1. LESSON 16: The Navigation Triangle Learning ObjectivesLearning Objectives –Comprehend the interrelationships of the terrestrial, celestial, and horizon coordinate systems in defining the navigation triangle. –Gain a working knowledge of the celestial and navigation triangles.

2. The Celestial Triangle The celestial, horizon, and terrestrial coordinate systems are combined on the celestial sphere to form the astronomical or celestial triangle.The celestial, horizon, and terrestrial coordinate systems are combined on the celestial sphere to form the astronomical or celestial triangle.

3. The Celestial Triangle

4. The three vertices of the triangle:The three vertices of the triangle: –observer’s zenith –position of the celestial body on the celestial sphere –celestial pole nearest the observer (referred to as the elevated pole) The Celestial Triangle

5. Two of the angles are also of concern:Two of the angles are also of concern: –meridian angle (t) –aximuth angle (Z) Meridian angle is simply a more convenient way of expressing LHAMeridian angle is simply a more convenient way of expressing LHA –if LHA<180 o, t=LHA (west) –if LHA>180 o, t=360 o -LHA (east)

6. The Celestial Triangle Likewise, azimuth angle is simply a more convenient way of expressing true azimuth (Z n )Likewise, azimuth angle is simply a more convenient way of expressing true azimuth (Z n ) The third angle is known as the parallactic angle and is not of use in our discussion.The third angle is known as the parallactic angle and is not of use in our discussion. Let’s take another look at the triangle….Let’s take another look at the triangle….

8. The Navigation Triangle When the celestial triangle is projected downward from the celestial sphere onto the earth’s surface, it becomes the navigation triangle.When the celestial triangle is projected downward from the celestial sphere onto the earth’s surface, it becomes the navigation triangle. The solution of this navigation triangle is the purpose of celestial navigation.The solution of this navigation triangle is the purpose of celestial navigation. Each of the three coordinate systems forms one side of the triangle.Each of the three coordinate systems forms one side of the triangle.

9. The Navigation Triangle

10. Now the vertices of the triangle areNow the vertices of the triangle are –our assumed position (AP) Corresponds to the observer’s zenith on the celestial triangleCorresponds to the observer’s zenith on the celestial triangle –the geographic position (GP) of the celestial body Corresponds to the star’s celestial positionCorresponds to the star’s celestial position –the earth’s pole (P n or P s ) Corresponds to the elevated poleCorresponds to the elevated pole

11. The Geographic Position (GP)

12. The Navigational Triangle The three sides of the triangle are used in determining the observer’s position on the earth. The length of each side is as follows:The three sides of the triangle are used in determining the observer’s position on the earth. The length of each side is as follows: –colatitude = 90 – latitude Connects observer’s position to pole’s positionConnects observer’s position to pole’s position –polar distance = 90 +/- declination Connects pole to the star’s geographic positionConnects pole to the star’s geographic position –coaltitude = 90 – altitude Connects star’s geographic position to observer’s positionConnects star’s geographic position to observer’s position

13. The Navigation Triangle Note that the polar distance may be greater than 90 degrees (if the GP and the elevated pole are on opposite sides of the equator) but coaltitude and colatitude are always less than 90 degrees.Note that the polar distance may be greater than 90 degrees (if the GP and the elevated pole are on opposite sides of the equator) but coaltitude and colatitude are always less than 90 degrees.

14. The Navigation Triangle The angles of the celestial and navigational triangles are the same:The angles of the celestial and navigational triangles are the same: –meridian angle (t) measured 0 o to 180 o, east or westmeasured 0 o to 180 o, east or west suffix E or W is used to indicate directionsuffix E or W is used to indicate direction –Azimuth angle (Z) measured 0 o to 180 omeasured 0 o to 180 o prefix N or S is used to indicate elevated poleprefix N or S is used to indicate elevated pole suffix E or W used to indicate on which side of the observer’s meridian the GP lies.suffix E or W used to indicate on which side of the observer’s meridian the GP lies.

15. Given:Given: LHA = 040 oLHA = 040 o Z = 110 o TZ = 110 o T Find: t Zn Consider this scenario:Consider this scenario: The Navigation Triangle

16. Solution Since the LHA<180 o, LHA and t are equivalent, thusSince the LHA<180 o, LHA and t are equivalent, thus t = LHA = 40 o W To determine Zn, it is usually helpful to draw a diagram, as shown on the next slide….To determine Zn, it is usually helpful to draw a diagram, as shown on the next slide….

17. Zn is the angle between the north pole and the GP, as seen by the observer.Zn is the angle between the north pole and the GP, as seen by the observer. Since Ps is the elevated pole and the GP is west of the observer, add 180 degrees to Z:Since Ps is the elevated pole and the GP is west of the observer, add 180 degrees to Z: Zn = S 290 o W

18. Only four possible combinations exist when you combineOnly four possible combinations exist when you combine –GP either east or west of the observer –elevated pole either north or south pole It is simple enough to come up with an equation for converting between Zn and Z for each case, or you can draw a picture as we just did.It is simple enough to come up with an equation for converting between Zn and Z for each case, or you can draw a picture as we just did. –Just draw the picture. LHA, t, Zn, and Z