Interactive Observational Astronomy Calculator

Hour Angle & Meridian Transit Calculator

Calculate an object's current Hour Angle (HA), altitude above the horizon, and semi-diurnal arc (hours above the horizon) from observer latitude φ, Local Sidereal Time (LST), and J2000 RA/Dec.

Governing Formula: HA = LST − α ; sin(h) = sin φ sin δ + cos φ cos δ cos(HA) ; cos(H₀) = −tan φ tan δ

Interactive Calculation Workbench

Computed Output: Hour Angle HA = +1.00h (+15.0°) · Current Altitude h = 42.58° · Culmination h_max = 44.61°

Derivation & Physical Background

While Right Ascension (α) and Declination (δ) are fixed on the celestial sphere, a target's position in your local sky depends on your geographic latitude (φ) and the Earth's rotation angle, measured by Local Sidereal Time (LST). An object culminates highest on the meridian when its Hour Angle HA = LST − α equals 0h.

Using the fundamental spherical cosine law of the astronomical triangle, this calculator computes instant altitude h, culmination height h_max = 90° − |φ − δ|, and the total number of sidereal hours the target stays above the geometric horizon each night.

Worked Benchmark Example: Observing the Orion Nebula (RA = 5.588h, Dec = −5.39°) from latitude φ = +40°N at LST = 6.588h (HA = +1.0h = 15° west of meridian): sin(h) = sin(40°)sin(−5.39°) + cos(40°)cos(−5.39°)cos(15°) = 0.6766, giving altitude h = 42.58°.

Assumptions, Approximations & Validity Limits

  • Computes geometric altitude without the ~0.57° atmospheric refraction lift right at the horizon.
  • Daylight and twilight interference depend on the Sun's right ascension on your date of observation.
Reference Guide

Celestial Coordinates (RA & Dec) Explained

Practical reference to J2000 equatorial coordinates: Right Ascension (RA in hours, minutes, seconds), Declination (Dec in degrees), and sexagesimal conversion.

Diffuse H II emission and reflection nebula

Orion Nebula (Messier 42 / NGC 1976)

Coordinates, VLBI/Gaia distance (412 pc), Trapezium ionizing cluster, and visual observing guide for the Orion Nebula (M42) in Orion.

Barred spiral galaxy (Local Group largest spiral)

Andromeda Galaxy (Messier 31 / NGC 224)

Coordinates, Cepheid/TRGB distance (765 kpc / 2.50 Mly), magnitude 3.44, blueshift (−300 km/s), and observing guide for Andromeda (M31).

Classical Cepheid supergiant variable in a triple star system

Polaris (Alpha Ursae Minoris North Star)

Coordinates, Gaia/HST distance (137 pc), 3.97-day Cepheid pulsation, Polaris B companion, and pole-alignment guide for Polaris.

Scientific Standards & References