0): V magnitude, otherwise B. A galaxy V fainter than its B by more than 0.3 mag is discarded as implausible (no galaxy has such colours). Common name: OpenNGC's first one, except on an object classed as a star (NGC 1990 “Alnilam”: the star layer names it) and where it belongs to another object: IC 434 “Flame Nebula” (NGC 2024), IC 4703 “Eagle Nebula”, which lies within M 16's 120′ field (app rule, build-dso.js).
• Globular clusters — integrated V and (B−V) of Harris (1996, AJ 112, 1487; 2010 edition, VizieR VII/202), B = V + (B−V), in preference to OpenNGC for every cluster it lists.
• Nebulae of our Galaxy (|b| < 30°, so not the H II regions of M 33, M 101 or the Magellanic Clouds) whose OpenNGC magnitude comes from SIMBAD (OpenNGC source code 2): SIMBAD has no integrated photometry of the nebulosity, so that magnitude is the one of its main object, the illuminating star or the embedded cluster (NGC 6888 at B = 7.44 is the Wolf-Rayet star HD 192163, IC 4592 at 3.9 is ν Sco). With a cluster identifier (MWSC), the magnitude stays and the object is judged as that cluster, which resolves into stars (IC 5146, IC 1805, M 16). Otherwise the nebula magnitude of W. Steinicke's Revised NGC/IC catalogue (SEDS “Revised NGC Data” pages) replaces it when it is fainter (NGC 6888: b = 10.0); with none, the object has no magnitude and no brightness verdict, the star's going to the central-star field. When a nebula's magnitude is Steinicke's and his extent is larger than OpenNGC's (often the bright part measured by Lynds 1965), his extent is used: NGC 7000, b = 4.0 over 120′ × 100′, not the 120′ × 30′ of LBN 373. Both rules only ever make an object fainter or larger. App rule (build-nebula-mags.js, checked by check-nebula-mags.mjs).
• Missing magnitudes — galaxies: total B of the RC3 (de Vaucouleurs et al. 1991, VizieR VII/155); compact groups: Hickson (1982, VizieR VII/213), red magnitude, shown but never used to judge.
• Galaxies in visible light — when only B is known, V = B − (B−V)T with the RC3 total colour; without it, the RC3 median colour of the same Hubble stage (E 0.99 … Sb 0.77 … Sc 0.64 … Im 0.46; 0.82 without type) — correct on average, not per galaxy.
• Checked by hand — M 20 V = 6.3 (SEDS; the nebula cannot be fainter than its own central stars); NGC 246, NGC 1933 and IC 142 lose an implausible B; NGC 2070 keeps V = 7.25.
• Kind of nebula — emission, reflection or dark, from the OpenNGC class (HII and EmN, RfN, DrkN). Where OpenNGC only says nebula or cluster with nebula, from the catalogue of its other names: Sharpless (1959) emission, van den Bergh (1966) reflection, Lynds (1962) and Barnard (1927) dark, a bright-nebula catalogue winning over a dark one; LBN (Lynds 1965) alone, or Sharpless with van den Bergh, leaves the kind unknown (app rule).
• Planetary nebula lines — [O III] 5007 Å and Hα intensities relative to Hβ from the Strasbourg-ESO catalogue (Acker et al. 1992, VizieR V/84), nearest nebula within 1.5′, for 144 of the 205 planetaries; a saturated 5007 measured on 4959 is multiplied by 2.98 (Storey & Zeippen 2000, MNRAS 312, 813), and spectra referred to Hα alone are not used. Long-slit spectra, often of the central region: for a large nebula (Helix, M 27) the ratio describes the part measured, not the whole object.

6. Stars and planets

• Stars — ATHYG database (HYG + Tycho-2 + Gaia DR3, astronexus); doubles from the WDS (Mason et al.), variables from the GCVS (Samus et al.).
• Effective temperature — from the spectral type, interpolated within each class; otherwise from B−V by Ballesteros (2012, EPL 97, 34008), T = 4600 [1/(0.92 (B−V) + 1.7) + 1/(0.92 (B−V) + 0.62)]. Radius R/R☉ = √L × (5772 K / T)².
• Deep-sky distances — measured whenever a catalogue gives one, with the source shown next to the value. Every source is matched by name, then checked by position.
  • Galaxies: first Cosmicflows-4 (Tully et al. 2023, ApJ 944, 94; VizieR J/ApJ/944/94), distance modulus of all methods, d = 10(DM − 25)/5 Mpc, kept when its uncertainty is at most 0.5 mag; otherwise the Updated Nearby Galaxy Catalog (Karachentsev, Makarov & Kaisina 2013, AJ 145, 101; VizieR J/AJ/145/101), any method except its Hubble-flow estimates. Position within 2′ or one eighth of the major axis. Failing both, Hubble's law cz / H₀ with H₀ = 70 km/s/Mpc, only beyond 1,000 km/s, where the expansion dominates the galaxy's own motion; below that, no distance is shown.
  • Globular clusters: distance from the Sun of Harris (1996, AJ 112, 1487; 2010 edition, VizieR VII/202).
  • Open clusters, and the nebulae that bear their cluster's number: median maximum-likelihood distance from the Gaia DR3 parallaxes of their members (Hunt & Reffert 2023, A&A 673, A114; VizieR J/A+A/673/A114), matched on any of the cluster's names, position within 5′ or 40 % of the diameter (the catalogue gives the densest point, not the catalogued centre). The Lagoon (M 8) takes the distance of its cluster NGC 6530.
  • Planetary nebulae: distance combining the Gaia EDR3 parallax of the central star and the Hα surface brightness–radius relation of Frew, Parker & Bojičić (2016, MNRAS 455, 1459), as given by Chornay & Walton (2021, A&A 656, A110; VizieR J/A+A/656/A110).
  • Published one by one: M 42 and M 43, 414 pc (VLBI parallax, Menten et al. 2007, A&A 474, 515); M 17, 1.98 kpc (maser parallax, Xu et al. 2011, ApJ 733, 25; the Gaia distance of its cluster NGC 6618, 606 pc, is rejected as foreground); M 1, 2.0 kpc (Trimble 1973, PASP 85, 579).
  Other diffuse nebulae, supernova remnants and dark nebulae have no distance: no homogeneous catalogue covers them. Real size = distance × angular major axis.
• Planet magnitudes — astronomy-engine: Hilton (2005, AJ 129, 2902) for Mercury and Venus, the classical almanac coefficients for the other planets, and Saturn with its rings after P. Schlyter's formulas.
• Shading of the globes — bodies covered with regolith follow the Lommel–Seeliger law I ∝ μ₀ / (μ₀ + μ) (Seeliger 1887; Hapke, Theory of Reflectance and Emittance Spectroscopy, 1993, §8; McEwen 1991, LPSC XXII, 1135, for the Moon); Venus and the gas giants keep Lambert's law with limb darkening.
• Moon at the terminator — selenographic colongitude c = 90° − longitude of the subsolar point; morning terminator at −c, evening at 180° − c. Feature names from the IAU nomenclature (USGS Gazetteer of Planetary Nomenclature); a feature is “at the terminator” when the Sun is less than 10° above its horizon.
• Jupiter — Galilean moons from the IMCCE L1.2 theory (via astronomy-engine); transits, shadows, occultations and eclipses use the edge of the shadow cone, without penumbra (IMCCE convention), checked against JPL Horizons. Elongations and conjunctions use Meeus's X coordinate (Astronomical Algorithms, ch. 44: offset from the disc centre along Jupiter's projected equator): a conjunction is X = 0, inferior when the moon is on the Earth side; a greatest elongation is an extremum of X taken as an angle on the sky, since the Earth–Jupiter distance changes across the flat peak (one minute for Europa). Sampled every 20 min, refined by bisection to the second; checked against JPL Horizons (conjunctions within 6 s, elongations within 37 s). Moments when the moon is occulted or eclipsed are not listed. Central meridian in System III (IAU W, checked against Horizons) and System II, W = 43.3° + 870.270°/day since J2000 (Explanatory Supplement to the Astronomical Almanac); Great Red Spot longitude from JUPOS / Project Pluto with its measured drift.

7. Comets and asteroids

• Orbits — two-body propagation in universal variables, with the Stumpff functions (Meeus, Astronomical Algorithms, 2nd ed., ch. 33; Vallado), from the Minor Planet Center elements (comets) and the JPL Small-Body Database (asteroids), one light-time iteration. Elements are flagged as ageing beyond thresholds measured against JPL Horizons.
• Comet magnitude — m₁ = H + 5 log Δ + K log r, the MPC parameters, without phase correction; often wrong by several magnitudes, which the info panel states.
• Asteroid magnitude — the IAU H, G phase function (Bowell et al. 1989, in Asteroids II).
• Which comets are drawn — a comet is drawn when its total magnitude, dimmed by atmospheric extinction, clears the sky background of the moment and the deep-sky limit of the view, min(slider, 8.2 + 4 log₁₀(60° / field)) — the limit a galaxy of the same brightness would meet. An app heuristic for a diffuse object, not a published threshold; the selected comet is always drawn.

8. Eclipses and occultations

• Search and local circumstances — astronomy-engine (global and local solar eclipses, lunar eclipses), with the local horizon applied to contacts.
• Fraction of the Sun hidden — area of the circular lens between the two discs, with linear limb darkening I(μ) = 1 − u (1 − μ), u = 0.6, a classical order of magnitude in the visible (compared with the fifth-order tables of Neckel & Labs 1994, Solar Physics 153, 91). Corona: Baumbach's radial profile (1937).
• Lunar eclipses — umbra and penumbra radii by Danjon's sum of parallaxes, Rp = k·Pm + Ss + Ps, Ru = k·Pm − Ss + Ps (Meeus, ch. 54), with Chauvenet's traditional enlargement (1891), k = 1.02.
• Planet close approaches — geocentric J2000 directions of Mercury to Saturn (astronomy-engine, aberration included) sampled every 6 h; each local minimum of separation is refined by golden-section search. Fixed targets at their catalogue J2000 positions, without proper motion: every deep-sky object of magnitude ≤ 7 within 7° of the ecliptic (23 objects, NGC 6530 excepted as part of M 8), and Aldebaran, Regulus, Spica, Antares. The Moon is tested against the brightest only (M 45, M 44, M 35, M 8, M 22, M 23, M 24, M 25), sampled hourly, topocentric when the site is known (its parallax reaches 1°). Checked against the Jupiter–Saturn conjunction of 21 December 2020 (6.1′). App thresholds, not a published rule: within 3° for two planets, 1.5° for a fixed target, and no planet within 15° of the Sun. Seen from a site, an approach is dated at the closest visible moment (planet above the local horizon, Sun below −6°, 10-min steps, up to 12 h away), before and after the minimum when the minimum itself falls in daylight, with the separation at that moment, kept only while it stays under the threshold.
• Lunar occultations — geocentric search hour by hour, then topocentric apparent positions every 2 minutes and bisection to 0.1 s; lunar radius k = 0.2725076 Earth equatorial radii (IAU value used by IOTA); dark or bright limb from the position angle of the brightest limb point (Meeus, eq. 48.5). Checked against IOTA timings (Aldebaran, 2017) and JPL Horizons.
• Is the star seen at the limb? — a star is listed only if it is brighter than the faintest star visible at one of its contacts, from the model of Schaefer, Bulder & Bourgeois (1992, Icarus 100, 60), adopted by IOTA, ported line by line from Schaefer's OCCVIS.BAS (Sky & Telescope, January 1993, p. 89). Inputs: Moon's sunlit fraction, its zenith distance, Sun altitude, cusp angle of the contact (positive on the dark limb, where scattered moonlight and earthshine set the background; negative on the bright limb, where the lunar surface does), aperture and magnification of the instrument set, and the site's extinction coefficient. Program defaults kept for the rest: B−V 0.7, seeing 1″, Moon on 20 % of the field, throughput 80 %, observer aged 30 with 20/20 acuity, average experience. Checked against the published LCROSS case (71 % sunlit, 254 mm at ×300, bright limb: 5.263). Planets are not filtered.

9. Satellites

• Positions — SGP4 (satellite.js; Vallado et al. 2006, “Revisiting Spacetrack Report #3”), from CelesTrak elements. For Starlink, SGP4 at intervals and a two-body extrapolation in between (WGS-72 gravitational constant), checked against SGP4.
• Launch — no position, pass or transit before a satellite's launch. ISS from the launch of its first module, Zarya (1998-11-20 06:40 UT); Tiangong from Tianhe (2021-04-29 03:23 UT); Starlink from the launch date in its type table; any other satellite from 1 January of the year in its international designator (the only date the elements carry). App rule: within that launch year, a satellite can still be shown before its actual launch.
• Sunlit — conical Earth shadow with its penumbra (Montenbruck & Gill, Satellite Orbits, 2000, §3.4.2): the fraction of the solar disc visible from the satellite, Sun and Earth as discs of their apparent radii. Sunlit when more than half the disc shows, which falls within 0.1 s of the former cylindrical shadow's edge; the ISS crosses the penumbra in about 10 s. Earth's atmosphere is ignored: in reality the station reddens and dims a few seconds before the geometric shadow.
• Magnitude — a rough estimate from a standard magnitude (at 1000 km and 90° phase: ISS −1.3, Tiangong −0.8, Hubble 2.5) scaled by range and a diffuse (Lambert) sphere phase function.
• Starlink magnitude — a magnitude at 1000 km per satellite type, scaled by range (+5 log₁₀(d / 1000 km)), with no phase function: the published values are averages over all observed geometries (A. Mallama et al.). First generation, the characteristic magnitudes of arXiv:2210.17268 (4.7, 6.2, 5.5 at ~550 km, i.e. 6.0, 7.5, 6.8 at 1000 km); V2 Mini in its dimmed attitude, 7.87 (arXiv:2306.06657); Direct-to-Cell, 6.47 (arXiv:2502.03651); below 357 km, while raising its orbit undimmed, 4.58 (arXiv:2405.12007). The type of each satellite comes from J. McDowell's GCAT. App rule: V2 Mini Optimized, with no published photometry, is taken as V2 Mini; a flat, oriented satellite departs from these averages by several magnitudes.
• Space station passes (ISS, NORAD 25544; Tiangong, its Tianhe core module, NORAD 48274) — SGP4 sampled every 30 s, a pass being any stretch above 10°; its peak refined by golden-section search to 0.1 s. Its instants are then refined by bisection to 0.1 s, sampling every 10 s from the peak: rise and set on the local horizon (profile or N/E/S/W heights, with the dip), entry into and exit from the shadow (half the solar disc), the Sun crossing −6°; the station is seen while above the horizon, sunlit and under a sky darker than that, and its appearance and disappearance are the first and last bounds of that stretch, each with its cause. App rule: visible when the Sun is below −6° at the peak and the station is sunlit at some point of the pass; magnitude at the brightest sunlit instant.
• ISS trajectory — while it covers the date, the ISS follows the ephemeris published by NASA (JSC Flight Operations Directorate, TOPO office; CCSDS OEM, EME2000, 4-min steps over 15 days, planned manoeuvres and dockings included), relayed by the site since NASA's server does not allow a web page to read it. Cubic Hermite interpolation on positions and velocities (under 100 m between samples); EME2000 to Earth-fixed by the engine's precession-nutation (true equator and equinox of date) and apparent sidereal time, both refreshed hourly. Checked against SGP4 at the epoch of fresh elements: 1.1 km off the track. Measured with scripts/measure-iss-tle.mjs (2026-10-13): within 4.3 s of fresh elements 5 to 7 days after its publication. The planned events listed under the passes are read from its header.
• Pass time quality — outside that trajectory, from the gap between the date and the epoch of the elements, measured with the same script on 60 days of Space-Track history (2026-08-14 to 10-12), elements against elements, worst gap on the time of a pass without / with a reboost in between: ISS 12 s / 2 min at 2–3 days, 71 s / 4 min 46 s at 5–7 days, 2 min 26 s / 6 min 54 s at 7–10 days, 4 min 45 s / 9 min 30 s at 10–14 days; Tiangong 5 s / 37 s, 26 s / 89 s, 60 s / 2 min 2 s, 2 min 9 s / 2 min 49 s. Up to 3 days the minute holds, and the Stations screen and the trajectory labels show seconds (as they do on the NASA trajectory); past that they show the minute only, never a second the data cannot back; up to 14 days the time is marked “≈” with a margin, the worst gap without reboost rounded up to the minute (ISS ± 2, ± 3, ± 5 min up to 7, 10, 14 days; Tiangong ± 1, ± 2, ± 3 min); beyond, no pass is predicted (the Stations screen still lists them, with the age of the elements). App rule: an unplanned reboost can exceed the margin.
• Transits of the Sun and Moon — for every pass of a station, daytime included, sampled every 30 s from the horizon up, the sample nearest the centre of the Sun or Moon is refined by golden-section search to 20 ms. Both directions are geometric (refraction moves the station and the body, on one line of sight, alike); the body is apparent and topocentric (astronomy-engine, the Moon’s parallax included), the station without annual aberration since it moves with the Earth. Apparent radius: IAU 2015 nominal solar radius, 695 700 km; Moon, 1 737.4 km at its topocentric distance. Transit when the minimum separation is below the radius; duration = chord 2√(r² − m²) ÷ the station’s angular speed relative to the body.
• Ground track of a transit — the signed miss distance (which side of the body’s centre the station passes) is recomputed for an observer moved 1 km north and 1 km east; it varies linearly with the place over a few tens of km, which gives the distance to the centre line (|miss| ÷ gradient), its direction (against the gradient) and the track width (2 r ÷ gradient) — longer near the horizon, where the track stretches as 1/sin h, which separation × station distance alone would miss. Moving the observer by the computed amount lands within 3 % of the centre line (0.7 km at 23 km, 2 km at 52 km). Transits are kept within 25 km of the centre line (5, 10, 25 or 50 km in the Ephemerides ⋯ menu). Checked against Orbital Radar’s published predictions of 2026-10-17 with Celestrak elements of the same morning (scripts/check-transits.mjs): Seattle, ISS across the Sun, 18:41:57.2 UTC against 18:41:58.0 (18:41:58.0 with the NASA trajectory), centre line 23.3 km NE against 24, track 9.3 against 10.0 km, 1.25 against 1.30 s; Seattle, Tiangong across the Moon, 06:33:54.1 against 54.9, centre line 20.7 km against 20.
• Transit time quality — a transit lasts 0.3 to 2 s and the station covers 7.7 km/s, so one second of error moves the track 7 km along it. Measured with scripts/measure-iss-tle.mjs (run of 2026-10-17: 206 Space-Track element sets from 2026-08-18 to 10-16, NASA trajectories of 13 and 14 October), worst gap on the time of an ISS pass: elements 0.5–1 day old 1.3 s (90 % under 0.2 s), 1–1.5 days 1.7 s, 2–3 days 12.2 s; across a reboost, 27 s from half a day; NASA trajectory against fresh elements, by the trajectory’s age, 0.8 s at 1–1.5 days, 1.2 s at 1.5–2 days, 1.8 s at 2–3 days. App rule: the time to the second, the distance to the centre line and the “Here” pill only from elements less than 1 day old, or from a NASA trajectory published less than 2 days before (worst measured gap under 1.5 s); for transits, a trajectory older than that gives way to elements less than a day old. Otherwise the time is preceded by “≈” and “to confirm the day before”, still given to the second (rounded) so that it reads against the simulation clock: a transit lasts about a second, and a time rounded to the minute could name the minute after the event. Beyond 14 days from the elements, where no pass time is predicted any more, a transit found up to 30 days out is kept as “Possible”, its time (to the second) and its distance to the centre line both preceded by “≈” — the figures today’s elements give, which an online prediction tool also shows; the time can be minutes off and the track hundreds of km. Checked on two such cases announced for Noisy-le-Grand at 15 and 19 days: same body position and chord, times 51 s and 32 s apart, well within the minutes of drift measured at that range.

10. Meteors and aurorae

• Shower activity — the IMO activity profile ZHR = ZHRmax · 10−b |Δλ☉| (International Meteor Organization).
• Rate you can expect — HR = ZHR · sin h · r−(6.58 − lm), with h the radiant altitude, r the population index and lm the true naked-eye limiting magnitude towards the radiant (section 2).
• Aurorae — the planetary Kp forecast of NOAA SWPC; geomagnetic latitude from the dipole pole of the World Magnetic Model 2025 (80.79° N, 72.76° W, NOAA NCEI); the Kp needed at a geomagnetic latitude from NOAA SWPC, “Tips on Viewing the Aurora”.

11. Planning a night

• Twilight — civil, nautical and astronomical at −6°, −12° and −18° (USNO, “Rise, Set, and Twilight Definitions”). An object needs the Sun below −6° when brighter than magnitude 1, below −12° when brighter than 3, and below −18° otherwise; the Moon needs nothing. The dark window falls back to −12° then −6° on summer nights that never get darker.
• Observing order — earliest deadline first (each target before it sets; Jackson's rule, 1955), then shortened by local moves that never lose a target and reduce the telescope's path.
• Companions in the night list — a deep-sky object lying within the catalogued extent of another at least three times larger is listed under it (M 32 and M 110 under M 31), provided the host spans 5° at most, about a binocular field; larger regions (Sh2-109, 18° of Cygnus) host nothing (app choices).
• Clouds over the night — the Open-Meteo hourly forecast over the dark window (or the Sun below −6° on nights without one), each hour judged on its most covered layer (low, middle, high): below 25 % every hour, “Clear”; above 75 % every hour, “Overcast”; cloudy hours in several spells making up a third of the night at most, “Passing clouds (n h of m)”; otherwise the hour clouds arrive, and the hour they leave if they do. The night chip and the Conditions banner use this one rule; rain, snow or a storm in any hour comes first. The night's heading reads “An overcast night” when overcast, “A cloudy night” when half the hours or more are cloudy, before what the sky would offer (app reading thresholds, no published standard).
• Dew — likely from the first hour of the night when the forecast air temperature is within 2 °C of the dew point: under a clear sky, a surface facing the sky cools 1 to 2 °C below the air (British Astronomical Association, “Dealing with dew”, 2021). The site’s sky section uses this whole-night answer, not a single instant.
• Chained night for a smart telescope — a timetable rather than a list, built greedily from the photo ranking's best twelve targets that have a common name or a Messier or Caldwell number (application heuristic, until the photographic verdict ranks objects in 3.7: the frame fill alone opened a Seestar S50's night in Paris on Sh2-132 and the dark nebula B 151; the others stay in the add list): at each step, among the targets that stay usable for their whole block from that moment, the one whose usable stretch ends first (Jackson's rule, as above), the ranking breaking ties; when none fits, the start moves on by 5 minutes. A target is usable above the night list's minimum altitude (20°) and the site's local horizon, and below the device's zenith limit: its published value, otherwise 80° (application heuristic: alt-az mounts cannot follow near the zenith, where the azimuth rate diverges). A block that would run into the zenith hole stops at its edge if at least half its length is left, otherwise the target waits until it is out or is left aside. Each block lasts a duration chosen by the user (30 minutes by default) until the photographic verdict computes it (3.7), and a target may be given its own under “Edit the programme”, which applies to it alone; a mosaic chosen there is the Suggested mosaic grid below, its time shared among the panels unless the user multiplies it (application choice); 5 minutes separate two blocks for slewing, plate solving and focusing (application value, not a measurement). Altitudes are checked minute by minute, local sidereal time interpolated at 360.9856° per day. Targets the user moves to the top keep their order and go first, each as early as it fits, provided its block can start within one block length of its turn (application value); one that cannot rejoins the other targets instead of holding the rest of the night until it rises again. The greedy rule resumes after them.
• Quality of a night (Night quality screen) — every 10 minutes from noon to noon, the sky background of a natural site — airglow, twilight and moonlight, without the site's light pollution — (section 2, the same functions as the limiting magnitude), the Moon's phase angle taken at each step; w = Bnat / B in each direction, the share of contrast kept against a moonless natural night sky, averaged over the sky above 30° altitude (36 directions on four 15° rings, each weighted by its solid angle). App rules: the zenith alone, the convention of SQM readings, missed that the Moon brightens mostly its own half of the sky; below 30°, extinction and the light dome leave a sky where faint objects are not sought; and judged against the site's own sky, a city background (Bortle 7, 18.25 mag/arcsec²) drowned an 82 % Moon and nautical twilight, so the Moon and twilight now count the same everywhere, the site's light pollution being shown apart (the Site sky factor). App thresholds, no published standard: a step is dark when w ≥ 0.75 (at most 0.31 mag/arcsec² above that reference, the order of an SQM reading's uncertainty and of the natural airglow's variation, 21.6–22.0, Benn & Ellison 1998) and grey when w ≥ 0.25 (1.5 mag, roughly rural to suburban on the Bortle scale). The verdict compares the dark hours with those of the same night without Moon or clouds: 75 % or more Excellent, 45 % Good, 20 % Fair, below Poor; “No dark night” when the season allows less than 30 minutes. Where the Open-Meteo forecast covers the night, a dark step under 30 % cloud or more no longer counts — cloud being the larger of the low and middle layers and half the high layer — and a night that clouds bring below 20 % is “Clouded out”. The forecast runs seven days; beyond three days (app choice) its clouds still count but are shown faded as a “distant forecast”, and the alerts ignore them. The strip of nights and the calendar fill each night's cell with the colour of the same verdict and its dark hours rounded to the hour; a night with no forecast at all keeps its colour with a dashed inner line, its verdict counting the Moon only.
• Night factors (summary of The night) — the dark hours taken by the Moon are the 10-minute samples dark without the Moon but not with it; those taken by clouds, the samples dark without Moon or clouds, not taken by the Moon, and covered by more than 30 %. Together they equal the possible dark night minus the dark night counted. A loss under 15 min is not shown. A Moon above the horizon that takes nothing is reported as without effect (low, or a thin crescent). Moon rise and set are where its upper limb crosses −0.833°, interpolated between samples. App rule (nightphrase.js#nightFactors, checked by check-night-quality.mjs).
• Measuring with the hand — a fist at arm's length covers about 10° (RASC Observer's Handbook; NASA/JPL), shown only next to degrees.

12. Astrophotography

• Focal length and f-number of a camera setup — a camera lens gives its own; at a telescope's focus, the optic's focal length F times the factor k of the reducer or Barlow fitted in the setup (k F, f-number k F / D). Field, sampling, frame and NPF exposure all use this focal length. Up to version 3.5 the factor was ignored, and a reducer changed nothing.
• Longest exposure without trails — NPF rule of Frédéric Michaud (Société astronomique du Havre), t = (35 N + 30 p) / f, with N the f-number, p the pixel size in µm and f the focal length in mm.
• Drift without tracking — 15.041″ per second (360° per sidereal day of 86,164 s), times cos δ.
• Field rotation on an alt-az mount — dq/dt = 15.041″/s × cos φ × cos A / cos h, derived from the parallactic angle (Meeus, ch. 14).
• Camera frame drawn on the sky — a rectangle of 2·atan(w / 2f) by 2·atan(h / 2f), its corners rotated in the tangent plane before projection by θ = s − q (towards the screen's left as θ grows). s is the sensor angle set on the camera setup (0° landscape, 90° portrait, plus a fine adjustment; −90° to +180° stored), which on an equatorial mount, keeping the sensor fixed to the sky, is the frame's position angle; q is the angle by which celestial north is turned to the right of the screen's up at the frame centre, measured in the drawing's own frame (the parallactic angle above on a view levelled on the zenith), subtracted only on an equatorial mount and 0 on any other mount or none. The frame's position angle thus stays at s as time runs. Up to version 3.5 the angle was s + q, which turned the frame the wrong way (position angle s + 2q, right on the meridian only). Application convention, not a published formula.
• Sensor view — with an imaging setup in use, the Telescope view shows the sensor's rectangle upright, the image's top at the screen's top, and rolls the sky instead: the camera takes the roll θ above (in the convention of the projection, where a positive roll turns the screen's up towards the left, like θ), with q measured at the view centre in the levelled view frame, so that the screen's up has position angle θ + q. A sensor turned between 45° and 135° (or a device delivering portrait images) is shown as a standing rectangle, its sides swapped, the remaining angle going to the roll. The view field fits the rectangle with 15 % margin in the free band of the screen; zoom runs from the rectangle alone to five fields across (the largest mosaic, 5 × 5 without overlap) plus the same margin. The rectangle on screen is the projection of the sensor's tangent-plane edges, the same camera as the drawing and the selection. Application convention, not a published formula.
• Position angle of a frame — the angle of the image's “up” side (its height), counted from celestial north through east. On a view levelled on the zenith, north lies q (parallactic angle) to the right of the screen's up and east 90° left of north, so a frame of position angle PA is drawn at θ = PA − q: a frame of PA 90° has its top towards the east.
• Mosaic panels — panel centres placed in the plane tangent to the mosaic centre, gnomonic (TAN) projection (Calabretta & Greisen 2002, A&A 395, 1077), at steps of (1 − r) · 2 tan(W/2) and (1 − r) · 2 tan(H/2) for an overlap r, rotated by the PA, then projected back to α, δ. Row 1 is on the image's top side, column 1 on its left, as in Telescopius's mosaic CSV. Each panel gets its own PA, the direction of the mosaic's “up” in the tangent plane measured against the panel's own north (meridians converge, ≈ Δα · sin δ near the equator, computed exactly here). Checked against Telescopius's published 1 × 2 example to its last written digit.
• Suggested mosaic grid — the smallest grid of panels containing the catalogue ellipse (major × minor axis at its position angle) plus 10 % of its size on each side, with the 20 % overlap, trying the image's up along the major axis and across it; fewest panels first, then least area; none beyond 5 × 5. Without a catalogue angle, an elongated object is covered as a circle of its major axis. Application heuristic, no published rule.
• Suggested mosaic grid at a fixed angle (smart telescope, framing strip and night programme) — such a device cannot turn its sensor, so the grid is counted at the device's own position angle instead of a chosen one: the angle set by turning it on a wedge, otherwise its sensor angle plus the parallactic angle (Meeus, ch. 14) at the start of the target's block (framing strip: at the time shown). The ellipse's extent along the image's top and across it, its major axis at φ = PA − angle from the top, is √((a cos φ)² + (b sin φ)²) and √((a sin φ)² + (b cos φ)²), with the same 10 % margin, 20 % overlap and 5 × 5 ceiling. Application heuristic.
• Frame fill (object panel, “On screen” tab with a photo setup; night programme in photo mode) — the catalogue major axis over the short side of the field, as a percentage. Application heuristic: it ignores the orientation, which the suggested mosaic above takes into account.
• Verdict card with a photo setup (object panel) — no contrast threshold is applied: the eye's detection threshold (Crumey 2014) does not describe a sensor that integrates for minutes. The card only checks that the object is in its slot (above 20° and the local horizon, in astronomical night) and reads “Visible now, on screen”. Application rule until the photographic verdict (3.7).
• Field of a smart telescope (Seestar, Dwarf, Vespera, Unistellar, Origin) — when the manufacturer publishes both sides of the field, the published field is used as is, and the rectangle above takes it in place of 2·atan(w / 2f): Unistellar crops the delivered image, so the full sensor would overstate it. Otherwise the field is computed from the sensor (pixel count × pixel size) and the focal length. A field published only as a diagonal checks the entry and never replaces the calculation. The image scale is always computed, 206265 × p / (1000 f) ″/px. Each catalogue entry cites its maker's sheet, and a published field more than 5 % away from the sensor's carries a written reason (crop, unpublished resolution). The frame takes s plus 90° when the device delivers its image in portrait, and subtracts q only when the device sits on an equatorial mount: alt-az mounts derotate while stacking, so the frame shows the orientation at the start of the session. Application convention, not a published formula.
• Target list for a smart telescope — no maker's application imports a target file, so the list is made to be typed in: each target under a Messier, NGC, IC, Caldwell or Sharpless (Sh2) designation, in that order of preference, never under a proper name alone; an object in none of them keeps its catalogue number (Barnard, Abell…). Whether every application finds Caldwell and Sharpless numbers is not yet checked. J2000 coordinates in clear, to the second of time and the minute of arc, rounded before splitting (59.6 s gives the next minute). The Telescopius CSV is written as published (header Pane, RA, DEC, Position Angle (East), Pane width (arcmins), Pane height (arcmins), Overlap, Row, Column; α to the second as 05hr 05' 30", δ to the arcsecond with the ordinal sign as -07º 03' 20"), one file and one Center line per target, north up (PA 0) — a mosaic planned in the framing strip (Lab) adds one Pane n line per panel after it, with its row, column and own PA, and its overlap as a whole percentage —, the device's field as the pane, its short side across for a portrait sensor. Not settled: the encoding the importers expect (UTF-8 without BOM here), the sign of a positive declination (none written, no published example) and the angle convention of third-party software.

13. Sky survey

• Photographic survey tiles — HEALPix (Górski et al. 2005, ApJ 622, 759), DSS2 colourised by the CDS (HiPS).
• Built-in deep-sky cutouts — in the Built-in background, 768 objects are drawn in the sky as a DSS2 cutout — the 134 that have a photo, every Herschel 400 object, and every object with a major axis of at least 5′ and a magnitude of 11 or brighter (scripts/build-dss-extracts.py, 36 MB in all): CDS hips2fits, survey CDS/P/DSS2/color, TAN projection, centred on the catalogue position, north up and east left, square field F = max(3 × major axis, 8′) (8′ when the catalogue has no major axis), side F / 1.7″ bounded to 384–1024 px. The plate background is removed per colour channel, its 35th percentile set to black (app choice), and a cutout whose blue plate is missing (99th percentile of blue below 0.3 × that of red: 12 cutouts towards the Galactic centre, such as M 25, M 26, M 62 and NGC 6231, all others being above 0.44) is made grey, the mean of its three channels, and the cutout is blended in screen mode like the online survey tiles. Its geometry is therefore exact by construction, with no measurement. The info panel keeps the original photo as an illustration; an object without one shows its cutout. Its calibration (scale, rotation and centre offset, 80 photos registered on the DSS2 by scripts/measure-dso-images.py on 2026-10-24, the others from the light moments, an app heuristic) is kept in the manifest and applies again only to an entry without a cutout.
• Survey contrast — the colourised DSS2 plates are dark (median luminance 0.035 to 0.08 on two Sagittarius tiles). On screen each pixel's luminance L (Rec. 709 weights) becomes ((L − 0.025) / 0.975)0.8, its colour scaled by the same factor so its hue is kept, then saturation × 1.15; a channel above 1 brings the other two down with it. WebGL path only: the Canvas 2D fallback shows the raw plates (application rule, set by eye).
• Catalogue stars over the online survey — the survey already shows the field's faint stars, so where a tile is painted the catalogue stars give way to it: a star more than 5 magnitudes brighter than the field's limit stays whole; at 2 magnitudes or less above it, it drops to 30 % opacity and two thirds of its radius, with a smooth step between. Never below that floor, so name, colour and selection remain; where a tile has not arrived, stars are drawn as usual (application rule, set by eye).

14. Alerts

• Same judge as the object panel — an alert's “visible from here” is the verdict the app already shows: the object panel's verdict for an object, the local circumstances for an eclipse (section 8), the pass verdict for a station (section 9), the visible-nights rule for a close approach. An alert cannot announce what the panel calls invisible.
• How far ahead each source is computed — eclipses 3 years; Moon phases, planet events, meteor showers and asteroids 1 year; comets 6 months; Moon–planet and planet–planet approaches and occultations 3 months; Jupiter's moons, twilight and an object's moments 1 month; clear nights and the sky summary 7 days; station passes and aurorae 3 days; Starlink trains 2 days. The schedule is rebuilt at least every 6 hours (application choices, matched to how long each prediction holds).
• Cloud condition — Open-Meteo total cloud cover at the hour of the event (its peak when it has one), read only within 3 days: further out, the forecast is too uncertain to decide anything, and the alert goes off. An alert sent while the app is closed checks that hour again on arrival; if the cover then exceeds the rule's limit, it arrives silently, marked “overcast forecast”, since a push must show something (application rules).
• Quiet hours — 23:00 to 07:00 by default: an alert whose warning falls in them is dropped, unless the event itself falls in them too (an occultation at 02:10 still warns at 02:00). An eclipse visible from here is never held back (application rule).
• Six events a night — beyond the cap (6 by default), the events kept are ranked by source: eclipse, aurora, station pass, meteor peak, occultation and clear night, planet event, comet and approach, Moon–planet pairing, Starlink and an object's moment, Moon phase, Jupiter and asteroid, twilight, then the sky summary, earliest first on a tie. Two warnings for one event take one place; an eclipse visible from here and the sky summary are outside the cap (application heuristic).
• Grouping — warnings less than 10 minutes after the first of a group go off together, at its time; a warning given less than 5 minutes ahead is never grouped, since grouping would bring it forward. The same event recomputed a few seconds later (a station's elements refreshed) is recognised within 15 minutes and does not go off twice (application choices).
• Late alerts — a warning missed by more than 2 minutes at planning time is skipped, one missed by more than 5 minutes at delivery (device asleep, app closed) is not sent; a relayed message waits at most 15 minutes in the browser's push service (application choices).
• Calendar file — iCalendar (RFC 5545): times in UTC, an instant event lasts 15 minutes, one reminder per warning. Station passes stop at 3 days, beyond which their orbital elements are no longer good to the minute; aurorae, clear nights and the sky summary, which depend on short forecasts, are left out. The link's position is rounded to 0.01° (about 1 km).
• App closed — each message is encrypted on the device for its own push subscription (RFC 8291, aes128gcm of RFC 8188), signed for the push service with a VAPID key (RFC 8292); the relay holds the next 7 days and never sees a location or a text.