6.6 Azimuth by Star Hour Angles.

From the transformation of Hour Angle celestial

coordinates (h, δ) to Horizon celestial coordinates (A,a), we have the equation:

sin h Equation 4
tan A=
sin φ cos h−tan δ cos φ

To solve this equation for A, we must know the latitude φ of the place of
observation. The declination of the observed star, δ, can be obtained from a star
catalogue, ephemeris, or almanac (e.g. FK4, AA, SALS) and updated to the epoch of
observation (T). The hour angle of a star, (h), cannot be observed, but it can be
determined if time is observed at the instant the star crosses the vertical wire of the
observer's telescope.

Assuming one observes zone time (ZT), then the hour angle is given by
combining the following equations:

LST =h+ a Equation 5

GSD=0.671+1.002737903 JD Equation 6

LAST =GAST + λ Equation 7

UT =ZT +∆ Z Equation 8

GMST =UT +¿) Equation 9

combined equation of 5-9

h
[h=ZT + ∆ Z +(am −12 )+ Eq . E .+ λ−a] Equation 10

in which:
ZT is observed
∆ Z∧λ are assumed known
h
Eq . E .∧a m−12 are catalogued ∈ AA
or ( am−12 ) + Eq . E . is catalogued ∈SALS
h

The minimization of the effects of systematic errors are important for any azimuth
determination. This can be done, in part, through the selection of certain stars for an
observing program. Assuming that the sources of systematic errors in (equation 4)
occur in the knowledge of latitude and the determination of the hour angle, we can
proceed as follows.

Differentiation of (equation 4) yields.

dA=sinA cotZ dφ+ cosφ ( tanφ−cosA cotZ ) dh ; Equation 11

Examination of (equation 10) yields the following:

(i) when A=0 °∨180 ° , the effects of dφ are eliminated,
(ii) when tanφ=cosA cotz , the effects of dh are "eliminated.

Thus, if a star is observed at transit (culmination), the effects of dφ are minimized,

while if a star is observed at elongation (parallactic angle ¿ 90 ° ), the effects of dh are
minimized. Of course, it is not possible to satisfy both conditions simultaneously;
however, the effects will be elliminated if two stars are observed such that

sin A1 cot z 1=−sin A 2 cot z2 , Equation 12

Equation 13
2 tanφ−(cos A 1 cot z 1 +cos A 2 cot z 2 )=0

As a general rule, then, the determination of astronomic azimuth using the hour
angle method must use a series of star pairs that fulfill conditions (equation 12) and
(equation 13).

A special case can be easily made for circumpolar stars, the most well-known of
which in the northern hemisphere is Polaris (Ursae' Minoris ). When φ> 15° Polaris is
easily visible and directions to it are not affected unduly by atmospheric refraction.
Since A=0 °, the error dφ is eliminated

Equation 14
 cosφ (tanφ−cosA cotz)=cos δ cosz cosp,

then when ~ = 90° I the effects of dh are eliminated. Then Polaris can be observed at
any hour angle for the determination of astronomic azimuth by the hour angle method.

A suggested observing sequence for Polaris is as follows [Mueller, 1969]:

(i) Direct on R.O., record H.C.R.,

(ii) Direct on Polaris, record H.C.R. and T,
(iii) Repeat (ii),
(iv) Repeat (i),
(v) Reverse telescope and repeat (i) through (iv).

The above observing sequence constitutes one azimuth determination; eight sets are
suggested. Note that if a striding level is 'used, readings of both ends of the level (e.g. e
and w for direct, e ’∧w ’ for reverse) should be made and recorded after the paintings on
the star. An azimuth correction, for each observed set, is then given by [Mueller, 1969]

∆ A = {d ¿ ((w+ w ' )−(e +e ' )) cotz Equation 15

4

in which d is the value in arc-seconds of each division of the striding level.

Briefly, the computation of astronomic azimuth proceeds as follows:

(i) for each of the mean direct and reverse zone time (ZT) readings of
each set on Polaris, compute the hour angle (h) using (equation 10),
(ii) using (equation 4), compute the astronomic azimuth A of Polaris for
each of the mean direct and reverse readings of each set,
(iii) using the mean direct and mean reverse H.C.R.'s of each set on
Polaris and the R.O., compute the astronomic azimuth of the terrestrial
line,
(iv) the mean of all computed azimuths (two for each observing sequence,
eight sets of observations) is the azimuth of the terrestrial line,
(v) computation.of the standard deviation of a single azimuth
determination and of the mean azimuth completes the computations.
computations may be shortened somewhat by (i) using the mean readings direct and
reverse for each set of observed times and H.C.R.'s (e.g. only 8 azimuth
determinations) or (ii) using the mean of all time and H.C.R.'S, make one azimuth
determination. If either of these approaches are used, an azimuth correction due to the
non-linearity of the star's path is required. This correction ( ∆ A c ) is termed a second
order c curvature correction [Mueller, 1969].

It is given by the expression

Equation 16
in which n is the number of observations
that have been mean (e.g. 2 for each observing sequence (direct and reverse), 16 for
the total set of observations (8 direct, 8 reverse). The term C A is given by

Equation 17

in which the azimuth A is the azimuth of the star (Polaris) computed without the
correction. The term miis given as

Equation 18
where

Equation 19

and

Equation 20

If, when observing Polaris, the direct and reverse readings are made within 2m ¿ 3m
, the curvature correction ∆ A c will be negligible and could therefore be neglected
[Robbins, 1976]. An example of azimuth determination by the hour angle of Polaris is
given in equation 4.
 Azimuth by hour angles is used for all orders of astronomic work. The main
advantages of this method are that the observer has only to observe the star as it
coincides with the vertical wire of the telescope and since no zenith distance
measurement is made, astronomic refraction has no effect. The main disadvantages are
the need for a precise time-keeping device and a good knowledge of the observer's
longitude.

6.7 World Geodetic System (WGS 84). WGS 84 is a commonly used worldwide datum
developed from satellite measurements of the earth. It is rapidly becoming the preferred
datum around the world. Satellite images are often published using this datum. Unlike
most datum, the origin for WGS 8he ellipsoid is also called WGS 84. Its ellipsoidal
parameters are:

a=6378137

2
e =0.00669438

6.8 The Luzon Datum. The Luzon Datum is the local datum used in the Philippines for
geodetic survey such as triangulation. Its reference ellipsoid is the Clarke 1866. Its
parameters are as follows:

a=6378206.4

1
=294.98
f

Its point of origin is in Balanacan station, Marinduque, Philippines and has the
following geographic position:

φ=13 °33 ' 41

λ=121 ° 52 ' 03

6.9 Convergence of Meridian. The angular convergence of two meridians is a function

of the distance between the meridians, the latitude, and dimensions of the reference
ellipsoid. The angular convergence in seconds is:
 d tan φ √(1−e sin φ)
2 2
θ= ρ
a

Where:

a=semi−major axis of the ellipsoid

e=eccentricity of the ellipsoid

ρ=206,265 /ra

6.10 Universal Transverse Mercator (UTM). The Universal transverse Mercator (UTM)
geographic coordinate system is a grid based method of specifying locations on the
surface of the earth that is a practical application of a two dimensional Cartesian
coordinate system. It is a horizontal position representation, i.e, it is used to identify
locations on the earth independently of vertical position, but differs from the traditional
method of latitude and longitude in several respects.

The UTM is not a single map projection. The system instead employs a series of
sixty zones, each of which is based on specifically defined secant transverse Mercator
projection.

6.11 UTM Zone. The UTM system divides the surface of earth between 80 ° S∧84 ° N
latitude into 60 zones, each 6 ° of longitude in width. Zone numbering increases
eastward to zone 60 that covers longitude 174 to 180 east.

Each of the 60 longitude zones in the UTM system uses a transverse Mercator
projection, which can map a region of large north-south extent with low distortion. By
using narrow zones of 6 ° of the longitude (up to 800 km) in width, and reducing the
scale factor along the central meridian by only 0.0004 to 0.9996 (a reduction of 1:2500),
the amount of distortion is held below 1 part in 1,000 inside each zone. Distortion of
scale increases to 1.0010 at the outer zone boundaries along the equator.

6.12 Universal Transverse Mercator (UTM) Projection Based on Luzon Datum.

Each state plane transverse Mercator projection has defining constants which are as
follows:
 λ 0=longitude of the central meridian

E0 =false easting∨X coordinate of the central meridian

Φ=latitude of the grid origin

N 0=northing∨Y coordinate of the grid origin

k 0=scale factor for the central meridian

In projecting Luzon Datum coordinates to UTM, the following constants and

formulas are used.
Constants:

In the determination of Easting, the following formula is used based on the book “Theory
and Practice"

Where: