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DETERMINATION OF AZIVUTH , LATITUDE

AND LONGITUDE BY ASTRONOMICAL OBS RVATIONS.

A
Thesis
Submitted to the Faculty
of
MICHIGAN AGRICULTURAL COLLEGE

By

M. H. Small E. I. Matson

Candidates for the
Degree of

BACHELOR OF SCIENCE

June 1922
THESIG
PREFACE

We wish to express our thanks
to Professor H. K. Vedder for his
aid and advice in the compilation

of the data used in this Thesis.

 

103801
DETERMINATION OF AZIMU™I, LATITUDE

AND LONGITUDE BY ASTRONOMICAL OBSERVATIONS.

 

To the average layman tne subject of astronomy
and the movements of all heavenly bodies is an unfathom-
able mystery into which he feels not the slightest interest
to delve. ‘To him astronomical research is merely a
pastime for the scientist who is possessed of sufficient
wealth as to be able to carry on his work for his personal
satisfaction. He does not realize that the field of
astronomical research has contributed greatly to the
store of human knowledge.

To the engineer and more especially the surveyor,
& thorough knowledge of practical astronomy is very
essential.

The locations of all land lines and other geog-
raphical and geodetic data obtained by the Government
engineers on the Geodetic Survey, is based on the exact-—
ness of astronomical observations.

The locations of all points and places on the
earth, the difference in longitude and latitude and the
exact time of day can be obtained by & surveyor by
observations on certain stars, and so it behooves the

young engineer to acquire a thorough grounding in the
basic and fundamental laws which control the movements of
the bodies of the celestial sphere.

In takins this subject for our ‘thesis, it wes our
intention to accomplish three thinss by astronomical ob-
servations. First - The exact latitude of Lest Iansing;
Second - Checking the Azimth of a line lzid out on the
compass; Third - The longitude of Last Lansing. we will
attempt furthermore to give a description of the subject
of practical astronomy that may be readily understood by
anyone outside of the field of science.

Practical astronomy treats of the tneory and use
of astronomical instruments and the methods of computing
the results obtained by observations.

That part of most importance to the surveyor is
that which deals witn the mthcds of locating points on
the earth's surface and of orienting the lines of a survey,
and includes the determination of; First, latitude -
second, time - Third, longitude - and Fourth, Azimuth.

It would be well here to give several definitions
of the meanings of terms and phrases to be used in this
Thesis so that it will be more readily understood. These
definitions are taken from Hosmer's work on “Practical
Astronomy” and because of the plain and exact lan guage
used are set dowm here verbatim. (See Plate 2)

VERTICAL LIVE: At any point on the earth's surface is
the direction of gravity at that point, md is mown by
the plumb line, or indirectly by means of the spirit level.
ZENITH“NADIR; If the vertical line at any point be pro-
longed upward it will pierce the sphere at a point called
the Zenith. This point is of great importance because

it is the point on the sphere wnich indicates the posi-
tion of the observer on the earth's surface. The point
where the vertical prolonged downward pierces the sphere
is called the NADIR.

HORIZON: The Horizon is the great circle on the celestial
sphere cut by a plane throush the center of the earth
perpendicular to the vertical. The Horizon is everywhere
90° from the Zenith and Nadir. It is evident that a
plane through the observer pernendicular to the vertical
cuts the sphere in this same great circle. The Visible
Horizon is the circle where the sky and sea seem to meet,
Projected onto the sphere it is @ small circle below

the true horizon and parallel to it. Its distance below
the true horizon depends upon the height of the observer's
eye above the water.

VERTICAL CIRCLES; Vertical Circles are great circles
passing through the Zenith and Nadir. they all cat the
horizon at right angles.

AIMUCANTERS: Parallels of latitude or almucanters are

 

smalil circles parallel to the horizon.

POLES: If the earth's axis of rotation be produced in-

-3-
definitely it will pierce the sphere in two points
called the celestial poles.

EQUATOR: The Celestial Equator is a great circle of
the celestial sphere cut by a plane tnroush the
center of the earth perpendicular to the axis of ro-
tation. It is everywhere 90° from the poles. A
parallel plane through the observer cuts the sphere
in the same circle.

HOUR CIRCLES: Hour Circles are great circles passing
through the north and south celestial poles.
PARALLELS OF DECLINATION : small circles parallel to

 

the plane of the equator are called parallels of
declination.

MERIDIAN: The Meridian is the great circle passing
through the Zenith and the poles. It is at once an
hour circle and a vertical circle. It is evident that
different observers will have different meridians.

The meridian cuts the horizon in the north and south
points. The intersection of the plane of the Meridian
with the horizontal plane through the observer is the
Meridian line used in surveying.

PRIME VERTICAL: Prime Vertical is the vertical circle
whose plane is perpendicular to the plane of the
Meridian. It cuts the horizon in the east and west

points.
 

 
LATITUDE: On the earth's surface is the angular dist-
ance of the point in cuestion north or south of the
equator.

LONGITUDE: The arc of the equator between the primary

 

meridian, which is usually taken through Greenwich, and
the meridian of the point in question.

AZIMUTH: (1) Azimuth is the angular distance on the
horizon between the meridian and the vertical circle
through the point.

AZIMUTH: (8) On the earth's survace, Azimuth is the
angular distance measured from the south in a clockwise
direction of the point in cuestion.

In makinz the observations in obtaining the
data used in this Thesis, we used the equator system
of coordinates on the celestial sphere.

The circles of reference in this system are
the equator and great circles throuch the poles, o9
hour circles. the first coordinate of a point is its
angular distance north or south of the equator. It is

called the DECLINATION.

 

The complement of the Declination is called

the POLAR DISTANCE.

 

The second coordinate of the point is the
arc of the equator between tne vernal equinox and the

foot of the hour circle through the point. It is called
RIGHT ASCH SION.

 

Risht Ascension is neasured fron the equinox
“eastward” to the hour circle throush the point in

question, it may be measured in desrees, minutes and
seconds of arc, or in hours, minutes and secords of
time.
HOUR ANGLE: The Hour Angle of a point is the are of
the equator between the observer's meridian and the
hour circle throush the point. It is masured from
the meridian "westward" from O hours to 24 hours, or
from 0° to 360°,

The instruments used in making the observ-
ations for this Thesis were a transit, with all its
accessories including a prismatic eyepiece, chrono-
meter and the data for the observations were firured
from the "NAUTICAL ALMANAC AND SOLER EPHMWERIS FOR
1922" issued by the Federal Government.

The line and point used in this work is a
line located on the drill field of the M. A. C.
CAMPUS, the south point of the line beings a@ copper
bolt set in a@ concrete monument vhich is set flush
with the surface of the ground. This south noint is
located by the follovinge witness: Si corner of
Armory - N 61° oo' EF 128.1' feet. WN W corner of
Gymnasium - N 88° 50' W 146.9" feet. Hydrant y 29 15:

W 168.4' feet.
The north point of the line is wn iron bolt
set in a concrete monunent flusn with the surface of
the ground, which is loceted by tne followins witnesces:
S E corner of Yean Bissell's house ! 69° 30' y 132.7!
feet. S w comer Professor Johnston's house NW 81° 16' E
108.4' feet. 30" Hard lavle Tree S 8° 30' EB 12.5' feet.

This line is exactly pérallel witn esnother
line already established. ‘The alternate interior angles
between the two different North and South points being
0° 43' 30" (See Plate 1).

In making the observations for Azimuth, Ursa
Minorus, Paloris was used. This is a star of the second
magnitude which means it is very bricht and easily seen
with the naked eye. It is & circumpolar star md has
avery small orbit being very close to the pole :nd is
the brightest of all the stars in this custellation.
This is the star most generally used in determining
Azimuth and the "NAUTICAL ATI/UAC" gives complete
tables necessary to the computing its position in the
heavens at any tim.

To get the Azimth of a line on the earth we
must get the true Azimuth from observation on som star
and by means of @ trensit project it onto the earth.

The true Azimutr of Polaris can be computed for any

time of the 24 hours.
From the "NAUTICAL ALMANAC" we first find
the RIGHT ASCENSION of the mean sun at Greenwich,
this must now be corrected for if. “4. C. This
correction amounts to 55.5¢€ seconds because If. A. %.
is on the 84°30' meridian of longitude and “Sreenwich
is on the zero meridian. Tnis correction is obtained
by mltiplying the variation per hour of the sun's
right ascension by the number of degrrees of lonsi-
tude between If. A. C. and Greenwich expressed in
hours - 15° of longitude being equal to one hour of
time. This correction is added to the risht ascen-
sion for Greenwich because we are Vest of Greenwich
which gives us the sun's risht ascension for mean
noon at I. A. Ce

The right ascension of Polaris for the day
of observation is then obtained from the table, no
correction being used as they ere very small, being
decimals of a second and would meke no difference in
the final result. Subtrect the Richt Ascension of
Polaris from the xisht Ascension of the Sun which
gives the Sidereal Interval. As the table for
Azimaths of Polaris at any hour ansle and degree of
latitude is in Solar time, this Sidereal Interval
must be changed into Solar time.

Now choose any convenient time for making

-8-
 
the observation in mean standard time, and as M.A.C.
is east of the 90th Meridian, the time interval must
be added which for 84°30' is 22 minutes. Wow add
the mean Solar time for observation as corrected,

to the Solor interval obtained above und we have the
hour angle of Polaris, the chosen time. From the
table of Azimuths already referred to, the Azimuths

can be readily obtained.
In making the observations for Azimuth, the
following method was used: Tne transit was set up
directly over the center of the south point of the
line already mentioned - the plates leveled up, the
instrument being set very firmly in the ground. This
is very essential for in makings the observation, any
little jar however small may cause the instrument to
go out of adjustment, thus causine an error in the
results obtained. The index error on the vertical
arc, if any, is very important and should be known
and allowances made for it. By means of a flashlight
the north point of the line whose Azimuth we were
determining, was located and the vertical cross-hair
set on it, the plates being set at zero.

The lower plate was now clamped tightly ad
the upper plate loosened so as to be moved with the
tangent screw as the time of observation approached.
The star was brouscht into the field of the telescope,
and by means of the tangent screw, the observer kept
the vertical cross-hair directly on the star, - his
assistant calling out the seconds and giving a tip
at the exact time. ‘The readine on the vernier should
now check the Azimuth as computed from the tables in
in the Almanac for the hour anzle assumed. "The n to

obtain the Azimuth of the assumed line, this reading

-10-
will be the number of degrees or minutes from true
North. Turn this off, thus obtaining true North,
and then the Azimutn of tne line in cuestion can be
readily obtained by readinz the ancle between tne

lines.

-ll-
LATITUDE.

In meakinz the observation for Latitude,
all the above precautions of set-up of instrument
and technic of observation should be followed. The
compiling and computing of the necessary data is
also sonewhat Similiar.

The Right Ascension of the man sun at
Greenwich being taken from the table, and this
corrected for M. *. C. The Risrht «scension of
Polaris is then obtained for the day of observation
and if the observation is made at culmination as was
done in the work for this Thesis, the tizht 4scensi on
of Polaris is subtracted from the corrected Risht
Ascension of the mean sun to which has been added 12
hours for lower Culmination. For upper Culmination,
the two Right Ascensions are added.

The result is then changed to mean Solar
time and then reduced to mean standard time for KM.A.C.
by subtracting the correction as above. This result
gives the exact time to observe,

The Declination of Polaris is then obtained
for that Risht Ascension from the table, and as lower
Culmination was used, this Declination is subtracted
from 90° and added to the observed readinz on the
vertical arc. The result is the exact latitude of
the place, i.e. IH. A. Ceo

-12-
LONGITUDE

 

In obtéinines lonzitude it is imperative that
the time be exact. The precision of tne results de-
pends entirely on the correctness of the chronometer.

AS we received our correct time fron Arlington
at nine o'clock, we cnose stars for tie observation
which transited very close to nine o'clock standard
time. Spica, and 1 of the constellation Virco were
used.

The time of transit of these stars being com-
puted from the "NAUTICAL AVL; AC" in the sane manner
as for Polsris. Tnis must be exnressed in mean Solor
time for M. A. C.

The transit was set uo at an intermediate
point on the meridian, and the exact standard time of
transit taken. ‘the difference between the observed
standard time and the Yolar time of transit cives the
difference in longitude between the 90° meridian and
M. A. CG. in minutes of time. The exact loncitude of
Me. A. C. was then found by reducing this difference

to degrees and subtracting it from 90°,

~13-
AZIMUTH (SAMPLE CO!MPUTATTOIS)
For May 22, 1922

 

 

 

 

 

Right Ascension mean sun gh 57m 19.08°
at Greenwich
Correction for M. A. C. 55.52
Right Ascension mean Sun ah 5gm 14.69
Qat Me. Ae Ce
Right Ascension of Polaris 1 32 350 246
Sidereal Interval 2 25 44.14
Correction Sidereal Time to Solar P4
Time of observation is 9:30 eh g5™ 90,8
Standard Time
9:30 - 22 = 9:25" 9 52
12h 47m ~~ 208

Agimuth of Polaris at M. A. C. for an hour angle of 12h

177 208= 6,.5' East of North
Angle read on Plate 6.5' East of North
LATITUDE (SAMPL"S COMPUTATIONS)

Observation on May &€2,1922

 

Right Ascension mean Sun at
Greenwich

Correction

Right Ascension mean Sun
Qt M. A. Ce

Right Ascension of Polaris
Me. Ae Ce

Add 12 hours for lown culmination

Sidereal Interval
Correction Sidereal to Solar
Solar Interval

Difference between M. Ae C.
and Standard =

Mean Solar Time of Transit
at Me Ae Ce

Declination of Polsris
May £2

Observed Ansile on Transit

Latitude of M. A. C.

-15-

30 577 19,@88

55.52

 

 

 

 

 

 

 

3 58 14.60
1 32 30.46
12
139 sem —s_ 30,468
3 58 14.16
9 34 15.86
1 34.04
9 B82 41.82
22 1.38
9h 10M = 40.44
90°
88° 53! 6,74
19 «#6 53.26
41 37
429 431 53!
LONGITUDE (SAMPLE COMPUTATIONS)

June 12, 1922

 

 

 

 

 

 

 

Spica Declination -10° 45" 26.57"
Right Ascension of mean Sun at 5h 20" 6,768
Greenwich June 12
Correction 55.52
Right Ascension of mean Sun at 5h em © 288
Me. A. Ce
Right Ascension of Spica h
at \iashineton June 12 13 e317 7.13°®
5 AL 2628
Sidereal Interval at MK. A. Cc. gh om 4.858
Sidereal to Solar 1 18 .64
Solar Time of Transit qn 58" 46,218
Watch Time of Transit gh 371 568
(Correction from Arlington
at 9-00 PeM. Standard Time 1 13
17 13° fast)
Standard Time of Trensit 7h 36" 438
7 58" 468
7h 360 428
Difference in Lonsitude between apm 38
M. Ae Ce. and Madison
90° 00? 00
5 30° 45
p27 38 --5° zor 45 g49 291 15" Longi-
tude M. A.C.

-16-
RESULTS OF OBSERVATIONS FOR AZIMUTH

Time of Observati on

Azimuth of Polaris Azimuth of Time
(Standard Tim)

0°(Exnressed in 0°

 

 

 

 

 

 

 

 

 

 

Minutes) Expressed in ‘Minutes

May 22 = 8:30 16.5 W Or

nF = 9:30 6.5" E OF
May 23 - 8:30 15.'O W e5' E

rm NH he 69230 B8.'O E oO!
May 26 - 8:30 10.5" W oO!

" w - 9:30 13.0! E 1.0! W
May 27 =- 8:30 9.0° VW Q'

yr Ne 69230 14.5 E oh
May 28 = 8:30 7.5' W QO!

" no - 9:30 15.5' E Or
May 29 - 9:00 5.5' E 2d W
y YW = 9230 16.0° E 5 W
May 320 = 9:00 7,.0° E O

" " = 9:30 17.0° & O
June 1 =— 9:00 10.5" E

" " - 9-30 23,0! E

Jane 6 = 9:00 15.0' E O

" noe 9:30 £8.5' E 1.0' E
June 7 = 93:00 16.5 E O

Ww wm 9°30 £9.0" E O

 

-17-
RESULTS OF OBSERVATION

FOR LATITM DE

 

Lower Culmination Correction to Observed Latitude of
on Polaris on: be added snale Mle Ae Ceo
May 22 1° 6 53! 419 37" 429 43" 53"
May 23 1° 6" 53! 41° 37" 42 43 53
May 26 1° 6 54" 41° 37" 42 43 54
May 27 1° 6' 54" 41° 37 42 43 54
May 2&8 1° 6 54" 41° 37 42 43 54
Mey 29 1° 6' 55" 41° 37 «42 43° 55
May 30 1° 6" 55! 41° 37 42 43 55
June 1 1° 6" 55! 41° 37 42 43 55
June 6 1° 6 57! 41° 37 42 43 57
June 7 1° 6! 57! 41° 37 42 43°C"

 

-18-
RESULTS

RESU>TS OF OBSEAVATTC.LS FOR LOPGrmrpn

From Observations on Spica (See Sample Computat-
ions on Longitude)

Longitude I. A. Ce g4° e9' 45"

From Observations on 1 Virgins.

Solar Time of Transit at M. A. %. gh 50™ 55

 

 

Standard Vine of l'ransit at " g” £8 53
epm 8
22m 28 of Solar Time is equal to 5° zo!
90° 0cO 00
5° 30" oo!
84° 30! Longitude of M. A.C.

-19-
le

3

BIBLIOGRAPHY

 

Text-Book on Practical Astronomy - By Hosvner.

Deternination of Time, Longitude, Latitude,
and Azimuth - Department of Commerce,
Ue Se. Coast and Geodetic Survey -
By wine Sowie.

Geodetic Survey - By Cory.

General Instructions for Field Work -~U. 5.
Coast and Geodetic Survey.

Geodesy - Clarke.

 

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MAP SHOW/NG
LOCATION
TAAL AACE a
MA SMALL. £.4 MATSON

 

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Thestar Shas the altitude RS, azimuth S'R, hour angle 4m,
right ascension Vm, and declination m5; the meridian is ZS’,

MH SMALL. £./ MATSON.

 

 

 

 
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