# Local Hour Angle

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 15o 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 (0o).  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 x15o ) + (3x 15’) =  30o 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 3045’ East and since LHA is measured westwards from the observer’s meridian, the LHA must also be 30o 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.

To summarise the above rules:

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

Long West, LHA = GHA – LONG (+ 360as necessar

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

Then LHA = GHA + LONG -360o

= 300o + 90o = 390o – 360o = 30o

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

LHA = 45o – 90o = -45o + 360o = 315o

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

Remember the rule:

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

GHA   =  39o  53’.8

LONG =  35o  46’.0E    (+)

LHA    =  75 o 39’.8   .

(Remember 60 minutes in 1 degree)

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

Long West, LHA = GHA – LONG (+360?)

GHA    =    243o 44’.7

LONG  =    125o 13’.0W   (-)

LHA     =    118o  31’.7

Example 5.

Longitude is 120oW. GHA is 70o.

What is the LHA?

LHA = GHA – LONG (+360o ?)

GHA   =    70o  00’.0

LONG =  120o  00’.0 W.  (-)

LHA   =  -50o 00’.0

360o  00’.0      (+)

LHA   =  310o  00’.0

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

What is the LHA?

LHA = GHA + LONG (-360o ?)

GHA   =  340o

LONG =   90oE     (+)

LHA   = 430o

360o            (-)

LHA   =  70o

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