Popular Science Monthly/Volume 58/December 1900/Chapters on the Stars VI
CHAPTERS ON THE STARS. |
Masses and Densities of the Stars.
THE spectroscope shows that, although the constitution of the stars offers an infinite variety of detail, we may say, in a general way, that these bodies are suns. It would perhaps he more correct to say that the Sun is one of the stars and does not differ essentially from them in its constitution. The problem of the physical constitution of the Sun and stars may, therefore, be regarded as the same. Both consist of vast masses of incandescent matter at so exalted a temperature as to shine by their own light. All may be regarded as bodies of the same general nature.
It has long been known that the mean density of the Sun is only one-fourth that of the earth, and, therefore, less than half as much again as that of water. In a few cases an approximate estimate of the density of stars may be made. The method by which this may be done can be rigorously set forth only by the use of algebraic formulæ, but a general idea of it can be obtained without the use of that mode of expression.
Let us in advance set forth an extension of Kepler's third law, which applies to every case of two bodies revolving around each other by their mutual gravitation. The law in question, as stated by Kepler, is that the cubes of the mean distances of the planets are proportioned to the squares of their times of revolution. If we suppose the mean distances to be expressed in terms of the earth's mean distance from the Sun as a unit of length, and if we take the year as the unit of time, then the law may be expressed by saying that the cubes of the mean distances will be equal to the squares of the periods. For example, the mean distance of Jupiter is thus expressed as 5.2. If we take the cube of this, which is about 140, and then extract the square root of it, we shall have 11.8, which is the period of revolution of Jupiter around the Sun expressed in the same way. If we cube 9.5, the mean distance of Saturn, we shall have the square of a little more than 29, which is Saturn's time of revolution.
We may also express the law by saying that if we divide the cube of the mean distance of any planet by the square of its periodic time we shall always get 1 as a quotient.
The theory of gravitation and the elementary principles of force and motion show that a similar rule is true in the case of any two bodies revolving around each other in virtue of their mutual gravitation. If we divide the cube of their mean distance apart by the square of their time of revolution, we shall get a quotient which will not indeed be 1, but which will be a number expressing the combined mass of the two bodies. If one body is so small that we leave its mass out of consideration, then the quotient will express the mass of the larger body. If the latter has several minute satellites moving around it, the quotients will be equal, as in the case of the Sun, and will express the mass of this central body. If, as in the case we have supposed, we take the year as a unit of time and the distance of the earth from the Sun as a unit of length, the quotient will express the mass of the central body in terms of the mass of the Sun. It is thus that the masses of the planets are determined from the periodic times and distances of their satellites,
and the masses of binary systems from their mean distance apart and their periods. To express the general law by a formula we put
a, the mean distance apart of the two bodies, or the semi-major axis of their relative orbit in terms of the earth's mean distance from the Sun;
P, their periodic time;
M, their combined mass in terms of the Sun's mass as unity.
Then we shall have:
Another conclusion we draw is that if we know the time of revolution and the radius of the orbit of a binary system, we can determine what the time of revolution would be if the radius of the orbit had some standard length, say unity.
We cannot determine the dimensions of a binary system unless we know its parallax. But there is a remarkable law which, so far as I know, was first announced by Pickering, by virtue of which we can determine a certain relation between the surface brilliancy and the density of a binary system without knowing its parallax.
Let us suppose a number of bodies of the same constitution and temperature as the Sun—models of the latter we may say—differing from it only in size. To fix the ideas, we shall suppose two such bodies, one having twice the diameter of the other. Being of the same brilliancy, we suppose them to emit the same amount of light per unit of surface. The larger body, having four times the surface of the smaller, will then emit four times as much light. The volumes being proportional to the cubes of their diameters, it will have eight times its volume. The densities being supposed equal, it will have eight times the mass. Suppose that each has a satellite revolving around it, and that the orbit of the satellite of the larger body is twice the radius of that of the smaller one.
Calling the radius of the nearer satellite 1, that of the more distant one will be 2. The cube of this number is 8. It follows from the extension of Kepler's third law, which we have cited, that the times of revolution of the two satellites will be the same. Thus the two bodies, A and B, with their satellites, C and C, form two binary systems whose proportions and whose periods are the same, only the linear dimensions of B are all double those of A. In other words, we shall have a pair of binary systems which may look alike in every respect, but of which one will have double the dimensions and eight times the mass of the other.
Now let us suppose the larger system to be placed at twice the distance of the smaller. The two will then appear of the same size, and, if stars, will appear of the same brightness, while the two orbits will have the same apparent dimensions. In a word, the two systems will appear alike when examined with the telescope, and the periodic times will be equal.
Near the end of the second chapter we have given a little table showing the magnitude that the Sun would appear to us to have were it placed at different distances among the stars. The parallaxes we have there given are simply the apparent angle which would have to be subtended by the radius of the earth's orbit at different distances. It follows that, were the stars all of similar constitution to the Sun, the numbers given in the last column of the table referred to would, in all cases, express the apparent distance from the star of a companion, having a time of revolution of one year. From this we may easily show what would be the time of revolution of any binary system of which the companions were separated by 1", if the stars were of the same constitution as the Sun.
Periods of binary systems whose components are separated by 1" and whose constitution is the same as that of the Sun.
Mag. | Period. y. |
Annual Motion. | |||
1 | 1 | .8 | 200 | ° | |
2 | 3 | .5 | 102 | ||
3 | 7 | .0 | 51 | ||
4 | 14 | .1 | 25 | ||
5 | 28 | .1 | 13 | ||
6 | 56 | .0 | 6 | ||
7 | 112 | . | 3 | .2 | |
8 | 223 | . | 1 | .6 |
It will be seen that the periods are very nearly doubled for each diminution of the brilliancy of the star by one magnitude. Moreover, the value of the photometric ratio for two consecutive magnitudes is a little uncertain, so that we may, without adding to the error of our results, suppose the period to be exactly double for each addition of unity to the magnitude. A computation of the period for any magnitude may be made with all necessary precision by the formula:
P = O^{y}.88 × 2^{m}; | |
or,log. | P = 9.994 + 0.3m. |
It will now be of interest to compare the results of this theory with the observed periods of binary systems with a view to comparing their constitution with that of our Sun. There are, however, two difficulties in the way of doing this with rigorous precision.
The first difficulty is that there are very few binary systems of which the apparent dimensions of the orbit and the periods are known with any approach to exactness. This would not be a serious matter were it not that the short, and, therefore, known periods belong to a special class, that having the greatest density. Hence, when we derive our results from the systems of known periods we shall be making a biased selection from this particular class of stars.
The next difficulty is that the theory which we have set forth assumes the mass of the satellite either to be very small compared with that of the star, or the two bodies to be of the same constitution. If we apply the theory to systems in which this is not the case, the results which we shall get will be, in a certain way, those corresponding to the mean of the two components. Were it a question of masses, we should get with entire precision the sum of the masses of the two bodies. The best we can do, therefore, is to suppose the two companions fused into one having the combined brilliancy of the two. Then, if the result is too small for one, it will be too large for the other.
To show the method of proceeding, I have taken the six systems of shortest period found in Dr. See's 'Researches on Stellar Evolution.' The principal numbers are shown in the table below.
The first column, a", after the name of the star, gives the apparent semi-major axis of the orbit in seconds of arc. The next column gives the period in years. Column Mag. gives the apparent magnitude which the system would have were the two bodies fused into one.
Column P gives the period in years as it would be were the radius of the orbit equal to one second. It is formed by dividing the actual period by A. The next column gives the period as it would be were the stars of similar constitution to the Sun. The last column gives the square of the ratio of the two bodies, which, if the stars had the same surface brilliancy as the Sun, would express the ratio of density of the stars to that of the Sun. Actually, it gives the product:
Density x (brilliancy).32
a." | Per. | Mag. | P. | Sun's Per. |
Star's Density. | |||||||
κ Pegasi | 0". | 42 | 11y. | 4 | 4. | 2 | 41. | 9 | 16. | 2 | 0. | 15 |
ζ Equulei | 0". | 45 | 11y. | 4 | 4. | 6 | 37. | 8 | 21. | 0 | 0. | 31 |
ξ Sagittarii | 0". | 69 | 18y. | 8 | 2. | 9 | 32. | 7 | 6. | 7 | 0. | 04 |
F9 Argus | 0". | 65 | 22y. | 0 | 5. | 5 | 42. | 0 | 39. | 7 | 0. | 90 |
42 Cornæ | 0". | 64 | 25y. | 6 | 4. | 4 | 50. | 0 | 18. | 5 | 0. | 14 |
β Delpirii | 0". | 67 | 27y. | 7 | 3. | 7 | 50. | 4 | 11. | 4 | 0. | 51 |
The numbers in the last column being all less than unity, it follows that either the stars are much less dense than the Sun or they are of much less surface brilliancy. Moreover, they belong to a selected list in which the numbers of the last column are larger than the average.
To form some idea of the result of a selection from the general average, we may assume that the average of all the measured distances between the components of a number of binary systems is equal to the average radius of their orbits, and that the observed annual motion is equal to the mean motion of the companion in its orbit. Taking a number of cases of this sort, I find that the number corresponding to the last number of the preceding table would be little more than one thousandth.
A very remarkable case is that of ξ Orionis. This star, in the belt of Orion, is of the second magnitude. It has a minute companion at a distance of 2".5. Were it a model of the Sun, a companion at this apparent distance should perform its revolution in fourteen years. But, as a matter of fact, the motion is so slow that even now, after fifty years of observation, it cannot be determined with any precision. It is probably less than 0°.1 in a year. The number expressing the comparison of its density and surface brilliancy with those of the Sun is probably less than .0001.
The general conclusion to be drawn is obvious. The stars in general are not models of our Sun, but have a much smaller mass in proportion to the light they give than our Sun has. They must, therefore, have either a less density or a greater surface brilliancy.
We may now inquire whether such extreme differences of surface brilliancy or of density are more likely. The brilliancy of a star depends primarily not on its temperature throughout, but on that of some region near or upon its surface. The temperature of this surface cannot be kept up except by continual convection currents from the interior to the surface. We are, therefore, to regard the amount of light emitted by a star not merely as indicating temperature, but as limited by the quantity of matter which, impeded by friction, can come up to the surface, and there cool off and afterward sink down again. This again depends very largely on internal friction, and is limited by that. Owing to this limitation, we cannot attribute the difference in question wholly to surface brilliancy. We must conclude that at least the brighter stars are, in general, composed of matter much less dense than that of the Sun. Many of them are probably even less dense than air and in nearly all cases the density is far less than that of any known liquid.
An ingenious application of the mechanical principle we have laid down has been made independently by Mr. Roberts, of South Africa, and Mr. Norris, of Princeton, in another way. If we only knew the relation between the diameters of the two companions of a binary system and its dimensions, we could decide how much of the difference in question is due to density and how much to surface brilliancy. Now this may be approximately done in the case of variable stars of the Algol and β Lyræ types. If, as is probably the most common case, the passage of the stars over each other is nearly central, the ratio of their diameter to the radius of the orbit may be determined by comparing the duration of the eclipse with the time of revolution. This was one of the fundamental data used by Myers in his work on β Lyræ, of which we have quoted the results. Without going into reasoning or technical details at length, we may give the results reached by Roberts and Norris in the case of the Algol variables:
For the variable star X Carinæ, Roberts finds, as a superior limit for the density of the star and its companion, one-fourth that of the Sun. It may be less than this is, to any extent.
In the case of S Velorum the superior limits of density are:
Bright star | 0.61 |
Companion | 0.03 |
In the case of RS Sagittarii the upper limits of density are 0.16 and 0.21.
It is possible, in the mean of a number of cases like these, to estimate the general average amount by which the densities fall below the limits here given. Roberts' final conclusion is that the average density of the Algol variables and their eclipsing companions is about one eighth that of the Sun.
The work of Russell was carried through at the same time as that of Roberts, and quite independently of his. It appeared at the same time.^{[1]} His formulæ and methods were different, though they rested on similar fundamental principles. Taking the density of the Sun as unity, he computes the superior limit of density for 12 variables, based on their periods and the duration of their partial eclipses. The greatest limit is in the case of Z Herculis and is 0.728. The least is in the ease of S Caneri and is 0.035. The average is about 0.2. As the actual density may be less than the limit by an indefinite amount, the general conclusion from his work may be regarded as the same with that from the work of Roberts.
The results of the preceding theory are independent of the parallax of the stars. They, therefore, give us no knowledge as to the mass of a binary system. To determine this we must know its parallax, from which we can determine the actual dimensions of the orbit when its apparent dimensions are known. Then the formula already given will give the actual mass of the system in terms of the Sun's mass.
There are only six binary systems of which both the orbit and the parallax are known. These are shown in the table below. Here the first two columns after the stars named give the semi-major axis of the orbit and the measured parallax. The quotient of the first number by the second gives the actual mean radius of the orbits in terms of the earth's distance from the Sun as unity. This is given in the third column, after which follow the period and the resulting combined mass of the system. The last column shows the actual amount of light emitted by the system, compared with that of the Sun.
a." | Par. | a. | Period. | Mass. | Light. | |||||||
" | " | y. | ||||||||||
η Cassiopiæ | 8. | 21 | 0. | 20 | 41. | 0 | 195. | 8 | 1. | 8 | 1. | 0 |
Sirius | 8. | 03 | 0. | 37 | 21. | 7 | 52. | 2 | 3. | 7 | 32. | 0 |
Procyon | 3. | 00 | 0. | 30 | 10. | 0 | 40. | 0 | 0. | 6 | 8. | 5 |
α Centauri | 17. | 70 | 0. | 75 | 23. | 6 | 81. | 1 | 2. | 0 | 1. | 7 |
70 Ophiuchi | 4. | 55 | 0. | 19 | 24. | 0 | 88. | 4 | 1. | 8 | 0. | 7 |
85 Pegasi | 0. | 89 | 0. | 05 | 17. | 8 | 24. | 0 | 9. | 0 | 2. | 2 |
Even in these few cases some of the numbers on which the result depends are extremely uncertain. In the case of Procyon, the radius of the orbit, can be only a rough estimate. In the case of 85 Pegasi the parallax is uncertain. In the case of η Cassiopiæ the elements are still doubtful.
So far as we have set forth the principles involved in the question, we do not get separate results for the mass of each body. The latter can be determined only by meridian observations, showing the motion of the brighter star around the common center of gravity of the two. This result has thus far been worked out with an approximation to exactness only in the cases of Sirius and Procyon. For these systems we have the following masses of the companions of these bodies in terms of the Sun's mass:
Companion of Sirius | 1.2 |
Companion of Procyon | 0.2 |
It will now be interesting to compare the brightness of these bodies with that which the Sun would have if seen at their distance. In a former chapter we showed how this could be done. The results are:
At the distance of Procyon the apparent magnitude of the Sun would be 2^{m}.8. At the distance of Sirius, it would be 2^{m}.3. Supposing the Sun to be changed in size, its density remaining unchanged, until it had the same mass as the respective companions of Procyon and Sirius, its magnitudes would be:
For companion of Procyon | 3.9 |
For companion of Sirius | 2.9 |
The actual magnitudes of these companions cannot be estimated with great precision, owing to the effect of the brilliancy of the star. From the estimate of the companion of Sirius, by Professor Pickering, its magnitude was about the eighth. It is probable that the magnitude of the companion of Procyon is not very different. It will be seen that these magnitudes are very different from those which they would have were the companions models of the Sun. What is very curious is that they differ in the opposite direction from the stars in general, and especially from their primaries. Either they have a far less surface brilliancy than the Sun or their density is much greater. There can be no doubt that the former rather than the latter is the case.
This great mass of the two companions as compared with their brilliancy suggests the question whether they may not shine, in part at least, by the light of their primaries. A very little consideration will show that this cannot be the case. A simple calculation will show that, to shine as brightly as they do, the diameter of the companion of Sirius would have to be enormous, at least 1-30 its distance from Sirius. Moreover, its apparent brightness would vary so widely in different parts of its orbit that we should see it almost as well when near Sirius as when distant from it. The most likely cause of the small brightness is the low temperature of the body.
Gaseous Constitution of the Stars.
The results of the last chapter point to the conclusion that the stars, or at least the brighter among them, are masses of gas, more or less compressed in their interior by the action of gravitation upon their more superficial parts. We have now to show how this result was arrived at, at least in the case of the Sun, from different considerations, before the spectroscope had taught us anything of the constitution of these bodies.
We must accept, as one of the obvious conclusions of modern science, the fact that the Sun and stars have, for untold millions of years, been radiating heat into space. If we refrain from considering the basis on which this conclusion rests, it is not so much because we consider it unquestionable, as because the discussion would be too long and complex for the present work.
One of the great problems of modern science has been to account for the source of this heat. Before the theory of energy was developed this problem offered no difficulty. In the time of Newton, Kant and even of La Place and Herschel, no reason was known why the stars should not shine forever without change. Now we know that when a body radiates heat, that heat is really an entity termed energy, of which the supply is necessarily limited. Kelvin compared the case of a star radiating heat with that of a ship of war belching forth shells from her batteries. We know that if the firing is kept up, the supply of ammunition must at some time be exhausted. Have we any means of determining how long the store of energy in Sun or star will suffice for its radiation?
We know that the substances which mainly compose the Sun and stars are similar to those which compose our earth. We know the capacity for heat of these substances, and we also have determined how much the Sun radiates annually. From these data, it is found by a simple calculation that the temperature of the Sun would be lowered annually by more than two degrees Fahrenheit, if its capacity for heat were the same as that of water. If this capacity were only that of the substances which compose the great body of the earth, the lowering of temperature would be from 5° to 10° annually. Evidently, therefore, the actual heat of the Sun would only suffice for a few thousand years' radiation, if not in some way replenished.
When the difficulty was first attacked, it was supposed that the supply might be kept up by meteors falling into the Sun. We know that in the region round the Sun, and, in fact, in the whole Solar System, are countless minute meteors some of which may from time to time strike the Sun. The amount of heat that would be produced by the loss of energy suffered by a meteor moving many hundred miles a second would be enormously greater than that which would be produced by combustion. But critical examination shows that this theory cannot have any possible basis. Apart from the fact that it could at best be only a temporary device there seems to be no possibility that meteors sufficient in mass can move round the Sun or fall into it. Shooting stars show that our earth encounters millions of little meteors every day; but the heat produced is absolutely insignificant.
It was then shown by Kelvin and Helmholtz that the Sun might radiate the present amount of heat for several millions of years, simply from the fund of energy collected by the contraction of its volume through the mutual gravitation of its parts. As the Sun cools it contracts; the fall of its substance toward the center, produced by this contraction, generates energy, which energy is constantly turned into heat. The amount of contraction necessary to keep up the present supply may be roughly computed; it amounts in round numbers to 220 feet a year, or four miles in a century.
Accepting this view, it will almost necessarily follow that the great body of the Sun must be of gaseous constitution. Were it solid, its surface would rapidly cool off, since the heat radiated would have to be conducted from the interior. Then, the loss of heat no longer going on at the same rate, the contraction also would stop and the generation of heat to supply the radiation would cease. Even were the Sun a liquid, currents of liquid matter could scarcely convey to the surface a sufficient amount of heated matter to supply the enormous radiation. Thus the reason of the case combines with observation of the density of the Sun to show that its interior must be regarded as gaseous rather than solid or liquid.
A difficult matter, however, presents itself. The density of the Sun is greater than we ordinarily see in gases, being, as we have remarked, even greater than the density of water. The explanation of this difficulty is very simple: the gaseous interior is subject to compression by its superficial portions. The gravitation on the surface being 27 times what it is on the earth, the pressure increases 27 times as fast when we go toward the center as it does on the earth. We should not have to go very far within its body to find a pressure of millions of tons on the square inch. Under such pressure and at such an enormous temperature as must there prevail, the distinction between a gas and a liquid is lost; the substance retains the mobility of a gas, while assuming the density of a liquid.
It does not follow, however, that the visible surface of the Sun is a gas, pure and simple. The sudden cooling which a mass of gaseous matter undergoes on reaching the surface may liquefy it or even change it into a solid. But, in either case, the sudden contraction which it thus undergoes makes it heavier and it sinks down again to be remelted in the great furnace below. It may well be, therefore, that the description of the Sun as a vast bubble is nearly true. It may be added that all we have said about the Sun may very well be presumed to apply to the stars. We have now to consider the law of change as a sun or star contracts through the loss of heat suffered by its radiation into space.
This subject was very exhaustively developed by Ritter some years since.^{[2]} It is not practicable to give even an abstract of Ritter's results at the present time, especially as every mathematical investigation of the subject must either rest on hypotheses more or less uncertain, or must, for its application, require data impossible to obtain. We shall, therefore, confine ourselves to a brief outline of the main points of the subject. A fundamental proposition of the whole theory is Lane's law of gaseous attraction, which is as follows:
When a spherical mass of incandescent gas contracts through the loss of its heat by radiation into space, its temperature continually becomes higher as long as the gaseous condition is retained.
The demonstration of this law is simple enough to be understood by any one well acquainted with elementary mechanics and physics, and it will also furnish the basis for our consideration of the subject.
We begin by some considerations on the condition of a mass of gas held together by the mutual attraction of its parts. This attraction results in a certain hydrostatic pressure, capable of being expressed as so many pounds or tons per unit of surface, say a square inch. This pressure at any point is equal to the weight of a column of the gas, having a section of one square inch and extending from the point in
question to the surface. It is a law of attraction in a sphere of which the density is the same at equal distances from its center, that if we suppose an interior sphere concentric with the body, the attraction of all the matter outside that interior sphere, on any point within it, is equal in every direction, and, therefore, is completely neutralized. A point is, therefore, drawn towards the center only by the attraction of the sphere on the surface of which it lies.
At every point in the interior the hydrostatic pressure must be balanced by the elastic force of the gas. In the case of any one gas this force is proportional to the product of the density into the absolute temperature. This condition of equilibrium must be satisfied at every point throughout the mass.
Let the two circles in the figure represent gaseous globes, of the kind supposed. The larger one represents the globe in a certain condition of its evolution; the second its condition after its volume has contracted to one half. The temperature in each case will necessarily increase from the surface to the center. The law of this increase is incapable of accurate expression, but is not necessary for our present purpose.
Let the inner circle, C D, represent a spherical shell, situated anywhere in the interior of the mass, but concentric with it. Let E F be the corresponding shell after the contraction has taken place. The case will then be as follows:
The two shells will by hypothesis have the same quantity of matter, both in their own substance and throughout their interior.
In case B the central attraction being as the inverse square from the center, will be four times as great for each unit of matter in the shell.
This force of attraction, tending to compress the shell, is, in case B, exerted on a surface one quarter as great, because the surface of a shell is proportional to the square of its diameter.
Hence the hydrostatic pressure per unit of surface is 16 times as great in case B as in case A.
The elastic force of a gas, if the two bodies were at the same temperature, would be 8 times as great in case B as in case A, being inversely as the volume.
The hydrostatic pressure being 16 times as great, while the elastic force to counterbalance it is only 8 times as great, no equilibrium would be possible. To make it possible, the absolute temperature of the gas must be doubled, in order that the elastic force shall balance the pressure.
That a mass can become hotter through cooling, may, at first sight, seem paradoxical. We shall, therefore, cite a result which is strictly analogous. If the motion of a comet is hindered by a resisting medium, the comet will continually move faster. The reason of this is that the first effect of the medium is to diminish the velocity of the object. Through this diminution of velocity, the comet falls towards the Sun. The increase of velocity caused by the fall more than counterbalances the diminution produced by the resistance. The result is that the comet takes up a more and more rapid motion, as it gradually approaches the Sun, in consequence of the resistance it suffers. In the same way, when a gaseous celestial body cools, the fall of its mass towards the center changes from a potential to an actual form an amount of energy greater than that radiated away.
The critical reader will see a weak point in this reasoning, which it is necessary to consider. What we have really shown is that if the mass, assumed to be in a state of equilibrium when it has the size A, has to remain in equilibrium when it has the size B, then its temperature must be doubled. But we have not proved that its temperature actually will be doubled by the fall. In fact, it cannot be doubled unless the energy generated by the fall of the superficial portions towards the center is sufficient to double the absolute amount of heat. Whether this will be the case depends on a variety of circumstances; the mass of the whole body, and the capacity of its substance for heat. If we are to proceed with mathematical rigor, it is, therefore, necessary to determine in any given case whether this condition is fulfilled. Let us suppose that in any particular case the mass is so small or the capacity for heat so considerable that the temperature is not doubled by the contraction. Then the contraction will go on further and further, until the mass becomes a solid. But in this case let us reverse the process. The body being supposed nearly in a state of equilibrium in position A, let the elastic force be slightly in excess. Then the gas will expand. In order that it be reduced to a state of equilibrium by expansion, its temperature must diminish according to the same law that it would increase if it contracted. When its diameter doubles, its temperature should be reduced to one half or less by the expansion, in order that the equilibrium shall subsist. But, in the case supposed, the temperature is not reduced so much as this. Hence, it is too high for equilibrium by a still greater amount and the expansion must go on indefinitely. Thus, in the case supposed, the hypothetical equilibrium of the body is unstable. In other words, no such body is possible.
This conclusion is of fundamental importance. It shows that the possible mass of a star must have an inferior limit, depending on the quantity of matter it contains, its elasticity under given circumstances and its capacity for heat. It is certain that any small mass of gas, taken into celestial space and left to itself, would not be kept together by the mutual attraction of its parts, but would merely expand into indefinite space. Probably this might be true of the earth, if it were gaseous. The computation would not be a difficult one to make, but I have not made it.
In what precedes, we have supposed a single mass to contract. But our study of the relations of temperature and pressure in the two masses assumes no relationship between them, except that of equality. Let us now consider any two gaseous bodies, A and B, and suppose that the body B, instead of having the same mass as that of A, is another body with a different mass.
Since the mass, B, may be of various sizes, according to the amount of attraction it has undergone, let us begin by supposing it to have the same volume as A, but twice the mass of A. We have then to inquire what must be its temperature in order that it may be in equilibrium. We have first to inquire into the hydrostatic pressure at any point of the interior. Referring once more to a figure like either of those in Fig. 2, a spherical shell like C D will now in the case of the more massive body have double the mass of the corresponding shell of A. The attraction will also be doubled, because the diameter of the spherical shell is the same, while the amount of matter within it is twice as great. Hence the hydrostatic pressure per unit of surface will be four times as great, or will vary as the square of the density. The elasticity at equal temperatures being proportional to the density, it follows that were the temperature the same in the two 'masses, the elasticity would be double in the case of mass B; whereas, to balance the hydrostatic pressure it should be quadrupled. The temperature of B must, therefore, be twice as great as that of A. It follows that in the case of stars of equal volume, but of different masses, the temperature must be proportional to the mass of density.
But how will it be if we suppose the density to be always the same, and, therefore, the mass to be proportional to the volume? In this case the attraction at a given point will be proportional to the diameter of the body. If, then, we suppose one body to have twice the diameter of the other, but to be of the same density, it follows that at corresponding points of the interior, the hydrostatic pressure will be twice as great in the larger body. The density being the same, it follows that the temperature must be twice as high in order that equilibrium may be maintained. It follows that the stars of the greatest mass will be at the highest temperature, unless their volume is so great that their density is less than that of the smaller stars.
Stellar Evolution.
It follows from the theory set forth in the last chapter that the stars are not of fixed constitution, but are all going through a progressive change—cooling off and contracting into a smaller volume. If we accept this result, we find ourselves face to face with an unsolvable enigma—how did the evolution of the stars begin? To show the principle involved in the question, I shall make use of an illustration drawn from a former work.^{[3]} An inquiring person wandering around in what he supposes to be a deserted building, finds a clock running. If he knows nothing about the construction of the clock, or the force necessary to keep it in motion, he may fancy that it has been running for an indefinite time just as he sees it, and that it will continue to run until the material of which it is made shall wear out. But if he is acquainted with the laws of mechanics, he will know that this is impossible, because the continued movement of the pendulum involves a constant expenditure of energy. If he studies the construction of the clock, he will find the source of this energy in the slow falling of a weight suspended by a cord which acts upon a train of wheels. Watching the motions, he will see that the scape wheel acting on the pendulum moves very perceptibly every second, while he must watch the next wheel for several seconds to see any motion. If the time at his disposal is limited, he will not be able to see any motion at all in the weight. But an examination of the machinery will show him that the weight must be falling at a certain rate, and he can compute that, at the end of a certain time, the weight will reach the bottom, and the clock will stop. He can also see that there must have been a point from above which the weight could never have fallen. Knowing the rate of fall, he can compute how long the weight occupied in falling from this point. His final conclusion will be that the clock must in some way have been wound up and set in motion a certain number of hours or days before his inspection.
If the theory that the heat of the stars is kept up by their slow contraction is accepted, we can, by a similar process to this, compute that these bodies must have been larger in former times, and that there must have been some finite and computable period when they were all nebulæ. Not even a nebula can give light without a progressive change of some sort. Hence, within a certain finite period the nebulæ themselves must have begun to shine. How did they begin? This is the unsolvable question.
The process of stellar evolution may be discussed without considering this question. Accepting as a fact, or at least as a working hypothesis, that the stars are contracting, we find a remarkable consistency in the results. Year by year laws are established and more definite conclusions reached. It is now possible to speak of the respective ages of stars as they go through their progressive course of changes. This subject has been so profoundly studied and so fully developed by Sir William and Lady Huggins that I shall depend largely on their work in briefly developing the subject.^{[4]}
At the same time, in an attempt to condense the substance of many folio pages into so short a space, one can hardly hope to be entirely successful in giving merely the views of the original author. The following may, therefore, be regarded as the views of Sir William Huggins, condensed and arranged in the order in which they present themselves to the writer's mind.
There is an infinite diversity among the spectra of the stars; scarcely two are exactly alike in all their details. But the larger number of these spectra, when carefully compared, may be made to fall in line, thus forming a series in which the passage of one spectrum into the next in order is so gradual as to indicate that the actual differences represent, in the main, successive epochs of star life rather than so many fundamental differences of chemical constitution. Each star may be considered to go through a series of changes analogous to those of a human being from birth to old age. In its infancy a star is simply a nebulous mass; it gradually condenses into a smaller volume, growing hotter, as set forth in the last chapter, until a stage of maximum temperature is reached, when it begins to cool off. Of the duration of its life we cannot form an accurate estimate. We can only say that it is to be reckoned by millions, tens of millions or hundreds of millions of years. We thus view in the heavens stars ranging through the whole series from the earliest infancy to old age. How shall we distinguish the order of development? Mainly by their colors and their spectra. In its first stage the star is of a bluish white. It gradually passes through white into yellow and red. Sir William gives the following series of stars as an example of the successive orders of development:
Sirius, α Lyræ.
α Ursae Majoris.
α Virginis.
α Aquilæ.
Rigel
α Cygni.
...
Capella—The Sun.
Arcturus.
Aldebaran.
α Orionis.
The length of the life of a star has no fixed limit; it depends entirely on the mass. The larger the mass, the longer the life; hence a small star may pass from infancy to old age many times more rapidly than a large one.
A remarkable confirmation of this order is found in the generally yellow or red color of the companions of bright stars in binary systems. The two stars of such a system naturally commenced their life history at the same epoch, but the smaller one, going through its changes more rapidly, is now found to be yellower than the other. Additional confirmation is afforded by the very great mass of the companions of Sirius and Procyon, notwithstanding the faintness of their light.
At the same time, up to at least the yellow stage, the star continually grows hotter as it condenses. A difficulty may here suggest itself in reconciling this order with a well-known physical fact. As a radiating body increases in temperature, its color changes from red through yellow to white, and the average wave length of its light continually diminishes. We see a familiar example of this in the case of iron, which, when heated, is first red in color and then goes through the changes we have mentioned. The ordinary incandescent electric light is yellow; the arc light, the most intense that we can produce by artificial means, is white. When the spectrum of a body thus increasing in temperature is watched, the limit is found to pass gradually from the red toward the violet end. It would seem, therefore, that the hotter stars should be the white ones and the cooler the yellow or red ones.
There are, however, two circumstances to be considered in connection with the contracting star. In the first place, the light which we receive from a star does not emanate from its hottest interior, but from a region either upon or, in most cases, near its surface. It is, therefore, the temperature of this region which determines the color of the light. In the next place, part of the light is absorbed by passing through the cooler atmosphere surrounding the star. It is only the light which escapes through this atmosphere that we actually see.
In the case of the Sun all the light which it sends forth comes from an extreme outer surface, the photosphere. The most careful telescopic examination shows no depth to this surface. It sends light to us, as if it were an opaque body like a globe of iron. This surface would rapidly cool off were it not for convection currents bringing up heated matter from the interior. It might be supposed that such a current would result in the surface being kept at nearly as high a temperature as the interior; but, as a matter of fact, the opposite is the case. As the volume of gas rises, it expands from the diminished pressure and it is thus cooled in the very act of coming to the surface.
In the case of younger stars, there is probably no photosphere properly so called. The light which they emit comes from a considerable distance in the interior. Here the effect of gravity comes into play. The more the star condenses, the greater is gravity at its surface; hence the more rapidly does the density of the gas increase from the surface toward the interior. In the case of the Sun, the density of any gas which may immediately surround the photosphere must be doubled every mile or two of its depth until we reach the photosphere. But if the Sun were many times its present diameter, this increase would be less in a still larger proportion. Hence, when the volume is very great the increase of density is comparatively slow; there being no well-defined photosphere, the light reaches us from a much greater depth from the interior than it does at a later stage.
The gradual passing of a white star into one of the solar type is marked by alterations in its spectrum. These alterations are especially seen in the behavior of the lines of hydrogen, calcium, magnesium and iron. The lines of hydrogen change from broad to thin; those of calcium constantly become stronger.
Of the greatest interest is the question—at what stage does the temperature of the star reach its maximum and the body begin to cool? Has our Sun reached this stage? This is a question to which, owing to the complexity of the conditions, it is impossible to give a precise answer. It seems probable, however, that the highest temperature is reached in about the stage of our Sun.
The general fact that every star has a life history—that this history will ultimately come to an end—that it must have had a beginning in time—is indicated by so great a number of concurring facts that no one who has most profoundly studied the subject can have serious doubts upon it. Yet there are some unsolved mysteries connected with the case, which might justify a waiting for further evidence, coupled with a certain degree of skepticism. Of the questions connected with the case the most serious one is: How is the supply of energy radiated by the Sun and stars kept up? Only one answer is possible in the light of recent science. It is that already given in the last chapter—the continual contraction of volume. The radiant energy sent out is balanced by the continual loss of potential energy due to the contraction.
On this theory the age of the Sun can be at least approximately estimated. About twenty millions of years is the limit of time during which it could possibly have radiated anything like its present amount of energy. But this conclusion is directly at variance with that of geology. The age of the earth has been approximately estimated from a great variety of geological phenomena, the concurring result being that stratification and other geological processes must have been going on for hundreds—nay, thousands of millions of years. This result is in direct conflict with the only physical theory which can account for the solar heat.
The nebulæ offer a similar difficulty. Their extreme tenuity and their seemingly almost unmaterial structure appear inadequate to account for any such mutual gravitation of their parts as would result in the generating of the flood of energy which they are constantly radiating. What we see must, therefore, suggest at least the possibility that all shining heavenly bodies have connected with them some form of energy of which science can, as yet, render no account. This suspicion cannot, however, grow into a certainty until we have either seen the nebulæ contracting in volume or have made such estimates of their probable masses that we can compute the amount of contraction they must undergo to maintain the supply of energy.
In the impressive words of Sir William Huggins:
"We conclude filled with a sense of wonder at the greatness of the human intellect, which from the impact of waves of ether upon one sense-organ, can learn so much of the Universe outside our earth; but the wonder passes into awe before the unimaginable magnitude of Time, of Space and of Matter of this Universe, as if a Voice were heard saying to man: 'Thou art no Atlas for so great a weight.'"