**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)*.

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 15^{o} 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.

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 360^{o}, we must modify the rule so that the result will be between 0^{o }and 360^{o}. So the rule now becomes:

**Long East, LHA = GHA + LONG (-360 ^{o} **

**as necessary).**

For example: If Long. is 90^{o}E. and GHA is 300^{o}

Then LHA = GHA + LONG -360^{o}

= 300^{o} + 90^{o} = 390^{o} – 360^{o} = 30^{o}

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 360^{o}, we must modify the rule so that the result will be between 0^{o }and 360^{o}. So the rule now becomes:

**Long West, LHA = GHA – LONG **(+ 360^{o }as necessary)

For example, if Long. is 90^{o}W. and GHA is 45,^{o} we have:

LHA = 45^{o} – 90^{o} = -45^{o} + 360^{o} = 315^{o}

**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 24^{th.}^{ }December. The examples following demonstrate how the daily pages are used to calculate GHA:

**
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 40

^{o}35.5’ South 32

^{o}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 240^{o} 07’.5. This is written as: **GHA (04 ^{h}) : 240^{o} 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.

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

This is written as:

**Inc. (32 ^{m} 04^{s}): 8^{o} 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 (04^{h}) 240^{o} 07’.5

Inc. (32^{m} 04^{s}) __ + 8__^{o}__ 01’.0__

GHA Sun 248^{o} 08’.5

** ****Step 4. Find LHA **

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

GHA 248^{o} 08’.5

Long: +__ 32__^{o}__ 13’.8 __East

LHA Sun 280^{o }22’.3

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

GHA Sun (04^{h}) 240^{o} 07’.5

Inc. (32^{m }04^{s}) __8__^{o}__ 01’.0__

GHA Sun 248^{o} 08’.5

Long __ 32__^{o}__ 13’.8 __E. (+)

LHA Sun 280^{o} 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

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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

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