| Distance from centre | Density | Pressure in 103 atmospheres | Temperature in 103° Cent. | |
|---|---|---|---|---|
| Hydrogen | Iron | |||
| 0 | 0.00844 | 852 | 2460 | 137,500 |
| 0.1 | 0.00817 | 807 | 2406 | 134,600 |
| 0.2 | 0.00739 | 683 | 2251 | 126,100 |
| 0.3 | 0.00623 | 513 | 2007 | 112,400 |
| 0.4 | 0.00488 | 342 | 1707 | 95,600 |
| 0.5 | 0.00354 | 200 | 1377 | 77,100 |
| 0.6 | 0.00233 | 100 | 1043 | 58,400 |
| 0.7 | 0.00136 | 40 | 728 | 48,800 |
| 0.8 | O.00065 | 12 | 445 | 24,900 |
| 0.9 | 0.00020 | 1.7 | 202 | 11,300 |
| 1.0 | 0.00000 | 0 | 0 | 0 |
Schuster’s calculation was really made for the sun—that is to say, for a celestial body whose diameter is ten times smaller, and whose specific gravity is therefore a thousand times greater than the above-assumed values. According to the laws of gravitation and of gases, the pressure must there be 10,000 times greater, and the temperature ten times higher, than those in our table. The density of the interior portions would, however, become far too large to admit of the application of the gas laws. I have therefore modified the calculations so as to render them applicable to a celestial body of ten times the radius of the sun or of 1080 times the radius of the earth; the radius would then represent one-twenty-second of the distance from the centre of the sun to the earth’s orbit, and the respective celestial body would have very small dimensions indeed if compared to a nebula.
The extraordinarily high pressure in the interior portions of the celestial body is striking; this is due to the great mass and to the small distances. In the centre of the sun the pressure would amount to 8520 million atmospheres, since the pressure increases inversely as the fourth power of the radius. The pressure near the centre of the sun is, indeed, almost of that order. If the sun were to expand to a spherical planetary nebula of a thousand times its actual linear dimensions (when it would almost fill the orbit of Jupiter), the specific gravity at its centre would be diminished to one-millionth of the above-mentioned value—that is to say, matter in this nebula would not, even at the point of greatest concentration, be any denser than in the highly rarefied vacuum tubes which we can prepare at ordinary temperatures. The pressure would likewise be greatly diminished—namely, to about six millimetres only, near the centre of the gaseous mass. The temperature, however, would be rather high near the centre—namely, 24,600° C., if the nebula should consist of atomatic hydrogen, and fifty-six times as high again if consisting of iron gas. Such a nebula would restrain gases with 1.63 times the force which the earth exerts. Molecules of gases moving outward with a velocity of about 18 km. (11 miles) per second would forever depart from this atmosphere.
The estimation of the temperature in such masses of gases is certainly somewhat unreliable. We have to presume that neither radiation nor conduction exert any considerable influence. That might be permitted for conduction; but we are hardly justified in neglecting radiation. The temperatures within the interior of the nebula will, therefore, be lower than our calculated values. It is, however, difficult to make any definite allowance for this factor.
If the mass of the celestial body should not be as presumed—for instance, twice as large—we should only have to alter the pressure and the density of each layer in the same proportion, and thus to double the above values. The temperature would remain unchanged. We are hence in a position to picture to ourselves the state of a nebula of whatever dimensions and mass.
Lane has proved, what the above calculations also indicate, that the temperature of such nebula will rise when it contracts in consequence of its losing heat. If heat were introduced from outside, the nebula would expand under cooling. A nebula of this kind presumably loses heat and gradually raises its own temperature until it has changed into a star, which will at first have an atmosphere of helium and of hydrogen like that of the youngest stars (with white light). By-and-by, under a further rise of temperature, the extremely energetic chemical compounds will be formed which characterize the interior of the sun, because helium and hydrogen—which were liberated when the nebula was re-formed and which dashed out into space—will diffuse back into the interior of the star, where they will be bound under the formation of the compounds mentioned. The atmosphere of hydrogen and of helium will disappear (helium first), the star will contract more and more, and the pressure and the convection currents in the gases will become enormous. There will be a strong formation of clouds in the atmosphere of the star, which will gradually become endowed with the properties which characterize our sun. The sun behaves very differently from the gaseous nebulæ for which the calculations of Lane, Ritter, and Schuster hold. For when the contraction of a gas shall have proceeded to a certain limit, the pressure will increase in the ratio 1: 16, while the volume will decrease in the ratio 8: 1, provided there be no change in the temperature. When the gas has reached this point and is still further compressed, the temperature will remain in steady equilibrium. At still higher pressures, however, the temperature must fall if equilibrium is to be maintained. According to Amagat, this will occur at 17° C. (290° absolute) in gases like hydrogen and nitrogen, which at this temperature are far above their critical points, and at a pressure of 300 or 250 atmospheres. When the temperature is twice as high on the absolute scale, or at 307° C., twice the pressure will be required.
We can now calculate when our nebula will pass through this critical stage, to which a lowering of the temperature must succeed. Accepting the above figures, we find that half the mass of the nebula will fill a sphere of a radius 0.53 of that of the nebula. If the mass were everywhere of the same density, half of it would fill a sphere of 0.84 of this radius. When will the interior mass cross the boundary of the above stage, while the exterior portions still remain below this stage? That will be at about the time when the nebula in its totality will pass through its maximum temperature. We will now base our calculations on the temperatures which apply to iron in the gaseous state; for in the interior of the nebula the mean molecular weight will at least be 56 (that of iron). We shall find that the pressure at the distance 0.53 will be about 177,000 atmospheres, and the temperature approximately 71 million degrees—i.e., 245,000 times higher than the absolute temperature in the experiments of Amagat. The specified stage will then be reached when the pressure will be 245,000 times as large as 250 atmospheres—viz., 61 million atmospheres. As, now, the pressure is only 177,000 atmospheres, our nebula will yet be far removed from that stage at which cooling will set in. We can easily calculate that this will take place when the nebula has contracted to a volume about three times that of our sun. The assertion which is so often made that the sun might possibly attain higher temperatures in the future is unwarranted. This celestial body has long since passed through the culminating-point of its thermal evolution, and is now cooling. As the temperatures which Schuster deduced were no doubt much too high, the cooling must, indeed, have set in already in an earlier stage. But stars like Sirius, whose density is probably not more than one per cent, of the solar density, are probably still in a rising-temperature stage. Their condition approximates that of the mass of gas of our example.
The planetary nebulæ are vastly more voluminous. The immense space which these celestial bodies may occupy will be understood from the fact that the largest among them, No. 5 in Herschel’s catalogue, situated near the star B in the Great Bear, has a diameter of 2.67 seconds of arc. If it were as near to us as our nearest star neighbor, its diameter would yet be more than three times that of the orbit of Neptune; doubtless it is many hundreds of times larger. This consideration furnishes us with an idea of the infinite attenuation in such structures. In their very densest portions the density cannot be more than one-billionth of the density of the air. In the outer portions of such nebulæ the temperature must also be exceedingly low; else the particles of the nebula could not be kept together, and only hydrogen and helium can occur in them in the gaseous state.
Yet we may regard the density and temperature of such celestial bodies as gigantic by comparison with those of the gases in the spirals of the nebulæ. There never is equilibrium in these spirals, and it is only because the forces in action are so extraordinarily small that these structures can retain their shapes for long periods without noticeable changes. It is, probably, chiefly in those parts in which the cosmical dust is stopped in its motion that meteorites and comets are produced. The dust particles wander into the more central portions of the nebulæ, into which they penetrate deeply, owing to their relatively large mass, to form the nuclei for the growth of planets and moons. By their collisions with the masses of gases which they encounter, they gradually assume a circular movement about the axis of rotation of the nebula. In this rotation they condense portions of the gases on their surface, and hence acquire a high temperature—which they soon lose again, however, owing to the comparatively rapid radiation.
So far as we know, spiral nebulæ are characterized by continuous spectra. The splendor of the stars within them completely outshines the feeble luminosity of the nebula. The stars in them are condensation products and undoubtedly in an early stage of their existence; they may therefore be likened to the white stars, like the new star in Perseus and the central star in the ring nebula of the Lyre. Nevertheless, it has been ascertained that the spectrum of the Andromeda nebula has about the same length as that of the yellow stars. That may be due to the fact that the light of the stars in this nebula, which we only seem to see from the side, is partly extinguished by dust particles in its outer portion, as was the case with the light of the new star in Perseus during the period of its variability.