**Chapter XIII – Supplementary and Mandatory Provisions**

**B. Astronomical Observations**

1. Time Determination

Tangent ½T = √ cos S sec (S-P) sin (S-H) csc(S-L) Where:

T = Hour angle of the observed celestial body S = ½ (P+H+L)

H = Observed altitude of the celestial body, corrected for refraction in case of stars and parallax and refraction in the case of the sun.

L = Latitude of the place of the observation.

P = North polar distance of the celestial body 2. Latitude Determination

a. By Polaris or Circumpolar star:

Lat = h-p cos t + ½ p² sin² t tan h sin 1“

Where:

h = corrected mean observed altitude

t = hour angle of Polaris or any circumpolar star (difference between sidereal time and the star’s right ascension)

p = polar distance in second of Polaris or any circumpolar star.

b. By Sun or star at meridian passage:

For the star’s and the sun’s center south of the zenith but north of the equator:

L = Z + D

For those south’s of the zenith but north of the equator:

L = Z - D

For those north of zenith and north of the equator:

L = D – Z

L = Latitude of observer

Z = Zenith distance of celestial body

D = Declination

3. Procedure for Solar Observation for azimuth determination:

a. Set up the instrument over the station and level the instrument.

b. Set the vernier plate or micrometer reading to 00°00’ 00’ and sight the rearward station, the azimuth of which is to be determined using the lower motion and lower tangent screw.

c. Screw the colored prismatic glass to the eyepiece and using the upper motion and upper tangent screw turn the telescope toward the sun.

d. Observe the sun in accordance with the following tangent positions and record the time, vertical angle or zenith distance and horizontal angle for each sighting as shown by the following example: (1) Telescope direct (5) Telescope reverse

(2) Telescope direct (6) Telescope reverse (3) Telescope reverse (7) Telescope direct (4) Telescope reverse (8) Telescope direct

e. After the eight sightings, turn the telescope to the rearward station using the upper motion then read and record the horizontal angle. The time interval between successive sightings shall in no case exceed two minutes. The eight sightings constitute one series of observation and shall be completed within twenty minutes otherwise the results shall be voided and another series shall be made.

f. The allowable horizontal a closure shall be within +/- 30 seconds of arc and must be distributed equally among the horizontal angle readings.

g. After the observations have completed, test the accuracy of observations by determining the rate of motion of the sun. This is to detect any mistake in the observations and to be able to make additional series as required.

The test for solar variation shall be made by horizontal and vertical variation methods as follows:

1) To find the variation in horizontal angle divide the difference in horizontal angle readings by the corresponding difference in time. Combinations are from any two horizontals readings from the left-side of the telescope vertical cross-hairs or any two readings from the right-side of the vertical cross-hairs.

HVar = (Hn – H m) (Tn – Tm)

(1) (4-1) (5) (8-4) (9) (7-2) (2) (5-1) (6) (8-5) (10) (6-3) (3) (8-1) (7) (3-2) (11) (7-3) (4) (5-4) (8) (6-2) (12) (7-6)

2) To find the variation of vertical angle divide the difference in vertical angle readings by the corresponding difference in time. Combinations are from any two vertical readings from the upper-side of the telescope horizontal cross-hair or any two readings from the lower-side of the horizontal cross-hair.

Vvar = (Vn – Vm) (Tn – Tm)

(1) (3-1) (5) (7-3) (9) (8-2) (2) (5-1) (6) (7-5) (10) (6-4) (3) (7-1) (7) (4-2) (11) (8-4) (4) (5-3) (8) 6-2) (12) (8-6)

In both cases, the results should not differ from the mean variation by more than two seconds of arc per second of time. Other combinations using the mean of a direct and reverse positions shall be used to determine the variation.

Additional safeguard against mistake in observations shall be determined by noting that the average apparent angular diameter of the suns is about thirty-two minutes of arc divided by secant of altitude in horizontal.

4. Observations on circumpolar star for azimuth at elongation:

Cos t = tan L cot D Sin Z = sin P sec L

Where:

t = hour angle of the star at western elongation, and t = 24 – t at eastern elongation

L = latitude of place D = declination of star

P = north polar distance of star

The time shall be determined from the following formulas:

S = αstar + tstar

S = (αsun + 12h) + T

Where:

S = side real time

αstar = right ascension of star T = local civil time

Tstar = hour angle of star

αsun= right ascension of sun

5. The procedure for observing stars for azimuth at elongation shall be as follows:

a. Find sidereal time of elongation:

1) Compute the hour angle (t) of the star of elongation cos t = tan L cot D

2) Add the hour angle for western or eastern elongation to the right ascension of the star to get sidereal time.

b. Find the local civil time at the same instant.

a) Determine the (αsun + 12h) corrected for longitude.

b) Subtract the result of (a) for the sidereal time determine in step (1). The result is sidereal interval which shall be converted to local civil time by the application of correction from sidereal into mean solar time.

c) The result shall be corrected for longitude difference to reduce to 120° E time.

c. Set the instrument over the station in position at least thirty minutes before the time of elongation.

d. Set the micrometer (vernier) reading to approximately 00°00 ‘00” and sight the azimuth mark.

e. About five minutes before elongation bisect the star with the cross wires and follow it as it moves toward elongation using the tangent screw.

f. At elongation, read and record the horizontal angle and time.

g. Reverse the telescope, bisect the star again and read and record the horizontal angle and time.

h. Return the telescope to the azimuth mark and check the closing reading.

6. Formula for determining azimuth of a line by observations on the sun or stars at any hour angle:

Cot ½ A = √ sec S sec (S-P) sin (S-H) sin (S-L) The result shall be checked using the formula:

Cos A = tanH tanL – cosP secH secL In both formulas:

A = angle between the celestial body and the south

H = observed altitude of the celestial body corrected for refraction in case of stars and for parallax and refraction in the case of the sun.

L = Latitude of the place of observation

P = north polar distance of the observed celestial body S = ½ (P+ H + L)

P = 90 - D when the declination is north, and P = 90 +D when the declination is south.

D = apparent declination of the observed celestial body. The signs of D should be reversed for south declinations.

In the morning the azimuth of sun is 360° - A and in the afternoon, the azimuth of sun is equal to A. If the station mark is left of the sun, the horizontal angle is to be subtracted from the azimuth of the observed celestial body; if to the right it is to be added.