Exercise 4 – Local Hour Angle.

This post continues the series of navigation related exercises which were requested by navigation course and nautical college teachers.  When the post expires, its contents will be included under the menu heading ‘Navigation Exercises’ on the home page.

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

In the diagram below, LHA is the angle BNU 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 (GP).

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 time.  So the angle ZPX can be measured in terms of time and for this reason, it is know as the Local Hour Angle.

LHA is measured westwards from the observer’s meridian and can be expressed in terms of either angular distance or time.

For example, at noon (GMT) the Sun’s GP will be on the Greenwich Meridian (0^o).  If the time at an observer’s position is 2 hours and 3 minutes after noon, then the angular distance between the observer’s meridian of longitude and the Greenwich Meridian must be (2 x15^o ) + (3x 15’) =  30^o 45’.  Because it is after noon at the observer’s position, the longitude of that position must be to the East of the Greenwich Meridian since the Earth rotates from West to East.  Therefore the observer’s longitude must be 30^o 45’ East and since LHA is measured westwards from the observer’s meridian, the LHA must also be 30^o 45’.  However, it should be noted that as the Earth continues to rotate eastwards, the GP of the Sun will continue to move westwards so the LHA at the observer’s position will be continually changing.

Rules for calculating LHA:

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

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

Example 1. : If Long. is 90^oE. and GHA is 300^o

Then LHA = GHA + LONG -360^o

= 300^o + 90^o = 390^o – 360^o = 30^o

Example 2: if Long. is 90^oW. and GHA is 45,^o we have:

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

Example 3:  If your longitude is 35^o 46’ East and the GHA of Mars is 39^o  53’.8.  What is the LHA?

Remember the rule:

Long East, LHA = GHA + LONG (-360^o ?)

GHA   =  39^o  53’.8

LONG =  35^o  46’.0E    (+)

LHA    =  75 ^o 39’.8   .

(Remember 60 minutes in 1 degree)

Example 4.  Your assumed longitude = 125^o 13’.0W.  The GHA of the Sun is 243^o 44’.7  What is the LHA?

Long West, LHA = GHA – LONG (+360^o ?)

GHA    =    243^o 44’.7

LONG  =    125^o 13’.0W   (-)

LHA     =    118^o  31’.7

Example 5.  

Longitude is 120^oW. GHA is 70^o.

What is the LHA?

LHA = GHA – LONG (+360^o ?)

GHA   =    70^o  00’.0

LONG =  120^o  00’.0 W.  (-)

LHA   =  -50^o 00’.0

360^o  00’.0      (+)

LHA   =  310^o  00’.0

Example 6.  Longitude is 90^oE. GHA is 340^o

What is the LHA?

LHA = GHA + LONG (-360^o ?)

GHA   =  340^o

LONG =   90^oE     (+)

LHA   = 430^o

360^o            (-)

LHA   =  70^o

Test Questions

Question 1. 

Position: 38^o 15’N. 30^o 38’.3 W,

GHA Sirius: 150^o 15’.4.

What is the LHA?

Question 2. 

Position:  33^o 12’.8S.  70^o 54’.5E,

GHA Sun: 45^o 23’.4

What is the LHA?

Question 3. 

Position:  52^o 42’.45N.  65^o 19’.67E

GHA Moon:  345^o 15’.0

What is the LHA?

Question 4. 

Position:  34^o 28’.45S.   172^o 25’.8W.

GHA Mars:  15^o 30’.0

What is the LHA?

 Click here for answers to test questions.

 A fuller explanation of this topic is given in the book ‘Astro Navigation Demystified’.

Useful Link: Local Hour Angle and Greenwich Angle Hour,     Keeping Time By The Sun 

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