A METHOD OF SIMULTANEOUS DETERMINATION
OF ASTRONOMICAL LATITUDE, AZIMUTH AND TIME
FROM OBSERVATIONS OF AN UNKNOWN STAR
by J. C. B h a t t a c h a r j i , F.R.A.S.
A B ST R A C T
This paper endeavours to evolve a m ethod of simultaneous determ ina­
tion o f astronom ical latitude, azimuth and tim e from the observed vertical
angles and differences of recorded tim es and horizontal angles o f an
unknown star about an hour before and after the tim e o f elongation, either
east or west.
T h e special feature o f the method is that it does not require actual
identification o f the star o f observation.
T h e determ inations being o f an order better than third order precision,
the m ethod m ay have ample applications in both topographic and hydrographic surveys including navigation.
IN TR O D U CTIO N
T o provide a geodetic fram ew ork in a country is the m ain responsibility
o f a geodetic organisation. W h a t is required th erefore o f a topographer or
a hydrographer is just to fill up the big gaps inside the existing geodetic
setup fa llin g w ith in his jurisdiction o f land or sea surveys, by fixin g therein
a m ore dense netw ork o f m inor control points for the purpose o f producing
accurate maps o f his area. Both topographic and hydrographic surveys
are thus concerned w ith second or third order determ inations only.
Since ordinary topographers and hydrographers generally have
difficu lty in id en tifyin g stars correctly, an attempt has been made in this
paper to m od ify the existing methods o f observation so that there is
practically no need for actual identification o f the stars o f observation.
SELECTION OF STARS
A n y isolated star which can easily be distinguished w ith a theodolite
is chosen for observation about an hour before its tim e o f elongation either
east or west, so that it m ay reappear in the field of view neither much
later nor much earlier than tw o hours afterwards, for observations fo r
the second time, after altering only the vertical circle reading o f the
theodolite. The follo w in g practical procedure may, however, be adopted
for selecting these stars w ith ou t much difficulty.
Intersect any distinctly visible star at the centre o f the theodolite,
note its apparent path o f m otion on the diaphragm by holding the theodolite
in the same position for some tim e and make sure that the apparent path
o f m otion o f the star makes an angle o f about 1 0 ° w ith the vertical w ire
o f the theodolite as shown in figures 1 and 2 below, so that the star o f
observation m ay be able to reappear in the field of view neither much later
nor much earlier than tw o hours afterwards.
East S t a r
n e a r elongation
F ig . 1
W e st Star
n e a r elongation
F ig . 2
If, however, the above angle is found to differ significantly from 10°,
the vertical circle reading should be increased in the case o f a star near its
east elongation or decreased in the case o f a star near its west elongation,
and the horizontal circle readings should be conveniently altered either
w ay in both the cases till after a little trial and error a star is soon obtained
whose apparent path o f m otion is found to make the desired angle with
the vertical w ire o f the theodolite.
O BSER VA TION EQ U IPM ENT A N D PROCEDURE
A theodolite o f the standard o f W ild T 2 fo r both horizontal and
vertical circle readings and a chronometer and a stopwatch fo r recording
instants o f intersections are all that are required during observations.
Provided the theodolite is carefu lly levelled, no bubble reading is actually
needed except slight adjustments at the tim e o f taking either horizontal
or vertical circle readings. Observations, however, should be taken on both
faces o f the theodolite so that any slight error in the collim ation or the
trunnion axis tilt w ill not affect the observational results appreciably.
The follow in g procedure m ay be follo w ed w ith advantage during
observations :
First Set
Star:
E/W
Circle reading
Horizontal
O bje ct
R.M.
Star
Star
Star
Star
R.M.
Instants
Vertical
Face
FL
FL
FR
FL
FR
FR
Second Set
Same as in the first set except that observations are to be taken only
after about an hour so that it becomes possible for the star observed in
the first set to reappear in the field o f v ie w after altering only the vertical
circle reading o f the theodolite.
Latitude formula
R eferrin g to figure 3 and using the usual notations, w e have :
cos A — sin h { • sin <p
cos
h [ ■ cos 9
cos A — sin hx ■ sin ç
cos /ix • cos
or
9
cos A (cos /ix — cos h [ ) = sin ç (cos /ij sin h { — sin h 1 cos h [ )
. . h'x
h-L
. hi
/ix
cos A s in ------------ • s in ------------
or
sin <p =
. hi — hr
hi — ht
s in ------------ • cos
2
A
~v"
cos —
2
where
2
F ig. 3
A lso
cos V = cos 2 A + sin 2 A • cos T
T = ti — h
where
or
or
cos V = cos2 A + sin2 A — 2 sin 2 A sin2
2 sin 2 A sin2 — = 1 — cos V
2
sin
( 12)
or
sm
Relation (12) can be used to obtain the computed value o f A or
8
.
Thus on com paring the computed value o f 8 w ith the values o f
declinations o f stars given in the Nautical Almanac, it is easy to id en tify
the star o f observation and to derive the actual value o f 8 and R.A. o f the
star o f observation from the Nautical Almanac.
Hence, from (11) w e have:
V . hi + h,
sin © = cos A ■ sec — s in ------------
Y
2
2
Azimuth formula
R eferrin g to figure 3 and using the usual notaitons, we have:
sin A
sin (Z S X P or Z S [ P )
(14)
sin A
cos <e
sin (Z S X P or Z S { P )
sin T
------------:---------------- — --------and
sin A
sin V
Hence, fro m (14) and (15) we have:
sin A =
sin 2 A
(15)
sin T
cos 9
sin V
V
sm 2 -—
2
sin T
sin A = ----------------------- sec <p,
fro m (12)
T
sin V
sin 2 —
2
V
T
sin A = tan — - cot — sec ç
or
or
2
(16)
2
or in the case o f close circum polar stars :
A = A j + R» neglecting smaller terms,
(17)
V
T
w here
A, = — - cot — sec <d
2
2
and
/ A?
A 1 V2\
R = t ^ -(----- — — J sin 2 1", a tabular qu antity as given b elow in the table.
Table of R in secs
V
F o r different values o f A and —
2
\
A\
\
V
—
1 000 "
2 000 "
3 000"
4 000"
5 000"
0"
0
0
0
1
1
2
0"
0
0
1
1
2
2
0"
0
0
1
1
2
0"
0
1
1
2
2
0"
1
1
1
2
3
4
5
7
9
3
4
3
3
5
3
4
5
7
11
13
16
19
23
27
31
36
42
48
54
61
6 000 "
2
2 000 "
3 000
4 000
5 000
6 000
7 000
8 000
9 000
10 000
11 000
12 000
13 000
14 000
15 000
16 000
17 000
18 000
19 000
20 000
21 000
22 000
23 000
24 000
25 000
6
11
6
8
10
12
8
10
12
14
16
14
17
15
18
20
20
21
23
27
32
37
42
48
55
62
24
28
33
38
43
49
56
63
25
29
34
39
44
51
57
64
7
9
3
4
5
6
7
9
11
13
16
19
23
26
31
35
40
46
52
59
66
1"
1
1
2
3
3
4
5
7
8
10
12
15
17
20
24
28
32
37
42
48
54
61
68
Longitude formula
R eferrin g to figure 3 and using the usual notations, we have :
sin t [
sin A
cos h i
sin A
or
sin A
sin t\ — --------- • cos h\
sin A
or
T
sin —
2
sin t { = --------- • sin A - cos h [
V
sm
~2
T
or
c o s ---2
cos h [
sin t { = --------- ■-------V
cos <p
c o s --2
(18)
whence we obtain the value o f tt .
N ow on knowing t { by form u la (18) and the R.A. o f the star o f
observation from the Nautical Alm anac, as stated before, we can easily
obtain the L.S.T. o f the observation and the chronometer error in order
ultim ately to derive the longitude o f the place o f observation by using
wireless tim e signals.
Obviously the form ulae (13), (14) and (18) are applicable only when St
and Si lie on the great circular arc through Z, and for the sake o f accuracy
it is desirable to have repeated observations on both faces o f the theodolite
near both St and SJ as already indicated w hile describing the procedure o f
observations. But the positions Sr and S^ (r = 1, 2, ...) may not all lie on
the great circular arc through Z, in which case the observations at Sr
or S' have to be reduced to those at S' or S,' so that Sr. and S^ or Sr
and S', m ay lie on the great circular arc through Z, either w ith the help
dA
dh
o f the d ifferen tia l relations : ------ and ------ deduced from the relevant
dt
dt
formulae, or simply by the method o f linear interpolation as described
below.
Let the observed star be intersected at Sr and SJ (r = 1 or 2) and the
corresponding values of horizontal and vertical circle readings as instants
o f intersections be H r , H ' ; hr, h r ' and tr , fr' . N ow in order that Sj , S(
and S2 , S2 may lie on the great circular arc through Z,
and H 2 are
required to be equal to
and H 2 respectively, either by applying corrections
o f Hj — H [ to HJ and H 2 — H'2 to H 2 in which case the values o f t ( ,
and h\ , h'2 have to be reduced to those o f t [ - , t£., and h ’v , h'T by applying
corresponding corrections o f A t [ to t { , A t o f2 ,
to h [ and A/i2' to
h'2 where
(¾ — t o (Hi — H i )
Af,' = -------------------r—----- »
(H 2 — H i)
m — t[
(H 2 — H i )
) (H2— H'2)
----------------------------(H 2 — H i)
A t2 =
“
(19)
(H 2 — H I)
or by applying corrections o f H i — H i to H j and H 2 —- H 2 to H 2 , in w hich
case the values o f fi , t2 and 7ix , h 2 have to be reduced to those o f t\ , t2
and h { , h 2 by applying corresponding corrections o f \ t t to tt , Af 2 to t2 ,
A/it to h t and Ah 2 to h 2 where
A t, =
Ah,
(t 2— fiX H , — H i)
---------------------------(H 2 — H , )
( h 2 — h j (H j -—•H i )
= ---------------------------- ,
(H 2 — H ^
A
U 2 — fi) (H 2 — W 2)
= ---------------------------(H 2 — H J
Ah2 =
(21)
(h 2 — * i ) ( H 2 — H £ )
-----------------------------(H 2 — H i)
(22)
The evaluation o f form ulae (19), (20), (21) and (22) is obviously a
simple m atter o f arithm etical calculations in volvin g hardly more than
2 figures.
ACCURACY
Latitude
D ifferen tiatin g form ula (13) w ith respect to 9 and V and assuming the
error in h { , to be equal to that in V, fo r the sake o f sim plicity, w e have :
1
A <p= — tan
9
2
1
= — tan
2
0
Y
/
*i+Ai
V\
j c o t --------------- \- tan — 1 . AV
\
2
2 /
h[ + *i
c o t ------------ . AV
2
T his shows that fo r an error o f 1" in V or h [ , there is a corresponding
1"
error o f the order o f on ly ------ in ©, a relation w hich is v ery satisfactory
2
fo r the determ ination o f
9.
Azimuth
D ifferen tiatin g form ula (16) w ith respect to A, V , T and 9 and assuming
that the errors in h [ and t[ are equal respectively to those in V and T
(w ith opposite sign), for the sake o f sim plicity, w e have :
tan A
tan A
, ,
A A -- ----------- A V H------;------ A T + tan A • tan <p • A <p
sin V
sin T
=
tan ^ • A V , in the case o f close circum polar stars,
sin V
tt
It is obvious from the example, appended at the end o f this article,
w here A = 4°, V = 2° and T = 31°, that fo r an error o f 1" in V there is
a corresponding error o f the order o f 2 " in A, and the effect o f an error o f
1 " in T and 9 is almost negligible in the case o f close circumpolar stars but
it becomes m ore and more significant for larger values o f A and at places
w here latitudes are considerably high.
Time
D ifferen tiatin g form ula (18) w ith respect to t[, V and 9 and assuming
the error in h [ to be equal to that in V, for the sake o f simplicity, w e have :
1 tan —
v \
I tan h { — —
)
^
2
2/
tan ç • A 0
A t i -------------------------------------------A V H--------------- --------------/
1
T\
/
1
T\
/ cot t [ ---- — t a n — j
f cot t [ ---- — tan — I
f
= — tan h [ ■ tan t[ • A V + tan ç ■ tan t [ ■ A <p .
F rom the above, it is quite clear that fo r an error o f 1" in V and <p,
the corresponding error in t [ is large for values o f t [ near 90°, as in the
case o f close circum polar stars, but is significantly small for sm aller
values o f t(, except at places w here latitudes are considerably high.
C O N C LU SIO N
The m ethod described in this article m ay prove to be o f value fo r the
simultaneous determ ination o f latitude, azimuth and tim e required in
topographic and hydrographic surveys. But since the determ ination o f tim e
obtained fro m observations o f close circum polar stars is much less precise,
the m ethod m ay be recomm ended fo r general use, provided close circum ­
polar stars are avoided w h ile determ ining time.
A n exam ple is given at the end o f this article in order to make the
method o f observation and com putation more clear.
A C K N O W LE D G E M E N T
I
owe special thanks to Messrs S. B. D a s and K. S. N a m d h a r i o f the
Survey o f India for their kind assistance in com puting the results o f the
table and the example given as an appendix, and also to Mrs. Shanti
B h a t t a c h a r j i for her keen interest in drawing the diagrams included in
this article.
REFERENCES
B h a t ta c h a rji,
J.C. (1959) : Azim uth from observations o f close circum polar
stars.
(1951) : Plane and geodetic surveying, P art II, 4th edition.
W illia m (1863) : Manual o f spherical and practical astronomy.
A.G.N. (1938) : A d m iralty Manual of hydrographic surveying.
C la r k , D .
D o o lit t le ,
W y a tt,
F o rm
1
Field book
Place
Instrument
O b se rv e r
O b je c t
Dehra Dun, Hunter Observatory
W ild Ta
X
Star : W
Horizontal
FIRST SET
R.M.
Star
Star
Star
Star
R.M.
FL
FL
FR
FL
FR
FR
SECOND SET
R.M.
FR
Star
FR
FL
Star
Star
FR
Star
FL
R.M.
FL
Instants
of intersections
Circle readings
Face
28° 52'
0 42
180 41
0 42
180 41
208 52
35"
28
56
05
33
24
208” 52' 26"
180 41 19
0 41 50
180 41 41
0 42 12
28 52 34
L.S.T.
Vertical
121 “
29' 00 "
31 59
27 48
33 08
5
5
5
5
50
50
51
51
60° 36' 42"
119 23 06
60 37 52
119 21 55
7
7
7
7
53
53
54
55
58
121
58
15.2.62
S.T.
Date
C hronom eter
10.2
45.5
20.8
56.2
16.8
52.2
27.5
02.8
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