RU2497076C1 - Method to determine astronomic azimuth and latitude by unknown stars - Google Patents

Abstract

FIELD: measurement equipment.

SUBSTANCE: method to determine astronomic azimuth and latitude by unknown stars includes measurement of zenith distances of the observed unknown star with a theodolite, as well as of horizontal directions at it and at an earth object, calculation of the north place and azimuth as the difference of the horizontal direction at the earth object and the north place. Measurements with the theodolite are carried out four times via time intervals of not more than 60 min., and the north place is calculated in accordance with the formula: tg M_N=A/B, and width is determined twice in accordance with the formulae: tg φ=[sin z₂ cos(N₂-M_N)-sin z₁(N₁-M_N]:(cos z₁-z₂. tg φ=[sin z₄ cos (N4-M_N)-sin z₃(N₃-M_N)]:(cos z₃-z₄).

EFFECT: expansion of functional capabilities and increased accuracy of joint determination of azimuth and latitude.

4 dwg

Description

The invention relates to the field of astronomical and geodetic measurements and can be used to determine the direction of landmarks by astronomical azimuths from stars to solve various problems of engineering geodesy.

A known method for determining the azimuth of an earthly object (Kolesnichenko AE Astronomical determination of the azimuth of an earthly object. Artillery Journal, No. 6, pp. 24-28, 1951), including measuring the zenith distances of the same star twice after a short period of time. Knowing the latitude of the place, they find the azimuth and the stars at the time of its observation by the formula:

A = O-N,

where O is the average value of the direction to the star,

N = C ₀ -b ₀ + Θ,

here C ₀ is the average count in a horizontal circle on a star.

b ₀ = arc tg [ΔZ ₀ / (tg Z ₀ tg ΔC ₀ )],

Θ = arc sin [(tg φ tg Z ₀ sin b ₀ ) / cos C ₀ ],

ΔZ ₀ = (Z ₂ -Z ₁ ) / 2; Z ₀ = (Z ₂ + Z ₁ ) / 2;

ΔC ₀ = (C ₂ -C ₁ ) / 2.

The accuracy of determining the azimuth depends on the accuracy of the angular measurements by theodolite and the accuracy of the knowledge of the latitude of the observation site.

The disadvantage is the need for knowledge of the geographical latitude of the theodolite.

There is a method of determining azimuth by an unknown star (Kolesnichenko A.E., Trofimenko VT. On the accuracy of determining azimuth by an unknown star. Geodesy and cartography, No. 6, pp. 14-17, 1988), including measurement twice through a small time span of zenith distances and directions to the same unknown star. Azimuth A of a local subject is calculated by the formula:

A = M-M _N ,

where M is the countdown in a horizontal circle on an earthly object,

M _N - place of the north.

M _N = Θ-b + N ₀ ,

where Θ = arc sin [(tg φ tg z ₀ sin b cos ΔN) / 2],

b = arc ctg [(tg z ₀ ctg Δz tg ΔN) / 2],

where z ₀ and Δz are the half-sum and the difference of the zenith distances z ₂ and z ₁ ;

N ₀ and ΔN are the half-sum and the difference of directions to the star.

The latitude of the place is removed from the topographic map. The disadvantage of this method is a preliminary knowledge of the latitude of the place.

There is a method of determining latitude and azimuth from a star with unknown coordinates (Pandul I.S. Determination of latitude and azimuth without a chronometer from a star with unknown coordinates. // Sat. "Notes of the Mining Institute", volume 156, SPGGI, 2004, p. .225-228), adopted as a prototype. The azimuth, measured from the north point, and latitude are determined by 4-fold measurements of the zenith distances of the same unknown star, horizontal directions to it and by measuring the difference in hour angles using an average stopwatch. The method includes, in addition to measuring horizontal directions and zenith distances, measuring the difference in the hour angles of a star using a stopwatch. Knowledge of the latitude of the observer’s position is not required. Azimuth A is calculated by the formulas

A = N ₀ -M _N ,

where N ₀ is the horizontal direction to the earthly object,

M _N - place of the north corresponding to the direction of the meridian on the ground,

M _N = N ₁ -A ₁ ,

where N ₁ - measured by theodolite horizontal direction to the first point of the diurnal parallel of a star,

And ₁ = arc cos [(sin δ - sin φ cos z ₁ ): sin z ₁ cos φ],

where δ = arc sin (sin φ cos z ₁ + cos φ sin z ₁ cos A ₁ ),

φ = arc sin (cos z ₂ sin δ + sin z ₂ cos δ cos q ₂ ),

q ₂ is the parallactic angle of the second parallactic triangle.

The disadvantage of this method is a significant decrease in the accuracy of the determined azimuth due to the simultaneous use of theodolite and stopwatch. The method is only suitable for rough definitions.

The technical result of the proposed method is to expand the capabilities and improve the accuracy of the joint determination of azimuth and latitude.

The technical result is achieved by the fact that in the method for determining the astronomical azimuth to an earthly object and its latitude, including theodolite measuring the zenith distances of an observable unknown star and horizontal directions to it and to an earthly object without using a stopwatch, calculating the north and azimuth as the difference of the horizontal direction to the earth the subject and places of the north, theodolite measurements are carried out four times at intervals of no more than 60 minutes, and the place of the north is calculated by the formula:

tg M _N = A / B,

where A and B are calculated by the formulas:

A = -b cos N ₁ + a cos N ₂ + d cos N ₃ - with cos N ₄ ,

B = + b sin N ₁ - a sin N ₂ - d sin N ₃ + with sin N ₄ ,

where the coefficients a, b, c, d are calculated by the formulas:

a = sin z ₂ : (cos z ₁ -z ₂ );

b = sin z ₁ : (cos z ₁ -z ₂ );

c = sin z ₄ : (cos z ₃ -z ₄ );

d = sin z ₃ : (cos z ₃ -z ₄ );

where N ₁ , N ₂ , N ₃ , N ₄ are the horizontal directions to the observable unknown star at four points of its diurnal parallel, measured along the horizontal circle of the theodolite,

z ₁ , z ₂ , z ₃ z ₄ - zenith distances at four points of the diurnal parallel of the observed unknown star, determined by the formula corresponding to this type of theodolite, according to the vertical circle of the theodolite, and latitude is determined twice by the formulas:

tg φ = [sin z ₂ cos (N ₂ -M _N ) -sin z ₁ (N ₁ -M _N )] :( cos z ₁ -z ₂ ),

tg φ = [sin z ₄ cos (N ₄ -M _N ) -sin z ₃ (N ₃ -M _N )] :( cos z ₃ -z ₄ ).

To justify and derive the formula for determining the astronomical azimuth of an earthly object, we use Fig. 1, where the parallactic triangle of the luminary σ ₁ is represented. Let the horizontal directions N ₁ and N ₂ and zenith distances z ₁ and z _{2 of} two points of the diurnal parallel of some star be measured with a theodolite after a certain period of time.

$$\cos \Delta =\cos z_{\mathrm{one}}\sin \varphi +\sin z_{\mathrm{one}}\cos \varphi \cos \left(N_{\mathrm{one}}-M_N\right)\begin{array}{cccc}& & & \end{array}\left(\mathrm{one}\right)$$

Here M _N is the place of the north, Δ is the polar distance, Δ = 90 ° -δ.

$$\cos M_{N}A=\sin M_{N}B.\mathrm{ABOUT}t\mathrm{to}\mathrm{at}d\mathrm{but}tg{M}_N=A/B.\begin{array}{cccc}& & & \end{array}\left(9\right)$$

Knowing the place of the north, it is always easy to get the astronomical azimuth of direction to an earthly object

$$A=N_0-M_N,\begin{array}{cccc}& & & \end{array}\left(10\right)$$

where N ₀ is the horizontal direction to the earthly object.

Latitude is determined by formulas (4) and (5).

The convergence of latitudes φ calculated by formulas (4) and (5) serves as a control of calculations. Formulas (9-10) should be used in determining the desired azimuth. The formulas are easily programmed using the standard Mathcad program and it takes very little time to calculate the observations made.

The method is as follows. Choose a star in the western or eastern part of the sky. Observations of the eastern star should begin, and the western - end low above the horizon. Observe any star that is noticeable and easily identifiable for repeated observations. In order not to lose the selected star, it must be hydrated, keeping the rough theodolite in sight all the time. The star should be no closer than 35 ° to the observer’s meridian, for which theodolite should be roughly oriented in the meridian by compass or by eye. The measured zenith distance is burdened by instrumental errors, a pointing error at a star, and an error in determining astronomical refraction. To reduce the effect of refraction error, a star should be observed if its zenith distance is less than 80 °. To account for astronomical refraction in each technique, it is necessary to measure the temperature t ° C and atmospheric air pressure B (in mm Hg).

The observation order is as follows. For observations, use theodolite (preferably with an overhead level), an outdoor thermometer, aneroid barometer, a flashlight and a light target. To ensure theodolite stability, the legs of the tripod should be installed on bricks or stakes driven into the ground.

The first half-reception is the circle to the left (circle to the right):

1) sighting on an earthly object: readings N ₀ in horizontal and L, P in vertical circles (with two positions of the vertical circle). Before each countdown in a vertical circle, the level bubble in the alidade of the vertical circle is brought to the nulpunk;

2) sighting on a star and taking samples N ₁ on horizontal and L ₁ on vertical circles;

3) after about an hour, secondary sighting on the same star and taking samples N ₂ in horizontal and L ₂ in vertical circles;

This is followed by a break in the observations when changing circles for about 10-15 minutes.

The second half-reception is the circle right (circle left):

4) sighting on the same star, readings N ₃ in horizontal and P ₃ in vertical circles;

5) after an hour, the secondary sighting on the star and readings N ₄ in horizontal and P ₄ in vertical circles;

6) repeated sighting on an earthly object: counts N ′ ₀ along the horizontal and P ′, L ′ along the vertical circles (at two positions of the vertical circle). The vertical circle when sighting on an earthly object is counted out for subsequent calculation of the zenith location MZ. A complete measurement takes no more than 135 minutes.

Method of sighting a star. After a rough capture of a star, a horizontal thread should be set in the path of the visible movement of the star in the field of view of the rough. Enter the image of the star in the bisector and hold it there until it contacts a horizontal thread. At this moment, it is necessary to stop the rotation of the guide screw of the alidade and sequentially count the vertical and horizontal circles of the theodolite, first making sure that the level bubble with the alidade of the vertical circle is in the nulpunk. When performing a reception in the intervals between observations of a star, measure the temperature and atmospheric air pressure necessary to calculate the astronomical refraction ρ. To calculate these corrections, take meteorological data for specific observations.

To reduce a number of errors, in particular, errors in determining latitude, it is desirable to observe another star in the opposite side of the sky according to the method described above.

Calculation technique.

The zenith position of the vertical circle MZ is calculated by the formula, for example, for theodolite Theo - 010

MZ = 0.5 (L + P ± 360 °),

where L, P - readings on the vertical dial at two positions (to the right and left of the observer). Then the apparent zenith distances z ₁ , z ₂ , z ₃ , z ₄ are calculated - the zenith distances at the four points of the diurnal parallel of the observed star, determined by the formula corresponding to this type of theodolite, according to the data of the theodolite’s vertical circle. For example, for Theodolite Theo - 010,

z = 0.5 (L-P ± 360 °),

z = L-MZ,

z = MZ-P.

True zenith distances z can be obtained as z = z '+ ρ, where ρ is the true astronomical refraction, which can be calculated by the formula

$$\rho =\mathrm{60,}{3}^{″}\frac{B}{760}\cdot \frac{{273}^{\circ }}{t^{\circ }+{273}^{\circ }}tg{z}^{\prime }$$

Then, using formulas (6, 7), a, b, c, d are calculated and, using formulas (9), A and B and the place of the north. The calculations of the north location M _N for each individual half-reception will differ due to the influence of the error in the determination of MZ, therefore, at the last stage of the calculations, the average value of the north location from the reception is taken.

By the formula (10), the astronomical azimuth of the direction to the earthly object A is calculated and by the formulas (4, 5) the latitude of the place φ.

Example. A field experiment was conducted. Astronomical observations of azimuth and latitude by the method were performed in the summer of 2011 at the Berezhansky geodetic range using the exact optical theodolite Theo-010. The observations were carried out along the line B-14 - AC-1 with a length of 500 m. The coordinates of the points, latitude and azimuth of the line were determined from astronomical observations of class 1 and taken as reference:

φ ₀ = 49 ° 27 '40 ”N, A ₀ = 131 ° 56' 43”. Theodolite Theo-010 was installed at point B-14, and the light target was at point AC-1. On different days 4 measurements were taken.

The results of measurements and calculations are presented in figure 2, figure 3 and figure 4. The mean square error of determining the azimuth from four tricks is m _A = ± 11 ”. The root-mean-square error of determining the latitude from four techniques is m _φ = ± 15 ”. A field experiment confirmed the reliability of the conclusion of the working formulas of the method.

Thus, the method allows you to determine the astronomical azimuth of the direction to an earthly object with an accuracy of ± 10 ”and latitude with an accuracy of ± 15” from some unknown star without knowing the geographical coordinates of the place of observation, exact stellar time, name and equatorial coordinates of the observed star.

Claims (1)

A method for determining the astronomical azimuth and latitude from unknown stars, including theodolite measuring the zenith distances of the observed unknown star and horizontal directions to it and to an earthly object, calculating the north and azimuth as the difference of the horizontal direction to the earthly object and the north's place, characterized in that theodolite measurement produced four times at intervals of not more than 60 minutes, and the place of the north is calculated by the formula:
tgM _N = A / B;
where A and B are calculated by the formulas:
A = -b cos N ₁ + a cos N ₂ + d cos N ₃ -c cos N ₄ ;
B = + b sin N ₁ -a sin N ₂ -d sin N ₃ + c sin N ₄ ,
where the coefficients a, b, c, d are calculated by the formulas:
a = sin z ₂ : (cos z ₁ -z ₂ ); b = sin z ₁ : (cos z ₁ -z ₂ );
c = sin z ₄ : (cos z ₃ -z ₄ ); d = sin z ₃ : (cos z ₃ -z ₄ );
where N ₁ , N ₂ , N ₃ , N ₄ are the horizontal directions to the observable unknown star at four points of the diurnal parallel, measured along the horizontal circle of the theodolite; z ₁ , z ₂ , z ₃ , z ₄ are the zenith distances at the four points of the diurnal parallel of the observed unknown star, determined by the formula corresponding to this type of theodolite, according to the vertical circle of the theodolite, and latitude is determined twice by the formulas:
tg φ = [sin z ₂ cos (N ₂ -M _N ) -sin z ₁ cos (N ₁ -M _N )] :( cos z ₁ -cos z ₂ );
tg φ = [sin z ₄ cos (N ₄ -M _N ) -sin z ₃ cos (N ₃ -M _N )] :( cos z ₃ -cos z ₄ ).

Images