9 SPECIAL OBSERVATIONS: MOON; POLARIS; - AZIMUTHS & AMPLITUDES
29 THE EARTH’S NATURAL SATELLITE
109. THE MOON
The Earth’s satellite, the Moon, is the closest celestial body to the Earth. Its diameter is about 2,000 miles which is approximately one-quarter of the Earth’s diameter and the Moon has the distinction of being the satellite whose size is most near to the size of its parent planet.
The Moon revolves around the Earth, or more strictly speaking, the Earth and Moon revolve around one another about the common centre of gravity of the Earth-Moon. This point is known as the barycentre and lies on a straight line, which joins the centres of the Earth and Moon and is situated about a thousand miles within the Earth. Fig. 29-1.
The Moon’s orbit is elliptical; its the nearest approach to the Earth is about 222,000 miles, and when it is at the point in its orbit, which is most remote, it is about 230,000 miles from the Earth.
The names of these points in the Moon’s orbit are perigee and apogee respectively, fig. 29-3. As the Moon moves in its orbit around the Earth, it describes a great circle on the celestial sphere against the background of fixed stars. It maybe observed to move about 13° per day to the eastwards. This comparatively rapid easterly motion across the sky can be observed in a short interval of time, the Moon moving a distance equal to its apparent diameter in one hour.
To complete a circuit around the celestial sphere with respect to the stars, the Moon takes about 360/13 days, that is about 27⅓ days, this period of time is called a sidereal period. The easterly motion relative to the stars causes the Moon to rise, culminate and set later each day, to the extent of an average interval of fifty minutes, this interval is sometimes known as the retardation of the Moon’s rising and setting.
The Moon’s orbit is inclined at an angle, of about 5¼° to the plane of the ecliptic. The points where the two planes intersect are referred to as the nodes. The point where the Moon’s path crosses the ecliptic when the Moon is travelling northwards (i.e. when it is changing its celestial Lat. from S. to N.) is known as the ascending node, the other node being called the descending node. The line joining the nodes has a retrograde motion along the ecliptic so that the nodes perform a complete revolution around the Moon’s orbit in 18.6 years. The effect of this nodal motion is that the limits of the Moon’s Dec. change throughout this period. Fig. 29-4
The limits of the Moon’s Dec. depend on the relative positions of the nodes and the equinoxes. When the ascending node is at the Spring equinox, the Moon’s Dec. will vary between (23½°+ 5¼°) = 28¾° N. and S. during the sidereal period. When the ascending node is at the position of the Autumn equinox, the Moon’s Dec. will vary between (23½°- 5¼°) = 18¼° N. and S during the sidereal period.
When the nodes coincide with the solstitial points, the Moon’s Dec. limits are the same as the Sun’s (i.e. 23½° N. and S.) during the sidereal period.
Whatever may be the relative position of the nodes and the equinoxes, the Moon’s Dec. changes very rapidly during its movement across the celestial sphere. This can be seen by inspection of the Moon ephemerides in the N.A.
110. THE PHASES OF THE MOON
The Moon is rendered visible by the reflected light of the Sun and the amount of the Moon’s illuminated hemisphere which is visible from the Earth at any time depends on the relative positions of the Earth, Sun and Moon. The changing shapes of the part of the Moon’s surface which is visible from the Earth are known as the Phases of the Moon. These are shown in fig. 29-5.
When the Moon is in conjunction, its illuminated hemisphere is directed away from the Earth and therefore nothing of the Moon’s hemisphere can be seen from the Earth. At this time the Moon and the Sun cross the observers’ upper celestial meridian at the same instant. When this occurs the Moon is said to be a New Moon and its age at this instant is 00.00 days.
When the Moon is in opposition, its illuminated surface is facing the Earth and, therefore, the whole of the Moon’s illuminated hemisphere is visible from the Earth. At this time the Moon appears as a large disc of light and is referred to as a Full Moon. The full moon occurs when the Sun is on the observer’s lower celestial meridian the Moon being on his upper meridian. The angle between the Sun and the Full Moon is 180° and, therefore, the Moon rises at the time of sunset, and sets at the time of sunrise.
From the time of the New Moon to the time of the Full Moon, the Moon completes half the cycle of phases and is said to be waxing. This means that the portion of the illuminated surface, which is visible, increases in size. Prom the time of the Full Moon to the time of the next New Moon, the portion of the illuminated surface, which is visible, decreases, during this period the Moon is said to be waning.
Because of the Earth’s orbital movement, the Sun moves eastwards across the celestial sphere to the extent of about 1° per day, and it was said above that the Moon moves eastwards across the celestial sphere at 13° per day. Thus, the daily separation of the Sun and Moon amounts to about 12°, and it follows that the Moon takes about 360/12 days approximately, to complete a circuit around the celestial sphere with respect to the Sun. The actual time taken is 29½ days and this interval of time is known as a lunation or as a synodic period. It is the time taken for the Moon to complete a cycle of phases.
The lunation or Lunar Month may be defined as the interval of time between successive New Moons, and is important in tidal prediction.
The Moon, then, makes one complete revolution about the Earth in 27⅓ mean solar days, if the Moon were new at the beginning of this period, when the Earth is at A in fig. 29-6, it would not be new at the end of the period because it would not lie in a straight line with the Sun and the Earth, which is now at B. To achieve this position, it must move along its orbit round the Earth, and, while this is happening, the Earth continues along its own orbit round the Sun to a position C. A further 2⅙ mean solar days (approximately) elapse before the Moon is again new and a lunation, lunar month or synodic period is therefore equivalent to 27⅓ + 2⅙) or 29½ mean solar days.
In this period the Moon must cross the meridian once less than the Sun, and this fact establishes the 50-minute difference between the mean solar day and the lunar day because: –
28½ lunar days = 29½ mean solar days
1 lunar day = 29½ mean solar days
or 1d. 0h. 50m.
As the angle between the Sun and the Moon increases from 0° at New Moon to 180° at Full Moon, the Moon’s visible shape appears successively as a crescent half moon, gibbous and full moon. During this half lunation, the Moon rises after the Sun. When the angle between them is 90°, the Moon is said to be at first quarter, and at this time the Moon will rise about six hours after the Sun has risen.
During the second half of the lunation, the Moon will rise before the Sun. When the Moon is midway between Full and New, it is said to be at third quarter, and at this time the Moon will rise about six hours before the Sun rises.
The age of the Moon is the number of days, which have elapsed since the last New Moon.
Age at New Moon = 00 days
Age at First Quarter = 07 days
Age at Full Moon = 15 days
Age at Third Quarter = 22 days
The age of the Moon on 01st Jan is referred to as this epact, this being used in the calculation of the date of Easter. Twelve lunation’s amount to 354 days, which is about 11 days short of a solar year. Thus, the epact increases by eleven days for successive years.
111. WINTER AND SUMMER MOONS AND THE HARVEST MOON
As the Sun moves in the ecliptic, so successive Full Moons take place in different parts of the celestial sphere, and this has an effect on the duration of Moonlight.
In the summer, when the Sun has northerly Dec., the Full Moon has southerly Dec. In the northern hemisphere, celestial bodies which have S. Dec. above the horizon for less than half the day. Therefore the summer Full Moons are below the horizon for more than half the lunar day.
In winter, when the Sun has southerly Dec., the Full Moons will have N. Dec. and will, therefore, be above the horizon for more than half the lunar day. Thus during the long winter nights the Full Moon rides high when most needed.
In spring when the Sun is near the F.P. of Aries, the Full Moon will be the vicinity of the First Point of Libra. In September, when the Sun is near Libra. The Full Moon will be near Aries. At these times. the Moon’s Dec. is changing most rapidly for that particular lunation, because it is crossing the ecliptic. In spring, when the Sun’s Dec. is changing from S. to N., the Full Moon’s Dec. is changing from N. to S. This has the effect of speeding up the of moonset. because, as stated above, celestial bodies which have S. Dec. above the horizon of a northerly observer for less than half a day and the further S. the Dec., the shorter will be the period the body is above the horizon.
In autumn, the Moon’s Dec. is changing from S. to N. and therefore the time of moonset is retarded for the same reason, and the time of moonrise advanced. The retardation of the Moon’s rising. due to its easterly track across the celestial sphere, will be offset by the northerly change in the Moon’s Dec. and, therefore, in the autumn the Moon will rise only slightly later each day, and the interval between sunset and moonrise is short for several days. Hence before darkness sets in the large Moon rises, and the reflected sunlight assists farmers in the labour of harvest. For this reason, the Full Moon which occurs nearest to the autumnal equinox is known as the Harvest Moon.
When the Moon is a crescent the portion of its surface, which is not illuminated, the Sun is facing the Earth. The reflected sunlight from the Earth, if the sky is clear, illuminates the Moon so that the un-illuminated portion of the Moon, which faces the Earth, is, therefore, just rendered visible by earth shine. This accounts for saying The Old Moon in the Young Moon’s arms.
112. THE MOON AS AN OBSERVABLE CELESTIAL BODY
From the foregoing description of the Moon, it will be clear that for long periods during any month, the Moon is visible in clear weather as a large, conspicuous and unmistakable celestial body sufficiently above the horizon to provide excellent sights. There are occasions during the day when a sight of the Moon can be observed at the same time as the Sun. On other occasions, the Moon may be up in the morning or evening twilight and can be observed at the same time as one or more stars or a planet.
Sights of the Moon are taken, and worked up, in a similar way to those of the Sun, but because of the special characteristics of the Moon already discussed, certain points require particular treatment:
- Because the Moon is relatively near the Earth and this distance is not constant, parallax is produced which varies with the altitude observed and with the Moon’s distance from the Earth. This is known as parallax in altitude and will shortly be explained.
- Again because of the Moon’s comparative nearness to the Earth, the semi-diameter of the Moon tabulated in the N.A. (which is the angle at the Earth’s centre subtended by the radius of the Moon) is always less than the actual semi-diameter at the observer’s eye and should be increased by an amount referred to as the augmentation of the Moon’s Semi-Diameter.
- Because of the oblate shape of the Earth, the Moon’s horizontal parallax for any Lat. other than the Equator is slightly less than the value tabulated in the N.A. (which is the angle at the Moon’s centre subtended by the equatorial radius of the Earth). This correction is known as the Reduction of the Moon’s H.P. for the Terrestrial Spheroid.
- The ‘v’ and ‘d’ corrections for the Moon in the N.A. alter so quickly that it is not sufficient to use only one figure for a three-day period (as for the planets). The ‘v’ and ‘d’ factors to be used is therefore given on the daily pages of the Almanac against every hour (SEE AN PAMPHLET EXTRACTS).
- Although, as for the Sun, either the lower limb or the upper limb can be observed, in the case of the Moon the limb to be observed will depend on which edge is fully visible. Unless the Moon is full, the observer usually has no choice but to measure the altitude of one limb, which may be either the upper or lower one because at any phase other than full, only a part of the Moon is visible and the shape of that part is either gibbous or crescent with, more often than not, a sloping diameter.
Fig. 29-7 (a) shows a gibbous Moon, only the lower limb of which can be observed, while fig. 29-7 (b) shows a crescent Moon, only the upper limb of which can be observed. In each figure ML is the semi-diameter, to be added to the altitude when the lower limb is observed and subtracted when the upper limb is observed.
When taking a Moon-sight it is important that the altitude of either the upper limb or the lower limb is observed, and not the altitude of a point on the Moon’s terminator. The terminator is the curved line, which separates the illuminated from the dark hemisphere of the Moon. On some occasions, it is not obvious from the appearance of the Moon which of the upper or lower limbs is the illuminated limb. On these occasions, the following points relating to the Moon should be considered when selecting the Moon’s limb. First, when the Moon’s age is between 0 days and 14 days, i.e., during the period between the times of the New and Full Moon, the western limb of the Moon is illuminated.
When the Moon’s age is between 14 and 28 days her eastern limb is illuminated. The age of the Moon at any time may be ascertained from the daily pages of the N.A. so that it is a simple matter to ascertain which is the illuminated side of the Moon.
The second factor to bear in mind is that the straight line which joins the ends of the terminator (see fig 29-7) is at right angles to the direction of the Sun from the Moon, and the Moon’s visible limb is always that nearest the Sun.
113. CORRECTION OF THE MOONS ALTITUDE
Apart from the Moon’s phase necessitating the occasional use of the upper limb, the only differences between correcting the altitude of the Moon and that of the Sun lie in the adjustments for parallax and semi-diameter. The corrections to be applied for refraction and dip are the same for all bodies, being due solely to terrestrial considerations.
The concept of parallax was explained in § 58, and a comparison between fig. 13-10 in (see Lesson 4) and the adjacent fig. 29-8 will show that the parallax of a celestial body is greatest when the body is on the horizon, it is then known as the horizontal parallax (H.P.) Although for the purposes of this study, it is not necessary to understand the mathematical explanations of the theories expounded, those students who are mathematically inclined will recognise from fig. 29-8 that the horizontal parallax (HP).
For the Moon, the horizontal parallax is tabulated for every hour on the daily pages of the N.A. and is about 60′, so that the correction for parallax to be applied to the Obs. Alt. is therefore appreciable.
The H.P. of the Moon tabulated in the N.A. is the angle at the Earth’s centre subtended by the equatorial radius of the Earth. However, because of the oblate spheroid shape of the Earth, the Moon’s H.P. for any other Lat. is slightly less than the tabulated value.
In fig. 29-9 E is an observer on the equator and M1 is the Moon on his sensible horizon so that the Moon’s H.P. is OM1E (this being the tabulated value of H.P.) P is an observer at the pole of the Earth and M2 is the Moon on his sensible horizon. In this case the Moon’s H.P. is OM2P, noticeably less than OM1E.
The Earth’s equatorial radius OE is greater than the polar radius OP, and the Moon’s H.P. is greatest when the Lat. of the observer is 0 and decreases as the Lat. of the observer increases. The reduction of the Moon’s H.P. for the figure of the Earth is tabulated, for all navigable latitudes in most nautical tables. An inspection of such a table, however, will show that this reduction is never more than about 12″.
Due to the Moon’s comparative nearness to the Earth, (the average distance between the two centres being only 240.000 miles) the distance of the Moon from an observer decreases as the Moon’s altitude increases. This is depicted in fig. 29-10.
Because OM1 is a greater distance than OM, the Moon’s semi-diameter is greater when the Moon is at M1 than it is when the Moon is at M1, and greater at M2 than it is at either M or M1. The value of the Moon’s semi-diameter (S.D.) which is tabulated in the N.A. is the angle at the Earth’s centre subtended by the radius of the Moon (LCM in fig. 29-10), but this is always less than the actual semi-diameter at the observer’s eye, because, so long as the Moon is above the horizon tangent plane, OM is less than CM.
The tabulated semi-diameter must, therefore, be increased by the amount which is referred to as the augmentation of the Moon’s semi-diameter.
The amount of this augmentation is given in most sets of nautical tables, from which it can be seen that the maximum value of the augmentation is about 18″. This occurs at the same time as the Moon has its maximum semi-diameter, that is when the altitude of the Moon is 90°. The greatest SD for a particular day occurs when the Moon is at meridian passage, that is when it has reached its maximum daily altitude.
The individual corrections to the Moon’s altitude referred to above may have given the impression that the Moon is a difficult body to deal with, but the altitude correction tables for the Moon found inside the contents of The Nautical Almanac (or pages 18-19 of the Extracts) correct for refraction, parallax in altitude, semi-diameter, augmentation of semi-diameter, and reduction of the Moon’s HP for the terrestrial spheroid, merely by entry into the first and second half of the tables.
The Moon’s total correction table in The Nautical Almanac is in three parts. The first part contains the Dip correction to be applied to the observed altitude in order to obtain the apparent altitude. The main correction tables are split between two pages, one page for apparent altitudes up to 35° and the other page for apparent altitudes between 35° and 90°.
Two corrections are extracted from the main correction tables, the first correction being taken from the top part of the table with the argument apparent altitude, and the second correction from the bottom part of table with the argument horizontal parallax (HP) using the same column from which the first correction was taken.
To convert a sextant altitude of the Moon to its true altitude, the procedure is, therefore, as follows. After first applying the index error (if any) to the sextant altitude in order to obtain the observed altitude, the dip correction for the appropriate height of eye of the observer is next applied to obtain the apparent altitude. The top part of the main correction table is then entered with the apparent altitude — degrees across the top in groups of five degrees per column and minutes (in 10′ intervals) down the sides — and the first correction extracted. From whichever column the first correction is extracted, the second correction should be taken from the lower part of the table horizontally opposite the horizontal parallax (HP) taken from the daily page of the Almanac; and under “L” if it was a lower limb observation or under “U” if it was an upper limb observation. Both corrections from this table are always added to the apparent altitude, but when an upper limb observation of the Moon was taken, 30′ must be subtracted from the altitude of the upper limb after the other corrections have been applied (an arbitrary 30′ having in fact been added to the second correction for ‘U’ in order to keep it positive and small).
114. ‘THE NAUTICAL ALMANAC’ ELEMENTS FOR THE MOON.
If two successive transits of the Moon were observed across the same meridian, the interval between them would be one lunar day, and, since the Moon is itself describing an orbit about the Earth in the same direction as the Earth’s spin, this interval would be longer than the mean solar day, as shown in fig. 29-11 . AA1 is a measure of the mean solar day, but while the Earth has moved from A to A1, the Moon has reached C and so the Earth will have to turn through a further angle approximately equal to B1A1C before it is on the observer’s meridian again. The average time taken to turn this extra angle is 50 minutes.
The units derived from the lunar day are lunar units. It is not necessary, however, to work in lunar units to find the hour angle of the Moon. The formula for the hour angle of a celestial body applies equally well to the Moon, and
GHA Moon = GHA Aries + SHA Moon
The Sidereal Hour Angle of the Moon can be predicted. It is combined with the Greenwich Hour Angle of Aries in The Nautical Almanac to give the GHA Moon for every hour of GMT on the daily pages.
Because the Moon’s Declination varies less uniformly and more quickly than that of the Sun and Planets, the hourly difference (‘d’) has to be given for every hour of GMT on the daily pages but the correction is obtained from the Increments table for the minutes of GMT in the same way as for other bodies. For the same reason, the ‘v’ correction for the Moon is also tabulated for every hour of GMT together with the HP (Horizontal Parallax).
The Nautical Almanac shows, in the Moon panel on the daily pages, against every hour, FIVE figures: GHA, v, Dec. d, and HP. It is advisable to make a habit of extracting and writing on your sight form all five figures in one operation. This saves time and removes the risk of looking on the wrong line (or even on the wrong page) if you omit, say, the HP figure and have to find it at a subsequent point in the calculation.
The ‘v’ correction is always plus, but (like the Sun) the ‘d’ correction may be plus or minus depending on whether the Moon’s Declination is increasing (+) or decreasing (—) each successive hour on the daily pages.
The Sight Reduction Form supplied with this Study has been designed for use in reducing sights of the Moon as well as those of the Sun and the planets, and so should be used for the sight reduction of all celestial bodies except the selected stars. Either Sight Reduction Tables AP327O Volumes 2 and 3 for Air Navigation, or any volume of NP40l for Marine Navigation may be used to find the calculated altitude and azimuth.
115. LATITUDE BY MERIDIAN ALTITUDE OF THE MOON.
There are a number of days during the lunar month when a meridian altitude of the Moon can be observed during daylight hours. This will provide the vessel’s Lat., and if this is combined with a simultaneous sight of the Sun or a planet, will fix the vessel’s observed position.
The times of upper and loser meridian passage of the Moon at Greenwich are given for each day of the year in the bottom right-hand corner of the daily pages of the N.A. In comparison with the Sun and stars, however, the rate of change of the Moon’s S.H.A. is large and cannot be neglected. It cannot be assumed that the Moon’s apparent motion to the westward is equal to 15° of Longitude per hour without introducing an appreciable error when its time of transit at other meridians other than that of Greenwich is required.
It was shown above that the lunar day is longer than a mean solar day by an average of 50 minutes. This means that the Moon crosses a stationary observer’s meridian later each day by 50 minutes on average and that once in a lunation it will not cross his meridian at all during the mean solar day. The exact difference in the times of transit at Greenwich is obtained by subtracting the time of transit on one day from the time of transit on the next. The difference varies between 42 minutes and 65 minutes. For example, the time of upper transit at Greenwich on 04th Jul 199X is 04:08 and on 05th Jul it is 04:54 and the difference here is 46m.
Fig. 29-12 shows the Moon and the Mean Sun on the Greenwich meridian at A and B is a place 75° W. of A. When the rotation of the Earth has brought B to A so that the Mean Sun is on the meridian of B. the Moon will have moved along its orbit to C.
While the whole 360° of Longitude is passing under the Mean Sun, the Moon reaches D, and AD is a measure of the tabulated daily difference taken from the N.A. AC will therefore be to AD in the ratio of the Longitude of B to 360° thus:-
AC = 75°
AC = 75° x (the daily difference)
If the daily difference is the 46m. already referred to, the Moon will cross the meridian of 75° W. five hours after it crossed the Greenwich meridian plus an amount:
The L.M.T. of the Moon’s transit at 75°W. will therefore occur 9½m. after the L.M.T. of the transit at Greenwich. Had B been 75°E. of A., the correction would have to be subtracted because Mer. Pass. would occur at B before it occurred at A and the daily difference would have to be taken for the day in question and the previous day.
To avoid this complication in calculating the time of the Moon’s Mer. Pass., the method described in Lesson 4) for calculating the time of Mer. Pass. of any celestial body can be used equally well for the Moon and is in fact more accurate, because the method described above, although practical, is based on the invalid assumption that the Moon’s diurnal motion on the celestial sphere is uniform. By taking the ‘v’ correction into consideration when finding the time of Mer. Pass. from the G.H.A. Moon, this motion is taken into consideration (see Ex. No. 2 below).
Apart from this slight difference in calculating the time of Mer. Pass., the possibility of having to use the upper limb according to the Moon’s phase, and the use of the Moon’s total correction tables as described above, the procedure for a Moon meridian altitude is exactly the same as that for a Sun meridian altitude (see Lesson 4).
Vessel’s Lat. by Moon’s Meridian Altitude = 20 50.7 N.