Distribution Of Scale Lengths Research Articles

Inner core freezing and melting: Constraints from seismic body waves

Hypotheses for the mechanisms of growth of Earth’s inner core can be tested from observations of lateral variations of elastic and anelastic properties near the inner core boundary. Differential travel times of PKIKP–PKiKP and the apparent travel time of P waves diffracted around the inner core are consistent with the existence of a stably stratified, globally uniform, region of reduced velocity gradient at the bottom of the outer core. This layer may be a zone of iron enrichment relative to the bulk of the outer core, representing a zone either of crystallization or melting of the inner core. Weak backscattered PKiKP in the equatorial eastern hemisphere can be made consistent with either a hypothesis of freezing or melting. In a melting hypothesis, the eastern hemisphere is texturally older and smoother, with larger crystals or smaller elastic contrasts across grain boundaries providing weaker backscatter to the coda of PKiKP. Additional observations consistent with higher homologous temperature and melting in this region are the existence of a thin low velocity layer in the uppermost inner core inferred from multipathing of PKIKP + PKiKP waves and a bright spot in the amplitude of PKIIKP waves observed at antipodal distances. Alternatively, a freezing hypothesis in the eastern hemisphere is an inner core that is growing radially outward with crystals stretched in the vertical direction. This hypothesis can simultaneously explain weak backscattered PKiKP coda and high attenuation of transmitted PKIKP from the loss of backscattered energy at angles perpendicular to vertically oriented crystals. Strong backscattering observed in the coda of PKiKP sampling the equatorial western hemisphere of the inner core agrees best with a heterogeneity having an isotropic distribution of scale lengths in the uppermost inner core.

Dark Matter and Stellar Mass in the Luminous Regions of Disk Galaxies

We investigate the correlations among stellar mass (M*), disk scale length (Rd), and rotation velocity at 2.2 disk scale lengths (V2.2) for a sample of 81 disk-dominated galaxies (disk/total ≥ 0.9) selected from the SDSS. We measure V2.2 from long-slit Hα rotation curves and infer M* from galaxy i-band luminosities (Li) and g - r colors. We find logarithmic slopes of 2.60 ± 0.13 and 3.05 ± 0.12 for the (forward fit) Li-V2.2 and M*-V2.2 relations, somewhat shallower than most previous studies, with intrinsic scatter of 0.13 and 0.16 dex, respectively. Our direct estimates of the total-to-stellar mass ratio within 2.2Rd, assuming a Kroupa IMF, yield a median ratio of 2.4 for M* > 1010 M☉ and 4.4 for M* = 109-1010 M☉, with large scatter at a given M* and Rd. The typical ratio of the rotation speed predicted for the stellar disk alone to the observed rotation speed at 2.2Rd is ~0.65. The distribution of scale lengths at fixed M* is broad, but we find no correlation between disk size and the residual from the M*-V2.2 relation, implying that the M*-V2.2 relation is an approximately edge-on view of the disk galaxy fundamental plane. Independent of the assumed IMF, this result implies that stellar disks do not, on average, dominate the mass within 2.2Rd. We discuss our results in the context of infall models where disks form in adiabatically contracted cold dark matter halos. A model with a disk-to-halo mass ratio md = 0.05 provides a reasonable match to the Rd-M* distribution for spin parameters λ ranging from ~0.04 to 0.08, and it yields a reasonable match to the mean M*-V2.2 relation. A model with md = 0.1 predicts overly strong correlations between disk size and M*-V2.2 residual. Explaining the wide range of halo-to-disk mass ratios within 2.2Rd requires significant scatter in md values, with systematically lower md for galaxies with lower M* or lower stellar surface density Σ*.

Numerical Study of Elastic Wave Scattering by Cracks or Inclusions Using the Boundary Integral Equation Method

In this paper, we use a 2-D elastodynamic boundary integral equation or boundary element method (BEM) to solve multiple scattering problems due to existence of cracks or inclusions. The method is based on the integral representation of a scattered wavefield by assuming a fictitious source distribution on the scattering objects or inclusions (i.e. mathematical description of Huygens' principle), and the fictitious source distribution can be found by matching appropriate boundary conditions at the boundary of the inclusions. The method is called indirect boundary element method. Three numerical examples are presented to demonstrate the versatility of the BEM method. The first example shows that different spatial arrangements of the same scatters lead to profound differences in scattering characteristics, in particular the frequency contents of the transmitted wavefields using the method of time-frequency analysis. The second example shows the effects of power-law or fractal distribution of scalelengths on transmitted wavefields, and we conclude that frequency characteristics, such as the frequency of the peak attenuation, can be related to spatial size parameters of the model. In the third example, we show that orientated inclusions with aspect ratio less than unity have strong effects on the amplitudes of transmitted waves, and this has an important implication in characterizing inclusions and fractures using azimuthal variation in amplitudes (or attenuation anisotropy).

NUMERICAL STUDY OF ELASTIC WAVE SCATTERING BY CRACKS OR INCLUSIONS USING THE BOUNDARY INTEGRAL EQUATION METHOD

In this paper, we use a 2-D elastodynamic boundary integral equation or boundary element method (BEM) to solve multiple scattering problems due to existence of cracks or inclusions. The method is based on the integral representation of a scattered wavefield by assuming a fictitious source distribution on the scattering objects or inclusions (i.e. mathematical description of Huygens' principle), and the fictitious source distribution can be found by matching appropriate boundary conditions at the boundary of the inclusions. The method is called indirect boundary element method. Three numerical examples are presented to demonstrate the versatility of the BEM method. The first example shows that different spatial arrangements of the same scatters lead to profound differences in scattering characteristics, in particular the frequency contents of the transmitted wavefields using the method of time-frequency analysis. The second example shows the effects of power-law or fractal distribution of scalelengths on transmitted wavefields, and we conclude that frequency characteristics, such as the frequency of the peak attenuation, can be related to spatial size parameters of the model. In the third example, we show that orientated inclusions with aspect ratio less than unity have strong effects on the amplitudes of transmitted waves, and this has an important implication in characterizing inclusions and fractures using azimuthal variation in amplitudes (or attenuation anisotropy).

Elastodynamic BEM modelling of multiple scattering of elastic waves by spatial distributions of inclusions

We use a 2D elastodynamic boundary integral equation or boundary element method (BEM) in this paper and apply it to solve crack scattering problems. The method is based on the integral representation of a scattered wavefield by assuming a fictitious source distribution on the scattering objects or inclusions (i.e. mathematical description of Huygens’ principle), and the fictitious source distribution can be found by matching appropriate boundary conditions at the boundary of the inclusions. We present two numerical examples to demonstrate the versatility of the BEM method. The first example shows that different spatial arrangements of the same scatters lead to profound differences in scattering characteristics, in particular the frequency contents of the transmitted wavefields using the method of time‐frequency analysis. The second example shows the effects of power‐law or fractal distribution of scalelengths on transmitted wavefields, and we conclude that frequency characteristics, such as the frequency of the peak attenuation, can be related to spatial size parameters of the model.

Anisotropy of heterogeneity scale lengths in the lower mantle from PKIKP precursors

SUMMARY The anisotropy of heterogeneity scale lengths in the lower mantle is investigated by modelling its eVect on the high-frequency precursors of PKIKP scattered by the heterogeneities. Although models having either an isotropic or an anisotropic distribution of scale lengths can fit the observed coda shapes of short-period precursors, the frequency content of broad-band PKIKP precursors favours a dominantly isotropic distribution of scale lengths. Precursor coda shapes are consistent with 1 per cent fluctuations in P velocity in the wavenumber band 0.05‐0.5 km’1 extending to 1000 km above the core‐mantle boundary, and with a D region open to circulation throughout the lower mantle. The level of excitation of PKIKP precursors observed in the frequency band 0.02‐2 Hz requires a power spectrum of heterogeneity that is nearly white or slowly increasing with wavenumber. Anisotropy of scale lengths may exist in a D layer having larger horizontal than vertical scale lengths and produce little or no detectable eVects on PKIKP precursors for P-velocity perturbations as high as 3 per cent when averaged over a vertical scale of several kilometres, and much higher when averaged over scales of hundreds of metres or less.

What Can We Learn from the Local Group About the Role of Interactions in Galaxy Formation

This talk is mainly about the halos and bulges of Local Group galaxies, in particular those of the Milky Way and M31, and what they can tell us about the role of interactions in galaxy formation. Bulges are potentially of particular interest for galaxy formation because they are widely suspected to be the seeds for galaxy formation. Sites of active star formation at high redshift are often regarded as spheroids in the process of formation. In this talk, I will say little about disks because their formation is conceptually understood, although many details remain uncertain. To form such flat systems, a fairly undisturbed dissipative process is needed. This process occurred fairly early: the disks appear to be in place by a redshift z = 1 (Ellis 1997), with a distribution of scalelengths that is similar to the distribution at z = 0.

Structural characteristics and stellar composition of low surface brightness disk galaxies

We present UBVI surface photometry of a sample of low surface brightness (LSB) disk galaxies. LSB disk galaxies are fairly well described as exponential disks with no preferred value for either scale length, central surface brightness, or rotational velocity. Indeed, the distribution of scale lengths is indistinguishable from that of high surface brightness spirals, indicating that dynamically similar galaxies (e.g., those with comparable Rv^2) exist over a large range in surface density. These LSB galaxies are strikingly blue. The complete lack of correlation between central surface brightness and color rules out any fading scenario. Similarly, the oxygen abundances inferred from HII region spectra are uncorrelated with color so the low metallicities are not the primary cause of the blue colors. While these are difficult to interpret in the absence of significant star formation, the most plausible scenario is a stellar population with a young mean age stemming from late formation and subsequent slow evolution. These properties suggest that LSB disks formed from low initial overdensities with correspondingly late collapse times.

An application of the propagator matrix of dynamic ray tracing: The focusing and defocusing of body waves by three-dimensional velocity structure in the source region

Summary. Since the dynamic ray tracing system can be written in the linear form dW/ds=SW, it permits the definition of fundamental matrix and propagator matrix solutions. The propagator matrix can be exploited to connect complex 3-D and 2-D regions to 1-D regions of a model. For example, if the three-dimensional variations of a model are confined only to the portion of the ray path between O’ and Oo, numerical integration of the linear system need only be performed between O’ and Oo to construct the propagator π(Oo, O'). Along the segment of the ray path between O’ and Os, traversing a 1-D or homogeneous portion of the model, analytic solutions exist for π(O', Os). Thus π(Oo, Os) can be simply obtained by multiplying the analytic and numerically obtained π matrices. With this procedure, quantities are calculated that are needed for the synthesis of teleseismic P waves by superposition of Gaussian beams. These synthetics are used to investigate the focusing and defocusing of teleseismic P-waves by 3-D structure in the vicinity of the source. A source region, having a fluctuation in velocity of 4 per cent over a characteristic scale length of 50 to 100 km, produces factors of two fluctuations in amplitude and several tenths of a second in travel time at teleseismic range in experiments in which either source location is varied at constant azimuth or in which azimuth is varied at constant source location. A model having a maximum fluctuation as small as 0.8 per cent is capable of producing caustics and multipaths at teleseismic range, depending on its distribution of scale lengths and its ratio of vertical with respect to horizontal scale length. The multipaths and caustics of such a model, however, occur over too small an area and are too closely spaced in arrival time to be resolved with seismograph systems in the 0.01 to 4 Hz band.