Caption: An explication of synodic period.
Features:
For variant definition of synodic period that amounts to much the same thing, see Wikipedia: Orbital period.
Below, we derive the needed formulae
for a useful simplified special case:
i.e., the
specialized formulae
for the synodic period.
Derivation:
where the 3rd line is the formula for
the relative
rotation rate
formula
and the
last line is the
synodic period
formula.
(see Wikipedia:
Geometric series: Closed-form formula).
which is only 0.011 s too small compared to the accepted value given above.
where we have used the
mean sideral month = 27.321661547 days (J2000)
and
sidereal year = 365.256363004 days (J2000).
The calculated value agrees with the accepted
mean lunar month = 29.530588853 days (J2000)
to 11 digit places which is all
the significant figures the accepted value has.
where the solutions for b and c follow by permuting the variables
and where we have
used the geometric series
to expand in terms of small b/c (i.e., b/c << 1)
(see Wikipedia:
Geometric series: Closed-form formula).
To be explicit,
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360° + θ_0 = (R_1 - R_2)*t + θ_0
360° = (R_1 - R_2)*t
R = 360°/t = (R_1 - R_2)
360°/t = (360°/t_1 - 360°/t_2)
1/t = 1/t_1 - 1/t_2
t = t_1*t_2/(t_2 - t_1) ,
t = t_1/(1 - t_1/t_2) = t_1*(1 + t_1/t_2 + ...) ,
= t_1*(1 + t_1/t_2) to 1st order
= t_1*(1 + t_1/t_2) to zeroth order
t = t_1*t_2/(t_2 - t_1) = 86399.9908 s
t = t_1*t_2/(t_2-t_1) = 27.321661547*365.256363004/(365.256363004-27.321661547)
= 29.530588853 days ,
0 = 1/a + 1/b + 1/c which has solution
a = -bc/(b + c) = -b/(1 + b/c) = to 1st order -b*(1-b/c) = to 0th order -b ,
t = -(-t_1)*t_2/[t_2+(-t_1)] .
(-t_1) = -t_2*t/[t+t_2] and t_2 = -t*(-t_1)/[(-t_1)+t] or
t_1 = t_2*t/[t+t_2] and t_2 = t*t_1/[t-t_1] .