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IAA Transactions, No. 8, ``Celestial Mechanics'', 2002
FCN--period dependence on dynamic characteristics
of a lunar core
A. Gusev 1 , N. Petrova 1 , N. Kawano 2 , and RISE group 2
1 Kazan State University, Kazan, Russia
2 National Astronomical Observatory, Mizusawa, Iwate, Japan
Two modes in a polar oscillation are obtained when a free rotation of the
two--layer Moon was studied:
1
PCW
=
2\Omega A
Am
p
fffi ¸
1
74yr
1
P FCN
=
\Gamma\Omega
A
2Am
`
C c \Gamma A c
A c
+ ( C c \Gamma B c
B c
' p
fffi ¸
1
144yr
The numerical values of the periods were obtained on the assumption, that
R c = 220 km and that the dynamic figure of the core is similar to the dynamic
figure of the mantle.
Assuming the uniform density distribution for the core ae c = 7 g
cm 3
and mantle
ae m = 3:3 g
cm 3
, the modelling of the FCN­period in dependence on the core radius
and on its dynamical figure has been carried out.
Changes of the model structure will be reflected mainly in different values of
A
Am and of dynamical flattenings ff; fi; fl. Taking into account the slow rotation of
the Moon and the smallness of its core, we do not consider the deformation of the
liquid core by the rotation and propose that dynamical figure of the core is like to
the figure of the mantle, i.e. of the total Moon. We have calculated the variation
of the FCN period as a function of the core radius 1
P FCN
¸ R 5
core (in a first order
relative to dynamical flattening of the core). The FCN period is slightly changed
from 52785 days for R = 220 km to 52243 days for R = 600 km, i.e. the FCN­
period is decreased by 542 days as the radius is increased by 2.7 times. Such great
value of the FCN period for the Moon (about the 144 years) in comparison with
the Earth (435 days --- Defraigne et al., 1994) and with the Mars (231--283 days ---
Van Hoolst, 2000) is explained by the slow lunar
rotation(\Omega lunar = 1rev
28days
, where
\Omega Earth = 1rev
day
,\Omega Mars ¸ 1rev
day ) and by the smallness of the lunar core ellipticity.
The relative small decreasing of the FCN (1%) in dependence on the core radius
83

can be attributed mainly to the small relative core of the Moon, and hence core
mass. Its mass is 1.4%--3.5% of the Moon's mass (Konopliv, 1998). This results to
the value of A
Am almost equal to unity ( A
Am = 1:000065). For the Earth A
Am = 1:12
and for the Mars A
Am = 1:03 (Van Hoolst, 2000).
Geometrical figure of the core was modelled by the formulas:
ff 1
= C \Gamma A
A
= fi(1 + fl)
1 \Gamma flfi
¸
a \Gamma c
R c
ff 2
= C \Gamma B
B
= fi \Gamma fl
1 + fl
¸
b \Gamma c
R c
The calculation shows that the FCN­period is significantly decreased and ranges
up to the observed values ( 90 years) for the certain geometrical parameters of the
core. But this geometry corresponds to the non--physical values of the dynamical
flattening fl and fi: in this case the core will have the form of the highly extended
or highly flattened ellipsoid. Because of this, the most probable value of the FCN--
period falls in the range 130--160 years. Therewith, the ellipticity increases when
the core radius increases too. The analysis carried out shows, that the geometry of
the core is determined by the values: (a \Gamma c) ¸ 200 \Gamma 300 m, (b \Gamma c) ¸ 150 \Gamma 250 m.
For comparison, the LLR data give (a \Gamma c) = 140 m (Dickey et al, 1994). With
these magnitudes the dynamical figure of the core most closely resembles to that
of the mantle.
These theoretical simulations may be tested only by the practical measure­
ments of the lunar core. The scientific objectives of a Moon­orbiting mission
SELENE (Selenological and Engineering Explorer) (Kawano et al, 2001) are to
study the origin and evolution of the Moon, in--situ measurement of the lunar en­
vironment, mapping of lunar topography and surface composition, measurement
of the gravity and magnetic fields. The mission can obtain the improved value
of the lunar moment of inertia with 0.1% accuracy, that can put a constraint
on the density of the lunar core with 15% uncertainty provided that the mean
crustal density with 3% accuracy and the radii of the crust--mantle and the core--
mantle boundaries will be known. This complex of the measurements will allow
to determine the lunar core parameters more reliable.
References
1. Defraigne P., Dehant V., Hinderer J. Stacking gravity tide measurement and
nutation observations in order to determine the complex eigenfrequency of
the nearly diurnal free wobble. J. Geophys. Res., 1994, 99, 9203--3213.
2. Dickey J. O., Bender P. L., Faller J. E. Newhall X X et al. Lunar Laser
Ranging: A continuing Legacy of the Apollo Program, Science, 1994, 265,
482--490.
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3. Konopliv A. S., Binder A. B., Hood L. L., Kusinskas A. B., Sjogren W. l.,
Williams J. G. Improved gravity field of the Moon from Lunar Prospector.
Science, 1998, 281, 1476.
4. Kawano N., Hanado H., Iwata T. Tsubokawa T. 2001, Japanese Lunar Ex­
plorer ``SELENE'' and geodetic observations of the Moon. Proc. of the Intern.
Conf. ``AstroKazan--2001'', Kazan, 2001, 157--161.
5. Van Hoolst T, Dehant V., Defraigne P. Computation of Mars normal modes,
including the free core, nutation and the Chandler Wobble, PEPI, 2000, 117,
397.
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