Learning Objectives:

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Lesson 17 The Navigational Triangle

• Learning Objectives
• Comprehend the interrelationships of the
  terrestrial, celestial, and horizon coordinate
  systems in defining the celestial and
  navigational triangles.
• Gain a working knowledge of the celestial and
  navigational triangles.
• Applicable reading Hobbs pp. 293-300.

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Navy Pilots...
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The Navigational Triangle

• The celestial triangle For the purposes of
  celestial navigation, the terrestrial, celestial,
  and horizon coordinate systems are combined on
  the celestial sphere to form the astronomical or
  celestial triangle. When the celestial triangle
  is related to the earth it becomes the
  navigational triangle. The solution of which is
  the basis of celestial navigation.

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The Navigational Triangle

• Each of the three coordinate systems examined in
  the celestial navigation section of this course
  is used to form one of the sides of the celestial
  triangle. As an illustration, consider the
  celestial triangle depicted below

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The Navigational Triangle

• The three vertices of the triangle are the
  celestial pole nearest the observer, the
  observers zenith, and the position of the
  celestial body.
• The side of the triangle connecting the celestial
  pole with the observers zenith is a segment of a
  projected terrestrial meridian the side between
  the pole and the celestial body is a segment of
  the hour circle of the body and the side of the
  triangle between the observers zenith and the
  position of the body is a segment of a vertical
  circle of the horizon system. The lengths of
  the sides of the celestial (later navigational)
  triangle are of paramount importance.
• The length of the side formed by the projected
  terrestrial meridian between the north celestial
  pole, Pn, and the observers zenith expressed as
  an angle is 90 degrees minus the observers
  latitude.
• The length of the side concurrent with the hour
  circle of the body is in this case 90o minus the
  declination, but if the body were south of the
  equator, this length would be 90o plus the
  declination.
• The length of the third side, which is measured
  along the vertical circle from the observers
  zenith to the body, is 90o(the altitude of the
  zenith) minus the altitude of the body.

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The Navigational Triangle

• Only two angles within the celestial triangle are
  of concern in celestial navigation
• Meridian Angle The angle measured east or west
  from the observers celestial meridian to the
  hour circle of the body. It is the angle marked
  in the figure at the north celestial pole. The
  meridian angle bears a close relationship to the
  local hour angle (LHA). If the LHA is less than
  180 degrees, the meridian angle t is equal to the
  LHA, and is west. If the LHA is greater than 180
  degrees, the meridian angle is equal to 360
  degrees minus the LHA, and is east.
• Azimuth Angle (abbreviated Z) The angle at the
  zenith between the projected celestial meridian
  of the observer and the vertical circle passing
  through the body it is measured from 0o to 1800
  either east or west of the observers meridian.
  Note It is important to distinguish this azimuth
  angle of the celestial triangle from the true
  azimuth of the observed body, which, as mentioned
  previously, can be likened to the true bearing of
  the body from the observer. Azimuth in the
  latter sense is always abbreviated Zn and with
  altitude forms the two horizon coordinates by
  which a celestial body is located with respect
  to an observer on the earths surface.

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The Navigational Triangle

• The third interior angle of the celestial
  triangle is called the parallactic angle it is
  not used in ordinary practice of celestial
  navigation.

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The Navigational Triangle

• The navigational triangle In the determination
  of position by celestial navigation, the
  celestial triangle must be solved to find a
  celestial line of position through the observers
  position beneath their zenith this is
  accomplished by the construction of a closely
  related navigational triangle. To form this
  triangle the observer is imagined to be located
  at the center of the earth, and the earths
  surface is expanded outward (or the celestial
  sphere compressed inward) until the surface of
  the earth and the surface of the celestial sphere
  are coincident.

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The Navigational Triangle

• Geographic Position (GP) After the earths
  surface has been expanded (or the celestial
  sphere compressed), the position of the celestial
  body being observed becomes a body on the
  earths surface.
• Every celestial body has a GP located on the
  earths surface directly beneath it
• As the celestial sphere rotates about the earth,
  all geographic positions of celestial bodies move
  from east to west across the earths surface.
  The GP of the sun is called the subsolar point,
  the GP of the moon is called the sublunar point,
  and the GP of a star the substellar point.
• In every case, the diameter of the body is
  considered to be compressed to a point on the
  celestial sphere, located at the center of the
  body. The GP of the observed celestial body
  forms one vertex of the navigational triangle.

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The Navigational Triangle

• Assumed Position (AP) The observers position on
  earth is not known, so the zenith of the observer
  in the celestial triangle becomes a hypothesized
  position of the observer. This assumed position,
  AP, is the second vertex of the navigational
  triangle.
• Elevated Pole, (abbreviated Pn or Ps) The
  remaining vertex of the navigational triangle.
  The elevated pole is the pole nearest the
  observers assumed position it is so named
  because it is the celestial pole above the
  observers celestial horizon. Thus, the assumed
  position of the observer and the elevated pole
  are always on the same side of the celestial
  equator, while the geographic position of the
  observed body may be on either side.

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The Navigational Triangle

• The three sides of the navigational triangle are
  called the colatitude, coaltitude, and the polar
  distance
• The colatitude is the side of the navigational
  triangle joining the AP of the observer and the
  elevated pole. Since the AP is always in the
  same hemisphere as the elevated pole, the length
  of the colatitude is always 90o (the latitude of
  the pole) minus the latitude of the AP.
• The coaltitude is the side of the navigational
  triangle joining the AP of the observer and the
  Gp of the body. Because the maximum possible
  altitude of any celestial body relative to the
  observers celestial horizon is 90o (the altitude
  of his zenith) the length of the coaltitude is
  always 900 minus the altitude of the body.
• The polar distance is the side of the
  navigational triangle joining the elevated pole
  and the GP of the body. For a body in the same
  hemisphere, the length of the polar distance is
  900 minus the declination of the GP for a body
  in an opposite hemisphere, is length is 900 plus
  the declination of the GP.

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The Navigational Triangle

• The interior angles of the navigational triangle
  bear the same names as the corresponding angles
  of the celestial triangle, with only the meridian
  angle, t, at the elevated pole and the azimuth
  angle, Z, at the AP of the observer being of any
  consequence in the solution of the triangle.
• The meridian angle is measured from 0 to 180
  degrees east or west from the observers
  celestial meridian to the hour circle of the body
  and is labeled with the suffix E (east) or W
  (west).
• The azimuth angle is always measured from the
  observers meridian toward the vertical circle
  joining the observers AP and the GP of the body.
  Since the angle between the observers meridian
  and the vertical circle can never exceed 180o,
  the azimuth angle must always have a value
  between 0o to 180o. It is labeled with the
  prefix N(north) or S(south) to agree with the
  elevated pole, and with the suffix E (east) or W
  (west) to indicate on which side of the
  observers meridian the GP lies. The suffix of
  the meridian angle and the azimuth angle Z will
  always be identical.

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The Navigational Triangle

• The azimuth angle must be converted to the true
  azimuth, or bearing, of the GP of the body from
  the AP of the observer for use in navigation.
• As an example of this conversion process,
  consider the navigational triangle shown below in
  which the south pole is the elevated pole

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The Navigational Triangle

• The south pole is the elevated pole because the
  AP of the observer is in the southern hemisphere.
  At the time of the observation of the body, its
  GP has been determined to be north of the equator
  and to the west of the observer. Thus the prefix
  for the azimuth angle, Z, is S (south), to agree
  with the elevated pole, and the suffix is W
  (west), identical with the suffix of the meridian
  angle. Hence, if the size of the azimuth angle
  were 110o,, the angle would be written S1100W.
  To convert this azimuth angle to a true azimuth,
  it is helpful to draw a sketch of the directional
  relationships involved

Ps (180T)
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The Navigational Triangle

• From the figure, we can see that to convert the
  azimuth angle S110oW to true azimuth, it is
  necessary only to add 180 degrees. Thus the true
  azimuth or bearing of the GP from the AP is case
  is 1801102900.
• By solution of the appropriate triangle,
  navigators can determine their position at sea,
  check compass accuracy, predict the rising and
  setting of any celestial body, and locate and
  identify bodies of interest.