Formulas For Solving Great Circle Sailing Problems
There are many formulas which can be used to solve the types of great circle sailing problems found on USCG license exams. It is very difficult to memorize all of them. Fortunately, you don't have to do this. But, I would suggest the formulas for determining great circle distance and initial course angle be memorized.
These formulas are:
For Distance (D) Cos D = Sin L1 X Sin L2 + Cos L1 X Cos L2 X Cos Dlo
For Initial Course Angle (C) Cos C = Distance must be found first in order to be used in the second equation.
The other formulas needed to solve USCG exam problems are found in Bowditch, Volume II, Chapter X, The Sailings. Remember, this book is available in the exam room so you can reference it for the formulas when you take your exam. Adapting the Bowditch formulas to a hand-held calculator requires basic level understanding of the relationships between the different functions of trigonometry (sine, cosine, tangent, cotangent, secant, and cosecant). This is because calculators only have keys for the Sine, Cosine, and Tangents of angles.
Secant (angle X) = 1 divided by cosine (angle X)
Cosecant (angle X) = 1 divided by sine (angle X)
Cotangent (angle X) = 1 divided by tangent (angle X)
Sine (angle X) = cos (90° - angle X)
Cos (angle X) = sine (90° - angle X)
Procedures For Solving Great Circle Sailing Problems
A hand-held calculator with trig functions and at least three memory spaces is a must for solving great circle sailing problems. There are long hand mathematical methods which can be used, such as with logarithms, but they are time consuming and you might make a simple math mistake. There are several different types of great circle sailing problems that you could be asked on your examination, but most are the type which ask for great circle distance (D) and initial course (Cn). The other types are:
1. Finding the latitude of the upper vertex (Lv).
2. Finding the longitude of the upper vertex (Lov).
3. Finding the distance of the upper vertex (Dv).
4. Determining the latitude and or longitude of point X (Lx and Lox) along the great circle track.
Note: Each USCG question should be read carefully because many have given more information than is needed to solve the problem. You need to have a format to go by. A format that you can use to tabulate values and record results. The use of a neat format for these problems will help you from making mistakes.
Great Circle Sailing Charts and Voyage Planning
Below is some information on gnomonic charts in planning voyages which incorporate great circle tracks for all or a portion of the voyage, and the procedures for answering typical questions found on USCG license exams which relate to gnomonic charts.
The Gnomonic Projection
The gnomonic chart projection is a geometrical projection in which the surface features of a certain portion of the earth are projected outward from the center of the earth to a tangent plane. There are three types of gnomonic charts depending on the location of the point of tangency. In equatorial gnomomic charts, the point where the plane is tangent is located at some point on the equator. A polar gnomonic chart is tangent at one of the poles. Oblique gnomonic charts have their point of tangency at some latitude and longitude between the equator and the poles. The land features of gnomonic charts become more distorted as the distance from the point of tangency increases. The most often used gnomonic charts are oblique charts with a point of tangency located in the center of an ocean basin. Gnomonic tracking charts WOXZC 5270 and WOXZC 5274, have their points of tangency in the North Pacific and North Atlantic Ocean basins.
The chief advantage of gnomonic charts is that a straight line between two points represents the great circle between the points. If you proceed by great circle, all you need to do is connect the points of departure and arrival with a straight line. The latitudes and longitudes of points at intervals along this line may be measured on the gnomonic chart and then transferred to a Mercator chart. When these points are connected with straight lines (rhumb lines), you have the "legs" of the great circle track. Once transferred to the Mercator chart, the course on each leg can be found and the sum of the distances along each leg gives the total great circle distance. The main disadvantages of the gnomomic projection are that courses, direction and distance can not be readily measured. For this reason gnomonic projections are not used for day to day navigation and plotting, and should be used as an aid to voyage planning.
Use of Gnomonic Charts for Track Planning
The use of gnomonic charts for voyage planning provides for quick checking of planned great circle tracks for intervening land, iceberg hazards, and other factors. Example, if you wished to determine the advantage of proceeding by great circle from Cape Flattery, at the entrance to Puget Sound, to the entrance to Tokyo Bay, Japan, you would draw a straight line between the two places on chart WOXZC 5270. Inspection of the chart would reveal that this track would require your vessel to pass through the Aleutian Chain twice, and part of the voyage would be in the Bering Sea. Knowledge of the average poor weather, violent storms, uncharted rocks, and fog in these northern waters makes the saving in distance of a great circle over a rhumb line less attractive.
The great circle chart can be a big help to plan composite sailing tracks. Composite sailing consists of proceeding from the point of departure until the great circle reaches a limiting latitude. You then proceed by rhumb line (parallel sailing) along that latitude to the point where a great circle from the point of arrival is tangent to the limiting latitude line. In a later blog I will show how you can solve Coast Guard Great Circle Sailing exam questions using Pub. 229.
Saturday, January 31, 2009
Thursday, January 29, 2009
Great Circle Sailing Definitions
Great Circle Definitions
Great Circle - the largest circle that can be placed around a sphere. On earth the primary great circles are the equator and the meridians infinite in number (1 minute of arc being equal to 1 nautical mile). On the celestial sphere there are celestial meridians and the equator, hour circles, and vertical circles that can be measured in arc only.
Declination - the celestial equivalent of latitude on earth.
Vernal Equinox - that point on the celestial equator (equinoctial) that the sun on its south to north transit crosses about March 21 in the establishment of the hour circle of Aries.
Equator - the great circle of the earth that lies midway between the extremities of the earth's axis. It is the baseline of latitude for north and south measurement.
Celestial Equator or Equinoctial - a projection from the earth's equator to the celestial sphere.
Greenwich Meridian - the meridian on earth that passes through the Greenwich Royal Observatory. It is the zero line of east / west direction.
Greenwich Celestial Meridian - a projection to the celestial sphere of the Greenwich meridian. It is the zero line of establishment of westerly direction.
Local Meridian - the meridian the observer on earth is on.
Local Celestial Meridian - a projection of the local meridian to the celestial sphere.
Hour Circle - a great circle on the celestial sphere that passes through the poles and a celestial body. It is in constant movement to circle the earth approximately once a day.
Vertical Circles - a great circle on the celestial sphere that passes through two established points (zenith and nadir) and a celestial body.
Hour Angles - angles measured from reference points of the Greenwich meridian, local meridian, and the hour circle of Aries in a westerly direction through 360° to another reference.
Meridian angle - the angle measured from the local meridian in an easterly or westerly direction through 180°.
Great Circle Sailing
The great circle distance between any two points on the assumed spherical surface of the earth and the initial great circle course angle can be found by relating the problems to the solution of the celestial triangle. By entering the tables with latitude of departure as latitude, latitude of destination as declination, and difference of longitude as LHA, the tabular altitude and azimuth angle can be extracted and converted to distance and course. The tabular azimuth angle becomes the initial great circle course angle, N or S for the latitude of departure, and E or W depending on the destination being east or west of point of departure.
If all the entering arguments are integral degrees, the altitude and azimuth angle are obtained directly from the tables without interpolation. If the latitude of destination is nonintegral, interpolation for the additional minutes of latitude is done as in correcting altitude for any declination increment, if either the latitude of departure or difference of longitude, or both, are nonintegral, the additional interpolation is done graphically. Since the latitude of destination becomes the declination entry, and all declinations appear on every page the great circle solution can always be extracted from the volume which covers the latitude of departure.
Great Circle solutions are in one of the four following cases:
Case 1 - Latitudes of departure and destination of same name and initial great circle distance less than 90°. Enter the tables with latitude of departure as latitude argument (Same Name), latitude of destination as declination argument, and difference of longitude as local hour angle argument. If the respondents as found on a right-hand page do not lie below the C-S Line,
Case III is applicable. Extract the tabular altitude which subtracted from 90° is the desired great circle distance. The tabular azimuth angle is the initial great circle course angle.
Case II - Latitudes of departure and destination of contrary name and great circle distance less than 90°. Enter the tables with latitude of departure as latitude argument (Contrary Name) and latitude of destination as declination argument, and with the difference of longitude as local hour angle argument. If the respondents do not lie above the C-S Line on the right-hand page, Case IV is applicable. Extract the tabular altitude which subtracted from 90° is the desired great-circle distance. The tabular azimuth angle is the initial great circle course angle.
Case III - Latitudes of departure and destination of same name and great circle distance greater than 90°. Enter the tables with latitude of departure as latitude argument (Same Name), latitude of destination as declination argument, and difference of longitude as local hour angle argument. If the respondents as found on a right-hand page do not lie above the C-S Line, Case I is applicable. Extract the tabular altitude which added to 90° gives the desired great-circle distance. The initial great-circle course angle is 180° minus the tabular azimuth angle.
Case IV - Latitudes of departure and destination of contrary name and great circle distance greater than 90°. Enter the tables with latitude of departure as latitude argument (Contrary Name), latitude of destination as declination argument, and difference of longitude as local hour angle argument. If the respondents as found on a right-hand page do not lie below the C-S Line, Case II is applicable. If the DLo is in in excess of 90°, the respondents are found on the facing left-hand page. Extract the tabular altitude which added to 90° gives the desired great circle distance. The initial great circle course angle is 180° minus the tabular azimuth angle.
Points along a Great Circle
If the latitude of the point of departure and the initial great circle course angle are integral degrees, points along the great circle are found by entering the tables with the latitude of departure as the latitude argument (Same Name), the initial great circle course angle as the LHA rgument, and 90° minus distance to a point on the great circle as the declination argument. The latitude the point on the great circle and the difference of longitude between that point and the point of departure are the tabular altitude and azimuth angle respondents.
Spherical Triangle Solutions
Of the six parts of the spherical astronomical triangle, these tables utilize three as entering arguments and tabulate two as respondents. The only remaining part of the triangle is the position angle, which is the angle between a body's hour circle and its vertical circle. The position angle can be found through simple interchange of arguments, effecting a complete solution.
The instructions are:
(a) When latitude and declination are of same name, enter the tables with the appropriate local hour angle, with the declination as latitude argument of the same name and the latitude as declination argument, and extract the tabular azimuth angle as the parallactic angle.
(b) When latitude and declination are of contrary name, enter the tables with the appropriate local hour angle and with the declination as latitude argument of contrary name and the latitude as declination argument, the tabular azimuth angle is then the supplement of the parallactic angle which equals 180° minus the azimuth angle). This method generally requires all the volumes of the series. An approximate value of the parallactic angle X, accurate enough for most navigational requirements, can be calculated from the formula, cos X = d / 60', where d is the difference between successive tabular altitudes for the desired latitude, local hour angle and declination.
Within the limitations of the tabular precision and interval, the tabular data of these tables include the solution of any spherical triangle, given two sides and the included angle. When using the tables for the general solution of the spherical triangle, the use of latitude, declination, and altitude in the tables instead of their corresponding parts of the astronomical triangle should be kept in mind. In general if any three parts of a spherical triangle are given, these tables can be used to find the remaining parts this will sometimes mean searching through the volumes to find, for example, the altitude in a particular latitude and a given LHA in order to find the corresponding azimuth angle and declination.
Great Circle - the largest circle that can be placed around a sphere. On earth the primary great circles are the equator and the meridians infinite in number (1 minute of arc being equal to 1 nautical mile). On the celestial sphere there are celestial meridians and the equator, hour circles, and vertical circles that can be measured in arc only.
Declination - the celestial equivalent of latitude on earth.
Vernal Equinox - that point on the celestial equator (equinoctial) that the sun on its south to north transit crosses about March 21 in the establishment of the hour circle of Aries.
Equator - the great circle of the earth that lies midway between the extremities of the earth's axis. It is the baseline of latitude for north and south measurement.
Celestial Equator or Equinoctial - a projection from the earth's equator to the celestial sphere.
Greenwich Meridian - the meridian on earth that passes through the Greenwich Royal Observatory. It is the zero line of east / west direction.
Greenwich Celestial Meridian - a projection to the celestial sphere of the Greenwich meridian. It is the zero line of establishment of westerly direction.
Local Meridian - the meridian the observer on earth is on.
Local Celestial Meridian - a projection of the local meridian to the celestial sphere.
Hour Circle - a great circle on the celestial sphere that passes through the poles and a celestial body. It is in constant movement to circle the earth approximately once a day.
Vertical Circles - a great circle on the celestial sphere that passes through two established points (zenith and nadir) and a celestial body.
Hour Angles - angles measured from reference points of the Greenwich meridian, local meridian, and the hour circle of Aries in a westerly direction through 360° to another reference.
Meridian angle - the angle measured from the local meridian in an easterly or westerly direction through 180°.
Great Circle Sailing
The great circle distance between any two points on the assumed spherical surface of the earth and the initial great circle course angle can be found by relating the problems to the solution of the celestial triangle. By entering the tables with latitude of departure as latitude, latitude of destination as declination, and difference of longitude as LHA, the tabular altitude and azimuth angle can be extracted and converted to distance and course. The tabular azimuth angle becomes the initial great circle course angle, N or S for the latitude of departure, and E or W depending on the destination being east or west of point of departure.
If all the entering arguments are integral degrees, the altitude and azimuth angle are obtained directly from the tables without interpolation. If the latitude of destination is nonintegral, interpolation for the additional minutes of latitude is done as in correcting altitude for any declination increment, if either the latitude of departure or difference of longitude, or both, are nonintegral, the additional interpolation is done graphically. Since the latitude of destination becomes the declination entry, and all declinations appear on every page the great circle solution can always be extracted from the volume which covers the latitude of departure.
Great Circle solutions are in one of the four following cases:
Case 1 - Latitudes of departure and destination of same name and initial great circle distance less than 90°. Enter the tables with latitude of departure as latitude argument (Same Name), latitude of destination as declination argument, and difference of longitude as local hour angle argument. If the respondents as found on a right-hand page do not lie below the C-S Line,
Case III is applicable. Extract the tabular altitude which subtracted from 90° is the desired great circle distance. The tabular azimuth angle is the initial great circle course angle.
Case II - Latitudes of departure and destination of contrary name and great circle distance less than 90°. Enter the tables with latitude of departure as latitude argument (Contrary Name) and latitude of destination as declination argument, and with the difference of longitude as local hour angle argument. If the respondents do not lie above the C-S Line on the right-hand page, Case IV is applicable. Extract the tabular altitude which subtracted from 90° is the desired great-circle distance. The tabular azimuth angle is the initial great circle course angle.
Case III - Latitudes of departure and destination of same name and great circle distance greater than 90°. Enter the tables with latitude of departure as latitude argument (Same Name), latitude of destination as declination argument, and difference of longitude as local hour angle argument. If the respondents as found on a right-hand page do not lie above the C-S Line, Case I is applicable. Extract the tabular altitude which added to 90° gives the desired great-circle distance. The initial great-circle course angle is 180° minus the tabular azimuth angle.
Case IV - Latitudes of departure and destination of contrary name and great circle distance greater than 90°. Enter the tables with latitude of departure as latitude argument (Contrary Name), latitude of destination as declination argument, and difference of longitude as local hour angle argument. If the respondents as found on a right-hand page do not lie below the C-S Line, Case II is applicable. If the DLo is in in excess of 90°, the respondents are found on the facing left-hand page. Extract the tabular altitude which added to 90° gives the desired great circle distance. The initial great circle course angle is 180° minus the tabular azimuth angle.
Points along a Great Circle
If the latitude of the point of departure and the initial great circle course angle are integral degrees, points along the great circle are found by entering the tables with the latitude of departure as the latitude argument (Same Name), the initial great circle course angle as the LHA rgument, and 90° minus distance to a point on the great circle as the declination argument. The latitude the point on the great circle and the difference of longitude between that point and the point of departure are the tabular altitude and azimuth angle respondents.
Spherical Triangle Solutions
Of the six parts of the spherical astronomical triangle, these tables utilize three as entering arguments and tabulate two as respondents. The only remaining part of the triangle is the position angle, which is the angle between a body's hour circle and its vertical circle. The position angle can be found through simple interchange of arguments, effecting a complete solution.
The instructions are:
(a) When latitude and declination are of same name, enter the tables with the appropriate local hour angle, with the declination as latitude argument of the same name and the latitude as declination argument, and extract the tabular azimuth angle as the parallactic angle.
(b) When latitude and declination are of contrary name, enter the tables with the appropriate local hour angle and with the declination as latitude argument of contrary name and the latitude as declination argument, the tabular azimuth angle is then the supplement of the parallactic angle which equals 180° minus the azimuth angle). This method generally requires all the volumes of the series. An approximate value of the parallactic angle X, accurate enough for most navigational requirements, can be calculated from the formula, cos X = d / 60', where d is the difference between successive tabular altitudes for the desired latitude, local hour angle and declination.
Within the limitations of the tabular precision and interval, the tabular data of these tables include the solution of any spherical triangle, given two sides and the included angle. When using the tables for the general solution of the spherical triangle, the use of latitude, declination, and altitude in the tables instead of their corresponding parts of the astronomical triangle should be kept in mind. In general if any three parts of a spherical triangle are given, these tables can be used to find the remaining parts this will sometimes mean searching through the volumes to find, for example, the altitude in a particular latitude and a given LHA in order to find the corresponding azimuth angle and declination.
Sunday, January 25, 2009
Celestial Navigation Information
Celestial navigation is a position fixing technique that was devised to help sailors cross the oceans without having to rely on dead reckoning to enable them to find land. Celestial navigation uses angular measurements (sights) between the horizon and a celestial object. You can use the Sun, Moon, Planets or one of the 57 navigational stars whose coordinates are tabulated in the nautical almanac.
How it Works
Celestial navigation is used to measure angles between celestial objects in the sky and the horizon to locate your position. At any given time, any celestial object the Sun, Moon, Planets, and Stars will be located directly over a particular geographic position on the Earth. Your location (latitude and longitude) can be found by referring to tables in the nautical almanac. The measured angle between the celestial object and the horizon is used to define a circle on the surface of the Earth called a celestial line of position (LOP). The size and location of this circular line of position can be found using mathematical or graphical methods. The LOP is significant because the celestial object would be observed to be at the same angle above the horizon from any point along its circumference at that instant.
Angular Measurement
Using a marine sextant to measure the altitude of the sun above the horizon has developed over the years. One method is to hold your hand above the horizon with your arm stretched out. The width of a finger is an angle just over 1.5 degrees. The need for more accurate measurements led to the development of a number of increasingly accurate instruments, including the kamal, astrolabe, octant and the sextant. The sextant and octant are more accurate because they measure angles from the horizon, eliminating errors caused by the placement of an instrument's pointers, and because their dual mirror system cancels relative motions of the instrument, showing a steady view of the object and horizon. Navigators measure distance on the globe in degrees, arc minutes and arc seconds. A nautical mile is defined as 1852 meters, it is also one minute of angle along a meridian on the earth. Sextants can be read accurately to within 0.2 arc minutes. So the observer's position can be determined within (theoretically) 0.2 miles, about 400 yards (370 m). Most ocean navigators, shooting from a moving platform, can achieve a practical accuracy of 1.5 miles (2.8 km), enough to navigate safely when out of sight of land.
Practical Navigation
Practical celestial navigation usually requires a marine chronometer to measure time, a sextant to measure the angles, an almanac giving the coordinates of celestial objects, a set of sight reduction tables to compute the height and azimuth, and a chart of your area. With sight reduction tables, the only math required is addition and subtraction. Most people can master simpler celestial navigation procedures after a day or two of instruction and practice, even using manual calculation methods. Modern practical navigators usually use celestial navigation in combination with satellite navigation to correct a dead reckoning track, that is, a course estimated from a vessel's position, angle and speed. Using multiple methods helps the navigator detect errors, and simplifies procedures. When used this way, a navigator will from time to time measure the sun's altitude with a sextant, then compare that with a precalculated altitude based on the exact time and estimated position of the observation. On the chart, you can use the straight edge of a plotter to mark each position line. If the position line shows you to be more than a few miles from the estimated position, you can take more observations to restart the dead-reckoning track. In the event of equipment or electrical failure, one can get to a port by simply taking sun lines a few times a day and advancing them by dead reckoning to get a crude running fix.
Latitude
Latitude was measured in the past either at noon (the "noon sight") or from Polaris, the north star. Polaris always stays within about 1 degree of celestial north pole. If a navigator measures the angle to Polaris and finds it to be 10 degrees from the horizon, then you are on a circle at about North 10 degrees of the geographic latitude. Angles are measured from the horizon because locating the point directly overhead, the zenith hard to do. When haze obscures the horizon, navigators use artificial horizons, which are bubble levels reflected into a sextant. Latitude can also be determined by the direction in which the stars travel over time. If the stars rise out of the east and travel straight up you are at the equator, but if they drift south you are to the north of the equator. The same is true of the day-to-day drift of the stars due to the movement of the Earth in orbit around the Sun; each day a star will drift approximately one degree. In either case if the drift can be measured accurately, simple trigonometry will reveal the latitude.
Longitude
Longitude can be measured in the same way. If you can accurately measure the angle to Polaris, a similar measurement to a star near the eastern or western horizons will provide the longitude. The problem is that the Earth turns about 15 degrees per hour, making the measurements dependent on time. A measure only a few minutes before or after the same measure the day before creates navigation errors. Before good chronometers were available, longitude measurements were based on the transit of the moon, or the positions of the moons of Jupiter. For the most part, these are too difficult to be used by anyone except professional astronomers.
Use of Time
The most popular method was, and still is to use an accurate timepiece to directly measure the time of a sextant sight. The need for accurate navigation led to the development of progressively more accurate chronometers in the 18th century. Today, time is measured with a chronometer, a quartz watch, a shortwave radio time signal broadcast from an atomic clock, or the time displayed on a GPS. A quartz wristwatch normally keeps time within a half-second per day. If it is worn constantly, keeping it near body heat, its rate of drift can be measured with the radio, and by compensating for this drift, a navigator can keep time to better than a second per month. Traditionally, three chronometers were kept in gimbals in a dry room near the center of the ship. They were used to set a watch for the actual sight, so that no chronometers were never risked to the wind and salt water on deck. Winding the chronometers was the duty of the navigator, logged as "chron. wound." for checking by line officers. Navigators also set the ship's clocks and calendar.
Modern Celestial Navigation
The celestial line of position was discovered in 1837 by Thomas Hubbard Sumner when, after one observation he computed and plotted his longitude at more than one trial latitude in his vicinity and noticed that the positions lay along a line. Using this method with two bodies, navigators were finally able cross two position lines and obtain their position in effect determining both latitude and longitude. Later in the 19th century came the development of the modern (Marcq St. Hilaire) intercept method, with this method the body height and azimuth are calculated for a convenient position, and compared with the observed height. The difference in arc minutes is the nautical mile "intercept" distance that the position line needs to be shifted toward or away. Two other methods of reducing sights are the longitude by chronometer and the ex-meridian method.
Celestial navigation is becoming increasingly redundant with the advent of inexpensive and highly accurate satellite navigation receivers (GPS), it was used extensively in aviation until 1960s, and marine navigation until recently. But since a prudent mariner never relies on any sole means of fixing his / her position, many national maritime authorities still require deck officers to show knowledge of celestial navigation in examinations, primarily as a back up for electronic navigation. One of the most common current usages of celestial navigation aboard large merchant vessels is for compass calibration and error checking at sea when no terrestrial references are available.
How it Works
Celestial navigation is used to measure angles between celestial objects in the sky and the horizon to locate your position. At any given time, any celestial object the Sun, Moon, Planets, and Stars will be located directly over a particular geographic position on the Earth. Your location (latitude and longitude) can be found by referring to tables in the nautical almanac. The measured angle between the celestial object and the horizon is used to define a circle on the surface of the Earth called a celestial line of position (LOP). The size and location of this circular line of position can be found using mathematical or graphical methods. The LOP is significant because the celestial object would be observed to be at the same angle above the horizon from any point along its circumference at that instant.
Angular Measurement
Using a marine sextant to measure the altitude of the sun above the horizon has developed over the years. One method is to hold your hand above the horizon with your arm stretched out. The width of a finger is an angle just over 1.5 degrees. The need for more accurate measurements led to the development of a number of increasingly accurate instruments, including the kamal, astrolabe, octant and the sextant. The sextant and octant are more accurate because they measure angles from the horizon, eliminating errors caused by the placement of an instrument's pointers, and because their dual mirror system cancels relative motions of the instrument, showing a steady view of the object and horizon. Navigators measure distance on the globe in degrees, arc minutes and arc seconds. A nautical mile is defined as 1852 meters, it is also one minute of angle along a meridian on the earth. Sextants can be read accurately to within 0.2 arc minutes. So the observer's position can be determined within (theoretically) 0.2 miles, about 400 yards (370 m). Most ocean navigators, shooting from a moving platform, can achieve a practical accuracy of 1.5 miles (2.8 km), enough to navigate safely when out of sight of land.
Practical Navigation
Practical celestial navigation usually requires a marine chronometer to measure time, a sextant to measure the angles, an almanac giving the coordinates of celestial objects, a set of sight reduction tables to compute the height and azimuth, and a chart of your area. With sight reduction tables, the only math required is addition and subtraction. Most people can master simpler celestial navigation procedures after a day or two of instruction and practice, even using manual calculation methods. Modern practical navigators usually use celestial navigation in combination with satellite navigation to correct a dead reckoning track, that is, a course estimated from a vessel's position, angle and speed. Using multiple methods helps the navigator detect errors, and simplifies procedures. When used this way, a navigator will from time to time measure the sun's altitude with a sextant, then compare that with a precalculated altitude based on the exact time and estimated position of the observation. On the chart, you can use the straight edge of a plotter to mark each position line. If the position line shows you to be more than a few miles from the estimated position, you can take more observations to restart the dead-reckoning track. In the event of equipment or electrical failure, one can get to a port by simply taking sun lines a few times a day and advancing them by dead reckoning to get a crude running fix.
Latitude
Latitude was measured in the past either at noon (the "noon sight") or from Polaris, the north star. Polaris always stays within about 1 degree of celestial north pole. If a navigator measures the angle to Polaris and finds it to be 10 degrees from the horizon, then you are on a circle at about North 10 degrees of the geographic latitude. Angles are measured from the horizon because locating the point directly overhead, the zenith hard to do. When haze obscures the horizon, navigators use artificial horizons, which are bubble levels reflected into a sextant. Latitude can also be determined by the direction in which the stars travel over time. If the stars rise out of the east and travel straight up you are at the equator, but if they drift south you are to the north of the equator. The same is true of the day-to-day drift of the stars due to the movement of the Earth in orbit around the Sun; each day a star will drift approximately one degree. In either case if the drift can be measured accurately, simple trigonometry will reveal the latitude.
Longitude
Longitude can be measured in the same way. If you can accurately measure the angle to Polaris, a similar measurement to a star near the eastern or western horizons will provide the longitude. The problem is that the Earth turns about 15 degrees per hour, making the measurements dependent on time. A measure only a few minutes before or after the same measure the day before creates navigation errors. Before good chronometers were available, longitude measurements were based on the transit of the moon, or the positions of the moons of Jupiter. For the most part, these are too difficult to be used by anyone except professional astronomers.
Use of Time
The most popular method was, and still is to use an accurate timepiece to directly measure the time of a sextant sight. The need for accurate navigation led to the development of progressively more accurate chronometers in the 18th century. Today, time is measured with a chronometer, a quartz watch, a shortwave radio time signal broadcast from an atomic clock, or the time displayed on a GPS. A quartz wristwatch normally keeps time within a half-second per day. If it is worn constantly, keeping it near body heat, its rate of drift can be measured with the radio, and by compensating for this drift, a navigator can keep time to better than a second per month. Traditionally, three chronometers were kept in gimbals in a dry room near the center of the ship. They were used to set a watch for the actual sight, so that no chronometers were never risked to the wind and salt water on deck. Winding the chronometers was the duty of the navigator, logged as "chron. wound." for checking by line officers. Navigators also set the ship's clocks and calendar.
Modern Celestial Navigation
The celestial line of position was discovered in 1837 by Thomas Hubbard Sumner when, after one observation he computed and plotted his longitude at more than one trial latitude in his vicinity and noticed that the positions lay along a line. Using this method with two bodies, navigators were finally able cross two position lines and obtain their position in effect determining both latitude and longitude. Later in the 19th century came the development of the modern (Marcq St. Hilaire) intercept method, with this method the body height and azimuth are calculated for a convenient position, and compared with the observed height. The difference in arc minutes is the nautical mile "intercept" distance that the position line needs to be shifted toward or away. Two other methods of reducing sights are the longitude by chronometer and the ex-meridian method.
Celestial navigation is becoming increasingly redundant with the advent of inexpensive and highly accurate satellite navigation receivers (GPS), it was used extensively in aviation until 1960s, and marine navigation until recently. But since a prudent mariner never relies on any sole means of fixing his / her position, many national maritime authorities still require deck officers to show knowledge of celestial navigation in examinations, primarily as a back up for electronic navigation. One of the most common current usages of celestial navigation aboard large merchant vessels is for compass calibration and error checking at sea when no terrestrial references are available.
Friday, January 23, 2009
Celestial Navigation Definitions
Altitude - the arc of a vertical circle between the horizon and a point or body on the celestial sphere. Altitude as measured by a sextant is called sextant altitude (hs). Sextant altitude corrected only for inaccuracies in the reading (instrument, index, and personal errors, as applicable) and inaccuracies in the reference level (principally dip) is called apparent altitude (ha). After all corrections are applied, it is called corrected sextant altitude or observed altitude (Ho). An altitude taken directly from a table is called a tabular or tabulated altitude (ht). Tabular altitude as interpolated for declination, latitude, and LHA increments as is called computed altitude (Hc).
Altitude Difference (d) - the first difference between successive tabulations of altitude in a latitude column of these tables.
Argument - one of the values used for entering a table or diagram.
Assumed Latitude (aL), Assumed Longitude (aʎ - geographical coordinates assumed to facilitate sight reduction.
Assumed Position (AP) - a point at which an observer is assumed to be located.
Azimuth (Zn) - the horizontal direction of a celestial body or point from a terrestrial point, the arc of the horizon, or the angle at the zenith, between the north part of the celestial meridian or principal vertical circle and a vertical circle through the body or point, measured from 000° at the north part of the principal vertical circle clockwise through 360°.
Azimuth Angle (Z) - the arc of the horizon, or the angle at the zenith, between the north part or south part of the celestial meridian, according to the elevated pole, and a vertical circle through the body or point, measured from 0° at the north or south reference eastward or westward through 180° according to whether the body is east or west of the local meridian. It is prefixed N or S to agree with the latitude and suffixed E or W to agree with the meridian angle.
Celestial Equator - the primary great circle of the celestial sphere, everywhere 90° from the celestial poles, the intersection of the extended plane of the equator and the celestial sphere. Also called EQUINOCTIAL.
Celestial Horizon - that circle of the celestial sphere formed by the intersection of the celestial sphere and a plane through the center of the Earth and perpendicular to zenith-nadir line.
Celestial Meridian - on the celestial sphere, a great circle through the celestial poles and the zenith. The expression usually refers to the upper branch, that half from pole to pole which passes through the zenith.
Course Angle - course measured from 0° at the reference direction clockwise or counterclockwise through 180°. It is labeled with the reference direction as a prefix and the direction of measurement from the reference direction as a suffix. Which would be course angle S21°E is 21 ° east of south, or true course 159°.
Course Line - the graphic representation of a ship's course.
Declination (Dec.) - angular distance north or south of the celestial equator, the arc of an hour circle between the celestial equator and a point on the celestial sphere, measured northward or southward from the celestial equator through 90°, and labeled N or S (+ or -) to indicate the direction of measurement.
Declination Increment (Dec. Inc.) - in sight reduction, the excess of the actual declination of a celestial body over the integral declination argument.
Double-Second Difference (DSD) - the sum of successive second differences. Because second differences are not tabulated in these tables, the DSD can be formed by subtracting, algebraically, the first difference immediately above the tabular altitude difference (d) corresponding to the entering arguments from the first difference immediately below. The result will always be a negative value.
Ecliptic - the apparent annual path of the Sun among the stars, the intersection of the plane of the Earth's orbit with the celestial sphere. This is a great circle of the celestial sphere inclined at an angle of about 23°27' to the celestial equator.
Elevalated Pole (Pn or Ps) - the celestial pole above the observer's horizon, agreeing in name with the observer's latitude.
First Difference - the difference between successive tabulations of a quantity.
First Point of Aries - that point of intersection of the ecliptic and the celestial equator occupied by the Sun as it changes from south to north declination on or about March 21. Also called VERNAL EQUINOX.
Geographical Position (GP) - the point where a line drawn from a celestial body to the Earth's center passes through the Earth's surface.
Great Circle - the intersection of a sphere and a plane through its center.
Great Circle Course - the direction of the great circle through the point of departure and the destination, expressed as angular distance from a reference direction, usually north, to the direction of the great circle. The angle varies from point to point along the great circle. At the point of departure it is called INITIAL GREAT CIRCLE COURSE.
Greenwich Hour Angle (GHA) - angular distance west of the Greenwich celestial meridian; the arc or the celestial equator, or the angle at the celestial pole, between the upper branch of the Greenwich celestial meridian and the hour circle of a point on the celestial sphere, measured westward from the Greenwich celestial meridian through 360°.
Hour Circle - on the celestial sphere, a great circle through the celestial poles and a celestial body or the vernal equinox. Hour circles are perpendicular to the celestial equator.
Intercept (a) - the difference in minutes of arc between the computed and observed altitudes (corrected sextant altitudes). It is labeled T (toward) or A (away) as the observed altitude is greater or smaller than the computed altitude, Hc greater than Ho, intercept is away (A), Ho greater than Hc, intercept is toward (T).
Line of Position (LOP) - a line indicating a series of possible positions of a craft, determined by observation or measurement.
Local Hour Angle (LHA) - angular distance west of the local celestial meridian; the arc of the celestial equator, or the angle at the celestial pole., between the upper branch of the local celestial meridian and the hour circle of a celestial body or point on the celestial sphere, measured westward from the local celestial meridian through 360°
Meridian Angle (t) - angular distance east or west of the local celestial meridian; the arc of the celestial equator, or the angle at the celestial pole, between the upper branch of the local celestial meridian and the hour circle of a celestial body, measured eastward or westward from the local celestial meridian through 180°, and labeled E or W to indicate the direction of measurement.
Nadir (Na) - that point on the celestial sphere 180° from the observer's zenith.
Name - the labels N and S which are attached to latitude and declination are said to be of the same name when they are both N or S and contrary name when one is N and the other is S.
Navigational Triangle - the spherical triangle solved in computing altitude and azimuth and greatcircle sailing problems. The celestial triangle is formed on the celestial sphere by the great circles connecting the elevated pole, zenith of the assumed position of the observer, and a celestial body. The terrestrial triangle is formed on the Earth by the great circles connecting the pole and two places on the Earth: the assumed position of the observer and geographical position of the body for celestial observations, and the point of departure and destination for great circle sailing problems. The term astronomical triangle applies to either the celestial or terrestrial triangle used for solving celestial observations.
Polar Distance (p) - angular distance from a celestial pole the arc of an hour circle between a celestial pole, usually the elevated pole, and a point on the celestial sphere, measured from the celestial pole through 180°.
Prime Meridian - the meridian of longitude 0°, used as the origin for measurement of longitude. Prime Vertical-the vertical circle through the east and west points of the horizon.
Prime Vertical - the vertical circle through the east and west points of the horizon.
Principal Vertical Circle - the vertical circle through the north and south points of the horizon, coinciding with the celestial meridian.
Respondent - the vaiue in a table or diagram corresponding to the entering arguments.
Second Difference-the difference between successive first differences.
Second Difference - the difference between successive first differences.
Sidereal Hour Angle (SHA) - angular distance west of the vernal equinox; the arc of the celestial equator, or the angle at the celestial pole, between the hour circle of the vernal equinox and the hour circle of a point on the celestial sphere, measured westward from the hour circle of the vernal equinox through 360°.
Sight Reduction - the process of deriving from a sight (observation of the altitude, and sometimes the azimuth, of a celestial body) the information needed for establishing a line of position.
SmalI Circle - the intersection of a sphere and a plane which does not pass through its center.
Vertical Circle - on the celestial sphere, a great circle through the zenith and nadir. Vertical circles are perpendicular to the horizon.
Zenith (Z) - that point on the celestial sphere vertically overhead.
Zenith Distance (z) - angular distance from the zenith, the arc of a vertical circle between the zenith and a point on the celestial sphere.
Altitude Difference (d) - the first difference between successive tabulations of altitude in a latitude column of these tables.
Argument - one of the values used for entering a table or diagram.
Assumed Latitude (aL), Assumed Longitude (aʎ - geographical coordinates assumed to facilitate sight reduction.
Assumed Position (AP) - a point at which an observer is assumed to be located.
Azimuth (Zn) - the horizontal direction of a celestial body or point from a terrestrial point, the arc of the horizon, or the angle at the zenith, between the north part of the celestial meridian or principal vertical circle and a vertical circle through the body or point, measured from 000° at the north part of the principal vertical circle clockwise through 360°.
Azimuth Angle (Z) - the arc of the horizon, or the angle at the zenith, between the north part or south part of the celestial meridian, according to the elevated pole, and a vertical circle through the body or point, measured from 0° at the north or south reference eastward or westward through 180° according to whether the body is east or west of the local meridian. It is prefixed N or S to agree with the latitude and suffixed E or W to agree with the meridian angle.
Celestial Equator - the primary great circle of the celestial sphere, everywhere 90° from the celestial poles, the intersection of the extended plane of the equator and the celestial sphere. Also called EQUINOCTIAL.
Celestial Horizon - that circle of the celestial sphere formed by the intersection of the celestial sphere and a plane through the center of the Earth and perpendicular to zenith-nadir line.
Celestial Meridian - on the celestial sphere, a great circle through the celestial poles and the zenith. The expression usually refers to the upper branch, that half from pole to pole which passes through the zenith.
Course Angle - course measured from 0° at the reference direction clockwise or counterclockwise through 180°. It is labeled with the reference direction as a prefix and the direction of measurement from the reference direction as a suffix. Which would be course angle S21°E is 21 ° east of south, or true course 159°.
Course Line - the graphic representation of a ship's course.
Declination (Dec.) - angular distance north or south of the celestial equator, the arc of an hour circle between the celestial equator and a point on the celestial sphere, measured northward or southward from the celestial equator through 90°, and labeled N or S (+ or -) to indicate the direction of measurement.
Declination Increment (Dec. Inc.) - in sight reduction, the excess of the actual declination of a celestial body over the integral declination argument.
Double-Second Difference (DSD) - the sum of successive second differences. Because second differences are not tabulated in these tables, the DSD can be formed by subtracting, algebraically, the first difference immediately above the tabular altitude difference (d) corresponding to the entering arguments from the first difference immediately below. The result will always be a negative value.
Ecliptic - the apparent annual path of the Sun among the stars, the intersection of the plane of the Earth's orbit with the celestial sphere. This is a great circle of the celestial sphere inclined at an angle of about 23°27' to the celestial equator.
Elevalated Pole (Pn or Ps) - the celestial pole above the observer's horizon, agreeing in name with the observer's latitude.
First Difference - the difference between successive tabulations of a quantity.
First Point of Aries - that point of intersection of the ecliptic and the celestial equator occupied by the Sun as it changes from south to north declination on or about March 21. Also called VERNAL EQUINOX.
Geographical Position (GP) - the point where a line drawn from a celestial body to the Earth's center passes through the Earth's surface.
Great Circle - the intersection of a sphere and a plane through its center.
Great Circle Course - the direction of the great circle through the point of departure and the destination, expressed as angular distance from a reference direction, usually north, to the direction of the great circle. The angle varies from point to point along the great circle. At the point of departure it is called INITIAL GREAT CIRCLE COURSE.
Greenwich Hour Angle (GHA) - angular distance west of the Greenwich celestial meridian; the arc or the celestial equator, or the angle at the celestial pole, between the upper branch of the Greenwich celestial meridian and the hour circle of a point on the celestial sphere, measured westward from the Greenwich celestial meridian through 360°.
Hour Circle - on the celestial sphere, a great circle through the celestial poles and a celestial body or the vernal equinox. Hour circles are perpendicular to the celestial equator.
Intercept (a) - the difference in minutes of arc between the computed and observed altitudes (corrected sextant altitudes). It is labeled T (toward) or A (away) as the observed altitude is greater or smaller than the computed altitude, Hc greater than Ho, intercept is away (A), Ho greater than Hc, intercept is toward (T).
Line of Position (LOP) - a line indicating a series of possible positions of a craft, determined by observation or measurement.
Local Hour Angle (LHA) - angular distance west of the local celestial meridian; the arc of the celestial equator, or the angle at the celestial pole., between the upper branch of the local celestial meridian and the hour circle of a celestial body or point on the celestial sphere, measured westward from the local celestial meridian through 360°
Meridian Angle (t) - angular distance east or west of the local celestial meridian; the arc of the celestial equator, or the angle at the celestial pole, between the upper branch of the local celestial meridian and the hour circle of a celestial body, measured eastward or westward from the local celestial meridian through 180°, and labeled E or W to indicate the direction of measurement.
Nadir (Na) - that point on the celestial sphere 180° from the observer's zenith.
Name - the labels N and S which are attached to latitude and declination are said to be of the same name when they are both N or S and contrary name when one is N and the other is S.
Navigational Triangle - the spherical triangle solved in computing altitude and azimuth and greatcircle sailing problems. The celestial triangle is formed on the celestial sphere by the great circles connecting the elevated pole, zenith of the assumed position of the observer, and a celestial body. The terrestrial triangle is formed on the Earth by the great circles connecting the pole and two places on the Earth: the assumed position of the observer and geographical position of the body for celestial observations, and the point of departure and destination for great circle sailing problems. The term astronomical triangle applies to either the celestial or terrestrial triangle used for solving celestial observations.
Polar Distance (p) - angular distance from a celestial pole the arc of an hour circle between a celestial pole, usually the elevated pole, and a point on the celestial sphere, measured from the celestial pole through 180°.
Prime Meridian - the meridian of longitude 0°, used as the origin for measurement of longitude. Prime Vertical-the vertical circle through the east and west points of the horizon.
Prime Vertical - the vertical circle through the east and west points of the horizon.
Principal Vertical Circle - the vertical circle through the north and south points of the horizon, coinciding with the celestial meridian.
Respondent - the vaiue in a table or diagram corresponding to the entering arguments.
Second Difference-the difference between successive first differences.
Second Difference - the difference between successive first differences.
Sidereal Hour Angle (SHA) - angular distance west of the vernal equinox; the arc of the celestial equator, or the angle at the celestial pole, between the hour circle of the vernal equinox and the hour circle of a point on the celestial sphere, measured westward from the hour circle of the vernal equinox through 360°.
Sight Reduction - the process of deriving from a sight (observation of the altitude, and sometimes the azimuth, of a celestial body) the information needed for establishing a line of position.
SmalI Circle - the intersection of a sphere and a plane which does not pass through its center.
Vertical Circle - on the celestial sphere, a great circle through the zenith and nadir. Vertical circles are perpendicular to the horizon.
Zenith (Z) - that point on the celestial sphere vertically overhead.
Zenith Distance (z) - angular distance from the zenith, the arc of a vertical circle between the zenith and a point on the celestial sphere.
Thursday, January 22, 2009
Sextant Choosing
When looking to buy a sextant your biggest decision is quality and purchase price. The two basic sextants to look at are plastic and metal, with the price ranges varying. Their is quite a bit of difference in the price range between the two, choose whats best suited for you and how much it will be used.
The plastic sextant's advantage is price, disadvantage is plastic will expand and contract with varying temperatures, the index correction (instrument error) is constantly changing. This can be partially compensated for by obtaining an index correction each time you take a set of sights. The navigator will find that even between the first and last sight during a twilight series, the change can be considerable. Remember, a minute of error in sextant altitude corresponds to one nautical mile on the plot.
Secondly, plastic sextants weigh less than a pound and some have considerable wind resistance, making it more difficult to hold the sextant vertical when sighting in windy conditions.
Thirdly, the quality of the components is less, the filters, the mirrors, the zero to three power viewing scopes. And last, the life of a plastic sextant is shorter, depending on the amount of use. Filters break off, the plastic gearing wears out, the micrometer drum develops slop.
Any of the plastic sextants make excellent teaching aids where principle, not accuracy, is important. Also, as a back-up sextant, the micrometer plastic sextants can be very valuable. As a primary sextant when a celestial fix is important, I would recommend investing in the better sextant. The advantages of a metal sextant are obvious after reading the disadvantages involved in using a plastic sextant. Index correction is always the same unless the sextant is dropped or mirror adjustments are made. The weight (2 to 4 pounds) and open work frame reduce windage problems, the better optics and filters give better accuracy and the life of the instrument is long if care is used in usage and storage.
The metal frame may be made of either brass or an aluminum alloy, lightening the weight from roughly four to three pounds. The size of the frame varies too, also changing the weight. The telescope power varies from three to eight power, the advantage of the higher power being mainly its ability to pick up the light of a star earlier in the twilight when the naked eye still cannot see it. The disadvantage of greater power is reduced field of view, and this is critical when the navigator is trying to keep the celestial body in the field while bouncing around on a small vessel. A four power scope is a good compromise. Lighting is another option, of course this isn't needed during the day but near the end of twilight, it is convenient to press a button or turn a switch to illuminate the arc and micrometer drum. The battery case, wires and bulb socket are all subject to corrosion at sea and batteries tend to wear down when when needed. Cases usually come with the sextant and are included in the price.
Second hand metal sextants are a rare and not much of a price bargain. Some sextants are sold as antiques and application of this prices them beyond their value. The true antiques, the vernier sextants or octants, are nice as display items but the difficulty of reading a vernier versus a micrometer drum is a disadvantage. Sometimes Navy surplus sextants can be found at reasonable prices. In any of these situations, instrument cleaning or mirror resilvering may be necessary but this cost will be minimal compared to the price of the instrument. The final choice of instrument to buy comes down to how much you can afford, how essential celestial navigation is to you, how comfortable the instrument is to use, and how experienced the navigator is in handling the sextant.
Wednesday, January 21, 2009
Longitude and Time
Longitude and time are related, without time longitude by normal methods cannot be established correctly. The earth is a circle 360° with a rotation of approx. 24 hours of time, one zone or hour of time is 15° of longitude, one hour of rotation of the earth is 15° angle of rotation of the earth. Reducing this even further, one second of time corresponds to one quarter of a minute of longitude, equal to one quarter of a mile at the equator, less of a distance as latitude increases. An error of time is a error of longitude.
A system of time zones has been established worldwide, most zones being one hour or 15° wide. Everyone within the zone keeps the same time called zone time or ZT. The centers of each zone have longitudes divisible by fifteen such as 0°, 15°, 30°, 45°, etc. The edges of the zones extend 7 1/2 to each side of the center. The zone whose center is 45° would include longitudes from 37 1/2 degree's to 52 1/2 degree's.
In celestial navigation, zone time must be converted to Greenwich Mean Time, time kept at the zone whose center is 0° or the Prime Meridian. This is done by the Zone Description or ZD, a number obtained by dividing your longitude by 15 and rounding the answer off to the nearest whole number. If the longitude is west, the ZD is positive. If the longitude is east, the ZD is negative. If your longitude was 40° W, dividing by 15 would give 2.66. Rounding to the nearest whole number would give you 3 as a ZD. It would be a +3 since your longitude is west. This means that GMT is three hours later than your zone time. If your longitude was 153° E, dividing by 15 would give you 10.2. Rounding to the nearest whole number would give you 10, a negative ZD since your longitude is east. If addition or subtraction of the ZD to your ZT puts your GMT over 24h or less than 00h, a date change must be made. If daylight time is used instead of standard time, a negative one hour (- l hour) is used to the ZD to get your GMT. For example, if ZD were 6h for standard time, it would be +5h for daylight time. If ZD were - 4h for standard time, it would be - 5h for daylight time.
Watch error (WE) is the error that your time varies from true zone time. If your chronometer is 12 seconds slow, the error should be added to the time to give true zone time. A fast error should be subtracted from the chronometer reading to give true zone time. A chronometer doesn't have to tell the exact time. If its rate of loss or gain is one second per day, and is consistent then the known error can be applied to get true zone time. If you don't have a chronometer a stopwatch can be used that is set to a radio time signal. In this case, the time the stopwatch is started is added to the stopwatch reading at the time of sextant sight to obtain true zone time. The watch time (WT) is the time read on your time piece to the nearest second at the time your sextant sight is taken.
Example
Watch Time (WT) is the reading of your watch at the instant you make the sextant reading. The watch is set to the standard time of the time zone your vessel is in.
Zone Time (ZT) is the time of the zone you are in.
Watch Error (WE) is the amount of time the watch is slow (S) or fast (F). If the watch is slow you add the error. If it's fast subtract.
A system of time zones has been established worldwide, most zones being one hour or 15° wide. Everyone within the zone keeps the same time called zone time or ZT. The centers of each zone have longitudes divisible by fifteen such as 0°, 15°, 30°, 45°, etc. The edges of the zones extend 7 1/2 to each side of the center. The zone whose center is 45° would include longitudes from 37 1/2 degree's to 52 1/2 degree's.
In celestial navigation, zone time must be converted to Greenwich Mean Time, time kept at the zone whose center is 0° or the Prime Meridian. This is done by the Zone Description or ZD, a number obtained by dividing your longitude by 15 and rounding the answer off to the nearest whole number. If the longitude is west, the ZD is positive. If the longitude is east, the ZD is negative. If your longitude was 40° W, dividing by 15 would give 2.66. Rounding to the nearest whole number would give you 3 as a ZD. It would be a +3 since your longitude is west. This means that GMT is three hours later than your zone time. If your longitude was 153° E, dividing by 15 would give you 10.2. Rounding to the nearest whole number would give you 10, a negative ZD since your longitude is east. If addition or subtraction of the ZD to your ZT puts your GMT over 24h or less than 00h, a date change must be made. If daylight time is used instead of standard time, a negative one hour (- l hour) is used to the ZD to get your GMT. For example, if ZD were 6h for standard time, it would be +5h for daylight time. If ZD were - 4h for standard time, it would be - 5h for daylight time.
Watch error (WE) is the error that your time varies from true zone time. If your chronometer is 12 seconds slow, the error should be added to the time to give true zone time. A fast error should be subtracted from the chronometer reading to give true zone time. A chronometer doesn't have to tell the exact time. If its rate of loss or gain is one second per day, and is consistent then the known error can be applied to get true zone time. If you don't have a chronometer a stopwatch can be used that is set to a radio time signal. In this case, the time the stopwatch is started is added to the stopwatch reading at the time of sextant sight to obtain true zone time. The watch time (WT) is the time read on your time piece to the nearest second at the time your sextant sight is taken.
Example
Watch Time (WT) is the reading of your watch at the instant you make the sextant reading. The watch is set to the standard time of the time zone your vessel is in.
Zone Time (ZT) is the time of the zone you are in.
Watch Error (WE) is the amount of time the watch is slow (S) or fast (F). If the watch is slow you add the error. If it's fast subtract.
Tuesday, January 20, 2009
Navigational Triangles
Solving Navigational Triangles
This will give you an idea on how to "solve" the navigational triangle to find computed altitude (Hc) and azimuth (Zn) for a celestial sight. By comparing (Hc) with observed altitude (Ho), you can find the altitude intercept (a). With the assumed position coordinates, azimuth, and intercept, a line of position can be established.
The two basic methods for solving a navigational triangle or other spherical triangle are by sight reduction table or by mathematical solution. A sight reduction table gives solutions of the navigational triangle for computation of the altitude and azimuth of a celestial body. The variable components are co-latitude, polar distance, and meridian angle "t". Their values depend on time of the observation and the observer's position. The parts of the triangle to find are computed co-altitude (in order to find Hc) and azimuth angle (in order to determine Zn). Mathematical solutions use the same formulas used to pre-compute the values found in the sight reduction tables. Final results, found through the correct use of either method, seldom differ by more than one-tenth minute of arc for intercept values, or one tenth degree for azimuth values.
Sight Reduction Table Pub. 229
The Sight Reductton Table for Marine Navigation (Pub 229) is a form of sight reduction table which is produced by the Defense Mapping Agency. They are designed to help in the solution of navigational triangles to find computed altitude (Hc) and azimuth (Zn). There are six volumes of Publication 229, each volume covering 15 degrees of latitude. All of the problems used in USCG license exams, which may be solved with Publication 229, are in latitudes 15 degrees to 30 degrees covered by Volume II of the tables. The entering arguments for 229 are assumed latitude, the whole degree LHA which results when assumed longitude is applied to GHA, and declination. The main part of the tables is divided in half. The first half of Volume II covers latitudes 15 degrees through 22 degrees, and the second half covers latitudes 23 through 30 degrees. The latitude which is used to enter the table may be either north or south. Each half covers LHA's from 0 degrees through 360 degrees.
There are two pages for each LHA value from 0 through 90 degrees, and 270 through 360 degrees. Left-hand pages are for situations where declination and latitude are both north or both south, they are said to have the "same name." Each "same name" page covers declinations from 0 through 90 degrees. The adjacent right-hand pages cover "contrary name" situations (meaning the declination is north and latitude south or vice versa) for the LHA values at the top of the page, and "same name" situations when the LHA values are between 90 degrees and 270 degrees located at the bottom of the page. The dividing line between "contrary" and "same name" situations is the step-like line across the right-hand page.
The step-like line across the right-hand page represents the horizon. Entry values of LHA, latitude, and declination (same or contrary) should never result in crossing this line, as this would mean the body is below the horizon. The inside front and back covers of each 229 are an interpolation table to be used for correcting the the altitude and azimuth angle obtained in the main part of the table for the declination increment. Declination increment is the minutes and tenths of minutes part of the declination of the body. For example, if the body's declination is N 43° 23.7', the declination increment is 23.7'. In other words, declination is tabulated on each page in whole degrees, and you must interpolate between those whole degrees for your actual declination value. Declination increments 00.0 through 31.9 minutes are located inside the front cover, and declination increments 28.0 through 59.9 minutes are inside the back cover.
This will give you an idea on how to "solve" the navigational triangle to find computed altitude (Hc) and azimuth (Zn) for a celestial sight. By comparing (Hc) with observed altitude (Ho), you can find the altitude intercept (a). With the assumed position coordinates, azimuth, and intercept, a line of position can be established.
The two basic methods for solving a navigational triangle or other spherical triangle are by sight reduction table or by mathematical solution. A sight reduction table gives solutions of the navigational triangle for computation of the altitude and azimuth of a celestial body. The variable components are co-latitude, polar distance, and meridian angle "t". Their values depend on time of the observation and the observer's position. The parts of the triangle to find are computed co-altitude (in order to find Hc) and azimuth angle (in order to determine Zn). Mathematical solutions use the same formulas used to pre-compute the values found in the sight reduction tables. Final results, found through the correct use of either method, seldom differ by more than one-tenth minute of arc for intercept values, or one tenth degree for azimuth values.
Sight Reduction Table Pub. 229
The Sight Reductton Table for Marine Navigation (Pub 229) is a form of sight reduction table which is produced by the Defense Mapping Agency. They are designed to help in the solution of navigational triangles to find computed altitude (Hc) and azimuth (Zn). There are six volumes of Publication 229, each volume covering 15 degrees of latitude. All of the problems used in USCG license exams, which may be solved with Publication 229, are in latitudes 15 degrees to 30 degrees covered by Volume II of the tables. The entering arguments for 229 are assumed latitude, the whole degree LHA which results when assumed longitude is applied to GHA, and declination. The main part of the tables is divided in half. The first half of Volume II covers latitudes 15 degrees through 22 degrees, and the second half covers latitudes 23 through 30 degrees. The latitude which is used to enter the table may be either north or south. Each half covers LHA's from 0 degrees through 360 degrees.
There are two pages for each LHA value from 0 through 90 degrees, and 270 through 360 degrees. Left-hand pages are for situations where declination and latitude are both north or both south, they are said to have the "same name." Each "same name" page covers declinations from 0 through 90 degrees. The adjacent right-hand pages cover "contrary name" situations (meaning the declination is north and latitude south or vice versa) for the LHA values at the top of the page, and "same name" situations when the LHA values are between 90 degrees and 270 degrees located at the bottom of the page. The dividing line between "contrary" and "same name" situations is the step-like line across the right-hand page.
The step-like line across the right-hand page represents the horizon. Entry values of LHA, latitude, and declination (same or contrary) should never result in crossing this line, as this would mean the body is below the horizon. The inside front and back covers of each 229 are an interpolation table to be used for correcting the the altitude and azimuth angle obtained in the main part of the table for the declination increment. Declination increment is the minutes and tenths of minutes part of the declination of the body. For example, if the body's declination is N 43° 23.7', the declination increment is 23.7'. In other words, declination is tabulated on each page in whole degrees, and you must interpolate between those whole degrees for your actual declination value. Declination increments 00.0 through 31.9 minutes are located inside the front cover, and declination increments 28.0 through 59.9 minutes are inside the back cover.
Monday, January 19, 2009
Circles of Equal Altitude
Circles Of Equal Altitude
Celestial bodies are so far away that rays of light from them reach earth as parallel lines. In solving celestial navigation problems you can assume that all bodies are at the same infinite distance from earth on the celestial sphere. Corrections for bodies close to earth are taken into account when determining the actual altitude of a body in a celestial sight.
The geographical position of a celestial body is the point on the earth directly below the body at any given instant. The geographical position is one of the three corners of the "navigational" triangle, and the side of the triangle between the GP and an observer's position is equal to 90° minus altitude, or co-altitude. Because the light rays from celestial bodies arrive on earth as parallel lines, you measure the same angle between the body's light rays and the horizon are located the same distance from the GP of the body. All observers who measure a celestial body at the same altitude at an instant of time are located on a common circle. This circle is called a circle of equal altitude, and the distance from the center to the circle (radius) is equal to co-altitude.
This circle of equal altitude is a celestial "line of position" like a radar range line of position from piloting. In this case the co-altitude, converted from degrees and minutes of arc to miles, is similar to the radar range. The geographical position (GP) of the body is the center from which you swing an arc with a pencil compass.
A celestial line of position can be established by:
1. Observing the altitude of a celestial body.
2. Recording the instant of time (GMT) of the observation.
3. Determining the GHA and declination of the body from the Nautical Almanac.
4. Locating the GP of the body on a chart using GHA and declination.
5. Subtracting altitude from 90° and converting the resulting co-altitude to minutes of arc or miles.
6. Swinging an arc with a radius equal to co-altitude in miles using the GP as the center.
A celestial fix may be obtained by crossing two or more circles of equal altitude based on observations of celestial bodies.
Plotting Fixes Based on Circles of Equal Altitude
In actual practice fixes based on circles of equal altitude are impractical and are seldom used in marine navigation. The main reason for not using them is that the radii of these circles is too large to plot the circular line of position on the chart unless very high altitude sights are used, and they are difficult to obtain. The altitude of the body must be very close to 90° (87° is the practical minimum) in order for the co-altitude to be small enough to be plotted on a normal celestial plotting sheet or navigation chart. In taking a sight on a moving ship, the navigator would have difficulty determining the actual point on the horizon, below the body vertically, from which to measure altitude when the body is nearly at the zenith. The only real advantage of high altitude sights is that the same body may be used for a fix because the location of the GP will move a considerable distance in only a few minutes, giving a good angle of intersection between the two arcs when plotted.
The USCG requires all unlimited license candidates to be proficient in solving high altitude "running" fixes using the sun. The following is an example solution of a high altitude running fix.
Example Problem
On 15 November 1981, your 1030 ZT DR position is Lat. 19° 41.0' S, Long. 41° 37.0' W. You are on course 239°T, speed 22 knots. You take the following observations of the Sun:
What was the 1200 ZT position?
Zone Time 1128
GHA 40° 50.4
Declination S 18° 33.6
Ho (observed altitude) 88° 18.4
Zone Time 1133
GHA 42° 05.4
DeclinationS 18° 33.6
Ho (observed altitude 88° 37.7
Step 1: Using a plotting sheet constructed for the proper latitudes, label three meridians from right to left 40°W, 41°W, and 42°W. Plot the ship's 1030 DR position and establish a DR based on the course and speed. Determine the 1200 DR position by measuring the distance which would be traveled between 1030 and 1200 along course 239° T at a speed of 22 knots. (Distance = speed x time / 60 or 22kts x 90m / 60m = 33 miles).
Step 2: Plot the geographical positions of the sun and label them with their observation times. Treat declination as latitude and since GHA is less than 180° it is equal to longitude (west). Next, advance each GP in the direction of 239° T at 22 knots to 1200. The first would be advanced from 1128 to 1200 (32m at 22 knots = 11.7 miles). The second would be advanced from 1133 to 1200 (27m at 22 knots = 9.9 miles).
Step 3: The next step is to determine the radius of the circles of equal altitude of the respective sights. To do this subtract each observed altitude from 90 degrees.
1128
89° 60.0 - 88°18.4 = 1° 41.6 co-altitude
1133
89° 60.0 - 88° 37.7 = 1° 22.3 co-altitude
Convert the co-altitudes just found to minutes (miles). Doing the conversion you should find that 1128 is 101.6 miles and 1133 is 82.3 miles. Using a compass, measure each distance on the latitude scale and then swing an arc of that radius from the advanced GP. The circles formed will intersect at two points, but the DR position indicates which is the correct position for 1200. The fix should be labeled and the coordinates you obtain should be close to Lat. 20° 01.0' S, Long. 42° 05.0' W.
Celestial bodies are so far away that rays of light from them reach earth as parallel lines. In solving celestial navigation problems you can assume that all bodies are at the same infinite distance from earth on the celestial sphere. Corrections for bodies close to earth are taken into account when determining the actual altitude of a body in a celestial sight.
The geographical position of a celestial body is the point on the earth directly below the body at any given instant. The geographical position is one of the three corners of the "navigational" triangle, and the side of the triangle between the GP and an observer's position is equal to 90° minus altitude, or co-altitude. Because the light rays from celestial bodies arrive on earth as parallel lines, you measure the same angle between the body's light rays and the horizon are located the same distance from the GP of the body. All observers who measure a celestial body at the same altitude at an instant of time are located on a common circle. This circle is called a circle of equal altitude, and the distance from the center to the circle (radius) is equal to co-altitude.
This circle of equal altitude is a celestial "line of position" like a radar range line of position from piloting. In this case the co-altitude, converted from degrees and minutes of arc to miles, is similar to the radar range. The geographical position (GP) of the body is the center from which you swing an arc with a pencil compass.
A celestial line of position can be established by:
1. Observing the altitude of a celestial body.
2. Recording the instant of time (GMT) of the observation.
3. Determining the GHA and declination of the body from the Nautical Almanac.
4. Locating the GP of the body on a chart using GHA and declination.
5. Subtracting altitude from 90° and converting the resulting co-altitude to minutes of arc or miles.
6. Swinging an arc with a radius equal to co-altitude in miles using the GP as the center.
A celestial fix may be obtained by crossing two or more circles of equal altitude based on observations of celestial bodies.
Plotting Fixes Based on Circles of Equal Altitude
In actual practice fixes based on circles of equal altitude are impractical and are seldom used in marine navigation. The main reason for not using them is that the radii of these circles is too large to plot the circular line of position on the chart unless very high altitude sights are used, and they are difficult to obtain. The altitude of the body must be very close to 90° (87° is the practical minimum) in order for the co-altitude to be small enough to be plotted on a normal celestial plotting sheet or navigation chart. In taking a sight on a moving ship, the navigator would have difficulty determining the actual point on the horizon, below the body vertically, from which to measure altitude when the body is nearly at the zenith. The only real advantage of high altitude sights is that the same body may be used for a fix because the location of the GP will move a considerable distance in only a few minutes, giving a good angle of intersection between the two arcs when plotted.
The USCG requires all unlimited license candidates to be proficient in solving high altitude "running" fixes using the sun. The following is an example solution of a high altitude running fix.
Example Problem
On 15 November 1981, your 1030 ZT DR position is Lat. 19° 41.0' S, Long. 41° 37.0' W. You are on course 239°T, speed 22 knots. You take the following observations of the Sun:
What was the 1200 ZT position?
Zone Time 1128
GHA 40° 50.4
Declination S 18° 33.6
Ho (observed altitude) 88° 18.4
Zone Time 1133
GHA 42° 05.4
DeclinationS 18° 33.6
Ho (observed altitude 88° 37.7
Step 1: Using a plotting sheet constructed for the proper latitudes, label three meridians from right to left 40°W, 41°W, and 42°W. Plot the ship's 1030 DR position and establish a DR based on the course and speed. Determine the 1200 DR position by measuring the distance which would be traveled between 1030 and 1200 along course 239° T at a speed of 22 knots. (Distance = speed x time / 60 or 22kts x 90m / 60m = 33 miles).
Step 2: Plot the geographical positions of the sun and label them with their observation times. Treat declination as latitude and since GHA is less than 180° it is equal to longitude (west). Next, advance each GP in the direction of 239° T at 22 knots to 1200. The first would be advanced from 1128 to 1200 (32m at 22 knots = 11.7 miles). The second would be advanced from 1133 to 1200 (27m at 22 knots = 9.9 miles).
Step 3: The next step is to determine the radius of the circles of equal altitude of the respective sights. To do this subtract each observed altitude from 90 degrees.
1128
89° 60.0 - 88°18.4 = 1° 41.6 co-altitude
1133
89° 60.0 - 88° 37.7 = 1° 22.3 co-altitude
Convert the co-altitudes just found to minutes (miles). Doing the conversion you should find that 1128 is 101.6 miles and 1133 is 82.3 miles. Using a compass, measure each distance on the latitude scale and then swing an arc of that radius from the advanced GP. The circles formed will intersect at two points, but the DR position indicates which is the correct position for 1200. The fix should be labeled and the coordinates you obtain should be close to Lat. 20° 01.0' S, Long. 42° 05.0' W.
Sunday, January 18, 2009
Celestial Navigation Sight Form Definitions
Here is the format that I use for basic celestial sights and the definitions.
Body - Name of the Body
Date - Local Date
DR Lat. - For the time of the sight
DR Long. - For the time of the sight
Hs - Sextant Altitude (Reading at the time of the sight)
IE - Index Error +/- "On the arc / Off the arc"
Hs - Sextant Altitude
Dip - Height of eye correction
Ha - Apparent Altitude
Alt. Corr. - Altitude Correction
Ho - Observed Altitude
ZT - Zone Time of the sight
ZD - Zone Description
GMT - Greenwich Mean Time and Date
CT - Chronometer Time
CE - Chronometer Error
CCT - Corrected Chronomete Time
GHA - Greenwich Hour Angle
M & S - Minutes and Seconds Correction (Increments)
SHA - Sideral Hour Angle (for stars only)
v - "v" Correction (for planets only)
GHA - Greenwich Hour Angle
A / Long. +E / - W - Assumed Longitude
LHA - Local Hour Angle
Tab. Dec. - Tabulated Declination
d corr. - "d" correction
Dec. - Declination
Body - Name of the Body
Date - Local Date
DR Lat. - For the time of the sight
DR Long. - For the time of the sight
Hs - Sextant Altitude (Reading at the time of the sight)
IE - Index Error +/- "On the arc / Off the arc"
Hs - Sextant Altitude
Dip - Height of eye correction
Ha - Apparent Altitude
Alt. Corr. - Altitude Correction
Ho - Observed Altitude
ZT - Zone Time of the sight
ZD - Zone Description
GMT - Greenwich Mean Time and Date
CT - Chronometer Time
CE - Chronometer Error
CCT - Corrected Chronomete Time
GHA - Greenwich Hour Angle
M & S - Minutes and Seconds Correction (Increments)
SHA - Sideral Hour Angle (for stars only)
v - "v" Correction (for planets only)
GHA - Greenwich Hour Angle
A / Long. +E / - W - Assumed Longitude
LHA - Local Hour Angle
Tab. Dec. - Tabulated Declination
d corr. - "d" correction
Dec. - Declination
Saturday, January 17, 2009
Universal Plotting Sheets and how to set up
Universal Plotting Sheets
In order to plot celestial lines of position you need a Universal Plotting Sheet, the first thing that probably sticks out on the sheet is the compass rose, just like conventional charts. All the compass rose bearings are true bearings. You will notice the vertical line running due north and south through the center of the compass rose. Next, are the horizontal lines which will represent your desired latitudes. And last on the lower right, a longitude scale, which will allow you along with your dividers a way to lay off minutes of longitude.
Your first observation that you make is the scale between any two horizontal latitude lines. It's a scale breaking down one degree of latitude into 60 minutes. If you designate the latitude line 28 degrees north and your indicated position is 28 degrees put your dividers on 28 you now have your latitude.
Now that you have an understanding of how to plot a latitude position without having to do anything to the sheet other than fill in the latitude and scale up the from the vertical compass rose, you only need to set up the sheet for longitude. Your assumed position is 28 degrees north latitude, sixty two degrees thirty six minutes west longitude. The first step is your latitude, whatever whole degree of latitude that you are currently located in or centered on as part of a plot will be indicated as the central latitude line, In this case 28 degrees. Since we are in North latitude the one below will be 27 degrees the one above 29 degrees. If you are in the Southern Hemisphere, it will be reversed. So now all you need to do is to account for your longitude.
Setting up your Longitude Lines
On the edge of the compass rose there are two tick marks just below 30, two tick marks the top thirty, the other tick mark below thirty.This is important the two tick marks below 30 and above 30 on the compass rose are actually on the 28 degree marks, the same 28 as our central latitude. Next, draw a vertical line connecting the two points. Looking at a plotting sheet you will see that this gives you your second longitude line and your longitudinal spacing. To get additional longitude lines, use your dividers one point on the compass rose central line the other on the constructed line and lay them off as needed.
Since the plot on the sheet required a longitude of 62 degrees West, pick one convenient to your position and labeled it.The next one that would have obvious chart significance is 63 degrees West. Your position is 62 degrees 36 minutes West. How do you lay off the minutes? Look to the longitude scale lower right corner. Draw a line between 25 degrees and 30 degrees latitude, approx. 28 with the construction of this line, you put one point of my dividers on 30, extending the other point back to the zero mark would indicate only 30 minutes. The lines to the right are further broken down into increments of two degrees. Use your dividers and walk out 6 more minutes this should give you 36. This sheet is prepared and published by the US Defense Mapping Agency, normally comes in a pad of 100, and you can use both sides. I like to save mine for a reference if I want to go back and check.
In order to plot celestial lines of position you need a Universal Plotting Sheet, the first thing that probably sticks out on the sheet is the compass rose, just like conventional charts. All the compass rose bearings are true bearings. You will notice the vertical line running due north and south through the center of the compass rose. Next, are the horizontal lines which will represent your desired latitudes. And last on the lower right, a longitude scale, which will allow you along with your dividers a way to lay off minutes of longitude.
Your first observation that you make is the scale between any two horizontal latitude lines. It's a scale breaking down one degree of latitude into 60 minutes. If you designate the latitude line 28 degrees north and your indicated position is 28 degrees put your dividers on 28 you now have your latitude.
Now that you have an understanding of how to plot a latitude position without having to do anything to the sheet other than fill in the latitude and scale up the from the vertical compass rose, you only need to set up the sheet for longitude. Your assumed position is 28 degrees north latitude, sixty two degrees thirty six minutes west longitude. The first step is your latitude, whatever whole degree of latitude that you are currently located in or centered on as part of a plot will be indicated as the central latitude line, In this case 28 degrees. Since we are in North latitude the one below will be 27 degrees the one above 29 degrees. If you are in the Southern Hemisphere, it will be reversed. So now all you need to do is to account for your longitude.
Setting up your Longitude Lines
On the edge of the compass rose there are two tick marks just below 30, two tick marks the top thirty, the other tick mark below thirty.This is important the two tick marks below 30 and above 30 on the compass rose are actually on the 28 degree marks, the same 28 as our central latitude. Next, draw a vertical line connecting the two points. Looking at a plotting sheet you will see that this gives you your second longitude line and your longitudinal spacing. To get additional longitude lines, use your dividers one point on the compass rose central line the other on the constructed line and lay them off as needed.
Since the plot on the sheet required a longitude of 62 degrees West, pick one convenient to your position and labeled it.The next one that would have obvious chart significance is 63 degrees West. Your position is 62 degrees 36 minutes West. How do you lay off the minutes? Look to the longitude scale lower right corner. Draw a line between 25 degrees and 30 degrees latitude, approx. 28 with the construction of this line, you put one point of my dividers on 30, extending the other point back to the zero mark would indicate only 30 minutes. The lines to the right are further broken down into increments of two degrees. Use your dividers and walk out 6 more minutes this should give you 36. This sheet is prepared and published by the US Defense Mapping Agency, normally comes in a pad of 100, and you can use both sides. I like to save mine for a reference if I want to go back and check.
Friday, January 16, 2009
Moon LOP Azimuth and Intercept Questions
Moon (Azimuth and Intercept)
On 25 February 1981, at 0622 ZT, you observe the upper limb of the Moon with a sextant altitude of 59° 58.6'. Your DR position is Lat. 30°28.3'8, Long. 102°39.3 E. The chronometer reading at the time of the sight is 11h 21m 18s and the chronometer is 48s slow. The height of eye is 59 feet and the index error is 2.5' on the arc. What are the azimuth (Zn) and intercept (a) of this sight using the assumed position?
A. Zn 305.4°, a 4.2 Towards
B. Zn 234.6°, a 4.2 Away
C. Zn 305.4°, a 1.5 Towards
D. Zn 305.4°, a 9.3 Towards
On 22 July 1981, at 0720 ZT, in DR position Lat. 20° 38.2'N, Long. 87°16.0'W, you observe the Moon's lower limb. The sextant altitude (hs) is 38°32.6, and the chronometer reads 01h 18m 14s. The chronometer is 01m 28s slow. The index error is 3.1' off the arc, and the height of eye is 68 feet. What is the azimuth (Zn) and intercept (a) of this sight from the assumed position?
A. Zn 291.4°, a 5.2 Away
B. Zn 111.4°, a 8.7 Away
C. Zn 248.6°, a 5.0 Towards
D. Zn 068.6°, a 6.5 Towards
On 25 February 1981, at 0622 ZT, you observe the upper limb of the Moon with a sextant altitude of 59° 58.6'. Your DR position is Lat. 30°28.3'8, Long. 102°39.3 E. The chronometer reading at the time of the sight is 11h 21m 18s and the chronometer is 48s slow. The height of eye is 59 feet and the index error is 2.5' on the arc. What are the azimuth (Zn) and intercept (a) of this sight using the assumed position?
A. Zn 305.4°, a 4.2 Towards
B. Zn 234.6°, a 4.2 Away
C. Zn 305.4°, a 1.5 Towards
D. Zn 305.4°, a 9.3 Towards
On 22 July 1981, at 0720 ZT, in DR position Lat. 20° 38.2'N, Long. 87°16.0'W, you observe the Moon's lower limb. The sextant altitude (hs) is 38°32.6, and the chronometer reads 01h 18m 14s. The chronometer is 01m 28s slow. The index error is 3.1' off the arc, and the height of eye is 68 feet. What is the azimuth (Zn) and intercept (a) of this sight from the assumed position?
A. Zn 291.4°, a 5.2 Away
B. Zn 111.4°, a 8.7 Away
C. Zn 248.6°, a 5.0 Towards
D. Zn 068.6°, a 6.5 Towards
Thursday, January 15, 2009
Wednesday, January 14, 2009
Celestial Observations of the Moon
Celestial Observations of the Moon
The moon is easy to identify and is often visible during the day. Yet, the navigator taking the sight must follow the moon across the horizon. It appears to move very fast, so the navigator must be quick in obtaining the sextant reading. The moon's proximity to the earth requires only some additional corrections (Parallax) to (Ha) to obtain (Ho). The rest of the sight reduction is the same process as an observation of the sun.
Parallax (P)
Parallax is the difference in the direction of an object at a finite distance when viewed simultaneously from two different positions. It enters into the sextant altitude corrections because (Hs) is measured from the earth's surface, but Ho is calculated from the earth's center. Since the moon is the celestial body nearest the earth, parallax has its greatest effect on lunar observations. If the moon is directly overhead, such as with an altitude of 90°, there is no parallax, as its direction is the same at the center of the earth as for the observer. As the moon decreases in altitude its direction from the observer begins to differ with its direction from the earth's center, and the difference in direction increases continuously until the moon sets. The same effect occurs in reverse when a body is rising. Parallax ranges from zero for a body with an altitude of 90° to a maximum when the body is on the horizon, with 0° altitude. At altitude 0°, it is called horizontal parallax (HP).
In addition to increasing as altitude decreases, parallax increases as distance to a celestial body decreases. Venus and Mars, when close to the earth, are also affected by parallax. The sun is slightly affected, the parallax correction for the sun being +0.1' from zero altitude to 65°. All other celestial bodies are too far from the earth to require correction for parallax when observed with the sextant. The correction for parallax is always positive. and is applied only to observations of the moon, sun, Venus, and Mars. When it is applied to (Hs), the sextant altitude is corrected to the value it would have if the observer were at the center of the earth. The value of horizontal parallax (HP) is found on the daily pages Nautical Almanac under Moon for every hour of GMT.
As with other observations the sextant corrections for dip and altitude are in the front of the Nautical Almanac. But for the Moon, the altitude and dip tables are in the back of the Nautical Almanac. Which means you cannot correct your (Hs) to (Ho) until you get the HP from the daily pages.This procedure goes against the format that I use, but you have to be flexible as you will see in the next blog titled "How to Compute the Intercept and Azimuth of the Moon".
The moon is easy to identify and is often visible during the day. Yet, the navigator taking the sight must follow the moon across the horizon. It appears to move very fast, so the navigator must be quick in obtaining the sextant reading. The moon's proximity to the earth requires only some additional corrections (Parallax) to (Ha) to obtain (Ho). The rest of the sight reduction is the same process as an observation of the sun.
Parallax (P)
Parallax is the difference in the direction of an object at a finite distance when viewed simultaneously from two different positions. It enters into the sextant altitude corrections because (Hs) is measured from the earth's surface, but Ho is calculated from the earth's center. Since the moon is the celestial body nearest the earth, parallax has its greatest effect on lunar observations. If the moon is directly overhead, such as with an altitude of 90°, there is no parallax, as its direction is the same at the center of the earth as for the observer. As the moon decreases in altitude its direction from the observer begins to differ with its direction from the earth's center, and the difference in direction increases continuously until the moon sets. The same effect occurs in reverse when a body is rising. Parallax ranges from zero for a body with an altitude of 90° to a maximum when the body is on the horizon, with 0° altitude. At altitude 0°, it is called horizontal parallax (HP).
In addition to increasing as altitude decreases, parallax increases as distance to a celestial body decreases. Venus and Mars, when close to the earth, are also affected by parallax. The sun is slightly affected, the parallax correction for the sun being +0.1' from zero altitude to 65°. All other celestial bodies are too far from the earth to require correction for parallax when observed with the sextant. The correction for parallax is always positive. and is applied only to observations of the moon, sun, Venus, and Mars. When it is applied to (Hs), the sextant altitude is corrected to the value it would have if the observer were at the center of the earth. The value of horizontal parallax (HP) is found on the daily pages Nautical Almanac under Moon for every hour of GMT.
As with other observations the sextant corrections for dip and altitude are in the front of the Nautical Almanac. But for the Moon, the altitude and dip tables are in the back of the Nautical Almanac. Which means you cannot correct your (Hs) to (Ho) until you get the HP from the daily pages.This procedure goes against the format that I use, but you have to be flexible as you will see in the next blog titled "How to Compute the Intercept and Azimuth of the Moon".
Tuesday, January 13, 2009
Monday, January 12, 2009
Celestial Navigation 3 Star Running Fix
A running fix is generally accepted, from piloting, as a fix where earlier LOP's are advanced to cross a later LOP. In celestial navigation this is also true up to a point. When several observations of different bodies are taken in a relatively short period of time (5 to 30 minutes), the resulting fix is regarded as a simultaneous fix rather than a running fix, even though the earlier LOP's are advanced to the final LOP time. Star fixes taken at morning or evening twilight are examples of these fixes.
"CLICK HERE TO VIEW"
"CLICK HERE TO VIEW"
Sunday, January 11, 2009
Saturday, January 10, 2009
Celestial Navigation 3 LOP Fix (Sun Only)
In their multiple sun line running fixes, the Coast Guard gives GHA, Ho, and Declination. This means you don't have to use the Nautical Almanac and cuts out several steps normally required in the solution for intercept and azimuth. In this page you will find the solution and a format on how to work these problems out.
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"CLICK HERE TO VIEW"
Friday, January 9, 2009
Celestial Navigation 2 LOP AM Sun and LAN Questions
To get a better feel for advancing and retarding lines of position for a running fix, the following problems cover this aspect of celestial navigation, AM Sun and LAN (2 LOP). These Coast Guard questions require attention to detail and being accurate.
1. On 30 August 1981 your 0554 zone time (ZT) position was Lat. 25° 39.0' S, Long. 31° 51.0' E. Your vessel was steaming on course 325° T at a speed of 15.0 knots. An observation of the Sun's lower limb was made at 0836 ZT. The chronometer read 06h 38m 36s and was fast 02m 24s. The observed altitude (Ho) was 30° 49.2'. LAN occurred at 1157 ZT. The observed altitude (Ho) was 56° 40.0'. What was the longitude of your 1157 ZT running fix?
A. 30° 59.8' E
B. 30° 57.6' E
C. 30° 55.9' E
D. 30° 52.5' E
2. On 8 February 1981, your 0800 zone time position is Lat. 21 ° 55.0' S, Long. 52° 27.0' W. Your vessel is on course 056° T at a speed of 17.5 knots. An observation of the Sun's lower limb is made at 0938 zone time, and the observed altitude (Ho) is 46° 06.5'. The chronometer reads 12h 37m 23s, and the chronometer error is 1m 24s slow. LAN occurs at 1243 zone time, and a meridian altitude of the Sun's lower limb is made. The observed altitude (Ho) for this sight is 83° 56.1'. Determine the vessel's 1200 zone time position.
A. Lat. 20° 57.0' S, Long. 51° 21.5' W
B. Lat. 20° 58.0' S, Long. 51° 25.5' W
C. Lat. 21° 04.0' S, Long. 51° 12.0' W
D. Lat. 21° 04.0' S, Long. 51° 21.5' W
3. On 4 May 1981, your 0500 zone time position was Lat. 24° 45.0' N, Long. 120° 18.0' W. Your vessel was steaming on course 315° T at a speed of 15.5 knots. An observation of the Sun's upper limb was made at 0830 ZT. The chronometer read 04h 31m 32s and was fast 01m 24s. The observed altitude (Ho) was 40° 11.8'. LAN occurred at 1204 zone time. The observed altitude (Ho) was 80° 05.0'. What was the longitude of your 1300 zone time running fix?
A. Long. 121° 59.2' W
B. Long. 121° 57.4' W
C. Long. 121° 53.5' W
D. Long. 121° 49.8' W
4. On 22 February 1981, your 0800 zone time position is Lat. 24° 16.0' S, Long. 95° 37.0' E. Your vessel is on course 126° T at a speed of 14 knots. An observation of the Sun's lower limb is made at 0945 zone time. The chronometer reads 03h 47m 22s, and the chronometer error is 02m 37s fast. The observed altitude (Ho) is 57° 02.1'. LAN occurs at 1148 zone time, and a meridian altitude of the Sun's lower limb is made. The observed meridian altitude (Ho) is 75° 22.3'. Determine the vessel's 1200 zone time position.
A. Lat. 24° 49.3' S, Long. 96° 24.0' E
B. Lat. 24° 49.3' S, Long. 96° 27.2' E
C. Lat. 24° 52.2' S, Long. 96° 24.0' E
D. Lat. 24° 52.2' S, Long. 96° 27.2' E
5. On 29 June 1981, your 0800 zone time fix gives you a position of Lat. 26° 16.0' S, Long. 61° 04.0' E. Your vessel is steaming a course of 079° T at a speed of 15.5 knots. An observation of the Sun's upper limb is made at 0905 zone time, and the observed altitude (Ho) is 25° 20.1'. The chronometer reads 05h 08m 12s, and the chronometer error is 02m 27s fast. Local apparent noon occurs at 1154 zone time, and a meridian altitude of the Sun's lower limb is made. The observed altitude (Ho) for this sight is 40° 44.2'. Determine the vessel's 1200 zone time position.
A. Lat. 26° 02.0' S, Long. 62° 05.0' E
B. Lat. 26° 02.0' S, Long. 62° 23.2' E
C. Lat. 26° 05.1' S, Long. 62° 06.3' E
D. Lat. 25° 56.0' S, Long. 62° 03.0' E
6. On15 August 1981, your 0512 zone time position was Lat. 29° 18.0' N, Long. 57° 24.0' W. Your vessel was steaming on course 262° T at a speed of 20.0 knots. An observation of the Sun's lower limb was made at 0824 Z1. The chronometer read 00h 22m 24s and was slow 01 m 34s. The observed altitude (Ho) was 38° 16.7'. LAN occurred at 1204 zone time. The observed altitude (Ho) was 74° 58.0'. What was the longitude of your 1204 zone time fix?
A. Long. 59° 52.0' W
B. Long. 59° 54.0' W
C. Long. 59° 58.5' W
D. Long. 60° 02.0' W
1. On 30 August 1981 your 0554 zone time (ZT) position was Lat. 25° 39.0' S, Long. 31° 51.0' E. Your vessel was steaming on course 325° T at a speed of 15.0 knots. An observation of the Sun's lower limb was made at 0836 ZT. The chronometer read 06h 38m 36s and was fast 02m 24s. The observed altitude (Ho) was 30° 49.2'. LAN occurred at 1157 ZT. The observed altitude (Ho) was 56° 40.0'. What was the longitude of your 1157 ZT running fix?
A. 30° 59.8' E
B. 30° 57.6' E
C. 30° 55.9' E
D. 30° 52.5' E
2. On 8 February 1981, your 0800 zone time position is Lat. 21 ° 55.0' S, Long. 52° 27.0' W. Your vessel is on course 056° T at a speed of 17.5 knots. An observation of the Sun's lower limb is made at 0938 zone time, and the observed altitude (Ho) is 46° 06.5'. The chronometer reads 12h 37m 23s, and the chronometer error is 1m 24s slow. LAN occurs at 1243 zone time, and a meridian altitude of the Sun's lower limb is made. The observed altitude (Ho) for this sight is 83° 56.1'. Determine the vessel's 1200 zone time position.
A. Lat. 20° 57.0' S, Long. 51° 21.5' W
B. Lat. 20° 58.0' S, Long. 51° 25.5' W
C. Lat. 21° 04.0' S, Long. 51° 12.0' W
D. Lat. 21° 04.0' S, Long. 51° 21.5' W
3. On 4 May 1981, your 0500 zone time position was Lat. 24° 45.0' N, Long. 120° 18.0' W. Your vessel was steaming on course 315° T at a speed of 15.5 knots. An observation of the Sun's upper limb was made at 0830 ZT. The chronometer read 04h 31m 32s and was fast 01m 24s. The observed altitude (Ho) was 40° 11.8'. LAN occurred at 1204 zone time. The observed altitude (Ho) was 80° 05.0'. What was the longitude of your 1300 zone time running fix?
A. Long. 121° 59.2' W
B. Long. 121° 57.4' W
C. Long. 121° 53.5' W
D. Long. 121° 49.8' W
4. On 22 February 1981, your 0800 zone time position is Lat. 24° 16.0' S, Long. 95° 37.0' E. Your vessel is on course 126° T at a speed of 14 knots. An observation of the Sun's lower limb is made at 0945 zone time. The chronometer reads 03h 47m 22s, and the chronometer error is 02m 37s fast. The observed altitude (Ho) is 57° 02.1'. LAN occurs at 1148 zone time, and a meridian altitude of the Sun's lower limb is made. The observed meridian altitude (Ho) is 75° 22.3'. Determine the vessel's 1200 zone time position.
A. Lat. 24° 49.3' S, Long. 96° 24.0' E
B. Lat. 24° 49.3' S, Long. 96° 27.2' E
C. Lat. 24° 52.2' S, Long. 96° 24.0' E
D. Lat. 24° 52.2' S, Long. 96° 27.2' E
5. On 29 June 1981, your 0800 zone time fix gives you a position of Lat. 26° 16.0' S, Long. 61° 04.0' E. Your vessel is steaming a course of 079° T at a speed of 15.5 knots. An observation of the Sun's upper limb is made at 0905 zone time, and the observed altitude (Ho) is 25° 20.1'. The chronometer reads 05h 08m 12s, and the chronometer error is 02m 27s fast. Local apparent noon occurs at 1154 zone time, and a meridian altitude of the Sun's lower limb is made. The observed altitude (Ho) for this sight is 40° 44.2'. Determine the vessel's 1200 zone time position.
A. Lat. 26° 02.0' S, Long. 62° 05.0' E
B. Lat. 26° 02.0' S, Long. 62° 23.2' E
C. Lat. 26° 05.1' S, Long. 62° 06.3' E
D. Lat. 25° 56.0' S, Long. 62° 03.0' E
6. On15 August 1981, your 0512 zone time position was Lat. 29° 18.0' N, Long. 57° 24.0' W. Your vessel was steaming on course 262° T at a speed of 20.0 knots. An observation of the Sun's lower limb was made at 0824 Z1. The chronometer read 00h 22m 24s and was slow 01 m 34s. The observed altitude (Ho) was 38° 16.7'. LAN occurred at 1204 zone time. The observed altitude (Ho) was 74° 58.0'. What was the longitude of your 1204 zone time fix?
A. Long. 59° 52.0' W
B. Long. 59° 54.0' W
C. Long. 59° 58.5' W
D. Long. 60° 02.0' W
Thursday, January 8, 2009
Wednesday, January 7, 2009
Tuesday, January 6, 2009
How to Advance or Retard a Line of Position in Celestial Navigation
In celestial navigation, lines of position are rarely obtained simultaneously. This is especially true during the day when the sun may be the only available celestial body. A celestial line of position may be advanced for 3 or 4 hours, if necessary, to obtain a celestial running fix. It may also be advanced by advancing the AP in the direction and distance an amount consistent with the ship's travel during the interval between two successive observations. In the latter procedure, the azimuth line is drawn through the advanced AP without any change in direction. The advanced LOP is drawn perpendicular to the azimuth, a distance from the AP equal to the intercept, and toward or away from the GP, as appropriate.Plotting the Celestial Fix
At morning and evening twilight, the navigator may succeed in observing the altitudes of a number of celestial bodies in a few minutes and establish a celestial fix. If 2 or more minutes elapse between observations, the navigator must consider:
1. Elapsed time
2. Speed of ship
3. Scale of the chart or plotting sheet
To determine whether or not a more accurate fix can be obtained by advancing AP's to a common time. It is possible during the day to obtain a celestial fix rather than a celestial running fix if two or more of the three following bodies are visible:
1. Sun
2. Moon
3. Venus
LOP from Celestial Observations
A ship has many possible locations on a line of position. In other words, the ship's position must be somewhere along that line. A fix, by position is the intersection of two or more lines of position, but this is not the ship's exact position, because one can always assume some errors in observation, plotting. The celestial navigator must establish the lines of position by applying the results of the observations of heavenly bodies. A line of position obtained at one time may be used at a later time. All you need to do is move the line parallel to itself, a distance equal to the run of the ship in the interim, and in the same direction as the run. Such a line of position cannot be as accurate as a new line, because the amount and direction of its movement can be determined only by the usual DR methods.
If two new lines cannot be obtained, however, a old line, advanced and intersected with a new one, may be the only possible way of establishing a fix. Naturally, the distance an old line may be advanced without a substantial loss of accuracy depends on how closely the run can be reckoned. In celestial navigation, as in piloting, you essentially are trying to establish the intersection of two or more lines of position. A single observation is insufficient to obtain a fix, however it can be used with a loran line, etc. to provide a fix.
Two Circles
Observation of two bodies at the same time gives the navigator two circles of equal altitude. The circles intersect each other at two points, and because the ship is somewhere on each one of them, she must be at one or the other points of intersection. A circle on the surface of the earth, on every point of which the altitude of a given celestial body is the same at a given instant, the pole of this circle is the geographical position of the body, and the great-circle distance from this pole to the circle is the zenith distance of the body.
Line of Position
In practice, you may not be able nor will you need to plot the whole of a circle of equal altitude. The position is usually known within 10 miles and possibly even less than that. Inside these limits, the curve of the arc of a circle of equal altitude is hardly perceptible, and the arc is plotted and regarded as a straight line. Such a line, comprising enough of the arc of a circle of equal altitude to cover the probable limits of a position, is called a Sumner line of position or just a line of position.
Two Lines of Position
The prefered method of establishing two lines of position is by observing two different bodies, although two lines may be obtained from the same body by observations taken at different times. As in piloting, the nearer the two lines approach at right angles to each other, the more accurate the fix. When two lines are determined by observing the same body, the first line established is brought forward the distance run on the course steered. For example, if a ship steams 27 miles on course 315° between the first and second observations, obviously the position is on a line parallel with the first one established, but drawn 27 miles away (to scale) on the course line 315°. Intersection of the line established by the second observation with the advanced line of the first observation is a fix. The fix progressively decreases in accuracy, depending on how far the first line is advanced. You should not advance such a time for more than 5 hours of a run.
Determining a Line of Position
At this point you might be entitled to complain that much has been said concerning what a line of position tells you, but very little has been said about how you should determine it in the first place. I will get to that part now. You probably have grasped the idea that what you want to find out is which circle of equal altitude you are on, and what this altitude is. To draw such a circle, you would need a chart covering an extensive area, unless the heavenly body's altitude approached 90° you do not determine the entire circle but only a portion of its arc, this is so small that it is plotted and regarded as a straight line. An assumed position (AP) is selected according to the rule of 30' of your DR position for the time of sight. Observation of a body provides sextant altitude. Sextant altitude is then corrected to obtain observed altitude (Ho). The body's altitude from the assumed position (called the computed altitude (Hc) and its azimuth angle are determined from tables. The azimuth angle is then converted to azimuth.
After selecting an AP, draw the azimuth through the AP. Along the azimuth measure off the altitude intercept (difference between the observed altitude and the computed altitude). At the end of this measurement, draw a perpendicular line, which is the LOP. You must know whether altitude intercept (a) should be measured from AP TOWARD the body or from AP AWAY from the body. Sometimes the initials for Coast Guard Academy (CGA) can be helpful. If the computed altitude is greater than the observed altitude, altitude intercept (a) is measured away from the body. In other words applying the CGA "memory aid," you have computed, greater away (CGA). Another memory aid is Ho Mo To when it is towards.
Selecting Bodies for Observation
Before going into problems and tables, mention should be made of a few items concerned with selecting astronomical bodies for observation. Observing two heavenly bodies in rapid succession is the most convenient method of finding two lines of position necessary to establish a fix. Noting three bodies gives three lines, and these three define the fix more accurately (as in piloting). Accuracy of the fix established by intersecting lines of position depends upon the angle between the lines. The nearer this angle approaches 90°, the more accurate the fix. Actually, sights seldom are taken on two or more bodies simultaneously. Instead, the navigator decides which bodies to observe, then takes a round of sights, each one timed exactly. Resulting lines of position are advanced or retarded in the amount of the ship's run between the time of observation and the time of the desired fix.
The ideal situation for lines of position established by observing three bodies would be that where the bodies lie 120° apart in azimuth. An ideal fix using four bodies would include two north / south lines and two east / west lines of position to form a box. Lines perpendicular to the course are valuable for checking the run. Those lines parallel to it are helpful in deciding the accuracy of the course made good. Concerning altitude, best results are obtained by observation of bodies whose altitudes are between 10° and 65° In general, observations are taken from bodies whose altitudes are between 10° and 80°.
Advancing and Retarding the LOP
Several methods may be used to advance a line of position.The most frequent method consists of advancing the AP in the direction of your course and for the distance of the run, and drawing the new LOP. To retard a LOP, just do the reciprocal of your course for the distance run.
Estimated Position by Celestial Navigation
If appreciable time has elapsed since the determination of the last fix of the ship's position at sea, the error in the DR plot may change where the ship's actual position is well away from the DR plot. A single line of position can be useful in establishing an estimated position. If an accurate line is obtained, the actual position is somewhere on this line. In the absence of better information, a perpendicular from the previous DR position or EP to the line of position establishes the new EP. The foot of the perpendicular from the AP has no significance in this regard, since it is used only to locate the line of position.
The establishment of a good EP is dependent upon accurate interpretation of all information available. Generally, such ability can be acquired only by experience. If, in the judgment of the navigator, the course has been made good, but the speed has been uncertain, the best estimate of the position might be at the intersection of the course line and the LOP. If the speed since the last fix is considered accurate, but the course is considered uncertain, the EP might be at the intersection of the line of position and an arc centered on the previous fix and of a radius equal to the distance traveled.
Monday, January 5, 2009
How to Use the Marine Sextant
The marine sextant is designed to measure angles, either horizontally or vertically. The most common use of the sextant is for celestial observations using vertical angles between celestial objects and the horizon. It can also be used to measure the horizontal angle between two terrestial objects to get a line of position.Sun Sights
For a Sun sight, hold the sextant vertically and direct the sight line at the horizon directly below the Sun. Move the shade glasses into the line of sight, move the index arm outward along the arc until the reflected image appears in the horizon glass near the direct view of the horizon. Rock the sextant slightly to the right and left to make sure it is perpendicular. As you rock the sextant, the image of the Sun appears to move in an arc. The sextant is vertical when the Sun appears at the bottom of the arc. This is the correct position for making the observation. The Sun’s reflected image appears at the center of the horizon glass, one half appears on the silvered part, and the other half appears on the clear part. Move the index arm with the drum or vernier slowly until the Sun's exactly on the horizon. This takes some practice and some people will get a error by bringing the sun too far down.Some navigators will let the body contact the horizon by its own motion, bringing it slightly below the horizon if rising, and above if setting. At the instant the body is touching the horizon, note the time. This is the uncorrected reading of the sextant.
Moon Sights
When observing the Moon, use the same procedure as for the Sun. Sights of the Moon are best made during daylight hours or twilight when it is easier to see it and the horizon. At night you can get false horizons below the Moon because the Moon illuminates the water below it.
Star and Planet Sights
Most people find the Sun and Moon are easy to find in the sextant, but the stars and planets are harder to locate because your field of view is so small. There are several ways to help you locate a star or planet.
1. Set the index arm and micrometer drum on 0° and direct the line of sight at the body to be observed. Then keep the reflected image of the body in the mirrored half of the horizon glass, swing the index arm out and rotate the sextant down. Keep the sighted image in the mirror until the horizon appears in the clear part of the horizon glass. This way is hard to do, if the body is lost while it is being brought down, you might not find it, which means you will have to start over again.
2. Direct the line of sight at the body while holding the sextant upside down. Slowly move the index arm out until the horizon appears in the horizon glass. Then invert the sextant and take the sight in the usual manner. This is my preferred method, you will have less of a chance of losing the body.
3. Determine in advance the approximate altitude and azimuth of the body by the starfinder 2102-D. Set the sextant at the indicated altitude and face in the direction of the azimuth. The image of the body should appear in the horizon glass with a little searching. When measuring the altitude of a star or planet, bring its center down to the horizon. Stars and planets have no upper or lower limb that you can see, you must observe the center of the point of light. Because stars and planets don't have a limb and because their visibility may be limited, the method of letting a star or planet intersect with the horizon by its own motion is not recommended. As with the sun and moon, rock the sextant to get the perpendicularity.
Taking a Sight
Get the altitudes and azimuths for several stars or planets when preparing to take celestial sights. Choose the stars and planets that will give you the best bearing spread. Try to select bodies with a altitude between 20° and 65°. Take sights of the brightest stars first in the evening, take sights of the brightest stars last in the morning. Sometimes weather or other ships can obscure the horizon directly below a body that you want to observe. You could try a back sight using the opposite point of the horizon as the reference. For this you must face away from the body and observe the supplement of the altitude. If the Sun or Moon is observed, what appears in the horizon glass to be the lower limb is the upper limb, and vice versa. In the case of the Sun, it is usually best to observe what appears to be the upper limb. The arc that appears when rocking the sextant for a back sight is inverted, that is the highest point indicates the position of perpendicularity. If more than one telescope is furnished with the sextant, the erecting telescope is used to observe the Sun.
The collar that the sextant telescope fits may be adjusted in or out in relation to the frame. When moved in, more of the mirrored half of the horizon glass is visible to the navigator, and a star or planet is best observed when the sky is bright. Near the darker limit of twilight, the telescope can be moved out, giving a broader view of the clear half of the glass, and making the less distinct horizon more easily seen. When measuring an altitude, have an assistant note and record the time, with a "stand-by" warning when the measurement is almost ready, and a "mark" at the moment a sight is made. If a flashlight is needed to see the comparing watch, your assistant should be careful not to interfere with the navigator’s night vision. If an assistant is not available to time the observations, the observer holds the watch in the palm of his left hand, leaving his fingers free to turn the tangent screw of the sextant. After making the observation, note the time as quick as possible. The delay between completing the altitude observation and noting the time should not be more than one or two seconds.
Saturday, January 3, 2009
LOP Altitude and Azimuth Questions
Line of Position - Sun (Computed Altitude and Azimuth)
1. At 0600 ZT, on 24 July 1981, your DR position is LAT 22°37.0' N, LONG 32°45.0' W. You are steering 185°T at a speed of 20.0 knots. Determine the computed altitude (Hc) and azimuth (Zn) for an observation of the Sun's lower limb taken at 1030 ZT. At this time the chronometer reads 00h 30m 16s and is 0m 31s slow.
A. Hc 64°27.5' - Zn 092.3°
B. Hc 64°30.8' - Zn 090.1°
C. Hc 64°41.7' - Zn 087.8°
D. Hc 64° 44.2' -Zn 094.7°
2. On 18 August 1981, at 0600 ZT, morning stars were observed, and the vessel's position was determined to be LAT 19°48'N, LONG 108° 34'W. Your vessel is steaming on course 166°T at a speed of 16 knots. An observation of the Sun's lower limb is made at 1036 ZT. The chronometer reads 05h 34m 48s and is slow 01m 24s. What is the computed altitude (Hc) and azimuth (Zn) for this 1036 ZT observation using the assumed position method?
A. Hc 65°18.5' - Zn 102.1°
B. Hc 65°14.8' - Zn 100.4°
C. Hc 65°11.3' - Zn 099.4°
D. Hc 65° 07.2' - Zn 101.2°
Line of Position - Sun (Observed Altitude and Azimuth)
3. On 8 August 1981, at 0545 ZT, morning stars were observed, and the vessel's position was determined to be LAT 26°16.0'S, LONG 94°16.0'E. Your vessel is steaming at 20.0 knots on a course of 346°T. A sextant observation of the Sun's lower limb is made at 0905 ZT. The chronometer reads 03h 02m 52s, and the sextant altitude (hs) is 38°07.5'. The index error is 5.2' off the arc, and the chronometer error is 2m 17s slow. Your height of eye on the bridge is 72 feet (22.0 meters). What is the observed altitude (Ho) and azimuth (Zn) of this sight using the assumed position?
A. 38°19.4' -048.4° T
B. 38°19.4' - 131.6° T
C. 38°54.9' -048.4° T
D. 38°54.9' -131.6° T
4. On 9 November 1981, at 0426 ZT, your position was LAT 25°17.0'S, LONG 154°16.0'E. Your vessel is steaming at 14.0 knots on course 066°T. A sextant observation ofthe Sun's lower limb is made at 0837 ZT. The chronometer reads 10h 35m 21s, and the sextant altitude (hs) is 50°26.9'. The index error is 1.5' on the arc, and the chronometer error is 01m 48s slow. Your height of eye on the bridge is 56.0 feet. What is the observed altitude (Ho) and azimuth (Zn) of this sight using the assumed position?
A. 50°18.1' - 086.3° T
B. 50°18.1' - 093.7° T
C. 50°33.5' - 085.9° T
D. 50°33.5' - 093.7° T
Line of Position - Sun (Azimuth and Intercept)
5. On 12 April 1981, at 0515 ZT, morning stars were observed, and the vessel's position was determined to be LAT 21°05'S, LONG 16°30'W. Your vessel is steaming at 19 knots on a course of 278°T. A sextant observation of the Sun's lower limb is made at 0930 ZT. The chronometer reads 10h 28m 25s, and the sextant altitude (hs) is 40° 15 .9'. The index error is 2.5' off the arc, and the chronometer error is 2m 15s slow. Your height of eye on the bridge is 57 feet. What are the intercept (a) and azimuth (Zn) from the assumed position of this sight?
A. Zn 057.7° - a 15.4' Towards
B. Zn 057.0° - a 17.7' Away
C. Zn 122.3° - a 17.7' Away
D. Zn 123.0° - a 22.7' Away
6. On 4 June 1981, at 0630 ZT, morning stars were observed, and the vessel's position was determined to be LAT 26°15'S, LONG 121°20'W. Your vessel is steaming at 13.0 knots on a course of 246°T. A sextant observation ofthe Sun's lower limb is made at 0915 ZT. The chronometer reads 05h 14m 27s, and the sextant altitude is 25°57.8'. The index error is 2.1' off the arc, and the chronometer error is 0m 53s slow. Your height of eye is 39.0 feet. What is the intercept (a) and azimuth (Zn) of this sight using the assumed position method?
A. Zn 044.6° - a 1.7' Away
B. Zn 044.6° - a 2.5' Towards
C. Zn 135.1° - a 1.7' Away
D. Zn 135.1° - a 2.5' Towards
1. At 0600 ZT, on 24 July 1981, your DR position is LAT 22°37.0' N, LONG 32°45.0' W. You are steering 185°T at a speed of 20.0 knots. Determine the computed altitude (Hc) and azimuth (Zn) for an observation of the Sun's lower limb taken at 1030 ZT. At this time the chronometer reads 00h 30m 16s and is 0m 31s slow.
A. Hc 64°27.5' - Zn 092.3°
B. Hc 64°30.8' - Zn 090.1°
C. Hc 64°41.7' - Zn 087.8°
D. Hc 64° 44.2' -Zn 094.7°
2. On 18 August 1981, at 0600 ZT, morning stars were observed, and the vessel's position was determined to be LAT 19°48'N, LONG 108° 34'W. Your vessel is steaming on course 166°T at a speed of 16 knots. An observation of the Sun's lower limb is made at 1036 ZT. The chronometer reads 05h 34m 48s and is slow 01m 24s. What is the computed altitude (Hc) and azimuth (Zn) for this 1036 ZT observation using the assumed position method?
A. Hc 65°18.5' - Zn 102.1°
B. Hc 65°14.8' - Zn 100.4°
C. Hc 65°11.3' - Zn 099.4°
D. Hc 65° 07.2' - Zn 101.2°
Line of Position - Sun (Observed Altitude and Azimuth)
3. On 8 August 1981, at 0545 ZT, morning stars were observed, and the vessel's position was determined to be LAT 26°16.0'S, LONG 94°16.0'E. Your vessel is steaming at 20.0 knots on a course of 346°T. A sextant observation of the Sun's lower limb is made at 0905 ZT. The chronometer reads 03h 02m 52s, and the sextant altitude (hs) is 38°07.5'. The index error is 5.2' off the arc, and the chronometer error is 2m 17s slow. Your height of eye on the bridge is 72 feet (22.0 meters). What is the observed altitude (Ho) and azimuth (Zn) of this sight using the assumed position?
A. 38°19.4' -048.4° T
B. 38°19.4' - 131.6° T
C. 38°54.9' -048.4° T
D. 38°54.9' -131.6° T
4. On 9 November 1981, at 0426 ZT, your position was LAT 25°17.0'S, LONG 154°16.0'E. Your vessel is steaming at 14.0 knots on course 066°T. A sextant observation ofthe Sun's lower limb is made at 0837 ZT. The chronometer reads 10h 35m 21s, and the sextant altitude (hs) is 50°26.9'. The index error is 1.5' on the arc, and the chronometer error is 01m 48s slow. Your height of eye on the bridge is 56.0 feet. What is the observed altitude (Ho) and azimuth (Zn) of this sight using the assumed position?
A. 50°18.1' - 086.3° T
B. 50°18.1' - 093.7° T
C. 50°33.5' - 085.9° T
D. 50°33.5' - 093.7° T
Line of Position - Sun (Azimuth and Intercept)
5. On 12 April 1981, at 0515 ZT, morning stars were observed, and the vessel's position was determined to be LAT 21°05'S, LONG 16°30'W. Your vessel is steaming at 19 knots on a course of 278°T. A sextant observation of the Sun's lower limb is made at 0930 ZT. The chronometer reads 10h 28m 25s, and the sextant altitude (hs) is 40° 15 .9'. The index error is 2.5' off the arc, and the chronometer error is 2m 15s slow. Your height of eye on the bridge is 57 feet. What are the intercept (a) and azimuth (Zn) from the assumed position of this sight?
A. Zn 057.7° - a 15.4' Towards
B. Zn 057.0° - a 17.7' Away
C. Zn 122.3° - a 17.7' Away
D. Zn 123.0° - a 22.7' Away
6. On 4 June 1981, at 0630 ZT, morning stars were observed, and the vessel's position was determined to be LAT 26°15'S, LONG 121°20'W. Your vessel is steaming at 13.0 knots on a course of 246°T. A sextant observation ofthe Sun's lower limb is made at 0915 ZT. The chronometer reads 05h 14m 27s, and the sextant altitude is 25°57.8'. The index error is 2.1' off the arc, and the chronometer error is 0m 53s slow. Your height of eye is 39.0 feet. What is the intercept (a) and azimuth (Zn) of this sight using the assumed position method?
A. Zn 044.6° - a 1.7' Away
B. Zn 044.6° - a 2.5' Towards
C. Zn 135.1° - a 1.7' Away
D. Zn 135.1° - a 2.5' Towards
Friday, January 2, 2009
Thursday, January 1, 2009
Celestial Navigation Observations
Complete Solution for Celestial Navigation Sights
1. After reading this you will be able to compute a complete celestial observation of the Sun using the Nautical Almanac and the Sight Reduction Tables for Marine Navigation, Pub 229 and plot a celestial line of position.
The December 2008 blogs have dealt with all the aspects needed for determining a line of position from an observation of a celestial body. This will give you the complete solution using the Nautical Almanac and the Sight Reduction Tables for Marine Navigation Pub 229. The steps involved will be covered in the order in which they are to be taken.
Pub. 229 Method
Pub. 229 Sight Reduction Tables for Marine Navigation is a set of six volumes of pre-calculated solutions for the computed altitude (Hc) and the azimuth angle (Z) of the navigational triangle. Entering arguments for the tables are local hour angle (LHA) expressed in whole degrees. This is done by using an assumed longitude, vice a DR longitude, assumed latitude in whole degree, and declination. Values of Hc and Z are tabulated for each whole degree of each of the entering arguments. Tables inside the front and back covers of each volume allow for interpolation of Hc and Z for the exact declination. No interpolation is necessary for LHA or assumed latitude.
Working Sights With Pub. 229
To work a sight with Pub. 229, you enter the tables by selecting the proper volume and turning to the page with the appropriate LHA. Using the assumed latitude and declination extract the tabulated values for Hc and Z. You then determine the exact value of Hc and Z corresponding to the time of observation by interpolation by using the interpolation tables or using the formula that the tables are based upon. For our purposes I will use the formula method.
To find the intercept distance (a), this final Hc is compared to the observed altitude (Ho). If the computed altitude Hc is greater than observed altitude Ho the intercept is AWAY from the direction of the GP of the body. If the Ho is greater than the Hc it is TOWARDS the direction of the GP of the body. When working out a celestial sight a form should be followed so that there will be less chance for leaving out any pertinent information.
Sight Reduction
You will need a Sextant, a Watch, Nautical Almanac the Tables HO 249 or HO 229 marine or air sight reduction tables, I prefer HO 229. For sighting stars, a Star-Finder No. 2101-D, Parallell Rulers or Triangles, Dividers, 0.5 mm Pencil with a good eraser, Scratch Paper, and Universal Plotting Sheets.
With the "intercept" method, you will be comparing the position you think you might be in from dead reckoning on a boat, with what you actually observe. Your observed altitude is compared to a calculated altitude, calculated to be what altitude you would get if you were actually at the position you chose as your assumed position. You have to observe an altitude with the sextant and put your figures on your worksheet and along with the tables what the altitude would be if seen from the assumed position.
The "v" is an extra correction for additional longitude movement of the body, and "d" is an extra correction for additional declination movement. The sun has no "v" correction and the stars have no "v" or "d" correction. The sun needs the "d", and the planets and moon need both "v" and "d".
These steps are for a SUN LINE only.
Step 1: Setup a plotting sheet, DR ahead and enter it in your format.
Step 2: Apply your IE, if it is on the arc, subtract it, if it is off the arc add.
Step 3: Using the DIP Table on the inside cover of the Nautical Almanac, enter with your height of eye. Dip correction is always a minus correction.
Step 4: On page A2 "Altitude Correction Tables 10° - 90° Sun, Stars, Planets" Enter with your Ha and find the Altitude Correction. Remember that Lower limb are always + corrections and upper limbs are - corrections for the sun.
Step 5: Compute your corrected chronometer time.
Step 6: Using GMT, and Greenwich date of observation, enter the Nautical Almanac and record tabulated hourly value of GHA and Tab. Dec. in your format.
Step 7: At the foot of each declination sub-column, get the"d corr". This number is called the "d" correction. This is the average over the three day period that the declination changes per hour of GMT. The "d" is recorded on the "d corr." line off to the side. This is a correction, as with any correction, it is either a + or - . If the declination is increasing (getting larger), then it is a plus ( + ) correction, if the declination is decreasing, then it is a minus ( - ) correction.
Step 8: Turn to the yellow pages of the Nautical Almanac, and find the minute page, enter with your seconds. Then under the Sun and Planets column find the increase in the sun's GHA since the last tabulated (hourly) value, this is your M & S correction and enter this in your format. Always add the GHA and "M & S ''tabulated value together to get the total GHA.
Step 9: While on the minute page under the "v" or "d" correction column, find the "d" on the left hand side. This will be the declination (dec) of the sun at the time of sight.
Step 10: In full sight reduction, LHA has to end in the whole degree and YOUR ASSUMED LONGITUDE has to be within 30' of your DR Longitude.
This is where it's easy to make a mistake.
EXAMPLE for East Longitude
Say your DR Longitude is 93° 58.0 E
GHA 224° 52.9
A Long +94° 07.1 E
LHA 319° 00.0
EXAMPLE for West Longitude
Say your DR Longitude is 93° 58.0 W
GHA 224° 52.9
A Long. - 94° 52.9 W
LHA 130° 00.0
GHA 224° 52.9
A Long. - 93° 52.9 W
LHA 131° 00.0
Enter Pub. 229 with LHA, ALat, and Dec. Then extract the tabulated Hc, base Z, Z for the next whole degree of declination, and the rule for converting Z to Zn and enter this into your format. Make sure to note the sign of the differences. Then using the formula:
Difference x declination increments ÷ 60 = correction, determine your corrections.
Step 12: Add or subtract your corrections to your Tab Hc and base Z to get the Hc and Z.
Step 13: Fill in your Ho. Find the difference between Hc and Ho, this will give your altitude intercept (a). Next you must determine if (a) is; (A-away or T- towards the bearing Zn). You say to yourself, Coast Guard Academy-Computed Greater Away, if not then it is towards, Ho Mo To.
Step 14: Compute the Zn by following the rule.
Step 15: Fill in your assumed latitude (ALat) and assumed longitude (ALong).
1. After reading this you will be able to compute a complete celestial observation of the Sun using the Nautical Almanac and the Sight Reduction Tables for Marine Navigation, Pub 229 and plot a celestial line of position.
The December 2008 blogs have dealt with all the aspects needed for determining a line of position from an observation of a celestial body. This will give you the complete solution using the Nautical Almanac and the Sight Reduction Tables for Marine Navigation Pub 229. The steps involved will be covered in the order in which they are to be taken.
Pub. 229 Method
Pub. 229 Sight Reduction Tables for Marine Navigation is a set of six volumes of pre-calculated solutions for the computed altitude (Hc) and the azimuth angle (Z) of the navigational triangle. Entering arguments for the tables are local hour angle (LHA) expressed in whole degrees. This is done by using an assumed longitude, vice a DR longitude, assumed latitude in whole degree, and declination. Values of Hc and Z are tabulated for each whole degree of each of the entering arguments. Tables inside the front and back covers of each volume allow for interpolation of Hc and Z for the exact declination. No interpolation is necessary for LHA or assumed latitude.
Working Sights With Pub. 229
To work a sight with Pub. 229, you enter the tables by selecting the proper volume and turning to the page with the appropriate LHA. Using the assumed latitude and declination extract the tabulated values for Hc and Z. You then determine the exact value of Hc and Z corresponding to the time of observation by interpolation by using the interpolation tables or using the formula that the tables are based upon. For our purposes I will use the formula method.
To find the intercept distance (a), this final Hc is compared to the observed altitude (Ho). If the computed altitude Hc is greater than observed altitude Ho the intercept is AWAY from the direction of the GP of the body. If the Ho is greater than the Hc it is TOWARDS the direction of the GP of the body. When working out a celestial sight a form should be followed so that there will be less chance for leaving out any pertinent information.
Sight Reduction
You will need a Sextant, a Watch, Nautical Almanac the Tables HO 249 or HO 229 marine or air sight reduction tables, I prefer HO 229. For sighting stars, a Star-Finder No. 2101-D, Parallell Rulers or Triangles, Dividers, 0.5 mm Pencil with a good eraser, Scratch Paper, and Universal Plotting Sheets.
With the "intercept" method, you will be comparing the position you think you might be in from dead reckoning on a boat, with what you actually observe. Your observed altitude is compared to a calculated altitude, calculated to be what altitude you would get if you were actually at the position you chose as your assumed position. You have to observe an altitude with the sextant and put your figures on your worksheet and along with the tables what the altitude would be if seen from the assumed position.
The "v" is an extra correction for additional longitude movement of the body, and "d" is an extra correction for additional declination movement. The sun has no "v" correction and the stars have no "v" or "d" correction. The sun needs the "d", and the planets and moon need both "v" and "d".
These steps are for a SUN LINE only.
Step 1: Setup a plotting sheet, DR ahead and enter it in your format.
Step 2: Apply your IE, if it is on the arc, subtract it, if it is off the arc add.
Step 3: Using the DIP Table on the inside cover of the Nautical Almanac, enter with your height of eye. Dip correction is always a minus correction.
Step 4: On page A2 "Altitude Correction Tables 10° - 90° Sun, Stars, Planets" Enter with your Ha and find the Altitude Correction. Remember that Lower limb are always + corrections and upper limbs are - corrections for the sun.
Step 5: Compute your corrected chronometer time.
Step 6: Using GMT, and Greenwich date of observation, enter the Nautical Almanac and record tabulated hourly value of GHA and Tab. Dec. in your format.
Step 7: At the foot of each declination sub-column, get the"d corr". This number is called the "d" correction. This is the average over the three day period that the declination changes per hour of GMT. The "d" is recorded on the "d corr." line off to the side. This is a correction, as with any correction, it is either a + or - . If the declination is increasing (getting larger), then it is a plus ( + ) correction, if the declination is decreasing, then it is a minus ( - ) correction.
Step 8: Turn to the yellow pages of the Nautical Almanac, and find the minute page, enter with your seconds. Then under the Sun and Planets column find the increase in the sun's GHA since the last tabulated (hourly) value, this is your M & S correction and enter this in your format. Always add the GHA and "M & S ''tabulated value together to get the total GHA.
Step 9: While on the minute page under the "v" or "d" correction column, find the "d" on the left hand side. This will be the declination (dec) of the sun at the time of sight.
Step 10: In full sight reduction, LHA has to end in the whole degree and YOUR ASSUMED LONGITUDE has to be within 30' of your DR Longitude.
This is where it's easy to make a mistake.
EXAMPLE for East Longitude
Say your DR Longitude is 93° 58.0 E
GHA 224° 52.9
A Long +94° 07.1 E
LHA 319° 00.0
EXAMPLE for West Longitude
Say your DR Longitude is 93° 58.0 W
GHA 224° 52.9
A Long. - 94° 52.9 W
LHA 130° 00.0
GHA 224° 52.9
A Long. - 93° 52.9 W
LHA 131° 00.0
Enter Pub. 229 with LHA, ALat, and Dec. Then extract the tabulated Hc, base Z, Z for the next whole degree of declination, and the rule for converting Z to Zn and enter this into your format. Make sure to note the sign of the differences. Then using the formula:
Difference x declination increments ÷ 60 = correction, determine your corrections.
Step 12: Add or subtract your corrections to your Tab Hc and base Z to get the Hc and Z.
Step 13: Fill in your Ho. Find the difference between Hc and Ho, this will give your altitude intercept (a). Next you must determine if (a) is; (A-away or T- towards the bearing Zn). You say to yourself, Coast Guard Academy-Computed Greater Away, if not then it is towards, Ho Mo To.
Step 14: Compute the Zn by following the rule.
Step 15: Fill in your assumed latitude (ALat) and assumed longitude (ALong).
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