Algorithms and accuracy

Every calculation in Sidralune is based on published methods. This page lists them by area, with the accuracy you can expect and the references. It is meant for users who want to know how far to trust a number, and for contributors.

Contents

• Time
• Coordinates and reference frames
• Sun, Moon and planets
• Physical ephemerides
• Rise, set, twilight and events
• Comets and asteroids
• Satellites
• Visibility and sky brightness
• Star detection and measurement
• Registration and plate solving
• Calibration and stacking
• Image processing
• Planetary imaging
• Photometry and time series
• Spectroscopy
• Radio astronomy
• Numerical methods
• How accuracy is tested

Time

Item	Method
Time scales	UTC, TAI, TT, UT1 and TDB. TDB uses the dominant periodic term (Fairhead & Bretagnon 1990), good to about 30 µs
Leap seconds	IERS table, 1972 to the present
ΔT (TT − UT1)	From leap seconds for 1972 onwards; Espenak & Meeus (2006) polynomials for other dates
Sidereal time	Earth rotation angle (IAU 2000) and IAU 2006 GMST; apparent sidereal time with the equation of the equinoxes
Julian dates	Meeus, Astronomical Algorithms, chapter 7

Coordinates and reference frames

Item	Method
Catalogue frame	ICRS / J2000, with proper motion applied to the date
Precession	IAU 2006, Fukushima-Williams angles with frame bias
Nutation	Leading terms of the IAU 1980/2000 series (about 0.01″ level)
Aberration	Annual aberration from the Earth's barycentric velocity
Light time	Iterated for solar-system bodies
Topocentric correction	Observer position on the WGS 84 ellipsoid
Refraction	Sæmundsson / Bennett formula scaled by temperature and pressure
Airmass	Pickering (2002), valid down to the horizon
Galactic and supergalactic coordinates	IAU rotation matrices

Sun, Moon and planets

Source	Method	Accuracy
JPL DE440s (when installed)	NAIF SPK reader, Chebyshev segment types 2 and 3, barycentric states	Milliarcseconds, 1849–2150
Analytic planets	Standish (JPL) Keplerian elements and rates, with Jupiter–Saturn perturbation terms for Jupiter	About 127″ RMS against JPL Horizons over 1990–2045 (the Saturn and Uranus terms made the agreement worse and are disabled)
Analytic Moon	ELP-2000/82 truncated series as in Meeus chapter 47	About 10″ in longitude, 4″ in latitude
Magnitudes	Mallama & Hilton (2018) planetary magnitude formulas, including Saturn's rings

Physical ephemerides

Item	Method
Planet rotation, orientation and radii	IAU WGCCRE (2015) pole positions, prime meridians and radii
Jupiter central meridians	Systems I, II and III
Saturn's rings	Ring-plane tilt B and position angle P (Meeus chapter 45)
Sun	P, B0, L0 and Carrington rotation (Meeus chapter 29)
Moon	Optical libration, position angle of the axis, selenographic colongitude (Meeus chapter 53)
Galilean moons (Tonight, Solar system, Radio)	Meeus chapter 44 (lower-accuracy method), with transit and occultation geometry
Planets' moons (sky chart)	65 moons of Mars, Jupiter, Saturn, Uranus, Neptune and Pluto: precessing ellipses fitted to JPL Horizons for 2000–2050, with tables of corrections along the orbit for moons that librate or share an orbit (Mimas, Tethys, Janus, Epimetheus, Helene, Telesto, Calypso, Atlas, Prometheus, Pandora); Hyperion and the irregular outer moons integrated from JPL states a year (Jupiter's, Hyperion) or four years apart, with the planet's and the Sun's gravity and, for Hyperion, Saturn's oblateness and Titan. Within 2000–2050 every moon is within 3″ of JPL Horizons (most within 1″); outside it they follow their mean ellipses, which keeps the regular moons close and the irregular ones rough

Rise, set, twilight and events

Item	Method
Rise, transit and set	Altitude sampling with root refinement, standard altitudes −0.833° (Sun), −0.567° (stars) and a parallax-dependent value for the Moon
Twilight	Sun at −6°, −12° and −18°
Moon phases, perigee and apogee	Root finding on elongation and distance
Conjunctions, oppositions, elongations	Root finding on the planets' ecliptic longitude difference from the Sun
Eclipses	Lunar: the Moon's distance from the Earth's shadow axis at each full Moon, with umbra and penumbra radii from Danjon's rule. Solar: geocentric Sun-Moon separation against the sum of their radii plus parallax at each new Moon
Meteor shower peaks	Solar longitude of the peak (IMO calendar values, on the J2000.0 ecliptic and equinox as the IMO gives them)

Comets and asteroids

Item	Method
Orbit formats	MPC packed designations and epochs, MPCORB, CometEls, JPL SBDB
Propagation	Two-body motion with universal variables (Stumpff functions), valid for elliptic, parabolic and hyperbolic orbits; light-time correction
N-body integration	Dormand-Prince 5(4) adaptive integrator with planetary perturbations, available to scripts
Asteroid magnitudes	IAU H, G system (Bowell et al. 1989)
Comet magnitudes	Total magnitude m = H + 5 log Δ + 2.5 K log r
Diameters	D = 1329 km / √albedo × 10^(−H/5)
Occultations	Close approaches of the asteroid's geocentric track to Gaia DR3 stars; shadow width from the estimated diameter; ground track and observer offset from the shadow's motion
MPC reports	MPC 1992 80-column format and ADES 2017 PSV

Accuracy is limited by the orbit: for well-observed numbered asteroids, arcseconds; for newly discovered comets and NEAs, possibly arcminutes or more. Refresh orbits regularly.

Satellites

SGP4 (Hoots & Roehrich 1980, as revised by Vallado et al. 2006, "Revisiting Spacetrack Report #3") with WGS-72 constants. Deep-space objects (period of 225 minutes or more) use the same secular model rather than full SDP4, which is adequate for visual pass predictions over a few days. Accuracy for low orbits is about 1 km at the TLE epoch, degrading by a few kilometres per day.

Visibility and sky brightness

Item	Method
Natural and artificial sky brightness	Site SQM value at the zenith, brightening with airmass (van Rhijn-like), dimmed by extinction
Moonlight	Krisciunas & Schaefer (1991)
Twilight	Empirical twilight brightness as a function of the Sun's altitude
Detection threshold	Schaefer (1990): threshold illuminance I = c₁ (1 + √(c₂ B))² for a background B, with Ricco/Piper spatial summation for extended objects
Eye pupil	7.8 mm at 20 years, decreasing about 0.45 mm per decade (after Watson & Yellott 2012)
Telescope	Light grasp, transmission, central obstruction, exit pupil, seeing and diffraction blur
Naked-eye limiting magnitude from SQM	NELM = 7.93 − 5 log₁₀(10^(4.316 − SQM/5) + 1)

The model predicts typical detectability, not certainty: individual eyes and nights vary by half a magnitude or more.

Star detection and measurement

Item	Method
Background	Mesh of robust (sigma-clipped) local medians with interpolation
Noise	Multiresolution support / difference noise estimators
Detection	Thresholding above the background noise with connected components
Centroid, FWHM, eccentricity	Adaptive (Gaussian-weighted) moments, as in Bernstein & Jarvis (2002); unbiased FWHM even for undersampled stars
Half-flux radius	Radius containing half the background-subtracted flux
PSF fitting	Gaussian and Moffat fits by Levenberg-Marquardt

Registration and plate solving

Item	Method
Star matching	Triangle-similarity invariants (Valdes et al. 1995), using each star's nearest neighbours (Beroiz et al. 2020)
Stars used	The brightest stars detected (at most 80). Saturated stars are left out while at least 20 others remain; in short exposures of bright fields, such as polar alignment frames, they are matched too
Robust fitting	RANSAC, then least squares on the inliers
Transforms	Translation, similarity, affine, homography, 2nd-order polynomial
World coordinates	TAN (gnomonic) projection with a CD matrix and SIP distortion polynomials (Shupe et al. 2005), iteratively refined
Blind solving	Field-size ladder from 0.25° to 60° and sky tiling, with mirrored parity tested
External solvers	ASTAP command line; nova.astrometry.net web API

Calibration and stacking

Item	Method
Masters	Row-band streamed combination to bound memory
Dark scaling and optimisation	Exposure-time scaling; golden-section search for the dark factor that minimises the calibrated frame's noise
Flats	Normalised to a mean of 1 per Bayer colour
Hot and cold pixels	From the master dark and by local-median deviation; replaced by same-colour neighbours
Debayering	Bilinear, super-pixel, Malvar-He-Cutler (2004) high-quality linear interpolation, and an edge-directed method
Frame score	Σ√SNR of stars ÷ (FWHM × (1 + eccentricity))
Rejection	Kappa-sigma, median kappa-sigma, Winsorized sigma (Huber 1981), linear fit clipping, generalised ESD (Rosner 1983), percentile clipping, auto-adaptive weighting (Stetson 1989), entropy-weighted average
Normalisation	Additive, additive with scaling, multiplicative, on robust location and scale
Weighting	Inverse noise variance, score, SNR
Drizzle	Variable-pixel linear reconstruction (Fruchter & Hook 2002) with a square drop kernel
Interpolation	Lanczos-3, bicubic, bilinear, nearest

Image processing

Item	Method
Wavelets	Starlet (à trous) transform with the B3 spline kernel (Starck, Murtagh & Fadili 2010)
Background extraction	Grid sampling with robust rejection; polynomial or thin-plate spline radial basis function surface
Auto stretch	Screen transfer function: shadows at median − 2.8 × normalised MAD, background mapped to 0.25 with the midtones transfer function
Generalised hyperbolic stretch	Payne & Cranfield (2021)
Arcsinh stretch	Lupton et al. (2004), colour preserving
Deconvolution	Richardson (1972) / Lucy (1974); with total-variation regularisation (Dey et al. 2006); Wiener; Van Cittert; Landweber; blind Richardson-Lucy (Fish et al. 1995)
PSFs	Empirical (stacked from stars), Gaussian, Moffat, Airy with central obstruction
Noise reduction	Wavelet soft thresholding at k × layer noise; non-local means (Buades et al. 2005)
Local contrast	CLAHE (Zuiderveld 1994)
Star removal	Star mask, harmonic (Laplace) inpainting and noise injection
Star reduction	Morphological erosion blended through a star mask
Colour calibration	Star-based white balance; photometric calibration against Gaia DR3 colours with a blackbody model
SCNR	Average-neutral, maximum-neutral and additive protection methods
PixelMath	Expression compiler to a stack machine, evaluated in parallel

Planetary imaging

Item	Method
Object detection	Otsu threshold, largest connected component, intensity-weighted centroid
Frame quality	Laplacian energy, gradient energy or local contrast, after a noise-robust blur
Global alignment	Phase correlation and centroid alignment
Local alignment	Multi-point alignment boxes with sub-pixel cross-correlation, multi-scale, per-point frame selection, smooth blending
Field rotation	Parallactic angle from site, time and target; or estimated from the images by polar cross-correlation
Limb fitting	Radial gradient edges, RANSAC and Kåsa algebraic circle fit
Sphere derotation	Re-projection on an oblate spheroid using the rotation period, sub-Earth latitude and pole angle from the ephemeris, with limb-darkening compensation

Photometry and time series

Item	Method
Aperture photometry	Circular apertures with partial-pixel weighting, sky annulus with sigma clipping and mode estimate, CCD noise equation (Merline & Howell 1995)
Ensemble differential photometry	Weighted mean of comparison-star zero points, optional colour transformation
Time corrections	HJD and BJD_TDB from the light-travel (Rømer) delay to the Sun and to the solar-system barycentre
Lomb-Scargle	Generalised (floating-mean, weighted) periodogram (Zechmeister & Kürster 2009), with false-alarm probability
Box least squares	Kovács, Zucker & Mazeh (2002)
Phase dispersion minimisation	Stellingwerf (1978)
Minimum timing	Kwee & van Woerden (1956)
Transit model	Quadratic limb darkening with Mandel & Agol (2002) geometry, numerically integrated
Fitting	Levenberg-Marquardt with parameter bounds and covariance errors
MCMC	Affine-invariant ensemble sampler (Goodman & Weare 2010; Foreman-Mackey et al. 2013)
AAVSO reports	AAVSO Extended File Format specification

Spectroscopy

Item	Method
Trace	Peak tracking in blocks along the dispersion axis, polynomial fit
Extraction	Box or optimal extraction (Horne 1986) with sky subtraction
Wavelength calibration	Lamp line identification by pattern matching from an approximate solution, polynomial dispersion fit
Continuum normalisation	Iterative asymmetric sigma-clipped polynomial fit
Lines	Centroid, FWHM, equivalent width and depth from a local continuum
Radial velocity	Doppler shift of lines; cross-correlation with a template; barycentric correction
Temperature	Planck fit and Wien's law
Instrument response	Smoothed ratio of observed to reference spectrum

Radio astronomy

Item	Method
Spectra	Windowed FFT (Blackman-Harris) averaging, DC spike suppression
Calibration	(ON − OFF) / OFF × Tsys; Y-factor method for Tsys
Velocity frames	LSR correction with the solar motion of Schönrich, Binney & Dehnen (2010)
Galactic rotation	Flat rotation curve, R₀ = 8.18 kpc (GRAVITY 2019), V₀ = 229 km/s (Eilers et al. 2019); kinematic distances and tangent points
HI column density	N = 1.823 × 10¹⁸ ∫ T_B dv (optically thin)
Jupiter	Io phase and System III CML source regions (Carr, Desch & Alexander 1983)
Pulsars	Epoch folding; incoherent dedispersion with the cold-plasma dispersion constant 4.148808 × 10³ s MHz² pc⁻¹ cm³
Radiometer equation	ΔT = Tsys / √(B τ)

Numerical methods

Fast Fourier transform (radix-2 and Bluestein), cubic splines, LU and QR linear algebra, polynomial fitting, Levenberg-Marquardt and Nelder-Mead optimisation, Brent root finding, golden-section search, and an ensemble MCMC sampler, all implemented in Sidralune.Core/Math. Image loops run in parallel on all processor cores with Parallel.For and unsafe pointer code.

How accuracy is tested

The test suite (Sidralune.Tests, about a thousand checks across every library, plus Sidralune.AppTests for the application itself) checks results against independent references, including:

• Julian dates, sidereal time, nutation, precession and Kepler's equation against Meeus' worked examples, and coordinate transforms, refraction, the Sun's orientation and the Moon's libration against Meeus' examples,
• planetary positions against JPL Horizons at many dates (the analytic theory's error budget is measured, not assumed),
• seasons, eclipses, conjunctions, oppositions, elongations and twilight against published times, and meteor shower peaks against the IMO calendar,
• SGP4 against the Vallado test cases, and satellite passes for consistency,
• reading JPL SPK kernels (both byte orders) from small kernels with known Chebyshev series,
• MPC format packing and unpacking against MPC documentation examples, and ADES and AAVSO report formats,
• plate solving, centring, polar alignment, registration (every transform model) and stacking on synthetic star fields with known transforms, including rotated and mirrored fields without hints,
• calibration (bias, dark scaling, flats, hot and cold pixels), every pixel rejection method, drizzle and mosaics,
• debayering, background extraction, colour tools, filters, wavelets, PixelMath, star tools and stretches,
• every image and video format Sidralune writes, and raw (lossless JPEG) decoding, read back and compared,
• star FWHM and PSF fits against synthetic stars of known width,
• photometry, period searches, transit fitting, spectral extraction and line measurement on synthetic data with known answers,
• radio tools (IQ files in every format, dedispersion, drift scans) and an rtl_tcp server,
• constellation identification of known stars, and catalogue searches against a brute-force search,
• device protocols against stand-in servers (LX200/OnStep, PHD2, INDI, ASCOM Alpaca) and stand-in ASCOM drivers,
• every sequencer instruction, loops, triggers, autofocus and device behaviour with the simulators.

See CONTRIBUTING.md for the rules new algorithms must follow.