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# Popular Science Monthly/Volume 60/January 1902/Comets' Tails, The Corona and the Aurora Borealis

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(1902)
Comets' Tails, The Corona and the Aurora Borealis by John Cox

 COMETS' TAILS, THE CORONA AND THE AURORA BOREALIS.

By Professor JOHN COX,

MCGILL UNIVERSITY.

THERE is undeniable fascination about a theory which includes within its sweep the time-honored problems of astronomy connected with comets' tails and the reason why they point away from the sun; the solar prominences and the corona; the source of the light by which the nebulæ shine; the origin and structure of meteor-swarms; and the aurora borealis; besides solving incidentally half a dozen minor outstanding mysteries of the heavens.

Such a theory has been advanced by Sweden's distinguished chemist and physicist, Svante Arrhenius, in a paper published in the 'Physikalische Zeitschrift' for November, 1900. Its main points were briefly mentioned with approval by no less an authority than Professor J. J. Thomson at the end of his captivating article on 'Bodies smaller than Atoms' in the August number of The Popular Science Monthly. All the physical principles on which Arrhenius relies, with one exception, are explained at length in that article, and are now very generally accepted. We may therefore say that the theory is based on 'veræ causæ,' and its accordance with known facts is so impressive when the comparison is made in detail that I venture to think the readers of Professor Thomson's article will be interested in a more complete statement of Arrhenius' views than time permitted him to give.

Let us begin by taking stock of the physical principles already to hand. We know (Professor Thomson's paper) that corpuscles, about 1,000 times smaller than hydrogen atoms, and each bearing a charge of negative electricity, are discharged with high velocity:

(1) from the negative electrode in a Crookes tube (kathode rays).

(2) from objects struck by kathode rays (Röntgen rays).

(3) from hot bodies, such as glowing metals.

(4) from cold metals under the influence of ultra-violet light.

(5) from the radio-active substance radium.

Again we know that these corpuscles, or ions, in passing through a gas produce other ions by collision with the molecules of the gas, and that the negatively charged ions are capable of serving as nuclei for the condensation of ordinary matter.

Comets' Tails.

The single new principle introduced by Arrhenius arises in connection with the problem of comets' tails. Astronomers have always felt that the phenomena exhibited by these strange objects could only be accounted for by making the sun the seat of a violent radial repulsive force, but were entirely at a loss to account for this repulsion. So long as light was supposed to consist of myriads of corpuscles discharged with a speed of 186,000 miles per second, it was easy, with Kepler, to regard the corpuscles as carrying with them in their rush the materials vaporized from the comet by the heat of the sun. But the establishment of the Wave-Theory of light put an end to this idea. Thus Newcomb says ('Popular Astronomy'): "If light were an emission of material particles, as Newton supposed it to be, this view would have some plausibility. But light is now conceived to consist of vibrations in an ethereal medium; and there is no known way in which they could exert any propelling force on matter!"

Now Arrhenius points out that according to the Electromagnetic Theory of light a ray of light does exert a pressure on any surface on which it impinges. Maxwell not only proved this in his original publication of the theory in 1873, but showed how to calculate its value. With the known constants of solar radiation he found that sunlight at the surface of the earth should exert a pressure of.592 X 10-10 grams on every square centimeter. This is too small a force to be detected, though it has been looked for.

But at the surface of the sun the pressure would mount up to 2.75 milligrams per sq. cm. On the other hand, a cubic centimeter of water, which weighs one gram at the surface of the earth, would weigh 27.47 grams at the surface of the sun, i.e., the attraction of the sun would draw it inwards with about 10,000 times the force with which the sun's light would tend to drive it away.

Very different is the case if, instead of a cubic centimeter, we consider a much smaller cube. The pressure on its base would fall off as the square of its edge, but the weight would diminish as the cube. There must come a point at which the pressure of the light would just balance the weight; and still smaller particles would be driven off with a force greater than their weight. They would behave, in fact, as if gravity had become negative.

For example, a cube of water measuring one-thousandth of a millimeter (10-4 cm.) in the edge would weigh 27.47${\displaystyle \times }$10-12 gms.; and the pressure of light on its base would be 2.75${\displaystyle \times }$10-3${\displaystyle \times }$10-8, ${\displaystyle =}$ 27.5 ${\displaystyle \times }$ 10-12 gms., i.e., slightly more than its weight.

In measuring wave-lengths of light physicists denote one-thousandth of a millimeter by the symbol μ. The critical value of the edge of a cube of water, i.e., the value for which its weight is exactly neutralized by the pressure of light at the sun's surface, is thus approximately μ. For a spherical drop the critical diameter may be calculated to be 1.5 μ for water. For other substances the critical value is inversely proportional to the specific gravity.

A similar effect of extreme minuteness is familiar to us as the explanation of the long time required by very small particles to settle through the atmosphere, amounting to many months in the case of the finely divided dust thrown up during the eruptions of Krakatoa. But the resistance to suspended dust particles can never exceed their weight, since it is only called forth by the motion produced by the weight itself. The pressure of light now considered may enormously exceed the weight provided the particles are small enough.

From the motions, and especially the curvature, of comets' tails the magnitude of the repulsive forces to which they are subject may be calculated. Thus Bredichin finds, in four instances, that the repulsion must have been about 18.5, 3.2, 2.0, and 1.5 times the sun's gravitational attraction. Now the vapors emitted by comets are largely hydrocarbons of specific gravity about .8. To account for these repulsions on Arrhenius' principle, the drops must have had diameters of 0.1μ, 0.59 μ, 0.94 μ, 1.25 μ respectively. In another case, where the tail curved towards the sun, Bredichin found the repulsion to be 0.3 times gravity. This would indicate particles of diameter 6 μ. Particles of this order of magnitude, and far smaller, are familiar enough to us, especially in combustion and in the early stages of condensation.

The theory suggested is then as follows: As the comet approaches the sun, the intense heat causes a violent eruption of hydrocarbon vapors on the side towards the sun. The hydrogen boils off, and the vapors condense into small drops of hydrocarbons with higher boiling-points, or ultimately solid carbon is thrown out, finely divided as in an ordinary flame. The largest of these particles fall back to the comet, or if they are not condensed till at a great distance from it, they form tails turned towards the sun. The smaller are driven rapidly from the sun by the pressure of its light, with a speed depending on their size, and form the ordinary tails pointing away from it. That particles of different sizes should be formed from the same comet is natural since the comet is likely to be formed of heterogeneous materials, and there must be great variety in the circumstances of condensation. Thus the comet of 1744 had no less than five tails of different curvature. Occasionally the calculated repulsion on the same tail is not found to follow exactly the law of the inverse square of the distance from the sun throughout its whole length. This puzzling circumstance is at once explained, if the particles should for any reason change their state of aggregation, and consequently their size, during their headlong career. In the light of this theory the following passages will be found very suggestive.

(Herschel, 'Outlines of Astronomy,' p. 376.)

"It is for the most part after thus passing the sun, that they shine forth in all their splendor, and that their tails acquire their greatest length and development; thus indicating plainly the action of the sun's rays as the exciting cause of that extraordinary emanation."

Again (p. 566).

"The tail of the great comet of 1680 immediately after its perihelion passage was found by Newton to have been no less than 20,000,000 leagues in length, and to have occupied only two days in its emission from the comet's body! a decisive proof this of its being darted forth by some active force, the origin of which, to judge by the direction of the tail, must be sought in the sun itself."

Now a particle with one-half the critical diameter would in the course of traveling from the sun's surface to a distance equal to his radius acquire a speed of 430 kilometers per second. With this velocity it would cross a space equal to the diameter of the sun, 865,000 miles, in less than an hour. In comets' tails we probably have to do with particles having less than one eighteenth of the critical diameter. Such particles would cover the same distance in less than four minutes. With a force many times the sun's attraction driving them into space, they would make little of 20,000,000 leagues in two days; whereas if this were to be accomplished against gravity the velocity of projection required might well stagger the astronomers.

Referring to Halley's comet, Herschel says (p. 381):

On the 2d of October (the very day of the first observed commencement of the tail) the nucleus, which had been faint and small, was observed suddenly to have become much brighter, and to be in the act of throwing out a jet or stream of light from its anterior part, or that turned towards the sun. This ejection after ceasing a while was resumed, and with much greater apparent violence, on the 8th, and continued with occasional intermittences so long as the tail itself continued visible. . . These jets, though very bright at their point of emanation from the nucleus, faded rapidly away, and became diffused as they expanded into the coma, at the same time curving backwards, as streams of steam or smoke would do, if thrown out from narrow orifices, more or less obliquely, in opposition to a powerful wind, against which they were unable to make way, and ultimately yielding to its force, so as to be drifted back and confounded in a vaporous train, following the general direction of the current. It seems impossible to avoid the following conclusions. 1st. That the matter of the nucleus of a comet is powerfully excited and dilated into a vaporous state by the action of the sun's rays, escaping in streams and jets at those points of the surface which oppose the least resistance. 2ndly. That this process chiefly takes place in that portion of the nucleus which is turned towards the sun; the vapour escaping chiefly in that direction. 3rdly. That when so emitted, it is prevented from proceeding in the direction originally impressed on it, by some force directed from the sun, drifting it back and carrying it out to vast distances behind the nucleus, forming the tail. 4thly. That this force, whatever its nature, acts unequally on the materials of the comet. 5thly. That the force thus acting on the materials of the tail cannot possibly be identical with the ordinary gravitation of matter, being centrifugal or repulsive, as respects the sun, and of an energy very far exceeding the gravitating force towards that luminary. This will be evident if we consider the enormous velocity with which the matter of the tail is carried backwards, in opposition both to the motion which it had as part of the nucleus, and to that which it acquired in the act of emission.

Again, describing the long straight tail of the great comet of 1843, from which a lateral tail, nearly twice the length of the regular one, was shot forth in a single day, Herschel says:

The projection of this ray, which was not seen either before or after the day in question, to so enormous a length (nearly 100°) in a single day conveys an impression of the intensity of the forces acting to produce such a velocity of material transfer through space, such as no other natural phenomenon is capable of exciting. It is clear that if we have to deal here with matter, such as we conceive it, viz., possessing inertia, at all, it must be under the dominion of forces incomparably more energetic than gravitation, and quite of a different nature.

And finally (p. 406):

There is beyond question some profound secret and mystery of nature concerned in the phenomenon of their tails. In no respect is the question as to the materiality of the tail more forcibly pressed on us for consideration than in that of the enormous sweep which it makes round the sim in perihelio, in the manner of a straight and rigid rod, in defiance of the law of gravitation, nay, even of the received laws of motion, extending (as we have seen in the comets of 1680 and 1843) from near the sun's surface to the earth's orbit, yet whirled round unbroken, in the latter case through an angle of 180° in little more than two hours. It seems utterly incredible that in such a ease it is one and the same material object which is thus brandished. If there could be conceived such a thing as a negative shadow, a momentary impression made upon the luminiferous ether behind the comet, this would represent in some degree the conception such a phenomenon irresistibly calls up. But this is not all. Even such an extraordinary excitement of the ether, conceive it as we will, will afford no account of the projection of lateral streamers; of the effusion of light from the nucleus of a comet towards the sun; and of its subsequent rejection; of the irregular and capricious mode in which that effusion has been seen to take place.

These passages give a vivid picture of the utter puzzledom of astronomers over difficulties which arise from precisely those phenomena which fit most naturally into the theory of Arrhenius.

The Prominences and the Corona.

At the moment when the sun's disc is obscured in a total eclipse enormous red flames, sometimes curving over towards the sun and sometimes floating like clouds at heights up to 40,000 miles above his surface, are seen projecting over the region of sunspots, where the sun's eruptive activity is greatest; and silvery streamers with a radial structure form a lens-shaped envelope about the same region, often extending to a distance of several times the sun's radius. These are known as the prominences and the corona.

The sun must itself project vapors into space. When these condense, the drops will, if larger than the critical size, fall back to the sun, giving rise to the curved prominences; and if smaller, they will be driven off into space, and be seen as the streamers of the corona. Since the eruptions will not always be perpendicular to the sun's surface, the prominences will often exhibit parabolic curves, and the streamers may not always be strictly radial, though the greater part of this effect is to be attributed to the foreshortening under which some of them are viewed from the earth.

Those particles which have approximately the critical diameter will float as clouds, sustained by the pressure of light. This point is specially interesting, since it has been difficult to account for the maintenance of the cloudlike prominences without assuming the existence of a considerable atmosphere about the sun. Yet the comet of 1843 described 300,000 miles within a distance of less than one-third of the sun's radius from his surface with a velocity of 350 miles per second, and came out without having suffered any visible damage or retardation.

The corona has been as great a stumbling-block to astronomers as the comet's tail. Thus Newcomb ('Popular Astronomy,' p. 263) says:

The corona is not a mass of foggy or milky light, but has a hairy structure like long tufts of flax. . . Of this appendage we may say with confidence that it cannot be an atmosphere, that is, a continuous mass of elastic gas held up by its own elasticity. . . What then is the corona? Probably detached particles partially or wholly vaporized by the intense heat to which they are exposed. . . The difficult question which we meet is, How are these particles held up? To this question only conjectural replies can be given.

Three conjectures are then mentioned, of which we may note the first.

That the matter of the corona is in what we may call a state of projection, being constantly thrown up by the sun, while each particle thus projected falls down again according to the law of gravitation. The difficulty we encounter here is that we must suppose velocities of projection rising as high as 200 miles per second constantly maintained in every region of the solar globe. The prominences are of two classes—the cloud-like and the eruptive. The first class presents the appearance of clouds floating in an atmosphere; but as no atmosphere dense enough to sustain anything can possibly exist there, we find the same difficulty in accounting for them, that we do in accounting for the suspension of the matter of the corona.

Professor Young is frankly despairing.

I do not know what to make of the corona. . . By what forces the peculiar radiated structure of the corona is determined I have no definite idea. The analogies of comets' tails and auroral streamers both appear suggestive; but on the other hand, the spectra of the corona, the aurora borealis, the comets and the nebulæ are all different, no two in the least alike. . . Nor have I any theory to propose to account for the certain connection between disturbances of the solar surface and of terrestrial magnetism.

The words we have underlined in this passage have almost a Sophoclean irony to a reader acquainted with the further developments of Arrhenius' theory to which we now turn.

The Zodiacal Light and the Gegenschein.

Not only is the sun the source of those eruptions of ordinary matter which form the prominences, but we have every reason to be believe that he must emit streams of electrically charged corpuscles both directly, as a hot body, and indirectly, since the electrical discharges which, according to all terrestrial analogies, must accompany the violent chemical actions going on near his surface, will, when they take place in the higher and rarer regions of his atmosphere, give rise to kathode rays, and these, in turn, to Röntgen rays. As Professor Thomson says: "As a very hot metal emits these corpuscles, it does not seem an improbable hypothesis that they are emitted by that very hot body, the sun."

Now the negatively charged corpuscles are preeminently fitted to serve as nuclei for the condensation of the ordinary matter. Hence those particles of the latter which, having more than the critical diameter, fall back to the sun, will carry back a negative charge to him; while those which have less than the critical diameter will carry a negative charge off into space. On both counts the corona will be left with a surplus of positive charge. The same arguments hold for the vapors emitted by the nucleus of a comet. Thus comets' tails should consist of negatively charged particles.

Let us follow the career of the particles launched into space. They proceed radially from the sun above the regions of sunspots with rapidly increasing speed, which, however, may be shown to approach a finite limit at a distance of about ten radii from the sun. If they encounter another body, such as the earth, they charge its outer atmosphere negatively, and when this charge reaches a certain value, it will begin to repel them. The oncoming rush will be deflected, and stream past the earth on each side in hyperbolic orbits. Far out in space they must sooner or later meet particles from other bodies, and, if by collision or aggregation they increase beyond the critical diameter, they will first lose speed and then drift back with ever increasing velocity past the earth, directly towards the sun. The space immediately behind the earth would be screened by her, and so be void of particles. Could we take our stand on the moon, we should thus see the earth attended by a faint double tail with a dark dividing line (so conspicuous a feature in comets), immediately behind it, pointing from the sun; and a similar, though perhaps fainter, tail, pointing towards him. Not only so, but the earth helps to form her own tail. For when the negative charge in the upper atmosphere is high enough, discharges are brought about by the powerful ultra-violet radiation from the sun, and particles are driven off radially from the earth on the side turned towards the sun, only to be drifted back with the other streams into the tail. The effect will be as if a sheaf of light projected from her towards the sun.

Compare with this the description of the Zodiacal Light (Newcomb, 'Popular Astronomy,' p. 416):

This object consists of a very soft faint column of light, which may be seen rising from the western horizon after twilight on any clear winter or spring evening; it may also be seen rising from the eastern horizon just before daybreak in the summer or autumn. It really extends out on each side of the sun, and lies nearly in the plane of the ecliptic. . . Near the equator, where the ecliptic always rises high above the horizon, the light can be seen about equally well all the year round. . . It is due to a lens-shaped appendage of some sort surrounding the sun, and extending out a little beyond the earth's orbit. The nature of the substance from which this light emanates is entirely unknown. . . Professor Wright of Yale College finds its spectrum to be continuous. Accepting this, we should be led to the conclusion that the phenomenon in question is due to reflected sunlight, probably from an immense cloud of meteorites, filling up the space between the earth and the sun.

The difficulty in this view is that the orbits of such swarms of meteorites as are known to us are distributed irregularly with regard to the ecliptic. On the other hand, Arrhenius' streams of particles, when near enough to be visible, necessarily lie in or near the ecliptic, as required by observation. More than this, the particles emitted by the earth herself should be most abundant over those regions which have been exposed for many hours to the sun. Now it has been observed that the zodiacal light is stronger on what Arrhenius calls the 'evening side' of the earth (i. e., that side which is in the act of turning away from the sun, and has the sun in the west) than on the 'morning side.'

Even at night, when the sun is below the horizon, faint reflections should reach us from the streamers behind the earth, and by an effect of perspective, these should have a maximum in the point opposite to the sun, where they will appear most dense. Let Professor Newcomb describe the Gegenschein:

Another mysterious phenomenon associated with the zodiacal light is known by its German appellation, the Gegenschein. It is said that in that point of the heavens directly opposite the sun there is an elliptical patch of light, a few degrees in extent, of such extreme faintness that it can be seen only by the most sensitive eyes, under the best conditions, and through the clearest atmosphere. This phenomenon seems so difficult to account for that its existence is sometimes doubted; yet the testimony in its favor is difficult to set aside.

How is it that the moon does not exhibit such tails? The moon has no atmosphere, so that the particles which reach her give up their negative charge to her directly, and it spreads equally all over her surface. When in turn she herself discharges the particles, it will be uniformly in all directions, and she should appear surrounded with a uniform sheath. Possibly this sheath of cosmical dust affords the reason that in a lunar eclipse the shadow of the earth can be traced a short distance beyond the limb of the moon on each side.

The Aurora Borealis.

Perhaps the most interesting application of Arrhenius' theory is his explanation of the Aurora. In a well-known experiment the streams of negative particles forming kathode rays in a Crookes tube are exposed to a magnetic field, when they are seen to describe helices round the lines of force. If the field is powerful enough, they may thus be bent into a complete circle inside a moderately large tube.

Now the negative particles discharged from the sun arrive most thickly over the equatorial regions of the earth, which are most directly exposed to him. Long before they reach any atmosphere dense enough to excite luminescence, they are caught by the lines of force of the earth's magnetic field, which are horizontal over the equator, and have to follow them, winding round them in helices whose radii are so much less than their height above us that the effect to a beholder on the earth is as if they moved along the lines of force. Over the equator there is little luminescence, for lack of atmosphere. But as the lines of force travel north and south, they dip downwards making for the magnetic poles, over which they stand vertical. Soon the particles find themselves in lower layers of the atmosphere, comparable in density with our highest artificial vacua, and begin to give out the darting and shifting lights of the kathode ray. But this can only be at the cost of absorption, and by the time the denser layers of air are reached, their energy is exhausted. Hence the dark circles round the magnetic poles from which, as from behind a curtain, the leaping pillars of the Aurora rise. From this point of view it is significant that Dr. Adam Paulsen, who has made a special study of the northern lights, found so many points of correspondence between them and kathode rays that in 1894 he was led to regard the aurora as a special case of the latter, though unable to give any account of their origin in the upper atmosphere, such as is supplied by Arrhenius' theory.

The most obvious test to which we can subject such a theory is to ask from it some explanation of the very remarkable periodic variations in the frequency of auroræ. If they are caused by streams of particles ejected from the sun, there should be some connection between the changes in the sun's activity, as indicated by the number of sunspots, and the number of auroræ observed. Again, since a negative charge in motion is (pace M. Cremieux) equivalent to a negative current, the passage of electrified particles through the upper atmosphere should affect magnetic instruments on the earth. Sunspots, auroræ, magnetic storms should therefore vary together.

It has long been known empirically that they do agree in a general way. Arrhenius' discussion of the mass of statistics of observed auroræ forms so striking an example of the 'Method of Concomitant Variations' that at the risk of wearying the reader we shall give it in some detail.

1. Slow secular periods.

(a) Both sunspots and auroræ show marked maxima at the middle of the eighteenth and the end of the nineteenth centuries.

(b) Sunspots, auroræ, and magnetic storms go through a simultaneous increase and decrease in the well-known period of 11.1 years.

The source of these slow variations must be looked for in the little understood variations of the sun's activity.

2. Annual period.

The number of auroræ is greatest in March and September, and least in June and December; and the mean frequency for both hemispheres is somewhat less in June than in December.

Now the sun's activity, as indicated by the number of sunspots, is a minimum at his equator, the spots occurring principally in belts about 15° north and south of his equator. Since the streams of particles issue radially from the sun, the earth will be most exposed to them, when she is most nearly opposite the active belts. But the earth stands opposite the sun's equator on June 4 and December 6, and is at her farthest north and south of it (7°), i. e., most nearly opposite the sunspot belts, on March 5 and on September 3. Moreover, she is somewhat nearer to the sun in December than in June.

As between the two hemispheres, the same conditions apply as those which regulate the seasons, viz., altitude of the sun above the horizon, and length of time during which he remains above it daily. Auroræ should therefore be more frequent in summer than in winter, a result which is verified by the records. And just as the highest daily temperature occurs from two to three hours after mid-day, so we ought to find a daily maximum of aurorae about 3 p. m. It is not possible to verify this directly, since auroræ are not visible in daylight. But Arrhenius remarks (1) that the majority of them occur before midnight and not after it, which is so far in general agreement with the theory; (2) that Carlheim-Gyllenskiöld, discussing the observations made at Cape Thordsen in Spitzbergen during the winter of 1882-1883, with a view to correcting the numbers recorded for the effect of daylight in concealing them, deduces a probable maximum for the number actually occurring at 2.40 p. m.

But though we cannot observe auroræ in daylight, we are not without resource, for even when invisible, they give notice of their presence by disturbing the ordinary course of the records photographically taken in our magnetic observatories. In 1899 van Bemmelen discussed the records of such magnetic storms taken in Batavia. He found that they show maxima in March and September, minima in January and June, and a daily maximum at 3 p. m., and minimum at 1 a. m.

3. Monthly Variations.

It is only recently (1898) that the collection of statistics of auroræ published by Eckholm and Arrhenius has brought to light two curious monthly variations in their number.

One of these, with a variation of 20% on each side of the mean, depends on the revolution of the moon in her orbit, showing in the northern hemisphere a maximum when the moon is farthest south of the equator, a minimum when she is farthest north: and vice versa for the southern hemisphere.

The explanation appears highly ingenious. It is as follows: The moon, being unprotected by an atmosphere, is charged by the streams of particles that reach her much as the outer layers of our own atmosphere are charged, and therefore, as we have good reason to believe, to a far higher negative potential than is observed at the surface of the earth. If so, she will seriously affect the number of auroræ at any place over which she stands, by lowering the potential gradient, and thus reducing the number of negative discharges in the highest regions of our atmosphere.

The other variation of some 10% each way has a period of 25.93 days and affects both hemispheres alike. At first sight it is natural to refer this to the synodical time of revolution of the sun on his axis as determined by observations of sunspots. But this is 27.3 days. Remembering that the earth never departs more than 7° from the sun's equator, we should rather take the time of revolution of the equatorial belt for comparison. As estimated by the motion of the faculæ, this is 26.06 days, the equator moving faster than the sunspot belts, and probably the time of revolution of the outermost layers, from which the particles stream, is yet a little shorter. The agreement with the period of the aurora (25.93 days) would thus be within the limits of error of the observations.

Atmospheric Electricity.

Let us now trace the effect of the auroræ on the earth's atmosphere. If they are really kathode rays on a grand scale, they must ionize the air, the negative ions will form centers for condensation, and sinking to the earth by gravitation, will charge it negatively, leaving the layers at moderate heights positively charged. This agrees with the results of recent observations made from balloons up to heights of 3,000 meters.

Since condensation will depend on the number of ions available for nuclei, we have at once an explanation of the curious fact that cloud-formation in the upper atmosphere is more copious in years of frequent auroræ than when they occur rarely. In this connection another odd coincidence may be mentioned. When sunspots are numerous, Jupiter shines with a white light; when they are few, his light has a reddish tinge. Now it is agreed that Jupiter is still at a high temperature. If, therefore, sunspots cause auroræ on Jupiter, and consequent cloud-formation, we must see less of the heated interior in sunspot years than we do when his cloud-layers are not so opaque.

In 1899 von Bezold showed that the daily variation of the compass over the earth's surface could be simply represented as follows. Imagine two points, one in latitude 40° N., and one 40° S., to move round with the sun. Then it is as if the north end of the compass needle were attracted towards the northerly point, and the south end towards the southerly. Remembering that the air immediately above the earth has a positive charge, we see that this effect would follow by Ampère's rule, if the sun's heat caused two air-whirls, one in the northern and one in the southern hemisphere, over the places of highest temperature, the former rotating counter-clockwise, the latter clock-wise. Such whirls would result from the sucking in of currents from the slower-moving north latitudes and the faster-moving south latitudes towards the mean latitude of 40°, in the northern hemisphere, and similarly for the southern. If this be the true explanation, then for a given frequency of sunspots, the amplitude of the diurnal variation should increase by the same fraction of itself for all parts of the earth. Thus if A° is the amplitude at a given place in a year of no sunspots, and A its value in a year for which Wolf's number expressing the relative frequency of spots is f, we ought to find A ${\displaystyle =}$ A° (1${\displaystyle +}$af). Now the value of the coefficient a comes out.0064 from whatever part of the world the observations be taken from which it is calculated.

Meteorites and Nebulæ.

To the man of science this discussion of terrestrial details will probably be the most convincing part of the evidence adduced by Arrhenius for his theory. But it is time to turn from it, and follow, with lagging imagination, the destinies of those particles, by far the greatest number, which miss the earth and the planets, and launch forth into interstellar space.

Many of them will meet similar streams ejected from other suns. and overcoming the mutual repulsion of their negative charges by their mighty velocities, will clash together, like Lucretius' atoms, and unite to form larger masses. But this aggregation must have an end. For if, in the void of space, they are unable to get rid of their electric charges, the potential of the growing mass must rapidly increase, since the charge increases as the cube of the radius, being proportional to the total number of particles, while the capacity for holding electricity only increases as the radius itself. To put this in popular language, each particle brings to the account the whole charge it can bear on its surface; but in the mass, since electricity flies to the surface, only the outer parts of those particles which are actually in the surface can be useful in harboring the accumulating charge, and hence the electric pressure rises. When it becomes intense enough to prevent fresh particles from approaching, accretion will cease. Space will thus be sown with masses of moderate size, formed irregularly, particle by particle, in spite of repulsive forces. These are the meteorites which blaze for a moment in the upper air, or in rare cases reach the earth to puzzle philosophers with their porous structure.

Another multitude of the particles will at last reach other suns. For if in their wanderings they have united with others till they are beyond the critical size, they will be drawn in, and raise the charge of the bodies they reach, till they in turn discharge their streams into space.

In these we see the 'greyhounds' of the abyss, engaged in distributing the materials of the universe, forever busied in a cosmic traffic by whose exchanges the stellar hosts are made more and more alike in constitution, whatever may have been their differences in the beginning.

For those myriads which are fated to escape all visible suns, far out in the 'flaming bounds of space' the Nebulæ lie in wait, spreading spider-like their impalpable webs across immeasurable breadths of sky. Ever since the spectroscope showed that many nebulæ are gaseous, and yet shine by their own light, two problems have vexed the astronomers. How can they be hot enough to send light to us, and yet be held together against the expansive force of the heated gas, by the feeble gravitation which such inconceivably diffuse masses can exert at their borders? If they are really at a temperature of not less than 500° C, so as to shine, or, indeed, if they are much above absolute zero, their own gravitation should not be able to prevent their speedy dissipation into space.

Again, why do they show the spectroscopic lines of so few gases, and those the lighter ones, such as hydrogen and helium?

According to Arrhenius the nebulæ are cold, with the cold of empty space. Their light is due to the rain of negatively charged particles which, plunging into their outermost regions, give rise to electric discharges and make their gases shine as the gases in a vacuum tube. To this the intense cold is no bar, for Stark has shown that the intensity of light excited in a vacuum tube is greater the lower the temperature at which the experiment is tried. And this process should take place at the surface of the nebula, where the lighter gases would be found, the heavier settling inwards. Hence the few lines found in the spectrum of a nebula, and the comparative brightness of the outlying parts, especially to be observed in the planetary and the ring nebuæ.

Such is Arrhenius' theory. It is too early, as yet, to pronounce any judgment upon it, but glancing back over the array of hitherto unexplained facts which fall into order, without forcing, at its touch, we must admit that it is at least plausible. It springs from a single principle, itself a necessary theoretical consequence of the accepted Electromagnetic Theory of light, viz., that light must exert a pressure which, in the case of small particles, may very greatly exceed their weight. By means of this principle in conjunction with recent views about the nature and properties of ions, which can all be experimentally verified, this theory gives a rational explanation of the astounding behavior of comets' tails; accounts for the 'hairy' structure of the corona; shows us how the prominences can float where the existence of a supporting atmosphere is inadmissible; what is. the origin of the zodiacal light and the Gegenschein; of 'the certain connection' between sunspots and magnetic storms; of the aurora, and why it is subject to such complicated periodical variations; why meteorites are porous and limited in size; how the nebulæ shine in the absolute cold of interstellar space, and yet hang together; and why their constituents appear to be so restricted, while the suns among which they are strewn give evidence of most of the elements known on earth.

A theory which sweeps the astronomical horizon of so many mysteries must not only arouse our profound interest, but claim the respectful consideration of men of science.


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# Understand the problem

Let $$S=\{x \in \Bbb R| x=tr(A)$$ for some $$A \in M_4(\Bbb R)$$ s.t $$A^2=A\}$$

#### Then which of the following will describe $$S$$:

1. $$S=\{0,2,4\}$$
2. $$S=\{0,1/2,1,3/2,2,5/2,3,7/2,4\}$$
3. $$S=\{0,1,2,3,4\}$$
4. $$S=[0,4]$$
##### Source of the problem
TIFR 2019 GS Part A, Problem 10
Linear algebra
Moderate
##### Suggested Book
Linear algebra by Friedbarg

Do you really need a hint? Try it first!

What can you say about the minimal polynomial of $A$?
Observe that min poly$(A)|x^2-x=x(x-1)$. Then the minimal poly could be..?
Can you construct the matrix now?
The matrix corresponding to $x, x-1$ is zero matrix and the identity matrix respectively and what are the matrices corresponding to $x(x-1)$?
Can you think about option 3)?

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# Can an even square exceed a cube by one?

Can an even square exceed a cube by one?

I can't find any examples so I assume it is false. How would I prove it?

I forgot to add that they must be positive.

• Well, there is $(-1)^3+1=0^2$. (There are no others.) – André Nicolas Dec 9 '13 at 3:28
• @JackAidley 1 is not even – Ivo Beckers Dec 9 '13 at 8:39

Here is a proof using elementary methods:

We want to find positive integer solutions to the equation $(2m)^2 = n^3 + 1$. This is equivalent to $$n^3 = (2m+1)(2m-1).$$ If $p$ is any prime in the prime factorization of $2m+1$ then from the equation above $p$ divides $n^3$. Because $n^3$ is a perfect cube, the exponent of $p$ in the prime factorization of $n^3$ must be divisible by 3. Then since $2m+1$ and $2m-1$ are relatively prime, $2m+1$ must contain all of $n^3$'s factors of $p$. So the exponent of $p$ in the prime factorization of $2m+1$ must be divisible by 3. This is true for every prime factor $p$ of $2m+1$, so $2m+1$ is a perfect cube.

Using a similar argument we see that $2m-1$ must also be a perfect cube.

So we have two perfect cubes whose difference is $(2m+1)-(2m-1) = 2$. But this is impossible, since the difference between two (distinct) positive perfect cubes is at least $2^3 - 1^3 = 7$.

Hence the equation $(2m)^2 = n^3 + 1$ has no positive integer solutions.

You are looking for positive integer solutions to $(2a)^2 = b^3+1$, or equivalently for $(2a-1)(2a+1)=b^3$.

Now we have two consecutive odd numbers on the LHS, so they are relatively prime (any common factor has to divide $2$, and they are odd). Hence they both must be cubes.

Now it is easy to show that two consecutive cubes $x^3$ and $(x+1)^3$ are separated by $3x^2+3x+1$ which would be more than $2$ for positive $x$.

If I remember correctly, the elliptic curve $y^2=x^3+1$ only has $(0,\pm 1)$, $(-1, 0)$ and $(2, \pm 3)$ as rational points (not including infinity). The only solution with $y$ even was already mentioned in the comments.

Edit': Thanks to Álvaro Lozano-Robledo for finding the reference, and Keith Conrad for informing me that the result is only quoted in the reference, not actually proven. Nonetheless, I am now sure that the result is true, and have an idea of the proof. Álvaro Lozano-Robledo suggested a strategy using $2$-descent to show that the rank of this curve is $0$, and then using the Nagell-Lutz theorem to find the points of finite order. I will return to this post in some finite amount of time after doing some more reading and see if I can supply the proof.

• Why? – chubakueno Dec 9 '13 at 4:52
• Why are these the only rational points? That's a great question and I don't have a good answer. I just remember reading (somewhere) that this curve had rank $0$, and that these were the only rational points. Hopefully an expert sees this and can remark on it or give a reference. – Dylan Yott Dec 9 '13 at 4:54
• My colleague Keith Conrad has some notes on this here: math.uconn.edu/~kconrad/blurbs/gradnumthy/mordelleqn1.pdf, see Example 3.6. – Álvaro Lozano-Robledo Dec 9 '13 at 14:14
• needs another $y^2 = x^3 \large\bf{+1}$ upswag – Stahl Dec 9 '13 at 17:18
• The treatment of integral solutions to $y^2 = x^3 + 1$ in Example 3.6 of the reference Alvaro links to is incomplete. It just turns that problem into the problem of solving $x^3 - 2y^3 = 1$ in integers. – KCd Dec 10 '13 at 4:38

Catalan's conjecture states that there are no two powers (with exponents $\geq2$) of natural numbers at distance $1$ from each other, apart from $8$ and $9$. The conjecture has been proven by Preda Mihailescu in 2002.


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# Student Name:9 Nomenclature of Alkynes HomeworkName the following alkynes following IUPAC system of nomenclature:Br

###### Question:

Student Name: 9 Nomenclature of Alkynes Homework Name the following alkynes following IUPAC system of nomenclature: Br

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