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Register a new userGeneration of potential/surface density pairs in flat disks Power law distributions
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We report a simple method to generate potential/surface density pairs in flat axially symmetric finite size disks. Potential/surface density pairs consist of a ``homogeneous pair (a closed form expression) corresponding to a uniform disk, and a ``residual pair. This residual component is converted into an infinite series of integrals over the radial extent of the disk. For a certain class of surface density distributions (like power laws of the radius), this series is fully analytical. The extraction of the homogeneous pair is equivalent to a convergence acceleration technique, in a matematical sense. In the case of power law distributions, the convergence rate of the residual series is shown to be cubic inside the source. As a consequence, very accurate potential values are obtained by low order truncation of the series. At zero order, relative errors on potential values do not exceed a few percent typically, and scale with the order N of truncation as 1/N**3. This method is superior to the classical multipole expansion whose very slow convergence is often critical for most practical applications.
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We analyze the Miyamoto--Nagai substitution, which was introduced over forty years ago to build models of thick disks and flattened elliptical galaxies. Through it, any spherical potential can be converted to an axisymmetric potential via the replacement of spherical polar $r^2$ with $R^2 + ( a + !sqrt{z^2+b^2} )^2$, where ($R,z$) are cylindrical coordinates and $a$ and $b$ are constants. We show that if the spherical potential has everywhere positive density, and satisfies some straightforward constraints, then the transformed model also corresponds to positive density everywhere. This is in sharp contradistinction to substitutions like $r^2 rightarrow R^2 + z^2/q^2$, which leads to simple potentials but can give negative densities. We use the Miyamoto--Nagai substitution to generate a number of new flattened models with analytic potential--density pairs. These include (i) a flattened model with an asymptotically flat rotation curve, which (unlike Binneys logarithmic model) is always non-negative for a wide-range of axis ratios, (ii) flattened generalizations of the hypervirial models which include Satohs disk as a limiting case and (iii) a flattened analogue of the Navarro--Frenk--White halo which has the cosmologically interesting density fall-off of (distance)$^{-3}$. Finally, we discuss properties of the prolate and triaxial generalizations of the Miyamoto-Nagai substitution.
We report an empirical determination of the probability density functions $P_{text{data}}(r)$ of the number $r$ of earthquakes in finite space-time windows for the California catalog. We find a stable power law tail $P_{text{data}}(r) sim 1/r^{1+mu}$ with exponent $mu approx 1.6$ for all space ($5 times 5$ to $20 times 20$ km$^2$) and time intervals (0.1 to 1000 days). These observations, as well as the non-universal dependence on space-time windows for all different space-time windows simultaneously, are explained by solving one of the most used reference model in seismology (ETAS), which assumes that each earthquake can trigger other earthquakes. The data imposes that active seismic regions are Cauchy-like fractals, whose exponent $delta =0.1 pm 0.1$ is well-constrained by the seismic rate data.
We consider a general stochastic branching process, which is relevant to earthquakes as well as to many other systems, and we study the distributions of the total number of offsprings (direct and indirect aftershocks in seismicity) and of the total number of generations before extinction. We apply our results to a branching model of triggered seismicity, the ETAS (epidemic-type aftershock sequence) model. The ETAS model assumes that each earthquake can trigger other earthquakes (``aftershocks). An aftershock sequence results in this model from the cascade of aftershocks of each past earthquake. Due to the large fluctuations of the number of aftershocks triggered directly by any earthquake (``fertility), there is a large variability of the total number of aftershocks from one sequence to another, for the same mainshock magnitude. We study the regime where the distribution of fertilities mu is characterized by a power law ~1/mu^(1+gamma). For earthquakes, we expect such a power-law distribution of fertilities with gamma = b/alpha based on the Gutenberg-Richter magnitude distribution ~10^(-bm) and on the increase ~10^(alpha m) of the number of aftershocks with the mainshock magnitude m. We derive the asymptotic distributions p_r(r) and p_g(g) of the total number r of offsprings and of the total number g of generations until extinction following a mainshock. In the regime gamma<2 relevant for earhquakes, for which the distribution of fertilities has an infinite variance, we find p_r(r)~1/r^(1+1/gamma) and p_g(g)~1/g^(1+1/(gamma -1)). These predictions are checked by numerical simulations.
Biorthonormal basis function expansions are widely used in galactic dynamics, both to study problems in galactic stability and to provide numerical algorithms to evolve collisionless stellar systems. They also provide a compact and efficient description of the structure of numerical dark matter haloes in cosmological simulations. We present a two-parameter family of biorthonormal double-power-law potential-density expansions. Both the potential and density are given in closed analytic form and may be rapidly computed via recurrence relations. We show that this family encompasses all the known analytic biorthonormal expansions: the Zhao expansions (themselves generalizations of ones found earlier by Hernquist & Ostriker and by Clutton-Brock) and the recently discovered Lilley, Sanders, Evans & Erkal expansion. Our new two-parameter family includes expansions based around many familiar spherical density profiles as zeroth-order models, including the $gamma$ models and the Jaffe model. It also contains a basis expansion that reproduces the famous Navarro-Frenk-White (NFW) profile at zeroth order. The new basis expansions have been found via a systematic methodology which has wide applications in finding further examples. In the process, we also uncovered a novel integral transform solution to Poissons equation.
We study mass models that correspond to MOND (triaxial) potentials for which the Hamilton-Jacobi equation separates in ellipsoidal coordinates. The problem is first discussed in the simpler case of deep-MOND systems, and then generalized to the full MOND regime. We prove that the Kuzmin property for Newtonian gravity still holds, i.e., that the density distribution of separable potentials is fully determined once the density profile along the minor axis is assigned. At variance with the Newtonian case, the fact that a positive density along the minor axis leads to a positive density everywhere remains unproven. We also prove that (i) all regular separable models in MOND have a vanishing density at the origin, so that they would correspond to centrally dark-matter dominated systems in Newtonian gravity; (ii) triaxial separable potentials regular at large radii and associated with finite total mass leads to density distributions that at large radii are not spherical and decline as ln(r)/r^5; (iii) when the triaxial potentials admit a genuine Frobenius expansion with exponent 0<epsilon<1, the density distributions become spherical at large radii, with the profile ln(r)/r^(3+2epsilon). After presenting a suite of positive density distributions associated with MOND separable potentials, we also consider the important family of (non-separable) triaxial potentials V_1 introduced by de Zeeuw and Pfenniger, and we show that, as already known for Newtonian gravity, they obey the Kuzmin property also in MOND. The ordinary differential equation relating their potential and density along the z-axis is an Abel equation of the second kind that, in the oblate case, can be explicitly reduced to canonical form.
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