MODULE 2: TRADITIONAL ASTRO-POSITION 1. Basic principle of the position line Using the sun and stars to find our way is a very old art. It is known the Vikings used the sun and the pole star to sail a line of latitude on their voyages to North America. However until the mid -18th century there was no certain method of fixing a ships position. Ocean navigation remained a hit and miss affair with explorers dying of thirst sailing to and from along lines of latitude looking for islands. In 1837 Captain Thomas Sumner of Boston, USA developed a method of establishing position lines from sights. Sumner's insight lead to the understanding that as a ship sails along a celestial body's azimuth toward (or away from) the point where a line between the body and the center of the earth intersects the earth's surface (the Geographical Position), then the navigator will find that his sextant altitudes increase (or decrease). History credits Commander Adolphe-Laurent-Anatole Marcq de Blonde de Saint-Hilaire of the French Navy for using Sumner's insight to introduce in 1875 the altitude-difference ("intercept") method of determining a line of position. His work triggered "the new navigation" (nouvelle navigation) that we use today. His understanding was the reverse of Sumner's insight, namely that "...the Line of Position moves away from the Assumed Position one mile along the azimuth of the sun for every one minute that the observed altitude is less than the altitude calculated for the Assumed Position" (Vanvaerenbergh and Ifland). Vanvaerenbergh and Ifland explain that his method was based on these principles, some of which were already understood at the time. Saint-Hilaire patiently and creatively crystallized and then extended earlier ideas, particularly: 1. Determine the altitude and azimuth of a Body for an estimated position for a given date and time. 2. Determine whether the observed altitude is greater or less than the calculated altitude. 3. Convert minutes of difference in altitudes to miles.

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4. Place a line perpendicular to the azimuth that many miles toward or away from the geographic position of the body, relative to the estimated position. Saint-Hilaire's contribution was not a single stunning insight like Sumner's, rather he carefully developed a practical method for plotting a celestial line of position, making important insights along the way. His key contribution "...was to recognize that the same concept (correction along the azimuth) could be applied to the difference between the observed altitude of a celestial body and the altitude calculated from an assumed position to determine the true line of position" (Vanvaerenbergh and Ifland). Saint-Hilaire was an extremely competent and experienced naval navigator and mathematician. He not only developed and explained the intercept method, but he also carefully worked out some of its mathematical sources of error. He rose to the rank of Rear Admiral, and then tragically died of infection in Algiers at age 57. The principle of the LOP

Each celestial body for a certain moment given will have a geographical position (GP) on the earth’s surface. Someone standing on that position will see this celestial body in his zenith. Furthermore, all observers sighting the celestial body with a same

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altitude but different direction , will stay on a circle (of position) having the GP of the celestial body as centre.

Since stars move with the celestial sphere, their GPs also move on the surface of the Earth. And they are fast. The Sun's GP, for example, travels a mile every four seconds. The GPs of other stars,

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closer to the celestial poles, move more slowly. The GP of Polaris moves very slowly, since it's very close to the North Pole. Because both Earth and Celestial equators are in the same plane, the latitude of the GP is equal to the declination of the star. The longitude of the GP is known as Greenwich Hour Angle - or GHA – with reference the prime meridian.. We can determine, using a Nautical Almanac, the GP of a star (it's GHA and declination) in any moment of time. But we must know the exact time of the observation. As we have seen, 4 seconds may correspond to one mile in the GP of a star. This shows the importance of having a watch with the correct time for the celestial navigation.

In the figure , the GP of the star is represented by X and the Zenith of the observer by Z. The distance XZ, from the GP of the star to the point Z of the navigator is called Zenithal Distance. This distance, as we have seen, can be expressed in miles or minutes of degrees, since it's an arc on the surface of the Earth ( mile = 1 minute of degree).

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The angle that XZ makes with the True North (i.e. the "bearing" of the star) is called Azimuth ( Az ). The stars are at a great distance from the earth and so the light rays coming from them that reach the Earth are parallel. Therefore, as illustrated in the figure below, we may say that the distance XZ (as an angle) is equal to the angle that the navigator observes between the star and the vertical. This is important. The distance XZ, measured as an angle, is equal to the angle that the navigator observes between the star and the vertical.

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It's a lot easier to measure the angle between the star and the horizon. This important angle for the celestial navigator is called altitude (h) of the star. The altitude of a star is taken with the sextant held in the vertical plane, measuring the angle between the horizon and the star. In the previous figure, we can see that the zenith distance (ζ) equals 90° less the altitude of the star. So we have a first correspondence:

ζ = 90° - h To obtain one’s position, we need at least 3 circles of position in order to distinguish the correct combination of intersection.

But drawing these big circles would require really big charts! The position obtained on such a chart of large scale would not be precise enough and be readable to some degrees ! In such we have to find a way to plot our position with great accuraccy and readable up to tenths of minutes in latitude and longitude. We work around this problem by making a guess of our position (DR). No matter how lost we are, we can always make a guess. Using this assumed position we can calculate the expected altitude for the star at a given time for that position. In such we 46

obtain two circles: one related to the observed altitude and one related to the expected altitude.

As we can not plot the GP of the celestial body on the chart (will be out of reach of the chart), we can not draw the correct position circles. Instead we will use the tangent to the circles which is a line of position (LOP) on which is our true current position at time of observation.

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2. the intercept

In traditional celestial navigation the determination of a Line of Position involves the computation of the GP of the star (GHA and declination) using the Nautical Almanac and the solution of the Navigation Triangle PXZ, formed by the terrestrial elevated pole (P), the GP of the star (X) and the assumed position of the navigator (Z). The altitude by observation is known by sextant and is called the true altitude (hv).

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The expected or also called the calculated altitude (hc) based upon our assumed position has to be calculated by using the navigation triangle. The difference between the true and calculated altitude is called the intercept ( ∆h ).

The navigation triangle has (see figure above): - the sides: polar distance (co-declination) ∆, the co-latitude col of the assumed position and the zenithal distance ζ;

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- the angles: azimuth Az (angle between vertical celestial body and meridian of assumed longitude) and Polar angle P (in the figure tW) (angle between meridian of assumed longitude and hour circle of celestial body).

In such we can use the formulae: Cos ζ = cos ∆ x cos col + sin ∆ x sin col x cos P Replacing the complements, we find: Sin hc = sin δ x sin la + cos δ x cos la x cos P la = assumed latitude δ = declination of celestial body P = polar angle hv – hc = ∆h (when intercept is positive : towards celestial body, if negative, away from celestial body) The final result will be:

The intersection of several position lines will result in a fix.

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Assumed position

Determination of the Astronomical Position It's not necessary to draw the lines of position when using Navigator software. But let's see how this is done using pencil and paper: 1. Plot your assumed position. 2. Using a parallel ruler, draw a line passing on the assumed position, in the direction of the Azimuth of the star. 3. Over this line, measure the intercept - in the direction of the star or contrary to it - according to the sign of the intercept. 4. Draw the line of position, orthogonal to the Azimuth, at this point.

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hc DR

hv TP

GP Az

DR= assumed position (dead reckoning) TP = true position DR-TP = ∆h GP = geographical position celestial body Detailed Nautical Charts are usually only available for places near the shore. When in high seas, we normally don't have charts with the adequate scale to plot our position. Special plotting paper is used instead only indicating the parallels and meridians and a compass rose . 3. azimuth To plot a position line demands 3 elements at the same time from the observer: - altitude measurement by sextant - reading of chronometer to obtain exact UT time of observation - compass bearing Unfortunately, the compass is not precise enough for celestial navigation. One error of just 3°, common when reading a compass, corresponds to tenth’s of miles of error ! Not an acceptable error. Therefore it is better to calculate the azimuth giving a true bearing up to parts of a degree. 52

hc DR

hv TP

GP Az

DR= assumed position (dead reckoning) TP = true position DR-TP = ∆h GP = geographical position celestial body Detailed Nautical Charts are usually only available for places near the shore. When in high seas, we normally don't have charts with the adequate scale to plot our position. Special plotting paper is used instead only indicating the parallels and meridians and a compass rose . 3. azimuth To plot a position line demands 3 elements at the same time from the observer: - altitude measurement by sextant - reading of chronometer to obtain exact UT time of observation - compass bearing Unfortunately, the compass is not precise enough for celestial navigation. One error of just 3°, common when reading a compass, corresponds to tenth’s of miles of error ! Not an acceptable error. Therefore it is better to calculate the azimuth giving a true bearing up to parts of a degree. 52

We start again from our assumed position (DR = dead reckoning). Even when we are miles away from our true position, the azimuth will not change that much as it is the case by observing the direction by compass.

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Pn ∆

P

col

Az ζ Z

In the above spherical triangle are known: - col (co-latitude) - polar distance ∆ - Polar Angle P We can use the formulae: Cos P x cos col = sin P x cotg Az + sin col x cotg ∆ Cos P x sin la = sin P x cotg Az + cos la x tg δ Cos P x sin la – cos la x tg δ = sin P x cotg Az

( : sin P)

Sin la cos la x tg δ -------- - ---------------- = cotg Az tg P sin P

( : cos la)

tg la tg δ cotg Az ------ - -------- = ----------tg P sin P cos la or A

+

B

=

C

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In the Norie’s Nautical Tables, pre-computed tables are available also called the ABC tables. Those tables have been made based upon above mentioned formulae. An extract of each tabel is given in next pages. We enter in table A with our assumed latitude and LHA of the celestial body and find a factor A; We enter table B with the declination and LHA of celestial body and find factor B; Add both factors and use the sign of each factor as per instructions of the tables and find factor C. Factor C is N or S. Enter table C with assumed latitude and factor C and read the corresponding azimuth angle ( N/S same sign as C, E or W sign of Polar angle). Example : A= B= C=

1.03 N 1.63 S 0.60 S

P = west

Assumed latitude: 18° N Az = S 60.3° W

v = 240.3°

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Another way to find the azimuth is by the sinus rule. In the same navigation triangle we know: - polar Angle P - polar distance ∆ (complement declination) - observed zenithal distance (90° - hv), ζ sin Az sin P -------- = ------sin ∆ sin ζ sin Az = sin P x cos δ x sec hv

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4. amplitudo The amplitudo is a special case of azimuth. If the celestial body is observed when its center is on the true horizon, the amplitude, which is the arc between the prime vertical and the body, can be calculated. This means that the amplitude (A) is quadrantal but starting from E/W towards N/S ( 90° difference with azimuth). Why this change? Z Az

col Pn

ζ = 90° ζζ = ∆

The navigation triangle ZPnstar has: - zenith distance = 90° - co-latitude - polar distance - azimuth The formulae we use is: Cos ∆ = cos col x cos ζ + sin col x sin ζ x cos Az

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In this formulae is: Cos ζ = cos 90° = 0 Sin ζ = sin 90° = 1 The formulae becomes: Cos ∆ = sin col x cos Az Sin δ = cos l x cos Az Cos Az = sin δ x sec l Sin A = sin δ x sec l A = amplitude δ = declination of celestial body l = latitude of observer A is E/W xx N/S Example: A = E 15° S

v = 90° + 15° = 105°

The Norie’s Nautical Tables have a precomputed table called “True Amplitudes”. An extract is given on next page. This table is made for a maximum of declination of 29°. Why? The celestial body has to stand with its center on the true horizon which is above the visible horizon (see course 1° Bach). We can predict for the Sun and the Moon their location respective to the visible horizon when their center is in the true horizon. For the other celestial bodies (planets and stars), we do not know exactly the reference of the true horizon at sight. For the sun : lower limb half diameter above visible horizon For the Moon: when the moon is 2/3 visible at the visible horizon (1/3 below visible horizon). As the declination of both celestial bodies never reaches 29° (Moon max. 28,5°), the table was made up till declination 29°.

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5. determination of compass errors The determination of compass errors can be done by several means. We consider in this chapter only the astro posibility. As a calculated azimuth and amplitude results in a true bearing, we can compare the compass bearing with the calculated true bearing to extract the total error. For the magnetic compass with the variation of the location of the observer, we find the deviation of the magnetic compass. After ample observations we can than draw the deviation curve for the magnetic compass. For the gyro compass, the total error is also called gyro deviation. The magnetic field of the earth does not influence the gyro compass. For a complete definition and understanding of the gyro deviation, we refer to the course nautical instruments.

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