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NEW FORMULATION OF THE EIGEN-VALUE PROBLEM
FOR GALACTIC DISCS
E. V. Polyachenko 1
Abstract. A new statement of the eigen-value problem to study of
small perturbations in arbitrary integrable selfgravitating systems is
presented. An example of such a system, 2D stellar disc is considered
in detail. The theory based on the general equation for disc eigenmodes allows to reveal mechanisms for formation and growth of global
galactic structures.
1
Introduction
In the previous paper (Polyachenko 2004, henceforth Paper I), a new approach
to studying spiral and bar-like structures was suggested. These structures were
considered as the low-frequency normal modes in a disc of orbits precessing at
different angular speeds. Then the problem of determining the normal modes was
reduced to a simple integral equation in the form of the classical eigen-value problem. Here we demonstrate that this integral equation can be obtained as a special
case of a general eigen-value problem, applicable to the study of small perturbations in any integrable system, i.e. its potential admits full separation of variables
in the Hamilton – Jacobi equation (see, e.g., Binney & Tremaine 1987). Using this
general approach, we then consider 2D axisymmetric stellar disc. Such a formulation is essentially analogous to the well-known matrix approach of Kalnajs (1977).
Its advantage is in systematic use of action-angle variables (Kalnajs used them
also, but he introduced the r-functions for the bi-orthogonal potential – surface
density system). Besides, our equation for 2D stellar disc has a form of the classical eigen-value problem (as the basic integral equation in Paper I), which allows
comparatively simple general analysis and ease in the numerical calculations.
1
Institute of Astronomy, Moscow 119017, Russia
c
°
2
2
Small perturbations of Hamiltonian integrable systems
In this section we consider the eigen-value problem for gravitating collisionless
systems in general. Let a self-consistent system is described by the Hamiltonian
H(I, w, t) = H0 (I) + Φ(I, w, t).
(2.1)
Here actions I and angles w are N -dimensional vectors, N is the number of degrees
of freedom. The unperturbed part H0 (I) describes a completely integrable system,
whose motion is quasi-periodic with N frequencies, Ωi = ∂H0 /∂Ii , or more briefly:
Ω(I) = ∂H0 /∂I. Φ(I, w, t) is the potential of perturbation.
A stellar distribution function f (I, w, t) obeys the collisionless Boltzmann equation
X ∂H ∂f
X ∂f ∂H
∂f
= [H, f ] ≡
−
.
(2.2)
∂t
∂wi ∂Ii
∂wi ∂Ii
i
i
For small perturbations of the system, the distribution function can be represented
as a sum
f = F0 (I) + F(I, w, t)
(2.3)
of the unperturbed and perturbed parts. Expanding the perturbations into following Fourier series:
F=
X
Fn (I) ein·w−iωt ,
(2.4)
Φn (I) ein·w−iωt ,
(2.5)
n
Φ=
X
n
(here n is the N -dimensional vector of integers, n = (n1 , ... nN ); n · w is the scalar
product of vectors n and w; ω is the frequency), we obtain after linearizing (2.2):
[−ω + n · Ω(I)]Fn (I) =
µ
¶
∂F0
n·
Φn (I).
∂I
(2.6)
To make the problem self-consistent, one needs another relation between Fn
and Φn , which comes from the Poisson equation
Z
F(r0 , v0 )
,
(2.7)
Φ(r) = −G dr0 dv0
r12
where G is the gravity constant and r12 = |r − r0 |; r, r0 are the radius-vectors. In
the action-angle variables, the latter equation can be rewritten as
Z
X
0
0
1 X
in·w
Φn (I) e
= −G dI0 dw0
Fn0 (I0 ) ein ·w ,
(2.8)
r
12
0
n
n
New Formulation of the Eigen-value Problem for Galactic Discs
3
Here we use the equality dr0 dv0 = dI0 dw0 , valid for canonically conjugate variables,
and assume that r12 is expressed as a function of I, I0 , w, w0 . Thus, the Fourier
terms Φn are obtained as follows:
Z
X
Φn (I) = −G dI0
Πn,n0 (I, I0 )Fn0 (I0 ),
(2.9)
n0
where
1
Πn,n0 (I, I ) =
(2π)N
0
Z
dwdw0 e−in·w
1 in0 ·w0
e
.
r12
(2.10)
Combining (2.6) and (2.9), one obtains a set of integral equations for small amplitude eigen perturbations of the system:
µ
¶Z
X
∂F0
[ω − n · Ω(I)]Fn (I) = G n ·
dI0
Πn,n0 (I, I0 )Fn0 (I0 ).
∂I
0
(2.11)
n
In operator form this equation turns into classical eigen-value problem
·
µ
¶ ¸
∂F0
Π̂ F.
ω F = n · Ω(I) E + G n ·
∂I
(2.12)
Here E is the identity operator; F = {Fn } are terms of the Fourier expansion (2.4)
of the perturbed distribution function and the eigen-value is the eigen-frequency
of the mode itself.
Not all of the solutions of (2.12) are physical. Since the integration in the
r.h.s. over real values of variables I is assumed, the equation is valid only for
unstable modes (Im ω > 0). To treat accurately neutral and damping modes, one
should choose a contour of integration passing below their eigen-frequencies in the
complex ω-plane.
The solution of the eigen-value problem can be obtained numerically. Using a
quadrature rule
Z
X
gn (I)dI →
gn (I)wn (I),
(2.13)
I
the problem results in the ordinary matrix eigen-value problem 1992:
¯
¯ µ
¶
¯
¯
∂F0 (I)
0
0
0
¯
Πn,n0 (I, I )wn0 (I ) + δn,n0 (I, I )(n · Ω(I) − ω)¯¯ = 0, (2.14)
det ¯G n ·
∂I
where δn,n0 (I, I0 ) being the Kroneker symbol.
In principal, equation (2.12) and formula (2.14) give a general solution of the
eigen-problem for any integrable gravitating system. In the rest of the paper,
we apply this approach to 2D stellar discs and find their eigen-frequencies and
corresponding eigen-functions.
4
3
Equation for flat axisymmetric disc
The Hamiltonian of the 2D stellar disc in the polar coordinates can be written as
H0 =
v2
+ Φ0 (r).
2
(3.1)
The unperturbed potential Φ0 (r) includes contributions from the stellar disc and
from an axisymmetric halo. This system possesses two integrals of motions: angular momentum L and energy E, and thus the system is integrable. Instead of
E, it is more convenient to use the radial action Ir :
rZmax
2
dr0 vr ,
Ir =
vr = (2[E − Φ0 (r0 )] − L2 /r0 )1/2 .
(3.2)
rmin
In the action–angle variables I = (Ir , L) and w = (w1 , w2 ), the Hamiltonian
is a function of the actions H0 = H0 (Ir , L). An unperturbed phase trajectory is
simply wi = Ωi (I)t + wi0 , where
Ω1 =
∂H0 (I)
,
∂Ir
Ω2 =
∂H0 (I)
.
∂L
(3.3)
Since modes with different azimuthal harmonics separate, one can consider in
(2.4, 2.5) only series for w1
X
Fl (I) eilw1 ,
(3.4)
F = eimw2 −iωt
l
Φ = eimw2 −iωt
X
Φl (I) eilw1 .
(3.5)
l
These equations describe perturbation with azimuthal number m. The equation
(2.10) then turns into
Z
0 0
1
0
Πl,l0 (I, I ) =
dw1 dw10 ψ(r, r0 )eil w1 −ilw1 eimδϕ .
(3.6)
2π
where δϕ = ϕ(I0 , w10 ) − ϕ(I, w1 ),
Zr
ϕ(I, w1 ) = Ω2 (I)
rmin (I)
and
dr0
−L
vr
rmin (I)
Z2π
0
ψ(r, r ) =
dθ
0
[r2
+
Zr
r0 2
dr0
.
r 0 2 vr
cos mθ
− 2rr0 cos θ]1/2
(3.7)
(3.8)
New Formulation of the Eigen-value Problem for Galactic Discs
5
(in the last equation r and r0 are understood as functions of their respective action–
angle variables: r = r(I, w1 ), r0 = r0 (I0 , w10 )). Substituting (3.6) into (2.10) we
obtain a eigen value equation for unstable modes of 2D stellar disc
Z
X
0
Fl (I)(ω − lΩ1 − mΩ2 ) = GF0,l
(I) dI0
Πl,l0 (I, I0 )Fl0 (I0 ).
(3.9)
l0
0
Here F0,l
(I) denotes
0
F0,l
(I)
¶
µ
∂F0 (I)
∂F0 (I)
∂F0
=l
+m
.
≡ m·
∂I
∂I1
∂L
(3.10)
Set of equations (3.9) can be used for searching unstable eigen-modes in any 2D
galactic stellar disc (with an arbitrary rotation curve and any amount of elongation
of star orbits).
4
Angular momentum transfer
From (3.9) it follows that
(Im ω) Lm = 0,
where
Lm = −
XZ
l
X
|Fl (I)|2
=−
dI 0
F0,l (I)
l
(4.1)
Z
dI
0
F0,l
(I)|Φl (I)|2
|lΩ1 + mΩ2 − ω|2
(4.2)
represents the angular momentum stored in the stars taking part in the wave
motion (Lynden-Bell & Kalnajs 1972).
Equation (4.1) shows that the total angular momentum of an unstable mode is
equal to zero. This result is anticipated: indeed, since the total angular momentum
remains constant, it can only be redistributed from one part of the disc to another.
Different summands in (4.1) correspond to contribution of the resonance terms
into the total angular momentum. Ability of the resonance stars to change angular
0
momentum depends on the sign of F0,l
(I). The latter for reasonable distribution
functions is opposite to the sign of the product lm. Thus stars at ILR drain, and
stars on CR, OLR, etc. gain the angular momentum, so it transfers outwards
(Lynden-Bell & Kalnajs 1972).
However, for bar-modes there is no ILR resonance. From where CR and OLR
stars get their angular momentum then? The answer is that the growth rate
should be large enough so that nonresonance stars in the disc center involve in the
wave motion. Otherwise, if Im ω → 0, all the contribution to (4.1) comes from
resonances only, and the total angular momentum of the wave cannot be zero.
0
In some cases, function F0,−1
(I) in the ILR term, can take negative values
1
somewhere in the phase space . Then, some of stars on the ILR can consume the
angular momentum, so that angular momentum transfer occurs between different
ILR stars.
1 See
Paper I for details. This function is called there the Lynden-Bell derivative.
6
5
Numerical solutions
Numerical solution of the integral equation (3.9) requires the choice of some
quadrature rule. In the examples below we use the trapezoidal rule2 . For distribution functions given in (E, L) variables, it is reasonable to change the variables
in (3.9):
dE 0 dL0
dI0 →
.
Ω1
In this section we demonstrate the efficiency of the equation (3.9) by considering
a Kalnajs model with parameters (6, 0, 1.0, 0.25) for the Plummer disc3 ,
Φ0 = − √
1
.
1 + r2
(5.1)
0
Recall that for this model the Lynden-Bell derivative, F0,−1
is positive everywhere
in the phase volume, so only bar-modes are possible (see Paper I for more detail).
Using the basic integral equation of Paper I, we determined the pattern speed
of the fastest mode; the result is reproduced in Table 1 (see Variant 1). For
comparison, in the 1st line of the Table, the “experimental” value of Athanassoula
& Sellwood (1986) obtained with help of N -body simulations is given.
To calculate this bar-mode using the integral equation (3.9), we must fix the
number of terms in the Fourier expansion (3.4). These terms involve characteristic
resonance denominators,
0
Φl F0,l
Fl =
.
(5.2)
lΩ1 + mΩ2 − ω
Looking at the denominator in equation (5.2), we understand the l = −1, l = 0
and l = 1 summands as the ILR, the CR and the OLR terms, respectively.
Table 1 summarizes the results of the computations, using only these three
terms. Taking different combinations, we can see how the eigen-frequency depends
on accounting for either resonances.
The Table shows only small variations (±6.5 %) of the pattern speed from one
variant to others, which means that its value is determined mainly by the inner
part of the disc. Moreover, the pictures of unstable modes in the complex ω-plane
resemble each other, provided that the ILR term is present. However, the situation
changes drastically when we drop the ILR term. Then we obtain the multitude of
unstable modes with the growth rates of the order of 10−3 and obviously unreasonable pattern speeds, which have nothing to do with the “experimental” values.
These results certainly confirms the conclusion about the dominating role of the
ILR term, argued in Paper I.
2 This rule has been used for the sake of simplicity. For smooth, non-singular kernels, such as
defined by Πl,l0 (I, I0 ), it is reasonable to use more efficient quadrature rules (see, e.g., Press et
al. 1992).
3 Kalnajs models is fully described in Athanassoula & Sellwood (1986); to specify the model
we use their notation.
New Formulation of the Eigen-value Problem for Galactic Discs
Variant
AS
1
2
3
4
Resonances
N -body
ILR
ILR, CR, OLR
ILR, CR
ILR, OLR
7
Eigen-frequency
0.465 + 0.066i
0.44
0.48 + 0.058i
0.43 + 0.015i
0.49 + 0.036i
Table 1. The eigen-frequency of the Kalnajs model (6, 0, 1.0, 0.25), obtained by employing basic integral equation from the Paper I (variant 1) and the equation (3.9), in which
various resonances are taken into account (variants 2 – 4), comparing with the result of
N -body simulation performed by Athanassoula & Sellwood (1986).
Variant 2 corresponds to the case, when all three terms are present. The agreement with the results of Athanassoula & Sellwood (1986) is satisfactory. Choosing
various combinations of the resonance terms, we can find out contributions of
different resonances into the growth rate of the mode. It is determined by the
interaction of the mode on resonances, first of all, CR and OLR. Variant 3 shows
the contribution of the CR to the growth rate (ILR and CR terms are taken
into account); Variant 4 – contribution of the OLR. One can see that the OLR
contribution exceed one of the CR in the total growth rate.
In general, equations (3.9) allow to consider any number of terms. However,
the higher harmonics (with |l| ≥ 2) normally cannot significantly influence the
large scale modes, which are the subject of our study. Computations confirm that
taking into account terms l = ±2 results only in small increasing of the pattern
speed and growth rate of the mode.
Fig.1 shows the pattern of the most unstable bar-mode computed from the
generalized equation (3.9), in which all three resonances are taken into account
(see Variant 2 in the Table). The spiral-like behaviour of the mode on the periphery
is due to spiral responses on the CR and OLR (Polyachenko 2002).
6
Discussion
The general approach elaborated in Section 2 formally solves the problem of eigenmodes in selfgravitating integrable systems. However, two obstacles are still there:
(i) A technical problem of computing Π-matrices for multi-dimensional systems
(N ≥ 3), which requires powerful computer facilities;
(ii) Lack of the self-consistent models for general integrable potentials4 .
In this sense, 2D disc models are lucky exceptions. On the one hand, they
admit separation of variables. On the other hand, 2D discs are the simplest models
4 To be more precise, here we mean sufficiently representative phase models: some artificial
models are available. Thus, Merritt (1990) and (1991) have studied the dynamical stability of
oblate and prolate galaxy models constructed from thin long-axis tube orbits (“shell orbits”).
8
4
3
2
1
0
−1
−2
−3
−4
−4
−2
0
2
4
Fig. 1. The picture of the surface density for the most unstable mode in the Kalnajs
model (6, 0, 1.0, 0.25) obtained from the generalized equation (3.9), in which three terms
(ILR, CR, and OLR) are taken into account. The dash-dotted lines show the locations
of CR and OLR resonances.
suitable for studying galactic structures.
There are plenty of disc models thoroughly studied by means of N -body simulations. Adaptation of our approach to study such models provides a better
insight to physical mechanisms of formation of galactic structures. In particular,
it allows in the most general form to establish a connection between patterns of
unstable modes and basic characteristics of the model (the rotation curve and the
unperturbed distribution function). The efficiency of mode calculations is proved
by comparison with earlier results, obtained both by N -body (Athanassoula &
Sellwood 1986) and modal analysis (Toomre 1977, and Polyachenko 2005).
Most of the physical mechanisms were described in Paper I, where galactic
structures are considered as the low-frequency normal modes in discs of precessing
orbits. The basic integral equation for normal modes in such discs coincides with
the system (3.9), in which only one ILR term is taken into account. From the point
of view of the present general theory, justification of the low-frequency approach
rests on two arguments:
i) Combination 2Ω(r) − κ(r), (Ω is the angular rotation velocity, κ is the
epicyclic frequency) is relatively small in a galaxy. Thus the phases in the exponents in the r.h.s. of (2.10) vary slowly when l and l0 are equal to −1 (we presume
bi-symmetric modes, m = 2). This leads to the dominance of Π−1,−1 (I, I0 ) over
others Πl,l0 .
ii) A perturbation is assumed to be localized in the centre of a galactic disc.
New Formulation of the Eigen-value Problem for Galactic Discs
9
The perturbation rapidly falls down with approaching the corotation. This assumption allows us to exclude the terms corresponding to the corotation and all
further resonances from the equation. At the same time, the mode interacts with
resonance stars by means of its long-range gravitational field, transferring the
angular momentum, that leads to growth of the mode.
Numerical analysis for models with the Plummer potential shows that the
basic integral equation of the Paper I provides 10% accuracy in determination of
pattern speeds of bar-modes. Calculation of the angular momentum exchange on
resonances determines the growth rate to an order of magnitude. In general, bar
modes occur in discs with smoothly rising rotation curves (maximum precession
rate is small enough), or if the disc is rather heavy.
In addition to bar-modes, the unified theory described in Paper I, shows the
possibility of centrally localized spiral modes, that can develop in discs with a
sufficiently hot central part or some peculiarities on the rotation curve. The necessary condition for excitation of these modes is the presence of a domain in the
phase space, where the Lynden-Bell derivative
0
F00 ≡ F0,−1
=−
∂F0
∂F0
+2
,
∂Ir
∂L
(6.1)
is negative. In this case, an unstable spiral mode consists of stars, supplying the
angular momentum, with positive Lynden-Bell derivative and stars, absorbing the
angular momentum, with negative Lynden-Bell derivative. This is a curious example when the ILR promotes a mode growth. It was pointed out in Paper I
that some of the modes obtained in N -body experiments by 1986 have just this
nature. From the analysis of the equation (4.2), we can suggest peripheric spiral
modes, with rather small amplitude at the ILR, and larger amplitude at the CR.
An example of the peripheric spiral mode is provided in Toomre (1977), see Polyachenko (2005) for details. Such modes are spread over the whole disc, have small
amplitudes in the disc centre, and can be obtained by using the general equation
(3.9) only.
In view of aforesaid, we understand a bar-mode as a mode without ILR. The
entire bar-mode consists of the bar itself and the adjacent spiral arms. The spiral
modes of both types grow due to the interaction on ILR, although they differ by
the roles of other resonances. For centrally localized spirals the released angular
momentum can be absorbed by other stars on the same ILR, while the peripheral
spirals give angular momentum released on ILR away to CR and OLR.
Acknowledgments
With thanks to V.L. Polyachenko for useful discussions. The work was supported
in part by Russian Science Support Foundation, RFBR grant No. 02-02-16878,
grant “Leading Scientific Schools” No. 925.2003.2, and the contract with Ministry
of Industry, Science, and Technology No. 40.022.1.1.1101 and 40.020.1.1.1167.
10
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