The Evolution of Worlds/Notes 1

by Percival Lowell

NOTES

I

Meteor Orbits

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

${\begin{aligned}&&v^{2}&=\mu \left({\frac {2}{1}}-{\frac {1}{2.85}}\right){\text{,}}\\&{\text{in which}}&\mu &=18.5^{2}{\text{ in miles per second}}\\&&&=342.25{\text{,}}\\&{\text{whence}}&v&=23.76{\text{ in miles per second.}}\end{aligned}}$

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 $\theta$ be the real angle of the meteor's approach reckoned from the Earth's quit; $\theta _{1}$ the apparent angle due to compounding the meteor's velocity-direction

with that of the Earth. Then those approaching it at any angle $\theta$ less than that which makes $\theta _{1}=90^{\circ }$ will be visible at sunset ; those at a greater angle, at sunrise. The angle $\theta _{1}$is given by the relation,

$\cos {\theta _{1}}=+{\frac {a}{x}}$,

in which $a$ is the Earth's velocity, $x$ the meteor's, and $\theta _{1}$ is reckoned from the Earth's quit.

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

$\int _{0}^{\theta _{1}}\int _{0}^{360^{\circ }}\sin {\theta }\cdot d\theta \cdot d\phi$,

where $\phi$ is the azimuth,

that at sunrise, $\int _{0}^{180^{\circ }}\int _{0}^{360^{\circ }}\sin {\theta }\cdot d\theta \cdot d\phi$.

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

for sunset, $\int _{0}^{\theta _{1}}\int _{0}^{360^{\circ }}\sin {\theta }\cdot d\theta \cdot d\phi$,

and for sunrise, $\int _{0}^{90^{\circ }}\int _{0}^{360^{\circ }}\sin {\theta }\cdot d\theta \cdot d\phi$.

The mean inclination at sunset is

${\frac {\int _{0}^{\theta _{1}}\int _{0}^{360^{\circ }}\theta _{1}\cdot \sin {\theta }\cdot d\theta \cdot d\phi }{\int _{0}^{\theta _{1}}\int _{0}^{360^{\circ }}\sin {\theta }\cdot d\theta \cdot d\phi }}$,

in which $\theta _{1}$ must be expressed in terms of $\theta$, 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 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.