Free oscillations for modern interior structure models of the Moon T.V. Gudkova S. Raevskiy Schmidt Institute of Physics of the Earth RAS

Seismic lunar models I/MR 2 =0.3931± “Lunar Prospector” (Konopliv et al., 1998) Crust thickness 60 km ( Nakamura et al.,1976; Goins et al.,1977) km ( reprocessing or inversion of the Apollo seismic data) (Chenet et al., 2000; Khan et al., 2000; Vinnik et al., 2001;Gagnepain-Beyneix et al. 2004; Chenet et al., 2004) ≈ 20 – 110 km ( Hood and Zuber, 2000 ) Core radius of the Moon Seismic data km ( Nakamura et al., 1974 ) Data on gravitational field and topography km ( Konopliv et al., 1998 ) Data obtained when the Moon passed through the Earth’s magnetospheric tail 340±90 km ( Hood et al., 1999 ) Weber et al. (2011) Solid inner core 240 km Liquid outer core 330 km Partial melt up to 480 km

Interior structure models used for calculations The models fit both geodetic (lunar mass, polar moment of inertia, and Love numbers) and seismological (body wave arrivals measured by Apollo network) data. MW model (dashed line) : Weber et al. (2011) reanalyzed Apollo lunar seismograms using array processing mehods to search for the presence of reflected and converted seismic energy from the core. The results suggest the presence of a solid inner (240 km) and fluid outer (330 km, 8 g/cm 3 ) core, overlain by a partially molten boundary layer (150 km thick). MG model (solid line) : Garsia et al. (2011) have constructed the Very Preliminary Reference Moon model and estimated the core radius by detecting core reflected S wave arrivals from waveform stacking methods. It has a core radius of 380  40 km and an average core mass density of 5.2  1.0 g/cm 3.

Free oscillations, if they are excited, are particularly attractive to probe beneath the surface of an extraterrestrial body into its deep interior. Interpretation of data on free oscillations does not require knowledge of the time or location of the source; thus, data from a single station are sufficient. Since the planet has finite dimensions and is bounded by a free surface, the study of the free oscillations is based on the theory of vibration of an elastic sphere. The planet reacts to a quake (or an impact) by vibrating as a whole, vibrations being the sum of an infinite number of modes that correspond to a set of frequencies. Free Oscillations

The free oscillations are divided into two types: (a) Torsional oscillations, whose displacement vector is perpendicular to the radius of a sphere - n T l (b) Spheroidal oscillations, whose displacement vector has components in both the radial and azimuthal direction - n S l The radial functions n W l of torsional and n U l spheroidal oscillations depend on two indices n and l, n - number of the overtone (the number of nodes along the radius in functions U and W), l – the degree of the oscillation.

Functions 0 W l proportional to the displacements for torsional oscillations for the fundamental mode, l =2 to 10 vesus relative radius r/R. The important feature of free oscillations is that they concentrate towards the surface with increasing the degree l. Therefore different regions of interiors are sounded by different frequency intervals. The fundamental modes sound to those depth in the interiors where its displacement  0.3 The horizontal line drawn at level =0.3 enables one to judge graphically which modes give information about one or another zone of the planet. 0 W l is normalized to unity at the surface.

The relative period difference for models MW and MG is about 1.5% for basic tones and about 2-4% for overtones of torsional oscillations. For spheroidal oscillations it is about 2% for basic tones and up to 10% for overtones. If these modes are detected, we could refine the models. Relative period difference  Т/Т(%) as a function of the oscillation number forfundamental mode (solid line) and two first overtones (dashed line- 1 and dot-dashed line - 2) of torsional (а) and spheroidal oscillations (b) for models by Weber et al. (2011) and Garsia et al. (2011).

The effect of the inner core rigidity on the structure of oscillations The model MW has the ‘core oscillation’ - FC. As the rigidity of the inner core increases, its period (42.96 min at μ=0) decreases up to 6.94 min at μ=4.23x10 11 dyne/cm 2, and its amplitude covers the mantle. At μ=0.5x10 11 dyne/cm 2 it looks like a regular oscillatiob R with a period of about 16 min. Besides the regular oscillation and its overtones O1,O2,..., there are ‘inner core’ oscillations, with the energy localised in the core. The curves are not crossed, they change a slope and a type of oscillation. Spheroidal oscillations, l =2. Period T as a function of shear modulus of the inner core (for the model by Weber et al. (2011)  =4.23х10 11 dyn/cm 2 ). R – regular oscillations, FC – basic core oscillation, IC – oscillations of the inner core and О – overtones of regular oscillations.

As the lunar core is rather small, the period difference of oscillations for the regular oscillation and the first overtone, for the models with inner core and without it, is very small (0.1 and 0.7%).It increases with the overtone number, and reachs 5-10% for the second and third overtones. The rigidity of the inner core influences mostly the periods of core oscillations, but their amplitudes are very small at the surface. The transition of FC type oscillation to R type and R type to FC type for the model by Weber et al. (2011), spheroidal oscillations, l =2

The amplitude spectrum for fundamental tones of torsional oscillations for l = 2, 3, 5, 7, 10, 20, 30, 40, 50 and their first overtones. X-axis - frequency  =2  /T and the period T (min), Y-axis - the amplitude of the horizontal component u N (cm). M 0 =1. Focal mechanism is 45 o, 45 o, 45 o for dip, strike and slip angles with a focal depth of 50 km. The epicenter coordinates are 29 o N, 98 o W, the seismometer coordinates are 0 o N, 45 o W, the epicentral distance is 58 o. M 0 =1: u N ≈ cm M 0 =10 22 dyn cm: u N ≈ cm l  7-10 (up to 500 km) M 0 =10 23 dyn cm: u N ≈ cm l  5 (up to 750 km)

The amplitudes of displacements u N, u E and u R for fundamental modes of spheroidal oscillations with l =2-20. M 0 =1. Focal mechanism is 45 o,45 o,45 o for dip, strike and slip angles at focal depths of 50 km (solid lines), 150 km (dot-dashed lines), 200 km (dashed lines). The epicenter coordinates are 29 o N, 98 o W, the seismometer coordinates are 0 o N, 45 o W, the epicentral distance is 58 o. For  =0.02s -1 : u R ≈ 6x x1.6x10 22 a R ≈ 0.4x two orders lower than the value that can be recorded

CONCLUSION Free oscillations, if they are excited, are indeed particularly attractive to probe beneath the surface of the Moon into its deep interiors. The spectrum of torsional modes n T l would allow noticeable progress to be made in constructing a global model of the Moon’s interior structure (up to about km depth), it was shown that the torsional modes with l > 5-7 can be recorded with current instruments. The seismic events on the Moon detected so far are too weak to excite free oscillations that could be recorded. The accuracy of seismometers has been improved, and the application of new methods for processing seismic data let us hope to identify harmonics with smaller amplitudes in the future.