- A synodic period
is a period of a rotation
(either orbital revolution or
axial rotation about a fixed axis)
relative to another rotation.
For variant definition of synodic period that amounts to much the same thing, see Wikipedia: Orbital period.

- For example, the period between
oppositions/conjunctions
(as seen from Earth)
is the synodic period
for a
planet
since it is the rotation period of
the planet's
orbital revolution
relative to the
Earth's orbital rotation.
- The term
synodic period
is usually
**NOT**used for rotations relative to inertial frames and when**NOT**, the rotation can be classified a geometrical rotation rather than a physical rotation since that is relative to an inertial frame. - As examples of
synodic periods,
see the animation
in the figure below
(local link /
general link: prograde_retrograde.html).
Each
moon
in the animation
has a set of
synodic periods
for its rotations relative to the other
moons and
the planet's
axial rotation.
- Synodic periods are
important because they are often direct observables when
physical rotation periods
are
**NOT**direct observables. This importance was particularly important in astronomy history in establishing the orbital periods of the planets. - One needs formulae to convert back
and forth between
synodic periods
and basic rotation periods
(e.g., those relative to
inertial frames).
Below, we derive the needed formulae for a useful simplified special case: i.e., the specialized formulae for the synodic period.

Derivation:

- We make the simplyfing assumption that
all the basic
rotations
are
uniform circular motions
in a common plane.
This assumption can be dispensed with, but that leads one to
formulae that are too complex
for educational purposes.
- For definiteness, we consider
counterclockwise/clockwise
rotations
as prograde/retrograde
rotations.
The prograde/retrograde
rotations
have positive/negative
rotation rates
and
rotation periods.
- Consider the case of two basic
rotations
around a common center.
This situation corresponds to, for example:
- Two planets orbiting a star (with uniform circular motions).
- A planet in axial rotation geometrically orbited by a star.
- A planet geometrically orbited by a star and (physically) orbited by a moon.

- Let the first
rotation rate
be R_1 and the second, R_2.
Let the corresponding
rotation periods be t_1 and t_2.
Without loss of generality, we set R_1 ≥ R_2 which implies
t_2 ≥ t_1.
- The two rotating objects (e.g.,
planet,
star,
moon,
reference point on a
planet
for an
axial rotation)
are separated by
angle θ
subtended
at the common center.
The initial separation angle is θ_0.
- Let R be the relative
rotation rate
(i.e., the "synodic rotation rate)
and t be the relative
rotation period
(i.e., the
synodic period).
- Note that
synodic period t
has been completed when θ has increased by 360° above
θ_0.
Thus, t satisfies:
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) ,

where the 3rd line is the formula for the relative rotation rate formula and the last line is the synodic period formula.

- Note if R_1 < R_2, then the
relative rotation
is retrograde
and the synodic period t < 0
(i.e., it is negative).
- Note if R_1 = R_2, then
relative rotation rate
is zero
and the
synodic period t = ∞.
In this case, the two objects
corotate.
- Note if |t_1/t_2| << 1, then we can expand the
synodic period
formula
in a
geometric series
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

(see Wikipedia: Geometric series: Closed-form formula).

- We make the simplyfing assumption that
all the basic
rotations
are
uniform circular motions
in a common plane.
This assumption can be dispensed with, but that leads one to
formulae that are too complex
for educational purposes.
- As example of the use of the
synodic period
formula,
we can calculate the
solar day = current mean value 86400.002 s
which is the
synodic period
for two
rotations
the sidereal year = 365.256363004 days (J2000)
the sidereal day = 86164.0905 s = 0.997269566 days
= 1 day - 4 m + 4.0905 s (on average).
By the formula, we get
t = t_1*t_2/(t_2 - t_1) = 86399.9908 s

which is only 0.011 s too small compared to the accepted value given above.

- As a second example of using the
synodic period
formula,
let's calculate the
mean lunar month which is
the synodic period
for the
Moon's orbit relative to the
geometrical orbit of the
Sun
around the Earth.
Note that the
"orbital period" of
the Sun
around the Earth equals
the true physical orbital period
of the Earth around the
Sun.
We get
t = t_1*t_2/(t_2-t_1) = 27.321661547*365.256363004/(365.256363004-27.321661547) = 29.530588853 days ,

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.

- The equation we solved for the
synodic period
formula is of the form
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 ,

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).

- Using the last item, we can now just write down the solutions for t_1 and t_2
just by permuting
t, (-t_1), and t_2 in
synodic period
formula form
t = -(-t_1)*t_2/[t_2+(-t_1)] .

To be explicit,

(-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] .

Caption: An explication of synodic period.

Features:

Image link: Itself.

Local file: local link: synodic_period.html.

File: Orbit file: synodic_period.html.