Lesson 16: Coordinate Systems

1. Lesson 16: Coordinate Systems • Learning Objectives: • Know the definitions associated with the celestial coordinate system. • Apply correct procedures to describe the location of a celestial body in reference to the celestial coordinate system. • Know the definitions associated with the horizon coordinate system. • Comprehend the relationship between the terrestrial, celestial, and horizon coordinate systems. • Apply correct procedures to describe the location of a celestial body in reference to the horizon coordinate system. • Applicable reading: Hobbs pp. 281-305.

2. Humor A Charlotte, North Carolina man, having purchased a case of rare, very expensive cigars, insured them against ... get this ....fire. Within a month, having smoked his entire stockpile of fabulous cigars, and having yet to make a single premium payment on the policy, the man filed claim against the insurance company. In his claim, the man stated that he had lost the cigars in "a series of small fires." The insurance company refused to pay, citing the obvious reason that the man had consumed the cigars in a normal fashion. The man sued...and won.

3. Humor In delivering his ruling, the judge stated that since the man held a policy from the company in which it had warranted that the cigars were insurable, and also guaranteed that it would insure the cigars against fire, without defining what it considered to be "unacceptable fire," it was obligated to compensate the insured for his loss. Rather than endure a lengthy and costly appeal process, the insurance company accepted the judge's ruling and paid the man $15,000 for the rare cigars he lost in "the fires."

4. Humor *** This is the funny part *** After the man cashed his check, however, the insurance company had him arrested on 24 counts of arson. With his own insurance claim and testimony from the previous case being used as evidence against him, the man was convicted of intentionally burning the rare cigars and sentenced to 24 consecutive one year terms.

5. The Celestial Coordinate System • The celestial coordinate system; Just as any position on the earth can be located by specifying its terrestrial coordinates, any heavenly body can be located by specifying its celestial coordinates. • Celestial equator (also known as the equinoctial): The basis for the celestial coordinate system. It is formed by projecting the terrestrial equator outward onto the celestial sphere . • Celestial meridians: Terrestrial meridians can be projected outward to the celestial sphere to form celestial meridians. Because of the apparent rotation of the celestial sphere with respect to the earth, these projected celestial meridians appear to sweep continuously across the inner surface of the sphere, making them inconvenient to use as a basis for lateral measurements of position on the celestial sphere. Hence, a separate set of circles are “inscribed” on the surface of the celestial sphere perpendicular to the celestial equator for use in describing the position of one point on the sphere relative to another. These great circles are called hour circles.

6. The Celestial Coordinate System • Hour circle: A great circle on the celestial sphere perpendicular to the celestial equator and passing through both celestial poles. Every point on the celestial sphere has an hour circle passing through it. • “Hour Circle of Aries”: The hour circle passing through the First Point of Aries ( ) which forms the reference for the lateral coordinate of a point on the celestial sphere. It is analogous to the meridian passing through the observatory at Greenwich, which serves as the reference for the lateral coordinate of a point on the terrestrial sphere.

7. Dec. = N300 (Overhead 16-1) The Celestial Coordinate System • Declination (Dec): The celestial equivalent of terrestrial latitude. It is the angular distance of a point on the celestial sphere north or south of the celestial equator measured through 90 degrees. Declination is labeled with the prefix N (north) or S (south) to indicate the direction of measurement; prefixes are used to differentiate declination from latitude. The figure below depicts the declination of a star located 30 degrees off the celestial equator.

8. The Celestial Coordinate System • Hour angle:The celestial equivalent of longitude. It is the angular distance measured laterally along the celestial equator in a westerly direction through 360 degrees. • Sidereal hour angle (SHA): Hour angles measured in a westerly direction from the hour circle of Aries to the hour circle of a particular body. • For the purposes of celestial navigation, it is not only desirable to locate a body on the celestial sphere relative to Aries but also to locate a body relative to a given position on earth at a given time. To do this, two terrestrial meridians are projected onto the surface of the celestial sphere for use as references for hour angle measurements - The Greenwich meridian and the observer’s meridian. The celestial meridians thus projected are termed the Greenwich celestial meridianand the local celestial meridian.

9. The Celestial Coordinate System • Greenwich hour angles (GHA): Hour angles measured relative to the Greenwich meridian. • Local hour angles (LHA): Hour angles measured with respect to the local celestial meridian. • Both Greenwich hour angles and local hour angles are measured westward from a projected terrestrial meridian to a celestial hour circle moving ever westerly with the rotating celestial sphere. Consequently, both GHA and LHA values are constantly growing larger with time, increasing from 0 to 360 degrees once each 24 hours. They relate the rotating celestial sphere to the meridians of the earth. • Sidereal hour angles are measured between two hour circles on the celestial sphere; although the value of the SHA of the star changes with time as the stars move through space relative to one another, the rate of change is extremely slow. Hence for purposes of celestial navigation, sidereal hour angles are considered to remain constant.

10. The Celestial Coordinate System • The hour circle of Aries and the projected Greenwich and observer’s meridians are shown in the following figure. The resulting sidereal, Greenwich, and local hour angles (SHA, GHA, AND LHA) of the star at a given time are indicated. • It can be seen from the figure below that the GHA of the star (GHA ) is equal to the sum of the GHA of Aries (GHA ) plus the SHA of the star (SHA ): GHA = GHA + SHA = Aries = star

11. Pn Celestial Equator G LHA  GHA SHA GHA Ps (Overhead 16-2) The Celestial Coordinate System

12. The Celestial Coordinate System • For some applications in celestial navigation, it is advantageous to use an alternative angle to LHA to express the angular distance from the observer’s meridian to the hour circle of a body. This is called the meridian angle (t). The meridian angle is defined as the angular distance between 0 degrees and 180 degrees, measured at the pole nearest the observer, from the observer’s meridian either easterly or westerly to the hour circle of the body. The meridian angle is always labeled with the suffix E (east) or W (west) to indicate the direction of measurement. The significance of the meridian angle will be discussed later when solving the celestial triangle.

13. The Horizon Coordinate System • The Horizon coordinate system: In order to obtain a celestial line of position by observation of a celestial body, a third set of coordinates, called the horizon system, is required. This coordinate system differs from the celestial coordinate system in that it is based on the position of the observer, rather than on the projected terrestrial equator and poles. • Celestial horizon : A plane passing through the center of the earth perpendicular to a line passing through the observer’s position and the earth’s center. This reference plane corresponds with the plane of the equator in the terrestrial and celestial systems . ZENITH

14. The Horizon Coordinate System • Zenith: The line passing through the observer and the center of the earth perpendicular to the celestial horizon extended outward from the observer to the celestial sphere defines a point on the sphere directly over the observer called the observer’s zenith. • The observer's zenith is always exactly 90 degrees of arc above the celestial horizon. • Nadir: The extension of the line through the center of the earth and the observer to the opposite side of the celestial sphere defines a second point directly below the observer is called the observer’s nadir . • The observer’s zenith and nadir correspond to the terrestrial and celestial poles, while the zenith-nadir line connecting the observer’s zenith and nadir corresponds to the axis of the celestial and terrestrial spheres.

15. The Horizon Coordinate System • Vertical circle: A great circle on the celestial sphere passing through the observer’s zenith and nadir, perpendicular to the plane of the celestial horizon. It is the equivalent of a meridian in the terrestrial system and an hour circle in the celestial system. • Prime vertical: The vertical circle passing through the east and west points of the observer’s horizon. • Principal vertical: The vertical circle passing through the north and south points the observer’s horizon. It is always coincident with the projected terrestrial meridian (i.e. the local celestial meridian) passing through the observer’s position. • Altitude: The angular distance of a point on the celestial sphere above a designated reference horizon, measured along the vertical circle passing through the point. It is the horizon system’s equivalent of latitude.

16. The Horizon Coordinate System • The reference horizon for the horizon coordinate system is the celestial horizon of the observer, defined previously as the plane passing through the center of the earth perpendicular to the zenith-nadir line of the observer. • Observed Altitude (abbreviated HO): Altitude measured relative to the celestial horizon. It is the angle formed at the center of the earth between the line of sight of the body and the plane of the observer’s celestial horizon. • Visible or Sea Horizon The other horizon used as a reference for altitude measurements. It is the line along which the sea and sky appear to meet. In practice, sextant altitudes must be converted to observed altitudes (Ho) to obtain an accurate celestial LOP.

17. The Horizon Coordinate System • True Azimuth (abbreviated Zn): The horizontal angle measured along the celestial horizon in a clockwise direction from 0000T to 3600T from the principle vertical circle to the vertical circle passing through a given point or body on the celestial sphere. True azimuth can be thought of as the true bearing of a celestial body from the observer’s position. It is the equivalent of longitude in the horizon system • The figure below illustrates the horizon coordinate system.

18. Zenith Principal Vertical Pn Circle N Altitude Vertical True azimuth Celestial Horizon Nadir (Overhead 16-3) The Horizon Coordinate System

19. Homework • Chapter 15: Section 1- 2,5,6,7,9,11 Section 2- 1,2,5,6,7,8,9,10 • Handout of Tides/Current