If the space of the solar system be equally filled with meteors throughout, or if they diminish as one goes out from the Sun according to any rational law, their average speed of encounter with the Earth would be nearly parabolic.

If they were travelling in orbits like those of the short-period comets, that is with their aphelia at Jupiter’s orbit and their perihelia at or within the Earth’s, their major axes would lie between 6.2 and 5.2. If we suppose their perihelion distances to be equally distributed according to distance, we have for the mean a major axis of 5.7. Their velocity, then, at the point where they cross the Earth’s track would be given by

v² = µ(2/1 - 1/2.85),

in which µ = 18.5² in miles per second = 342.25,
whence v = 23.76 in miles per second.

Suppose them to be approaching the Earth indifferently from all directions.

At sunset the zenith faces the Earth’s quit; at sunrise the Earth’s goal. Let ? be the real angle of the meteor’s approach reckoned from the Earth’s quit; ?1 the apparent angle due to compounding the meteor’s velocity-direction with that of the Earth. Then those approaching it at any angle 0 less than that which makes ?1 = 90° will be visible at sunset; those at a greater angle, at sunrise. The angle 01 is given by the relation,

		a
cos ?1	=	+ —— ,
		x

in which a is the Earth’s velocity, x the meteor’s, and ?1 is reckoned from the Earth’s quit.

The portion of the celestial dome covered at sunset is, therefore,

(	?1	(	360°
?		?		sin ?·d?·df,
)	0	)	0

where f is the azimuth,

that at sunrise,

(	180°	(	360°
?		?		sin ?·d?·df.
)	?1	)	0

If the meteors have direct motion only, ? can never exceed 90°, and the limits become,

for sunset,

(	?1	(	360°
?		?		sin ?·d?·df,
)	0	)	0

and for sunrise,

(	90°	(	360°
?		?		sin ?·d?·df.
)	?1	)	0

The mean inclination at sunset is

(	?1	(	360°
?		?		?1 · sin ?·d?·df,
)	0	)	0

???? ,

(	?1	(	360°
?		?		sin ?·d?·df,
)	0	)	0

in which ?1 must be expressed in terms of ?, etc.

From this it appears that the relative number of bodies, travelling in all directions and at parabolic speed, which the Earth would encounter at sunrise and sunset respectively would be:—

sunrise	5.8
sunset	1.0

and with the speed of the short-period comets,

sunrise	8.0
sunset	1.0

If, however, the bodies were all moving in the same sense as the Earth, i.e. direct, the ratios would be:—

	Parabolic Speed	Speed of Short Period Comets	Speed of Actual Short-Period Period Comets about Jupiter
Sunrise	2.4	3.5	3.3
Sunset	1.0	1.0	1.0

As the actual number encountered is between 2 and 3 to 1, we see that the greater part must be travelling in the same sense as the Earth, since they come indifferently at all altitudes from the plane of her orbit.

The densities of the principal planets, so far as we can determine them at present, the density of water being unity, are:—

Mercury	3.65
Venus	5.36
Earth	5.53
Moon	3.32
Mars	3.93
	———	mean 4.36
Jupiter	1.33
Saturn	0.72
Uranus	1.22
Neptune	1.11
	———	mean 1.09
Sun	1.38

The second decimal place is not to be considered as anything but an indication.

In the case of a body reflecting light, the shift differs from that for a body emitting it. If the planet be on the further side of the Sun, the approaching rim advances both toward the Sun and toward the Earth, thus doubling the shift. The receding rim recedes in like manner. At elongation the rims approach or recede with regard to the Earth, but not the Sun, and the shift is single as for emission. At inferior conjunction rotational approach to the Earth implies rotational recession from the Sun, and the two effects cancel.

The tilts of the plane of rotation of the Sun and of the orbits of the several planets to the dynamical plane of the system tabulated are:—

Sun	7°
Mercury	6° 14'
Venus	2° 4'
Earth	1° 41'
Mars	1° 38'
Asteroids	various
Jupiter	20'
Saturn	56'
Uranus	1° 2'
Neptune	43'

where, in the determination of that plane, the latest values of the masses of the planets and the rotations of the Sun, Jupiter, and Saturn have been taken into account.

These tilts suggest something, doubtless, but it is by no means clear what it is they suggest. They are just as compatible with a giving off from a slowly condensing nebula as with an origin by shock. The greater inclinations of Mercury and Venus may be due to their late birth from the central mass without the necessity of a cataclysm, the rotation of that central mass out of the general plane being caused by the consensus of the motions of the particles from which it was formed. The accordance of the larger planetary masses with the dynamical plane of the system would necessarily result from their great aggregations. So that this, too, is quite possible without shock.

If we compute the speeds of satellites about their primaries in the solar system and compare them with the velocities in their orbits of the planets themselves, a striking parallelism stands displayed between the several systems. This is shown in the following table of them:

		Mean Speed, Miles a Second		Parabolic Speed at Orbit	Ratio Speed Sat. about Primary to Planet’s Speed in Orbit
		of Primary in Orbit V	of Satellite about Primary v	Miles a second
Jupiter		8.1		11.5
Sat. 1			10.7		1.32
2			8.5		1.05
3			6.7		0.83
4			5.1		0.63
Saturn		6.0		8.5
1			9.0		1.50
2			7.9		1.31
3			8.2		1.36
4			8.2		1.36
5			5.3		0.89
6			3.5		0.59
8			2.0		0.34
Uranus		4.2		5.9
1			3.5		0.82
2			2.9		0.70
3			2.3		0.54
4			2.0		0.47
Neptune		3.4		4.8
1			2.7		0.81

The relations here disclosed are too systematic to be the result of chance.

The orbits of all these satellites have no perceptible eccentricity independent of perturbation except Iapetus, of which the eccentricity is about .03.

In view of the various cosmogonies which have been advanced for the genesis of the solar system it is interesting to note what these speeds imply as to the effect upon the satellites of the impact of particles circulating in the interplanetary spaces at the time the system evolved. To simplify the question we shall suppose—which is sufficiently near the truth—that the planets move in circles, the interplanetary particles in orbits of any eccentricity.

Taking the Sun’s mass as unity, the distance R of any given planet from the Sun also as unity, let the planet’s mass be represented by M and the radius of its satellite’s orbit, supposed circular, as r. We have for the space velocity of the satellite on the sunward side of the planet, calling that of the planet in its orbit V and that of the satellite in its orbit round the planet v,

V - v =	v	1/R	-	v	M/r.

For a particle, the semi-major axis of whose orbit is a1 and which shall encounter the satellite, the velocity is

v1 = (2/(R-r) - 1/a1)^½.

That no effect shall be produced by the impact of these two bodies, their velocities must be equal, or

v	1/R	-	v	M/r.	=	v	2/(R-r) - 1/a1

As R-r = a1(1 + e) for the point of impact if the particle be wholly within the orbit of the planet and e the eccentricity of its orbit, we find

e =	2	v	MR/r	- RM/r	approx.

for the case of no action, the other terms being insensible for the satellites in the table, since in all r< R/400.

Supposing, now, the particles within the orbit of the planet to be equally distributed according to their major axes, then as the velocity of any one of them, taking R-r = R approx. as unity, is

v1 = (2/1 - 1/a1)^½,

the mean velocity of all of those which may encounter the satellite is, at the point of collision,

(¹
?	((2a1 - 1)½ / a1½) da1
)½

?????

(¹
?	da1
)½

1
= 2[	(2a1² - a1)½ - 1/v(2) log{(2a1 - 1)½ + v(2a1)}	]
½
= 0.754;

that is, just over three-quarters of the planet’s speed in its orbit.

If we suppose the particles to be equally distributed in space, we shall have more with a given major axis in proportion to that axis, and our integral will become

(¹
?	(2a1 - 1)½ a1½ da1
)½

?????

(¹
?	a1 da1
)½

1
= 8/3[	(4a1-1)/8 (2a1² - a1)½ - (1/16v2 ) log[(2a1² - a1)½ + v2 · a1 - 1/2v2]	]
½
= 0.792 of the planet’s orbital speed.

The speed v, then, at which a satellite must be moving round the planet to have the same velocity as the average particle within the planet’s orbit, is

V - v1 = v.

This velocity is, for the several planets:—

	Distribution of Particles as their Major Axes	Distribution of Particles Equal in Space
	Miles a second	Miles a second
Jupiter	2.0	1.6
Saturn	1.5	1.2
Uranus	1.0	0.9
Neptune	0.8	0.7

If the satellite be moving in its orbit less fast than this, its space-speed will exceed that of the average particle; it will strike the particle at its own rear and be accelerated by the collision. If faster, the particle will strike it in front and retard it in its motion round its primary.

From the table it appears that all the large satellites of all the planets have an orbital speed round their primaries exceeding those in either column. In consequence, all of them must have been retarded during their formation by the impact of interplanetary particles and forced nearer their primaries than would otherwise have been the case; and this whether the particles were distributed more densely toward the Sun, as 1/a1, or were equally strewn throughout.

For interplanetary particles whose orbits lie without the particular planet’s path the mean speed is the parabolic at the planet’s distance, given in the third column of the table. This is the case on either supposition of distribution. The orbital speed of the satellite which shall not be affected by collisions with them is, for the several planets:—

	Miles a second
Jupiter	3.4
Saturn	2.5
Uranus	1.7
Neptune	1.4

All the satellites but Iapetus have orbital speeds exceeding this, and consequently are retarded also by these particles.

For particles crossing the orbit (2) the mean velocity would be practically parabolic, 1.4, even if the distribution were as 1/r', r' being the distance from the Sun. The effect would depend upon the angle of approach and in the mean give a greater velocity for the particle than for the satellite within the orbit, a less one without; retarding the satellite in both cases. Thus the total effect of all the particles encountering the large satellites is to retard them and to tend to make them hug their primary.

For retrograde satellites the velocities of impact with inside and outside particles moving direct are respectively:

	Inside	Outside
Jupiter	2.0 + v	v + 3.4
Saturn	1.5 + v	v + 2.5
Uranus	1.0 + v	v + 1.7
Neptune	0.8 + v	v + 1.4

In both cases the impact tends to check the satellite.

Comparing with these the velocities of impact for direct satellites in a direct plenum:—

	Inside	Outside
Jupiter	2.0 - v	3.4 - v
Saturn	1.5 - v	2.5 - v
Uranus	1.0 - v	1.7 - v
Neptune	0.8 - v	1.4 - v

the signs being taken positive when the motion is direct, we see that retrograde satellites would be more arrested than direct ones with the same orbital speed round the primary.

In a plenum of direct moving particles, then, the force tending to stop the satellite and bring it down upon the planet is greater for retrograde satellites than for direct ones.

If, therefore, the positions of the satellites have been controlled by the impact of interplanetary particles, the retrograde satellites should be found nearer their planets than the direct ones.

Since the moment of momentum is the velocity into the perpendicular upon its direction, in the time dt it is:—

vp dt = h dt = r²dT.

The whole moment of momentum from perihelion to perihelion is therefore:—

(360°
?	r²dT = a²·(1-e²)² / 1-e²
)0

360°
[	-e sin T	+	2	tan?¹	(	v	1-e	·tan	T	)	]
	1+e cos T		(1-e²)½				1+e		2
0

= 2pa² · (1 - e²)^½,

which is twice the area of the ellipse.

The energy in the ellipse during an interval dt is

1	mv²dt=	1	mµ	(	2	-	1	)	dt,
2		2			r		a

from the well-known equation for the velocity in a focal conic. The integral of this for the whole ellipse is

(T	1	mv²dt=	(360°	1		mµ	(	2r-	r²	)	dT
)0	2		)0	2		h			a

= mµ^½pa^½.

Since

(

rdT=

(

a·1-e²

dT=

2a·1-e²

tan?¹

(

v

1-e

tan

T

)

)

)

1+ecosT

(1-e²)½

1+e

2

and ?r²dT is given above.

By collision a part of this energy is lost, being converted into heat. The major axis, a, is, therefore, shortened. But from the expression 2pa²·(1-e²)^½ for the moment of momentum we see that this is greatest when e is least. If, therefore, a is diminished, e must also be diminished, or the moment of momentum would be lessened, which is impossible.

See has recently shown (Astr. Nach. No. 4341-42) that a particle moving through a resisting medium under the attraction of two bodies revolving round one another in circles may eventually be captured by one of them though originally under the domination of both. The argument consists in introducing the effect of a resisting medium upon the motion in the space permitted by Jacobi’s integral, following Darwin’s examination of this space. In the actual case of nature the effect is much more complicated, and at present is not capable of exact solution for masses other than indefinitely small, even supposing circular orbits for the chief bodies. It may, however, explain the curious relation shown in the arrangement of the direct and retrograde movement of satellites.

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