Page:DeSitterGravitation.djvu/15

The equation (28) can then also be written in the form given by Wacker (l.c., page 55),

$\mathrm {E} ={\frac {k^{2}\mathrm {M} }{r}}+{\text{const.}},$

where

$\mathrm {E} ={\frac {1}{\sqrt {1-\phi ^{2}}}}-1={\tfrac {1}{2}}{\bigg [}\left({\frac {dx}{dt}}\right)^{2}+\left({\frac {dy}{dt}}\right)^{2}{\bigg ]}\left(1+{\tfrac {3}{4}}\phi ^{2}+\dots \right).$

9. The equations (24) of course give, taking the orbital plane as plane of ($x,y$),—

$x=a(\cos \ u-e),$ $y=a{\sqrt {1-e^{2}}}\sin \ u,$

$r=a(1-e\ \cos \ u),$

(30)

where

$u-e\ \sin \ u=n\tau ,\quad a^{3}n^{2}=k^{2}\mathrm {M} .$

(31)

To introduce heliocentric time we have

${\frac {dt}{d\tau }}={\sqrt {1+(\phi )^{2}}}=1+{\tfrac {1}{2}}{\frac {\lambda ^{2}}{a}}{\frac {1+e\ \cos \ u}{1-e\ \cos \ u}},$

$d\tau ={\frac {1-e\ \cos \ u}{n}}du,$

therefore

$ndt=\left(1+{\tfrac {1}{2}}{\frac {\lambda ^{2}}{a}}\right){\bigg (}1-{\frac {1-{\frac {1}{2}}{\frac {\lambda ^{2}}{a}}}{1+{\frac {1}{2}}{\frac {\lambda ^{2}}{a}}}}e\ \cos \ u{\bigg )}du$

or if we put

${\begin{aligned}n'&=n\left(1-{\tfrac {1}{2}}{\frac {\lambda ^{2}}{a}}\right)={\frac {k^{2}\mathrm {M} }{a^{3/2}}}\left(1-{\tfrac {1}{2}}{\frac {\lambda ^{2}}{a}}\right),\\e'&=e\left(1-{\frac {\lambda ^{2}}{a}}\right),\end{aligned}}$

(32)

we have

$u-e'\ \sin \ u=n't.$

(33)

Consequently for the law I. the coordinates of a planet of infinitesimal mass are expressed by the ordinary formulæ (30) of elliptical motion, but to express the excentric anomaly in heliocentric time we must in the equation of Kepler (33) use a slightly different excentricity, and also Kepler’s third law is no longer quite exact. The deviations from pure Keplerian motion are periodic and have $\lambda ^{2}$ as a factor. Taking the semi-major axis of the Earth’s orbit as unit of length, the value of $\lambda ^{2}$ is

$\lambda ^{2}=9.85\times 10^{-9}.$

All these periodic terms are thus entirely insensible. The length of the year appears to be different when measured in