Unstable and Stable Galaxy Models

TL;DR

This paper investigates the stability of isotropic spherical galaxy models, deriving a criterion for linear instability and proving the nonlinear stability of the King model under spherical perturbations.

Contribution

It introduces a new sufficient criterion for linear instability and demonstrates the nonlinear stability of the King model, advancing understanding of galaxy model stability.

Findings

Derived a criterion linking operator negativity to instability.

Proved the King model's operator is positive definite.

Established nonlinear stability of the King model under spherical perturbations.

Abstract

To determine the stability and instability of a given steady galaxy configuration is one of the fundamental problems in the Vlasov theory for galaxy dynamics. In this article, we study the stability of isotropic spherical symmetric galaxy models $f_{0}(E)$, for which the distribution function $f_{0}$ depends on the particle energy $E$ only. In the first part of the article, we derive the first sufficient criterion for linear instability of $f_{0}(E):$ $f_{0}(E)$ is linearly unstable if the second-order operator \[ A_{0}\equiv-\Delta+4\pi\int f_{0}^{\prime}(E)\{I-\mathcal{P}\}dv \] has a negative direction, where $\mathcal{P}$ is the projection onto the function space $\{g(E,L)\},$ $L$ being the angular momentum [see the explicit formula (\ref{A0-radial})]. In the second part of the article, we prove that for the important King model, the corresponding $A_{0}$ is positive definite. Such a positivity leads to the nonlinear stability of the King model under all spherically symmetric perturbations.

Equations

$$A_0\equiv -\Delta +4\pi \int f_0^{\prime }(E)\{I-\mathcal{P}\}dv$$

$${\partial }_{t}f+v\cdot {\nabla }_{x}f-{\nabla }_{x}U\cdot {\nabla }_{v}f=0,\Delta U=4\pi {\int }_{{\mathbf{R}}^3}f(t,x,v)dv,$$

$$f_0(x,v)\equiv f_0(E,L^2),$$

$$E=\frac{1}{2}|v{|}^2+U_0(x),L^2={\left|x\times v\right|}^2,$$

$$f_0(x,v)\equiv f_0(E).$$

$$\mathcal{H}(f)\equiv \int \int Q(f)+\frac{1}{2}\int \int |v{|}^{2}f-\frac{1}{8\pi }\int |{\nabla }_x{U}_f{|}^2,$$

$$Q_0^{\prime }(f_0(E))\equiv -E$$

$${\mathcal{H}}_{f_0}^{(2)}[g]\equiv \frac{1}{2}\int {\int }_{\left\{f_0>0\right\}}\frac{g^2}{-f_0^{\prime }(E)}dxdv-\frac{1}{8\pi }\int |{\nabla }_x{U}_g{|}^{2}dx$$

$${\mathcal{H}}_{f_0}^{(2)}[Dh]=\int \int \frac{{\left|Dh\right|}^2}{\left|f_0^{\prime }(E)\right|}dxdv-\frac{1}{4\pi }\int {\left|\nabla {\psi }_h\right|}^{2}dx$$

$$D=v\cdot {\nabla }_x-{\nabla }_x{U}_0\cdot {\nabla }_v,$$

$$||h|{|}_{|f_0^{\prime }|}^2\equiv \int \int |f_0^{\prime }(E)|h^{2}dxdv.$$

$$(A_0\varphi ,\varphi )=\int |\nabla \varphi {|}^{2}dx+4\pi \int \int f_0^{\prime }(E){\left(\varphi -\mathcal{P}\varphi \right)}^{2}dxdv,$$

$$\ker D=\left\{g\left(E,L^2\right)\right\},$$

$$(A_0{\varphi }_0,{\varphi }_0)<0,$$

$$f_0=[e^{E_0-E}-1{]}_{+}$$

$$Q_0(f)=(1+f)\ln (1+f)-1-f,$$

$${\mathcal{S}}_{f_0}\equiv \left[f(t,r,v_r,L)\ge 0:\int Q(f,L)=\int Q(f_0,L),\text{ for }Q\in C_c^{\infty }\text{ and }Q(0,L)\equiv 0.\right].$$

$$\underset{0\le t<\infty }{\sup }d\left(f\left(t\right),f_0\right)<\epsilon ,$$

$$f_0\left(x,v\right)=f_0(E)$$

$$\Delta U_0=4\pi \int f_{0}dv.$$

$${\partial }_{t}f+v\cdot {\nabla }_{x}f-{\nabla }_x{U}_0\cdot {\nabla }_{v}f={\nabla }_x\varphi \cdot {\nabla }_v{f}_0,\Delta \varphi =4\pi \int f(t,x,v)dv.$$

$$\lambda f+v\cdot {\nabla }_{x}f-{\nabla }_x{U}_0\cdot {\nabla }_{v}f=f_0^{\prime }v\cdot {\nabla }_x\varphi .$$

$$\left\{\begin{array}[c]{c}\frac{dX(s;x,v)}{ds}=V(s;x,v)\\ \frac{dV(s;x,v)}{ds}=-\nabla_{x}U_{0}\end{array}\right.$$

$$f(x,v)$$

$$=f_0^{\prime }(E)\varphi (x)-f_0^{\prime }(E){\int }_{-\infty }^0\lambda e^{\lambda s}\varphi (X(s;x,v))ds.$$

$$-\Delta \varphi +[4\pi \int f_0^{\prime }(E)dv]\varphi -4\pi \int f_0^{\prime }(E){\int }_{-\infty }^0\lambda e^{\lambda s}\varphi (X(s;x,v))dsdv=0.$$

$$A_{\lambda }\varphi \equiv -\Delta \varphi +[4\pi \int f_0^{\prime }(E)dv]\varphi -4\pi \int f_0^{\prime }(E){\int }_{-\infty }^0\lambda e^{\lambda s}\varphi (X(s;x,v))dsdv.$$

$$K_{\lambda }\varphi =-4\pi [\int f_0^{\prime }(E)dv]\varphi +4\pi \int f_0^{\prime }(E){\int }_{-\infty }^0\lambda e^{\lambda s}\varphi (X(s;x,v))dsdv.$$

$$f_0^{\prime }\left(x,v\right)=f_0^{\prime }\left(X(s;x,v),V(s;x,v)\right)$$

$$\left(K_{\lambda }\varphi \right)\left(x\right)$$

$$=-4\pi [\int f_0^{\prime }(E)dv]\varphi +4\pi \int {\int }_{-\infty }^0\lambda e^{\lambda s}\left(f_0^{\prime }(E)\varphi \right)(X(s;x,v))dsdv$$

Full text

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has a negative direction, where $\mathcal{P}$ is the projection onto the function space $\{g(E,L)\},$ $L$ being the angular momentum [see the explicit formulae (27) and (26)]. In the second part of the article, we prove that for the important King model, the corresponding $A_{0}$ is positive definite. Such a positivity leads to the nonlinear stability of the King model under all spherically symmetric perturbations.

1. Introduction

A galaxy is an ensemble of billions of stars, which interact by the gravitational field which they create collectively. For galaxies, the collisional relaxation time is much longer than the age of the universe ([8]). The collisions can therefore be ignored and the galactic dynamics is well described by the Vlasov - Poisson system (collisionless Boltzmann equation)

[TABLE]

where $\left(x,v\right)\in\mathbb{R}^{3}\times\mathbb{R}^{3}$, $f(t,x,v)$ is the distribution function and $U_{f}\left(t,x\right)$ is its gravitational potential. The Vlasov-Poisson system can also be used to describe the dynamics of globular clusters over their period of orbital revolutions ([11]). One of the central questions in such galactic problems, which has attracted considerable attention in the astrophysics literature, of [7], [8], [11], [31] and the references there, is to determine *dynamical stability *of steady galaxy models. Stability study can be used to test a proposed configuration as a model for a real stellar system. On the other hand, instabilities of steady galaxy models can be used to explain some of the striking irregularities of galaxies, such as spiral arms as arising from the instability of an initially featureless galaxy disk ([7]), ([32]).

In this article, we consider stability of spherical galaxies, which are the simplest elliptical galaxy models. Though most elliptical galaxies are known to be non-spherical, the study of instability and dynamical evolution of spherical galaxies could be useful to understand more complicated and practical galaxy models . By Jeans’s Theorem, a steady spherical galaxy is of the form

[TABLE]

where the particle energy and total momentum are

[TABLE]

and $U_{0}(x)=U_{0}\left(\left|x\right|\right)$ satisfies the self-consistent Poisson equation. The isotropic models take the form

[TABLE]

The cases when $f_{0}^{\prime}(E)<0$ has been widely studied and these models are known to be linearly stable to both radial ([9]) and non-radial perturbations ([2]). The well-known Casimir-Energy functional (as a Liapunov functional)

[TABLE]

is constant along the time evolution. If $f_{0}^{\prime}(E)<0,$ we can choose the Casimir function $Q_{0}$ such that

[TABLE]

for all $E.$ By a Taylor expansion of $\mathcal{H}(f)-\mathcal{H}(f_{0})$, it follows that formally the first variation at $f_{0}$ is zero, that is, $\mathcal{H}^{(1)}(f_{0}(E))=0$ (on the support of $f_{0}(E)$), and the second order variation of $\mathcal{H}$ at $f_{0}$ is

[TABLE]

where $Q^{\prime\prime}(f_{0})=\frac{1}{-f_{0}^{\prime}(E)},\ g=f-f_{0}$ and $\Delta U_{g}=\int gdv$. In the 1960s, Antonov ([1], [2]) proved that

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is positive definite for a large class of monotone models. Here

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$\ h(x,v)$ is odd in $v$ and $-\Delta\psi=\int Dhdv$. He showed that such a positivity is equivalent to the linear stability of $f_{0}(E)$. In [9], Doremus, Baumann and Feix proved the radial stability of any monotone spherical models. Their proof was further clarified and simplified in [10], [37], [22], and more recently in [33], [21]. In particular, this implies that any monotone isotropic models are at least linearly stable.

Unfortunately, despite its importance and a lot of research (e.g., [20], [5], [6], [13]), to our knowledge, no rigorous and explicit instability criterion of non-monotone models has been derived. When $f_{0}^{\prime}(E)$ changes sign, functional $\mathcal{H}_{f_{0}}^{(2)}$ is indefinite and it gives no stability information, although it seems to suggest that these models are not energy minimizers under symplectic perturbations. In this paper, we first obtain the following instability criterion for general spherical galaxies. For any function $g$ with compact support within the support of $f_{0}(E),$ we define the $\left|f_{0}^{\prime}(E)\right|-$weighted $L^{2}\left(\mathbf{R}^{3}\times\mathbf{R}^{3}\right)$ space $L_{\left|f_{0}^{\prime}\right|}^{2}$ with the norm $\left\|\cdot\right\|_{\left|f_{0}^{\prime}\right|}$ as

[TABLE]

Theorem 1.1**.**

Assume that $f_{0}(E)$ has a compact support in $x$ and $v,$ and $f_{0}^{\prime}$ is bounded. For $\phi\in H^{1},$ define the quadratic form

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where $\mathcal{P}$ is the projector of $L_{\left|f_{0}^{\prime}\right|}^{2}$ to

[TABLE]

and more explicitly $\mathcal{P}\phi$ is given by (18) for radial functions and (26) for general functions. If there exists $\phi_{0}\in H^{1}$ such that

[TABLE]

then there exists $\lambda_{0}>0$ and $\phi\in H^{2},$ $f\left(x,v\right)$ given by (14), such that $e^{\lambda_{0}t}[f,\phi]$ is a growing mode to the Vlasov-Poisson system (1) linearized around $\left[f_{0}(E),U_{f_{0}}\right].$

A similar instability criterion can be obtained for symmetry preserving perturbations of anisotropic spherical models $f_{0}\left(E,L^{2}\right)$, see Remark 2. We note that the term $\mathcal{P}\phi$ in the instability criterion is highly non-local and this reflects the collective nature of stellar instability. The proof of Theorem 1.1 is by extending an approach developed in [25] for 1D Vlasov-Poisson, which has recently been generalized to Vlasov-Maxwell systems ([26], [28]). There are two elements in this approach. One is to formulate a family of dispersion operators $A_{\lambda}$ for the potential, depending on a positive parameter $\lambda$. The existence of a purely growing mode is reduced to find a parameter $\lambda_{0}$ such that the $A_{\lambda_{0}}$ has a kernel. The key observation is that these dispersion operators are self-adjoint due to the reversibility of the particle trajectories. Then a continuation argument is applied to find the parameter $\lambda_{0}$ corresponding to a growing mode, by comparing the spectra of $A_{\lambda}$ for very small and large values of $\lambda$. There are two new complications in the stellar case. First, the essential spectrum of $A_{\lambda}$ is $[0,+\infty)$ and thus we need to make sure that the continuation does not end in the essential spectrum.This is achieved by using some compactness property due to the compact support of the stellar model. Secondly, it is more tricky to find the limit of $A_{\lambda}$ when $\lambda$ tends to zero. For that, we need an ergodic lemma (Lemma 2.4) and use the integrable nature of the particle dynamics in a central field to derive an expression for the projection $\mathcal{P}\phi$ appeared in the limit.

In the second part of the article, we further study the nonlinear (dynamical) stability of the normalized King model:

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motivated by the study of the operator $A_{0}.$ The famous King model describes isothermal galaxies and the core of most globular clusters [24]. Such a model provides a canonical form for many galaxy models widely used in astronomy. Even though $f_{0}^{\prime}<0$ for the King model, it is important to realize that, because of the Hamiltonian nature of the Vlasov-Poisson system (1), linear stability fails to imply nonlinear stability (even in the finite dimensional case). The Liapunov functional is usually required to prove nonlinear stability. In the Casimir-energy functional (2), it is natural to expect that the positivity of such a quadratic form $\mathcal{H}_{f_{0}}^{(2)}[g]$ should imply stability for $f_{0}(E)$. However, there are at least two serious mathematical difficulties. First of all, it is very challenging to use the positivity of $\mathcal{H}_{f_{0}}^{(2)}[g]$ to control higher order remainder in $\mathcal{H}(f)-\mathcal{H}(f_{0})$ to conclude stability [38]. For example, one of the remainder terms is $f^{3}$ whose $L^{2}$ norm is difficult to be bounded by a power of the stability norm. The non-smooth nature of $f_{0}(E)$ also causes trouble here. Second of all, even if one can succeed in controlling the nonlinearity, the positivity of $H_{f_{0}}^{(2)}[g]$ is only valid for certain perturbation of the form $g=Dh$ [22]. It is not clear at all if any arbitrary, general perturbation can be reduced to the form $Dh$. To overcome these two difficulties, a direct variational approach was initiated by Wolansky [39], then further developed systematically by Guo and Rein in [14], [15], [17], [18], [19]. Their method avoids entirely the delicate analysis of the second order variation $\mathcal{H}_{f_{0}}^{(2)}$ in (3), which has led to first rigorous nonlinear stability proof for a large class of $f_{0}(E).$ The high point of such a program is the nonlinear stability proof for every polytrope [18] $f_{0}(E)=(E_{0}-E)_{+}^{k}$. Their basic idea is to construct galaxy models by solving a variational problem of minimizing the energy under some constraints of Casimir invariants. A concentration-compactness argument is used to show the convergence of the minimizing sequence. All the models constructed in this way are automatically stable.

Unfortunately, despite its success, the King model can not be studied by such a variational approach. The Casimir function for a normalized King model is

[TABLE]

which has very slow growth for $f\rightarrow\infty.$ As a result, the direct variational method fails. Recently, Guo and Rein [21] proved nonlinear radial stability among a class of measure-preserving perturbations

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The basic idea is to observe that for perturbations in the class $\mathcal{S}_{f_{0}}$, one can write $g=f-f_{0}$ as $Dh=\left\{h,E\right\}$. Therefore, $\mathcal{H}_{f_{0}}^{(2)}[g]$ $=\mathcal{H}_{f_{0}}^{(2)}[Dh]$, for which the positivity was proved in [22] for radial perturbations. To avoid the difficulty of controlling the remainder term by $\mathcal{H}_{f_{0}}^{(2)}[g]$, an indirect contradiction argument was used in [21].

As our second main result of this article, we establish nonlinear stability of King’s model for general perturbations with spherical symmetry:

Theorem 1.2**.**

The King’s model $f_{0}=[e^{E_{0}-E}-1]_{+}$ is nonlinearly stable under spherically symmetric perturbations in the following sense: given any $\varepsilon>0$ there exists $\varepsilon_{1}>0$ such that for any compact supported initial data $f(0)\in C_{c}^{1}$ with spherical symmetry, if $d\left(f\left(0\right),f_{0}\right)<\varepsilon_{1}$ then

[TABLE]

where the distance functional $d\left(f,f_{0}\right)$ is defined by (35).

For the proof, we extended the approach in [27] for the $1\frac{1}{2}D$ Vlasov-Maxwell model. To prove nonlinear stability, we study the Taylor expansion of $\mathcal{H}(f)-\mathcal{H}(f_{0})$. Two difficulties as mentioned before are: to prove the positivity of the quadratic form and to control the remainder. We use two ideas introduced in [27]. The first idea is to use any finite number of Casimir functional $Q_{i}\left(f,L^{2}\right)$ as constraints. The difference from [21] is that we do not impose $Q_{i}\left(f,L^{2}\right)=Q_{i}\left(f_{0},L^{2}\right)$ in the perturbation class, but expand the invariance equation $Q_{i}\left(f\left(t\right),L^{2}\right)-Q_{i}\left(f_{0},L^{2}\right)=Q_{i}\left(f\left(0\right),L^{2}\right)-Q_{i}\left(f_{0},L^{2}\right)$ to the first order. In this way, we get a constraint for $g=f-f_{0}$ in the form that the coefficient of its projection to $\partial_{1}Q_{i}\left(f_{0},L^{2}\right)$ is small. Putting these constraints together, we deduce that a finite dimensional projection of $g$ to the space spanned by $\left\{\partial_{1}Q_{i}\left(f_{0},L^{2}\right)\right\}$ is small. To control the remainder term, we use a duality argument. Noting that it is much easier to control the potential $\phi$, we use a Legendre transformation to reduce the nonlinear term in $g$ to a new one in $\phi$ only. The key observation is that the constraints on $g$ in the projection form are nicely suited to the Legendre transformation and yields a non-local nonlinear term in $\phi$ only with the projections kept. By performing a Taylor expansion of this non-local nonlinear term in $\phi$, the quadratic form becomes a truncated version of $(A_{0}\phi,\phi)$ defined by (6), whose positivity can be shown to be equivalent to that of Antonov functional. The the remainder term now is only in terms of $\phi$ and can be easily controlled by the quadratic form. The new complication in the stellar case is that the steady distribution $f_{0}\left(E\right)$ is non-smooth and compactly supported. Therefore, we split the perturbation $g$ into inner and outer parts, according to the support of $f_{0}$. For the inner part, we use the above constrainted duality argument and the outer part is estimated separately.

2. An Instability Criterion

We consider a steady distribution

[TABLE]

has a bounded support in $x$ and $v$ and $f_{0}^{\prime}$ is bounded, where the particle energy $E=\frac{1}{2}|v|^{2}+U_{0}(x).$ The steady gravitational potential $U_{0}(x)$ satisfies a nonlinear Poisson equation

[TABLE]

The linearized Vlasov-Poisson system is

[TABLE]

A growing mode solution $(e^{\lambda t}f(x,v),e^{\lambda t}\phi(x))$ to (1) with $\lambda>0$ satisfies

[TABLE]

We define $[X(s;x,v),V(s;x,v)]$ as the trajectory of

[TABLE]

such that $X(0;x,v)=x,~{}$and $V(0;x,v)=v.$ Notice that the particle energy $E$ is constant along the trajectory. Integrating along such a trajectory for $-\infty\leq s\leq 0$, we have

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Plugging it back into the Poisson equation, we obtain an equation for $\phi$

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We therefore define the operator $A_{\lambda}$ as

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Lemma 2.1**.**

Assume that $f_{0}(E)$ has a bounded support in $x$ and $v$ and $f_{0}^{\prime}$ is bounded. For any $\lambda>0$, the operator $A_{\lambda}:H^{2}\rightarrow$ $L^{2}$ is self-adjoint with the essential spectrum $[0,+\infty)\,.$

Proof.

We denote

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Recall that $f_{0}\left(x,v\right)=f_{0}(E)$ has a compact support $\subset S\subset\mathbb{R}_{x}^{3}\times\mathbb{R}_{v}^{3}$. We may assume $S=S_{x}\times S_{v}$, both balls in $\mathbb{R}^{3}$. Let $\chi=\chi\left(|x|\right)$ be a smooth cut-off function for the spatial support of $f_{0}$ in the physical space $S_{x}$; that is, $\chi\equiv 1$ on the spatial support of $f_{0}$ and has compact support inside $S_{x}$. Let $M_{\chi}$ be the operator of multiplication by $\chi$. Then $K_{\lambda}=K_{\lambda}M_{\chi}=M_{\chi}K_{\lambda}=M_{\chi}K_{\lambda}M_{\chi}$. Indeed,

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because of the invariance of $E$ under the flow. So

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First we claim that

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Indeed, the $L^{2}$ norm for the first term in $K_{\lambda}$ is easily bounded by $4\pi\left|\int f_{0}^{\prime}(E)dv\right|_{\infty}$. For the second term, we have for any $\psi\in L^{2},$

[TABLE]

Moreover, we have that $K_{\lambda}$ is symmetric Indeed, for fixed $s,$ by making a change of variable $(y,w)\rightarrow(X(s;x,v),V(s;x,v)),$ so that $(z,v)=(X(-s;y,w),V(-s;y,w)),$ we deduce that

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Here we have used the fact $[X(s;y,w),V(s;y,w)]=[X(-s;y,-w),-V(s;y,-w)]$ in the last line. Hence

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Since $K_{\lambda}=K_{\lambda}M_{\chi}$ and $M_{\chi}$ is compact from $H^{2}$ into $L^{2}$ space with support in $S_{x}$, so $K_{\lambda}$ is relatively compact with respect to $-\Delta$. Thus by Kato-Relich and Weyl’s Theorems, $A_{\lambda}:H^{2}\rightarrow$ $L^{2}$ is self-adjoint and $\sigma_{\text{ess}}(A_{\lambda})=\sigma_{\text{ess}}(-\Delta).$ ∎

Lemma 2.2**.**

Assume that $f_0^{}$