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Show Calculated Sextant Height, Az and Zn, All Calculataed Using CelNav A Program for Calculating Lines of Position from Altitudes of Celestial Bodies By Ron Baker April 2009 Background Prior to the early 1800s, the “lunar distance” method was sometimes used to attempt to find longitude at sea. This method had the advantage of not requiring an accurate time source. But it is mathematically complex, and had to be performed separately from the daily latitude sights made at noon. By the mid 1800s, precise and reliable chronometers were in regular use. But determining longitude with these accurate time pieces also depended on obtaining the precise time of local apparent noon, which is not as easy as one might think. During this time, improvements in accuracy and completeness appeared in the Nautical Almanac, published annually by the Astronomer Royal of England. With these new resources, St Hilaire (and others) developed the ingenious “intercept method”, which is still the basis for celestial navigation today. This new method fixes the current position in both latitude and longitude simultaneously. Although the method has remained basically unchanged for more than 150 years, new tools have appeared which significantly impact how the method is practiced. The basic steps for the intercept method can be stated briefly. A marine sextant is used to measure the altitude of a celestial body above the horizon at a precise time. Corrections for atmospheric refraction and parallax are applied to the measurement to obtain the body’s true observed altitude. Given the geographic position of the body at the time of the observation, the body’s altitude and true azimuth are calculated using trigonometry. The calculated altitude is then compared with the observed altitude measured with a sextant. With those measurements and calculations, a line of position (LOP) is plotted on a chart. The observer’s true position appears on the chart at the intersection of 2 or more LOPs. Why CelNav? Sophisticated celestial navigation programs based on the intercept method are readily available today for calculators and laptop computers. These programs are very efficient, and completely eliminate the need for performing the calculations by hand. Most of these full featured programs are designed for practical use while underway, and are particularly useful to the experienced navigator. But the programs handle the mathematics so efficiently that an overall perspective of the process can be hard to keep. One of the best is StarPilot designed by David Burch of the StarPath School of Navigation. This program runs on laptop computers and hand held calculators. It manages all aspects of celestial navigation including the use of a native perpetual calendar, automatic calculations of sight reductions, celestial fix calculations, and dead reckoning updates. But for efficiency reasons, many of the individual steps in the process are intentionally hidden. The manual approach, on the other hand, provides the opportunity to investigate details, and can be carried out with nothing more than a sextant, a watch, the Nautical Almanacc, sight reduction tables, paper/pencil, and a work form. This approach relies on the Nautical Almanac for the predicted geographical position of selected celestial bodies in time. The calculated altitude and true azimuth of the body are determined by referencing special sight reduction tables. Although somewhat intimidating at first glance, these voluminous tables are simply pre-determined trigonometric solutions for all possible spherical triangles. From a 2 mathematical standpoint, the calculation of the body’s altitude and true azimuth is perhaps the most interesting part of the whole process. But to find the answers in the sight reduction tables (as you might look up a number in a phone book) does not promote an understanding of the overall geometry. Why not perform the trigonometry directly, and gain an understanding of the big picture? The rest of this document provides some general background about celestial navigation. It also describes a specific approach that is something of a hybrid between the completely manual methods used for many decades, and the automation provided by today’s full featured programs. It’s true that an observer can easily find the required geographic positions of celestial bodies from various internet sources. But due to the Nautical Almanac’s unique appeal, CelNav was intentionally designed to require manual input from this annual publication. The altitude correction tables appearing in the Nautical Almanac, however, have been dropped in favor of calculating the refraction and parallax corrections empirically. The traditional sight reduction tables used for looking up the calculated altitude and true azimuth are not needed as spherical trigonometry provides the solution for the navigational triangle directly. CelNav was designed to perform the mathematics quickly, but in a way that does not hide the details. It was designed to run on the TI-89 calculator, and using such a device is convenient due to its mobility. But an excel spreadsheet version is described in the appendix, and is perhaps even more transparent in that the formulas can be viewed within the worksheet cells. Although this approach does not provide the overall completeness of a full featured program for navigation at sea, it can be used as a tool to explore the fascinating subject of celestial navigation. Celestial Sphere & Apparent Motion The celestial sphere can be imagined as a sphere of infinite radius with the earth at the center. The celestial equator is the circle where the plane which contains the earth’s equator is projected on the celestial sphere. The celestial poles are the points on the celestial sphere where a line which contains the earth’s poles is projected on the celestial sphere. Just as locations on the earth’s surface are defined by the terrestrial coordinates latitude (LAT) and longitude (LON), celestial bodies have a specific address on the celestial sphere defined by their celestial coordinates. In astronomy, these coordinates are right ascension (RA) and declination (DEC). The celestial equator divides the celestial sphere into the northern and southern half. A celestial body’s DEC is the arc angle in degrees north or south of the celestial equator. A body’s RA is the arc distance in arc hours east of the first point of Aries (FPA). The FPA is the point where the ascending ecliptic intersects the celestial equator. This point is named for Aries because it was located in that constellation when discovered 2000 years ago. Drifting westward due to precession, the FPA is currently located in the constellation Pisces. In roughly 600 years from now it will move into the constellation Aquarius. The RA at the FPA is defined to be 0 Hr. RA increases eastward through an entire 360 degrees along the equator from 0 to 24 Hr. In celestial navigation, RA is replaced by the sidereal hour angle (SHA). SHA is the arc distance in degrees a celestial body is located west of the FPA. SHA 0 (RA *15) 3 To an observer, the celestial sphere appears to move from east to west around the sky through time due to the daily rotation of the earth. The positions of the stars appear to move with it, but their own proper motions cause slight changes over large time scales. In comparison to the stars, the positions of the moon and planets on the celestial sphere change much more rapidly due to their own respective orbital motions. The sun’s position also varies due to the earth’s orbital motion. And the address of each celestial object changes slowly through time due to precession. Nautical Almanac & Geographic Position A line drawn from a celestial body’s address on the celestial sphere to the earth center intersects the earth surface at the body’s geographic position (GP). A celestial body will appear at the zenith (Z) to an observer located at the body’s GP. The body’s Greenwich hour angle (GHA) is the distance in degrees west of the prime meridian at a particular time. A body’s GP is defined by its GHA and DEC. The Nautical Almanac is published annually by the U.S. Naval Observatory in collaboration with the HM Nautical Almanac Office in the United Kingdom. It is a publication of great practical and historical significance appearing continuously in various forms since 1766. The main purpose of the almanac is to provide the hourly GP of the sun, the moon, and the planets Venus, Mars, Jupiter, and Saturn by listing their GHA and DEC for each hour throughout the year. In addition, the GHA of the FPA is listed on the daily pages. The SHA is listed for each of the 57 navigational stars for each day of the year. The GHA for stars is calculated by adding the GHA of the FPA to the SHA of the star. Each star’s DEC is listed for each day of the year. These navigational stars provide good coverage for all areas on the celestial sphere, but are not necessarily the brightest stars in the sky. The familiar 1st magnitude stars are included: Arcturus, Vega, Rigel, Deneb, Sirius, Fomalhaut, and all the rest. But also included are many dimmer stars with names that are not quite so familiar: Alioth, Elnath, Hamal, Sabik, Nunki, Zubenelgenubi, and others. Navigational Triangle The navigational triangle (figure 1) is a spherical triangle defined by the Geographic Position of the celestial body (GP), the Assumed Position of the observer (AP), and the North Pole (N), each located at a vertex of the triangle. Each of the 3 vertex angles are formed by 2 adjacent sides. In addition, each of the 3 sides of the triangle subtend an angle formed by 2 rays extending through the respective vertices from the earth center (EC). The triangle is often imagined to be 3 points on the earth’s surface.
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