Nautical Astronomy

a branch of practical astronomy that answers the needs of navigation. Nautical astronomy is concerned with developing methods for determining from celestial bodies and navigational, artificial earth satellites the position of a ship at sea and corrections for course-indication instruments. Nautical astronomy is part of the science of navigation.

The position of a ship at sea, that is, its geographic latitude ϕ and longitude λ, is determined by measuring the altitudes of celestial bodies over the visible sea horizon or over the plane of an artificial horizon created on the ship by different methods. The use of angle-measuring devices with an artificial horizon has expanded the possibilities for determining ship positions by astronomical means and has also increased the precision in measuring altitudes of celestial bodies.

Each value h of the true altitude of a celestial body yields one equation for determining the ship’s coordinates, so that at least two measurements of altitudes of celestial bodies are required to determine a ship’s position at sea. The solution of the spherical triangle with vertices at the celestial pole, the zenith of the observer, and the position of the star—that is, the parallactic or astronomical triangle—leads to the equation

(1) sin h = sin ø . sin δ + cos ø . cos δ . cos(t_Gr + λ)

where ø and t_Gr are the declination and the Greenwich hour angle of the celestial body, respectively. The values of δ and t_Gr are selected from a marine astronomical almanac for the moments of observation. The longitude λ is measured eastward from the Greenwich meridian: t_Gr + λ = t_loc is the local hour angle of the celestial body. When the celestial body is on the meridian of the observer in upper culmination (t_loc = 0), equation (1) yields the solution ϕ = δ ± (90° - H), where H is the altitude of the celestial body in upper culmination, called the meridian altitude. The minus sign is taken for a transit of the celestial body northward from the zenith.

If equation (1) is solved for t ioc, then we obtain the equation

(2) cos t_loc = sin n . sec ø . sec δ - tan ø . tan δ

Knowing the latitude ϕ of the position, we can also obtain the longitude λ = t_loc — t_Gr using equation (2).

It is possible to determine both the latitude and longitude of a position from two measurements of altitude. With a larger number of measurements the accuracy of the determination can also be evaluated. Using what is called the estimated ship position, that is, the coordinates (ϕ_e, λ_e) of the position found graphically or analytically with respect to the course angle and distance covered (dead reckoning), we can represent each of the equations obtained in the form of an error equation or geometrically interpret each equation as an altitude position line. The position-line equation has the form

(3) Δh = Δø . cos A + ΔW . sin A

In order to construct the position line, the estimated ship position (ϕ_e,λ_e) is made the origin of coordinates (see Figure 1), with the latitude increment Δϕ plotted along one axis and the corresponding range increment ΔW = Δλ . cos ϕ along the other axis. If the difference Δ h = h — h_e between the altitude of the celestial body found by observation and the estimated altitude calculated from the estimated coordinates is plotted from the estimated position in the direction determined by the azimuth A of the celestial body, then a point K is found, called the intercept. The position line passes through the intercept in the direction perpendicular to the azimuth of the celestial body.

Figure 1

The ship position is determined by the intersection point of the position lines of two stars that are continually observable. For a large number of observations, the position lines as a rule do not intersect at a single point but form an error figure. The most probable ship position can be found from this figure either by a graphical method or analytically.

The correction for course indicators is determined by comparing the observed bearing of a celestial body and the azimuth A of this body, calculated from its known declination 8, hour angle t_loc = t_Gr + λ, and the latitude of the observation position. The azimuth A can be calculated from the equation

(4) cot A = cos ø . tan δ . cos t_loc - sin ø . cot t_loc

Whenever the altitude of a celestial body is measured simultaneously with the bearing of the body, the azimuth can be calculated using either of the equations

(5) sin A = cos δ . sin tloc . sec h

(6) cos A = sec ø . sin δ . sec h - tan ø . tan h

Special tables for the calculation of the azimuth of a celestial body have been published.

The altitude of a celestial body over the visible sea horizon is measured by a sextant.

To determine the altitude h of a celestial body over the true horizon the reading obtained on the scale of the sextant is corrected by introducing the instrumental correction of the sextant, an index correction, and corrections that take into account the dip of the visible horizon, refraction, and the half-diameter and the parallax of the celestial body.

Historical survey. The positions of celestial bodies were already used in remote antiquity for orientation at an unknown position and for determination of the direction of travel. The growth of industry and trade and the expansion of navigation associated with this growth resulted, beginning in the 15th century, in the development of methods and instruments for determining a ship’s position on the open sea. Astronomical instruments suited for shipboard observations of stars, including angle rods, reflecting quadrants, astrolabes, and armillary spheres, became widespread. Ephemerides of the sun and planets, which are necessary for carrying out observations, were calculated. At this time, only the latitude of a position could be determined from astronomical observations. In the 16th and 17th centuries, methods were proposed for determining longitude based on observations of the angular distances between the moon and stars and observations of the eclipses of Jupiter’s satellites. A precise method for determining the longitude of a position based on the calculation of the difference between the local hour angle of a celestial body and its value at the moment of observation for the Greenwich meridian (λ = t_loc — t_Gr) was introduced in nautical astronomy only in the second half of the 18th century, with the construction of the chronometer.

A theory for the combined determination of latitude and longitude was developed in the early 19th century. In 1808 the German mathematician K. Gauss proposed a method requiring the solutions of five equations. In 1824 the Russian geodesist F. F. Shubert published a new method for the joint determination of ϕ and λ. These methods, however, proved to be unsuitable for practical application. In 1843 the American seaman T. Sumner published a method for determining a ship’s position based on the fact that the position circle corresponding to the value of a measured altitude, that is, the circle of equal altitudes, can be represented over a short distance by a straight line on a map. He constructed altitude position lines by means of the points at which these lines intersect two parallel lines near the parallel of latitude of the estimated position. The Russian seaman A. A. Akimov proposed a different method, published in 1849, for constructing the position line using the single point of intersection of the position line with the estimated parallel of latitude and the direction of the position line. The perpendicularity of the altitude position line and the direction to the star was first used in this method. In 1875 the French seaman M. Saint-Hilaire proposed a method for drawing the altitude position line through a specific point and perpendicular to the direction to the star. This method continues to be used in the 20th century. The Soviet scientists N. N. Matusevich and V. V. Kavraiskii contributed much to the development of modern methods of nautical astronomy and the systematic application of the generalized method of position lines to the solution of astronomical problems.