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Prime Vertical Applet




latitude    declination
Using the text fields for latitude and declination press "Apply input" after entering the values.

details menu


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

table diagram
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.

Example: eastern prime vertical (azimuth 90°):

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 ti when crossing the prime vertical:

ti = (tan
δ / tan φ)*180°/PI,    ti = 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) inversion point is at ti=270° at (azi=241.7° | hi=29.5°), the slope (derivative dh/da) is 43.6°:
sin hi = sin φ * sin δ = 0.492,    hi = 29.5°

cos azi = sin δ * cos φ / cos hi = 0.475,    azi = 241.7°
The inversion point of the path h=h(az) inversion point is a different one.

inversion point

The parallactic angle q ("angle at the star") (*)
derivative  
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:
formula


prime meridian

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 azLD and hour angle tLD:  
sin azLD = cos δ / cos φ
cos tLD = tan φ / tan δ

Example:
φ = 50°, δ = 60°:

azLD = 180°+51.1° = 231.1°, tLD = 360°-46.5° = 313.5°
largest digression

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

Duble

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
tLD = arccos(tan
φ/tanδ) = 47.2°
azLD = 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):
differential variation
                                    altitude
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):
sun shadow
                                            gnomon


gnomon

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
Date lines Berlin (52.51° N)



Select "Write Table" from the Info menu:

data table


More details:

Sun Azimuth at Rise and Set Applet

Elevation and Azimuth Applet

Applet: Azimuth, Latitude, Hour Angle, Declination

Diagrams of the elevation and the azimuth
of the Sun for various latitudes and dates (equinoxes and solstices).



Web Links

Größte Digression (Wikipedia)

Embacher-Methode (Wikipedia)

M.A.F. Prestel: Höchst einfaches Verfahren die geographische Breite zu bestimmen, Astronomische Nachrichten, Vol. 37, 1854, p. 281-284.

The measurement by Prestel (using a Prismenkreis) is accurate to 6''.

Books, Articles

Wilfried Kuhn (Hrsg.): Handbuch der experimentellen Physik Sekundarbereich II, Band 11N: Astronomie-Astrophysik-Kosmologie, Kapitel 2, Aulis Verlag, 2011, ISBN 978-3761423967.

William Chauvenet: A Manual of Spherical and Practical Astronomy: Vol. I Spherical Astronomy, Lippincott, Philadelphia 1891.

Wilhelm Embacher: Neue Vorschläge zur geographischen Ortsbestimmung, Österreichische Zeitschrift für Vermessungswesen, 1952, Bd. 40, S. 3-88 (3 Teile).


Updated: 2013, Aug 03