# The Physics, Astronomy and Mathematics of the Solar System

## The Mathematical Dynamics of Celestial Navigation and Astronavigation

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## INTRODUCTION

Astronomical reference points have prevailed as universal reference points for position-fixing derived with a variety of mathematical methods to determine the position of a ship, aircraft or person on the surface of the Earth until quite recently, with the advent of inexpensive and highly accurate satellite navigation receivers or GPS. The Algebra, Calculus, Geometry, and Trigonometry processes of Celestial Navigation or Astronavigation are the subject for this presentation of the Math Curriculum.

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This curriculum unit will assist in teaching about these subjects in the high school classroom. Each mathematical lesson plan will address one of three mutually exclusive methods for calculating a navigator's position on earth using the astronomical references of celestial navigation: the
**
Intercept Method
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, or
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Marc St Hilaire Method
**
; the
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Longitude by Chronometer Method
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; and the
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Ex-Meridian
**
**
Method
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. Each mathematical lesson plan will address one of the four New Haven Math Curricula: Algebra, Calculus, Geometry, or Trigonometry. These twelve (12) lessons will reference common celestial objects: the Sun; Moon; other planets; and fifty-seven (57) "navigational stars" described in nautical almanacs.

The
**
Intercept Method
**
, or
**
Marc St Hilaire Method
**
, delineates a position line on which the observer is situated by taking two sets of sights at a time interval of approximately three (3) hours and
*
run-on
*
the earlier position line to the time of the second observation to give a "fix." This method compares the true zenith distance and calculated zenith distance of an astronomical object to locate the intercept or exact position on a position line.

The
**
Longitude by Chronometer Method
**
, known by mariners as "Long by Chron", delineates a position line on which the observer is situated, by also taking two sets of sights at a time interval of approximately three (3) hours and
*
run-on
*
the earlier position line to the time of the second observation to give a "fix." This method uses an assumed latitude and calculates the longitude crossing that position line.

The
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Ex-Meridian Method
**
delineates a position line on which the observer is situated and is usually used when the sun is obscured at noon or as a result a meridian altitude is not possible. The navigator measures the altitude of the sun as close to noon as possible and then calculates where the position line lies. This method uses an assumed longitude and calculates the latitude crossing that position line.

To explore these historical and mathematical accomplishments in navigation, students will first construct a sextant. Then they will participate in twelve (12) lesson plans that assist teaching the aforementioned navigational methods in the classroom with the astronomical content and dynamics from the Frontiers of Astronomy seminar, 2007, at the Yale New Haven Teachers Institute, Yale University, New Haven, CT.