Work in progress. This part of the site is still under construction.
More results including graphics and animations will appear very soon.
The Accretion Disc Boundary Layer in Cataclysmic Variables
NEW: Preliminary results.
Accretion onto a star, for a relatively hot gas, therefore
demanding less resolution. Here the ratio H/r = 0.15, is about
3 times larger than in Cataclysmic Variables, and is therefore
more relevant to accretion onto a pre-main sequence star in
Young Stellar Objects. At this stage there is no heating or
cooling in the energy equations (no radiation and the heating
due to viscous dissipation is not added).
A short movie
shows the evolution of this
preliminary models. The density is shown on a logarithmic color
scale. The disk is shown edge-on, and only the first quarter
(upper right) of the r-theta plane is shown. The star is rotating
at only a fraction (0.3) of the Keplerian speed. The matter spreads
quickly towards the pole to eventually englobe the whole star.
(the brown-yellow-green spot in the upper rigth corner of the
picture is for color reference).
The same movie is shown when the
initial density profile has been substracted. This enables to see
from where in the disk, the matter is initially released to be accreted
onto the surface of the star. This movie is slightly shorter than the
previous one and stops a little bit before the matter spread completely
to the poles.
1. OBJECTIVES
I propose to carry out a numerical study of gaseous astrophysical discs
as found in some binary systems (Cataclysmic Variables - CVs),
in which a star is losing mass that is transferred to its companion (a white
dwarf). This matter forms a flat disc around the white dwarf and, due to
viscous dissipation, the matter in the disc is slowly accreted onto the
surface of the white dwarf. The main unsolved problem in these systems is the
boundary layer region between the fast rotating inner edge of the disc and
the slowly rotating stellar surface. In this region the disc is expected to
slow down until its velocity matches the velocity of the stellar surface.
The disc looses kinetic energy which is expected eventually to be emitted by
the inner disc. The luminosity of the inner edge of the disc predicted by
one-dimensional simulations[8,17,22,23,25] is large, however, two
decades of observations have not shown any evidence of a luminous (and hot)
boundary layer[4,18]. I plan here to carry out two-dimensional
simulations
of the boundary layer region to solve the long standing problem of the
missing boundary layer.
2. SIGNIFICANCE, IMPACT AND OUTCOMES
The way the matter
is accreted from the disc onto the star has direct consequences
on other important processes.
The accreted material, after it accumulates onto
the surface of the white dwarf, undergoes a thermonuclear runaway
- TNR (nova). In order to model correctly the TNR event
one has to know whether the material from the
disc has spread toward the poles of the star or has remained in the
equator[26].
It is not known either whether
the equatorial material on the star is accelerated
(as some observations suggest[2,5]) or not.
Convective currents due to Eckman pumping
(or even turbulence due to the shear)
in the outer envelop of the star (and in the
accreted envelop) could take place,
and would mixed the freshly accreted
material (Hydrogen-H) with heavier elements
(e.g. P,Al,..) found in the star. During a TNR the heavy elements would
be ejected in the interstellar medium and affect the composition
of the interstellar gas.
The problem of the BL is therefore important not only to
explain the observations of CV systems, but also to predict the TNR
events,
the mixing of the elements in the outer envelop of the white dwarf and the
heavy elements ejected in the interstellar medium[26], and most
importantly this would have significant implications in the theory
of STAR FORMATION.
In this picture, the heating of the star due to BL radiation
and advection of energy would also affect the TNR event[24].
3. PRELIMINARY STUDIES
I have been working on the boundary layer between the accretion
disc and the accreting star since I started my PhD[9].
I carried out one-dimensional (time-dependent)
simulations of the boundary layers in many star-discs system, ranging
from discs around white dwarf in Cataclysmic Variables[8,10] to
discs around Young stars[11,13] and around low mass stars[12].
I have shown that the energy in the boundary layer is radiated radially
(therefore, heating the star)
and consequently the temperature of the boundary layer is lower
than first expected[8].
However, the observed overall luminosity (energy released
by radiation) of the boundary layer is still much less than in these
theoretical models.
I also found[15]
that the BL energy is advected into the star (again resulting
in heating of the star) when the rate
of the mass accretion is high. This last result explains partially
why in high mass accretion rate CVs system the BL is not observed. However,
a theoretical model is still missing, for low mass accretion rate systems.
I have already modeled and written many two-dimensional hydrodynamic
codes[6,7,14], and I have also developed and
tested a two-dimensional multigrid method[19,20,21]
that will be the basis for
a new numerical code for the boundary layer problem (more details on
the code and the reason to chose the multigrid method are given in
the methodology section).
4. METHODOLOGY}
4.1 The numerical modeling
The main task of the research will be to solve numerically the equations
governing the boundary layer region between the star and the disc.
The equations consist of the Navier-Stokes equations, an
energy equation, an equation of state
and eventually an equation for the diffusion of the elements. The equations
will be written in spherical coordinates (r,theta,phi). I will
assume the flow to be axisymmetric around the vertical axis of rotation
of the star, and consequently there will be no dependence on phi.
I will solve the equation in the first quarter of the plane (r,theta)
with the disc located in the theta = pi/2 plane (with a finite thickness).
The radius will extent initially from the stellar surface (eventually the outer
layer of the degenerate star will be included) up to an outer radius of the
order of a few stellar radii. The equations will be written in finite
difference form (second order)
for the space derivatives, while the temporal scheme
will be implicit (a Crank-Nicholson solver will be used).
The implicit equations (for the diffusive terms) will then
be solved using a multigrid solver for the following reason.
In order to solve the system, one usually needs to invert a matrix,
which is computer-time consuming. A two-dimensional numerical simulation
of the boundary layer was carried out[16] but the model could be
followed only for a relatively short time scale, because of the inversion
of the matrix.
The multigrid method[1] (a variant of the relaxation method)
is faster than the matrix inversion
by 1-3 orders of magnitude[19] (depending on the number of dimensions
of the problem and the number of grid points - the larger the problem,
the faster the relative implementation).
Moreover, the multigrid method can easily be implemented
to work on a parallel computer and achieves an additional 1-2 order
of magnitude in speed[19] over the usual (non-parallel) matrix inversion.
Even for a modest performance of the multigrid code, I expect
to be able to solve the problem of the boundary layer on the long
(physical) evolution time scale, using much less computer time than
any matrix solver.
4.2 Plans of the Research
First I will develop a purely hydrodynamical two-dimensional
time-dependent code, using a simple polytropic approach (P=P(rho) only,
neglecting the energy equation and the diffusion of elements).
This will allow to check the formation of an Eckman
layer, and check the possibility of an Eckman pumping,
an accretion belt, circulation, and maybe even turbulence due to the
shear in the boundary layer. This simplistic approach should already
be able to show whether the matter has a tendency to spread towards the poles
of the star, or if it is confined to the equator due to the centrifugal
force[3].
Later, I will include an energy equation with
a source term (due to viscous dissipation) and a leak term (radiation),
with an ideal gas equation of state. Here I should already get a final
answer to the question of the spreading of the matter towards the poles,
the formation of an accretion belt, the thickening of the inner disc,
and mainly whether the flow is hydrodynamically unstable and produces turbulence
and/or short period oscillations (dwarf nova oscillations) in CVs?
Eventually, I will take into account the diffusion of elements
and I will include the outer layer of the stellar surface (the outer layer
of the white dwarf is degenerate). In this final stage I expect to answer
the following: what is the depth of the accretional heating
in the outer envelope of the white dwarf
versus latitude, and what is its thermal time scale?
how much spinup of the star results from angular momentum transfer to
the star by the accreted matter?
what is the belt rotational velocity?
and mainly does convective mixing and dredgeup due to shear mixing occur? \\
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