35 research outputs found
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&
A key-formula to compute the gravitational potential of inhomogeneous discs in cylindrical coordinates
We have established the exact expression for the gravitational potential of a
homogeneous polar cell - an elementary pattern used in hydrodynamical
simulations of gravitating discs. This formula, which is a closed-form, works
for any opening angle and radial extension of the cell. It is valid at any
point in space, i.e. in the plane of the distribution (inside and outside) as
well as off-plane, thereby generalizing the results reported by Durand (1953)
for the circular disc. The three components of the gravitational acceleration
are given. The mathematical demonstration proceeds from the "incomplete version
of Durand's formula" for the potential (based on complete elliptic integrals).
We determine first the potential due to the circular sector (i.e. a pie-slice
sheet), and then deduce that of the polar cell (from convenient radial scaling
and subtraction). As a by-product, we generate an integral theorem stating that
"the angular average of the potential of any circular sector along its tangent
circle is 2/PI times the value at the corner". A few examples are presented.
For numerical resolutions and cell shapes commonly used in disc simulations, we
quantify the importance of curvature effects by performing a direct comparison
between the potential of the polar cell and that of the Cartesian (i.e.
rectangular) cell having the same mass. Edge values are found to deviate
roughly like 2E-3 x N/256 in relative (N is the number of grid points in the
radial direction), while the agreement is typically four orders of magnitude
better for values at the cell's center. We also produce a reliable
approximation for the potential, valid in the cell's plane, inside and close to
the cell. Its remarkable accuracy, about 5E-4 x N/256 in relative, is
sufficient to estimate the cell's self-acceleration.Comment: Accepted for publication in Celestial Mechanics and Dynamical
Astronom
Approaching the structure of rotating bodies from dimension reduction
International audienceWe show that the two-dimensional structure of a rigidly rotating self-gravitating body is accessible with relatively good precision by assuming a purely spheroidal stratification. With this hypothesis, the two-dimensional problem becomes one-dimensional, and consists in solving two coupled fixed-point equations in terms of equatorial mass density and eccentricity of isopycnics. We propose a simple algorithm of resolution based on the self-consistent field method. Compared to the full unconstrained-surface two-dimensional problem, the precision in the normalized enthalpy field is better than in absolute, and the computing time is drastically reduced. In addition, this one-dimensional approach is fully appropriate to fast rotators, works for any density profile (including any barotropic equation of state), and can account for mass density jumps in the system, including the existence of an ambient pressure. Several tests are given
Rotation Curves of Galactic Disks for Arbitrary Surface Density Profiles: A Simple and Efficient Recipe
International audienceThere is apparently a widespread belief that the gravitational field (and subsequently the rotation curve) ``inside'' razor-thin, axially symmetric disks cannot be determined accurately from elliptic integrals because of the singular kernel in the Poisson integral. Here we report a simple and powerful method to achieve this task numerically using the technique of ``density splitting.'