CC BY
0MODELS OF SMALL-SCALE STRUCTURES IN DISK-LIKE SELF-
GRAVITATING OBJECTS
Ganiev J.M.
National University of Uzbekistan. https://doi.org/10.5281/zenodo.11166893 Mathematical modeling is used in all areas of modern science for the study of various dynamic processes. Modeling is also a universal tool for studying non-linear evolutionary processes, observed objects of the Universe [1]. With the help of mathematical modeling, many problems of computational astrophysics are solved, such as collision problems [2], galaxy evolution [3], star collapse processes [4], and kinetic processes in galaxies [5]. Also relevant is the study of magnetic fields in the arms of spiral galaxies [6], in the numerical simulation of space plasma [7], star formation processes [8], etc. The observable astrophysical objects of the Universe have different mechanisms of origin and specific methods of mathematical modeling are used to describe the physics of these objects.
At present, there are a number of non-linear non-stationary isotropic [9-11] and anisotropic [12-14] models for studying large-scale formations of disk subsystems of self-gravitating systems. The corresponding non-stationary dispersion equations for each perturbation mode are obtained. However, the role of small-scale modes in disks, especially against the background of non-stationary disks, has not been studied by anyone so far. A detailed and thorough analysis of the problems of their origin in various non-stationary flat formations has also not been completed, and there is also no corresponding non-linear theory of their formation. In particular, it is not clear under what criteria the observed small-scale formations can form in disklike systems, and what is the primary mechanism of this phenomenon. This implies the relevance of the problem of constructing a mathematical model for studying small-scale instabilities in non-stationary disk-like self-gravitating objects against the background of an anisotropic model.
Anisotropic model. Mathematical modeling of the structure and evolution of disk-like self-gravitating systems requires an analysis of the stability of various small-scale disturbances. The study of the corresponding mechanisms of formation of small-scale structural formations requires the construction of a mathematical model of a self-gravitating disk with an anisotropic velocity diagram. This model is based on the well-known isotropic nonlinearly pulsating model of Nuritdinov [12]
i
2 "x( R - r). (1)
r,Vr, v±, t) =
G„
2% nVi-Q2
1 -Q2
n2
2
n2
(vr - V, )2 -(VX- V )2
Here ct0 = a(0; 0), Q is a dimensionless parameter that characterizes the degree of rotation of the solid disk (0<Q< 1). vr and v± are the radial and tangential
velocities of the particles, the function n(t) has the meaning of the stretching
i ?\-3/2
coefficient and n(t) = (1 + A cos y) ■ (1 -A2)-1, t = (y + A sin y)-(1 -A2) ,
-A 2(r sin y/n2), vb =Qr/n2. The model pulsates with an amplitude
A = 1 - (2T /)0, where (2T /)0 is the virial parameter.
Using (1), one can construct an anisotropic model by averaging the phase density over the rotation parameter Q:
+1
\wizot P(Q)d a
¥ = --
ntll'7 I 1
(2)
Jp(a)d a
-1
m =
Aniz _
n
Here p(Q) is a weight function of Q. We have constructed an anisotropic model,
taking the weight function in the form p(Q) = 2/Vl-Q2(l + QQ'). After
integrating by Q in the interval -1 to +1 and some transformations, we obtain a nonlinearly pulsating model with an anisotropic velocity diagram:
;i+Q'- (xv, - yvx)] ■ X ((l - rVn2) (1 - n2v2) - n2 (Vr - vfl )2).
(3)
For model (3) we have found the nonstationary dispersion equation (NDE) in general form according to the general principle:
a(y) = Y Nm (N -1)( N2 + N - m2)D~N + iQm(N2 + N- m2)dN.
(4)
¥
Here YNm = (N +m - ^ - "l- 1)11 . DN =J E^/^ „
(N + m)!!(N- m)!! 4 d (cosh)
¥
r 1+ A cos y,
d„ = | El^y 1, W = -jJ, /n =wn-
1 d P P
1 d 1 N+1 1 N
N + 2 d(cosh)2 d(cosh) _
2
tgh,
t- rf/ \ / \ t, (cosy + A)(cosy1 + A) + (1 -A )sinysiny1
E = n (y1)S(y,y1)a(y1), cosh = ^—^— \ ; v-^-^
(1 + A cos y )(1 + A cos y1)
PN(cosh) is the Legendre polynomial, N is basic perturbation index, m is azimuth wavenumber.
On the basis of found NDE, recently, we have calculated the instability criteria for some sectorial [15] and tesseral [16] small-scale oscillation modes. Critical diagrams "virial parameter ^ degree of rotation" are obtained for each of these oscillation modes and the corresponding instability increments are calculated. Here we will consider the small-scale perturbation with two-arm nature m=2 for the spiral galaxies.
m=2; N=10
0.5 -1-1-1-1-1-1-1-1-1-
0.4
zz 0.2723 13. 0.2712
P] 0.2329 0.2
0.1
0
0 0.1 02 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Omega
Fig. 1. The critical dependence of the virial ratio on the rotation parameter for m=2; N=10.
The results of calculations. In this work, we present the results of calculations of the disk model for small-scale perturbation modes, in particular, for wave numbers m=2; N=10. For calculation of gravitational instability of mode m=2; N=10 we use the NDE (4).
The NDE for m=2; N=10 has the following form:
+ + + L =-49-t"q"TK2.10(/ ) + iM2.10(lT)1
d^2 L 32768• (1 + Acos^)18 L 2 2 J
_ (5)
where X is the amplitude of the disk pulsation and l = 0;9. The expressions for K2;10(lT) and M2.10(/r) are given in the Appendix.
The numerical calculation of the NDE shows that for the m=2; N=10 oscillation mode, the instability starts from the value of the virial parameter (2T/|U|)0~0.217 and reaches 0.413. Between the values (2T/|U|)0~0.2174^0.2178 there is a very
narrow stability region, and between the value (2T/|U|)0~0.2379^0.2312 and (2T/|U|)0~0.375^0.397 there is clearly noticeable stability island (Fig. 1). Hence, we note that with an increase in the value of the rotation parameter, the region of instability clearly increases. The shaded zone in the figure defines the region of instability.
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45
(2T/|U|)0
Fig. 2. Comparison of the instability increments of mode m=2; N=10 for different values of the rotation parameter.
We have also obtained and compared the instability increments for small-scale oscillation mode m=2, N=10 for different values of the virial ratio and degree of rotation. Narrow stability regions and the corresponding critical values of the intervals are found. It can be seen from Fig. 2 that the curves of the instability increments of the studied perturbation mode are arranged in the order of increasing rotation degree.
References:
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Appendix. We present the expressions for the two functions in (5):
K2;10(/T) = 1591 55c9 -495c7e2b2 + —c5e4b4 -—c3e6b6 + — ce8b8 \l0 2;10T 11 4 16 128 \
1485e2bc8 - 6930e4b3c6 + ^^eVc4 - ^VbV + ^e10b9 \
8 8 128
2^e4b2c7 -^e6b4c5 -495c9e2 + ^^W -^«Vc \\ 2 16 32 8 \ 2
234465 e6b3c6 -^Z865e8b5c4 - 6930e4bc8 + 5^e10b7c2 -\
4 8 32 16
7494795 ^45 557865 ^„3 53014^ 5197^ 3465 9 4 \
-e b c--e b c--e b c +--e b c +--c e \ +
64 8 16 8 4 \
■ Z49^e'0b5c< -530I45e.2b7c2 -«7«!ieW++^e'bc* \
64 16 8 4 8
234465 12l 6 3 14l 8 557865 10l 4 5 509355 8 2 7 5775 9 6
+ |-e 12b6c3 - 6930e14b8c--e10b4c5 +-e8b2c7--c9e6 \ +
4 8 32 16 \ 6
+ ^e»bV -495.W - 530M5,.2bV + ^¿W - 10^e»bc» \
2 16 32 8
1485e16b8c-6930e14b6c3 + ^e12b4c5 -^e10b2c7 + ^c9e8 \ +
8 8 128
+ 1 55e18b9 -495e16b7c2 + — e14b5c4 -— e12b3c6 + —e10bc8 \ k
4 16 128
M2;10(lr) = 140i^e 11 -495bc8 + U^eW - ^peVc4 + ^bV -I^b9 \ + +1 -^eV+<9^+"75^4 -93555c3e6b6 + ^«eV \ +
2 4 32 32 512
+ 1 -308385e4b3c6 + + 696465e6b5c4 -620235eVc2 + 3465e"V \\
16 2 32 128 32 \ 2
I 1867635 6 4 5 308385 4l 2 7 4163775 8l 6 3 1155 9 2 93555 10l 8 \,
+--e6b4c5 +-e4bc' +-e8b6c3--cV--e10b8c l, +
32 16 128 2 32
+ 1 - 21417165 e8b5c4 + ^e6b3c6 + 6?646VbV - "75^ -^e12b9 \4 +
256 32 32 32 32
1 1867635 10l6 3 21417165 8 4 5 197505 12l 8 696465 6 2 7 17325 9 4 \,
+--e b6c3 +-e8b4c5 +-eb8c--e6bV +-c9e4 l, +
32 256 32 32 32
308385 12,7 7 1867635 ^5. 1155 M 9 4]63775 93555 6 \
+--e b c +--e b c +--e b--e b c +--e bc +
16 32 2 128 32
I 5445 M „ 308385 12 6 3 69^65 ,^,, 620235 ^,7 3465 ,6 \
+--e b c +--e b c--e b c +--e b c--c e \/7 +
2 16 32 128 32
+ 1 -i?Vb9 + ^¿W - + 93^e!0b3c6 -\ +
4 2 32 32 512 1 s
+ 1 '^Vc - U^eW + ^e12bV - ^«W + \ k
4 2 32 32 512
Here c = X + cosy, b = siny, e = V 1 - X
