Abstract
In this work, we consider small-scale sectorial perturbation modes in a disk-like model of a radially nonstationary spiral galaxy in order to study the gravitational instabilities of these modes. Calculations of horizontal sectorial small-scale perturbation modes, such as
1 Introduction
Gravitational instabilities of disk-like models of self-gravitating systems are of great interest not only for galactic disks but also for accretion ones studied by Lodato (2007), Forgan and Rice (2011a, b), Lodato (2012), Rice (2016), Kratter and Lodato (2016), Paneque-Carreño et al. (2021), and Bethune and Latter (2021).
Both large-scale and various small-scale formations are observed in disk-like galaxies. Small-scale formations of disk-like galaxies are mainly open star clusters (OSC) and molecular clouds (MC). Formation and origin of these structural formations in disk-like self-gravitating systems occur due to gravitational instabilities (Mayer et al. 2016, Sharma 2016, Inoue and Yoshida 2018, Roshan and Rahvar 2019). They determine the global structure of the disk of galaxies. However, there is still no analysis of the problems of their origin and no one has studied in detail small-scale perturbations against the background of nonlinearly non-stationary models of disk-like self-gravitating systems, in particular, for our Galaxy. This raises a number of new questions. Can their global distribution be explained by the corresponding formation theory? It is not clear under what physical conditions these objects can form in disk-like systems, and what are the characteristic times of these phenomena? In addition, it is not clear which objects or structural formations of disk-like subsystems of galaxies are directly related to small-scale disturbances. These questions arise primarily due to the lack of knowledge of small-scale high-order perturbations in specific stationary and non-stationary models.
In the disks under study, self-gravity plays the main role in the studies of Rice et al. (2011), Trova et al. (2014), Meru (2015), and Young and Clarke (2015). The gravitational interaction between different parts of the system compresses matter. This process is called gravitational or Jeans instability, and it was studied by Fridman and Khoprskov (2013). It leads to a redistribution of mass, i.e., in one area of the system, the density increases, in another area, respectively, decreases. Clumps of MC are formed due to gravitational instabilities of gas disk-like galaxies in the studies of Forgan and Rice (2011a, b), Dipierro et al. (2014), and Tsukamoto et al. (2015).
Nowadays, many types of instabilities have already been identified for both equilibrium models (Binney and Tremaine 2008) and nonlinear nonstationary states of disk subsystems, which was considered by Nuritdinov (1993) and Mirtadjieva and Nuritdinov (2012), and the corresponding nonstationary dispersion equations (NDEs) were obtained for each perturbation mode.
2 Pulsating anisotropic model
This model is based on the well-known isotropic nonlinear – pulsating model of Nuritdinov (1993)
Here
The model pulsates with an amplitude
Using Eq. (1), it is possible to make an anisotropic model by averaging the phase density of the rotation parameter
where
Next, we can find the NDE of model (3). It can be in the following form:
where
where
3 Calculation results
In this section, we explore the results of calculations for six values of the sectorial small-scale oscillation modes taking
For example, NDE for the case
Here
where
Numerical calculation of the NDE shows that for the oscillation mode
For the perturbation mode
It is shown in Figure 3 for the case
Instability in the fourth case
For the oscillation mode
The instability in the sixth case,
The shaded area defines the instability regions for the indicated disturbance modes in the figures (Figures 1–6). It can be seen from the figures that as the speed of rotation of the model grows, the region of instabilities always grows, and with increasing values of
We also calculated and compared the increments of instabilities of small-scale oscillation modes for different values of the parameters (Figures 7 and 8). In Figures 7 and 8 show the increments of instabilities for all small-scale oscillation modes at different values of the virial ratio and degree of rotation. It can be seen from the figures that the curves of the growth rates of instabilities of the studied perturbation modes are arranged in descending order and it should be noted that with an increase in the value of the wavenumbers, the increments of instabilities also increase. The curved lines intersect each other, merging at some values of the increment of instabilities, except for the value of the rotation parameter
4 Conclusion
The gravitational instabilities are deeply studied for oscillation modes
From a comparison of the graphs (Figures 7 and 8), we can see that the intersection of the curves at some values of the instability increments except for the case
Acknowledgements
This work was partly supported by the Ministry of Innovative Development of the Republic of Uzbekistan.
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Funding information: The authors state no funding involved.
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Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.
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Conflict of interest: The authors state no conflict of interest.
Appendix
Substituting all the expressions of the functions in (4) and calculating each of them step by step, we find that
where
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© 2022 Jakhongir Ganiev and Salakhutdin Nuritdinov, published by De Gruyter
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