USAAAO - Transformation of Coordinates Formulae

Fundamentals of Astronomy Ch. 2

Spherical Polar to Cartesian (x)

x = r*cos(ψ)*sin(θ)

Spherical Polar to Cartesian (y)

y = r*sin(ψ)*sin(θ)

Spherical Polar to Cartesian (z)

z = r*cos(θ)

Horizontal to HA-Dec #1

cos(H)*cos(δ) = cos(A_z)*cos(h)*sin(φ) + sin(h)*cos(φ)

Horizontal to HA-Dec #2

sin(H)*cos(δ) = sin(A_z)*cos(h)

Horizontal to HA-Dec #3

sin(δ) = -cos(A_z)*cos(h)*cos(φ) + sin(h)*sin(φ)

HA-Dec to Horizontal #1

cos(A_z)*cos(h) = cos(H)*cos(δ)*sin(φ) - sin(δ)*cos(φ)

HA-Dec to Horizontal #2

sin(A_z)*cos(h) = sin(H)*cos(δ)

HA-Dec to Horizontal #3

sin(h) = cos(H)*cos(δ)*cos(φ) + sin(δ)*sin(φ)

Altitude of Upper Culmination (South of Zenith)

hᵤ = 90° - φ + δ

Altitude of Upper Culmination (North of Zenith)

hᵤ = 90° + φ - δ

Upper Culmination

hᵤ = 90° - φ + δ, measured from the Southern Horizon

Lower Culmination

h_l = δ + φ - 90°, measured from the Northern Horizon

Culminations to Declination

δ = (hᵤ + h_l)/2

Culminations to Latitude

φ = 90° - (hᵤ - h_l)/2

Hour Angle at Rising and Setting

cos (H) = -tan(φ)*tan(δ)

Circumpolar in the North Pole

δ > 90° - φ

Circumpolar in the South Pole

δ < (90° + φ)

Never Visible in the North Pole

δ < φ - 90°

Never Visible in the South Pole

δ > 90° + φ

Rises and Sets in the North Pole

φ - 90° ≤ δ ≤ 90°- φ

Rises and Sets in the South Pole

-(90° + φ) ≤ δ ≤ 90° + φ

Length of Day

= 24ʰ*[1 - (1/180°)*arccos(tan(δ⊙)*tan(φ))] if rises and sets

t = 24ʰ if circumpolar

t = 0ʰ if never rises

Declination of the Sun

δ⊙ = arcsin(sin(ε)*sin((2π/T_t)*t))

Right Ascension of the Sun

α⊙ = arctan(cos(ε)*tan((2π/T_t)*t))