this value and the speed of the ship, obtain-from the table "Reducing Altitudes to a Single Zenith"-the rate of change in altitude per minute of time (in the astronomical calculation forms it is designated by the symbol Ah); multiplying it by the time interval AT between the mean moment of measurement of the altitudes of this star and the mean moment of the measurement of the last star, obtain the correction Ah, in reducing to a single zenith (it is positive if the vessel is proceeding toward the star, and negative if the vessel is proceeding away from the star); adding it to the "True h” value, obtain the reduced altitude of the star "Red. h".

In correcting altitudes of the moon, in addition to the sextant index correction i and instrument correction s, we must take into account the correction for dip of the visible horizon Aha, total correction for the altitude of the lower or upper limb of the moon-which is the sum Ah, + hp ± R-and the correction for air temperature and atmospheric pressure.

2. Calculating the DR Hour Angle and Declination of a

Celestial Body

First the approximate Greenwich time is calculated and the Greenwich date of the observations determined (this is necessary so that there will be no 12-hour error later on in calculating the exact Greenwich time or the day in determining the date):

Approx. TGr = Approx. Tc ± Now,

(116)

approximate time of observations according to ship's clock; number of the time zone in which they were recorded.

If in calculating from Eq. (116) it develops that T No. 24 hrs, then we must subtract 24 hours from the result and take as the Greenwich date the following day. If in calculating from this equation we had to "borrow" several days in the date, then the Greenwich date of the previous day is taken. Then the precise Greenwich time of observations is obtained

where

TGr

= T + u + (12 hrs),

(117)

T – mean moment in the series of altitudes according to the clock; u clock correction;

-

12 hrs added, if the value of T + u differs from the approximate Green

wich time by 12 hrs.

A selection from the Nautical Almanac of celestial bodies, as well as their interpolation corrections, is made from the number of hours, minutes and seconds in Greenwich Time TGr

The westward hour angle and declination of the sun, moon and planets are calculated from the equations:

where T, 8T - Greenwich hour angle and declination, selected from the diurnal table for the total number of hours in Greenwich time; A1t interpolation correction, selected from the basic interpolation table, corresponding to the total number of minutes in Greenwich time, from the number of seconds and conventional symbol for the celestial body;

A2t, Ad - interpolation corrections, selected from the same interpolation table in the "Correction" column from quasidifference A and difference A presented in the diurnal tables.

Local sidereal time (local hour angle of the point of Aries) tm is calculated from the equations:

where * is a sidereal parameter selected together with the declination of the star from the Nautical Almanac table "Stars, visible positions for 19...."

If the eastward hour angle of the celestial body exceeds 180°, its eastward angle is calculated:

to
= 360°

tw.

(124)

3. Calculating DR Altitudes and Azimuths of Celestial Bodies

The DR altitude ha, and azimuth Adr are calculated using tables VAS-58, TVA-57 or the MT-63 tables of logarithms of trigonometric functions, from the

arguments:

- DR latitude: in determining from the sun-at the mean moment in the series of altitudes; in determining position from the stars-at the mean moment in measuring the altitudes of the last star;

8 - declination of the heavenly body;

t - local hour angle: westward (if it does not exceed 180°) or eastward.

In calculations using the VAS-58 tables, the tabulated values of altitude h, azimuth A and auxiliary angle q are selected from the tabulated values of the arguments close to the given values; the differences A, A8 and At between the given and tabulated values of the arguments are calculated; from the interpolation tables the interpolation corrections Ah (AA) are selected, the sum of which is added to the tabulated altitude (azimuth) values. The process of entry into the tables is depicted in Figs. 84-86.

The calculations are completed by calculating the differences h - har be tween true and DR altitude or, in determining position from the stars, the differences Red. h har between the reduced and DR altitude. These differences, together with the DR azimuths, are the elements of the astronomical lines of position.

4. Plotting Lines of Position and Calculating

the Coordinates of a Fix

Lines of position are plotted on a track chart or on the reverse side of an astronomical calculation form, the center O of which is taken as the DR position of the ship at the moment of measurement of the last series of altitudes (Fig. 87). Using the degree divisions on the edge of the form, draw a straight line from this point, forming an angle with the meridian line equal to the DR azimuth Adr of the star (direction to the illuminated pole). Plot the value h-har along this straight line, assuming that the side of one maneuvering board square is equal to 1' (1 nautical mile). If the value of h - har is negative, it is plotted on the opposite side (opposite the direction to the illuminated pole). From obtained point K draw a straight line perpendicular to the azimuth line-it will be the sought line of position. Plot the other lines of position in a similar manner.

If among the other celestial bodies the altitude of Polaris was measured, then-having measured the true altitude "True h" (see Section 44, Paragraph 1) and having added to it corrections obtained from the Nautical Almanac tables "Latitude from the Altitude of Polaris”—we must determine the observed latitude. Compute the difference 40 - dr. If it is positive, it is plotted from the center of the maneuvering board to the north; if it is negative-to the south. The line of position perpendicular to the meridian (line 0-180° on the form) is drawn through the obtained point.

The observed position is at the point of intersection of the lines of position constructed on the form. If they do not intersect at one point, the observed position must be noted visually in the middle of the error diagram formed by the

lines of position. If the error triangle is large, the observed position is considered the point of intersection of the anti-medians (straight lines, symmetrical to the medians with respect to the bisector of the triangle). In the error quadrangle, the point of intersection of the straight lines connecting the centers of the opposite sides is taken as the observed position. The difference in latitude Ay and departure between the center of the maneuvering board and the observed position (as in plotting the differences h hdr, one square of the maneuvering board corresponds to 1') are obtained from the form. Adding A to the DR latitude used for calculations of the last line of position (if the sign of Ay is opposite the latitude of the position, it is assumed to be negative), we obtain the observed latitude 0.

In order to determine the difference between the longitudes of the DR and observed position, an additional plot must be made in the lower left corner of the form (Fig. 88): draw auxiliary straight line AC at angle dr to the horizontal frame of the form. Along the horizontal frame mark the departure w and draw a vertical line from obtained point B to its point of intersection with auxiliary line AC. The length of the diagonal AC in nautical miles (number of squares on the maneuvering board) will be numerically equal to the difference between the longitudes of the DR and observed position. Adding this to the longitude of the DR position, we obtain the longitude of the observed position.

The mean quadratic error in astronomical observation:
1) Determined from two simultaneous lines of position

where Mar is the mean quadratic DR error for the time interval between altitude

measurements.