Local stability of self-gravitating disks in $f(R)$ gravityNote: Not to appear in Nonlearned J., 45.

In this section we derive the Poisson equation in the weak field limit of metric $f(R)$ gravity. For this purpose, it is convenient to combine equations (8) and (9) as a single equation. The result is the analouge the Poisson equation in Newtonian gravity. Using equations (8) and (9), one can show that

$\displaystyle\nabla^{2}\Psi=-\nabla^{2}\Phi+\frac{8\pi G}{f^{\prime}_{0}}\rho$

(14)

Now, substituting equation (14) into (9), we find

$\displaystyle(\nabla^{2}-m_{0}^{2})\nabla^{2}\Phi=-4\pi G\alpha\left(m_{0}^{2}\rho-\frac{4}{3}\nabla^{2}\rho\right)$

(15)

where $\alpha=1/f^{\prime}_{0}$ and $m_{0}$ is defined as

$\displaystyle m_{0}^{2}=-\frac{f^{\prime}_{0}}{3f^{\prime\prime}_{0}}$

(16)

in fact, equation (15) is the modified version of the Poisson equation. Therefore, loosely speaking, we are dealing with a theory with two free parameters $\alpha$ and $\beta$. In the limit $m_{0}^{2}\rightarrow\infty$ and $\alpha\rightarrow 1$, the standard Poisson equation is recovered, i.e. equation (15) takes the form $\nabla^{2}\Phi=4\pi G\rho$. Solution of this fourth order equation (15) can be written as follows

$\displaystyle\begin{split}\Phi(\mathbf{r})=-4\pi G\alpha&\int\int G_{1}(\mathbf{r}^{\prime\prime},\mathbf{r}^{\prime})G_{2}(\mathbf{r},\mathbf{r}^{\prime\prime})\\ &\times\left(m_{0}^{2}\rho(\mathbf{r}^{\prime})-\frac{4}{3}\nabla^{2}\rho(\mathbf{r}^{\prime})\right)d^{3}\mathbf{r}^{\prime}d^{3}\mathbf{r}^{\prime\prime}\end{split}$

(17)

where $G_{1}$ and $G_{2}$ are the Green functions:

$\displaystyle\begin{split}&G_{1}(\mathbf{r}^{\prime\prime},\mathbf{r}^{\prime})=-\frac{1}{4\pi}\frac{e^{-\sqrt{m_{0}^{2}}|\mathbf{r}^{\prime}-\mathbf{r}^{\prime\prime}|}}{|\mathbf{r}^{\prime}-\mathbf{r}^{\prime\prime}|}\\ &G_{1}(\mathbf{r},\mathbf{r}^{\prime\prime})=-\frac{1}{4\pi}\frac{1}{|\mathbf{r}-\mathbf{r}^{\prime\prime}|}\end{split}$

(18)

it is natural to expect that the free parameters $\alpha$ and $m_{0}$ should be fixed by relevant observations. Also, in order to prevent the oscillatory solutions, $m_{0}^{2}$ should be positive, i.e. $m_{0}^{2}>0$.

It is worth noting that the gravitational potential of a point source takes the following form

$\displaystyle\Phi(r)=-\frac{GM\alpha}{r}\left[1+\frac{1}{3}e^{-m_{0}r}\right]$

(19)

This result can be straightforwardly derived from equation (17) just by substituting the matter density as $\rho(\mathbf{r})=M\delta(\mathbf{r})$, where $M$ is the mass of the point particle.

In this section, we derive the collisionless Boltzamnn equation in the weak field limit of metric $f(R)$ gravity. This equation is the main equation governing the dynamics of a stellar system. Assume that $f(x^{\mu},u^{\mu})$ is the particle’s phase space distribution function. Where $u^{\mu}=\frac{dx^{\mu}}{d\tau}$ is the four velocity of the particle. In Newtonian dynamics the Boltzmann equation can be expressed as $\hat{L}[f]=0$, where $\hat{L}$ is the Liouville operator. It is worth remembering that in metric $f(R)$ gravity non-spinning test particles move on the geodesics. Therefore with the aid of the geodesic equation, we obtain the covariant, relativistic generalization of the Boltzmann equation

$\displaystyle\hat{L}[f]=u^{\mu}\frac{\partial f}{\partial x^{\mu}}-\Gamma^{\mu}_{\alpha\beta}u^{\alpha}u^{\beta}\frac{\partial f}{\partial u^{\mu}}=0$

(20)

It is straightforward to linearize this equation using the perturbation introduced in Eq. (6). Keeping in mind that in the first order Newtonian limit $\Gamma^{i}_{00}\simeq\frac{\partial\Phi}{\partial x^{i}}$ and $u^{i}\simeq v^{i}=\frac{dx^{i}}{dt}$, we have

$\displaystyle\frac{\partial f}{\partial t}+\mathbf{v}\cdot\nabla f-\nabla\Phi\cdot\frac{\partial f}{\partial\mathbf{v}}=0$

(21)

as expected, only metric potential $\Phi$ appears in the Boltzmann equation. Mathematically, this equation is the same as the Boltzmann equation in Newtonian dynamics. However, we know that the field equation governing $\Phi$ is different from the standard Poisson equation.

In this section, we analyze the behavior of a tightly wound density wave in the framework of metric $f(R)$ gravity. First, we use the modified Poisson equation (15) in order to calculate the gravitational potential of a tightly wound surface density. Furthermore, we linearize the relevant equations in order to find the dispersion relation for such a density wave.

The continuity equation (12), the Euler equation (13) and the modified Poisson equation (15) are the required equation for studying the gravitational stability of the system. It is important to remember that we also need the equation of state of the fluid to make a complete description. We assume that the background disk is axisymmetric and is placed in $z=0$ plane. Using the cylindrical coordinates $(R,\phi,z)$, the continuity equation reads (note that hereafter $R$ is a coordinate and should not be confused with the Ricci scalar)

$\displaystyle\frac{\partial\Sigma}{\partial t}+\frac{1}{R}\frac{\partial}{\partial R}\left(\Sigma Rv_{R}\right)+\frac{1}{R}\frac{\partial}{\partial\phi}\left(\Sigma v_{\phi}\right)=0$

(24)

where $\Sigma$ is the surface density and $v_{R}$ and $v_{\phi}$ are the velocity components in the radial and azimuthal directions respectively. Since the disk is located in the $x-y$ plane, there are only two components for the Euler equation. These components can be , respectively, expressed as:

	$\displaystyle\frac{\partial v_{R}}{\partial t}+v_{R}\frac{\partial v_{R}}{\partial R}+\frac{v_{\phi}}{R}\frac{\partial v_{R}}{\partial\phi}-\frac{v_{\phi}^{2}}{R}=-\frac{\partial}{\partial R}\left(\Phi+h\right)$		(25)
	$\displaystyle\frac{\partial v_{\phi}}{\partial t}+v_{R}\frac{\partial v_{\phi}}{\partial R}+\frac{v_{\phi}}{R}\frac{\partial v_{\phi}}{\partial\phi}+\frac{v_{\phi}v_{R}}{R}=-\frac{1}{R}\frac{\partial}{\partial\phi}\left(\Phi+h\right)$		(26)

where we have chosen a simple barotropic equation of state as $p=K\Sigma^{\gamma}$, where $K$ and $\gamma$ are constant real parameters. Also, $h$ is the specific enthalpy defined as:

$\displaystyle h=\int\frac{dp}{\Sigma}$

(27)

Now, let us perform the following perturbations: $\Sigma=\Sigma_{0}+\Sigma_{1}$, $v_{R}=v_{R0}+v_{R1}=v_{R1}$, $v_{\phi}=v_{\phi 0}+v_{\phi 1}$, $\Phi=\Phi_{0}+\Phi_{1}$ and $h=h_{0}+h_{1}$. We denote the equilibrium quantities with a ”$0$” subscript and the perturbations with a ”$1$” subscript. After linearization, the fluid equations (24) and (26) can be written as :

$\displaystyle\frac{\partial\Sigma_{1}}{\partial t}+\frac{1}{R}\frac{\partial}{\partial R}\left(\Sigma_{0}Rv_{R1}\right)+\Omega\frac{\partial\Sigma_{1}}{\partial\phi}+\frac{\Sigma_{0}}{R}\frac{\partial v_{\phi 1}}{\partial\phi}=0$

(28)

$\displaystyle\frac{\partial v_{R1}}{\partial t}+\Omega\frac{\partial v_{R1}}{\partial\phi}-2\Omega v_{\phi 1}=-\frac{\partial}{\partial R}\left(\Phi_{1}+h_{1}\right)$

(29)

$\displaystyle\frac{\partial v_{\phi 1}}{\partial t}+\Omega\frac{\partial v_{\phi 1}}{\partial\phi}+\frac{\kappa^{2}}{2\Omega}v_{R1}=-\frac{1}{R}\frac{\partial}{\partial\phi}\left(\Phi_{1}+h_{1}\right)$

(30)

where $\Omega(R)$ is the circular frequency and the epicyclic frequency $\kappa$ is

$\displaystyle\kappa^{2}(R)=R\frac{d\Omega^{2}}{dR}+4\Omega^{2}$

(31)

Also, the modified Poisson equation can be linearized as

$\displaystyle\begin{split}(\nabla^{2}-m_{0}^{2})\nabla^{2}\Phi_{1}=&-4\pi G\alpha\\ &\times\left(m_{0}^{2}\Sigma_{1}\delta(z)-\frac{4}{3}\nabla^{2}\Sigma_{1}\delta(z)\right)\end{split}$

(32)

Now, consider an arbitrary spiral surface density perturbation mode as follows

$\displaystyle\Sigma_{1}(R,\phi,t)=H(R)e^{i(F(R)+m\phi-\omega t)}$

(33)

where $\omega$ is the frequency of the mode, $H$ is a slowly varying function of $R$ and $F(R)$ is the shape function. Also, $m$ is a positive integer, and the perturbation has $m$-fold rotational symmetry. The WKB approximation requires that $\frac{|kR|}{m}\gg 1$, where $k(R)=\frac{dF}{dR}$. In this approximation the pitch angle of the spiral arms is very small at any radius. Therefore, one can conclude that the spiral arms are tightly wound.

Let us find the gravitational potential of the tightly wound spiral perturbation (33) in the neighborhood of a point $(R_{0},\phi_{0})$. Using the properties of WKB approximation, one can expand $\Sigma_{1}$ as follows (for more detail see Binney & Tremaine 2008)

$\displaystyle\Sigma_{1}(R,\phi,t)\sim\Sigma_{a}(R_{0})e^{i(k(R_{0})R-\omega t)}$

(34)

where

$\displaystyle\Sigma_{a}(R_{0})=H(R_{0})e^{i(F(R_{0})+m\phi_{0}-k(R_{0})R_{0})}$

(35)

the spiral perturbation (34) is reminiscent of a plane wave with wavenumber $\mathbf{k}=k\mathbf{e}_{R}$, where $\mathbf{e}_{R}$ is the unit vector in the radial direction. The potential of a plane wave in a thin disk in the framework of the $f(R)$ gravity has been derived in Appendix A. Using equation (A5), the potential can be expressed as

$\displaystyle\Phi_{1}=-\frac{2\pi G\alpha}{|k|}\frac{m_{0}^{2}}{k^{2}+m_{0}^{2}}\Sigma_{1}$

(36)

Regarding the form of $\Sigma_{1}$ and $\Phi_{1}$, any solution of equations (28)-(30) can be expressed as

	$\displaystyle v_{R1}(R,\phi,t)=v_{Ra}(R)e^{i(m\phi-\omega t)}$		(37)
	$\displaystyle v_{\phi 1}(R,\phi,t)=v_{\phi a}(R)e^{i(m\phi-\omega t)}$		(38)
	$\displaystyle h_{1}(R,\phi,t)=h_{a}(R)e^{i(m\phi-\omega t)}$		(39)

In Newtonian gravity, using the properties of the WKB approximation, equations (28)-(30) take the following forms respectively (for more detail see Binney & Tremaine 2008 )

$\displaystyle(m\Omega-\omega)\Sigma_{a}+k\Sigma_{0}v_{Ra}=0$

(40)

	$\displaystyle v_{Ra}=\frac{(m\Omega-\omega)k(\Phi_{a}+h_{a})}{\Delta}$		(41)
	$\displaystyle v_{\phi a}=-\frac{2iBk(\Phi_{a}+h_{a})}{\Delta}$		(42)

where coefficient functions $\Delta$ and $B$ (the Oort constant of rotation) are:

	$\displaystyle\Delta=\kappa^{2}-(m\Omega-\omega)^{2}$		(43)
	$\displaystyle B(R)=-\left(\Omega+\frac{1}{2}R\frac{d\Omega}{dR}\right)$		(44)

It is important noting that mathematical form of the Euler equation in the weak field limit of $f(R)$ gravity is the same as the Newtonian gravity. The only difference between these theories appears in the way by which the disk responds to the perturbation. On the other hand, the Poisson equation determines the response or the behavior of the disk to the perturbation. Since, for obtaining equations (41), the Poisson equation has not been used, so we can conclude that they are also true in the weak filed limit of $f(R)$ gravity. This is the case also for continuity equation (40). Therefore, we skip the details of the calculations for deriving these equations, (40) and (41), and refer the reader to chapter 6 of Binney & Tremaine 2008.

Now, we substitute $v_{Ra}$ from equation (41) into equation (40) and use equation (36). Also, taking into account that $h_{a}=v_{s}^{2}\frac{\Sigma_{a}}{\Sigma_{0}}$, we find the following dispersion relation

$\displaystyle(m\Omega-\omega)^{2}=\kappa^{2}-2\pi G|k|\Sigma_{0}\frac{\alpha m_{0}^{2}}{k^{2}+m_{0}^{2}}+k^{2}v_{s}^{2}$

(45)

We shall derive the local stability criterion from this dispersion relation. As expected, in the limit $m_{0}\rightarrow\infty$ and $\alpha\rightarrow 1$, the dispersion relation for WKB modes in Newtonian gravity can easily be recovered. In this paper, we restrict ourselves to the axisymmetric disturbances, i.e. $m=0$. Since the right-hand-side of (45) is real, then the disk is stable against local axisymmetric disturbances if $\omega^{2}>0$ and unstable if $\omega^{2}<0$.

It is worthwhile to mention that Toomre’s criteria (22) and (23) have also been derived for axisymmetric perturbations. However, it cannot be concluded strongly that they are not applicable for local nonaxisymmetric stability. In fact, no general criterion for nonaxisymmetric stability is known in Newtonian gravity.

Now, we consider the general case by adopting non-zero speed of sound. In this case, using the dispersion relation (45), the stability criterion $\omega^{2}>0$ can be rewritten in the dimensionless form

$\displaystyle X^{4}+(1+\beta^{2})X^{2}-\frac{2\beta\alpha}{Q_{g}}|X|+\beta^{2}>0$

(46)

where

$\displaystyle X=\frac{k}{m_{0}}~~~~,~~~~\beta=\frac{\kappa}{m_{0}v_{s}}$

(47)

The schematic behavior of the left-hand side (LHS) of equation (46) is shown in Figure 1. It is clear from this figure that, if the minimum value of the LHS is positive, then the condition $\omega^{2}>0$ will hold for all wavenumbers and consequently throughout the region that we explore, all solutions that we will find represent stable modes. In other words, the disk is stable against all axisymmetric tightly wound perturbations, if the minimum value of the LHS is positive. It is easy to check that the minimum value of the LHS occurs at $X_{min}$ given by

$\displaystyle X_{min}=\pm\left[-\frac{1+\beta^{2}}{3}A^{1/3}+\frac{1}{2}A^{-1/3}\right]$

(48)

where $A$ is defined as

$\displaystyle A=\frac{Q_{g}}{2\beta\alpha\left(1+\sqrt{1+\frac{2(1+\beta^{2})^{3}Q_{g}^{2}}{27\beta^{2}\alpha^{2}}}\right)}$

(49)

The location of this point varies as the parameters $\alpha$ and $\beta$ change. Finally the local stability criterion can be expressed as

$\displaystyle Q_{g}>\frac{2\beta\alpha|X_{min}|}{X_{min}^{4}+(1+\beta^{2})X_{min}^{2}+\beta^{2}}$

(50)

This is the main result of this section. In fact, equation (50) is the generalized version of the Toomre’s local stability criterion for a fluid disk, and should be compared with the standard case given by (22). Since $X_{min}$ contains $Q_{g}$, criterion (50) is very complicated than the standard Toomres criterion (i.e. $Q_{g}>1$). However, it is straightforward to show that in the limit $\beta\rightarrow 0$ and $\alpha\rightarrow 1$, the right-hand side (RHS) of equation (50) equals $1$, and the standard Toomre’s criterion is recovered.

It is also instructive to plot the neutral stability curves $\omega=0$ for axisymmetric tightly wound waves in a fluid disk. These curves specify the boundary of the stable and unstable modes/waves. To do so, let us introduce a critical wavelength $\lambda_{crit}=2\pi/k_{crit}$. Using this definition and the dispersion relation (45), the line separating stable and unstable modes is given by

$\displaystyle Q_{g}(y)=\alpha\sqrt{\frac{4\alpha y}{1+\left(\frac{\beta Q_{g}}{2y\alpha}\right)^{2}}-4y^{2}}$

(51)

where $y=\lambda/\lambda_{crit}$ and $\lambda=2\pi/k$. We have plotted the neutral curve for various values of $\alpha$ and $\beta$ in Figure 2. The dashed curves corresponds to Newtonian gravity where $(\alpha,\beta)=(1,0)$.

Let us first consider the other curves with $\beta=0$, i.e. curves corresponding to $(\alpha,\beta)=(1.2,0)$ and $(\alpha,\beta)=(0.5,0)$. In this case one can readily show that the Toomre’s criterion (50) is $Q_{g}>\alpha$. Consequently, if $\alpha>1$ (as in the case ($\alpha,\beta)=(1.2,0)$) then a lager $Q_{g}$ parameter compared with Newtonian gravity is needed to overcome the local gravitational instability. Keeping in mind the physical interpretation of the Jeans instability, this means that the disk needs more gas pressure than it would in Newtonian gravity in order to overcome the gravitational collapse. On the other hand, if $\alpha<1$ then smaller gas pressure is needed for preventing the collapse.

For the general case $\beta\neq 0$ see the curves $(\alpha,\beta)=(1,0.3)$ and $(\alpha,\beta)=(0.8,0.3)$ in Figure 2. In this case the neutral curves corresponding to $\beta\neq 0$ lie below the dashed curve. This can not be explained with the above mentioned simple interpretation and one should consider it with more care. In order to describe this result, for the sake of simplicity and with no loss of generality we assume that $\alpha=1$ and $\beta\ll 1$. We shall see that in reality $\beta$ is small and our assumption here is a suitable one. Therefor, the RHS of the criterion (50) can be expanded in terms of $\beta$. Consequently, we have

$\displaystyle Q_{g}>1-\beta^{2}-\frac{3}{2}\beta^{4}+O(\beta^{5})$

(52)

Thus in the case $\alpha=1$, although the gravitational force is supposed to be stronger than the Newtonian case, the required value of $Q_{g}$ for local stability is always smaller than that of the Newtonian gravity. This situation is reminiscent of the difference between Jeans mass in modified gravity and Newtonian gravity. In fact in modified theories of gravity for which the gravitational force is stronger than that of Newtonian gravity, the Jeans mass is smaller than the standard case, for example see [18, 3].

Therefore, comparing the two cases $\beta=0,~\alpha>1$ and $\alpha=1,~\beta>0$ (note that in both cases the gravitational force is stronger than the Newtonian gravitational force), we can conclude that in theories where the gravitational force in the weak field limit is stronger than that of Newtonian gravity, the required value of Toomre’s parameter $Q_{g}$ for preventing gravitational collapse can be larger or smaller than the corresponding value in Newtonian gravity. In other words, $Q_{g}$ in these theories is not necessarily larger than that of Newtonian gravity.

In Figure 3 we show the fluid disk dispersion relation for growing perturbations for various values of $Q_{g}$ and $\beta$. The solid curves correspond to $Q_{g}=0.3$ and $\beta=0.1$ to $0.5$. Furthermore, the dashed curves correspond to $Q_{g}=0.5$ and $\beta=0.1$ to $0.5$. Also, the dotted curves ($\beta=0$) are the corresponding Newtonian dispersion relations for $Q_{g}=0.3$ and $Q_{g}=0.5$. As expected, the growth rates tends to increase with decreasing $Q_{g}$. Also, it is clear form Figure 3 that the effect of parameter $\beta$ in equation (45) is to stretch the range of instability to small wavenumber while larger $\beta$ leads to smaller growth rate.

Similar to the case of the fluid disk, it is straightforward to find the dispersion relation of tightly wound waves in a stellar disk. Since most of the calculations are similar to that of the fluid case, we have briefly derived the dispersion relation in Appendix B. The final result is given by equation (B5). In the case of axisymmetric perturbations, we may write

$\displaystyle\omega^{2}=\kappa^{2}-2\pi G|k|\Sigma_{0}\frac{\alpha m_{0}^{2}}{k^{2}+m_{0}^{2}}\mathfrak{F}(\frac{\omega}{\kappa},\frac{k^{2}\sigma_{R}^{2}}{\kappa^{2}})$

(53)

For a given mode of perturbation, if $\omega^{2}<0$, then the mode is unstable. Therefore, one may expect that the system is stable against all axisymmetric disturbances if equation (53) does not have a solution with $\omega^{2}<0$. In order to consider this expectation, we write down equation (53) as follows

$\displaystyle 1=\frac{2\pi G|k|\Sigma_{0}\alpha m_{0}^{2}}{(k^{2}+m_{0}^{2})(\kappa^{2}-\omega^{2})}\mathfrak{F}(\frac{\omega}{\kappa},\frac{k^{2}\sigma_{R}^{2}}{\kappa^{2}})$

(54)

Now, if $\omega^{2}<0$, i.e. $\omega=i\gamma$ where $\gamma$ is a real number, then the right-hand side (RHS) of equation (54) takes the form

$\displaystyle\frac{2\pi G|k|\alpha m_{0}^{2}\Sigma_{0}\kappa^{-2}}{(k^{2}+m_{0}^{2})\sinh\pi s^{\prime}}\int_{0}^{\pi}e^{-\chi(1+\cos\tau)}\sinh s^{\prime}\tau\sin\tau d\tau$

(55)

where $s^{\prime}=is=\gamma/\kappa$. Also we have used equation (B7) for the shape function. The schematic behavior of (55) in terms of $\omega^{2}$ is illustrated in Figure 4. From this figure we see that the maximum value occurs at $\omega^{2}=0$. Therefore, if the RHS of equation (53) for $\omega^{2}=0$ is smaller than $1$, then there is no tightly wound spiral mode with negative $\omega^{2}$ which satisfies the dispersion relation (54). Consequently, the self-gravitating disk will be stable to all WKB perturbations. Therefore, the criterion for stability to all wavelengths can be written down as

$\displaystyle\frac{2\pi G|k|\Sigma_{0}\alpha m_{0}^{2}}{(k^{2}+m_{0}^{2})\kappa^{2}}\mathfrak{F}(0,\frac{k^{2}\sigma_{R}^{2}}{\kappa^{2}})<1$

(56)

this equation can be represented in a more simplified form as

$\displaystyle\frac{\sigma_{R}\kappa}{2\pi G\Sigma_{0}\alpha}>\frac{1}{(1+\beta^{2}\chi)}\frac{\left(1-e^{-\chi}I_{0}(\chi)\right)}{\sqrt{\chi}}$

(57)

where $I_{0}$ is the modified Bessel function of order $0$. Also we have used the following identity

$\displaystyle\mathfrak{F}(0,\chi)=\frac{1}{\chi}\left(1-e^{-\chi}I_{0}(\chi)\right)$

(58)

Equation (57) is the main result of this section. In fact, at every location $(R_{0},\phi_{0})$ on the disk, this inequality should be satisfied in order to have local stability to all spiral axisymmetric tightly wound perturbations. As $\alpha\rightarrow 1$ and $\beta\rightarrow 0$, this criterion approaches to the so-called Toomre’s local stability criterion for stellar disk given by (23), again as expected. In Newtonian gravity the RHS of (57) is always larger than that of metric $f(R)$ gravity. This means that in the context of $f(R)$ gravity smaller value of Toomre’s parameter is required for local stability of rotating disks. As we discussed in the previous subsection, this result also is the case for a fluid disk.

Although equation (57) is our final stability criterion for every perturbation with wavenumber $k$, if we find the maximum value of the RHS of equation (57), then we will find a criterion which guarantees the stability at every location on the disk. Even in Newtonian gravity (where $\beta=0$), one has to use numerical analysis to find the maximum value of the RHS (57). Therefore, we need the numeric value of $\beta$ to find required criterion. However, using an excellent approximation for reduction factor [8], we can analytically find the maximum value of RHS of (57). The above mentioned approximation is

$\displaystyle\mathfrak{F}(0,\chi)\simeq\frac{1}{1+\chi}$

(59)

In the Figure 5, the solid line is the exact value of $\mathfrak{F}$ and the dashed line corresponds the approximate value. As it is clear from the figure, the deviation between the two functions is small. Therefore, with the aid of this approximation, we rewrite (57) as follows

$\displaystyle\frac{\sigma_{R}\kappa}{2\pi G\Sigma_{0}\alpha}>\frac{\sqrt{\chi}}{(1+\beta^{2}\chi)(1+\chi)}$

(60)

the maximum value of the RHS of this inequality occurs

$\displaystyle\chi=\frac{-1-\beta^{2}+\sqrt{1+14\beta^{2}+\beta^{4}}}{6\beta^{2}}$

(61)

stability criterion takes the following form

$\displaystyle Q_{s}>\frac{6\sqrt{6}\alpha\sqrt{g(\beta)-(1+\beta^{2})}}{(g(\beta)+5-\beta^{2})(g(\beta)+5\beta^{2}-1)}$

(62)

where $g(\beta)=\sqrt{1+14\beta^{2}+\beta^{4}}$. The criterion (62) is the final result of this section. As expected, the free parameters of the theory, i.e. $\alpha$ and $m_{0}$, appear in the stability criterion. If we use the same approximation in Newtonian gravity, then we will get the following criterion

$\displaystyle\frac{\sigma_{R}\kappa}{3.14G\Sigma_{0}}>1$

(63)

this criterion has a little difference with the exact criterion (23). However let us write (63) as $Q_{s}>1$. As we discussed previously, we expect that $\beta$ be small in real situations, so the stability criterion can be expanded with respect to $\beta$ as

$\displaystyle Q_{s}>\alpha-\alpha\beta^{2}+3\alpha\beta^{4}+O(\beta^{5})$

(64)

this criterion can be compared with the corresponding criterion for a fluid disk (52). The similarity between the stability criterion of fluid and stellar disk is obvious. Our discussion in the last paragraph of section III.1 considering the parameter $Q_{g}$ for a fluid disk, also holds here.

In order to make a comparison between the growth rates in fluid and stellar disks, we have shown the stellar disk dispersion relation (53) in Figure 6. The solid curves correspond to $Q_{s}=0.3$ and $\beta=0.1$ to $0.5$. The dashed curves correspond to $Q_{s}=0.5$ and $\beta=0.1$ to $0.5$. Also, the dotted curves are the corresponding Newtonian dispersion relations for $Q_{s}=0.3$ and $Q_{s}=0.5$ where $\beta=0$. As for a fluid disk, the growth rates increase with decreasing parameter $Q_{s}$, and the effect of parameter $\beta$ in equation (53) is to stretch the range of instability to small wavenumber. Also larger $\beta$ leads to smaller growth rate. Comparing Figures 3 and 6, one can conclude that the the growth rate of perturbation in the stellar disk is smaller than that of the fluid disk. In other words, if we consider two patches with the same Toomre’s parameter, one in a fluid disk and one in a stellar disk, the the growth rate in the patch on the stellar disk will be smaller than that of the fluid disk.