# The Solar System/Chapter 6

Present the outcome of the past. After the present, the past. The forces that the we have found to be moulding the system to-day must be those that fashioned it earlier. Given, therefore, the condition at the moment, if we apply to it the forces now at work reversed, we shall get the condition that was.
Similarly, we can cast its horoscope for the future, by Taylor's theorem. Unfortunately, the problem is so complicated that no solution, even approximately satisfactory, has yet been obtained; but that the mystery baffles us renders it all the more fascinating.
Striking relations in the solar system. In the solar system, as we find it to-day, are several remarkable congruities which are quite independent of gravitation, and bespeak a cause.
I. The central body is much larger than its attendants.
II. The planets move in orbits nearly circular.
III. They travel nearly in one plane.
IV. And in the same sense (direction).
As for the planets themselves—
​V. Their planes of rotation nearly coincide with their orbital planes (except Uranus and Neptune).
VI. They rotate also in the same direction that they revolve, counter-clockwise, all of them (except Uranus and Neptune).
VII. Their satellites revolve nearly in the planes of their primaries' equators (so far as we can see).
VIII. And in the same direction.
IX. They rotate in the same plane (so far as we can see).
X. In the same direction (so far as we can see).
Kant's nebular hypthoesis. Immanuel Kant was the first to suggest something approaching a rational explanation of this very curious and elegant state of things. He made the error, however, of supposing that rotation of the whole could be produced by collisions of its parts; but no moment of momentum can be caused by the interaction of parts of a system, since internal forces occur in pairs and their moments round any line are equal and opposite. We will consider this in detail a little further on. Laplace, who appears not to have known of Kant's writing, himself some years later developed a somewhat similar theory, but with more mathematical foundation. He assumed an original rotation and got the credit for the nebular hypothesis. He had a faculty of getting credit for things which was only second to his ability.
​Laplace's nebular hypthoesis.To account for so orderly an arrangement Laplace supposed:—
a. That the matter now composing our solar system was once in the form of a nebula.
b. That this original nebula was very hot, a fire-mist.
c. That it possessed initially a slow rotation.
d. That as it contracted under its own gravity and thus, from the principle of conservation of moment of momentum, rotated faster as it shrank, it rotated always like a solid body with the same angular velocity throughout, until its outer portions, which went the fastest, came to go so fast that the centrifugal tendency overcame the centripetal force and they were left behind as a ring.
e. That this ring revolved as a whole until it broke, rolled back upon itself and made a planet; the outer parts of the ring having the swiftest motions, the resulting planet rotated in the same sense that it revolved.
f. The planet thus formed gave birth in like manner to its satellite system.
Physical error in Laplace's hypothesis.The prestige of Laplace gave this explanation hypothesis a mental momentum which has carried conviction nearly to the present day. But it is erroneous for all that, nor can it be made to work by any additions or slight alterations as some text-books will ​tell you. For it was founded on what it has now foundered on: one fundamental mistake. Laplace assumed that his nebula would revolve, as he saw the air around the Earth to revolve, of a piece. But he forgot that friction due to the pressure alone produces this, and that in particles moving freely no pressure exists. Under the pull of a central mass each layer of the nebula would revolve at its own appropriate rate, or as $\scriptstyle r^{-{\frac {1}{2}}}$. So that his beautiful explanation of the agreement in direction of the rotations and the revolutions—the vital point of the theory falls to the ground.
Faye's nebular hypothesis.Faye first definitely pointed out this fatal fallacy in Laplace's hypothesis in 1886, in his "Origine du Monde," in which, after reviewing the previous history of the subject, he brought forward a new theory of his own, both elegant and ingenious.
He begins by assuming a nebulous mass of particles, roughly uniform throughout, but with local condensations. He supposes this nebula cold, for the heat can be trusted to come of itself. With uniform density throughout, the speed of rotation would also be uniform, thus giving the same result that Laplace got, but for a very different reason. In a spherical mass of matter of uniform density, a particle at any point is attracted only ​Force originaly as δr. by the sphere within it. It is therefore pulled by the force $\scriptstyle {\frac {m}{r^{2}}}={\frac {\delta r^{3}}{r^{2}}}=\delta r$, where $\scriptstyle \delta$ is the density.
Since the force is thus linear it may be resolved into two harmonic motions and becomes motion in an ellipse with the acceleration directed to the centre, or elliptic harmonic motion whose equation is expressed in vector coördinates:—
whence: $\scriptstyle {\begin{cases}\rho &=a\cos {(nt+e)}+b\sin {(nt+e)}\\\rho '&=n[asin{(nt+e)}-b\cos {(nt+e)}]\\\rho ''&=n^{2}[a\cos {(nt+e)}+b\sin {(nt+e)}]=-n^{2}p\\\end{cases}}$
The form of the ellipse depends upon the amount and direction of the initial velocity of the particle.
This equation shows, first, that the period of rotation is the same for all the particles; and second, that the angular speed in such different nebulae is as the square root of their densities.
Subsquently as $\scriptstyle {\frac {m}{r^{2}}}$When the mass has practically collected in the centre, the force is $\scriptstyle {\frac {m}{r^{2}}}$, or the ordinary law of gravitation, giving elliptic motion with acceleration directed to the focus, or elliptic motion par excellence.
At any intermediate stage of the process he supposes the force to be represented by $\scriptstyle f=\alpha r+{\frac {\beta }{r^{2}}}$ $\scriptstyle \alpha$ gradually dying out and $\scriptstyle \beta$ increasing as centralization goes on. ​Planets given off under the first state of things would rotate in the same direction in which they revolved; under the last in the opposite way. He, therefore, supposes the terrestrial planets to be the older; the outer planets the younger members of the system. His theory makes the order of birth the exact contrary of Laplace's.
More recently Lieutenant-Colonel R. du Ligondés has evolved another cosmogony. Ligondés's general theory is ingenious, but to me not convincing. His first point is unqualifiedly good. He starts out by calling attention to the evidence offered by the moment of momentum of the solar system upon the early history of that system. He shows that to produce a single star system like ours, the original motions of the several parts of the nebula must have been nearly balanced, the plus motions almost canceling the minus ones.
Moment of momentum.It now becomes of interest for us to consider this question. Conservation of moment of momentum is as fundamental in mechanics as the conservation of energy. The momentum of a body is its mass into its velocity, and the moment of momentum is this mass-velocity multiplied by the perpendicular upon its direction from the point ​or line around which the moment is taken. The moment of momentum is thus twice the area swept out by the moving body about the fixed one in unit time.
When two bodies collide, the amount of motion is not changed. This truth is the result of experiment, and was first determined by Newton. If the two are perfectly inelastic, they move on after the collision as one mass with a loss of kinetic energy. If perfectly elastic, they rebound in such a manner that not only the amount of motion, but the kinetic energy remains unchanged. Now probably no bodies are perfectly inelastic, just as no bodies are perfectly elastic. In the case, therefore, of the bodies in nature, while the amount of motion is never altered, a part of the kinetic energy is lost by the shock. It is transformed into heat energy.
Moment of momentum constant.Now the moment of a velocity, and therefore of a momentum, clearly remains constant when unacted upon by any force, for its direction continues the same, and a perpendicular upon it from any point measures out the same area in the same time, as the perpendicular, too, is constant.
The like is true, if it be acted upon by a force constantly directed to the same point; for in that case the force can generate no velocity except along the perpendicular upon the line which ​sents the body's momentum, and therefore cannot change the area swept out.
When two bodies collide, therefore, they each bring an eternal definite amount of motion to the collision; this amount is unaffected by the shock.
Nor can the mutual attraction of the two bodies themselves alter it; for, since a force is measured by the amount of velocity it can generate in a given time, the velocities generated must be as the opposite masses, and therefore the momentum produced in each be the same.
Let $\scriptstyle m$ and $\scriptstyle m_{1}$ be the masses.
$\scriptstyle f_{m}t={m_{1}}{v_{m_{1}}}$
$\scriptstyle f_{m_{1}}t=mv_{m_{1}}$
$\scriptstyle {\frac {v_{m}}{v_{m_{1}}}}={\frac {m}{m_{1}}}$
$\scriptstyle m_{1}v_{m}=mv_{m_{1}}$
$\scriptstyle Aa=-Bb$
$\scriptstyle Aa={\frac {v_{m}}{m}}$ and $\scriptstyle Bb={\frac {v_{m_{1}}}{m_{1}}}$
* FIG. XVII
​Moreover, it is directed in both cases along the same line. Whence its moment in the two cases about any point is the same in amount, the perpendicular from the point being common to both, but opposite in direction. The two moments thus destroy one another. From which we see that the internal forces of a system are unable to change the moment of momentum of the system. Similarly they are incapable of having created it to begin with.
The present moment of momentum of the solar system can be calculated. It is found to be nearly the least possible. It must, therefore, always have been so. It was predestined by internal motions to make a single star.
So far, he is admirable, but from this point I lose him; I cannot see the cogency of all his succeeding steps. They lead him to the conclusion that everything is as it should be, and incidentally that Jupiter or Neptune is the oldest planet, Uranus the next, then Saturn, Mars, the Earth, Venus, and Mercury. The importance of the order will appear shortly.
Trowbridge's explanation of direct and retrograde motion.With regard to the retrograde rotations of the outer planets and the direct rotations of the inner ones, Trowbridge suggested that uniform density, or a density increasing toward the centre, would ​account for it. Suppose, first, the density uniform, or nearly so. Then the inner parts of the mass that went to form the planet would be traveling fastest, and their momentum would prevail over that of the outer particles and give a retrograde rotation to the whole. Suppose, however, that the density increased toward the inner side of the mass. Then the centre of inertia would be so far shifted toward the inner edge, say to N, that the sum of the moments about it of the particles from without would, owing to their distance from it, surpass that of those within and a direct rotation result.
Laws of force graphically shown.The attraction, and thence the velocities in the Laws of force different parts of the nebula, may be well shown graphically.
Faye's laws of attraction in condensing nebula.
Fig. XVIII.
​Faye's equation holds only when α and β are functions of r as well as of t. It, therefore, fails to give a good representation of what occurs throughout at a given moment. Furthermore, the equations do not hold up to the axis of y, as a discontinuity occurs so soon as we enter the central mass.
A better picture is the following, somewhat changed from Ligondés. As the matter gets drawn into the central mass, the attraction at the outer parts of the original nebula grows less and less, therefore C sinks to F, and the successive curves of the attraction become OC, DD, EE, FF.
* Fig XIX
The velocities at different distances follow a similar law.
This shows, as Ligondés points out, that there ​Effect on rotation.is a maximum velocity somewhere in the centre of the nebula, which degrades on both sides, so that we should have a plan of velocities for outside and inside portions of the nebula, thus :—
* Fig. XX
Supposing the density either the same throughout or to increase toward the centre, we should have, if the various planets were formed simultaneously, a retrograde rotation for the outer, a direct rotation for the inner ones.
New congruities since Laplace.In addition to the ten congruities known in the time of Laplace, we must now add others from knowledge acquired since, to wit :—
XI. All the satellites turn the same face to their primaries (so far as we can judge).
XII. Mercury and probably Venus do the same to the Sun.
XIII. One law governs position and size in the solar system, and in all the satellite systems.
​XIV. Orbital inclinations in the satellite systems increase with distance from the primary.
XV. The outer planets show a greater tilt of axis to orbit-plane with increased distance from the Sun (so far as detectable).
XVI. The inner planets show a similar relation.
Tidal friction in partial explanation.Tidal friction explains xi. and xii. ; xiii., xiv., xv.,explanation. and xvi. are as yet unexplained.
Tidal friction fails with axial inclinations.Tidal friction would account for xiv., but only on the supposition that the outer satellites were given off first. This is contrary to Faye's theory, largely so to Ligondés's, and is not championed by any other, for Laplace's supposition with regard to this point cannot stand.
Not only must the outer satellite have been given off the first, but very long before the next inner one, and so on for all; for tidal friction is potent as the inverse sixth power of the distance.
A similar objection holds against the attempt to explain the increased tilt of rotation axis to orbit planes, as distance from the Sun increases both for the outer and the inner planets. This increased tilt with increased distance is well worth particular notice. It may be seen in the following table. ​
| Planet. | Inclination of Equator to Orbit-plane. | Inclination of Equator to Orbit-plane. |
| Neptune | 145 | °(?) |
| Uranus | 98 | °(?) |
| Saturn | 27 | ° |
| Jupiter | 3 | ° |
| Mars | 25 | ° |
| Earth | 23 | ½° |
| Venus | 0 | °(?) |
| Mercury | 0 | ° |
We cannot be certain of Uranus and Neptune because we cannot see their surfaces well enough to be sure of the position of their axes, but the planes in which their satellites revolve makes the value given altogether likely.
The tidal friction explanation of this would make Neptune very much the oldest planet, Uranus very much the next so, and so on. But the explanation is not satisfactory.
Small amount of momentum of solar systemOur solar system has, as I have said, a very small relative moment of momentum; only the one thousandth part of what it might have as exemplified in the system of α Centauri.
Explicable by collision of two suns.One supposition will account for the small moment of momentum of the system, without supposing the individual motions so nearly balanced at the start. The moment of momentum would be small if the principal mass were initially collected in the centre of the nebula. Now this ​would be the case if the present system had been formed by the collision of two bodies. For, when dealing with such masses, the elasticity may be considered small, and, in default of elasticity, the matter after the collision would be found chiefly near the scene of the catastrophe if the impact were in the line joining their centres. The collision in space of two bodies happening head on is, of course, one of which the chances are very small, and, were it not for another fact, might be dismissed from reasonable consideration.
Physical condition of meteroites sustains this idea.This fact is the present constitution of the unattached particles of the system, the meteorites. As we saw in a preceding lecture, these fragments betray a previous habitat. Their character shows that they came from the interior of a great cooled mass which once had been intensely heated. They are therefore proof of the prior existence of a great sun, and that they should be now strewn in space makes the theory of a subsequent collision far less improbable.
Distribution of density after collisionIf such a collision occurred, the fragments would be scattered more sparsely according to their distance from the scene of the catastrophe, and we may perhaps assume the law governing this sparseness to be the curve of probability,
$y={\frac {h}{\sqrt {\pi }}}e^{-h^{2}x{2}}.$
$x$
$x+dx$
$\textstyle {\frac {h}{\sqrt {\pi }}}e^{h^{2}x^{2}}dx$
$x$
$y$
$y$
$x$
$y$
$y$
$dy$
$\textstyle {\frac {h}{\sqrt {\pi }}}e^{h^{2}y^{2}}dy$
The probable amount, therefore, in the rectangle $dx$ $dy$ is $\textstyle {\frac {h^{2}}{\pi }}e^{h^{2}(x^{2}+y{2})}dxdy={\frac {h^{2}}{\pi }}e^{-h^{2}r^{2}}a$ where $a$ $dx$ $dy$, and $r$ denotes its distance from the origin, or, in this case, the centre of the Sun.
For the amount in a ring at distance r, we have $a=rdr$.
Effect on rotation.Consequently it is evident that there is less relative variation in the density with the distance as one goes out. A fortiori, therefore, when the planetary masses do not increase in like proportion, the two ends, the outer and the inner, of the strip or bunch of matter that went to make each up, vary less in density inter se. In the resultant rotation, the speed of the separate particles counts for more, relatively, than their density, and, in consequence, for the outer planets we should get a retrograde rotation; for the inner, a direct one.
Inner planets later formed.That the inner planets were not formed early in the system's development seems pointed at pretty ​conclusively by their several masses. Present mechanical conditions of the matter inside Jupiter's orbit appear to point to the pre-existent influence of Jupiter upon it before birth. Not only do the amounts of matter in the several terrestrial planets indicate this, but the lack of formation of a planet in the gap occupied by the asteroids seems well-nigh conclusive on the point.
Shown by axial rotation A glance at the axial inclinations of the outer shown by and the inner planets betrays a break in the symmetry of their arrangement. Each, taken by itself, evinces a gradual righting of the axis as one approaches the Sun. This appears strikingly from the table of the inclinations of the equators of the several planets to the planes of their orbits.
Jupiter's action the causeThis, too, seems to point to the action of Jupiter. On the whole it appears probable that Jupiter existed before any of the small planets within its orbit, and profoundly modified them prenatally.
Conclusion. We thus come to a conclusion in which nothing is concluded : but we need not regret that. The subject becomes the more exciting for remaining yet a mystery. We now know of relations so systematic and singular that we are sure some law underlies them, and it is rather pleasant than otherwise to have that law baffle our first attempts at discovery. ​Future of the systemBut though we cannot as yet review with the mind's eye our past, we can, to an extent, foresee our future. We can with scientific confidence look forward to a time when each of the bodies composing the solar system shall turn an unchanging face in perpetuity to the Sun. Each will then have reached the end of its evolution, set in the unchanging stare of death.
Then the Sun itself will go out, becoming a cold and lifeless mass; and the solar system will circle unseen, ghostlike, in space, awaiting only the resurrection of another cosmic catastrophe.