J/A+A/584/A129 Matrix Methods code (Fouvry+, 2015)
Secular diffusion in discrete self-gravitating tepid discs.
II. Accounting for swing amplification via the matrix method.
Fouvry J.-B., Pichon C., Magorrian J., Chavanis P.-H.
<Astron. Astrophys. 584, A129 (2015)>
=2015A&A...584A.129F 2015A&A...584A.129F (SIMBAD/NED BibCode)
ADC_Keywords: Models
Keywords: galaxies: evolution - galaxies: kinematics and dynamics -
galaxies: spiral - diffusion - gravitation
Abstract:
The secular evolution of an infinitely thin tepid isolated galactic
disc made of a finite number of particles is investigated using the
inhomogeneous Balescu-Lenard equation expressed in terms of
angle-action variables. The matrix method is implemented numerically
in order to model the induced gravitational polarization. Special care
is taken to account for the amplification of potential fluctuations of
mutually resonant orbits and the unwinding of the induced swing
amplified transients. Quantitative comparisons with N-body
simulations yield consistent scalings with the number of particles and
with the self-gravity of the disc: the fewer particles and the colder
the disc, the faster the secular evolution. Secular evolution is
driven by resonances, but does not depend on the initial phases of the
disc. For a Mestel disc with Q∼1.5, the polarization cloud around each
star boosts up its secular effect by a factor of the order of a
thousand or more, promoting accordingly the dynamical relevance of
self-induced collisional secular evolution. The position and shape of
the induced resonant ridge are found to be in very good agreement with
the prediction of the Balescu-Lenard equation, which scales with the
square of the susceptibility of the disc.
In astrophysics, the inhomogeneous Balescu-Lenard equation may
describe the secular diffusion of giant molecular clouds in galactic
discs, the secular migration and segregation of planetesimals in
proto-planetary discs, or even the long-term evolution of population
of stars within the Galactic centre. It could be used as a valuable
check of the accuracy of N-body integrators over secular timescales.
Description:
Matrix Methods: complements to the paper.
For the sake of reproducibility of the results presented in the paper,
we distribute the 2D Mathematica linear matrix reponse code which was
implemented for this paper. If ever you were to use this package,
please cite our paper, where the details of the implementation are
presented. The shared files are:
+ A Mathematica package MatrixMethod2D.m and its notebook version
MatrixMethod2D.nb. (Both have the same content.) This library allows
the computation of the response matrix in the 2D case of an infinitely
thin disc.
+ A Mathematica test-case notebook MestelMode.nb, which shows how the
figure C.1 in the paper was obtained. As the computation can be quite
expensive to run, one will also find the associated data file,
Data_MestelMode.zip, which when unzipped allows not to have to
re-perform the expensive calculations needed by MestelMode.nb.
In a nutshell, the library MatrixMethod2D allows for:
+ the determination as a function of (rp,ra) of the orbits quantities:
E, L, J_r, Ω_1 and Ω_2
+ the construction of the 2D basis from Kalnajs (1976, ApJ, 205, 745)
+ the computation of the Fourier transform of the basis elements w.r.t.
the angles
+ the calculation of the 2D response matrix
To load the library in a Mathematica notebook, execute
≪ "/path/to/MatrixMethod2D.m"
To see all the implemented functions, execute
?MatrixMethod2D`*
(Clicking on a function's name gives then more detail about its content.)
An example of use of the package is given in the notebook
MestelMode.nb. This example, along with the inserted comments, should
be self-explanatory on how to proceed to the evaluation of the 2D
response matrix.
All files are also available on the website :
http://www.iap.fr/users/pichon/MatrixMethod/
For any questions/remarks, please do not hesitate to contact
fouvry(at)iap.fr or pichon(at)iap.fr
File Summary:
--------------------------------------------------------------------------------
FileName Lrecl Records Explanations
--------------------------------------------------------------------------------
ReadMe 80 . This file
files/* . 4 Matrix Methods code files
--------------------------------------------------------------------------------
Acknowledgements:
Jean-Baptiste Fouvry, jbfouvry(at)gmail.com
(End) Patricia Vannier [CDS] 21-Sep-2015