Kepler's Music: Pitch Ranges of the Planets
Each planet has a warbling pitch, varying according
to the solar angular velocity of the planet, nu-dot, as
the planet moves around its elliptical orbit.
Pitch varies according to the 2nd law
According to the 2nd law, nu-dot
varies inversely to the square of the distance
from the sun, rho:
[4] nu-dot = K/rho2
The constant of proportionality, K,
depends on the units chosen. And for Kepler, the
units of choice were arc-minutes per mean-earth-day,
or more simply, minutes per day,
[4 mean] K1 = 21600/P minutes per day
where P is the period of the planet in mean-earth-days.
Pitch ratios
The highest pitch for a planet occurs, therefore,
when the motion is most rapid, and hence when
rho is at its minimum, rho-min, at perihelion.
Similary, the lowest pitch will occur when the
motion is slowest, and hence when
rho is at its maximum, rho-max, at aphelion.
The ratio of the highest pitch to the lowest is thus,
high / low = {K/rho-min2} / {K/rho-max2}
= rho-max2 / rho-min2
= (rho-max / rho-min)2
But rho-max = 1+e, and rho-min = 1-e, for a planet
with orbit eccentricity, e.
So the expression above becomes,
high / low = = (1+e / 1-e)2
depending on eccentricity only.
And so we may easily construct this table
of high-to-low pitch ratios, taking modern
values of e again from (Moore/Hunt, 1983; p. 440).
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