Finding North Direction and Time From the Moon, P2: Using Horn Line

by tonytran2015 (Melbourne, Australia)

Finding North direction from the Moon cannot not be as accurate as from the Sun. There are many causes for this:

1/- The Moon does not always rise on the east.

2/- We cannot work out by heart the Moon's declination (up to +/- 5.1 degrees to the ecliptic, and 23.5+5.1 degrees to the Celestial equator ).

3/- We cannot easily work out when the Moon reaches its highest elevation angle. The Moon does not often cast strong shadows for shadow sticks to work.

Here I describe my improved horn line method to find out North direction and time from the Moon. The method uses position of the Moon, shape (phase) of the Moon, solar declination and user’s latitude to work out North direction and rough local time.

Figure 1: The Moon on the Celestial sphere. Figure 2: Moon phase chart for a Solar declination of (-20) deg (South).

The Moon is a satellite of the earth. Everyday Moon-rise and Moon-set time is retarded by about 50 minutes. This allows the Sun to travel further on its journey every subsequent night. Therefore after full moon the partial bright side stays on the East (trailing) side and dark crescent appears on the West and dark area gets fatter daily until the whole moon is dark. Similarly, from new Moon a bright crescent appears on the West and grows fatter and bright area gets fatter daily until full Moon is reached. From the shape of the Moon, it is easy to say how late the Moon is trailing the Sun (new Moon trails by 0 degree and 0 hour, new half-Moon by 90 degree and 6 hours, full-Moon by 180 degrees and 12 hours, and late half-Moon by 270 degrees and 18 hours .). The shape and the position of the Moon allow some guessing of its trajectory for the night.

The Moon completes its orbit in space in 27.321 days, and it completes one full revolution on the Celestial sphere in that time. Its angular velocity on that sphere is 1/27.321 (rev/day) = 0.036601 (rev/day) .

The Sun apparent travel on the ecliptic takes 365.256 days. Its angular velocity on the elliptic is 1/365.256 (rev/day). The Moon revolves on the Celestial sphere faster than the Sun by an angular velocity of 1/((1/27.321)-(1/365.256)) = 1/29.530 (rev/day). When the Sun is one full turn ahead of it, the Moon will catch up, and they will be both in the same direction again on the Celestial sphere after a period of about 29.530 days. So the intervals between consecutive full Moons will be something like a pattern of (30days, 29days).

By keeping records of previous full Moon nights people know it is a WAXING or WANING Moon.

The simple Waxing-Waning rule is that the bright side of the Moon is on the West for waxing and East for waning Moon.

It is natural for people to desire to use the horn line, which is the line connecting the two horns of the Moon, to draw the North South direction. However it has been found that the intersection between the horn line and the horizon does not accurately give the North direction.

Here we find out the reasons for that inaccuracy and show a more accurate method of using Moon’s horn-line.

It may be easier for some readers to first read step 5 then come back to read steps 2, 3 and 4.

CAUTION: The horn line of the partial Moon can point far away from the terrestrial principal North or South directions.

Estimating the current declination.

The whole Celestial spherical shell rotates around its two Celestial poles. The Sun moves slowly on that sphere on a great circle called the ecliptic. Its distance to the two opposite Celestial poles varies periodically, and its distance to the Celestial equator is call the declination of the Sun.

You can make a rough sketch of this declination from the principal values and estimate the declination for the current day.

Figure: Panoramic view of the travel of a partial Moon in the sky. The Celestial axis is always at right angle to the path. This picture is for the winter, with the Sun in the other hemisphere and the bright side of the Moon tilts toward the ground. In the summer, it tilts toward the sky.

The horn line is at right angle to the plane containing the very slender triangle formed by the Earth, the Moon and the Sun but the Celestial axis is at angle of (90-23.5) degrees to that plane. So the horn line usually form an angle of that size to the Celestial axis.

Near to half-moons the horn line is easily defined, and it is also easy to see that the projection of the Celestial axis onto the half-moon makes with it an angle equal to solar declination.

On top of those complication, the Moon also has its own declination and an observer has additional difficulty working out the direction of the horn line as it is usually not at right angle to his line of view.

Figure 1: Moon phase chart for a Solar declination of (-20) deg (South).

At half Moon, it is possible to twist the horn line about the line of view to generate a line KL parallel to the Celestial axis. The amount of twisting is opposite to the declination angle of the Sun. When the angle Sun-Moon-Earth is about 90°+/-30° (=120° or 60°) the amount of required twisting is about 0.85*declination of the Sun.

The required twisting varies with the phase of the Moon as in the following:

Half Moon requires twisting by full Solar declination,

15% or 85% bright area requires twisting by half Solar declination,

full or no-Moon requires no twisting.

Remember that if the Sun is into your hemisphere (in your summer) the bright side of the half Moon has to be twisted downwards (toward the Celestial equator) by an angle equal to the solar declination angle.

The opposite should be done in your winter.

The amount of required twisting is given in the last line in figure for Moon phase in step 2.

This correction here already causes a difference between the results from the old horn line method and the current method. There will be another difference caused by drawing the "spear line" parallel to Celestial axis in the next section.

Note:

The required twisting on the horns of the Moon varies sinusoidally with time and peaks to the values of minus/plus Solar declination when the Moon is half-full.

Observing a Celestial axis KML drawn on the Moon: U is observer, E center of the Earth, N terrestrial North pole, S terrestrial South pole, M Moon, UV local vertical, KL a line parallel to Celestial axis, EQ normal to plane UKL, UP a line parallel to Celestial axis. The red circle through U is the intersection between the plane UKL and the Earth's surface.

Suppose that there is on the Moon M a line KL parallel to the Celestial axis, as illustrated in the figure. We draw a plane through the observer containing the line KL. On that plane UKL draw a line PU nearly in the direction of KL, descending into the ground at latitude angle.That is

(|angle /(PU,KL)| <90° ) and (angle /VUP = 90 deg. - latitude angle).

The line UP is then parallel to the Celestial axis, and its projection on the ground gives the local North South (meridian) direction.

When the Moon is high in the sky and the plane UKL is steeply inclined to the local horizontal, the last condition is the satisfied by

( angle /MUV < angle /MUP ) and (angle /VUP = 90 deg. - latitude angle).

The line UP is then parallel to the Celestial axis, and its projection on the ground gives the local North South (meridian) direction.

For Southern latitudes draw UP' close to the direction of LK (PUP' is a straight line).

Figure: Finding out North direction more accurately using horn line.

On the half-Moon nights twist the horn line to generate on the Moon the line KL parallel to the Celestial axis. When properly done, the plane UKL always makes with the horizontal plane an angle not less than the latitude angle. Consider KL the first approximation to the Celestial axis. Draw a spear line PU co-planar to UKL, nearly in the direction of KL and spearing the ground at an angle equal to latitude angle. The spear line PU so obtained is then pointing more exactly along the Celestial axis, towards the lower Celestial pole. The projection of PU onto the ground gives the terrestrial North-South line. Direction is then found.

When the Moon is high in the sky and the plane UKL is steeply inclined to the local horizontal, the lower end U of the spear line is nearer to the line KL than its upper end P.

Note:

The intersection line between the plane PKL and the ground surface is generally NOT in the North-South direction unless the observer is on the terrestrial equator circle. The drawing of the spear line PU cannot be by-passed in this method.

To apply this method, the navigator has to go through all the following steps:

a. Working out the ROUGH (+/- 45 degrees) principal North-Sout-East-West directions, from the Waxing-Waning rule and the knowledge that the horn line direction is being close to an RA arc,

b. Estimating the Solar declination for the current time of the year,

c. Working out the proportion of Solar declination used for twisting the horns of the Moon from the Moon phase,

d. Twisting the horns of the Moon by the required angle and waving a stretched arm along the adjusted horn line to establish the plane through the Earth, Moon containing the direction of the Celestial axis,

e. Drawing a spear line inside that plane, inclined by latitude angle, aligned roughly to the ROUGH direction of the lower Celestial pole to establish its exact direction, and

f. Projecting the Celestial axis onto the ground surface to obtain the terrestrial North-South direction.

New users to this method should initially only use it as a double check for another method using the position of the hidden Sun, given earlier as part 1, in a previous Instructables article [2]. The agreement between the two methods would make users confident on their results.

Figure: A rare (or not so rare?) failure of the simplistic, traditional horn line method. The horn line intercepts the horizon near to the terrestrial North (wrong by 180 degrees!) in a place in Northern hemisphere. The traditional horn line rule really requires amendments.

At latitude less than 28 degrees, the horn line may point to Northern skyline at high Moon although most of the times it points to Southern skyline ! There is a peril of being sent astray by 180 degrees for unreserved navigators.

At high Moon the horn line only gives North South direction near to equinox times ! The error can be due East or West by the declination value of the Sun.

If the steps for using this method seem to be too complex, the Moon navigators may have to accept reduced accuracy from the Moon and use only Waxing-Waning rule in combination with the stars to navigate !

There is a third part to this topic, with the title "Finding North direction and time using the Moon p3- Moon surface features".

References

[1]. Tristan Gooley, How to navigate using the Moon, the natural navigator, http://www.naturalnavigator.com/find-your-way-usi... 2015, accessed 2015Jun12

[2]. tonytran2015, Finding North direction and time from the Moon, Instructables,

https://www.instructables.com/id/Finding-North-dire... posted on Jun 9, 2015.