Simon Newcom. Works in Ephemeris Astronomy

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Historical introduce

sun moon correction acceleration

In all theories of the moon before the beginning of the last century the mean motion of that body was supposed to be uniform. The first inequality discovered was the secular acceleration. While the general proposition that a comparison of ancient and modern eclipses shows the mean motion of the moon to have increased since the time of Polemy is no doubt due to Halley, I believe the first careful determination pares his tables of the moon with the following eclipses:

Those of Tycho Brahe in his Progymnasmata;

Those of Walther and Regiomonstanus(A.D. 1478-90);

The of the Cario eclipses (A.D. 997 and 978);

The eclipse of Theon (A.D. 364);

The eclipses of Polemy.

The first of these series of eclipses was too near his epoch, and the second too unreliable, to predicate anything certain upon. From an examination of the others, he concludes that the observed times will be best satisfied by supposing a secular acceleration of 10" in a century.

Soon afterward, Tomas Mayer deduced an acceleration of 7" from the eclipses of the Almagest, which value he is said to have used in his earlier tables of the moon.

The subject is next discussed by Laland in the Memoirs of the French Academy of Sciences for year 1757. Like Bullialdus and others of his countrymen, he has grave doubts of the honesty with which Polemy has given the times of his eclipses, and therefore uses only the first of the series that of 720. He adds the two eclipses observed at Cario by Ebn Jounis, A.D. 997 and 978, and reported in the introduction to the Historia Coelestis of Tycho Brahe and thense concludes that the secular acceleration is about 9".886 per century.

The next event in the history of the problem is the discovery by Laplace of the physical cause of the acceleration, and his calculation of its amount, which he fixed at very nearly 10". The exact agreement of this results, and also that of Plana, with those derived by Donthorne and Lalande from observations, seems to have satisfied the next two generations of astronomers that no more exhaustive discussion of the adopted in all the Lunar Tables between those of Lalande and Hasses. I am not aware of any investigation having in view a definitive determination of the secular acceleration from observations alone during the century following Lalande's paper. We have, it is true, two important papers by Zech in a series of memoirs published at Leipsic under the general title.

The two papers are:

III J. Zech, Astronomishe Untersuchungen uber die Mondfinsternisse des Almagest. Leipzing 1851.

IV J. Zech, Astronomishe Untersuchungen uber die wichtigren Finsternisse, welche von den Shriftstellern des classichen Alterhums erieahut warden. 1853.

The first of these papers has formed the basis of all the late disscussions of Polemy's eclipses; but the author finds these eclipses inadequate to give any determination of the moon's mean motion as well as of its secular acceleration in his equation of condition. If we determine the mean motion, not from the modern observations alone, but from a comparison of the latter with those of Polemy it is ovident that we shall have no accurate data remaining with which to determine the secular acceleration.

In 1853 appeared the celebrated paper of Adams which showed that the theoretical value of the secular acceleration found by his predecessors needed a large diminuthion. This was followed by several accurate calculations of its amount by Adams himself and by Delaunay the letter finally fixing it at 6".176. I conceive that no rational doubt can remain that this result represents the true effect of the gravitation of the planets within a small fraction of a second.

In constructing his Lunar Tables Hansen introduced the coefficient 12".18 founded on a theoretical computation. A revision of his calculation , leading to a slightly greater result, namely, 12".557, is given in his Darlegung der teoretischen Berechuung der in den Mondtafeln angewandten Storungen. About the time of publication of this work, Hassen wrote that he had never disputed the correctness of the result of Adams and Delaunay and defended his result rather on the ground of its representing ancient observation than on its theoretical correctness. It can therefore scarcely by cited as tending to invalidate the results reached by these investigators.

It has long been recognized that there was no necessity for an agreement between the values of the acceleration derived from theory and from observation, because a retardation in the earth's motion of rotation would produce an apparent acceleration in the motion of the moon, and the friction of the tides must produce such a retardation.

A comparison of Ptolemy's series of lunar eclipses, as discussed by Zech, with Hansen's Tables, has been made by Hartwig, and published in the Astronomishe Nachrichten, Bd. 60. A clear tabular summary of his results is printed in the Monthly Notices of the Royal Astronomical Society. The nineteen eclipses indicate a sensible negative correction to the secular acceleration, the mean being - 1".9. Only there out of the nineteen give the correction positive; and, if we regard the series as consisting of observations really independent, the probable error of this result cannot be more than 0".4, and its reality would therefore be beyond doubt. The result of these eclipses may therefore be regarded, from this point of view, as incompatible with that derived by Atry from eclipses of the sun; but the steps of the investigation are not given with sufficient fullness to enable us to judge of the liableness of any conclusions which might be draw from it.

It will be seen from the foregoing that the only approach to a definitive answer to the question the question what value, &e., what value of the secular acceleration is deducible from observations, is to be found in the papers of Professor Airy. If we accept the three most ancient eclipses which he has discussed as all undoubtedly total, then scareely any deviation from Hansen's value of the secular acceleration seems admissible. But I cannot conceive that the historic evidence bearing on the subject places the phenomena of totality so far beyond doubt that a discussion of other data is unnecessary.

Such a discussion in the more necessary because it has been known, since the time of Laplace, that, in addition to the uniform acceleration of which we have spoken, the mean motion of the moon is apparently affected by inequalities of long period, in the satisfactory explanation of which geometers and astronomers have always found difficulty. The first discussion of such an inequality being inferred from observations which showed that the mean motion of the moon during the second half of the eighteenth century was greater than during the first half. It was then assumed that the inequality was periodic one, due to the fact that twice the motion of the moon's node, plus that of its perigee, is a very small quantity. The value of the coefficient concluded from the observation was 47".51, and the expression for the resulting inequality was

47".51(=15".39) sin (2 ? ? + р ? - 3 р _)

Using Hansen's notation for the lunar elements, namely, щ for the distance of the moon's perigee from its mode, and щ? for the distance of the sun's perigee from the same node, the inequality would be

The following table shows how the observation on which the inequality was predicated were found by Laplace to be represented by it:



1. Summary Of The Data Now Are Our Disposal For Determining The Apparent Secular Acceleration Of The Moon From Observation Alone

It has long been tacitly assumed that we are dependent solely on the accounts of eclipses transmitted to us by history for the data necessary to prosecute the investigation in question. This view has undoubtedly been correct in times past. The effect of the cause sought increasing as the square of the time, the extreme roughness of the ancient observations has been more than counterbalanced by their remoteness. For, instance if the mean motion of the moon at the present epoch were accurately known the secular acceleration could be determined equally well from an observation one, century back and from an observation twenty centuries back affected with an error four hundred times as great. As there must be a long series of modem observations to determine the mean motion, itself any error in which will affect the comparisons by which the secular acceleration is to be determined by an amount increasing as the simple time, a still farther advantage is thus given to the ancient obtentions. We may see this advantage in the strongest possible light by reflecting that, with a value of the secular acceleration one second in error, the motion of the moon during a period of two centuries might still be represented without an error of more than half a second.

Notwithstanding these disadvantages, I think the time has arrived when the observations made between the epoch of the invention of the telescope and the year 1750 are entitled at least to consideration as a means of determining the element in question.

As a guide toward determining what observations are to be included in this discussion, and how they are to be used it is proposed to give a brief summary of all the data at our disposal for determining positions of the moon before the year 1750, and to estimate the accuracy with which the secular acceleration can be found from each class or series of determinations supposing the necessary favorable conditions to be fulfilled.

Among these conditions must be included a theory of the inequalities of long period which shall accurately represent observations without any empirical correction, a desideratum as, which, we have shown astronomy does not yet possess. The observations will be, divided into classes or series each class or series presenting some common feature by which the data are to be judged. We begin with

I. Statements of ancient historians from which it is inferred that the shadow of the moon passed over certain points of the earth's surface during certain total eclipses of the sun, If there were even a few cases in which this inference could be drawn without reasonable doubt this class of observations would doubtless furnish ns the most accurate data we possess for our present object. Considering only the eclipses at Larissa and Stiklastad it appears, from the investigations of AIRY just described, that the limits of the value of the secular acceleration within which both eclipses will be very narrow being only a small fraction of a second. But it seems to me that be total there is in nearly all these descriptions of phenomena too much vagueness to inspire observative confidence that any given eclipse was really total at the supposed point on. Reserving for the special discussion of each eclipse the difficult which are peculiar to it, I shall here mention some of a general nature.

The first difficulty is to be reasonably sure that a total eclipse was really the phenomenon observed. Many of the statements supposed to refer to total eclipse are so vague that they may be referred to other less rare phenomena. It must never be forgotten that we are dealing with an age when accurate observations and descriptions of natural phenomena were unknown and when mankind was subject to be imposed so vague that they may be referred to other less rare phenomena. It must never be forgotten that we are dealing with an age when accurate observation and descriptions of natural phenomena were unknow, and when mankind was subject to be imposed upon by imaginary wonders and prodigies. The circumstance which we should regard as most unequivocally marking a total eclipse is the visibility of the stars during the darkness. But even this can scarcely be regarded as conclusive, because Venus may be seen when there is no eclipse. The exaggeration of a single object into a plural is in general very easy.

Another difficulty is to be sure of the locality where the eclipse was total. It is commonly assumed that the description necessarily refers to something seen where the writer flourished, or where he locates his story. It seems to me that this cannot

be safely done unless the statement is made in connection with some battle or military movement, in w case we may presume the phenomena to have been seen by army

II. The series of lunar eclipses recorded by Ptolemy in the Almagest, and used by as the foundation of his lunar theory.

These are nineteen in number. They were observed at Babylon, Rhodes, and Alexandria affected only with the accidental errors of observation the comparisons HANSEN'S.

Tables made by Hartwig seem to indicate that the, probable error of each recorded time is between fifteen and twenty minutes. The probable error of a mean epoch derived from all the observations will then be about four minutes and the corresponding probable error of title moon's mean longitude will be 2 '. But there are two circumstances which prevent our assigning quite this degree of accuracy to PTOLEMY's record.

The first in applicable to all observations of the beginning and end of eclipses.

It is that the first contact is never seen and end of eclipses visible until a sensible interval after the time of real contact. We must expect that, as a general rule, the recorded times of the beginning of eclipses will be too late by a certain sensible amount, and those of the end too small by an amount somewhat less.

If we knew that the observers had always been on the alert for the eclipse, and alive to the necessity of seeing it at the earliest moment, and of noting its time immediately some estimate of the intervals in question might be made, and the results corrected accordingly. But, in these observations, we cannot safely apply any such estimate, and must determine the sum of the two errors from the discordances between beginning and end. In the case of eclipses in which only. The time of the middle is given, we have no means of knowing whether this time is a mean of observed times of and beginning ending, or whether, in the case of partial eclipses, it was the time when the observer thought the eclipse had reached its greatest phase. Happily, where beginnings and endings are both observed, the errors will be in opposite directions, and will partially eliminate each other. The only remaining doubt will arise from our ignorance of the amount by which the error of the beginning exceeds that of the end: in general, I should think the ratio would lie between 1.5 and 2. n range which reduces the outstanding uncertainty to a quite small amount. The other circumstance is that the observations which have reached us are not complete series but only a selection made for the foundation of a theory possibly a preconceived theory. In fact, PTOLEMY has been strongly suspected of selecting such observations from the records as would make the results fit his theory Bullialdu's. The other circumstance Is that the observations which have reached us are not a complete series, but only a selection made for the foundation of a theory possibly a preconceived theory. In fact, PTOLEMY has been strongly suspected of selecting such observations from the records as would make the results fit his theory. BOLLIALDUS founds this accusation upon PTOLEMY'S own statement that HIPPARCHUS employed a different interval between two of his eclipses from that calculated by himself. But its floes not seem probable that one who had dishonestly altered the records in his possession would have thus frankly stated the result of his alteration It seems more likely that there was something in the calculation of HIPARCHUS which PTOLEMY failed to understand, a circumstance not at all improbable.

My own judgment of the liableness of PTOLEMY'S lunar eclipses is founded on these considerations. First, they are not to be accepted without question, because the fact that Ptolemy deduced from a comparison of his own equinoxes with those found by Hipparchus the same erroneous value of the equinoctial year 365^d5^h55^m12' quantity too great by 6^m26 which HIPPARCHUS himself deduced, leads to a very strong suspicion that his observations might be in some way made or selected to fit a preconceived theory.

Yet all of PTOLEMY'S Almagest seems to me to breathe an air of perfect sincerity. We must remember that the scientific logic to which a selection of observations is opposed bad then no existence in men's minds. The question arises whether we have any strong reason to fear that the observations quoted by PTOLEMY were selected to confirm some preconceived theory of the moon s motion; and if so whether such a selection would be wikiing the moon's mean longitude systematically incorrect during the through which the observations extend. Expressing no op into on former, I am inclined to answer in question the latter in the negative.

Even if there was such a selection, it was probably made in favor of a theory of the moon's mean motion on other observations now founded last, and therefore entitled of itself in PTOLEMY sought to determine from observations were so numerous that it seem question's does likely that the mean longitude of the error thought moon be systematically neon's out the whole series. I consider that, on the whole, the observations in question are much more reliable that the accounts of supposed total eclipses, and yet that their confirmation by independent data is very desirable.

III. Passing over, for present a number of isolated observations, all deficient in precision, we reach the observations of the Arabian astronomers. We have already remarked that both Dunthorne and Lalande, in determining the secular acceleration, made use of two eclipses observed at Cairo in the tenth century.

2. Made of Deducing of errors of the Lunar elements from observations of eclipses and occultation

The method of computing eclipses and occultations may generally be divided though not perhaps with entire sharpness, into two classes: in the one, the position of the observer relatively to the cone, or cylinder, which circumscribe the moon and the occulted object is computed by geometric method, and the condition of an occultation or of the beginning or end of an eclipse is that the observed shall be on the surface of this cone; in the second class, the apparent position and magnitude of the two bodies, as seen by the observer, are computed, and the corresponding condition is, that the apparent distance of centers shall be equal to the sum of apparent semidiameters. The first method is preferable on the score of elegance of treatment and of general certainty and convenience in cases where the phenomenon has been observed from several station. It requires, however, that the positions of both bodies be known before the computations of the phenomenon are commenced. This requirement has prevented its use in the present investigation, because it was desirable to postpone the final determination of the positions of the stars to the latest practicable moment in order that the best available data might be used. The method adopted is, therefore, to determine separately and independently the apparent positions of the moon and of the sun or star, and then to deduce equations of condition from the difference between the computed distance of centers and the sum of the semi-diameters; or, in the case of solar eclipses, from the difference between the observed and computed phases.

The investigation of the formulae actually used is as follows

The value of л and в are obtained from the observer's geocentric longitude and local sidereal time by changing the right ascension and declination of his geocentric zenith into longitude and latitude. If we take the earth's equatorial radius for unity of distance, the value of с may be taken at once from geodetic tables, or computed from well-known formulae. Putting

Compute ъ and k? from the formulae

Then, с cos в, с sin в, and л are given by

A practical check on the accuracy of the computations may be obtained by computing sin в and cos в, and noting that the two correspond to the same angle; but, as this quantity is not needed separate from с, I have preferred to depend on duplicate computation by different computers. It may be remarked that 5-place logarithms are always ample in this computation, and that the errors from neglecting the nutation of the obliquity of the ecliptic can never exceed 0".15 in the apparent place of the moon.

Taking the earth's equatorial radius as unity, which gives

We have the there equations:

Let б be any artbitaly angle. If we transform the first two equations into the two others,

They will be:

If we suppose л=б, they will be:

Apart from the number of decimals required, these equations, are the simplest. They give R cos b? and l?-л, while the third of equations (1) gives R sin b?, whence b? and R are obtained. They require, however, the full number of decimals requisite to determine a large angle with the required degree of accuracy. Since л may be known only to 0?.1, while is subtracted from l, and then added to l?-л.

If we suppose б=l, the equations will be:

These equations have the advantage, of requiring, only 6-place logarithms.

Having thus, obtained R, b?, l?-l or l?-л, and thence l?, the apparent semidiameters of the moon, or s?, is found from the equation



k-being the ratio of the diameter of the moon to that of the earth. The semidiameter, s', is so minute that we may suppose it equal to its sine, making the equation for its determination, in seconds,

The value of k which we shall adopt is that of Oudemans, 0.27264. This will give:

Apparent Place of the Sun or Star.

If the phenomenon is an eclipse of the sun, the position of the sun is derived immediately from the tables. The must, however, be corrected for parallax. Owing to the minuteness of this correction, and be near approach of the centers of the sun and moon during any phase of an eclipse, it will be sufficient to subtract the horizontal parallax of the sun from that of the moon, to use this difference instead of р in the preceding formulae, and then to apply no correction to the sun an account of parallax.

In the case of a star, the longitude and latitude are to be reduced to the date of the observation by applying precession, proper motion nutation and aberration.

If we put

The resulting rate of change of the longitude and latitude of a star arising from the motion of the ecliptic alone will be:

These expressions being independent of the equinox of reference, we may, in them, suppose both L and г to be referred to a fixed equinox.

In reducing the star-places to the ecliptic, Hansen's obliquity will be used, the value of which is:

Adopting this change of obliquity, where the motion of the pole of the ecliptic in the direction of the vernal equinox of 1850 is 5”.43 T'^2, and that in the direction of 90° greater longitude is 46”.78 T' - 0”.06 T'^2, we find, taking the century as the unit of time, and counting from 1800,

г- being here counted from the fixed equinox of 1850. Counting from the equinox of 1800, the expression will be:

Taking the expressions just given for the motion in longitude and latitude, when the equinox is fixed, namely,



we find, by differentiating and substituting the numerical values of the centennial variations of ? and г,

In the case of occulted stars, the maximum value of this expression is about 0”.04, and the corresponding effect on the longitude of the star is 0”.02 T^2, it may, therefore, be entirely neglected. For the secular variation of the motion in latitude, we have, neglecting insensible terms,

The proper motion in longitude and latitude may be derived from those in right ascension and declination by the well-known formulae for converting changes of the one system of coordinates into those of the other. If we put

the proper motions in right ascension, declination, longitude, and latitude respectively; and E, the complement of the angle at the star of the triangle formed by the star and the poles of the equator and ecliptic, we shall have:

Owing to the extreme slowness with which the position of the ecliptic changes, and may be supposed constant, which is not the case with м and м'.



Collecting the various terms in the motion of the star which we have deduced, and including precession, we find that its longitude, at the epoch T centuries after 1800, referred to the equinox, is:

Where we put

If we count from the equinox and ecliptic of 1850, and T from the epoch 1850.0 the same expressions will hold by putting

Having thus obtained the mean position of the star for the required epoch, the apparent position is obtained by applying nutation and aberration. But, if the former correction be omitted from the place of the moon, it may be omitted from that of the star also. This course has been adopted. The correction for aberration is:

The symbol representing the sun's true longitude.

Distance of Centers of the Two Bodies

Having found, by the preceding methods,

The distance of centers, D, and the angle of position, m, of the line joining, the centers are given by the equations:

The angle m- being counted from the south point of the moon's disk toward the west. We have also

Since the last term of this equation can never amount to 1/10000, we may substitute in the first of equation (6). We may also determine D and m with all necessary accuracy from the approximate equations,

(7)

The error in this determination of m will be of no importance, because this angle is never observed with such accuracy as to be used as a datum for correcting the moon's place, while the error in D is so small as to be entirely unimportant. In fact, if we represent by D' the approximate value of D derived from (7), we have:

While developing the signs of in the rigorous equation to quantities of the third order, we have:

Substituting these values in the rigorous equations, and taking the sum of the squares of the two equations, we find

Where we have put, for brevity,

Substituting the above value of D^2, we have

Showing that the maximum value of D - D' is 1/48D^3, or less than 0”.01. The equations (7) are therefore exact enough for all practical purposes.

Equations of Correction.

If all the elements of reduction were correct, we should have, in case of an occultation, the value of D from (7) equal to that of s' from (4). We have now to find the equation of condition which must subsist among the corrections to the lunar elements in order that we may have D=s'. Owing to the minuteness of these corrections, their coefficient need not be accurate to more than two significant figures; we may therefore suppose to be equal to unity, since its minimum value exceeds 0.995. If then we put, for brevity,

From which

Let us now refer to the equations (1) and (3). If we put, for brevity,

We have from (3) and (1)

The angle l' - l, or the parallax in longitude, is so small that we may suppose it equal to its tangent, while the denominator, cos b-p cos (l-л), is always contained between the limits 0.98 and unity. Again, the quantity R is always contained between the limits 0.982 and unity. We may then put, without an error of more than one hundredth in the coefficient,

From these equations we obtain

and, putting cos b and cos b' equal to unity,

Owing to the minuteness

of p, q,the factor 1.01 may be entirely omitted in the above expressions. We have next, from (9), putting cos р equal to unity:

The longitude and latitude of the observer's geocentric zenith, л and в, are functions of his latitude and of the local sidereal time. The former must be supposed to be known; but the variation of the latter may be taken into account in order to determine the effect of an error in the time of observation upon the lunar elements. The simplest formulae for expressing errors of longitude and latitude, determine the angle E between 0° and 180° from the equation

The last term in each equation is included only for the sake of completeness in writing. The substitution of this value in дс and дq, neglecting дц', gives:



The correction to the tabular ecliptic longitude is represented by дl. For the sake of completeness, we shall suppose the local mean time of observation to require the correction дt, and the west longitude of the plane to require the correction дл'. We shall then have, for the total correction to the moon's geocentric longitude and latitude, which are to be substituted for дl and дb in (11).



3. Effect of Changes in Lunar Elements Upon the Path of The Central Line of An Eclipses



Bibliography

1. Researches on the motion of the moon, made at the United States Naval Observatory, Washington [microform]: part I: reduction and discussion of observations of the moon before 1750: Simon Newcomb

2. Historical Introduce page 9-12

3. Summary of the Data Now are Our Disposal for Determining the Apparent Secular Acceleration of the moon from Observation Alone. Page 17-20

4. Made of Deducing errors of The Lunar elements from Observation of eclipses and Occultation. Page 55-67

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