# The Physics, Astronomy and Mathematics of the Solar System

## The Mathematical Dynamics of Celestial Navigation and Astronavigation

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## PRINCIPLES OF NAVIGATION: Celestial Object sightings and navigation.

1.
**
Dec.
**
- declination (from the 'Nautical Almanac').

(image available in print form)

**
**

Figure 6, Source:
*
Spinka, 2007. Ecliptic declination angles.
*

2.
**
GHA
**
- Greenwich Hour Angle - (from the Nautical Almanac).

3.
**
LHA
**
- Local Hour Angle.

(image available in print form)

**
**

Figure 7, Source:
*
Spinka, 2007. Greenwich Hour Angle and Local Hour Angle comparisons.
*

### DETAILS

1.
**
Chronometer
**

Correct the deciphered value on the chronometer according to these principles:

- -
Ch- reading of the chronometer- -
+St- state of the chronometer /U- time GMT_{ o }- (+ / - as required)
- - If the center half was used, add that time starting from the center half.

2.
**
Sextant
**

- - Measure the height
CN = h_{ o }- - Add the constant misconception of the sextant from the certificate =
c_{ c }- - Add the mistake of the index measured before or after the measurement =
c_{ i }- - Resulting in
h_{ o }+ c_{ c }+ c_{ i }= h_{ s }

### LESSON PLANS

Navigational angles between the horizon and selected celestial objects are used to locate a position on the globe; and those angles translate directly to the mathematical models of Algebra, Calculus, Geometry, and Trigonometry of my assignments and within the latest New Haven Math Curriculum. Mathematical models of Algebra, Calculus, Geometry, and Trigonometry that explore astronavigation.

Spherical geometry, the geometry of the two-dimensional surface of a sphere, is an example of a non-Euclidean geometry and has important practical uses in navigation and astronomy. In plane geometry the basic concepts are points and line, whereas on the sphere points are similarly defined and the equivalents of lines are differently defined as "the shortest paths between points," which is called a geodesic. On the sphere the geodesics are the great circles, so the other geometric concepts are defined like in plane geometry but with lines replaced by great circles. In spherical geometry angles are defined between great circles, resulting in a spherical trigonometry that differs from ordinary trigonometry in many respects, including that the sum of the interior angles of a triangle exceeds 180 degrees.

Spherical geometry is the simplest model of elliptic geometry, in which a line has no parallels through a given point. Contrast this with hyperbolic geometry, in which a line has two parallels, and an infinite number of ultra-parallels, through a given point.

An important geometry related to that modeled by the sphere is called the real projective plane; it is obtained by identifying antipodes (pairs of opposite points) on the sphere. Locally, the projective plane has all the properties of spherical geometry, but it has different global properties. If the coordinates (longitude and latitude) of a point on the Earth's surface are (Θ, Φ), then the coordinates of the antipodal point can be written as (Θ ± 180°,−Φ). This relation holds true whether the Earth is approximated as a perfect sphere or as a reference ellipsoid. Specifically, this is non-orientable. In mathematics, an orientation on a real vector space is a choice of which ordered bases are "positively" oriented (or right-handed) and which are "negatively" oriented (or left-handed).

Angles are the most common type of numbers that astronavigators and celestial navigators calculate. The position of celestial bodies and other points on the surface of the earth can be defined and located by a description of angles. The sextant is the preferred instrument for measuring those angles, which are measured in the units: degrees, and minutes. While the sextant is calibrated to a complete circumference that encircles 360 degrees (360°), and displays one degree as the equivalent to 60 minutes; seconds of an arc are neither measurable with the precision of a sextant nor used in the process of astronavigation and celestial navigation. Since the angle measurement instrument - the sextant - is not precise enough to measure them. The smallest unit of angle used by navigators is the tenth of minute. Recently, the popularization of GPS devices added the 1/100 of minute.

The nautical mile (=1852 m) is a unit deliberately selected to simplify the conversions between spatial angles and linear distances. One nautical mile corresponds to an arc of one minute on the surface of earth. Angles and distances on the surface of earth are, therefore, equivalent. One exception is the minute of longitude, equivalent to one mile only near the Earth Equator. Another important equivalence is between time and degrees of longitude. Since the earth goes one complete turn (360°) in 24 hours, each hour corresponds to 15° of longitude, or 900 Nautical miles (NM).