Page:Practical astronomy (1902, John Wiley & Sons).djvu/21

Then after substituting ${\frac {1+\cos 2\theta }{2}}$ for $\cos ^{2}\theta$, we have

(3)

$ndt=d\theta ^{\prime }-2e\cos \theta d\theta +{\tfrac {3}{4}}e^{2}\cos 2\theta d(2\theta )+{\text{etc.}}$

Integrating we have

(4)

$nt+C=\theta ^{\prime }-2e\sin \theta +{\tfrac {3}{4}}e^{2}\sin 2\theta +{\text{etc.}}$

The earth's orbit is, however, not entirely undisturbed. Due to the perturbating action of other bodies of the solar system the earth is never exactly in the place which it would occupy in an undisturbed orbit. Moreover the line of apsides has a direct motion, i.e., in the direction in which longitudes are measured, of about $11^{\prime \prime }.7$ per annum, and the vernal equinox an irregular retrograde motion whose mean value is about $50^{\prime \prime }.2$ per annum.

Therefore (Fig. 1), let the line from which $\theta ^{\prime }$ is estimated be that drawn through the sun and the position of the mean vernal equinox $V$ at some fixed instant, called the epoch. Then when $\theta$ is zero, $\theta ^{\prime }$ will be the longitude of perihelion, estimated from this point. Let this be denoted by $l_{p}$, and the time of perihelion passage by $t_{p}$; then from (4) we have,

(5)

$nt_{p}+C=l_{p}$.