Page:Practical astronomy (1902, John Wiley & Sons).djvu/20

In Mechanics^[1] it was shown that the Earth's undisturbed orbit is an ellipse, having one of its foci at the sun's center, and that the earth's angular velocity is

(551)

${\frac {d\theta ^{\prime }}{dt}}={\frac {h}{r^{2}}}$;

its radius vector

(610)

$r={\frac {a\left(1-e^{2}\right)}{1+e\cos \theta }}$;

its constant double sectoral area,

(615)

$h={\sqrt {\mu ^{\prime }a\left(1-e^{2}\right)}}$;

and its periodic time,

(616)

$\tau =2\pi {\sqrt {\frac {a^{3}}{\mu ^{\prime }}}}={\frac {2\pi }{n}}$.

In these expressions $\theta ^{\prime }$ is the angle made by the earth's radius vector with any assumed right line drawn through the sun's center, $\theta$ that included between the radius vector and the line of apsides estimated from perihelion, and $n$ is the mean motion of the earth in its orbit.

From (551), (615) and (616), we have

${\frac {d\theta ^{\prime }}{dt}}={\frac {d\theta }{dt}}={\frac {{\sqrt {\mu ^{\prime }a\left(1-e^{2}\right)}}(1+e\cos \theta )^{2}}{a^{2}\left(1-e^{2}\right)^{2}}}$

(1)

$={\sqrt {\frac {\mu ^{\prime }}{a^{3}}}}{\frac {(1+e\cos \theta )^{2}}{\left(1-e^{2}\right)^{\frac {3}{2}}}}=n{\frac {(1+e\cos \theta )^{2}}{\left(1-e^{2}\right)^{\frac {3}{2}}}}$:

and therefore

(2)

$ndt=\left(1-e^{2}\right)^{\frac {3}{2}}(1+e\cos \theta )^{-2}d\theta ^{\prime }$.

Since $e$ varies but little from $0.01678$ (see Art. 185, Young^[2]), we may omit all terms containing the third and higher powers of $e$ in the development of the second member of the preceding equation.

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1. Michie's Mechanics, 4th Edition.
2. Young's General Astronomy.