(SECOND PAPER)
In the number of the PROCEEDINGS for December, 1905 (Vol. XXXI, No. 4), there was included a paper which, under the title of "A New Method in Nautical Astronomy," gave a description of a new process for deducing hour angle of body, and, therefore, longitude of place, from the altitude of a star situated near the prime vertical by means of tables.
The paper in question was limited to a general account of the principles of the method, with a practical example of its application in the case of the star Antares for latitude 55° S. Now that the tables have been published1 a slightly more detailed description of their construction, and of the objects which it is proposed to accomplish by their aid, may perhaps be acceptable to readers of this journal.
It may be well first to briefly recapitulate the principles on which the method is based.
Near the prime vertical a heavenly body moves in altitude with a velocity which, with certain limitations, may be regarded as uniform, and which, in a given latitude, is independent of declination, and is, therefore, the same for all bodies. If, therefore, we have given the hour angle and zenith distance of a particular body upon the prime vertical, and also the rate of change of altitude appropriate to the particular latitude, we may find the hour angle corresponding to a given zenith distance, or conversely the zenith distance corresponding to a given hour angle, by an
1 The work is entitled "Position-Line Star Tables," by H. B. Goodwin, R. N., published by Mr. J. D. Potter, Admiralty Agent for Charts, 145 Minories, London E. Price, five shillings.
application of the principle of proportion. The work of finding hour angle or zenith distance near prime vertical, therefore, becomes similar in character to that of correcting an element taken from the Nautical Almanac.
The accuracy of the method may be relied upon, when only the first or principal correction is employed, within a limit of one minute of arc, or four seconds of time, up to thirty or forty minutes of hour angle on each side of prime vertical according to the latitude of place. When, however, a "Second Correction" given in the tables is taken into account this limit is extended, for all latitudes as high as 60°, to about 70 minutes on each side of prime vertical, and in low latitudes to a limit much larger still. Generally, the tables will be available for a period of upwards of two hours when the bearing of star is nearly East, and again for a similar period when it is nearly West, i.e., for about four hours and a half in all for each star.
It should be pointed out here that near the equator, so soon as the declination of the body equals in value the latitude of place, the star will not pass the prime vertical at all. When declination of star exceeds latitude of place, however, the body will at some point of its diurnal path be in the position of maximum azimuth, the angle at the body, or "Angle of Position," being then a right angle. Here also, as upon the prime vertical, the altitude changes at a uniform rate, so that by tabulating hour angles and zenith distances for this position of maximum azimuth we may extend to low tropical latitudes the same principle of proportion which in higher latitudes is available near the prime vertical.
There are included altogether five tables, but only two of them, viz., Tables II and III, have direct reference to the new processes for stars near prime vertical. The five tables are as follows:
Table 1.—This is subdivided into Tables I (A), I (B), I (C), and I (D), which give the sidereal times of transit over prime vertical, circle of maximum azimuth, and meridian. Since the sidereal time at which in a given latitude a particular star passes one of these circles is practically constant, the approximate sidereal time, which is easily obtained by adding ship mean time to right ascension of mean sun, seems to offer a convenient argument for a table of this nature, the object of which is to find what stars are conveniently situated for observation at a particular time.
Table II—This table again is subdivided into three parts: Table 11(A), Table 11(B), and Table 11(C). Table II (A) gives the hour angles and zenith distances on prime vertical for use in finding the first or principal correction. The limit of latitude is 60°.
The following extract from the table for Aldebaran (Dec. 16° 19' N.) will illustrate the method of arrangement:
The correction for change of declination given in the third column, the nature of which was described in the earlier paper, is intended to compensate the effect of the difference between the actual value, and that for which the table was calculated, which in the case of Aldebaran was 16° 19' N. The effect of "Annual Variation" in declination for a very considerable period is thereby provided for.
Table II (B)—This gives the " Second Correction," additive to "First Correction," in finding hour angle from zenith distance. The arrangement for latitude 60° is as follows:
Table II (C)—The "Second Correction," subtractive from “First Correction," for use in finding zenith distance from hour angle.
Table III—This table is similar in arrangement to Table 11(A) and gives hour angles and zenith distances for the several stars when at maximum azimuth. No subdivision of this table is required, because Tables II (B), II (C) furnish the values of "Second Correction," if declination of star, instead of latitude of place, is taken as an "argument."
Table IV is not entirely new, nor is it immediately concerned with our present purpose. It gives within the compass of a few pages of tabular matter the means of reducing an altitude observed on a small bearing to the meridian, employing the arguments of azimuth and approximate latitude, and was first published in 1903 through Messrs. Griffin of Portsmouth. It is included here so that the navigator may have within the limits of a single volume a concise method of reducing a second star to the meridian, and thus obtaining a complete “fix " in latitude as well as longitude.
Table V—This is divided into two parts: Table V (A), and Table V (B). Its object is a simple one, viz., to assist in readily expressing a portion of rc given in minutes and seconds as the decimal of a degree, or a given portion of time expressed in seconds as the decimal of a minute. Thus, suppose that we have to deal with a difference of zenith distance of 7° 17' 30". Reference to the table shows that 17' 30" is .292 of a degree, and we avoid a troublesome arithmetical process, and write down at once 7°.292 as the value required.
So, again, if we have to deal with a difference of hour angle such as 41m 27s, from Table V (B) we find that 27 seconds is .45 of a minute, and we write down 41m .45 as the value. Unnecessary labor is thereby' avoided, and risk of error greatly diminished.
To illustrate practically the use of the tables let us suppose that on the present date, which happens to be August 6, 1906, it is required to find what star or stars are available for observation near prime vertical in evening twilight, say one hour after sunset, first in latitude 50° N., then in latitude 10° N.
FOR LATITUDE 50° N
From the almanac we have mean time of sunset approximately 7h 31m, R. A. mean sun 15h Va. The sum of these quantities, or
2A New Table for Solving the Ex-Meridian Problem on Kinematic Principles, by H. B. Goodwin, R. N. Griffin and Co., 2 The Hard, 1?ortsmouth. Price, one shilling.
22h 32m is, therefore, sidereal time of sunset. One hour later sidereal time is 23h 32m.
For the two latitudes we have from Table I (A):
APPROXIMATE SIDEREAL TIME AT WHICH STARS PASS THE PRIME
VERTICAL IN NORTH LATITUDE
Thus we see that either of the fine stars Aldebaran or Betelguese are immediately available to the eastward, while Altair will be crossing the prime vertical to the westward in less than two hours, and will, therefore, come within our limit of 70 minutes very shortly.
Let us suppose that Aldebaran's zenith distance was observed about this time or a little later, and found to be 62° 4' 24". Required the hour angle. Declination of star 16° 22' N.
From Table 11(A) we have:
And from Table 11(B) “Second Correction " for latitude 50°, approximate distance in hour angle 40m, is 75s. Thus we have:
Tabular hour angle 5h 3m 21s
Sum of corrections 40 0
Hour angle 4 23 21 (East) or 19h 36m 39s, which agrees exactly with logarithmic calculation.
Next, let us suppose that with the same data from tables, the hour angle 4h 23m 21s being assumed, it is required to find zenith distance.
Corrected hour angle (from table) 5h 3m 21s
Assumed hour angle 4 23 21
Difference 40 0
First correction = 9'.64 X 40=385'.6=6° 25' 36".
Second correction (from Table II C) for latitude 50°, distance in hour angle 40m = 1' 9". Difference of corrections = 6° 24' 27".
Whence zenith distance required = 68° 29' 0" — 6° 24' 27" = 62° 4' 33".
This application of the tables to the problem of finding the zenith distance corresponding to a given hour angle is of particular importance, because it is the calculation of zenith distances which forms the basis of the system of navigation known by the name of Marcq St. Hilaire, which is extensively practiced both in the English and French navies, and which promises sooner or later to come into very general use.
Reverting to the particular problem of finding stars available near prime vertical for the second assumed latitude, viz., 10° N., we have from the Almanac mean time of sunset on August 6 6h 18m. Adding R. A. mean sun 15h 1m we obtain sidereal time of sunset 21h 19m, and one hour later sidereal time will be 22h 19m.
From the extract from Table I (A) given above it will be seen that in latitude 10° N., nearly all the stars tabulated fail to cross the prime vertical, or do so at an altitude greater than 60°, the limit adopted for the tables. In the present case, however, we find Altair passing the prime vertical to westward at 21h 49m with a zenith distance of 30° 22', and at any sidereal time between 22h and 23h will be available for observation.
But in these low latitudes we have not to trust solely or mainly to stars near the prime vertical, for it is here that the position of “Maximum Azimuth" can be turned to practical account. We now resort to Table I (C), the extract from which will be as follows:
APPROXIMATE SIDEREAL TIME OF MAXIMUM AZIMUTH
(Angle of Position = 90°)
From this table it appears that between sidereal times 22h Om and 23h 0m we have a Arietis and Aldebaran available near maximum azimuth to eastward, in addition to Altair near prime vertical as previously mentioned.
The actual procedure for finding hour angle from observed zenith distance, or zenith distance corresponding to assumed hour angle, is the same in the case of the maximum azimuth, as for the prime vertical, and no additional example need, therefore, be given. It should be pointed out that no correction is necessary to hour angle for change of declination, since the effect is inappreciable at maximum azimuth.
With respect to the title " Position-Line Star Tables," which has been selected, it may be observed that this was adopted on account of the readiness with which the tables adapt themselves to the position-line methods regarded as indispensable in modern navigation. In the process which has hitherto been most usual, in the case of bodies observed upon large bearings, for finding hour angle of the body, and thence longitude of place, much time and trouble are saved by the use of the tables, but when the St. Hilaire method of calculated altitudes is employed, and it is quite likely that this may some day be the generally accepted method, the simplification effected by use of the tables is even more considerable. It should be noticed, too, that the tables being calculated for intervals of 20' in latitude no interpolation is required, since the latitude by account will never differ by more than to' from a value given in the tables.
The publication of the earlier paper in the PROCEEDINGS OF THE UNITED STATES NAVAL INSTITUTE in December, 1905, has enabled the author to submit his notions to some of the most distinguished living authorities on Nautical Astronomy in England and other countries, with the result of eliciting a most gratifying consensus of opinion as to the probable utility of the tables, and the soundness of the principles on which they are based. These favorable judgments inspire some hope that the work will commend itself also to the practical navigator. Should this prove to be the case the objects which have been kept in view, of diminishing labor of computation, and extending the practice of star observations, will have been fully accomplished, and the time and trouble devoted to the calculation of the tables will not be without result.