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Register a new userAxisymmetric and triaxial MOND density-potential pairs
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L. Ciotti Dept. ofn Astronomy
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We present a simple method, based on the deformation of spherically symmetric potentials, to construct explicit axisymmetric and triaxial MOND density-potential pairs. General guidelines to the choice of suitable deformations, so that the resulting density distribution is nowhere negative, are presented. This flexible method offers for the first time the possibility to study the MOND gravitational field for sufficiently general and realistic density distributions without resorting to sophisticated numerical codes. The technique is illustrated by constructing the MOND density-potential pair for a triaxial galaxy model that, in the absence of deformation, reduces to the Hernquist sphere. Such analytical solutions are also relevant to test and validate numerical codes. Here we present a new numerical potential solver designed to solve the MOND field equation for arbitrary density distributions: the code is tested with excellent results against the analytic MOND triaxial Hernquist model and the MOND razor-thin Kuzmin disk, and a simple application is finally presented.
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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.
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.
Efficient expansions of the gravitational field of (dark) haloes have two main uses in the modelling of galaxies: first, they provide a compact representation of numerically-constructed (or real) cosmological haloes, incorporating the effects of triaxiality, lopsidedness or other distortion. Secondly, they provide the basis functions for self-consistent field expansion algorithms used in the evolution of $N$-body systems. We present a new family of biorthogonal potential-density pairs constructed using the Hankel transform of the Laguerre polynomials. The lowest-order density basis functions are double-power-law profiles cusped like $rho sim r^{-2 + 1/alpha}$ at small radii with asymptotic density fall-off like $rho sim r^{-3 -1/(2alpha)}$. Here, $alpha$ is a parameter satisfying $alpha ge 1/2$. The family therefore spans the range of inner density cusps found in numerical simulations, but has much shallower -- and hence more realistic -- outer slopes than the corresponding members of the only previously-known family deduced by Zhao (1996) and exemplified by Hernquist & Ostriker (1992). When $alpha =1$, the lowest-order density profile has an inner density cusp of $rho sim r^{-1}$ and an outer density slope of $rho sim r^{-3.5}$, similar to the famous Navarro, Frenk & White (1997) model. For this reason, we demonstrate that our new expansion provides a more accurate representation of flattened NFW haloes than the competing Hernquist-Ostriker expansion. We utilize our new expansion by analysing a suite of numerically-constructed haloes and providing the distributions of the expansion coefficients.
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.
Cuspy triaxial potentials admit a large number of chaotic orbits, which moreover exhibit extreme stickiness that makes the process of chaotic mixing surprisingly inefficient. Environmental effects, modeled as noise and/or periodic driving, help accelerate phase space transport but probably not as much as in simpler potentials. This could mean that cuspy triaxial ellipticals cannot exist as time-independent systems.
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