366 research outputs found
The Newtonian potential of thin disks
The one-dimensional, ordinary differential equation (ODE) by Hur\'e & Hersant
(2007) that satisfies the midplane gravitational potential of truncated, flat
power-law disks is extended to the whole physical space. It is shown that
thickness effects (i.e. non-flatness) can be easily accounted for by
implementing an appropriate "softening length" . The solution of this
"softened ODE" has the following properties: i) it is regular at the edges
(finite radial accelerations), ii) it possesses the correct long-range
properties, iii) it matches the Newtonian potential of a geometrically thin
disk very well, and iv) it tends continuously to the flat disk solution in the
limit . As illustrated by many examples, the ODE,
subject to exact Dirichlet conditions, can be solved numerically with
efficiency for any given colatitude at second-order from center to infinity
using radial mapping. This approach is therefore particularly well-suited to
generating grids of gravitational forces in order to study particles moving
under the field of a gravitating disk as found in various contexts (active
nuclei, stellar systems, young stellar objects). Extension to non-power-law
surface density profiles is straightforward through superposition. Grids can be
produced upon request.Comment: Accepted for publication in A&
The potential of discs from a "mean Green function"
By using various properties of the complete elliptic integrals, we have
derived an alternative expression for the gravitational potential of axially
symmetric bodies, which is free of singular kernel in contrast with the
classical form. This is mainly a radial integral of the local surface density
weighted by a regular "mean Green function" which depends explicitly on the
body's vertical thickness. Rigorously, this result stands for a wide variety of
configurations, as soon as the density structure is vertically homogeneous.
Nevertheless, the sensitivity to vertical stratification | the Gaussian profile
has been considered | appears weak provided that the surface density is
conserved. For bodies with small aspect ratio (i.e. geometrically thin discs),
a first-order Taylor expansion furnishes an excellent approximation for this
mean Green function, the absolute error being of the fourth order in the aspect
ratio. This formula is therefore well suited to studying the structure of
self-gravitating discs and rings in the spirit of the "standard model of thin
discs" where the vertical structure is often ignored, but it remains accurate
for discs and tori of finite thickness. This approximation which perfectly
saves the properties of Newton's law everywhere (in particular at large
separations), is also very useful for dynamical studies where the body is just
a source of gravity acting on external test particles.Comment: Accepted for publication in MNRAS, 11 page
Generation of potential/surface density pairs in flat disks Power law distributions
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.Comment: Accepted for publication in Astronomy & Astrophysics 7 pages, 8
figures, F90-code available at
http://www.obs.u-bordeaux1.fr/radio/JMHure/intro2applawd.htm
Self-gravity at the scale of the polar cell
We present the exact calculus of the gravitational potential and acceleration
along the symmetry axis of a plane, homogeneous, polar cell as a function of
mean radius a, radial extension e, and opening angle f. Accurate approximations
are derived in the limit of high numerical resolution at the geometrical mean
of the inner and outer radii (a key-position in current FFT-based Poisson
solvers). Our results are the full extension of the approximate formula given
in the textbook of Binney & Tremaine to all resolutions. We also clarify
definitely the question about the existence (or not) of self-forces in polar
cells. We find that there is always a self-force at radius except if the
shape factor a.f/e reaches ~ 3.531, asymptotically. Such cells are therefore
well suited to build a polar mesh for high resolution simulations of
self-gravitating media in two dimensions. A by-product of this study is a newly
discovered indefinite integral involving complete elliptic integral of the
first kind over modulus.Comment: 4 pages, 4 figures, A&A accepte
A substitute for the singular Green kernel in the Newtonian potential of celestial bodies
The "point mass singularity" inherent in Newton's law for gravitation
represents a major difficulty in accurately determining the potential and
forces inside continuous bodies. Here we report a simple and efficient
analytical method to bypass the singular Green kernel 1/|r-r'| inside the
source without altering the nature of the interaction. We build an equivalent
kernel made up of a "cool kernel", which is fully regular (and contains the
long-range -GM/r asymptotic behavior), and the gradient of a "hyperkernel",
which is also regular. Compared to the initial kernel, these two components are
easily integrated over the source volume using standard numerical techniques.
The demonstration is presented for three-dimensional distributions in
cylindrical coordinates, which are well-suited to describing rotating bodies
(stars, discs, asteroids, etc.) as commonly found in the Universe. An example
of implementation is given. The case of axial symmetry is treated in detail,
and the accuracy is checked by considering an exact potential/surface density
pair corresponding to a flat circular disc. This framework provides new tools
to keep or even improve the physical realism of models and simulations of
self-gravitating systems, and represents, for some of them, a conclusive
alternative to softened gravity.Comment: Accepted for publication in A&A; 7 pages, color figure
The global structure of thin, stratified "alpha"-discs and the reliability of the one layer approximation
We report the results of a systematic comparison between the vertically
averaged model and the vertically explicit model of steady state, Keplerian,
optically thick "alpha"-discs. The simulations have concerned discs currently
found in three different systems: dwarf novae, young stellar objects and active
galactic nuclei. In each case, we have explored four decades of accretion rates
and almost the whole disc area
(except the narrow region where the vertically averaged model has degenerate
solutions). We find that the one layer approach gives a remarkably good
estimate of the main physical quantities in the disc, and specially the
temperature at the equatorial plane which is accurate to within 30% for cases
considered. The major deviations (by a factor < 4) are observed on the disc
half-thickness. The sensitivity of the results to the "alpha"-parameter value
has been tested for 0.001 < alpha < 0.1 and appears to be weak. This study
suggests that the ``precision'' of the vertically averaged model which is easy
to implement should be sufficient in practice for many astrophysical
applications.Comment: 4 pages, PostScript. Accepted in Astronomy & Astrophysic
AGN disks and black holes on the weighting scales
We exploit our formula for the gravitational potential of finite size,
power-law disks to derive a general expression linking the mass of the black
hole in active galactic nuclei (AGN), the mass of the surrounding disk, its
surface density profile (through the power index s), and the differential
rotation law. We find that the global rotation curve v(R) of the disk in
centrifugal balance does not obey a power law of the cylindrical radius R
(except in the confusing case s = -2 that mimics a Keplerian motion), and
discuss the local velocity index. This formula can help to understand how, from
position-velocity diagrams, mass is shared between the disk and the black hole.
To this purpose, we have checked the idea by generating a sample of synthetic
data with different levels of Gaussian noise, added in radius. It turns out
that, when observations are spread over a large radial domain and exhibit low
dispersion (standard deviation less than 10% typically), the disk properties
(mass and s-parameter) and black hole mass can be deduced from a non linear fit
of kinematic data plotted on a (R, Rv 2)-diagram. For a deviation higher than
10%, masses are estimated fairly well from a linear regression (corresponding
to the zeroth-order treatment of the formula), but the power index s is no
longer accessible. We have applied the model to 7 AGN disks whose rotation has
already been probed through water maser emission. For NGC3393 and UGC3789, the
masses seem well constrained through the linear approach. For IC1481, the
power-law exponent s can even be deduced. Because the model is scale-free, it
applies to any kind of star/disk system. Extension to disks around young stars
showing deviation from Keplerian motion is thus straightforward.Comment: accepted for publication in A&
Self-gravity in thin discs and edge effects: an extension of Paczynski's approximation
As hydrostatic equilibrium of gaseous discs is partly governed by the gravity
field, we have estimated the component caused by a vertically homogeneous disc,
with a special attention for the outer regions where self-gravity classically
appears. The accuracy of the integral formula is better than 1%, whatever the
disc thickness, radial extension and radial density profile. At order zero, the
field is even algebraic for thin discs and writes at disc surface, thereby correcting Paczynski's formula by a multiplying
factor , which depends on the relative distance to the
edges and the local disc thickness. For very centrally condensed discs however,
this local contribution can be surpassed by action of mass stored in the inner
regions, possibly resulting in . A criterion setting the limit
between these two regimes is derived. These result are robust in the sense that
the details of vertical stratification are not critical. We briefly discuss how
hydrostatic equilibrium is impacted. In particular, the disc flaring should not
reverse in the self-gravitating region, which contradicts what is usually
obtained from Paczynski's formula. This suggests that i) these outer regions
are probably not fully shadowed by the inner ones (important when illuminated
by a central star), and ii) the flared shape of discs does not firmly prove the
absence or weakness of self-gravity.Comment: Accepted for publication in A&
Algorithmic trading in a microstructural limit order book model
We propose a microstructural modeling framework for studying optimal market
making policies in a FIFO (first in first out) limit order book (LOB). In this
context, the limit orders, market orders, and cancel orders arrivals in the LOB
are modeled as Cox point processes with intensities that only depend on the
state of the LOB. These are high-dimensional models which are realistic from a
micro-structure point of view and have been recently developed in the
literature. In this context, we consider a market maker who stands ready to buy
and sell stock on a regular and continuous basis at a publicly quoted price,
and identifies the strategies that maximize her P\&L penalized by her
inventory. We apply the theory of Markov Decision Processes and dynamic
programming method to characterize analytically the solutions to our optimal
market making problem. The second part of the paper deals with the numerical
aspect of the high-dimensional trading problem. We use a control randomization
method combined with quantization method to compute the optimal strategies.
Several computational tests are performed on simulated data to illustrate the
efficiency of the computed optimal strategy. In particular, we simulated an
order book with constant/ symmet-ric/ asymmetrical/ state dependent
intensities, and compared the computed optimal strategy with naive strategies.
Some codes are available on https://github.com/comeh
A local prescription for the softening length in self-gravitating gaseous discs
In 2D-simulations of self-gravitating gaseous discs, the potential is often
computed in the framework of "softened gravity" initially designed for N-body
codes. In this special context, the role of the softening length LAMBDA is
twofold: i) to avoid numerical singularities in the integral representation of
the potential (i.e., arising when the relative separation vanishes), and ii) to
acount for stratification of matter in the direction perpendicular to the disc
mid-plane. So far, most studies have considered LAMBDA as a free parameter and
various values or formulae have been proposed without much mathematical
justification. In this paper, we demonstrate by means of a rigorous calculus
that it is possible to define LAMBDA such that the gravitational potential of a
flat disc coincides at order zero with that of a geometically thin disc of the
same surface density. Our prescription for LAMBDA, valid in the local,
axisymmetric limit, has the required properties i) and ii). It is mainly an
analytical function of the radius and disc thickness, and is sensitive to the
vertical stratification. For mass density profiles considered (namely, profiles
expandable over even powers of the altitude), we find that LAMBDA : i) is
independant of the numerical mesh, ii) is always a fraction of the local
thickness H, iii) goes through a minimum at the singularity (i.e., at null
separation), and iv) is such that 0.13 < LAMBDA/H < 0.29 typically (depending
on the separation and on density profile). These results should help us to
improve the quality of 2D- and 3D-simulations of gaseous discs in several
respects (physical realism, accuracy, and computing time).Comment: accepted in A&A, 7 pages, 7 figures, web link for the F90 code and
on-line calculations :
http://www.obs.u-bordeaux1.fr/radio/JMHure/intro2single.ph
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