# Exercise 4 – 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.

Rules for calculating LHA:

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

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

Test Questions

Question 1.

Position: 38o 15’N. 3038’.3 W,

GHA Sirius: 15015’.4.

What is the LHA?

Question 2.

Position:  33o 12’.8S.  70o 54’.5E,

GHA Sun: 45o 23’.4

What is the LHA?

Question 3.

Position:  52o 42’.45N.  65o 19’.67E

GHA Moon:  345o 15’.0

What is the LHA?

Question 4.

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

GHA Mars:  15o 30’.0

What is the LHA?

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

### 4 Responses to Exercise 4 – Local Hour Angle.

1. Hye Kan Chu says:

do all these rules work for finding local hour angle also work on mars as mentioned in step D-5 on this website found here? http://www.giss.nasa.gov/tools/mars24/help/algorithm.html
“…and H the hour angle, Λ – Λs…”

• Jack Case says:

Thank you for your interest. To answer your question, yes, Mars is one of the so called navigation planets and as such its GHA is listed in the Nautical Almanac for any hour of the day. Since the LHA is simply the angle between the meridian of an observer on the Earth’s surface and the meridian of the geographical position on the Earth’s surface of a celestial body, the rules, as explained in my book, will work for Mars just as they will for any other celestial body.

• Jack Case says:

Sorry, it has just dawned on me what your question is really about. Since the document to which you refer concerns time systems for Mars, I presume that you are asking if the system for LHA that is used on Earth could be applied to Mars. I can see no reason why not but it would require a system of longitude to be devised for that planet and it would also require calculations of the equivalent of the GHA of other celestial bodies as they would relate to Mars and for all of this data be collated into a collection of tables such as the nautical almanac.