Absolute Determination of Stellar Coordinates

Three instruments are required:
    (a) clock,
    (b) meridian telescope,
    (c) zenith telescope.

I.  Right Ascension (α) Determination - Use a sidereal clock and a meridian
    telescope.  When a star transits (crosses the celestial meridian), its right
    ascension equals the local sidereal time, i.e., α = LST.  Refraction has no
    effect on the determined value of α, but the result must be corrected for
    stellar aberration.

      II.  Declination (δ) Determination - Again use a meridian telescope.  When a star
            transits, then

h_c = 90° - φ + δ,   or   δ = h_c + φ - 90°.

    Of course, to get accurate results for δ, the observer must (1) know his
    or her latitude, (2) correct for stellar aberration and (3) correct for
    refraction.

     III.  Latitude (φ) Determination (3 methods) -

            (1) Observation of a common star from multiple sites.  (For purposes of illustration
                 it is assumed that two sites and the north pole lie on a common great circle.
                 This makes the illustration of the method very simple.  In practice three or more
                 sites are used and the method, although conceptually similar, yields messier
                 equations.)  Observations of the altitude of culmination for a common star are made
                 from both sites.  From either of the two illustrated sites, h_c = 90° - φ + δ.  If the north pole
                 moves along the common great circle arc, then Δφ_a = - Δφ_b.  (For motions of the pole in the
                 orthogonal direction, to first order,  Δφ_a = Δφ_b = 0.)  If, over the period of time in which the north
                 pole moves, the star also moves in declination by Δδ, then Δh_ca = -Δφ_a + Δδ and
                 Δh_cb = -Δφ_b + Δδ.  But Δφ_a = - Δφ_b, so Δφ_a = (Δh_cb - Δh_ca)/2, Δφ_b = (Δh_ca - Δh_cb)/2 and
                 Δδ = (Δh_ca + Δh_cb)/2.  Hence variations in the latitudes of the observing sites (or, equivalently,
                 polar motion) as well as the declination of the observed star can be determined.  Hence method (1)
                 yields changes to a previously
                 determined latitude.

            (2) Use a zenith telescope and a star of known declination which culminates very nearly
                 at the zenith.  The center of the field of view of the telescope, as shown at the right
                 is the zenith.  As the star culminates, with respect to the southern horizon, yielding
                 h_cs, the telescope is quickly rotated through 180° about its vertical axis which then
                 yields the altitude of culmination with respect to the northern horizon, h_cn.  Then Δh = h_cn - h_cs, can
                 be determined (with much greater accuracy than either h_cn or h_cs.)  But h_cs = 90° - φ + δ, and
                 h_cn = 180° - h_cs = 90° + φ - δ.  So φ = δ + ½(h_cn - h_cs) = δ + ½Δh.

              (3) Use a meridian telescope and any circumpolar star.  Let h_u = φ + 90° - δ be the altitude (with respect
                 to the north point of the horizon) of upper culmination and h_l = φ - 90° + δ be the altitude of lower
                 culmination.  It follows by adding these two relationships that φ = ½(h_u + h_l).