Learning Objectives:

1
Lesson 18

• Learning Objectives
• Comprehend the concept of of the circle of equal
  altitude as a line of position.
• Become familiar with the concepts of the circle
  of equal altitude.
• Comprehend the altitude-intercept method of
  plotting a celestial LOP.
• Comprehend the use of two or more celestial LOPs
  derived by the altitude-intercept method in
  forming a celestial fix.
• Applicable reading Hobbs pp. 300-315.

2

• The circle of equal altitude To illustrate the
  basic concepts involved in obtaining a celestial
  line of position, suppose a steel pole
  perpendicular to a level surface were raised, and
  a wire stretched from its top to the surface,
  such that the angle formed by the wire and the
  surface were 60 degrees. If the end of the wire
  were rotated around the base of the pole, a
  circle would be described

(Overhead 18-1)
3

• At any point on this circle, the angle between
  the wire and the surface would be the same, 60
  degrees. Such a circle is termed a circle of
  equal altitude.
• If the end of the pole were extended to an
  infinite distance, the angle formed by the wire
  anywhere on the flat surface would approach 90
  degrees, since the wire would be nearly parallel
  with the pole. If the surface were spherical,
  however, and the measurement were made relative
  to a tangent plane, the angle would vary from 90
  degrees at the base of the pole to 0 degrees at
  all points on the spherical surface 90 degrees
  away from the location of the base

(Overhead 18-2)
4

• The figure below depicts two concentric circles
  of equal altitude inscribed on such a spherical
  surface, centered on the location of the base -
  the GP- of the pole. At all points on the
  circumference of the smaller circle, the angle
  formed by the wire and the tangent plane is 60
  degrees, while the angles measured along the
  larger circle are all 30 degrees.

5

• In celestial navigation, the situation is
  analogous to that in the preceding figure. For
  instance, suppose a celestial body were observed
  at altitude of 60 degrees above the observers
  celestial horizon, and its GP at the moment of
  observation were 100 South latitude and 300 West
  longitude. Additionally , the AP of the observer
  (determined by DR) were 100 North latitude, 100
  West longitude. A navigational triangle could
  be constructed by plotting the coordinates of the
  GP and AP

(Overhead 18-3)
6

• A circle of equal altitude about the GP of the
  body, from which circle an altitude of 60
  degrees, could be observed. To do this, assume
  that the observers assumed position coincided
  with their actual position. If this were the
  case, the radius of the circle of equal altitude
  would have the same length as the coaltitude of
  the observers navigational triangle expressed
  as an angle, this is 90 degrees (the observers
  zenith) minus 60 degrees (the altitude of the
  body), or 30 degrees. Since the coaltitude is a
  segment of a vertical circle that is itself a
  great circle, the important assumption that one
  degree of a great circle equals 60 nautical
  miles on the earths surface can be used to find
  the linear length of the coaltitude it is 3060
  1800 miles. Thus the circle of equal altitude
  for this altitude can be drawn by swinging an arc
  of radius 1800 miles about the GP of the body.
  This circle, depicted in the following diagram
  represents a locus of all points, including the
  observers actual position, from which it is
  possible to observe an altitude of 60 degrees for
  this body at the time of observation

7

• If the AP of the observer was in fact a small
  distance from his actual position, as is usually
  the case in practice, the AP will lie off the
  circle as shown in the following diagram

(Overhead 18-4)
8

• In this situation, the AP will probably lie off
  of the circle, either closer to or farther away
  from the GP of the body. Thus, if the
  observation of the body had been made from the
  AP, a different altitude than that observed and
  therefore a different coaltitude (radius of the
  circle of equal altitude) would have been
  obtained.
• To find the exact position of the observer, i.e.
  the celestial fix, the observer could observe a
  second celestial body and plot a second circle of
  equal altitude around its GP as shown below

9

• Because of the large scale chart that would be
  necessary to plot a celestial fix of meaningful
  accuracy, the celestial fix is not normally
  plotted in the manner shown previously.
• A modified method of plotting a celestial LOP has
  been devised wherein only a small portion of the
  coaltitude and circle of equal altitude for each
  body observed is plotted. This method is known as
  the altitude intercept method.

10

• The Altitude-Intercept Method The
  altitude-intercept method eliminates the
  disadvantages of the the circle of equal
  altitude. This method utilizes daily tabulated
  data in either of one of two almanacs, the
  Nautical Almanac or the Air Almanac, in
  conjunction with a sight reduction table to
  produce a computed altitude (Hc) to a body being
  observed from an assumed position (AP) of the
  observer. The computed altitude Hc is compared
  to the observed altitude Ho to determine the
  position of the celestial LOP.
• The altitude intercept method only plots a small
  segment of the radius of the circle of equal
  altitude.
• A line drawn from the AP of the observer toward
  the GP of the body in the direction of the true
  azimuth then represents a segment of the radius
  of the circle of equal altitude a small segment
  of the circumference of the circle is positioned
  along the true azimuth (radius) line by comparing
  the computed altitude with the observed altitude
  actually obtained.

(Overhead 18-5)
11

• As the distance between the ground position and
  the observers apparent position increases, the
  portion of the circumference plotted on the chart
  will appear increasingly less curved, finally
  approaching a straight line as the distance to
  the GP reaches a few hundred miles
• The LOP in the altitude-intercept method is
  always represented by a short, straight line.
  Since the radius of the the circle of equal
  altitude always lies in the direction of the GP,
  the LOP can be thought of as being drawn
  perpendicular to the bearing of the true azimuth
  (Zn) of the GP of the body. It is the
  positioning of the LOP along the true azimuth
  line that constitutes the basic problem solved by
  the altitude-intercept method.

12

• As previously mentioned, the computed altitude of
  a celestial body Hc can be determined for the
  assumed position of an observer AP using a
  nautical almanac in conjunction with a sight
  reduction table.
• If the observed altitude Ho were the same as the
  computed altitude Hc, the circles of equal
  altitude would be coincident and both would pass
  through the AP of the observer.
• If Ho were greater than Hc, the radius of the
  circle of equal altitude corresponding to Ho
  would be smaller than the radius of the circle
  for Hc. In this case the observer would be
  located closer to the GP of the body than the
  assumed position
• If Ho were less than Hc, the radius of the
  circle of equal altitude corresponding to Ho
  would be larger than the radius of the circle for
  Hc. The observer would be farther from the GP
  of the body than the assumed position

13

• The figure below illustrates the relationship
  between Ho and Hc

(Overhead 18-6)
14

• The distance from the observer to the AP can be
  calculated by finding the difference between Hc
  and Ho (hence the equation Hc-Hodistance between
  AP and observer).
• For every minute of arc difference, the intercept
  point of the LOP with the true azimuth line is
  moved one nautical mile from the AP, either
  toward the GP of the body if Ho is greater than
  Hc, or away from the GP if Ho is less than Hc.
• Memory aid
• Coast Guard Academy (Computed Greater Away)
• HoMoTo (Ho More than Hc, Toward)
• Intercept Distance The distance between the
  intercept point and the AP. It is symbolized by
  a lowercase a.

15

• Example The figure below illustrates a
  calculation of an intercept distance, in which an
  observed altitude is Ho is greater than the
  computed altitude Hc.

16

• The star was observed to have an altitude of
  45000. For the AP at the time of observation,
  an Hc of 44045.5was computed as well as the true
  azimuth of the GP from the AP. To plot the
  celestial LOP corresponding to the observed
  altitude, the intercept point must be advanced
  along the true azimuth toward the GP of the body
  (since Ho is greater than Hc). The intercept
  distance is given by the following computation
• Ho 450 00.0
• Hc 440 45.5
• a 14.5 T
• The computed intercept distance is labeled with a
  T to indicate that it must be moved toward the
  GP of the body. If the intercept distance were
  to be moved away, it would be labeled with an A.
• The LOP is plotted perpendicular to the true
  azimuth through a point 14.5 miles from the AP
  toward the GP.
• The observers position lies somewhere on the
  LOP. A second celestial LOP obtained at the same
  time would provide a celestial fix.
• The AP is always chosen in such a way that the
  intercept distance is kept fairly short. As the
  intercept distance increases the error in the
  calculation will increase.

17

• Plotting the celestial LOP
• The celestial LOP can be plotted on a fairly
  large scale chart, but the usual procedure is to
  use a scaled plotting sheet that is marked with
  the appropriate latitude for the area of
  interest.
• From the AP, a line is drawn in the direction of
  the true azimuth ,Zn, in the case of toward
  intercepts, or 180 degrees from Zn in the case of
  away intercepts. The length of the intercept
  is then laid off along this line, and a point is
  inscribed on the line to indicate the length of
  the intercept distance. To complete the plot,
  the LOP is drawn through this point,
  perpendicular to the intercept line.

18

• Example An observation of Venus for an assumed
  position of 34000 N and 163008.4 E gave an
  intercept of 14.8 miles and a true azimuth of
  095.10 T

(Overhead 18-8)