Investigation of the halo effect in the evolution of a nonstationary disk of spiral galaxies

1 Introduction

Problems of the formation and evolution of galaxies and their subsystems are actively developing areas of extra galactic astronomy. It is known that different types of Hubble tuning fork galaxies have clearly different subsystems that were formed at certain stages of their evolution. Among them, e.g., spiral galaxies are distinguished by the greatest variety of subsystems: a core, a bulge, a disk with spiral arms, a halo, and an extensive corona; they also often have a bar (Seigar 2017, Binney and Tremaine 2008, Surdin 2013, Malcolm 1998, Díaz-García et al. 2016, Federico et al. 2016). In these galaxies, one more circumstance must also be taken into account, namely, during the collapse, the size of the gas cloud in the form of a somewhat rotating protogalaxy decreases unevenly along and across the rotation axis (see, e.g., Herrmann et al. 2009a,b, Suchkov 1988). Therefore, old stars that were born before centrifugal forces became significant formed an almost nonrotating spherical subsystem – a halo, and stars that were born later than all formed a rapidly rotating thin stellar disk (Kamphuis 2008, Hodges-Kluck et al. 2016, Henriksen 2017). Since the disk subsystem was formed after the halo, the processes of its formation and evolution proceeded against the background of the halo (see, e.g., Herrmann et al. 2009b, Kamphuis 2008, Hodges-Kluck et al. 2016, Henriksen 2017, Chakrabarty and Dubinski 2007).

Proceeding from this, in this article, we have considered the problem of the evolution of the disk subsystem of galaxies with the halo taken into account. It should be noted that quite a lot of works have been published in the literature on the construction of equilibrium disk models taking into account the halo, as well as on the study of the stability of linear oscillations of the simplest models to reveal the role of the halo (see, e.g., Osipkov and Kutuzov 1987, Bisnovaty-Kogan 1983, Kutuzov 1991, Polyachenko and Shukhman 1979, Abramyan and Kaplan 1974, Nuritdinov 1981). However, an analysis of a number of modern astrophysical observations shows that large-scale structures of galaxies begin to form precisely at the nonlinear, nonstationary stage of their evolution, when the processes of gravitational collapse of the medium, as well as its radial and nonradial global oscillations, are still ongoing. With this in mind, the problems of constructing a nonlinear and nonradially oscillating model of a nonstationary disk, taking into account a passive ellipsoidal halo with a uniform density, and obtaining the equation for the evolution of a disk subsystem of galaxies were considered in (Nuritdinov 1977, Mirtadjieva and Nuritdinov 1997, Mirtadjieva 2000, 2009). It should be noted that the problem we are considering presents a certain difficulty, which is associated with the issue of self-consistency, and in this case, from a methodological point of view, there are two ways to halo couple. In the first of them, one should begin with compiling the initial state taking into account the halo (Bisnovaty-Kogan 1983), only then imposing a nonlinear perturbation on the disk in the form of nonradial oscillations. In the second method, the perturbation and the halo are simultaneously turned on. When “freezing” the imposed perturbation, we must obtain a different initial state compared to the noted equilibrium state. These two approaches ultimately give the same results, and therefore, for the sake of diversity, the second method was used in some studies (Nuritdinov 1977, Mirtadjieva and Nuritdinov 1997, Mirtadjieva 2000, Mirtadjieva 2009). We also note that the problem statement and preliminary results of the analysis of nonlinear nonradial oscillations of a two-dimensional model without a halo in the general case are given in the study by Kirbijekova and Nuritdinov (1989).

2 Basic relations and equations

To derive the evolution equations for a self-gravitating disk, we use the generating function method. Since the dependence between the unperturbed and perturbed Lagrangian phase coordinates can be assumed to be linear and canonical, the generating function w must be a second-degree polynomial. According to the theory, it can be represented as follows (Nuritdinov 1977).

Moreover, N i ( t ) are two-dimensional square matrices, of which the matrices N 1 ( t ) and N 3 ( t ) are symmetric, and N 2 ( t ) is an arbitrary matrix, r → determines the perturbed coordinates of the particle, and finally, v 0 → , is the velocity vector in the nominal state. Then, using Eq. (1), one can obtain

where ∗ denotes transposition. Using Eq. (2), we have

As we noted above, in Nuritdinov (1977), for the first time, a nonlinear and nonradially oscillating disk model with the following surface density was constructed:

This model is surrounded by a passive ellipsoidal halo with a gravitational potential

where a and b are the semiaxes of the disk depending on time t, σ 0 ( t ) = 3 M d / ( 2 π a b ) , A 1 ( e ) and A 2 ( e ) are well-known functions of the eccentricity of the halo meridional section (Abramyan and Kaplan 1974). The corresponding phase density of this model has the form

where Ω is the disk rotation parameter, R 0 is its radius in the initial nominal condition, χ is the Heaviside function, and

and

and

Here, E is the identity matrix. In a perturbed state, an elliptical disk has a gravitational potential

where A and B are functions of a ( t ) and b ( t ) (Abramyan and Kaplan 1974).

Since the oscillations we are considering occur precisely in the plane of the disk itself, the gravitational potentials can be represented as follows:

and

where G is the gravitational constant and ρ 0 = const is the spatial density of the halo.

In an arbitrarily nonstationary state, the Hamilton–Jacobi equation is valid

By putting Eqs. (1), (5) and (7) into Eq. (10) and collecting the terms separately with the products of only r → , only the components v 0 → , and both, one can obtain the disk evolution equations. However, there is an inconvenience associated with the fact that in the initial state t = 0 , the elements of the matrices N 1 , N 2 , and N 3 are discontinuous functions (Nuritdinov 1977). Therefore, we turn to continuous functions – matrices U 1 , U 2 , U 3 , and U 4 , which we define by the following relations:

Then, introducing the normalization ( 4 π / 3 ) G ρ 0 + ( π 2 / 2 R 0 ) G σ 0 = 1 and the dimensionless parameter p = M h / M d as the ratio of halo and disk masses, we obtain the following evolution equation for the disk model under study:

Thus, we have obtained a system of equations for the evolution of a self-gravitating disk, taking into account the halo, in the form of matrix differential equations. Next, we numerically solve the Cauchy problem with the given initial conditions. It should be noted that these equations can, in particular, also describe the evolution of radial pulsations ( a = b ) of a circular disk in the presence of a halo.

3 Analysis of the evolution equation of disk (6) with the halo taken into account

To integrate the system of matrix differential Eqs. (12), it is necessary to set physically acceptable initial conditions. We introduce an auxiliary matrix

where the value μ is the ratio of the perturbed semimajor axis to the unperturbed one ( μ = a / a 0 ) , i.e., characterizes nonlinear deviations from the unperturbed equilibrium state, for which μ = 1 . Then, the relationship between perturbed and unperturbed coordinates and velocities r , v and r 0 , v 0 can be represented as follows:

Taking into account Eq. (2), we substitute into Eq. (14) the expressions r → and v → through r → 0 , v → 0 and also compare the coefficients for the same terms. Therefore, at the initial time,

Taking into account (11) and (15), we obtain the following initial values:

For the major and minor semiaxes of the disk, we have the following values:

Let us now indicate the specific form of the function of the eccentricity of the meridional section of the halo (Abramyan and Kaplan 1974):

where e = ( a 2 − b 2 ) / a . When e = 0 , there is uncertainty in the form 0/0. It is easy to uncover this indeterminacy by assuming

and calculating their derivatives:

Then, with the help of Eq. (17), we find

To determine the characteristics of the deformation of the disk model, we introduce the concept of statistical, average amplitude (the standard deviation of the area of the elliptical disk from the initial one):

where T is the integration interval, S is the disk area ( S = π a b ) , and S 0 is the area of the initial unperturbed disk, the radius of which can be taken equal to unity. Note that in numerical calculations, it is more convenient to use the following notation for the statistical amplitude η :

This amplitude η can take values from zero (in the absence of deformations, i.e., when the area of the perturbed disk remains equal to the area of the unperturbed disk during evolution) up to infinitely large values (at large deformations that destroy the disk, i.e., when the semiaxes of the disk a and b → ∞ at t → ∞ ).

With the help of a computer program compiled by us, numerical calculations were carried out for various values of the parameter p , the initial perturbation μ ∈ [ 1 ; 2 ] , and the circular speed of disk rotation Ω ∈ [ 0 ; 1 ] . It should be noted here that numerical calculations can be controlled using simple, well-known invariants of the nonlinear evolution of the system. For various values of p , μ , and Ω , we have determined the statistical amplitude η , which characterizes the degree of deformation of the system. It should be noted that the values of the statistical amplitude η have a fairly wide range from 1 0 − 7 to 1 0 5 , so it is more expedient to switch to a logarithmic scale. The results obtained show that no dependence is felt at large values of disturbances μ ≥ 1.5 , which confirms the instability of such oscillations. At small values of perturbations, the deformations change insignificantly in the interval Ω ∈ [ 0 ; 0.3 ] , and in the interval Ω ∈ [ 0.4 ; 0.5 ] , we have a zone of reduced deformations, but at Ω = 0.5 , there is a sharp increase in η , and the value of the statistical amplitude increases by several orders of magnitude.

When numerical calculations were carried out for different values of the disk rotation parameter Ω and initial perturbation μ , in particular, for the case p = 0 (Figures 1, 2, 3, 4, 5), it was found that the oscillation of the disk model is periodic up to μ ∗ ≈ 1.79 if Ω = 0 , and for Ω = 0.2 we have μ ∗ ≈ 1.77 , and so on. As can be seen from the figures, at μ ≥ 1.8 , the oscillations of the disk model become unstable regardless of the value of the rotation parameter, since, in this case, the semiaxes of the disk a ( t ) and b ( t ) grow linearly with time. Here, we note that as the integration interval increases, the overall picture changes insignificantly. Separate numerical calculations were carried out to determine the boundary between stable and unstable oscillations in terms of the disk rotation parameter.

Figure 1

Dependences of the major and minor semiaxes of the disk on time for different values of the initial perturbation μ for the rotation parameter Ω = 0.0 .

Figure 2

Dependences of the major and minor semiaxes of the disk on time for different values of the initial perturbation μ for the rotation parameter Ω = 0.2 .

Figure 3

Dependences of the major and minor semiaxes of the disk on time for different values of the initial perturbation μ for the rotation parameter Ω = 0.4 .

Figure 4

Dependences of the major and minor semiaxes of the disk on time for different values of the initial perturbation μ for the rotation parameter Ω = 0.6 .

Figure 5

Dependences of the major and minor semiaxes of the disk on time for different values of the initial perturbation μ for the rotation parameter Ω = 0.8 .

Thus, the obtained results showed a qualitative change in the nature of oscillations in stable and unstable zones of values for the parameters μ and Ω . In the case of stable oscillations, Ω < 0.5 , the semiaxes of the disk a ( t ) and b ( t ) are strictly periodic functions of time with a constant or varying pulsation amplitude for small perturbations μ . At Ω = 0 with increasing μ up to μ = 1.5 and at Ω = 0.4 up to μ = 1.3 , the behavior of the disk semiaxes does not differ from the behavior of a ( t ) and b ( t ) in the instability zone Ω > 0.5 . In these cases, although certain disc pulsations can be traced, they do not have a strict periodicity, and the pulsation amplitude is several times greater than the initial perturbation μ .

Using the results of numerical calculations, we also constructed the critical dependence of the initial perturbation μ on the rotation parameter Ω for various values of p (Figure 6). This figure shows the boundaries of the stable region in the parameter space. Here, the stable region for each case is below the corresponding critical dependence line. As can be seen from Figure 6, with an increase in the value of the parameter p , the stability regions corresponding to them increase, and the disk rotation parameter plays a destabilizing role.

Figure 6

Critical dependence of the initial perturbation μ on the rotation parameter Ω for different values of p .

Figure 7 shows the dependences of the major and minor semiaxes of the disk on time for both p = 1 and p = 0 in the form of different types of lines (dashed and continuous lines) for a visual comparison of cases with and without a halo at different values of the initial perturbation and disk rotation parameter.

Figure 7

Behavior of the semiaxes of the disk, taking into account the halo, for various values of the main parameters of the model. Here, dotted lines indicate the behavior of the semiaxes for the case p = 1 , and continuous lines for p = 0 .

4 Conclusion

Let us list the main results obtained by us in this work:

1. A system of equations for the evolution of nonlinear nonradial oscillations of a self-gravitating disk is obtained, taking into account the halo, in the form of matrix differential equations, and a method for its numerical analysis is developed.

2. Dependences of the major and minor semiaxes of the disk on time are obtained for various values of the system parameters.

3. The critical values of the ratio of the mass of the halo to the mass of the disk p and the disk rotation parameter Ω are determined, at which the halo stabilizes the nonlinear and nonradial oscillations of the self-gravitating disk.

4. The statistical amplitude η , which characterizes the degree of deformation of the system, is determined. It is found that in the interval of the rotation parameter Ω ∈ [ 0 ; 0.3 ] with small perturbations of the deformation, the values of η change insignificantly, and in the interval Ω ∈ [ 0.4 ; 0.5 ] , there is a zone of reduced deformations, but at Ω = 0.5 , there is a sharp increase in η , and the value of η increases by several orders of magnitude.

5. It is proved that, for μ ≥ 1.8 , the semiaxes of the disk grow linearly with time and the disk breaks up, i.e., dissipates when the halo is absent.

6. It has been established that at p = 1 and 0 ≤ Ω ≤ 1 , the halo stabilizes nonlinear and nonradial disk oscillations, but starting from p ≤ 0.001 , it ceases to stabilize these oscillations.