Increment of the sun’s hour angle

This page increment of the sun’s hour angle is designed to calculate the GHA (Greenwich Hour Angle) for the exact moment of a sextant observation by initially acquiring the GHA from the beginning of the whole hour as specified in the nautical almanac.

After reading this page remember the most important page to start with is the LaRochelle exercise.

Increment of the sun's hour angle image The hour angle of the sun increases 15° every hour
The hour angle of the sun increases 15° every hour

(0,25° per minute)


Cape Palliser lighthouse. (New Zealand) / Author image: Vince traveller

Increment of the sun’s hour angle

The increment of the sun’s hour angle exhibits a linear increase at a rate of 0.25 degrees per minute, known as the ‘time to arc.’ This expression captures the relationship between time and the corresponding solar angle. In our context, precise calculations necessitate determining the increase of this angle (arc) with accuracy down to the second

Increment of the sun's hour angle, image time to arc
For the 0h 16m 07s see example below

Increment of the sun’s hour angle

In conclusion we see in this daily page that the hour angle increases 15°00’,1 this day

(193°51’,2 – 178°51’,1 = 15° 00’,1) In fact we neglect the 0’,1

The Sun’s hour angle doesn’t precisely increase by 15 degrees every hour due to factors like the Earth’s elliptical orbit and axial tilt, resulting in a slight variation known as the equation of time. This leads to a non-uniform rate of change in the Sun’s position in the sky.

For practical purposes, we can use the increase in the Sun’s hour angle of exactly 15 degrees per hour for the calculation of time to arc.

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Increment of the sun’s hour angle

example: time to arc

Firstly we start with the scientific calculator

For instance what is the the GHA at 02h 16m 07s UT ?

02 h 00m 00s UT ⇨ GHA = 208°51′,3

02 h 16m 07s UT ⇨ GHA = —-°—‘,-

03 h 00m 00s UT ⇨ GHA = 223°51′,4

In fact, (223°51’,4 – 208°51’,3 = 15° 00’,1) We neglect the 0’,1

As a result we start the increment calcul or pp (time to arc)

Whereas pp = proportional part of 15°

Accordingly, the next step in the process involves utilizing the aforementioned scientific calculator method with the key [DMS] or [° ‘ “].


Increment of the sun’s hour angle:

Firstly we consider 16 minutes et 07 secondes

Therefore, by inputting the given values of

0 h [° ‘ “],

16 m [° ‘ “],

07 s [° ‘ “]

into the scientific calculator using the [DMS] or [° ‘ “] key and multiplying the resulting angle in decimal form by 15, we arrive at the solution of 4.029167.

0 [° ‘ “] 16 [° ‘ “] 07[° ‘ “] x 15 = 4,029167,

Also don’t forget the first zero!

After obtaining the decimal answer, we convert it into degrees, minutes, and tenths of a minute using the inverse (DMS) button.

4,02967 [INV]  [° ‘ “]  ⇨  4°01’ 45” ⇨ 4° 01′, 8

In conclusion, the sun’s hour angle has increased  4° 01′, 8 in 16minutes  and 07 seconds

Additionally see the use of calculator. how to round the seconds!

Finally, calculate GHA at 02h 16m 07s UT

GHA = 208° 51’,3   (UT: 02h00m00s )

+incr.  =   4° 01′, 8. (pp)

_______________

GHA =  212°53,1  (UT:02h 16m 07s)  

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Increment of the sun’s hour angle

exercises: time to arc
Increment of the sun's hour angle, (pp)exercises: time to arc

Also see the use of calculator. how to round the seconds!


Increment of the sun’s hour angle:

Increment and corrections tables

Second method time to arc: Instead of using a scientific calculator we use the permanent table of the nautical almanac

For instance we find for each minute a table with 60 seconds

Also see below that the pp (increment) for 16 minutes and 07 seconds

As a result pp = 4°01′, 8

21 pages for the increment tables