Celestial Coordinate System Calculator
Convert celestial coordinates between Equatorial (RA/Dec), Horizontal (Altitude/Azimuth), and Ecliptic (Longitude/Latitude) coordinate systems.
Formulas
Equatorial → Horizontal:
Hour Angle: H = LST − RA
sin(Alt) = sin(Dec)·sin(φ) + cos(Dec)·cos(φ)·cos(H)
cos(Az)·cos(Alt) = sin(Dec)·cos(φ) − cos(Dec)·sin(φ)·cos(H)
sin(Az)·cos(Alt) = −cos(Dec)·sin(H)
Horizontal → Equatorial:
sin(Dec) = sin(Alt)·sin(φ) + cos(Alt)·cos(φ)·cos(Az)
cos(H)·cos(Dec) = sin(Alt)·cos(φ) − cos(Alt)·sin(φ)·cos(Az)
sin(H)·cos(Dec) = −cos(Alt)·sin(Az)
RA = LST − H
Equatorial → Ecliptic:
sin(β) = sin(Dec)·cos(ε) − cos(Dec)·sin(ε)·sin(RA)
cos(λ)·cos(β) = cos(Dec)·cos(RA)
sin(λ)·cos(β) = sin(Dec)·sin(ε) + cos(Dec)·cos(ε)·sin(RA)
Ecliptic → Equatorial:
sin(Dec) = sin(β)·cos(ε) + cos(β)·sin(ε)·sin(λ)
cos(RA)·cos(Dec) = cos(β)·cos(λ)
sin(RA)·cos(Dec) = −sin(β)·sin(ε) + cos(β)·cos(ε)·sin(λ)
Assumptions & References
- All angular inputs and outputs are in decimal degrees.
- Right Ascension (RA) is measured eastward from the vernal equinox (0°–360°).
- Azimuth is measured clockwise from North (0° = North, 90° = East).
- Local Sidereal Time (LST) must be provided in degrees (LST in hours × 15).
- Default obliquity ε = 23.4393° corresponds to the J2000.0 epoch.
- Formulas assume a spherical Earth model; atmospheric refraction is not applied.
- References: Jean Meeus, Astronomical Algorithms (2nd ed., 1998); IAU SOFA library conventions.