Newtonian Fractional-Dimension Gravity and Disk Galaxies
Gabriele U. Varieschi
备注：
23 pages, including 6 figures: references added and other minor changes after peer review. Final version to be published in The European Physical Journal Plus, Springer
This paper continues previous work on a novel alternative model of gravity, based on the theory of fractional-dimension spaces applied to Newton's law of gravitation. In particular, our Newtonian Fractional-Dimension Gravity is now applied to axially-symmetric structures, such as thin/thick disk galaxies described by exponential, Kuzmin, or other similar mass distributions.
As in the case of spherically-symmetric structures, which was studied in previous work on the subject, we examine a possible connection between our model and Modified Newtonian Dynamics, a leading alternative gravity model, which accounts for the observed properties of galaxies and other astrophysical structures without requiring the dark matter hypothesis.
By relating the MOND acceleration constant $a_{0} \simeq 1.2 \times 10^{ -10}\mbox{m}\thinspace \mbox{s}^{ -2}$ to a natural scale length $l_{0}$ of our model, namely $a_{0} \approx GM/l_{0}^{2}$ for a galaxy of mass $M$, and by using the empirical Radial Acceleration Relation, we are able to explain the connection between the observed radial acceleration $g_{obs}$ and the baryonic radial acceleration $g_{bar}$ in terms of a variable local dimension $D$. As an example of this methodology, we provide a detailed rotation curve fitting for the case of the field dwarf spiral galaxy NGC 6503.
We expand our analysis of Newtonian Fractional-Dimension Gravity (NFDG), an extension of the classical laws of Newtonian gravity to lower dimensional spaces, including those with fractional (i.e., non-integer) dimension. We apply our model to four rotationally supported galaxies (NGC 5033, NGC 6674, NGC 5055, NGC 1090), in addition to other three galaxies (NGC 7814, NGC 6503, NGC 3741) which were analyzed in previous studies. NFDG is able to fit the rotation curves of all these galaxies without any dark matter component. We also investigate the possible violation of the strong equivalence prin…
We continue our analysis of Newtonian Fractional-Dimension Gravity, an extension of the standard laws of Newtonian gravity to lower dimensional spaces including those with fractional (i.e., non-integer) dimension. We apply our model to three rotationally supported galaxies: NGC 7814 (Bulge-Dominated Spiral), NGC 6503 (Disk-Dominated Spiral), and NGC 3741 (Gas-Dominated Dwarf). As was done in the general cases of spherically-symmetric and axially-symmetric structures, which were studied in previous work on the subject, we examine a possible connection between our model and Modified Newtonian Dy…
Fractional Newtonian gravity, based on the fractional generalization of Poisson's equation for Newtonian gravity, is a novel approach to Galactic dynamics aimed at providing an alternative to the dark matter paradigm through a non-local modification of Newton's theory. We provide an in-depth discussion of the gravitational potential for the Kuzmin disk within this new approach. Specifically, we derive an integral and a series representation for the potential, we verify its asymptotic behavior at large scales, and we provide illuminating plots of the resulting equipotential surfaces.
We evaluate general-relativistic effects in motion of stationary accretion disks around a Schwarzschild black hole, assuming the first post-Newtonian (1PN) approximation. There arises an integrability condition, that leads to the emergence of two types of general-relativistic corrections to a Newtonian rotation curve. The well known geometric dragging of frames accelerates rotation but the hitherto unknown dynamic term, that reflects the disk structure, deccelerates rotation. The net result can diminish the Newtonian angular velocity of rotation in a central disk zone but the geometric draggin…
This paper introduces a possible alternative model of gravity based on the theory of fractional-dimension spaces and its applications to Newtonian gravity. In particular, Gauss's law for gravity as well as other fundamental classical laws are extended to a $D$-dimensional metric space, where $D$ can be a non-integer dimension. We show a possible connection between this Newtonian Fractional-Dimension Gravity (NFDG) and Modified Newtonian Dynamics (MOND), a leading alternative gravity model which accounts for the observed properties of galaxies and other astrophysical structures without requirin…
We investigated hydrodynamical post-Newtonian models of selfgravitating stationary black-hole-disk systems. The post-Newtonian scheme presented here and also in our recent paper is a continuation of previous, purely Newtonian studies of selfgravitating hydrodynamical disks rotating according to the Keplerian rotation law. The post-Newtonian relativistic corrections are significant even at the 1PN level. The 1PN correction to the angular velocity can be of the order of 10% of its Newtonian value. It can be expressed as a combination of geometric and hydrodynamical terms. Moreover, in contrast t…
In this article we investigate mathematically the variant of post-Newtonian mechanics using generalized fractional derivatives. The relativistic-covariant generalization of the classical equations for gravitational field is studied. The equations match to (i) the weak Newtonian limit on the moderate scales, (ii) deliver a potential higher, that Newtonian, on certain, large-distance characteristic scales. The perturbation of the gravitational field results in the tiny secular perihelion shift and exhibits some unusual effects on large scales. The general representation of the solution for fract…
We present a pseudo-Newtonian potential for accretion disk modeling around the rotating black holes. This potential can describe the general relativistic effects on accretion disk. As the inclusion of rotation in a proper way is very important at an inner edge of disk the potential is derived from the Kerr metric. This potential can reproduce all the essential properties of general relativity within 10% error even for rapidly rotating black holes.
We study the global gravitational stability of a gaseous self-gravitating Maclaurin disk in the absence of a halo. Further, we replace Newtonian gravity with the specific Modified gravity theory known as MOG in the relevant literature. MOG is an alternative theory for addressing the dark matter problem without invoking exotic dark matter particles, and possesses two free parameters $α$ and $μ_0$ in the weak field limit. We derive the equilibrium gravitational potential of the Maclaurin disk in MOG and develop a semi-analytic method for studying the response of the disk to linear non-axysymmetr…
The dark matter constraint imposed by the recent WMAP experiment on gravitational theories is analyzed. Using the non-linearity of the vacuum Einstein's equations, it is shown that when the slow motion condition is applied to the geodesic equations, the resulting nearly Newtonian gravitational field describes nearly flat velocity curves for rotating stars in the vicinity of thin disk galaxies.
We study the global stability of a self-gravitating disk in the context of Modified Gravity (MOG) using N-body simulations. This theory is a relativistic scalar-tensor-vector theory of gravity and presented to address the dark matter problem. In the weak field limit MOG possesses two free parameters $α$ and $μ_0$ which have been already determined using rotation curve data of spiral galaxies. The evolution of a stellar self-gravitating disk and more specifically the bar instability in MOG is investigated and compared to a Newtonian case. Our models have exponential and Mestel-like surface dens…
Two pseudo-Newtonian potentials, which approximate the angular and epicyclic frequencies of the relativistic accretion disk around rotating (and counter rotating) compact objects, are presented. One of them, the Logarithmically Modified Potential, is a better approximation for the frequencies while the other, the Second-order Expanded potential, also reproduces the specific energy for circular orbits in close agreement with the General Relativistic values. These potentials may be included in time dependent hydrodynamical simulations to study the temporal behavior of such accretion disks.
Recent analytic results concerning stationary, self-gravitating fluids in Newtonian theory are discussed. We give a theorem that forbids infinitely extended fluids, depending on the assumed equation of state and the rotation law. This part extends previous results that have been obtained for static configurations. The second part discusses a Sobolev bound on the mass of the fluid and a rigorous Jeans-type inequality that is valid in the stationary case.
We prescribe a pseudo-Newtonian vector potential for studying accretion disks around Kerr black holes. The potential is useful to study the inner properties of disk not confined to the equatorial plane where general relativistic effect is indispensable. Therefore, we incorporate the essential properties of the metric at the inner radii through the pseudo-Newtonian potential derived from the general Kerr spacetime. The potential, reproducing most of the salient features of the general-relativity, is valid for entire regime of Kerr parameter. It reproduces the last stable circular orbit exactly …
Recently, it has been suggested that the phenomenology of flat rotation curves observed at large radii in the equatorial plane of disk galaxies can be explained as a manifestation of General Relativity instead of the effect of Dark Matter halos. In this paper, by using the well known weak field, low velocity gravitomagnetic formulation of GR, the expected rotation curves in GR are rigorously obtained for purely baryonic disk models with realistic density profiles, and compared with the predictions of newtonian gravity for the same disks in absence of Dark Matter. As expected, the resulting rot…
We present a method for constructing equilibrium disks with net angular momentum in general relativity. The method solves the relativistic Vlasov equation coupled to Einstein's equations for the gravitational field. We apply the method to construct disks that are relativistic versions of Newtonian Kalnajs disks. In Newtonian gravity these disks are analytic, and are stable against ring formation for certain ranges of their velocity dispersion. We investigate the existence of fully general relativistic equilibrium sequences for differing values of the velocity dispersion. These models are the f…
The gravitational stability of a two-dimensional self-gravitating and differentially rotating gaseous disk in the context of post-Newtonian (hereafter PN) theory is studied. Using the perturbative method and applying the second iterated equations of PN approximation, the relativistic version of the dispersion relation for the propagation of small perturbations is found. We obtain the PN version of Toomre's local stability criterion by utilizing this PN dispersion relation. In other words, we find relativistic corrections to Toomre's criterion in the first PN approximation. Two stability parame…
A binary system composed of an oscillating and rotating coplanar dusty disk and a point mass is considered. The conservative dynamics is treated on the Newtonian level. The effects of gravitational radiation reaction and wave emission are studied to leading quadrupole order. The related waveforms are given. The dynamical evolution of the system is determined semi-analytically exploiting the Hamiltonian equations of motion which comprise the effects both of the Newtonian tidal interaction and the radiation reaction on the motion of the binary system in elliptic orbits. Tidal resonance effects b…
The gravitational potential and the gravitational rotation field generated by a thin-disk mass distribution with exponential density are considered in the case when the force between any two mass elements is not the usual Newtonian one, but some general central force. We use an approximation such that in the Newtonian case the gravitational field generated by the disk reduces to the familiar expression that results from applying the Gauss' law. In this approximation, we invert the usual integral relations in such a way that the elemental interaction (between two point-like masses) is obtained …
We present a new solution for the rotation curves of galactic disks with gravitational potential of the Yukawa type. We follow the technique employed by Toomre in 1963 in the study of galactic disks in the Newtonian theory. This new solution allows an easy comparison between the Newtonian solution and the Yukawian one. Therefore, constraints on the parameters of theories of gravitation can be imposed, which in the weak field limit reduce to Yukawian potentials. We then apply our formulae to the study of rotation curves for a zero-thickness exponential disk and compare it with the Newtonian cas…
We consider a geometrically thin, Keplerian disk in the orbital plane of a binary black hole (BHBH) consisting of a spinning primary and low-mass secondary (mass ratio q < 1). To account for the principle effects of general relativity (GR), we propose a modification of the standard Newtonian evolution equation for the (orbit-averaged) time-varying disk surface density. In our modified equation the viscous torque in the disk is treated in full GR, while the tidal torque is handled in the Newtonian limit. Our GR-hybrid treatment is reasonable because the tidal torque is concentrated near the orb…
Gas-induced gravitational-wave dephasing and accretion periodicities of live post-Newtonian massive black hole binaries: warm disk
Center for Cosmology and Particle Physics, Physics Department, New York University（纽约大学物理系宇宙学与粒子物理中心）；Dipartimento di Fisica A. Pontremoli, Università degli Studi di Milano（米兰大学A. Pontremoli物理系）；DiSAT, Università degli Studi dell’Insubria（因苏布里亚大学DiSAT）；INFN, Sezione di Milano-Bicocca（意大利国家核物理研究所米兰-比可卡分部）
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