Line of Position — The Moon

I've saved the Moon for last as it's the most complicated of all. Unlike a star, the Moon moves rapidly through the heavens. In addition, the Moon is very close to the earth and thus has a large Horizontal Parallax. The Moon's orbit is elliptical, which means that the Horizontal Parallax varies over time.

Many of these issues are dealt with the the altitude correction tables, which for the Moon, are quite elaborate.

Computing a Line of Position takes these steps:

1. Measure the object's altitude with a sextant, note exact time of sight, correct sextant altitude for various errors.
2. Use almanac and correction reduction tables to compute Geographic Position (GP) of the Moon.
3. Pick an Assumed Position.
4. Use sight reduction tables to compute an azimuth (Z) to the object and an expected altitude (Hc).
5. Plot

Measure the Moon's altitude with a sextant

Leg 57 gives this information for the sighting of the Moon:

• Object: Moon Lower Limb.
• Time: 1999-03-24, 23:40:01
• Sextant Altitude: 15°04.9'
• Height of eye: 9 feet
• Index error: -2.5'

With this information, we can now fill in the first part of our worksheet:

Altitude Correction

Now we're ready to add altitude corrections. If you're working with the Nautical Almanac, 1999 Commercial Edition as I am, you'll notice that the altitude correction tables in the front of the book cover the Sun, planets and stars, but not the Moon. There's a simple reason for this: the tables for the Moon wouldn't fit.

The altitude correction tables for the Moon are in the back of the book, after the Increments and Corrections tables.

First, go to the Almanac for 1999-03-24 and look up the information for the moon at 23h:

• GHA: 65°41.2'
• v: 7.0'
• Dec: N19°38.6'
• d: 0.4'
• HP: 58.4'

Now, go to the altitude correction table for the moon in the back of the book. The table is divided into two halves. The top half lists the basic corrections (note that many of them are over a degree!). The bottom half further refines the correction obtained from the top half.

Note that the table operates in 10' intervals. We round 14°59.6 to 15°00'.

Following the directions given for the table, we look up 15°00 and get an altitude correction of +62.8'

Next, we travel down the column from 15° and find the Horizontal Parallax of 58.5' (closest to the almanac value of 58.4). This gives an additional correction of 6.0' for the lower limb. Our total correction is +66.8'

Use Almanac and Correction Tables

GHA: 65°41.2' — the Greenwich Hour Angle of the Moon at 1999-03-24, 23:00. The sighting was 40 minutes after the hour, so we need to correct.

v: 7.0' — an index into a correction table, see below. In detail: The rate of change of the Moon's GHA varies with its orbit. The Increments and Corrections table in the back of the book accounts for the Moon's minimal rate of change in GHA (14°19'/hour, according to Bowditch, ch.19). The v term indicates how much faster than this base value the Moon is moving.

In plain English: The moon is always moving across the sky as the earth turns and the Moon orbits the Earth. This value is never less than just over 14 degrees per hour. The 'v' term is how much faster the Moon is moving right now. In this case, the Moon is moving 14°26' per hour.

Dec: N19°38.6' — the declination of the Moon at 1999-03-24, 23:00

d: 0.4' — an index into a correction table, see below. In detail: this is the difference between this declination and the next hour's declination. IMPORTANT: this number is a signed quantity, but the sign isn't printed. Examine the next hour's declination to determine the sign. In this case, the declination at 1999-03-25, 00h is south of 19°38.6', so d is actually -0.4.

In plain English: The Moon's declination is moving south at 0.4 minutes per hour.

HP: 58.4' — Horizontal Parallax is essentially the Earth's semidiameter as viewed from the moon. It is combined with the Moon's altitude to compute a correction factor for the Moon's observed altitude. This correction factor is part of the altitude correction tables for the Moon.

Our worksheet now looks like this:

Now we use the minutes and seconds of the observation time. We now go to the Increments and Corrections tables in the back of the Almanac and find the page for 40 minutes past the hour. We look down the seconds column for 01 seconds and examine the Moon column to get a correction factor of 9°47.2'

Next, we use the value of 'v = 7.0' to find the extra correction factor. In this case, this is 4.8'.

We do the same for d and get a correction of 0.3

Our worksheet now looks like this:

Choose an Assumed Position

From here on down, it's the same as for any other celestial body. Let's choose an Assumed Position of 75°33.2; that will make the math easy.

Sight Reduction Tables

Now we go back to the tables to compute expected altitude (Hc) and azimuth (Z).

Go to the sight reduction tables and look up latitude 54°, declination contrary name as latitude. Find declination 19°, hour angle 0°

The table gives these values:

• Hc: 17°00
• d': -60
• Z: 0°

This Hc value is within a degree of Ho, so we're probably on the right track. We still have corrections to factor in.

We now look for the Correction Table, and enter d (-60') and the minutes value of the declination (38.9', rounded = 39'). The table gives 39':

Our line of position is 14.6 nm further from the moon than our assumed position.

Plot

For an enlarged view of the chart, see the final plot.

Next: The answers