Stability of two-fluid galactic disc under the influence of an external   tidal field

K. Aditya

arXiv:1905.10746·astro-ph.GA·May 28, 2019

Stability of two-fluid galactic disc under the influence of an external tidal field

K. Aditya

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TL;DR

This paper analyzes the stability of a two-fluid galactic disc composed of stars and gas under external tidal forces, deriving a modified dispersion relation and stability criterion to assess its susceptibility to instabilities.

Contribution

It introduces a new linear stability analysis framework for two-fluid galactic discs influenced by external tidal fields, including a modified dispersion relation and stability criterion.

Findings

01

Derived a coupled differential equation governing disc instabilities.

02

Formulated a modified dispersion relation for stability analysis.

03

Discussed potential applications of the stability criterion.

Abstract

We consider the dynamics of rotationally supported thin galactic disc composed of stars and gas under the influence of external tidal field and derive the coupled differential equations governing the evolution of instabilities. Further linearising the governing equation a modified dispersion relation and stability criterion for appraising the stability of the two fluid galactic disc under the influence of external tidal field is obtained. Possible applications and method for the same are discussed.

Tables4

Table 1. Table 1: Above table describes the values of maximum values of Q T subscript 𝑄 𝑇 Q_{T} obtained for two fluid galactic disc under influence of external tidal fields by varying T 0 subscript 𝑇 0 T_{0} and ϵ italic-ϵ \epsilon

$R \to$	0.0	+0.5	+2.5	-0.1	-0.5
$ϵ ↓$
0.05	1.794	2.098	2.693	-	1.44
0.1	1.675	1.88	2.515	-	1.280
0.3	1.306	1.599	2.55	1.88	-

Table 2. Table 2: Above table describes the values of mimimum values of Q T subscript 𝑄 𝑇 Q_{T} obtained for two fluid galactic disc under influence of external tidal fields by varying T 0 subscript 𝑇 0 T_{0} and ϵ italic-ϵ \epsilon

$R \to$	0.0	+0.5	+2.5	-0.1	-0.5
$ϵ ↓$
0.05	0.994	1.198	1.693	-	0.647
0.1	0.875	0.81	1.515	-	0.480
0.3	0.406	0.699	1.134	0.388	-

Table 3. Table 3: Above table depicts the values of ω − ( m i n ) 2 superscript subscript 𝜔 𝑚 𝑖 𝑛 2 \omega_{-(min)}^{2} at constant value of Q s = Q g = 1 subscript 𝑄 𝑠 subscript 𝑄 𝑔 1 Q_{s}=Q_{g}=1 .

$R \to$	0.0	+0.5	2.5	-0.5	-2.5
$ϵ ↓$
0.05	-0.232	0.267	2.26	-0.73	-2.73
0.3	-2.35	-1.86	0.140	-2.85	-4.85

Table 4. Table 4: Above table depicts the values of ω − ( m i n ) 2 superscript subscript 𝜔 𝑚 𝑖 𝑛 2 \omega_{-(min)}^{2} at constant value of Q s = Q g = 2 subscript 𝑄 𝑠 subscript 𝑄 𝑔 2 Q_{s}=Q_{g}=2 .

$R \to$	0.0	+0.5	2.5	-0.5	-2.5
$ϵ ↓$
0.05	0.69	1.19	3.19	0.19	-1.80
0.3	0.16	0.66	2.66	-0.377	-2.33

Equations70

Q = \frac{κσ}{π G Σ}

Q = \frac{κσ}{π G Σ}

\frac{\partial u _{i}}{\partial t} + Ω_{0} \frac{\partial u _{i}}{\partial θ} - 2 Ω_{0} v_{i} + \frac{c _{i}^{2}}{Σ _{0 i}} \frac{\partial Σ _{i}}{\partial R} + \frac{\partial ( ϕ _{g} + ϕ _{s} )}{\partial R} + \frac{Σ _{i}}{Σ _{0 i}} \frac{\partial ϕ _{e x t}}{\partial R} = 0

\frac{\partial u _{i}}{\partial t} + Ω_{0} \frac{\partial u _{i}}{\partial θ} - 2 Ω_{0} v_{i} + \frac{c _{i}^{2}}{Σ _{0 i}} \frac{\partial Σ _{i}}{\partial R} + \frac{\partial ( ϕ _{g} + ϕ _{s} )}{\partial R} + \frac{Σ _{i}}{Σ _{0 i}} \frac{\partial ϕ _{e x t}}{\partial R} = 0

\frac{\partial v _{i}}{\partial t} - 2 B u_{i} + Ω_{0} \frac{\partial v _{i}}{\partial θ} + \frac{1}{R} \frac{\partial ( Σ _{s} + Σ _{g} )}{\partial θ} = 0

\frac{\partial v _{i}}{\partial t} - 2 B u_{i} + Ω_{0} \frac{\partial v _{i}}{\partial θ} + \frac{1}{R} \frac{\partial ( Σ _{s} + Σ _{g} )}{\partial θ} = 0

\frac{\partial Σ _{i}}{\partial t} + Ω_{0} \frac{\partial Σ _{i}}{\partial θ} + \frac{\partial ( Σ _{0 i} u _{i} )}{\partial R} + \frac{u _{i} Σ _{0 i}}{R} + \frac{Σ _{0 i}}{R} \frac{\partial v _{i}}{\partial θ} = 0

\frac{\partial Σ _{i}}{\partial t} + Ω_{0} \frac{\partial Σ _{i}}{\partial θ} + \frac{\partial ( Σ _{0 i} u _{i} )}{\partial R} + \frac{u _{i} Σ _{0 i}}{R} + \frac{Σ _{0 i}}{R} \frac{\partial v _{i}}{\partial θ} = 0

\frac{1}{R} \frac{\partial}{\partial R} R \frac{\partial ( ϕ _{s} + ϕ _{g} )}{\partial R} + \frac{\partial ^{2} ( ϕ _{s} + ϕ _{g} )}{\partial z ^{2}} = 4 π G (Σ_{s} + Σ_{g}) δ (z)

\frac{1}{R} \frac{\partial}{\partial R} R \frac{\partial ( ϕ _{s} + ϕ _{g} )}{\partial R} + \frac{\partial ^{2} ( ϕ _{s} + ϕ _{g} )}{\partial z ^{2}} = 4 π G (Σ_{s} + Σ_{g}) δ (z)

\frac{\partial u _{i}}{\partial t} - 2 Ω_{0} v_{i} + \frac{c _{i}^{2}}{Σ _{0 i}} \frac{\partial Σ _{i}}{\partial R} + \frac{\partial ( ϕ _{s} + ϕ _{g} )}{\partial R} + \frac{Σ _{i}}{Σ _{0 i}} \frac{\partial ϕ _{e x t}}{\partial R} = 0

\frac{\partial u _{i}}{\partial t} - 2 Ω_{0} v_{i} + \frac{c _{i}^{2}}{Σ _{0 i}} \frac{\partial Σ _{i}}{\partial R} + \frac{\partial ( ϕ _{s} + ϕ _{g} )}{\partial R} + \frac{Σ _{i}}{Σ _{0 i}} \frac{\partial ϕ _{e x t}}{\partial R} = 0

\frac{\partial v _{i}}{\partial t} - 2 B u_{i} = 0

\frac{\partial v _{i}}{\partial t} - 2 B u_{i} = 0

\frac{\partial Σ _{i}}{\partial t} + Σ_{0 i} \frac{\partial u _{i}}{\partial R} = 0

\frac{\partial Σ _{i}}{\partial t} + Σ_{0 i} \frac{\partial u _{i}}{\partial R} = 0

\frac{\partial ( ϕ _{s} + ϕ _{g} )}{\partial R} = - 2 π i G (Σ_{s} + Σ_{g})

\frac{\partial ( ϕ _{s} + ϕ _{g} )}{\partial R} = - 2 π i G (Σ_{s} + Σ_{g})

\frac{\partial u _{s}}{\partial t} - 2 Ω_{0} v_{s} + \frac{c _{s}^{2}}{Σ _{0 s}} \frac{\partial Σ _{s}}{\partial R} + \frac{\partial ( ϕ _{s} + ϕ _{g} )}{\partial R} + \frac{Σ _{s}}{Σ _{0 s}} \frac{\partial ϕ _{e x t}}{\partial R} = 0

\frac{\partial u _{s}}{\partial t} - 2 Ω_{0} v_{s} + \frac{c _{s}^{2}}{Σ _{0 s}} \frac{\partial Σ _{s}}{\partial R} + \frac{\partial ( ϕ _{s} + ϕ _{g} )}{\partial R} + \frac{Σ _{s}}{Σ _{0 s}} \frac{\partial ϕ _{e x t}}{\partial R} = 0

\frac{\partial v _{s}}{\partial t} - 2 B u_{s} = 0

\frac{\partial v _{s}}{\partial t} - 2 B u_{s} = 0

\frac{\partial Σ _{s}}{\partial t} + Σ_{0 s} \frac{\partial u _{s}}{\partial R} = 0

\frac{\partial Σ _{s}}{\partial t} + Σ_{0 s} \frac{\partial u _{s}}{\partial R} = 0

\frac{\partial ^{2} Σ _{s}}{\partial t ^{2}} - c_{s}^{2} \frac{\partial ^{2} Σ _{s}}{\partial R ^{2}} - 4 Ω_{0} B Σ_{s} + 2 π i G Σ_{0 s} \frac{\partial}{\partial R} (Σ_{s} + Σ_{g}) - Σ_{s} \frac{\partial ^{2} ϕ _{e x t}}{\partial R ^{2}} = 0

\frac{\partial ^{2} Σ _{s}}{\partial t ^{2}} - c_{s}^{2} \frac{\partial ^{2} Σ _{s}}{\partial R ^{2}} - 4 Ω_{0} B Σ_{s} + 2 π i G Σ_{0 s} \frac{\partial}{\partial R} (Σ_{s} + Σ_{g}) - Σ_{s} \frac{\partial ^{2} ϕ _{e x t}}{\partial R ^{2}} = 0

\frac{\partial ^{2} Σ _{g}}{\partial t ^{2}} - c_{g}^{2} \frac{\partial ^{2} Σ _{g}}{\partial R ^{2}} - 4 Ω_{0} B Σ_{g} + 2 π i G Σ_{0 g} \frac{\partial}{\partial R} (Σ_{s} + Σ_{g}) - Σ_{g} \frac{\partial ^{2} ϕ _{e x t}}{\partial R ^{2}} = 0

\frac{\partial ^{2} Σ _{g}}{\partial t ^{2}} - c_{g}^{2} \frac{\partial ^{2} Σ _{g}}{\partial R ^{2}} - 4 Ω_{0} B Σ_{g} + 2 π i G Σ_{0 g} \frac{\partial}{\partial R} (Σ_{s} + Σ_{g}) - Σ_{g} \frac{\partial ^{2} ϕ _{e x t}}{\partial R ^{2}} = 0

Σ_{s} = \frac{2 π G k Σ _{0 s} Σ _{g}}{( ω ^{2} - c _{s}^{2} k ^{2} - κ ^{2} + 2 π G Σ _{0 s} k - T _{0} )}

Σ_{s} = \frac{2 π G k Σ _{0 s} Σ _{g}}{( ω ^{2} - c _{s}^{2} k ^{2} - κ ^{2} + 2 π G Σ _{0 s} k - T _{0} )}

Σ_{g} = \frac{2 π G k Σ _{0 g} Σ _{s}}{( ω ^{2} - c _{g}^{2} k ^{2} - κ ^{2} + 2 π G Σ _{0 g} k - T _{0} )}

Σ_{g} = \frac{2 π G k Σ _{0 g} Σ _{s}}{( ω ^{2} - c _{g}^{2} k ^{2} - κ ^{2} + 2 π G Σ _{0 g} k - T _{0} )}

(ω^{2} - c_{s}^{2} k^{2} - κ^{2} + 2 π G Σ_{0 s} k - T_{0}) (ω^{2} - c_{g}^{2} k^{2} - κ^{2} + 2 π G Σ_{0 g} k - T_{0}) =

(ω^{2} - c_{s}^{2} k^{2} - κ^{2} + 2 π G Σ_{0 s} k - T_{0}) (ω^{2} - c_{g}^{2} k^{2} - κ^{2} + 2 π G Σ_{0 g} k - T_{0}) =

(2 π G Σ_{0 s} k) (2 π G Σ_{0 g} k)

α_{s} = κ^{2} + c_{s}^{2} k^{2} - 2 π G Σ_{0 s} k + T_{0}

α_{s} = κ^{2} + c_{s}^{2} k^{2} - 2 π G Σ_{0 s} k + T_{0}

α_{g} = κ^{2} + c_{g}^{2} k^{2} - 2 π G Σ_{0} k + T_{0}

β_{s} = 2 π G Σ_{0 s} k

β_{g} = 2 π G Σ_{0 g} k

ω^{4} - ω^{2} (α_{s} + α_{g}) + (α_{s} α_{g} - β_{s} β_{g}) = 0

ω^{4} - ω^{2} (α_{s} + α_{g}) + (α_{s} α_{g} - β_{s} β_{g}) = 0

ω_{\pm}^{2} = \frac{1}{2} (α_{s} + α_{g}) \pm \frac{1}{2} ((α_{s} + α_{g})^{2} - 4 (α_{s} α_{g} - β_{s} β_{g}))^{\frac{1}{2}}

\frac{2 π G Σ _{0 s} k}{κ ^{2} + k ^{2} c _{s}^{2} + T _{0}} + \frac{2 π G Σ _{0 g} k}{κ ^{2} + k ^{2} c _{g}^{2} + T _{0}} = F

\frac{2 π G Σ _{0 s} k}{κ ^{2} + k ^{2} c _{s}^{2} + T _{0}} + \frac{2 π G Σ _{0 g} k}{κ ^{2} + k ^{2} c _{g}^{2} + T _{0}} = F

k^{3} (4 c_{s}^{2} c_{g}^{2}) - 3 k^{2} (2 π G Σ_{0 s} c_{s}^{2} + 2 π G Σ_{0 g} c_{g}^{2})

k^{3} (4 c_{s}^{2} c_{g}^{2}) - 3 k^{2} (2 π G Σ_{0 s} c_{s}^{2} + 2 π G Σ_{0 g} c_{g}^{2})

+ 2 k ((κ^{2} + T_{0}) (c_{g}^{2} + c_{s}^{2})) - (2 π G Σ_{0 s} + 2 π G Σ_{0 s g}) (κ^{2} + T_{0}) = 0

\frac{2}{1 + Q _{T}^{2}} = \frac{( 1 - ϵ )}{X ( 1 + \frac{( 1 - ϵ ) ^{2} Q _{s}^{2}}{4 X ^{2}} + \frac{T _{0}}{κ ^{2}} )} + \frac{ϵ}{X ( 1 + \frac{ϵ ^{2} Q _{g}^{2}}{4 X ^{2}} + \frac{T _{0}}{κ ^{2}} )}

\frac{2}{1 + Q _{T}^{2}} = \frac{( 1 - ϵ )}{X ( 1 + \frac{( 1 - ϵ ) ^{2} Q _{s}^{2}}{4 X ^{2}} + \frac{T _{0}}{κ ^{2}} )} + \frac{ϵ}{X ( 1 + \frac{ϵ ^{2} Q _{g}^{2}}{4 X ^{2}} + \frac{T _{0}}{κ ^{2}} )}

\frac{2}{1 + Q _{T}^{2}} = \frac{( 1 - ϵ )}{X ^{'} ( 1 + \frac{( 1 - ϵ ) ^{2} Q _{s}^{'2}}{4 X ^{'2}} )} + \frac{ϵ}{X ^{'} ( 1 + \frac{ϵ ^{2} Q _{g}^{'2}}{4 X ^{'2}} )}

\frac{2}{1 + Q _{T}^{2}} = \frac{( 1 - ϵ )}{X ^{'} ( 1 + \frac{( 1 - ϵ ) ^{2} Q _{s}^{'2}}{4 X ^{'2}} )} + \frac{ϵ}{X ^{'} ( 1 + \frac{ϵ ^{2} Q _{g}^{'2}}{4 X ^{'2}} )}

ω^{2} = (κ^{2} + T_{0}) k^{(0)} + c^{2} k^{(2)} - 2 π G Σ k^{(1)}

ω^{2} = (κ^{2} + T_{0}) k^{(0)} + c^{2} k^{(2)} - 2 π G Σ k^{(1)}

1 + \frac{Q ^{'2}}{4} \frac{k ^{2}}{k _{T}^{'2}} - \frac{k}{k _{T}^{'}} = 0

1 + \frac{Q ^{'2}}{4} \frac{k ^{2}}{k _{T}^{'2}} - \frac{k}{k _{T}^{'}} = 0

Q = 2 [ζ (1 - ζ (1 + \frac{T _{0}}{κ ^{2}}))]^{\frac{1}{2}}

Q = 2 [ζ (1 - ζ (1 + \frac{T _{0}}{κ ^{2}}))]^{\frac{1}{2}}

ω_{-}^{2} = \frac{1}{2} (α_{s} + α_{g}) - \frac{1}{2} ((α_{s} + α_{g})^{2} - 4 (α_{s} α_{g} - β_{s} β_{g}))^{\frac{1}{2}}

ω_{-}^{2} = \frac{1}{2} (α_{s} + α_{g}) - \frac{1}{2} ((α_{s} + α_{g})^{2} - 4 (α_{s} α_{g} - β_{s} β_{g}))^{\frac{1}{2}}

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TopicsAstrophysics and Star Formation Studies · Stellar, planetary, and galactic studies · Solar and Space Plasma Dynamics

Full text

Stability of two-fluid galactic disc under the influence of an external tidal field.

K. Aditya

Indian Institute of Science Education and Research, Tirupati 517507, India

E-mail : [email protected]

Abstract

We consider the dynamics of rotationally supported thin galactic disc composed of stars and gas under the influence of external tidal field and derive the coupled differential equations governing the evolution of instabilities. Further linearising the governing equation a modified dispersion relation and stability criterion for appraising the stability of the two fluid galactic disc under the influence of external tidal field is obtained. Possible applications and method for the same are discussed.

K****eywords :

hydrodynamics-instabilities :galaxies:kinematics and dynamics- galaxies:structure-galaxies:star formation

1 Introduction

Local stability of the galactic disc is a subtle balance between the mass contained in the fluid packet, the random velocity dispersion and the differential rotation. The competetion between the stabilising agents i.e differential rotation and the random velocity dispersion and the destabilising agent i.e the mass content is classically quantified by the stability criterion proposed by Toomre (1964).

[TABLE]

Where $\rm\kappa$ is the epicyclic frequency, $\rm\sigma$ is the radial velocity dispersion, and $\rm\Sigma$ the surface mass density. Value of $\rm Q\geq 1$ quantifies the stability of galactic disc against axis symmetric perturbations. The the above criterion appraises the stability a fluid disc consisting of either gas or stars at a time, but a real galactic disc consists of stars and gas which is also gravitationally important, more so in case of gas rich low surface brightness galaxies (LSBs) De Blok et al. (1996). The dispersion relation pertaining to the stability of gravitationally coupled two-fluid disc (stars+gas) has been extensively studied by Jog and Solomon (1984) and criterion for appraising the stability of two-fluid disc has been derived and studied by Elmegreen (1995), Jog (1996), and the same for multi-component disc has been studied by Rafikov (2001). Stability criterion for multi-component galactic disc with finite thickness has been studied by Romeo and Falstad (2013), which is useful when the galactic disc consists of say mutiple stellar disc (thin disc+thick disc) and HI disc. The role of external tidal field on the stability of gravitating, infinite,homogenous systems was investigated by Jog (2013) and modified Toomre stability criterion due to external tidal field also derived by Jog (2014). In this work we consider the simple case of an gravitationally interacting axis-symmetric isothermal fluid disc of stars and gas supported by differential rotation and the random velocity dispersion, under the influence of an external tidal field. We will derive the basic differential equations governing the growth of density perturbation and further obtain stability criterion for the two-component fluid disc in presence of external tidal field. The paper is organised as follows: in section 2 we will formulate the basic equations and derive the governing differential equation, dispersion relation modified due to external tidal fields, and an effective stability parameter, in section 3 we will discuss possible applications and conclude in section 4.

2 Formulation and derivation of basic equations

The stars and gas in the galactic disc togather constitute isothermal fluids interacting with each other gravitationally. The fluid disc is supported by random presure and rotation. The problem is completely described by understanding the flow of perturbed fluid components constituting a thin cylindrical disc, i.e lying in z=0 plane. The dynamics of the perturbed fluid and the conservation of mass density are dictated by the linearised force equation (eq 2,3) and the linearised continuity equation (eq 4) for the two -fluid system in cylindrical coordinates. The presence of an external tidal field is included by adding an extra force $\frac{-\partial\phi_{ext}}{\partial r}$ in the linearised force equation. The dynamics of the fluid in force equation is governed by the self gravity of the two-fluid disc, i.e each component of the perturbed fluid say ’gas’ moves under the combined potential of stars + gas, thus the right hand side of the linear force equation is supplimented with the linearised Poisson equation (eq 5) describing the combined mass densities of the two-fluid system and an isothermal equation of state $\rm P_{i}=\Sigma_{i}c_{i}^{2}$ is used to describe the presure of the perturbed fluids, where $\Sigma_{i}$ denotes the surface density and $\rm c_{i}$ denotes the isothermal speed of sound of the of the $i^{th}$ fluid component respectively. Throughout this work $\rm{}^{\prime}i^{\prime}$ will index stars $(s)$ and gas $(g)$ .

The linearised force equations under influence of an external tidal force are;

[TABLE]

Linearised continuity equation reads;

[TABLE]

Linearised Poisson equation reads;

[TABLE]

Here, $u_{i}$ and $v_{i}$ are the perturbed velocities in radial and tangential direction, $\Sigma_{i}$ and $\phi_{i}$ are the perturbed surface densities and the potentials of the $i^{th}$ components. The quantities $\Sigma_{0i}$ , $\Omega_{0}$ and $\phi_{0i}$ are the unperturbed surface densities, angular velocity and potential respectively.

Considering axis-symmetric case $\frac{\partial}{\partial{\theta}}\rightarrow 0$ , the perturbed equations read;

[TABLE]

and the Poisson equation for the thin disc assumes the form Toomre (1964);

[TABLE]

Equations 6,7,8,9 will be basis for deriving the dynamic equations governing growth of perturbations and also the associated dispersion relation. Indexing equation 6,7,8,9 for stars;

[TABLE]

Combining the equations (9), (10), (11), and (12) we obtain the following equation for stars;

[TABLE]

3 Results/Discussion

To gain better insight into on impact of tidal forces on two fluid disc we firstly investigate the impact of tidal force on a one-component disc. The dispersion relation for a one-component disc in presence of tidal field is;

[TABLE]

In the above equation at large value of $\rm k$ , $k^{2}$ will dominate thus presure stabilises the galactic disc at small scales, at small $\rm k$ i.e $\rm k^{(0)}$ the differential rotation and the tidal field stabilise the disc at large scales, and at intermdiate $\rm k$ self-gravity of the galactic disc becomes important. Tidal field comes in two flavours ’compressive’ $\rm T_{0}<0$ and ’disruptive’ $\rm T_{0}>0$ . Classic example of disruptive tidal field $\rm T_{0}>0$ is due to point mass. Renaud (2010), appendix B, has an exhaustive compilation of tidal fields due to all the important potential-density pairs. A disruptive tidal field adds up with the differential rotation and presure to stabilise the disc, whereas a compressive tidal field adds up with the self gravity of the galactic disc and aids in destabilising the galactic disc. Next we proceed to inspect the marginal stability of the one-fluid galactic disc. Putting $\omega^{2}=0$ in equation (25) can be recast to obtain a quadratic equation in $\rm k$ ,

[TABLE]

Where, $Q^{\prime}=Q(1+\frac{T_{0}}{\kappa^{2}})^{\frac{1}{2}}$ , $k^{\prime}_{T}=k_{T}(1+\frac{T_{0}}{\kappa^{2}})$ , $k_{T}=\frac{\kappa^{2}}{2\pi G\Sigma_{0}}$ and defining $\zeta^{\prime}=\frac{k^{\prime}_{T}}{k}=\frac{k_{T}}{k}(1+\frac{T_{0}}{\kappa^{2}})$ , i.e $\zeta^{\prime}=\zeta(1+\frac{T_{0}}{\kappa^{2}})$ . With above substitutions equation (26) can be shown to assume;

[TABLE]

The marginal stability curve (Fig 1) is plotted fixing $\frac{T_{0}}{\kappa^{2}}$ at a fixed value of 0.5. It can be seen that for for value of $Q<1$ in absence of tidal field the disc becomes unstable, whereas disruptive tidal increase this value, so the disc is less succeptible to external perturbations. And compressive tidal field lowers the value such that the disc becomes more prone to axis-symmetric perturbations.

Now we will try to address the role of compressive and disruptive tidal fields on a two fluid galaxy disc. We plot contours of $Q_{T}$ , for the given values of $Q_{s},Q_{g},\epsilon,$ and $\frac{T_{0}}{\kappa^{2}}$ . For the same range of values spanning $X_{s}=\frac{1}{2}(1-\epsilon)Q_{s}^{2}$ and $X^{\prime}_{g}=\frac{1}{2}\epsilon Q_{g}^{2}$ , and are computed over range of $Q_{s}$ and $Q_{g}$ at a constant value of $\epsilon$ and $\frac{T_{0}}{\kappa^{2}}$ . $Q_{T}$ is evaluated by finding minimum $X_{min}$ and maximum $X_{max}$ in range between $X_{s}$ and $X_{g}$ , and finally $Q_{T}$ is evaluated over a range of $X_{max}$ and $X_{min}$ , with the idea that $X$ lies in between $X_{max}$ and $X_{min}$ . Procedure and applications of two-component stability formula without external tidal are discussed in greater detail in Jog (1996). For studying impact of tidal force we plot contour diagrams fixing gas-fraction $\epsilon$ and $R=\frac{T_{0}}{\kappa^{2}}$ and for the same gas fraction we will vary $R$ to understand the impact of the external force on two-fluid disc. In table (1) and table (2) are summarised the values of maximum and minimum values of $Q_{T}$ obtained for varying values of gas-fraction $\epsilon$ and strength of external tidal field.

The main results of the work can be inferred directly from the table;

$\rm 1)$ Increasing gas fraction destabilises the disc,

$\rm 2)$ A disruptive tidal field tends to increase the stability of the disc even at high gas-fraction, whereas compressive tidal field destabilises the galaxy disc.

It can be see that by increasing the value of $\epsilon$ from 0.05 to 0.3 changes $Q_{Tmax}$ from 1.794 to 1.306. By applying a disruptive tidal field of intermediate strength $R=+0.5$ the value of $Q_{Tmax}$ for $T_{0}=0$ increases from 1.79 to 2.098 and for an intense tidal field $R=2.5$ the maximum value of stability $Q_{T}$ increases to 2.69, which is much higher than either of just stellar or gas disc. Even at very high gas fraction when the disc by itself is unstable the disruptive tidal forces tends to stabilise the disc as an example (see table 2) at a gas fraction of 0.3 without tidal forces the values of lowest stability contour is 0.406 and application of $R=+2.5$ makes the disk just stable $Q_{Tmin}=1.012$ . Compressive tidal force has a destabilising effect on the galactic disc, it can be seen that even a tidal field of intermediate strength $R=-0.1$ can destabilise the disk at high gas fraction, see table 1, at gas fraction of about 0.3 when $R=0$ , $Q_{Tmax}=1.306$ and at $R=-0.1$ , $Q_{Tmax}=1.188$ . And similarly when $T=0$ , for gas fraction $\epsilon=0.3$ , $Q_{Tmin}=0.406$ and under influence of an external tidal field $R=-0.1$ , $Q_{Tmin}=0.388$ .

As a simple possible application we will study the impact of gas-fraction and tidal forces on regulating the growth rate of the instabilities ( $\omega_{-}^{2}$ ) in the two-fluid disc. The growth rate of instabilities in two-fluid galactic disc is (see equation 19);

[TABLE]

where (see equation 18)

[TABLE]

The above quantities can be expressed as

[TABLE]

Here, $k_{T}=\frac{\kappa^{2}}{2\pi G(\Sigma_{0s}+\Sigma_{0g})}$ , for each case we will fix the values of $Q_{s}$ , $Q_{g}$ and the gas fraction $\epsilon$ and vary the value of $R$ to see the effect of the tidal forces on the two-fluid disc.

From Table 4 and Table 5 we can infer that a disruptive tidal fields tends to makes $\omega_{-(min)}$ more positive,thus making disc less prone to growth of instabilities, whereas a compressive tidal field tends to make $\omega_{-(min)}$ more negative and thus making the disc more susceptible to the growth of axis-symmetric perturbations. From table 4 we can see that growth rate $\omega_{-(min)}$ is positive for $Q_{s}=Q_{g}=2$ , but intense compressive tidal of order R= -2.5 at even very small gas fraction can make $\omega_{-(min)}$ negative. The growth rate of instabilities is enhanced immensely in case of galactic disc with high gas fraction aided by external compressive tidal field. Finally from figure 12 it can be seen that under influence of intense compressive tidal fields even the $\omega_{+(min)}$ can become negative thus further enhancing the growth of instabilities in galactic disc.

4 Conclusion

In this work differential equations governing the growth of instabilities in two-fluid galactic disc under external tidal field have been derived and a criterion for appraising the stability of the two-fluid disc is presented. The modified stability criterion is applied to the understand the influence of the compressive and disruptive tidal fields, along with role of gas fraction on the stability of the galactic disc. We have also derived modified dispersion relation and studied the effect of the tidal fields on growth rate of instabilites in galactic disc, and interestingly found out that the $\omega_{+}^{2}$ can also become negative and can further destabilise the disc. Some possible applications of the results presented here are: understanding the role of tidal fields in enhancing or quenching of star formation rates in interacting galaxies, understanding the role of dark matter haloes in stabilising gas rich low surface brightness galaxies,which without the tidal effects of the dark matter haloes will be unstable owing to large gas-fraction.

Bibliography11

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