UNCLASSIFIED U. OF ZOWA81-2 ML.

Transcription

1 AD-AO9 626 IONA WNI IONA CITY DEPT OF PH4YSICS AND ASTRONOMY F/6 17/7 AN ANALYTICAL. SOLUTION OF THE TWO STAR SIGHT PROBLEM OF CELESTI-ETC(Ul FEB 81.J A VAN ALLEN N C-OO16 UNCLASSIFIED U. OF ZOWA81-2 ML.

2 IIIJIL25 MICROCOPY RESOLUIIION TESI CHARI

3 U.Of Iowa '- THE UNIVEIY OF OW IoaCtIw 24

4 / U. of Iowa An Analytical Solution of the Two Star Sight Problem of Celestial Navigatione by JAMES A. VAN ALLENt Department of Physics and Astronomy The University of Iowa Iowa City, Iowa / February 1981 Temporary Address: National Air and Space Museum, Smithsonian Institution, Washington, D. C I BUT1ON S5 Approved for public releca*6 Dlstributiofl Uniyuted

5 UNCLASSIFIED 1ECU00,TY CLASSIFICATION OF THIS PAGE rflhou Dole Ented) REPORT DOCUMENTATION PAGE READ INSTRUCTIONS BEFORECOMPLETINGFORM 1. RE1PORT NUMBER 2Z. GOVT ACCESSION No. 3. RECgPIENT'S CATALOG NUMBER U. of Iowa 81-2 v C 14. TITLE (end Subtitle) S. TYPE OF REPORT & PERIOD COVERED An Analytical Solution of the Two Star Progress, February 1981 Sight Problem of Celestial Navigation 6. PERFORMING ORO. REPORT NUMBER 7. AUTOR(4). CONTRACT OR GRANT NUMBER(S) JAMVES A. VAN ALLEN NOOO14-76-C PtRFrORMING ORGANIZATION NAME AND ADDRESS 10. PROGRAM ELEMENT. PROJECTTS Department of Physics and Astronomy./AE! OKUI UBR S The University of Iowa Iowa City, Iowa It. CONTROLLING OFFICE NAME AND ADDRESS 12. REPORT DATE Office of Naval Research February 1981 Electronics Program Office IS. NUMBER OF PAGES Arlington, Virginia MONITORING AGENCY NAME & ADDRESS(Il differmnt from, Controlling office) IS. SECURITY CLASS. (of this report) I6. DISTRIBUTION STATEMENT (of this Report) UNCLASSIFIED ISDECILASSI FICATION/ DOWNGRADING SCHEDULE Approved for public release; distribution unlimited. 17. DISTRIBUTION STATEMENT (of the abstract mted In Block20 it different fro~m Repotf) IS. SUPPLEMENTARY NOTES To be published in Navigation. It. KEY WORDS (Continue an reverse. sideit neesary arad )~lyai by block nmmbaq) Celestial Navigation Two Star Sight Problem 20. ABSTRACT (Continueaon reverse aide It necesaran d Idmnlfy by block Mmbear) [See page following.] DD, JNt 'EDITION OFr I Nov so is OBSOLETE NLSSFE S,'W 44601SECURITY 0102.LF.0 CLASSIFICATION Of TIS PAGE (01hem wf e

6 2 NABSTRACT The classical problem of celestial navigation, in its simplest form, is the determination of an observer's longitude and latitude from the altitudes of two identified stars, observed at a known Greenwich Mean Time on a known date. A novel solution of this problem in closed analytical form is given herein. The solution yields the two possible positions of the observer without any prior knowledge of his position, without any dependence on tables of computed altitudes and azimuths for an assumed position, and without any graphical work. The basic two-fold ambiguity is resolved to yield a single, unique position by repeating the calculation using the altitudes of a third star and of one of the two previous stars. The full analytical solution is given as are some artificial numerical examples, readily performed on a hand- 4 *held, programable calculator. - I... B.! - D/ KI

7 3 I. INTRODUCTION AND SUMMARY The classical problem of celestial navigation, in its simplest form, is the determination of an observer's longitude and latitude from the altitudes of two identified stars, observed at a known Greenwich Mean Time on a known date. Numerous ingenious solutions to this problem by essentially trial-and-error methods using the concept of the "assumed position" have been devised and reduced to practice (Bowditch, 1962). But, to the author's knowledge, no treatise on practical navigation draws attention to the fact that the basic problem can be solved in closed analytical form without reference to tables of computed altitudes and azimuths, graphical approximations, etc. At least two analytical solutions to this problem have been published (Kotlaric, ) (Watkins and Janiczek, ). The solution given herein appears to have merit because of its geometric simplicity, its freedom from trigonometric ambiguities, and its ready adaptability to a hand-held, programmable calculator. 4 Data from one pair of star sights yield two possible positions, usually widely separated and easily distinguished. Data from a second pair of star sights (i.e., using the altitudes a third star and of one of the two previous stars) resolve the basic ambiguity because only one of the two positions from the second calculation is identical to one of the two positions from the first calculation. No prior knowledge of the approximate position of the observer is required.

8 2. STATEMENT OF THE POBLEM Observed Data h 1, h 2 t Altitudes (corrected for atmospheric refraction, etc.) of stars 1, 2 Greenwich Mean Time of observation (on a known date) For simplicity, it is assumed that the two observations are simul- *taneous. %I, %2 Data Derived from Observations and from the AIR ALMANAC East longitudes of substellar points of stars 1, 2 4; 8, 62 Declinations of stars 1, 2 C2 Zenith distances of stars 1, 2 * 4Problem To find the Longitudes, A (East + or West -), and Latitudes, (North + or South -) of both intersections (P and Q) of the two small circles of position on a sphere of unit radius (Figure 1). j

9 3. AN ANALYTIC SOLUTION (a) Each circle of position is the intersection of the circular cone of half angle C with the unit sphere (Figure 2). The center of the circle has the spherical coordinates of the star X, 6 (Figure 3). The equation of the plane containing the circle of position is of the form x cos a + y cos 0 + z cos 7 - P = 0 (1) where ci, 3, and 7 are the angles to the X, Y, and Z axes, respectively, of the normal to the plane, this normal being through the origin of coordinates 0, and p being the distance along that normal from 0 to the plane (Figures 2 and 4). The adopted XYZ coordinate system is a right-handed one with 0 at the center of the earth, with Z the earth's polar axis (+Z to the.4 North Pole), with the XY plane the equatorial plane, and with +X through the Greenwich meridian. Note that p cos *and that y = 90-8 Thus

10 Cosc =CosX Cos cos f3 = sin X cos8 cos 7 sin 8 After substituting the above quantities in equation (1), the equation of the plane is found to be: x (Cos%. Cos 89+ y (sin %. Cos z sin8 - Cos.= 0, ILet i = 1j,2. (2) (b) The two planes intersect in the line PQ. a. =Cos X. Cos 8. b. sin X. Cos 6.i c.= sin 8.i P. Cos The equation of this line of intersection is obtained by the simultaneous solution of equations (2) for i =1 and i = 2 to - 4 give: X =B+ -EY+F() where in A S(albs - a 2 bl) B (b2 cl - bica) C (b 2 P2. - bip2) D M(al C2 - a2cl) E (b2 - b 2 c 1 ) F (C2P1 - CIp 2 )

11 7 Equations (3) may be regarded as parametric equations of the line PQ with x the parameter that specifies the position of a point on that line. The other two coordinates of that point are given by: _ F - Dx E " C - Ax (4) (c) The two possible positions of the observer, P and Q, are the intersections of the above line with the unit sphere; that is, they are the two points for which x + 2 = (5) (d) Substitutions from Equations (4) into Equation (5) yield a quadratic equation in x, which has two real roots, xp and xq (e) Equations (4) are then used to find the other two coordinates of P and Q: S4 P Q (xp, yp, zp) (xq, yq, ZQ) (f) Finally, the rectangular coordinates of P and Q are converted to spherical coordinates to yield the desired longitude and latitude of each point, viz. P : Ap, 0p Q : AQ, Q.E.D. A

12 8 4. ARTIFICIAL EXAMPLES 0000 GMT of 1 September 1975 (a) Input Data for A = -91!532, = Arcturus = -125?915 8 = C = Altair = = C = Antares ~ = C = 68.o45 Vega x = - 60?520 8 = C = (b) Calculated Positions from Pairs of Star Sights Arcturus-Altair A = = A = Q= - 2.1)48 Arcturus -Antares A = - 91:532 P= A = -157?841 Q = P

13 9 Arcturus-Vega = P = A = - 86?950 Q P Vega-Antares - 91?532 P = A= - 42: Vega-Altair Ap P= A - 55?550 = Altair-Antares A p = - 91?532 P= A = - 11?087 = (c) Remarks It is clear that one of the two calculated positions in each of the six cases is identical to the position that was used in the 4 construction of these examples. This is the only solution that is camon to two or more cases. The method is, of course, applicable to observations of the sun, moon, and planets as well as to observations of stars.

14 * ACKNOWLEDGEMENT The author thanks Mark M. Baumback for helpful discussion and the U. S. Office of Naval Research for support. REFERENCES Nathaniel Bowditch, American Practical Navigator, 1962 (Washington: U. S. Government Printing Office, 1966). S. Kotlaric, New short method tables (K 11) for direct finding of a two star fix without use of altitude difference method, Navigation, Vol. 18, , Winter R. Watkins and P. M. Janiczek, Sight reduction with matrices, Navigation, Vol. 25, j8, Winter mo"wo

15 CAPTIONS FOR FIGURES Figure 1. This is a sketch of two circles of position on a unit sphere whose center is the center of the earth. The centers 1 and 2 of the two circles of position have the spherical coordinates X1, 61 and X2, 62 of the substellar points of stars 1 and 2, respectively. The line PQ is the intersection of the two planes containing the respective circles of position. The intersections of this line with the unit sphere yield the two possible positions of the observer, P and Q. Figure 2. This diagram shows a cross-section of the unit sphere that contains the origin 0 and the substellar point S of an observed star. The circle of position is the intersection with the sphere of the circular cone whose axis is OS and whose half angle is C. The normal to this plane OR has length p. 4 Figure 3. The east longitude X and declination 6 (or latitude) of the substellar point are shown relative to the coordinate system in which +Z is toward the north celestial pole, the XY plane is the earth's equatorial plane, and the +X axis lies in the Greenwich meridian. Figure 4. In the same coordinate system as that in Figure 3, a, B, and 7 are the angles the line from 0 to the star makes with the X, Y, and Z axes, respectively.

16 A-G NORTH 4 POLE SOUTH POLE Figu~re 1

17 A-G *PSI POSITION Figure 2

18 A- G SO Figure 3

19 A-G Figure ~4