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