- Email: [email protected]
Structures formation through self-organized accretion on cosmic strings
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Chaos, Solitons and Fractals 41 (2009) 583–586 www.elsevier.com/locate/chaos
Structures formation through self-organized accretion on cosmic strings R. Murdzek Physics Department, ‘‘Al.I.Cuza’’ University, Boulevard Carol I, Nr. 11, 700506 Iassy, Romania Accepted 18 February 2008
Communicated by Prof. L. Marek-Crnjac
Abstract In this paper, we shall show that the formation of structures through accretion by a cosmic string is driven by a natural feed-back mechanism: a part of the energy radiated by accretions creates a pressure on the accretion disk itself. This phenomenon leads to a nonlinear evolution of the accretion process. Thus, the formation of structures results as a consequence of a self-organized growth of the accreting central object. Ó 2008 Elsevier Ltd. All rights reserved.
1. Introduction In the cosmic strings model there are three mechanisms for structure formation: accretion by long, slowly moving strings, accretion by wakes created by fast moving strings and accretion by loops. Cosmic string loops have the same average field as a point source with mass M(R) = 2pRl, in which R is the radius of the loop and l is the linear density of the string. Loops will be seeds for accretion of matter. Long straight strings also create conditions for accretion of matter: even the metric near a straight string is flat, string presence deforms the gravitational field of a particle in such a way that the particle is attracted by the string. Accretion theory was initiated by Lust in 1952 and developed by Sakura and Sunyaev (1973), Lynden-Bell and Pringle (1974) and it was used at first as a theory of star formation. The most simply way to consider a theory with accretion is by neglecting the viscosity and the angular momentum of the accreting matter. Then, the matter falls directly on the central object, with spherical symmetry. The gravitational energy is transforming in kinetic energy and the acceleration lead to emission of radiation. This radiation is intercepted by the accreting matter and plays the role of a damping force. The bigger the accretion rate, the bigger the luminosity of the radiation and the damping force. The dynamics reaches _ Edd c2 ¼ 4pGc=r, where r is the scattering cross-section. equilibrium at Eddington limit, LEdd ¼ M But, in general, matter accreting on to a central mass forms a disk with specific angular momentum and presents viscosity. In the thin disk approximation, the height of the disk, H, is much smaller than its radius and the vertical structure is almost hydrostatic and decouples from the horizontal structure which can be described in terms of its surface density R. If the disk is axially symmetric, mass and angular momentum conservation imply that the latter obeys a E-mail address: [email protected] 0960-0779/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2008.02.025
584
R. Murdzek / Chaos, Solitons and Fractals 41 (2009) 583–586
nonlinear diffusion equation. The main actors here are then the density surface and the viscosity. Many accreting disk were observed to vary strongly. In order to describe this process, the basic model is the instability picture [1]. The fundamental idea is that in a certain range of mass transfer rates, the disk can exist in either two states: a hot, high viscosity state and a cool, low viscosity state. As a function of accretion rate and viscosity, we can expect the disk to evolve between the two states, but because of the fact that the viscosity is varying with the luminosity of accretions, we can also expect this behavior to be nonlinear.
2. Nonlinear accretion by cosmic string The simplest model for cosmic strings is a straight, infinite long string. The space–time in the vicinity of such object is described by the metric, ds2 ¼ dt2 þ dr2 þ dz2 þ ð1 8Gl0 Þr2 dh2
ð1Þ
By defining a new angular coordinate / = (1 8Gl0)h, we can immediately see that the metric (1) corresponds to a flat space–time with a deficit angle given by D = 8pGl0. For this metric, the Ricci scalar is zero. The space–time is flat, but it is easy to show that a particle with mass m in the string’s vicinity will suffer a self-interaction force proportional to m2, acting as an attraction by string. If the string is moving slowly into a background with matter then the matter will be accreted by the string with an approximate axially symmetry in the accretion cloud. Any section of constant z and t will show a perfectly thin accretion disk around the core of the string. Let us consider an accretion thin disk, with angular momentum J. Because of the acceleration, the accreting matter becomes to radiate. Part of the energy emitted in the central and inner region of the disk is intercepted by the disk itself and leads to the heating of the disk surface. If the heating of the surface is large enough then the thermal convection, which is responsible for the viscosity, could drop to zero. The decrease of the viscosity at big radius provides perturbation in the accretion rate and conduct to a decrease of the emitted energy. This mechanism sets a feed-back cycle between the accretion rates in the central region of the disk on one hand and the viscosity of the matter at large radius on the other hand. The matter flow through the inner region of the disk is determined by the matter flow at large radius: the matter falls on the disk form outside region and is transported in the inner region. One could expect a feed-back cycle if the matter flow at large radius is affected by the radiation emitted in the inner region. When the mass flow in central and inner region is big, the intense heating of the surface leads to the decrease of the mass transfer from the outside to the inner region. As the disk is going empty, the accretion rate and the accretion luminosity is decreasing. The decrease of the radiation conducts to a revival of the accretion flow from the outside. The effect is the stabilization of the disc through a global feed-back. In order to describe this process we consider that: the accretion disk is in rotation, axially symmetric and geometrically thin. The mass flow is then described by the diffusion equation [2], o 3 o 1=2 o Rðr; tÞ ¼ r Rðr; tÞmðr; tÞr1=2 þ SðrÞ ot r or or
ð2Þ
where S(r) is the mass amount from outside, r is the radius and m(r, t) is the viscosity. Let cs be the sound speed in the disk and H the thickness of the disk. Then [3] mðr; tÞ ¼ acs H ;
a 2 ð0; 1Þ
ð3Þ
By substitution x = r1/2, (1) becomes o 3 o2 Rðr; tÞ ¼ 3 2 Rðx; tÞmðx; tÞx þ SðxÞ ot 4x ox
ð4Þ
and, from the equation of continuity, we have _ ¼ 2prRvr ¼ 6pr1=2 o ðmRr1=2 Þ M or
ð5Þ
where vr is the radial velocity of matter flow. Finally, from Eqs. (3)–(5) we obtain _ o 1 oM R¼ þ SðxÞ 3 ot 4px ox
ð6Þ
R. Murdzek / Chaos, Solitons and Fractals 41 (2009) 583–586
585
and _ ¼ 3p o ðRmxÞ M ox
ð60 Þ
3. Numerical scheme and results Eqs. (6) and (60 ) can be written in the recurrent form: " # 1 _ jþS þM Rjþ1 ¼ Rj 1 a _ nj 1þM
ð7Þ
Fig. 1. Regimes of accretion by cosmic strings for different values of mass amount (S) and sensitivity to feed-back (n).
S=1.2, n=12
1.30
1.4
S=1.2, n=15
accretion rate
accretion rate
1.25
1.20
1.3
1.2
1.15
1.1 1.10 0
5
10
15
20
25
30
0
10
20
time
30
40
time
S=1.2, n=40 1.6
1.5
S=1.2, n=60
accretion rate
accretion rate
1.4
1.4
1.2
1.3 1.2 1.1
1.0 1.0
0.8 -200
0.9
0
20
40
60
80
time
100
120
140
160
0
5
10
15
time
Fig. 2. Accretion regimes for S = 1.2 and different values of sensitivity.
20
25
30
586
R. Murdzek / Chaos, Solitons and Fractals 41 (2009) 583–586
_ j ð1 2aÞ þ a2 Rj _ jþ1 ¼ M M
1 _ nj 1þM
ð70 Þ
where n describe the disk sensitivity to feed-back [4] and the terms in Eq. (6) were normalized to Eddington limit. The essential parameters in Eqs. (7) and (70 ) are the sensitivity to feed-back, n and the mass amount from the outside, S. So, we can obtain the accretion rate as a function of time for different values of S and n. We have obtained five possible regimes for accretion: Stationary regime. If S is smaller than 1 then the accretion is stationary for any value of n and we can see why: the accretion rate is too small and the disk is not sensitive to the feed-back. Oscillatory regime. By increasing the mass amount, the disk become sensitive to the radiation and heating and the accretion rate starts to oscillate. Oscillations occur if the exponent n is between 10 and 13 for any value of mass amount and for n < 50 if the mass amount is slightly over the Eddington limit. Quasi-periodic regime. At large values of S and n the accretion disk becomes stable at perturbations and evolves through a quasi-periodic behavior characterized by multiple periods. The accretion process is driven by the viscosity. Chaotic regime. If the mass amount is well over the Eddington limit and the disk is sensitive to feed-back then the matter heating and cooling occur in a chaotic manner. The turbulences in the mass flow become dominant. Mixed regime. For large mass amount and large sensitivity the accretion rate presents a complex behavior characterized by intervals of periodicity followed by chaos. Fig. 1 presents a synthesis of the above discussed regimes. Fig. 2 shows the accretion rate as a function of time for different values of sensitivity n. one can observe the transition from oscillations, through chaos and mixed regime, to quasi-periodic regime. Note here that the time unit used corresponds to the diffusion time through the disk.
4. Conclusions The formation of the observable structures in the universe may be the result of the accretion on cosmic strings, regularized by a feed-back process between the accretion rate and the radiation heating effect. This is, in fact, a self-organizing process. The recent successes of the physics of self-organizing systems allow us to reconsider some of among a plethora of problems in cosmology, simply because it is nontrivial that the universe organizes as it does [5,6]. We also know that self-organizing systems are often critical systems, with structures spread in space over all scales [7]. The fact that the matter structures in the universe presents links with cosmic strings and has a nonlinear evolution towards a higher organized state may be connected with El Naschie’s E-infinity theory [8–11].
References [1] [2] [3] [4] [5] [6] [7] [8]
Spruit HC. Accretion Disks; 2000.. Pringle J. ARA&A 1981;19:137. Landau L, Lifchitz E. Mecanique des fluides. Moscova: Mir; 1971. Filho CM. Ap J 1991;378. Murdzek R. A direct link between the large scale structure and cosmic strings. Chaos, Solitons & Fractals 2007;33:748–53. Murdzek R. Hierarchical Cantor set in the large scale structure with torus geometry. Chaos, Solitons & Fractals 2008;38:1269–73. Mercader JP; 1996. astro-ph/9608168. El Naschie MS. A review of E infinity theory and the mass spectrum of high energy particle physics. Chaos, Solitons & Fractals 2004;19:209–36. [9] Sigalotti LDG, Mejias A. On El Naschie’s conjugate complex time, fractal E-(infinity) space–time and faster-than-light particles. Int J Nonlinear Sci Numer Simul 2006;7(4):467–72. [10] Iovane G. E1 Naschie E-Infinity Cantorian spacetime and lengths scales in cosmology. Int J Nonlinear Sci Numer Simul 2006;7(2):155–62. [11] Murdzek R, Harabagiu A, Girtu M, Agop M. Towards a self-organising universe. In: Proceedings of the second symposium on nonlinear dynamics. Shanghai, China; 27–30 October 2007.
Chaos, Solitons and Fractals 41 (2009) 583–586 www.elsevier.com/locate/chaos
Structures formation through self-organized accretion on cosmic strings R. Murdzek Physics Department, ‘‘Al.I.Cuza’’ University, Boulevard Carol I, Nr. 11, 700506 Iassy, Romania Accepted 18 February 2008
Communicated by Prof. L. Marek-Crnjac
Abstract In this paper, we shall show that the formation of structures through accretion by a cosmic string is driven by a natural feed-back mechanism: a part of the energy radiated by accretions creates a pressure on the accretion disk itself. This phenomenon leads to a nonlinear evolution of the accretion process. Thus, the formation of structures results as a consequence of a self-organized growth of the accreting central object. Ó 2008 Elsevier Ltd. All rights reserved.
1. Introduction In the cosmic strings model there are three mechanisms for structure formation: accretion by long, slowly moving strings, accretion by wakes created by fast moving strings and accretion by loops. Cosmic string loops have the same average field as a point source with mass M(R) = 2pRl, in which R is the radius of the loop and l is the linear density of the string. Loops will be seeds for accretion of matter. Long straight strings also create conditions for accretion of matter: even the metric near a straight string is flat, string presence deforms the gravitational field of a particle in such a way that the particle is attracted by the string. Accretion theory was initiated by Lust in 1952 and developed by Sakura and Sunyaev (1973), Lynden-Bell and Pringle (1974) and it was used at first as a theory of star formation. The most simply way to consider a theory with accretion is by neglecting the viscosity and the angular momentum of the accreting matter. Then, the matter falls directly on the central object, with spherical symmetry. The gravitational energy is transforming in kinetic energy and the acceleration lead to emission of radiation. This radiation is intercepted by the accreting matter and plays the role of a damping force. The bigger the accretion rate, the bigger the luminosity of the radiation and the damping force. The dynamics reaches _ Edd c2 ¼ 4pGc=r, where r is the scattering cross-section. equilibrium at Eddington limit, LEdd ¼ M But, in general, matter accreting on to a central mass forms a disk with specific angular momentum and presents viscosity. In the thin disk approximation, the height of the disk, H, is much smaller than its radius and the vertical structure is almost hydrostatic and decouples from the horizontal structure which can be described in terms of its surface density R. If the disk is axially symmetric, mass and angular momentum conservation imply that the latter obeys a E-mail address: [email protected] 0960-0779/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2008.02.025
584
R. Murdzek / Chaos, Solitons and Fractals 41 (2009) 583–586
nonlinear diffusion equation. The main actors here are then the density surface and the viscosity. Many accreting disk were observed to vary strongly. In order to describe this process, the basic model is the instability picture [1]. The fundamental idea is that in a certain range of mass transfer rates, the disk can exist in either two states: a hot, high viscosity state and a cool, low viscosity state. As a function of accretion rate and viscosity, we can expect the disk to evolve between the two states, but because of the fact that the viscosity is varying with the luminosity of accretions, we can also expect this behavior to be nonlinear.
2. Nonlinear accretion by cosmic string The simplest model for cosmic strings is a straight, infinite long string. The space–time in the vicinity of such object is described by the metric, ds2 ¼ dt2 þ dr2 þ dz2 þ ð1 8Gl0 Þr2 dh2
ð1Þ
By defining a new angular coordinate / = (1 8Gl0)h, we can immediately see that the metric (1) corresponds to a flat space–time with a deficit angle given by D = 8pGl0. For this metric, the Ricci scalar is zero. The space–time is flat, but it is easy to show that a particle with mass m in the string’s vicinity will suffer a self-interaction force proportional to m2, acting as an attraction by string. If the string is moving slowly into a background with matter then the matter will be accreted by the string with an approximate axially symmetry in the accretion cloud. Any section of constant z and t will show a perfectly thin accretion disk around the core of the string. Let us consider an accretion thin disk, with angular momentum J. Because of the acceleration, the accreting matter becomes to radiate. Part of the energy emitted in the central and inner region of the disk is intercepted by the disk itself and leads to the heating of the disk surface. If the heating of the surface is large enough then the thermal convection, which is responsible for the viscosity, could drop to zero. The decrease of the viscosity at big radius provides perturbation in the accretion rate and conduct to a decrease of the emitted energy. This mechanism sets a feed-back cycle between the accretion rates in the central region of the disk on one hand and the viscosity of the matter at large radius on the other hand. The matter flow through the inner region of the disk is determined by the matter flow at large radius: the matter falls on the disk form outside region and is transported in the inner region. One could expect a feed-back cycle if the matter flow at large radius is affected by the radiation emitted in the inner region. When the mass flow in central and inner region is big, the intense heating of the surface leads to the decrease of the mass transfer from the outside to the inner region. As the disk is going empty, the accretion rate and the accretion luminosity is decreasing. The decrease of the radiation conducts to a revival of the accretion flow from the outside. The effect is the stabilization of the disc through a global feed-back. In order to describe this process we consider that: the accretion disk is in rotation, axially symmetric and geometrically thin. The mass flow is then described by the diffusion equation [2], o 3 o 1=2 o Rðr; tÞ ¼ r Rðr; tÞmðr; tÞr1=2 þ SðrÞ ot r or or
ð2Þ
where S(r) is the mass amount from outside, r is the radius and m(r, t) is the viscosity. Let cs be the sound speed in the disk and H the thickness of the disk. Then [3] mðr; tÞ ¼ acs H ;
a 2 ð0; 1Þ
ð3Þ
By substitution x = r1/2, (1) becomes o 3 o2 Rðr; tÞ ¼ 3 2 Rðx; tÞmðx; tÞx þ SðxÞ ot 4x ox
ð4Þ
and, from the equation of continuity, we have _ ¼ 2prRvr ¼ 6pr1=2 o ðmRr1=2 Þ M or
ð5Þ
where vr is the radial velocity of matter flow. Finally, from Eqs. (3)–(5) we obtain _ o 1 oM R¼ þ SðxÞ 3 ot 4px ox
ð6Þ
R. Murdzek / Chaos, Solitons and Fractals 41 (2009) 583–586
585
and _ ¼ 3p o ðRmxÞ M ox
ð60 Þ
3. Numerical scheme and results Eqs. (6) and (60 ) can be written in the recurrent form: " # 1 _ jþS þM Rjþ1 ¼ Rj 1 a _ nj 1þM
ð7Þ
Fig. 1. Regimes of accretion by cosmic strings for different values of mass amount (S) and sensitivity to feed-back (n).
S=1.2, n=12
1.30
1.4
S=1.2, n=15
accretion rate
accretion rate
1.25
1.20
1.3
1.2
1.15
1.1 1.10 0
5
10
15
20
25
30
0
10
20
time
30
40
time
S=1.2, n=40 1.6
1.5
S=1.2, n=60
accretion rate
accretion rate
1.4
1.4
1.2
1.3 1.2 1.1
1.0 1.0
0.8 -200
0.9
0
20
40
60
80
time
100
120
140
160
0
5
10
15
time
Fig. 2. Accretion regimes for S = 1.2 and different values of sensitivity.
20
25
30
586
R. Murdzek / Chaos, Solitons and Fractals 41 (2009) 583–586
_ j ð1 2aÞ þ a2 Rj _ jþ1 ¼ M M
1 _ nj 1þM
ð70 Þ
where n describe the disk sensitivity to feed-back [4] and the terms in Eq. (6) were normalized to Eddington limit. The essential parameters in Eqs. (7) and (70 ) are the sensitivity to feed-back, n and the mass amount from the outside, S. So, we can obtain the accretion rate as a function of time for different values of S and n. We have obtained five possible regimes for accretion: Stationary regime. If S is smaller than 1 then the accretion is stationary for any value of n and we can see why: the accretion rate is too small and the disk is not sensitive to the feed-back. Oscillatory regime. By increasing the mass amount, the disk become sensitive to the radiation and heating and the accretion rate starts to oscillate. Oscillations occur if the exponent n is between 10 and 13 for any value of mass amount and for n < 50 if the mass amount is slightly over the Eddington limit. Quasi-periodic regime. At large values of S and n the accretion disk becomes stable at perturbations and evolves through a quasi-periodic behavior characterized by multiple periods. The accretion process is driven by the viscosity. Chaotic regime. If the mass amount is well over the Eddington limit and the disk is sensitive to feed-back then the matter heating and cooling occur in a chaotic manner. The turbulences in the mass flow become dominant. Mixed regime. For large mass amount and large sensitivity the accretion rate presents a complex behavior characterized by intervals of periodicity followed by chaos. Fig. 1 presents a synthesis of the above discussed regimes. Fig. 2 shows the accretion rate as a function of time for different values of sensitivity n. one can observe the transition from oscillations, through chaos and mixed regime, to quasi-periodic regime. Note here that the time unit used corresponds to the diffusion time through the disk.
4. Conclusions The formation of the observable structures in the universe may be the result of the accretion on cosmic strings, regularized by a feed-back process between the accretion rate and the radiation heating effect. This is, in fact, a self-organizing process. The recent successes of the physics of self-organizing systems allow us to reconsider some of among a plethora of problems in cosmology, simply because it is nontrivial that the universe organizes as it does [5,6]. We also know that self-organizing systems are often critical systems, with structures spread in space over all scales [7]. The fact that the matter structures in the universe presents links with cosmic strings and has a nonlinear evolution towards a higher organized state may be connected with El Naschie’s E-infinity theory [8–11].
References [1] [2] [3] [4] [5] [6] [7] [8]
Spruit HC. Accretion Disks; 2000.