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 30o 45’ 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.
Greenwich Hour Angle (GHA). As discussed above, the angle between two meridians of Longitude can be expressed as an hour angle. 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 as shown in the following diagram.
O represents the longitude of an observer; G represents the Greenwich meridian; X represents the celestial meridian of a celestial body such as the Sun.
In this case it can be seen that LHA is equal to the observer’s longitude plus the GHA of the celestial body. However, the method of calculating LHA depends on whether the observer’s longitude is east or west of the Greenwich meridian. A fuller explanation of this topic is given in the book ‘Astro Navigation Demystified’ but the rules below will help you to make the calculations without necessarily having a full understanding.
Rules for calculating LHA:
Long East, LHA = GHA + LONG (- 360o as necessary)
Long West, LHA = GHA – LONG (+ 360o as necessary)
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 (-360o )
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 (+360o ?)
GHA = 243o 44’.7
LONG = 125o 13’.0W (-)
LHA = 118o 31’.7
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
LHA = 70o
A fuller explanation of this topic is given in the book ‘Astro Navigation Demystified’.