the considerations to be here developed have scarcely as yet drawn the attention of astronomers; there is hardly anything to cite except an ingenious idea of lord kelvin’s, which has opened a new field of research, but still waits to be followed out. nor have i original results to impart, and all i can do is to give an idea of the problems presented, but which no one hitherto has undertaken to solve. every one knows how a large number of modern physicists represent the constitution of gases; gases are formed of an innumerable multitude of molecules which, at high speeds, cross and crisscross in every direction. these molecules probably act at a distance one upon another, but this action decreases very rapidly with distance, so that their trajectories remain sensibly straight; they cease to be so only when two molecules happen to pass very near to each other; in this case, their mutual attraction or repulsion makes them deviate to right or left. this is what is sometimes called an impact; but the word impact is not to be understood in its usual sense; it is not necessary that the two molecules come into contact, it suffices that they approach sufficiently near each other for their mutual attractions to become sensible. the laws of the deviation they undergo are the same as for a veritable impact.
it seems at first that the disorderly impacts of this innumerable dust can engender only an inextricable chaos before which analysis must recoil. but the law of great numbers, that supreme law of chance, comes to our aid; in presence of a semi-disorder, we must despair, but in extreme disorder, this statistical law reestablishes a sort of mean order where the mind can recover. it is the study of this mean order which constitutes the kinetic theory of gases; it shows us that the velocities of the molecules are equally distributed among all the directions, that the rapidity of these velocities varies from one molecule to another, but that even this variation is subject to a law called maxwell’s law. this law tells us how many of the molecules move with such and such a velocity. as soon as the gas departs from this law, the mutual impacts of the molecules, in modifying the rapidity and direction of their velocities, tend to bring it promptly back. physicists have striven, not without success, to explain in this way the experimental properties of gases; for example mariotte’s law.
consider now the milky way; there also we see an innumerable dust; only the grains of this dust are not atoms, they are stars; these grains move also with high velocities; they act at a distance one upon another, but this action is so slight at great distance that their trajectories are straight; and yet, from time to time, two of them may approach near enough to be deviated from their path, like a comet which has passed too near jupiter. in a word, to the eyes of a giant for whom our suns would be as for us our atoms, the milky way would seem only a bubble of gas.
such was lord kelvin’s leading idea. what may be drawn from this comparison? in how far is it exact? this is what we are to investigate together; but before reaching a definite conclusion, and without wishing to prejudge it, we foresee that the kinetic theory of gases will be for the astronomer a model he should not follow blindly, but from which he may advantageously draw inspiration. up to the present, celestial mechanics has attacked only the solar system or certain systems of double stars. before the assemblage presented by the milky way, or the agglomeration of stars, or the resolvable nebulae it recoils, because it sees therein only chaos. but the milky way is not more complicated than a gas; the statistical methods founded upon the calculus of probabilities applicable to a gas are also applicable to it. before all, it is important to grasp the resemblance of the two cases, and their difference.
lord kelvin has striven to determine in this manner the dimensions of the milky way; for that we are reduced to counting the stars visible in our telescopes; but we are not sure that behind the stars we see, there are not others we do not see; so that what we should measure in this way would not be the size of the milky way, it would be the range of our instruments.
the new theory comes to offer us other resources. in fact, we know the motions of the stars nearest us, and we can form an idea of the rapidity and direction of their velocities. if the ideas above set forth are exact, these velocities should follow maxwell’s law, and their mean value will tell us, so to speak, that which corresponds to the temperature of our fictitious gas. but this temperature depends itself upon the dimensions of our gas bubble. in fact, how will a gaseous mass let loose in the void act, if its elements attract one another according to newton’s law? it will take a spherical form; moreover, because of gravitation, the density will be greater at the center, the pressure also will increase from the surface to the center because of the weight of the outer parts drawn toward the center; finally, the temperature will increase toward the center: the temperature and the pressure being connected by the law called adiabatic, as happens in the successive layers of our atmosphere. at the surface itself, the pressure will be null, and it will be the same with the absolute temperature, that is to say with the velocity of the molecules.
a question comes here: i have spoken of the adiabatic law, but this law is not the same for all gases, since it depends upon the ratio of their two specific heats; for the air and like gases, this ratio is 1.42; but is it to air that it is proper to liken the milky way? evidently not, it should be regarded as a mono-atomic gas, like mercury vapor, like argon, like helium, that is to say that the ratio of the specific heats should be taken equal to 1.66. and, in fact, one of our molecules would be for example the solar system; but the planets are very small personages, the sun alone counts, so that our molecule is indeed mono-atomic. and even if we take a double star, it is probable that the action of a strange star which might approach it would become sufficiently sensible to deviate the motion of general translation of the system much before being able to trouble the relative orbits of the two components; the double star, in a word, would act like an indivisible atom.
however that may be, the pressure, and consequently the temperature, at the center of the gaseous sphere would be by so much the greater as the sphere was larger since the pressure increases by the weight of all the superposed layers. we may suppose that we are nearly at the center of the milky way, and by observing the mean proper velocity of the stars, we shall know that which corresponds to the central temperature of our gaseous sphere and we shall determine its radius.
we may get an idea of the result by the following considerations: make a simpler hypothesis: the milky way is spherical, and in it the masses are distributed in a homogeneous manner; thence results that the stars in it describe ellipses having the same center. if we suppose the velocity becomes nothing at the surface, we may calculate this velocity at the center by the equation of vis viva. thus we find that this velocity is proportional to the radius of the sphere and to the square root of its density. if the mass of this sphere was that of the sun and its radius that of the terrestrial orbit, this velocity would be (it is easy to see) that of the earth in its orbit. but in the case we have supposed, the mass of the sun should be distributed in a sphere of radius 1,000,000 times greater, this radius being the distance of the nearest stars; the density is therefore 1018 times less; now, the velocities are of the same order, therefore it is necessary that the radius be 109 times greater, be 1,000 times the distance of the nearest stars, which would give about a thousand millions of stars in the milky way.
but you will say these hypothesis differ greatly from the reality; first, the milky way is not spherical and we shall soon return to this point, and again the kinetic theory of gases is not compatible with the hypothesis of a homogeneous sphere. but in making the exact calculation according to this theory, we should find a different result, doubtless, but of the same order of magnitude; now in such a problem the data are so uncertain that the order of magnitude is the sole end to be aimed at.
and here a first remark presents itself; lord kelvin’s result, which i have obtained again by an approximative calculation, agrees sensibly with the evaluations the observers have made with their telescopes; so that we must conclude we are very near to piercing through the milky way. but that enables us to answer another question. there are the stars we see because they shine; but may there not be dark stars circulating in the interstellar spaces whose existence might long remain unknown? but then, what lord kelvin’s method would give us would be the total number of stars, including the dark stars; as his figure is comparable to that the telescope gives, this means there is no dark matter, or at least not so much as of shining matter.
before going further, we must look at the problem from another angle. is the milky way thus constituted truly the image of a gas properly so called? you know crookes has introduced the notion of a fourth state of matter, where gases having become too rarefied are no longer true gases and become what he calls radiant matter. considering the slight density of the milky way, is it the image of gaseous matter or of radiant matter? the consideration of what is called the free path will furnish us the answer.
the trajectory of a gaseous molecule may be regarded as formed of straight segments united by very small arcs corresponding to the successive impacts. the length of each of these segments is what is called the free path; of course this length is not the same for all the segments and for all the molecules; but we may take a mean; this is what is called the mean path. this is the greater the less the density of the gas. the matter will be radiant if the mean path is greater than the dimensions of the receptacle wherein the gas is enclosed, so that a molecule has a chance to go across the whole receptacle without undergoing an impact; if the contrary be the case, it is gaseous. from this it follows that the same fluid may be radiant in a little receptacle and gaseous in a big one; this perhaps is why, in a crookes tube, it is necessary to make the vacuum by so much the more complete as the tube is larger.
how is it then for the milky way? this is a mass of gas of which the density is very slight, but whose dimensions are very great; has a star chances of traversing it without undergoing an impact, that is to say without passing sufficiently near another star to be sensibly deviated from its route! what do we mean by sufficiently near? that is perforce a little arbitrary; take it as the distance from the sun to neptune, which would represent a deviation of a dozen degrees; suppose therefore each of our stars surrounded by a protective sphere of this radius; could a straight pass between these spheres? at the mean distance of the stars of the milky way, the radius of these spheres will be seen under an angle of about a tenth of a second; and we have a thousand millions of stars. put upon the celestial sphere a thousand million little circles of a tenth of a second radius. are the chances that these circles will cover a great number of times the celestial sphere? far from it; they will cover only its sixteen thousandth part. so the milky way is not the image of gaseous matter, but of crookes’ radiant matter. nevertheless, as our foregoing conclusions are happily not at all precise, we need not sensibly modify them.
but there is another difficulty: the milky way is not spherical, and we have reasoned hitherto as if it were, since this is the form of equilibrium a gas isolated in space would take. to make amends, agglomerations of stars exist whose form is globular and to which would better apply what we have hitherto said. herschel has already endeavored to explain their remarkable appearances. he supposed the stars of the aggregates uniformly distributed, so that an assemblage is a homogeneous sphere; each star would then describe an ellipse and all these orbits would be passed over in the same time, so that at the end of a period the aggregate would take again its primitive configuration and this configuration would be stable. unluckily, the aggregates do not appear to be homogeneous; we see a condensation at the center, we should observe it even were the sphere homogeneous, since it is thicker at the center; but it would not be so accentuated. we may therefore rather compare an aggregate to a gas in adiabatic equilibrium, which takes the spherical form because this is the figure of equilibrium of a gaseous mass.
but, you will say, these aggregates are much smaller than the milky way, of which they even in probability make part, and even though they be more dense, they will rather present something analogous to radiant matter; now, gases attain their adiabatic equilibrium only through innumerable impacts of the molecules. that might perhaps be adjusted. suppose the stars of the aggregate have just enough energy for their velocity to become null when they reach the surface; then they may traverse the aggregate without impact, but arrived at the surface they will go back and will traverse it anew; after a great number of crossings, they will at last be deviated by an impact; under these conditions, we should still have a matter which might be regarded as gaseous; if perchance there had been in the aggregate stars whose velocity was greater, they have long gone away out of it, they have left it never to return. for all these reasons, it would be interesting to examine the known aggregates, to seek to account for the law of the densities, and to see if it is the adiabatic law of gases.
but to return to the milky way; it is not spherical and would rather be represented as a flattened disc. it is clear then that a mass starting without velocity from the surface will reach the center with different velocities, according as it starts from the surface in the neighborhood of the middle of the disc or just on the border of the disc; the velocity would be notably greater in the latter case. now, up to the present, we have supposed that the proper velocities of the stars, those we observe, must be comparable to those which like masses would attain; this involves a certain difficulty. we have given above a value for the dimensions of the milky way, and we have deduced it from the observed proper velocities which are of the same order of magnitude as that of the earth in its orbit; but which is the dimension we have thus measured? is it the thickness? is it the radius of the disc? it is doubtless something intermediate; but what can we say then of the thickness itself, or of the radius of the disc? data are lacking to make the calculation; i shall confine myself to giving a glimpse of the possibility of basing an evaluation at least approximate upon a deeper discussion of the proper motions.
and then we find ourselves facing two hypotheses: either the stars of the milky way are impelled by velocities for the most part parallel to the galactic plane, but otherwise distributed uniformly in all directions parallel to this plane. if this be so, observation of the proper motions should show a preponderance of components parallel to the milky way; this is to be determined, because i do not know whether a systematic discussion has ever been made from this view-point. on the other hand, such an equilibrium could only be provisory, since because of impacts the molecules, i mean the stars, would in the long run acquire notable velocities in the sense perpendicular to the milky way and would end by swerving from its plane, so that the system would tend toward the spherical form, the only figure of equilibrium of an isolated gaseous mass.
or else the whole system is impelled by a common rotation, and for that reason is flattened like the earth, like jupiter, like all bodies that twirl. only, as the flattening is considerable, the rotation must be rapid; rapid doubtless, but it must be understood in what sense this word is used. the density of the milky way is 1023 times less than that of the sun; a velocity of rotation √1025 times less than that of the sun, for it would, therefore, be the equivalent so far as concerns flattening; a velocity 1012 times slower than that of the earth, say a thirtieth of a second of arc in a century, would be a very rapid rotation, almost too rapid for stable equilibrium to be possible.
in this hypothesis, the observable proper motions would appear to us uniformly distributed, and there would no longer be a preponderance of components parallel to the galactic plane.
they will tell us nothing about the rotation itself, since we belong to the turning system. if the spiral nebul? are other milky ways, foreign to ours, they are not borne along in this rotation, and we might study their proper motions. it is true they are very far away; if a nebula has the dimensions of the milky way and if its apparent radius is for example 20′′, its distance is 10,000 times the radius of the milky way.
but that makes no difference, since it is not about the translation of our system that we ask information from them, but about its rotation. the fixed stars, by their apparent motion, reveal to us the diurnal rotation of the earth, though their distance is immense. unluckily, the possible rotation of the milky way, however rapid it may be relatively, is very slow viewed absolutely, and besides the pointings on nebul? can not be very precise; therefore thousands of years of observations would be necessary to learn anything.
however that may be, in this second hypothesis, the figure of the milky way would be a figure of final equilibrium.
i shall not further discuss the relative value of these two hypotheses since there is a third which is perhaps more probable. we know that among the irresolvable nebul?, several kinds may be distinguished: the irregular nebul? like that of orion, the planetary and annular nebul?, the spiral nebul?. the spectra of the first two families have been determined, they are discontinuous; these nebul? are therefore not formed of stars; besides, their distribution on the heavens seems to depend upon the milky way; whether they have a tendency to go away from it, or on the contrary to approach it, they make therefore a part of the system. on the other hand, the spiral nebul? are generally considered as independent of the milky way; it is supposed that they, like it, are formed of a multitude of stars, that they are, in a word, other milky ways very far away from ours. the recent investigations of stratonoff tend to make us regard the milky way itself as a spiral nebula, and this is the third hypothesis of which i wish to speak.
how can we explain the very singular appearances presented by the spiral nebul?, which are too regular and too constant to be due to chance? first of all, to take a look at one of these representations is enough to see that the mass is in rotation; we may even see what the sense of the rotation is; all the spiral radii are curved in the same sense; it is evident that the moving wing lags behind the pivot and that fixes the sense of the rotation. but this is not all; it is evident that these nebul? can not be likened to a gas at rest, nor even to a gas in relative equilibrium under the sway of a uniform rotation; they are to be compared to a gas in permanent motion in which internal currents prevail.
suppose, for example, that the rotation of the central nucleus is rapid (you know what i mean by this word), too rapid for stable equilibrium; then at the equator the centrifugal force will drive it away over the attraction, and the stars will tend to break away at the equator and will form divergent currents; but in going away, as their moment of rotation remains constant, while the radius vector augments, their angular velocity will diminish, and this is why the moving wing seems to lag back.
from this point of view, there would not be a real permanent motion, the central nucleus would constantly lose matter which would go out of it never to return, and would drain away progressively. but we may modify the hypothesis. in proportion as it goes away, the star loses its velocity and ends by stopping; at this moment attraction regains possession of it and leads it back toward the nucleus; so there will be centripetal currents. we must suppose the centripetal currents are the first rank and the centrifugal currents the second rank, if we adopt the comparison with a troop in battle executing a change of front; and, in fact, it is necessary that the composite centrifugal force be compensated by the attraction exercised by the central layers of the swarm upon the extreme layers.
besides, at the end of a certain time a permanent régime establishes itself; the swarm being curved, the attraction exercised upon the pivot by the moving wing tends to slow up the pivot and that of the pivot upon the moving wing tends to accelerate the advance of this wing which no longer augments its lag, so that finally all the radii end by turning with a uniform velocity. we may still suppose that the rotation of the nucleus is quicker than that of the radii.
a question remains; why do these centripetal and centrifugal swarms tend to concentrate themselves in radii instead of disseminating themselves a little everywhere? why do these rays distribute themselves regularly? if the swarms concentrate themselves, it is because of the attraction exercised by the already existing swarms upon the stars which go out from the nucleus in their neighborhood. after an inequality is produced, it tends to accentuate itself in this way.
why do the rays distribute themselves regularly? that is less obvious. suppose there is no rotation, that all the stars are in two planes at right angles, in such a way that their distribution is symmetric with regard to these two planes.
by symmetry, there would be no reason for their going out of these planes, nor for the symmetry changing. this configuration would give us therefore equilibrium, but this would be an unstable equilibrium.
if on the contrary, there is rotation, we shall find an analogous configuration of equilibrium with four curved rays, equal to each other and intersecting at 90°, and if the rotation is sufficiently rapid, this equilibrium is stable.
i am not in position to make this more precise: enough if you see that these spiral forms may perhaps some day be explained by only the law of gravitation and statistical consideration recalling those of the theory of gases.
what has been said of internal currents shows it is of interest to discuss systematically the aggregate of proper motions; this may be done in a hundred years, when the second edition is issued of the chart of the heavens and compared with the first, that we now are making.
but, in conclusion, i wish to call your attention to a question, that of the age of the milky way or the nebul?. if what we think we see is confirmed, we can get an idea of it. that sort of statistical equilibrium of which gases give us the model is established only in consequence of a great number of impacts. if these impacts are rare, it can come about only after a very long time; if really the milky way (or at least the agglomerations which are contained in it), if the nebul? have attained this equilibrium, this means they are very old, and we shall have an inferior limit of their age. likewise we should have of it a superior limit; this equilibrium is not final and can not last always. our spiral nebul? would be comparable to gases impelled by permanent motions; but gases in motion are viscous and their velocities end by wearing out. what here corresponds to the viscosity (and which depends upon the chances of impact of the molecules) is excessively slight, so that the present régime may persist during an extremely long time, yet not forever, so that our milky ways can not live eternally nor become infinitely old.
and this is not all. consider our atmosphere: at the surface must reign a temperature infinitely small and the velocity of the molecules there is near zero. but this is a question only of the mean velocity; as a consequence of impacts, one of these molecules may acquire (rarely, it is true) an enormous velocity, and then it will rush out of the atmosphere, and once out, it will never return; therefore our atmosphere drains off thus with extreme slowness. the milky way also from time to time loses a star by the same mechanism, and that likewise limits its duration.
well, it is certain that if we compute in this manner the age of the milky way, we shall get enormous figures. but here a difficulty presents itself. certain physicists, relying upon other considerations, reckon that suns can have only an ephemeral existence, about fifty million years; our minimum would be much greater than that. must we believe that the evolution of the milky way began when the matter was still dark? but how have the stars composing it reached all at the same time adult age, an age so briefly to endure? or must they reach there all successively, and are those we see only a feeble minority compared with those extinguished or which shall one day light up? but how reconcile that with what we have said above on the absence of a noteworthy proportion of dark matter? should we abandon one of the two hypotheses, and which? i confine myself to pointing out the difficulty without pretending to solve it; i shall end therefore with a big interrogation point.
however, it is interesting to set problems, even when their solution seems very far away.
