Part 13 (1/2)
The exception we have mentioned is the celebrated comet of Halley, whose period is also about seventy-five years. In reasoning on the resistance of the ether, we must consider that the case can have very little a.n.a.logy with the theory of projectiles in air; nor can we estimate the inertia of an infinitely divisible fluid, from its resisting influence on atomic matter, by a comparison of the resistance of an atomic fluid on an atomic solid. a.n.a.logy will only justify comparisons of like with like. The tangent of a comet's...o...b..t, also, can only be tangential to the circular motion of the ether at and near perihelion, which is a very small portion of its period of revolution. As far as the tangential resistance is concerned, therefore, it matters little whether its motion be direct or retrograde. If a retrograde comet, of short period and small eccentricity, were discovered moving also near the central plane of the vortex, it would present a very serious objection, as being indicative of contrary motions in the nascent state of the system. There is no such case known. So, also, with the inclinations of the orbits; if these be great, it matters little whether the comet moves in one way or the other, as far as the tangential current of the vortex is concerned.
Yet, when we consider the average inclination of the orbit, and not of its plane, we find that the major axes of nearly all known cometary orbits are very little inclined to the plane of the ecliptic.
In the following table of all the periodical comets known, the inclination of the major axis of the orbit is calculated to the nearest degree; but all cometary orbits with very few exceptions, will be found to respect the ecliptic, and never to deviate far from that plane:
+--------------------------------------------------------------------+ | Designations | Periodic | Inclination | Motion | Planetary | | of the Comets. | times. | of | in Orbit. | Intervals. | | | | Major Axes | | | |--------------------------------------------------------------------| |Encke | 1818 | 3 years. | 1 | Direct |Mars & Ceres.| |--------------------------------------------------------------------| |De Vico | 1814 | | 2 | Direct | | |Fayo | 1843 | | 4 | Direct | Ceres | |De Avrest| 1851 | From | 1 | Direct | | |Brorsen | 1846 | five | 7 | Direct | and | |Messier | 1766 | to | 0 | Direct | | |Clausen | 1743 | six | 0 | Direct | Jupiter. | |Pigott | 1783 | or | 4 | Direct | | |Pous | 1819 | seven | 3 | Direct | | |Biela | 1826 | years. | 9 | Direct | | |Blaupain | 1819 | | 2 | Direct | | |Lexell | 1770 | | 1 | Direct | | |--------------------------------------------------------------------| |Pous | 1812 | | 17 | Direct | | |Olbers | 1816 | about | 40 | Direct | Saturn | |De Vico | 1846 | 75 | 13 | Direct | and | |Brorsen | 1847 | years. | 12 | Direct | Ura.n.u.s. | |Westphal | 1852 | | 21 | Direct | | |Halley | 1682 | | 16 | Retrograde| | +--------------------------------------------------------------------+
From which it appears, that the objection arising from the great inclination of the _planes_ of these orbits is much less important than at first it appears to be.
Regarding then, that a comet's mean distance depends on its mean atomic density, as in the case of the planets, the undue enlargement of their orbits by planetary perturbations is inadmissible. In 1770 Messier discovered a comet which approached nearer the earth than any comet known, and it was found to move in a small ellipse with a period of five and a half years; but although repeatedly sought for, it was the opinion of many, that it has never been since seen. The cause of this seeming anomaly is found by astronomers in the disturbing power of Jupiter,--near which planet the comet must have pa.s.sed in 1779, but the comet was not seen in 1776 before it pa.s.sed near Jupiter, although a very close search was kept up about this time. Now there are two suppositions in reference to this body: the comet either moved in a larger orbit previous to 1767, and was then caused by Jupiter to diminish its velocity sufficiently to give it a period of five and a half years, and that after perihelion it recovered a portion of its velocity in endeavoring to get back into its natural orbit; or if moving in the natural orbit in 1770, and by pa.s.sing near Jupiter in 1779 this...o...b..t was deranged, the comet will ultimately return to that mean distance although not necessarily having elements even approximating those of 1770. In 1844, September 15th, the author discovered a comet in the constellation Cetus, (the same previously discovered by De Vico at Home,) and from positions _estimated with the naked eye_ approximately determined the form of its...o...b..t and its periodic time to be very similar to the lost comet of 1770. These conclusions were published in a western paper in October 1844, on which occasion he expressed the conviction, that this was no other than the comet of 1770. As the question bore strongly on his theory he paid the greater attention to it, and had, previously to this time, often searched in hopes of finding that very comet. Since then, M. Le Verrier has examined the question of ident.i.ty and given his decision against it; but the author is still sanguine that the comet of 1844 is the same as that of 1770, once more settled at its natural distance from the sun. This comet returns to its perihelion on the 6th of August, 1855, according to Dr. Brunnow, when, it is hoped, the question of ident.i.ty will be reconsidered with reference to the author's principles; and, that when astronomers become satisfied of this, they will do him the justice of acknowledging that he was the first who gave publicity to the fact, that the ”Lost Comet”
was found.
That comets do experience a resistance, is undeniable; but not in the way astronomers suppose, if these views be correct. The investigations of Professor Encke, of Berlin, on the comet which bears his name, has determined the necessity of a correction, which has been applied for several returns with apparent success. But there is this peculiarity about it, which adds strength to our theory: ”The Constant of Resistance” requires a change after perihelion. The necessity for this change shows the action of the radial stream. From the law of this force, (reckoning on the central plane of the vortex,) there is an outstanding portion, acting as a disturbing power, in the sub-duplicate ratio of the distances inversely. If we only consider the mean or average effect in orbits nearly circular, this force may be considered as an ablat.i.tious force at all distances below the mean, counterbalanced by an opposite effect at all distances above the mean. But when the orbits become very eccentrical, we must consider this force as momentarily affecting a comet's velocity, diminis.h.i.+ng it as it approaches the perihelion, and increasing it when leaving the perihelion. A resolution of this force is also requisite for the comet's distance above the central plane of the vortex, and a correction, likewise, for the intensity of the force estimated in that plane. There is also a correction necessary for the perihelion distance, and another for the tangential current; but we are only considering here the general effect. By diminis.h.i.+ng the comet's proper velocity in its...o...b..t, if we consider the attraction of the sun to remain the same, the general effect _may_ be (for this depends on the tangential portion of the resolved force preponderating) that the absolute velocity will be increased, and the periodic time shortened; but after pa.s.sing the perihelion, with the velocity of a smaller orbit, there is also superadded to this already undue velocity, the expulsive power of the radial stream, adding additional velocity to the comet; the orbit is therefore enlarged, and the periodic time increased. Hence the necessity of changing the ”Constant of Resistance” after perihelion, and this will generally be found necessary in all cometary orbits, if this theory be true. But this question is one which may be emphatically called the most difficult of dynamical problems, and it may be long before it is fully understood.
According to the calculations of Professor Encke, the comet's period is accelerated about 2 hours, 30 minutes, at each return, which he considers due to a resisting medium. May it not rather be owing to _the change of inclination of the major axis of the orbit, to the central plane of the vortex_? Suppose the inclination of the _plane_ of the orbit to remain unchanged, and the eccentricity of the orbit also, if the longitude of the perihelion coincides with that of either node, the major axis of the orbit lies in the ecliptic, and the comet then experiences the greatest mean effect from the radial stream; its mean distance is then, _ceteris paribus_, the greatest. When the angle between the perihelion and the nearest node increases, the mean force of the radial stream is diminished, and the mean distance is diminished also. When the angle is 90, the effect is least, and the mean distance least. This is supposing the ecliptic the central plane of the vortex.
When Encke's formula was applied to Biela's comet, it was inadequate to account for a tenth part of the acceleration; and although Biela moves in a much denser medium, and is of less dense materials, even this taken into account will not satisfy the observations,--making no other change in Encke's formula. We must therefore attribute it to changes in the elements of the orbits of these comets. Now, the effect of resistance should also have been noticed, as an acceleration of Halley's comet in 1835, yet the period was prolonged. To show, that our theory of the _cause_ of these anomalies corresponds with facts, we subjoin the elements in the following tables, taken from Mr. Hind's catalogue:
THE ELEMENTS OF ENCKE'S COMET.
Date of Longitude of Longitude of Difference of Perihelion. Perihelion. nearest Node Longitude.
1822 157 11' 44? 154 25' 9? 2 46' 35?
1825 157 14 31 154 27 30 2 47 1 1829 157 17 53 154 29 32 2 48 21 1832[42] 157 21 1 154 32 9 2 41 52 1835 157 23 29 154 34 59 2 48 30 1838 157 27 4 154 36 41 2 50 23 1842 157 29 27 154 39 10 2 50 17 1845 157 44 21 154 19 33 3 24 48 1848 157 47 8 154 22 12 3 24 56 1852 157 51 2 154 23 21 3 27 41
In this we see a regular increase of the angle, which ought to be attended with a small acceleration of the comet; but the change of inclination of the orbit ought also to be taken into consideration, to get the mean distance of the comet above the plane of the vortex, and, by this, the mean force of the radial stream.
In the following table, the same comparison is made for Biela's comet:--
ELEMENTS OF BIELA'S COMET.
Date of Longitude of Longitude of Difference of Perihelion. Perihelion. nearest Node. Longitude.
1772 110 14' 54? 74 0' 1? 36 14' 53?
1806 109 32 23 71 15 15 38 17 8 1826 109 45 50 71 28 12 38 17 38[43]
1832 110 55 55 68 15 36 41 45 19 1846 109 2 20 65 54 39 43 7 41
Between 1832 and 1846, the increase of the angle is twice as great for Biela as for Encke, and the angle itself throws the major axis of Biela 10 above the ecliptic, whereas the angle made by Encke's major axis, is only about 1; the cosine of the first angle, diminishes much faster therefore, and consequently the same difference of longitude between the perihelion and node, will cause a greater acceleration of Biela; and according to Prof. Encke's theory, Biela would require a resisting medium twenty-five times greater than the comet of Encke to reconcile observation with the theory. Halley's comet can scarcely be considered to have had an orbit with perfect elements before 1835. If they were known accurately for 1759, we should no doubt find, that the angle between the node and perihelion _diminished_ in the interval between 1750 and 1835, as according to the calculations of M. Rosenberg, the comet was six days behind its time--a fact fatal to the common ideas of a resisting medium; but this amount of error must be received as only approximate.
No comet that has revisited the sun, has given astronomers more trouble than the great comet of 1843. Various...o...b..ts have been tried, elliptical, parabolic and hyperbolic; yet none will accord with all the observations. The day before this comet was seen in Europe and the United States, it was seen close to the body of the sun at Conception, in South America; yet this observation, combined with those following, would give an orbital velocity due to a very moderate mean distance.
Subsequent observations best accorded with a hyperbolic orbit; and it was in view of this anomaly, that the late Sears C. Walker considered that the comet came into collision with the sun in an elliptical orbit, and its _debris_ pa.s.sed off again in a hyperbola. That a concussion would not add to its velocity is certain, and the departure in a hyperbolic orbit would be contrary to the law of gravitation. This principle is thus stated by Newton:--”In parabola velocitas ubiquo equalis est velocitati corporis revolventis in circulo ad dimidiam distantiam; in ellipsi minor est in hyperbola major.” (Vid. Prin. Lib.
1. Prop. 6 Cor. 7.)
But as regards the _fact_, it is probable that Mr. Walker's views are correct, so far as the change from an ellipse to an hyperbola is considered. The Conception observation cannot be summarily set aside, and Professor Peirce acknowledges, that ”If it was made with anything of the accuracy which might be expected from Captain Ray, it exhibits a decided anomaly in the nature of the forces to which the comet was subjected during its perihelion pa.s.sage.” The comet came up to the sun almost in a straight line against the full force of the radial stream; its velocity must therefore necessarily have been diminished. After its perihelion, its path was directly _from_ the sun, and an undue velocity would be kept up by the auxiliary force impressed upon it by the same radial stream; and hence, the later observations give orbits much larger than the early ones, and there can be no chance of identifying this comet with any of its former appearances, even should its...o...b..t be elliptical. This unexpected confirmation of the theory by the observation of Capt. Ray, cannot easily be surmounted.
We must now endeavor to explain the physical peculiarities of comets, in accordance with the principles laid down. The most prominent phenomenon of this cla.s.s is the change of diameter of the visible nebulosity. It is a most singular circ.u.mstance, but well established as a fact, that a comet contracts in its dimensions on approaching the sun, and expands on leaving it. In 1829, accurate measures were taken on different days, of the diameter of Encke's comet, and again in 1838. The comet of 1618 was also observed by Kepler with this very object, and also the comet of 1807; but without multiplying instances, it may be a.s.serted that it is one of those facts in cometary phenomena, to which there are no exceptions. According to all a.n.a.logy, the very reverse of this ought to obtain. If a comet is chiefly vaporous, (as this change of volume would seem to indicate,) its approach to the sun ought to be attended by a corresponding expansion by increase of temperature. When the contrary is observed, and invariably so, it ought to be regarded as an index of the existence of other forces besides gravitation, increasing rapidly in the neighborhood of the sun; for the disturbing power of the sun's attraction would be to enlarge the diameter of a comet in proportion to its proximity. Now, the force of the radial stream, as we have shown, is as the 2.5th power of the distances inversely. If this alternate contraction and expansion be due to the action of this force, there ought to be an approximate correspondence of the law of the effect with the law of the cause. Arago, in speaking of the comet of 1829, states, ”that between the 28th of October and the 24th of December, the volume of the comet was reduced as 16000 to 1, the change of distance in the meantime only varying about 3 to 1.” To account for this, a memoir was published on the subject by M. Valz, in which he supposes an atmosphere around the sun, whose condensation increases rapidly from superinc.u.mbent pressure; so that the deeper the comet penetrates into this atmosphere the greater will be the pressure, and the less the volume. In this it is evident, that the ponderous nature of a resisting medium is not yet banished from the schools. In commenting on this memoir, Arago justly observes, that ”there would be no difficulty in this if it could be admitted that the exterior envelope of the nebulosity were not permeable to the ether; but this difficulty seems insurmountable, and merits our sincere regret; for M. Valz's ingenious hypothesis has laid down the law of variation of the bulk of the nebulosity, as well for the short-period comet as for that of 1618, with a truly wonderful exactness.” Now, if we make the calculation, we shall find that the diameter of the nebulosity of a comet is inversely as the force of the radial stream. This force is inversely as the 2.5 power of the distances from the axis, and not from the sun: it will, therefore, be in the inverse ratio of the cosine of the comet's heliocentric lat.i.tude to radius, and to this ratio the comet's distance ought to be reduced. But, this will only be correct for the same plane or for equal distances above the ecliptic plane, considering this last as approximately the central plane of the vortex.
From the principles already advanced, the radial stream is far more powerful on the central plane than in more remote planes; therefore, if a comet, by increase of lat.i.tude, approaches near the axis, thus receiving a larger amount of force from the radial stream in that plane than pertains to its actual distance from the sun, it will also receive a less amount of force in that plane than it would in the central plane at the same distance from the axis. Now, we do not know the difference of force at different elevations above the central plane of the vortex; but as the two differences due to elevation are contrary in their effects and tend to neutralize each other, we shall make the calculation as if the distances were truly reckoned from the centre of the sun.
The following table is extracted from Arago's tract on Comets, and represents the variations of the diameter of Encke's comet at different distances from the sun,--the radius of the orbis magnus being taken as unity.
Times of observation, Distances of the Real diameters 1828. comet from the sun. in radii of the earth.
Oct. 28 1.4617 79.4 Nov. 7 1.3217 64.8 Nov. 30 0.9668 29.8 Dec. 7 0.8473 19.9 Dec. 14 0.7285 11.3 Dec. 24 0.6419 3.1
In order the better to compare the diameters with the force, we will reduce them by making the first numbers equal.