Page:The Solar System - Six Lectures - Lowell.djvu/126

108

The Solar System

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The critical angle.If however, its entering speed and approaching angle, which we will call $\scriptstyle \omega$, are such that $\scriptstyle \cos {\omega }<{\frac {1}{2}}.{\frac {v}{v_{1}}},$ where $\scriptstyle v$ is its actual velocity,$\scriptstyle v_{1}$ that of the planet ; then its relative velocity ,$\scriptstyle v_{o}$, can never be greater than $\scriptstyle v_{1}$ and the resulting orbit never can become retrograde. This angle we will call the critical angle, and designate it by the symbol $\scriptstyle \chi$.

Now $\scriptstyle \omega$ we can calculate for each of the comets of Jupiter's family from their known present paths. Furthermore, since Jupiter's only effect is to swing the outgoing asymptote of the relative orbit round $\scriptstyle v_{0}$ can never be changed, and the future possible values of $\scriptstyle \omega$ have a superior limit $\scriptstyle \omega '$, which they can never pass. This also we can calculate. Doing this, and calculating also the value of $\scriptstyle \chi$ for each comet, we find the table on the opposite page.

ω and ω' both less than χ in all comets of Jupiter's family.

From the table, it appears that in every one of comets of Jupiter's family, $\scriptstyle \omega$ is within the critcal angle.

Furthermore, that $\scriptstyle \omega '$, the maximum value which $\scriptstyle \omega$ may attain under the perturbative effect of the planet, owing to the swing of the asymptotes of the hyperbolic relative orbit of the planet, is also always within $\scriptstyle \chi$