Page:The Solar System - Six Lectures - Lowell.djvu/50

32

The Solar System

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Libration in longitude is a necessaryCause of libration in longitude. consequence of the planet's moving in a focal conic. The moment of rotation of a body of Mercury's mass is so great that it would take more than the Sun's might to suddenly alter it. The planet turns upon its axis, therefore, with a uniform spin. But its angular speed in its orbit is not uniform. Since the radius vector sweeps out equal areas in equal times, the angular velocity near perihelion exceeds that near aphelion. The revolution gains on the rotation here, and at the end of a certain time reaches its maximum ; after which the rotation gains on the revolution, and the deficiency is made up again at aphelion.

Maximum point of libration in longitude.

To determine what the maximum is, and where, wehave : that the mean angular velocity of revolution in the ellipse is the angular velocity of a body supposed to be describing a circle in the time occupied by the planet in the ellipse. The area of the ellipse being $\pi ab$, and the period T, the areal velocity in the ellipse, which is constant, is

$\scriptstyle {\frac {\pi ab}{T}}\,\!$

This is the areal velocity in a circle of radius $\scriptstyle {\sqrt {ab}}\,\!$ supposed described in the same time.

To find, therefore, the point on the ellipse where the radius has the value corresponding to the mean angular velocity, we must take the expression for r of the ellipse referred to its focus as a pole,