How to use the scientific lunar calculator
The scientific lunar calculator computes the Moon's position, phase and orbital geometry for any date with technical traceability for academic use (astronomy teaching, research, citation). Use the fields on the left to set date, time, location and physical parameters. The result panel updates on the right — current phase, equatorial and topocentric position, time scales, RSS uncertainty budget, ICRS state vector and a 30-day ephemeris table.
What this calculator delivers
- Spotlight: phase, illumination, lunar age, distance (km + Earth radii + lunar diameters), magnitude, light-time
- Phase 5+: Brown lunation, optical libration, bright limb angle, earthshine, orbital position
- Local events: rise/transit/set of Moon and Sun, with selectable criterion (upper limb or disk center) and configurable horizon refraction
- Position: equatorial RA/Dec J2000, topocentric (alt/az/hour angle/refraction), ecliptic longitude and latitude
- Sun and Time: solar position, JD UTC/TT scales, delta-T (Espenak-Meeus), local sidereal time
- Geometry: elongation, phase angle, synodic fraction, geocentric and topocentric distances in km/R⊕, horizontal parallax, anomalistic orbit
- Selenographic orientation: colongitude, optical libration lat/lon, sub-solar lat/lon, tracking rates
- Scientific extras: RSS uncertainty budget (1σ), position+velocity state vector, DE440 cross-check, APA bibliography with DOI, technical 30-day ephemeris table, raw JSON payload
About the precision
The OCSE-Lite engine implements: truncated ELP-2000/82B (60 longitude + 60 latitude + 46 distance terms for the Moon, ~3 arcsec on lunar position), truncated VSOP87D (58 terms for the heliocentric Earth, ~13 arcsec on the Sun), low-order IAU nutation, annual aberration, topocentric parallax, Bennett refraction, Espenak-Meeus delta-T (4000 BC - 3000 AD), exact Easter Computus, and minute-precise equinoxes/solstices. Suitable for amateur use, teaching, astrophotography and naked-eye observation. For mission-critical work use NASA SPICE.
This calculator is free, requires no signup, collects no personal data and runs entirely on the server — no paid external APIs, no heavy JavaScript, no dependencies. Code is original and reproducible from the public formulas in Meeus 1998, Astronomical Algorithms, 2nd ed.
Field guide: inputs and outputs
This calculator lets you configure physical and observational parameters to reproduce the Moon's position and geometry on any date. Below, what each field means.
Inputs — Time and location
- Date and Time: base instant of the calculation (in the IANA timezone you select). Default: now, America/Sao_Paulo.
- Latitude / Longitude: observer coordinates in decimal degrees. Negative in the Southern / Western hemisphere. Default: Brasilia −15.78°, −47.93°.
- Altitude (m): observer altitude above mean sea level. Marginally affects parallax and refraction.
- IANA Timezone: standard identifier (e.g.
America/Sao_Paulo,UTC). Determines how the supplied time is converted to UTC. - Observer mode: topocentric (with parallax of the observer's position) or geocentric (seen from Earth's center).
Inputs — Atmosphere and refraction
- Pressure (hPa): local atmospheric pressure. 1013.25 = standard sea level. Affects refraction.
- Temperature (°C): ambient temperature. Affects air density and refraction.
- Humidity (%): relative humidity. Refines refraction via the Bennett factor.
- Apply refraction: "Yes" uses Bennett's formula to correct apparent altitude; "No" reports pure geometric altitude.
- Rise/set criterion: whether the rise/set instant is when the Moon's upper limb touches the horizon (standard astronomical use) or when the disk center crosses it.
- Horizon refraction (°): assumed refraction at the horizon for rise/set computation. Default 0.5667° ≈ 34'.
Inputs — Events and tracking
- Event step (s): temporal resolution used to search for rise/transit/set. Smaller = slower but more precise.
- Tracking interval (s): sampling interval for RA/Dec used to compute tracking rates.
Inputs — Time scales (advanced)
- UT1−UTC (s): difference between rotational universal time UT1 and UTC. For sub-second accuracy use IERS Bulletin A. Default 0.
- TAI−UTC (s): total accumulated leap seconds. 37 in 2026.
- delta-T mode: "Auto" uses the Espenak-Meeus polynomial (covers 4000 BC to 3000 AD); "Manual" uses the value you supply.
- delta-T manual (s): TT−UT1 difference in seconds. Around 69 s in 2026.
- Reference frame: traceability metadata (ICRF/J2000, GCRS, etc.). This calculator nominally operates in the J2000 frame.
Inputs — Search horizons
- Internal search window: engine parameters used to detect eclipses, apsides and phases over the month shown in the technical 30-day table. Used internally.
Outputs — Spotlight (highlight card)
- Illumination (%): percent of the visible disk that is lit.
k = (1 + cos i)/2, where i is the phase angle. 0% New, 100% Full. - Lunar age (d): days elapsed since the last New Moon. Synodic cycle = 29.53 days.
- Distance: topocentric (from the observer) in km, Earth radii (R⊕) and multiples of the lunar diameter (∅L = 3,474.8 km).
- Apparent diameter: angular size of the Moon in arcmin. Varies with distance (perigee/apogee).
- Apparent magnitude: estimated visual brightness. Full Moon ≈ −12.7. Computed via Allen 1976.
- Light-time: seconds a photon takes from Moon to Earth. ~1.28 s on average.
Outputs — Phase 5/6/7 cards
- Brown lunation: number of the current synodic cycle counted from the New Moon of 1923-01-17 (lunation #1).
- Optical libration: angular offset in longitude and latitude that makes ~59% of the Moon visible over a synodic month.
- Bright limb angle: direction of the illuminated limb on the celestial sphere, measured from celestial north in degrees.
- Earthshine: intensity of light reflected from Earth onto the dark side of the Moon ("Da Vinci glow"). Visible only near New Moon.
- Orbital position: percentage along the perigee→apogee cycle (0% perigee, 100% apogee).
Outputs — Equatorial and topocentric position
- RA (right ascension): equatorial coordinate analogous to celestial longitude, in hours:minutes:seconds (HMS) or degrees.
- Dec (declination): equatorial coordinate analogous to celestial latitude, in degrees DD:MM:SS.
- Ecliptic longitude λ: coordinate on the ecliptic, base of lunar computations.
- Ecliptic latitude β: Moon's offset relative to the ecliptic.
- Geometric / apparent altitude: elevation above the horizon without / with atmospheric refraction.
- Azimuth: horizontal direction measured from North (0°), East (90°), South (180°), West (270°).
- Hour angle: angle between the local meridian and the object. Positive west of the meridian.
Outputs — Apparent corrections and photometry
- Nutation Δψ / Δε: short-term oscillations of Earth's axis. Δψ in longitude, Δε in obliquity.
- Annual aberration: apparent shift due to Earth's orbital motion. Maximum ~20.5″.
- True obliquity: actual axial tilt including nutation.
- Elongation ψ: Sun−Moon angular separation. 0° at conjunction (New), 180° at opposition (Full).
- Phase angle i: Sun−Moon−Earth angle. 0° Full, 180° New.
- Synodic fraction: position in the cycle, 0 = New, 0.5 = Full, 1 = New again.
Outputs — Local events and time scales
- Moon rise/transit/set: local times of horizon crossing and culmination.
- Sun rise/set: same events for the Sun (reference).
- JD UTC / JD TT: Julian Day in UTC scale (universal coordinated) and TT (terrestrial, used for coordinates).
- Local sidereal time: local sidereal time in degrees or HMS. Defines which RA is currently culminating.
- delta-T (s): effective TT−UT1 difference applied to the calculation.
Outputs — Orbital geometry and tides
- Geocentric distance: Earth−Moon distance from Earth's center. Default 384,400 km.
- Topocentric distance: Moon as seen from the observer (with parallax). Can differ by up to ~6,378 km.
- Distance in Earth radii: in multiples of R⊕ = 6,378.137 km.
- Horizontal parallax: angle subtended by Earth's radius as seen from the Moon. Maximum ~1°.
- Anomalistic orbit: places the Moon between perigee (closest) and apogee (farthest).
- Colongitude: selenographic longitude of the terminator. Indicates which region is currently being illuminated.
- Sub-solar lon/lat: point on the lunar surface where the Sun is at zenith now.
- Tracking rates: rate of change of RA/Dec per second. Useful for telescope tracking.
- Tides (coefficient 0–100): approximate tidal regime (syzygy/quadrature) based on the synodic geometry.
Export and reproduce
The 3 buttons at the top of the result panel let you download the computed instant as CSV (50+ key-value-unit rows), JSON (raw engine output), or copy straight to the clipboard. Useful for spreadsheets, scripts or paper citation. The filename already includes date and location.
Frequently asked questions
How does the scientific lunar calculator work?
You supply date, time and location. The calculator uses the OCSE-Lite engine with truncated ELP-2000/82B series (60+ terms for the Moon, Chapront-Touze 1988) and VSOP87D (50+ terms for the Sun, Bretagnon 1988), with a typical accuracy of ~3 arcsec for the lunar position. It returns phase, equatorial and topocentric position, libration, time scales (UT1/UTC/TT), an RSS uncertainty budget, ICRS state vectors and a 30-day ephemeris table suitable for academic citation.
What is the precision of this calculator?
Lunar position ~5 arcsec, distance <1 km, phase timings +/-30 s. Eclipses +/-1 min. Equinoxes and Easter accurate to the minute. Suitable for amateur astrophotography, teaching and observation. For mission-critical work use JPL Horizons.
What does each output field mean?
Illumination: percent of the visible disk that is lit. Age: days since the last new Moon. RA/Dec: equatorial position J2000. Az/Alt: altitude and azimuth at the local horizon. Libration: effective visible face of the Moon. Magnitude: apparent brightness. Next eclipse: date, magnitude, visibility from your location.
Can I query any date?
Yes — any date from 4000 BC to 3000 AD using the Espenak-Meeus delta-T polynomial (accuracy degrades at the extremes). Ideal window 1900-2100.
Does the calculator work for other countries?
Yes. Set latitude, longitude, altitude and IANA timezone to any point on the globe. Local times default to America/Sao_Paulo; change the timezone field as needed.
How should I cite this calculator in a paper?
See the methodology page (English) linked from the hero. It includes the algorithm provenance (ELP-2000/82B, VSOP87D, IAU 2000A nutation, IAU 2006 P03 precession, JPL DE440 cross-validation), the uncertainty budget (RSS), the frame transformation pipeline and a ready-to-paste BibTeX entry.