A METHOD OF DETERMINING ASTRONOMICAL LATITUDE AND LONGITUDE BY OBSERVING ONLY TIME AND HORIZONTAL

ANGLES BETWEEN PAIRS OF STARS

by J. C. BHATTACHARJI, Survey of India

(Communicated by C. Mahadevan, F.N.I.)

(Received June 13; read October 2, 1959)

ABSTRACT

The present paper is a modification of my previous paper under the title of: 'A method of determination of astronomical latitude and longitude when only time and horizontal angles are observed'.

An attempt has been made in this paper to further simplify the procedure of observation and obtain better accuracy in astrodeterminations by removing the restrictions of observing pairs of stars only in the same vertical plane.

INTRODUCTION

Longitude.-The astronomical longitude (time) is determined from obsen-ations of transits of stars lying in the same vertical plane, by a transit instrument mounted in the meridian of the place. The minimum number of stars required in one set of observation is three of which one is a slow-moving azimuth star and the others are fast-moving time stars, situated north and south of the zenith of the observer. The instrument being reversible in azimuth, the effect of instrumental errors are reduced to minimum, thus giving an accuracy of very high order. Various equations of the instrument have been deduced by Bessel, Hansen and Mayor. The only disadvan-tage ",ith this instrument is that it is too heavy to be easily carried from place to place which has rendered it quite unsuitable for field-determination.

There are other methods as well of determining longitude in the field by observing transits of stars in the same vertical plane. According to DoHen's method, the number of stars required is only two of which one is either oc or S Ursm minoris and the other an equatorial time star while, according to Chanuenet, any circumpolar star may be used in place of only oc or S Ursm minoris. G. T_ Mc Caw's method requires any two stars situated north and south of the zenith of the observer; selec-tion of a circumpolar star in place of a north star, according to it, is necessary only when azimuth is also to be determined along with longitude. The differences among these methods lie mainly in the comparative advantages or disadvantages in the procedure of observations, but as regards accuracy all are equally unsuitable when the light instruments like glass-arc theodolites are used in their practical applications because of the accumulative effect of the instrumental errors arising from observations being taken in the same vertical plane by keeping the horizontal circle clamped during observations.

The present method of determining longitude as suggested in this paper is an improvement on the methods described above. According to it, the observation of a star-pair does not require to be restricted to their transits in the same vertical circle by keeping the horizontal circle of a theodolite clamped during observation; on the other hand, the observer is here free to use the theodolite in any azimuth-position and determine the horizontal readings of star-positions of any high altitude star-pair situated near about north and south of the zenith of the observer, corres-ponding to the zero line of the diaphragm scale, on anyone face of the theodolite.

VOL. 26. A. No.2.

196 J. C. BHATTACHARJI: A METHOD OF DETERMINING

The main advantage of this method is that the effect of instrumental errors, e.g. collimation and trunnion axis tilt errors, are reduced to minimum by selecting a star pair of altitudes sufficiently near to each other and observing them .Qn the same face of the theodolite. The Geodetic Tavistock fitted with shutter eye~piece has been utilized in the practical applications of the above method to the determination of astronomical longitude (time) with the accuracy of geodetic standard.

Latitude.-The application of the transit instrument to the determination of the latitude of the place of observation by observing transits of prime vertical stars lying in both east and west of the meridian is due to Bessel, although the prime vertical transit for the determination of the latitude of a place was used by Reomer more than a hundred years earlier. This method has been considerably nsed by the astronomers of Europe and to a less extent in America.

The main difficulty with Bessel's method is that the sufficient number of prime vertical stars are not always available at all latitudes; it thus keeps the observer waiting indefinitely in order to obtain the minimum number of star pairs necessary for the correct determination of latitude. The difficulty becomes particularly greater, if the instrument used is not reversible in azimuth, in which case the number of star-pairs for observations required is not only more but the effect of instrumental errors also gets increased. This method is, therefore, now entirely superseded by the use of the zenith telescope or the prismatic astrolabe.

The present method of determining astronomical latitude as suggested in this paper is an improvement on Bessel's method. According to this method, the obser vation of a star-pair does not require to be restricted to their transits in the same (prime) vertical plane by keeping the horizontal circle of a theodolite clamped during observation; on the other hand, the observer is here free to use the theodolite in any azimuth position and determine the horizontal readings of star.positions of any high altitude star.pair situated near about east and west of the zenith of the observer, corresponding to the zero-line ofthe diaphragm scale, on anyone face ofthe theodolite.

As a result of this arrangement, not only the difficulty of obtaining more readings of star.positions of the same star.pair or of different star.pairs is considerably removed, but more important, the effect of instrumental errors is also reduced to minimum by selecting a star-pair of altitudes sufficiently near to each other and by observing them on the same face of the theodolite. The Geodetic Tavistock fitted with shutter eye.piece has been utilized in the practical application of the above method for the determination of the astronomical latitude of the place of observation.

GENERAL PRINCIPLE OF THE METHOD

Both longitude (time) and latitude are obtained from the timings and the horizontal readings of star'positions of two high altitude star.pairs corresponding to the zero.line of the diaphragm scale, one star.pair for longitude situated near about the north and south of the zenith and the other for latitude situated near about the east and west ofthe zenith ofthe observer, by determining the error in the diff~rence of the computed azimuths at the corresponding instants of observations of star positions after comparing them against the observed difference of azimuths of the respective star.positions corrected for all instrumental errors. Provided the observed angle between the star-pair is correctly determined, the error in the differ ence of their computed azimuths can be attributed only to the error either in the chronometer time (L.S.T.) or in the assumed latitude of the place or both. For stars near the meridian, small error in the assumed 4ttitude has practically no effect on the difference of their computed azimuths, and for stars near the prime vertical, if the error in the chronometer time is very small or corrected from time correction observation, the error in the difference of computed azimuths can be due only to the error in the assumed latitude of the place of observation. Thus two simple relations

ASTRONOMICAL LATITUDE AND LONGITUDE 197

are establlshed"'-One between the error in the computed azimuths of the star.pair near about the meridian and the error in the chronometer time (L.S.T.), and the other between the error in the computed azimuths of the star-pair near about the prime vertical and the error in the assumed latitude of the place. From these rela. tions it is easy to deduce the chronometer error (and ultimately longitude) and the latitude of the place of observation.

The accuracy of the method thus depends entirely on the accuracy with which the observed angle between the star-pair is determined. The observed angle ii'! improved upon in three ways:

(i) By using a glass-arc theodolite of more improved pattern. The Geodetic Tavistock fitted with shutter eye-piece as devised by Dr. J. De

Graaf Hunter is more suitable for this purpose. The special feature of this instru-ment is that it is provided with a striding level to determine the trunnion axis tilt and the dislevelment error, and a multi-graticule scale on which the positions of a star can be read at an interval of three complete seconds. The latter obviates completely the doubts of time-recording so common with other methods.

(ii) By observing the star-pair of altitudes sufficiently near to each other on the same face of the theodolite, thereby reducing to minimum the effect of instrumental errors due to the horizontal collimation and the trunnion axis tilt which otherwise do substantially affect the required result, and

(iii) By avoiding errors arising from any movement of the horizontal tangent-screw in intersecting the star-pair, by means of simply aligning the theodolite with its vertical circle set at the particular altitude-reading of the star-pair, approximately in the direction of the star and then its horizontal circle clamped so that its reading corresponding to the particular azimuth-position of the theodolite may be obtained without being required to use the horizontal tangent-screw at all.

This is possible only with Geodetic Tavistock theodolite fitted with shutter eye-piece where star-appearances on the graticule scale are read without being required to intersect the star with the help of the horizontal tangent-screw.

LoNGITUDE

Selection of stars.-Two fast-moving stars of magnitudes varying generally from 25 to 50, one situated near about the north and the other near about the south of the zenith of the observer, are selected for time correction observation. In addition, the altitudes of the above star-pa,ir require to be as high and as near to each other as conveniently possible.

Programme.-A carefully made programme last!ng for a period of about 30 minutes showing, for the selected star-pair, the names, magnitudes, aspects, altitudes correct only to tenth of a degree and times (L.S.T.) correct only to 10 seconds, of passages corresponding to the various azimuth-positions of the star-pair including observations on both faces of the theodolite, is always found useful.

Observations with Geodetic Tavistock fitted with shutter eye-piece.-With the help of an approximate value (correct only to 10 seconds) of the chronometer error and the observation of a known star, the line of collimation of the theodolite can be very easily set approximately in the meridian of the place of observation. The theodolite, then, requires to be levelled as accurately as possible with the help of the striding level provided. It is presumed that the value of the division of the level is determined in a bubble tester either before or after the season's observations. It is, however, very essential to determine the horizontal collimation of the theodolite i~mediately before or after the night's observation; Any precisely defined luminous object (at least half a mile away from the station of observation) may be used as a reference mark. Alternately Polaris at elongation may also be used as a reference mark when available. .

The routine of observation is that the theodolite is first set at the altitude

198 J. C. BHATTACHARJI: A METHOD OF DETERMINING

reading and the azimuth reading of the star and then the readings of the striding level in both positions are recorded in the manner as described later. Now as soon as the star enters the field of view, the vertical arc tangentscrew is manipulated to bring the star close to the horizontal line of the diaphragm scale and the observer starts calling out 'miss' for each shutter 'click'. The recorder is soon able to synchronize the call' miss' or the shutter' click' with the chronometer time which is either a complete second or more usually a half second and without any difficulty record the hours and minutes followed by the complete or half seconds corresponding to the call' yes' of the observer for the particular shutter ' click' for which the observer has estimated a corresponding scale 'reading of a star appearance on the diaphragm scale. Once the star is picked up and the first scale reading is read, the scale readings of a series of star. appearances at progressively different positions of the scale at regular intervals of three seconds can be easily read and recorded. It is to be noted here that at each whole minute the make and break mechanism of the break circuit chronometer omits to give an impulse to the shutter relay system thus making it necessary to allow for this omission in the computation. For each' miss' of the scale readings of the star appearances the recorder enters a dash in the record and as soon as the star passes the zero line of the diaphragm scale, the observer calls out' line' whereupon the recorder enters a long line instead of a dash. The instant of the last star. appearance is noted by the observer by means of a stop.watch.

There should be no gaps in the record and the number of scale readings on either side of the zero.line of the diaphragm scale requires to be equal. If, however, any intermediate reading is missed, the gaps in the record require to be filled up by interpolation or the corresponding reading omitted from the other side of the' line'. The observer does not require to give the full reading of the star.position with reference to the zeroline of the scale, which is afterwards reduced by the applica tion of the 'group correction' allocated to the readings of the star.appearances with reference to the recording of 'line' in the column of star.readings. From these corrected readings a star. position corresponding to the mean time of the star. appearances is derived. This, being expressed in scale units with proper sign, can be easily reduced to seconds in time by means of the mean of the factors determined from the scale readings of the star and numerically added to or subtracted from the mean time of the star appearances according as the mean star.position in scale units is negative or positive, in order to obtain the observed chronometer time corresponding to the star position on the zero-line of the diaphragm scale. It is to be noted that the readings of the diaphragm scale on the side qf the zero-line in which the star makes its first appear-ance are considered as negative and those on the other side of the diaphragm scale as positive. Now after taking the horizontal circle reading in this position, the hori. zontal circle is unclamped and the theodolite is turned towards right without dis-turbing the vertical circle till the star comes again in the field of view taking care that it is never overshot. The theodolite is then again clamped and the process of observation repeated except that the striding level is now to be read in one position only. Mter setting the vertical circle at the altitude reading of the second star, the theodolite is moved to pick up the second star for observation in exactly the same manner as in the previous case. Now the face of the theodolite is changed and the process of observation is repeated in the following order: first the second star and then the first star. .

Both in the beginning as well as at the end of the observations of the star'pair on each face of the theodolite, the readings of the striding level require to be taken in both positions with the following details:

Leftend reading Rightend rea.ding Micro left Micro right

ASTRONOMICAL LATITUDE AND LONGITUDE 199

But for all intermediate observations of the star-pair, the readings of the striding level in anyone position (e.g. one left-end and one right-end with the micro-position) is enough.

Now Gomparing the difference of any two angles between the two stars after having corrected them for horizontal collimation, aberration, trunnion axis tilt and dislevelment errors, against the difference of the computed azimuth of the two corresponding star-positions, the chronometer time is easily obtained with the accuracy of geodetic standard from the relation established between the change of azimuth and the chronometer error.

Observation for longitude difference. For longitude difference, however, it is necessary to determine the difference of correct local time at two stations of some event which can be perceived at both. The event is usually provided by some wireless rhythmic time signals giving the seconds beats of the emitting local clock. These seconds beats are compared against those of the local clock at the receiving station, whose error and rate have already been known, to determine the required longitude difference in time after having corrected for propagation of signal and a small error to the reputed time of emission of signal which is regularly published in the Admiralty Notices to Mariners and in the Bulletin Horaire. For an accurate determination of this longitude difference in time, the shutter is utilized in the following way.

The insulated contact of the shutter mechanism is cOl1l1ected to the wireless receiver head-phone circuit. The shutter is then set to open at every chronometer second. The actual signal, which is preceded by preliminary warning signals, con-sists of a morse dash (length 04 sec.) commencing on each whole minute, followed by morse dots (length 01 sec.) commencing at intervals of 60/61 mean time seconds, repeated for five minutes. The interval between the successive chronometer seconds differ from that between successive time signal dots. For a chronometer regulated to give sidereal time, the chronometer seconds beats (i.e. shutter openings) coincide with time signal dots at regular intervals of 72 seconds. The period of shutter opening is adjusted to be about llO m. sees., i.e. purposely longer than the duration of a signal dot. The effect is that the extinction of the signal is complete at the time of coincidence and extends over a period of about five dots. The actual time of coin-cidence is then the mean of the times of the begil1l1ing and the end of the extinction of the signal.

In practice it is convenient to reduce the chronometer time (correct to half a second) of the commencement of the time signal by noting with a stop-watch the third dot following each long dash signifying complete minutes of G.M.T., and also the mean chronometer times of the successive coincidences of the chronometer beats with the time signal dots, are reduced to the first mean coincidence time by noting with a stop-watch the third dots after each period of silence. Obviously three complete seconds require to be subtracted from the above observed chronometer times of both the commencement of the signal and the coincidences. The mean of the first mean coincidence time and the reduced coincidence times is accepted as the chronometer time of the first coincidence. From the difference between the approx-imate chronometer time of commencement of the time signal and the accepted chronometer time of the first coincidence in chronometer seconds, the actual chrono-meter time of commencement of signal is determined after applying corrections for the propagation and emission of signal and then compared against the sidereal time of the emitting station to obtain the longitude difference between the receiving station and'the emitting station correct to about 001 sec.

Formula.-Let a denote the azimuth angle between the elevated pole and the star at an hour angle t and of declination 8

Then with the usual notations, -sint

tan a = . (1) cos 4> . tan 8-eos t. sm 4>

200 J. C. BHATTACHARJI: A METHOD OF DETERMINING

For stars near culmination, a and t being small, we have a" = 15. t. cos 8 . cosec (t/J-8), t in seconds in time

Differentiating (1) with respect to a and t, we have

lia" = 15. (sin22a . cot t-sin2 a sin t/J) . h,t,

t:J. f in sf'conds in time and a generally> 5 or

11": sin2a . d" d 11 "k5 5. But

h,a - t:J.a = b. a -a = h,H = a -a - H -H " " ("") "("") (" ") s n S If S 11- S n

(3)

(3a)

whf're Hs = horizontal circle reading to south star corrected for horizontal collima-tion, trunnion axis tilt and dislevelment error, and aberration,

H" = horizontal circle reading to north star corrected for errors as for H. and b.H = error in the horizontal angle between the two stars due to error b.t in the

chronometer time. It is to be noted here that the effect of the graduation error in the mean

horizontal angle obtained from two faces of the Geodetic Tavistock theodolite may be considered negligible as the error is always expected to be much less than one second.

Hence we have

H" 15. t:J.t (. 2 . 2 t) h, = --2- sm as cot fs-sm a". co I", h,t in seconds in time and a generally ::t> 5 or

15 2 h,t [(2 sin t/J (sin2 a l1 -sini as)+(sin 2as cot fs-sin 2a,,: cot tn)], t:J.t in seconds in time and a generally> 5. That is,

t:J.t (in seconds in time) = 2 ~:" / [sin 2as . cot ts-sin 2a" . cot ttl], (4) a generally ::t> 5.

ASTRONOMICAL LATITUDE AND LONGITUDE 201

or

2~:"j (2 sin q, (sin2 a n-sin2 as) + (sin 2as . cot ts-sin 2an cot til)]' ... (4a) a generally> 50. Since AH and At are both small, At can be easily evaluated by using only 3 or 4 figures for sines and cotangents in relation (4) or (4a) Accuracy.-(i) Effect of q, on a :

Differentiating the formula (J) with respect to q, and a, we haw' A"a = sin a. tan It. A Hq,

or

" '" ("") (. h' h) A",J. A as- A an = A as -all = 81n as tan s-sm a. tan n U 'I' (5) Since a is small, it is clear from relation (5) that the effect of error of q, on an or

as is inappreciable. Thus the error of (as-an) is of the order of less than half a second if q, is known correct only to 2 minutes. Therefore At can be determined from relation (4) or (4a) correct to about hundredth of a second in time. Hence with the help of the wireless rhythmic signals, longitude difference also can be obtained correct to about hundredth of a second in time.

(ii) Effect of error of 0 on a: Differentiating the formula (1) with respect to a and 8, we have

Alia = sin a (tan h-tan 0). A"o (6) From relation (6) ,it is dear that the effect of error in 8 on a is also inappreciable.

LATITUDE

Selection of stars.-A star-pair of magnitudes varying generally from 25 to 50, one situated near about the east and the other near about the west of the zenith of the observer, are selected for latitude observation. In addition, the altitudes of the above star-pair require to be as high and as near to each other as conveniently possible .

Programme.-A carefully made programme lasting for a period of about 45 minutes showing, for the selected star-pair, the names, magnitudes, aspects, alti-

tudes correct only to 2 minutes, computed bv the formula h = cos-1(Sin ~. cos 8), . ~ ~a

and times (L.S.T.) correct only to a minut,e, of passages corresponding to the various azimuth-positions of the star-pair including observations on both faces of the theo-dolite, is very essential.

Observations with Geodetic Tavistock fitted with shutter eye-piece.-For latitude observations, the whole procedure is exactly similar to that used for the star-pair observed for time correction except that the theodolite in this case has got to be turned 90 to the right from the meridian position fixed in the manner as described before and the vertical arc tangent-screw constantly manipulated to keep the star close to the horizontal line of the diaphragm scale while taking readings of the star-positions .

Finally, comparing the difference of any two angles between the two stars, preferably as near to 1800 as conveniently possible, after having corrected for hori-zontal collimation, aberration, trunnion axis tilt and dislevelment errors, against the difference of the computed azimutln;; of the two corresponding star-positions, the error in the assumed value of q, is easily obtained with the accuracy of geodetic standard from the relation established between the change of azimuth and the latitude error. The final result, however, requires another small correction, e.g. -0"'0000521. sin 2q, for the height of the station of observation.

202 J. C. BHATTAOHARJI: A METHOD OF DETERMINING

Formula.-From formula (1), we have t

cos cp tan o-cos t. sin cp co a = .

-smt (7)

Differentiating the formula (7) with respect to a and cp, we have Aa" = tan h . sin at:J. IIcp .. (8)

If the suffixes E and W refer to the stars situated near about the east and the west respectively, then, we have

But t:J. "aw- A "aE = t:J. (a;v-a~) = A"H = (a~-a~)-(H~v:"'H~)

where H w = horizontal circle reading to west star corrected for horizontal colli-mation, trunnion axis tilt, dislevelment error and aberration

HE = horizontal circle reading to east star corrected for errors as for Hw

and t:J.H = error in the horizontal angle between the two stars due to error Acp in cpo

It is to be noted here that the effect of the graduation error in the mean hOl'i-zontal angle obtained from two faces of the Geodetic Tavistoek theodolite may be considered negligible as the error is always expected to 1:e much less than one I'-:econd. Hence we have

or (9)

Since t:J.cp and t:J.H are both small, Acp can be easily evaluated by using only 3 or 4 figures for sines and tangents in relation (9).

Accuracy.-(i) Effect of error of t on cpo Differentiating the formula (7) with respect to t and a, we have

" 15 ( . 2 ..1.. siu 2a) . d" A a= , sm a.sm'f'+-2-.-.cott .t:J.t, 6tmsecon smtlme. Hence:

t:J. N H = 15. [Sin cp (sin2 aw-sin2 aE)+ (Sin :aw . cot tw- siu 22aE . cot tE)] ., (10) Since a is small and(aw-aE)is nearly 1800 , the errol' in the difference of computed

azimuths due to small error in t can be hardly half a second which according to the relation (9) cannot affect the value of cp by more than 01 second. However, if the error in t is large, t can be corrected from time correction observation before it is used in the relation (9).

(ii) Effect of error of 0 on 4>. Differentiating the formula (7) with respect to cp and 8, we have

cos cp Acp= 8' h,AO (ll)

cos . sm

Since the altitudes of the stars are high, and 4> and 0 do not differ much from each other, the factor c~s ~ h is never much greater than unity and, therefore, the

cos . sm effect of any possible error of 8 on cp is always less than 0'1 second,

AS'l'RONOliICAL LATITUDE AND LONGITUDE 203

EXPERIMENTAL RESULTS

Observations were carried out with the Geodetic Tavistock theodolite fitted with shutter eye-piece in the Hunter Observatory at Dehra Dun on 26.11.55. The results of these observations have been given in tabular form as below:

Observed values

h m s 5121175

5 12 1177

Longitude in time

Mean observed

values

Probable error of

the mean Old

value

Latitude

Old Observed values

~Iean observed values

I Probable II the mean value

I error of

--------!---------

I 0 , " I ' 30 18505 I ,---------'30 18 503 I ,------

5 12 11-80 I) ~ I~ Il~77 0~005 ~ ~ Il~77\130 18 503 1,3~ 1~ 5~'311 __ ~~13~ 1'8 5~'4 5121179 30 18 509 I

-------\301849.9\ I 1----5 12 1l74 -5-12-1-1'-7-4-1-------- 130 18 50~1-----1----1----'

From the table it appears that the results are very satisfactory. These may, for all practical purposes, be considered free from personal equations, which may be further confirmed from observations by different observers. Elimination of personal equations is surely a very great advantage in all astro-determinations of geodetic standard.

Apart from that the method has many practical advantages too. The observa-tional equipment is very light. One Geodetic Tavistock theodolite fitted with shutter eye-piece, one chronometer with a relay box, a 6-volt accumulator and a wireless set are all that are required. For azimuth determination also the same theodolite may be used with advantage and no extra equipment is needed for the purpose. This i& never possible with Prismatic astrolabe or any other method. The working prin-ciple is also extremely simple and is more or less the same for both longitude and latitude. Above all it takes comparatively much less time for the entire programme to complete. For longitude, for instance, the programme does not usually last for more than 30 minutes. The latitude programme also may be completed within only 45 minutes, exceeding one hour in very rare cases. Even the universally adopted Talcott method, which might, otherwise, seem equally useful, is in fact no comparison to it, because of its usually long and tiresome programme.

The present method is, thus, likely to prove more useful than any other method for astro-determinations in the field, particularly when both longitude and latitude are wanted for the computation of deflections at a large number of stations for geodetic purposes.

ACKNOWLEDGEMENTS

I express my grateful thanks to Mr. B. L. Gulatee, M.A. (Cantab), F.N.I., retired Director, Geodetic and Research Branch, SUlYey of India, for his kind help in affording necessary facilities for the experimental observations. I am also very much indebted to Messrs. L. S. Sharma and S. B. Das of the same Branch for their valuable assistance in recording and checking of computations appended at the end. In addition it is a pleasure to thank Mr. Hari Singh, Officer Surveyor of Survey of India, who read the draft typescript of this article for valuable comments and suggestions.

204 J. C. lIHA1'1'ACHARJ"I: A :ME'l'HOD oF' DETERMINING

Place: HWlter Observatory, Dehra DWl

Observed by: J. C. B. Recorded by : L. S. Sharma

FORM 1. ANOLE BOOK Instrument: Geodetic Tavistock No. V0528

S.T. Chrono. No. 18372

Fast: I m. 30 secs. Date: 26-11-55 Slow:

Star: Observed scale readings Time and scale readings Aspect: corrected for groups Mean of

First half' Second half each pair of

readings Altitude: h before after First half corrected Horiz. reading: H' crossing crossing arranged in Second half for chrono. Level reading: L & R central wire cent.ral wire reverse order breaks

value: -ve value: +ve

,

h 32 Vnlp. W I h m s m s 22 3442'5 22 34 43,') 33 - I - - -It: 6T 50' 36 - I - - -:39 12 (-0'7) 12 (0'25)

H': FL 270c 01' 218' 42 1'.:; I 10 1-

ASTRONOMICAL LA'rlTUDli: AND LONGITUDg 205

FORM l---

206 J. C. BHATTACH.ARJI: A METHOD OF DETERMINING

FORM I~contd.

Star: Observed scale readings Time and scale readings Aspect: corrected for groups Mean of each pair of

First half Second half First half readings Altitude: h before after arranged corrected Horiz. reading: H' crossing crossing in reverse Second half for chrono. Level reading: L & R central wire central wire

order breaks value: -ve value: +ve

hm s hm s Pegasi W 032435 032525

- 12 (0-2) 12 (050) h: 57 10' 20 18 07 1-8 055

24 22 12 22 050 H': }

Star: Aspect:

Altitude: h Horiz. reading: H' Level reading: L & R

41 Arietis E

h: 66 00'

H': FR 271 0 05' 487"

L R -08 96

h: 64 57'

H':FR 3590 25' 381" L R

5.5 33

-----,-----:---"--v Piscium S

h: 64 57'

H' : FR 2 59' 493"

L R 57 3-0

I

I

-----------

82 Pegasi S

h: 70 23'

H':FL 1760 15' 252"

L R 30 59

8

.A.S1'RONOMIC.A.L LATITUDE .A.ND LONGI'l'trDE

I Time and scale readings

Observed scale readings ___ co_r_r_ec_t_e~d-:~_o_r_gr_o_u_p_s __ First half Second half I

before after crossing crossing

central wire central wire value: -ve value: +ve

10 09 12 11 17 lg 20 21 - -

28 28 32 32 36 36 40 39 - 13 - -

- -

- -

10 10 13 12 ]8 17 22 2] :27 :H 30 2!l 3-3 33 38 37 41 4-(J

}Iean time:

-

41 30 08 -27 -44 33

Mean time:

I 1 I

I

------ ------

10 -- 20 - 37 13 -32 23 - 40 1-7 09 33 -

! I Mean time:

First half arranged in reverse

order

hm s I 00 505

(-05) 09 12 17 20 23 28 32 37 40 -

-

.-

(56) 60 6-4 68 72

(7-6) 80 83 88

-9-0

10055'0

23 42 59

(77)

1 39400

145415 1-7 3-3

(5-0) 68 87

(105) (12,2) 140

1 4543'0

I I

i

-------

Second half

hm s 1 01 005

09 1-1 18 21

(2-4) 28 3-2 37 3-9 4-3 -

-

-

60 6-2 67 71 74 7-9 8-3 87 90

(9-4) Mean

scale reading:

30 (4-8) (6'5) g3

}Iean scale reading:

I 45445 (03) 2-0 37

(5-5) 73 9-0

109

1 - }lean I scale reading:

\

I ! I

I 2346455 I 2346485 42 I-l (-06) l-l

1-0 29 2-4 29 26 44 40 4-4 44 14 -;)8 64

Mean Mean time: 2346470 scale reading:

207

Mean of each pair of

readings corrected

for chrono. breaks

(014) 0-04 024 014

(0,14) 0-19 014 0]9 004 009

;) 035

(030) (030) (030)

+032

(-0'70) -0-65

(-065) (-065) -070

(-048) (-0-38)

-

-060

025 025 020 0-30 025

025

-

208 J. C. :lJHAT'rACHARJl: A METHOD OF DETER:M:INING

Star: Aspect:

,

Altitude: h Horiz_ reading: H' Level reading: L & R

\ 82 Pegasi S I

h: 70 23'

H' : FL 179" 49' 530" 1_ R

23 5-4

----.-....-----.--:--.--

FORl'oT 2. COMPUTA'l'ION OF CORRECTIONS TO HORIZONTAL ANGLES

Place: Hunter Observatory, Dehra Dun Instrument: Geodetic Tavistock No. VO.'j28 S.T. Chrono. No, 18372 Fast: 1 m. 30 secs. Rate: 018 sec. per hour. Date: 26-11-55

Latitude:", = 30 19' Instrumental constants: Value of 1 division of bubble: d = 3"

Value of 1 division of scale: F = 25" Computed by: J. C. B. Horizontal collimation: C = 15" (-I-ve when FL > FR) Checked by: S. B. Das Transit axis tilt: T ,= 20" (+ve when L.H.S. is higher than R.H.S.)

Observation for time or latitude Face of instrument: FL/FR Stars: N IS or E/W cos", (1) sec h (1) tan h (1) Correction for dislevelment : L; R . d. tan h Correction for transit axis tilt: T. tan h Correction for horizontal collimation:

O.sec.h Correction for diurnal aberration:

-032 cos'" cos a sec h Value of mean scale reading in arc:

S' . F. sec h -

Horizontal circular reading to stars: II' (2) Corrected horizontal reading: H

Observed chrono. time Correction for assumed error and rate Observed time corrected for assumed error

and rate

(1) Correct to three figures only. (2) Change sign for north stars only.

Latitude FL

32 Vulp. W 0863 265 :H5 8.511

-490

397

00

166

27001 218 27001 376 h m s

2234440 -1 3249

2233 1151

Latitude FL

v Pisco E 0863 229 207

-276" -414

343

00

406

8839 32,0 88 39 379 h m "

23 10 290 -1 3260

23085640

Latitude FL

v Pisco E 0863 229 207

-276" -414

343

00

189

9009 135 9008 577 h m s 2322440

-1 3263

2321 1137

Latitude FR

,Pegasi'V 0863 186 157

285" 31-4

-279

00

J86

8857 243 8858 149 hm 8 02443'0 -1 3282

023 1018

Latitude FR

,Pegasi '"V 0863 184 155

307" 310

-- 276

0'0

207

no 00 138 11001 086 hm H 032480 -1 3284

o 31 J 516

J.atitude FR

35 Arietif! E 0863 2'46 2'25

-32-4" 45'0

-369

0'0

332

26937 363 26937452 hm s 051 100 -I 3290

0493710

Latitude FR

4L Arietis E 0863 246 225

-351" 450

-369

00

80

271 05487 271 05 297 hm s I 00550 -1 3293

059 2207

j Z ~

~ t"'

~ ~ I;j t>;J

~ ~. ~ t>;J

l:-:) ~

Observation for time or latituue Face of instrument : FLjFR Sta.rs: NIS or E/W COB rf> (I) sec h (I) tan h (1) Correction for dislevelment : L; R . d. tan h Correction for transit axis tilt: T . tan h Correction for horizontal collimation:

C.sec.h Correction for diurnal aberration:

-032 cos'" cos a sec h Value of mean scale reading in arc:

S'. F. sec h

Horizontal circular reading to starR: H' (2) Corrected horizontal reading: H

Observed chrono. time Correction for a.ssumed error and rate Observed time corrected for assumed

and rate

(1) Correct to three figures only. (2) Change sign. for north stars only.

error

Time FL

82 Pegasi S 0863 298 281

-122" -562

447

-08

186

17615 252 176 15 193 h m R 2346470

-1 3040

2345 1660

FORM 2-concld.

Time FL

82 Pegasi S 0863 298 281

-173" -562

447

-08

104

17949 530 17949 338 h m s 2351 400 -I 3041

23500959

Time FL

.p Andro. N 0863 366 352

232" -704

549

-]0

-174

351 36 583 351 36 476 h m s 2358 500

-1 3043

23 57 1957

Time FR

f.L Cassio. N 0863 242 221

--43" 442

-363

-07

-375

17858 440 17858 094 h m s 1 10010 -1 3065

1 08 3035

Time FR

51 Andro. N 0863 322 306

-46" 612

-483

-09

56

181 58 582 181 59 112 h m s 1 33 150 -1 3072

1 31 4428

Time FR

v Pisco S 0863 236 214 71"

428

-354

-07

189

35925381 359 26 108 h m s 139400 -I 3074

I 38 0926

Time FR

1/ Pisco S 0863 236 214

87" 42.8

-354

-07

-378

259493 259269 h m s 145430 -I 3076

1 44 1224

w ....

o

:-<

~ t:d ; ::: II> Iii: t>;J

~ o t:1

~ t:1 t>;J

~ ~ Z

~

ASTRONOMICAL LATITUDE AND LONGITUDE 211

FORM 3. COMPUTATION OF TIME CORRECTION AND LONGITUDE

Place: Hunter Observatory, DehraDun

Latitude: if> = 30 18' 50"

Instrument: Geodetic Tavistock No. V0528 S.T. Chrono. No. 18372

Date: 261155 Formula: (i) tan a = -sin tf[cos if> tan II-sin if> cos t]

(ii) At= AHu/[eos lis. cosec (if>-'-Ils)-cos Iln.cosec (if>-1ln)].15, L::.t in seconds Computed by : J. C. B. Checked by : S. B. Das

Face of instrument: FLfFR Stars: N/S Observed time corrected for

assumed error and rate R.A.ofstar Hour angle: t .. t in arc Declination: II cos 1l cosec (if>-1l) sin t cos t sin if> cos tan 1l tan a (from formula (i a H (from Form 2) as-a", (I) Hs-H .. (1) (as-a,,)-(Hs-H.) = L::.H" cos 1l . cosec (-11).15 (cos 1ls cosec (if>-Ils)-cos II ...

cosec (if>-Il .. .15 Aqfrom fo="'" (ii)) 1 Observed cmono. time of

reception of time signal (2) Correct L.S.T. of reception of

time signal .. . . G.S.T. of emission of time

signal (3) Longitude in time

FL FL 82 Pegasi S 82 Pegasi S h m s h m s 23451660 23500959 23502348 23502348 -050688 -001389

-1 16432 -003284 1042205 10 42205

0983 0983 2980 2980

0022316 0001011 0999751 1000000 0504737 0504737 0863274 0863274 0189055 0189055 0065365 0002960

< , H

176 15366 17949494 176 15 193 17949338 18441388

I 188 15516

184 38317 188 12462 307-1 I 3 054

439 ! 439

819 819 -228 -226

h m s h m s 21 314833 21 314833

21 314605 21 314607

16 193430 16193430 5 12 n'75 5 12 1177

(1) Suffixes 8 and n refer to south star and north star. (2) Corrected for assumed error and rate. (3) Corrected for error of emission and propagation of time signal.

in time

FL .p Andro N h m s 23 57 1957 23435217 00132740

3 21 510 4610540

0692 3658

0058682 0998877 0504737 0863274 1042122 0148273

351 33 578 351 36 476

-380

212 J. C. BHATTACHARJI: A METHOD OF DETERMINING

Face of instrument: FL/FR Stars: NIS Observed time corrected for

assumed error and rate R.A.ofstar Hour angle: t t in arc Declination: 8 cos 8 cosec (",-8) sin t cos t sin '" cos tan 8 t,an a (from formula (i (1. H (from Form 2) as-a .. (1) Hs-H .. (1) (as-a .. )-(Hs-H,,) = t;,.H". cos 8. cosec (-8) . 15 .. (cos Os. cosec (-8s)-cos8".

cosec (-8,,.15 .. t;,.t (from formula (ii)) Observed chrono. Hme of re-

ception of time signal (2) Correct L.S.T. of reception of I

time signal . . . . G.S.T. of emission of time

signal (3) . . .. I Longitude in time ..

FORM 3--collcZd.

FR fL Cassio. N

h m s I 08 393.'5 1 052206

030829 o 47044

5442373 0578 2421

0013692 0999907 0504737 0863274 1412892 0019149

358 54 ]06 178 5809'4 180 30070 1802801'4

2056 210

563 -223

h m s 21 314833

21 31 4610

16 193430 5 12 1I80

FR 51 Andro N

h m s 1 314428 1 35 1919

033496 -0 53437 4824376

-- 0664 3219

0015632 0999878 0504737 0863274 1-126741 0033401

1 54469 181 59 1I2 177 29307 177 26596

231-1 32-1

674 -224

h m s 21 314833

21 31 4609

16 193430 5 121179

(I) Suffixes 8 and n refer to south star and north star. (2) Corrected for assumed error and rate.

FR Pisco S

h m s 1 380926 139 0999 -010073

-0 15 1I'0 5 16041

0996 2-362

0004416 0999990 0504737 0863274 0092200 0010387

179 24176 35926108 184 03265 184 01 175

2090 353

563 -229

h m s 21 31 4833

21 314604

16 193430 5 12 1l74

(3) Corrected for error of emission and propagat,ion of time signal.

FR v Pisco S h m s 1 441224 1 390999

050225 I 15338 5 16041

0996 2362

0021979 0999737 0504737 0863274 0092200 0'051713

182 57 371 259269

181 02502 181 00 157

2345 353

674 -229

h m s 21 :n 4833

21 314604

16 193430 5 12 1l74

AstRONOMICAL LATI'rUDE AND LONGITUDE 213

FORM 4. COMPUTATION OF IJATITUDEl

Place: Hunter Observatory, Instrument: Geodetic Tavistock No. V0528 Dehra Dun S.T. Chrono. No. 18372

Formula: (i) cot a = - (cos.,. tan il-sin t/> cos t)/sin t (ii) D.t/>n = -D.Hn/(cot tw-cot tE ). sec t/>

Computed by: J. C. B. Checked by: S. B. Das

Face of instrument: FL/FR St&rs:EJW Correct L.S. T. of observa-

tion R.A. of star

\

I I

Hour angle: t .. i . I {marc .. .. I

Latitude: ' (assumed value) I Declination: il .. sin t . . . . cos t sin 4> co" t/> tan 8 cot a (from formula (i a H (from Form 2) aw-0-z;; (1) HJV-HE (1) . (aw-aE)-(Hw-HE) = t:.H"1 sec t/> .. cot t . . .. cot t . sec q, . . .. I (cot tw-cot tE ). sec.p .. t:..pn (from formula (ii ., I Correction for height of sta- I

tion (Hfeet)=-0.0000521

H. sin 24>' .. . . .p' + D..V + height correction I

= latitude: '" .. i I

FL 32 Vulp. W h m s 22 33 1151 20524099

1 403052

25 07 37'S 301900 2753379

0'424629 0905367 0504779 0863249 0529336 0000145

2695930'0 27001376 181 21 192 lSI 21 597

-405 H5S 2132 247 432

-94

-0,1 o

30 IS 505

Date: 261]-55

FL v Pisco E

h m s 23 OS 56'40

I 170430 21 51 52'10

327 [;S 615 30 1900 2702133

0530407 0847743 0504779 0S63249 0510339 0023805

883810'8 8839379

179 51 56'0 179 52399

-439

159S l'S5 456

-9,6

-01

30 18 503

FL v Pisco E

h m 8 23 21 1137

1 170430 22040707

331 01 461 301900 2702133

0484360 0874869 0504779 0863249 0510339 0002201

9007340 90 OS 577

1806 2'0!)

(J) Suffixes E and W refer to east star and west star.

214 J. C. BHATTACHARJI: ASTRONOMICAL LA'1'ITUDll AND LONQITtmll

Face of instrument: FLjFR Star:EJW Correct L.S.T. of observa-

tion R.A.ofstar Hour angle: t ..

t in arc Latitude: 4>' (assumed value) Declination: /) sin t cos t Hin 4> cos '" tan /) cot a (from formula (i n H (from Form 2) 0W-oE (1) HW-HE (1) (ow-aE)-(Hw-HE) = b.H" sec '" cot t eot t . sec '" (cot fW-cot tEl . ~ee r/> b..p" (from formula (ii Correction for height of station

(H feet) = -0000052. H . sin 2r/>'

r/>' + b. 4>" height correction = latitude: r/>

FORM 4-,concld.

FR t Pegasi W h m s 0231018

22045812 2 18 1206 o

3433009 301900 2508028

0567129 0823629 0504779 0863249 0469161 0018951

o

26854515 88 58 149

179 19540 179 20297

-357

1452 168 373

-96

-01

30 18503

FR Pegasi W h m s o 31 1516

22045812 2261704

3634156 301900 2508028

0595819 0803119 0504779 0863249 0469161 0000663 o

269 57433 9001 086

177 52095 177 52452

-357

1348 156 395

-9,0

-01

30 18509

(1) Suffixes E and W refer to east star and west star.

REFERENCES

FR 35 Arietis E h m s 0493HO 2405396

220843-14 o

332 1047-1 301900 27 31 354

0466699 0884416 0'504779 0863249 0521094 0007284 o

26934575 26937452 18022458 18023234

-376

1895 219 375

-100

-01

30 18499

FR 41 Arietis E h m s 0592207 2472542

22 11 5665 o

332 59098 30 1900 270458)

0454208 0890896 0504779 0863249 0511347 0018241

271 02420 271 05297 17855013 178 55389

-376

1961 227 383

-98

-0,1

30 18501

Albrecht, Th. (1894). ]'ormeln und Hulfstafeln fur Geographiche ortsbestiinmungen. Publisher: Verlag von Wilhelm Engelmann, Leipzig.

Bhattacharji, J. C. (1958). A method of determination of astronomical latitude and longitude when only time and horizontal angles are observed. Emp. Survey Review, 14, 110, 352-363, October, 1958.

Uhauvenet, "William (1863). Manual of Spherical and Practical Astronomy. Publisher: J. B. Lippincott Co., Philadelphia.

Doolitt.le, C. L. (1896). A Treatise on Practical Astronomy as applied to Geodesy and Naviga-tion. Publisher: Chapman and Hull Ltd., London.

GraaffHunter, J. de (1947). The Hunter-shutter eyepiece for longitude and azimuth. Emp. Survey Review, 9, 63. 20-24, January, 1947.