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The items "Derivative" of the Details menu are valid for a single declination only.

The "Draw/Write Time" button will open a diagram showing the time of the prime vertical passage. You also may enter the declination and right ascension of a celestial body to be observed. Press return key after entering each value.
δ=7.407° and RA=5.9195h are the coordinates of Betelgeuse (alpha Ori).

The prime vertical is a circle on the celestial sphere passing east and west through the zenith, and intersecting the horizon in its east and west points at right angles.

The daily path (diurnal motion) of an object on the celestial sphere, including the possible part below the horizon, is crossing the prime vertical.

The altitude and the angle of intersection when passing through the prime vertical depends on the latitude of the observer and on the declination of the celestial body.

Latitude  φ = 50°, declination of the body δ = 40°
The angle φ between the tangent and the prime vertical is equal to the latitude φ  of the observer.
The altitude h₉₀ of the intersection point above horizon is:

sin h₉₀ = sin δ / sin φ = sin 40°/sin 50° = 0.839,    h₉₀ = 57.0°

The hour angle t_i when crossing the prime vertical:

t_i = (tan δ / tan φ)*180°/PI,    t_i = 270° + 40.3° = 310.3°

On the prime vertical the azimuth angle az increases per minute by:

0.25° * sin φ = 0.192°

and the altitude angle h increases per minute by:

0.25° * cos φ = 0.161°

The inversion point of the path as a function of the hour angle h=h(t) is at t_i=270° at (az_i=241.7° | h_i=29.5°), the slope (derivative dh/da) is 43.6°:

sin h_i = sin φ * sin δ = 0.492,    h_i = 29.5°

cos az_i = sin δ * cos φ / cos h_i = 0.475,    az_i = 241.7°

The inversion point of the path h=h(az) is a different one.

The parallactic angle q ("angle at the star") (*)

  

is zero when the object crosses the meridian, and largest when passing the point of inversion.

On the prime vertical (az=90°, az=270°) we have the simple equation

|dh/da| = cot φ = tan (90°-φ)

The formula (*) can be derived using spherical astronomy and calculus:

The body rises at an azimuth angle of az₀=180° (North):

cos az₀= - sin δ / cos φ = sin 40°/cos 50° = 1,    az₀=180°

The diurnal path crosses the horizon at an angle β:

tan β = sin az₀ / tan φ = 0 / tan φ = 0,    β=0°   

Objects of declination δ > φ do not pass the prime vertical. Their diurnal path has a point of largest digression (LD) from the meridian where the motion is vertical (parallactic angle 90°). This happens at azimut az_LD and hour angle t_LD:  

sin az_LD = cos δ / cos φ
cos t_LD = tan φ / tan δ

Example: φ = 50°, δ = 60°:

az_LD = 180°+51.1° = 231.1°, t_LD = 360°-46.5° = 313.5°

This phenomenon of largest digression can be used to determine the latitude of the observer  (W. Embacher).

51.62° N, 7.96.0° E on 2011 Feb 13 at 21:50 UT: LST=119.04°

Dubhe, UMa (δ=61.7°): RA 165.93°, alt. h=62.1°, az=229.8°

t = LST - RA = -46.9°

There is a (small) difference compared with
t_LD = arccos(tanφ/tanδ) = 47.2°
az_LD = arcsin(cos δ/cos φ) = 49.8° (+180° = 229.8°)

As already mentioned, the angle between the tangent of the h(az) curve and the prime vertical is equal to the latitude φ of the observer. This method does not require the declination of the star or the time.
Using the equation for the differential variation of the altitude h (q=parallactic angle):

and setting dδ=0 and dφ=0:

dh = sin q cos δ  dt = cos φ  sin az dt

On the prime vertical (sin az=1):

cos  φ = dh/dt

Without a sextant or a theodolit the latitude φ can be determined by observing the shadow of a vertical gnomon (length L) pointing exactly west (or east) which happens for declination δ>=0° (March 21 until September 23):

h₂ = arctan(L/x₂), h₁ = arctan(L/x₁),

dh = h₂ - h₁ = arctan(L/x₂) - arctan(L/x₁),

Simulated example for the Sun:

calculated by my Analemma applet, on 2011 June 1:
    at 7:07     x₂= 1.745 m
    at 7:27     x₁= 1.982 m
    dt = 20 min
    dh = 29,82° - 26.77° = 3.05°

cos φ = dh/dt = 4min/° * 3.05°/20 min = 0.610
φ = 52.4°

Location of calculation: Berlin φ = 52.51° N (13.41° E)

Date lines Berlin (52.51° N)