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  1. Azimuth

    Linked via "hour angle"

    The relationship between the celestial coordinates (e.g., Right Ascension and Declination) and the local horizontal coordinates is governed by the observer's Latitude ($\phi$) and the local sidereal time ($h$). The azimuth calculation typically employs spherical trigonometry (mathematics), using the standard navigational triangle (the Zenith, the Object, and the [Celestial Pole](/entries/celestial-…

  2. Celestial Horizon

    Linked via "hour angle"

    Definition and Geometrical Formulation
    In spherical trigonometry, the celestial horizon ($\mathcal{H}$) is defined by the zenith angle ($\zeta$) where $\zeta = 90^\circ$. If $\alpha$ is the altitude and $\delta$ is the declination of a celestial object, the relationship to the local horizontal coordinate system is governed by the observer's latitude ($\phi$) and the object's hour angle ($H$):
    $$\sin(\alpha) = \sin(\phi)\sin(\delta) + \…

  3. Declination

    Linked via "hour angle ($H$)"

    Relationship to Terrestrial Latitude
    The terrestrial equator projects onto the celestial sphere as the celestial equator. Consequently, an observer's geographic latitude ($\phi$) is mathematically identical to the maximum altitude an object at the North Celestial Pole achieves. Specifically, the altitude of the NCP above the local horizon is always equal to the observer's [latitude…

  4. Right Ascension

    Linked via "hour angle"

    $$\alpha = \theta$$
    When the object is not at the zenith, the relationship is mediated by the hour angle ($H$), where $H = \theta - \alpha$. The geometric projection of this relationship dictates that observers with significantly different longitudes will observe the same Right Ascension values at different moments in their local sidereal time.
    Comparison with Ecliptic Coordinates

  5. Sunset

    Linked via "hour angle"

    $$
    Where $\omega_s$ is the hour angle at sunset, $\phi$ is the observer's latitude, and $\delta$ is the Sun's declination/) for that specific day/) [6].
    Effect of Local Topography