The Demystified Astro Navigation Course Unit 2


Unit 2 – Local Hour Angle and Greenwich Hour Angle

 Local Hour Angle (LHA).  In astro navigation, we need to know the position of a celestial body relative to our own position.

(The following diagram illustrates the explanations given below).

PZX TRIANGLE

LHA is the angle ANU on the Earth’s surface which corresponds to the angle ZPX in the Celestial sphere.   In other words, it is the angle

between the meridian of the observer and the meridian of the geographical position of the celestial body.

Due to the Earth’s rotation, the Sun moves through 15o of longitude in 1 hour and it moves through 15 minutes of arc in 1 minute of mean time.  So the angle ZPX can be measured in terms of time and for this reason, it is known as the Local Hour Angle.

Greenwich Hour Angle (GHA).  The hour angle between the Greenwich Meridian and the meridian of a celestial body is known as the Greenwich Hour Angle.

The Local Hour Angle between an observer’s position and the geographical position of a celestial body can be found by combining the observer’s longitude with the GHA. This is demonstrated in the following diagram.

GHA and LHA

O represents the longitude of an observer;

X represents the meridian of a celestial body;

G represents the Greenwich Meridian.

Because, in this case, the observer’s longitude is east and because LHA is measured westwards from the observer’s meridian to the meridian of the celestial body, LHA is equal to the GHA plus the longitude.

So we have the rule:  Long East, LHA = GHA + LONG

Note.  If the result is greater than 360o, we must modify the rule so that the result will be between 0o and 360o.  So the rule now becomes:

Long East, LHA = GHA + LONG (-360o as necessary).

For example: If Long. is 90oE. and GHA is 300o

Then LHA = GHA + LONG -360o

= 300o + 90o = 390o – 360o = 30o

If the longitude is to be west then the rule will change so that LHA would equal GHA minus Longitude. In this case, the rule is:

Long West, LHA = GHA – LONG

Note.  If the result is greater less than 360o, we must modify the rule so that the result will be between 0o and 360o.  So the rule now becomes:

Long West, LHA = GHA – LONG (+ 360o as necessary)

For example, if Long. is 90oW. and GHA is 45,o we have:

LHA = 45o – 90o = -45o + 360o = 315o

Calculating the Greenwich Hour Angle. The Nautical Almanac contains tables of raw data concerning the Greenwich Hour Angle for the Sun, the Moon, the navigational planets and selected stars.

The extract shown below, is of the Nautical Almanac daily page displaying hourly values of the Greenwich Hour Angle and the Declination of the Sun and the Moon for the 24th.  December.  The examples following demonstrate how the daily pages are used to calculate GHA:

na_2009-249 copycropped2
Example 1.  To find the LHA and Declination of the Sun.  
At 04 hours, 32 minutes, 04 seconds GMT on 24 December 2009, the assumed position of your yacht is 40o 35.5’ South 32o 13.8’ East.   Find the LHA and Declination of the Sun.

 Calculating the Local Hour Angle (LHA).  Before we can calculate the LHA, we need to find the Greenwich Hour Angle of the Sun.  

 Step 1.  Find the GHA for 0400 GMT.   From the daily page extract, we find that the GHA of the Sun for 24 December at 0400 is 240o 07’.5.  This is written as:  GHA (04h) : 240o 07’.5

Step 2.  Calculate the increment for 32 minutes, 04 seconds:  In this step, we find the increase in GHA for minutes and seconds of GMT.

The Nautical Almanac contains tables of corrections for increments of time from 0 minutes to 59 minutes.  The following extract shows the increments and corrections tables for 32 minutes.

32 33cropped2

From the extract we see that, in the table for 32m, the increment for 04 seconds is 8o 01’.0

This is written as:

Inc. (32m 04s):  8o 01’.0

Note.  Since GHA is always increasing, the increment correction is always added.

 Step 3.  Calculate GHA at 04 hr 32 min 04 sec GMT

GHA (04h)             240o 07’.5

Inc. (32m 04s)         + 8o 01’.0

GHA Sun               248o 08’.5

 Step 4.  Find LHA 

In this step, we combine the GHA with the longitude to calculate the LHA:

GHA                       248o 08’.5

Long:                     +  32o 13’.8 East

LHA Sun                 280o 22’.3

The full procedure for calculating the LHA of the Sun can be summarised in the following format:

GHA Sun (04h)       240o 07’.5

Inc. (32m 04s)            8o 01’.0

GHA Sun               248o 08’.5

Long                        32o 13’.8 E.  (+)

LHA Sun                280o 22’.3

Click here for an exercise in this topic

 (Note. This topic is covered in greater depth in the book ‘Astro Navigation Demystified’).

Note.  Units of this course are issued weekly.

Watch for unit 3

Where to buy books of the Astro Navigation Demystified series:

Astro Navigation Demystified at Amazon.com

Astro Navigation Demystified at Amazon.uk

Applying Mathematics to Astro Navigation at Amazon .com

Applying Mathematics to Astro Navigation at Amazon .uk

Astronomy for Astro Navigation at Amazon.com

Astronomy for Astro Navigation at Amazon.uk

web: http://www.astronavigationdemystified.com

e: astrodemystified@outlook.com

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