Vol.3, No.1, 65-68 (2011)
doi:10.4236/ns.2011.31009
Natural Science
Entropy changes in the clustering of galaxies in an expanding univer
Naer Iqbal1,2*, Mohammad Shafi Khan1, Tabasum Masood1
1Department of Physics, University of Kashmir, Srinagar, India; *Corresponding Author:
2Interuniversity Centre for Astronomy and Astrophysics, Pune, India.
Received 19 October 2010; revid 23 November 2010; accepted 26 November 2010.
ABSTRACT
In the prent work the approach-thermody- namics and statistical mechanics of gravitating systems is applied to study the entropy change in gravitational clustering of galaxies in an ex-panding univer. We derive analytically the expressions for gravitational entropy in terms of temperature T and average
density n of the par-ticles (galaxies) in the given pha space cell. It is found that during the initial stage of cluster-ing of galaxies, the entropy decreas and fi-nally ems to be increasing when the system attains virial equilibrium. The entropy changes are studied for different range of measuring correlation parameter b. We attempt to provide a clearer account of this phenomena. The entropy results for a system consisting of extended mass (non-point mass) particles show a similar behaviour with that of point mass particles clustering gravitationally in an expanding uni-ver.
Keywords:Gravitational Clustering; Thermodynamics; Entropy; Cosmology
1. INTRODUCTION
Galaxy groups and clusters are the largest known gravitationally bound objects to have arin thus far in the process of cosmic structure formation [1]. They form the denst part of the large scale structure of the uni-ver. In models for the gravitational formation of struc-ture with cold dark matter, the smallest structures col-lap first and eventually build the largest structures; clusters of galaxies are then formed relatively. The clus-ters themlves are often associated with larger groups called super-clusters. Clusters of galaxies are the most recent and most massive objects to have arin in the hiearchical structure formation of the univer and the study of clusters tells one about the way g
alaxies form and evolve. The average density n and the temperature T of a gravitating system discuss some thermal history of cluster formation. For a better larger understanding of this thermal history it is important to study the entropy change resulting during the clustering phenomena be-cau the entropy is the quantity most directly changed by increasing or decreasing thermal energy of intraclus-ter gas. The purpo of the prent paper is to show how entropy of the univer changes with time in a system of galaxies clustering under the influence of gravitational interaction.
Entropy is a measure of how disorganid a system is. It forms an important part of cond law of thermody-namics [2,3]. The concept of entropy is generally not well understood. For erupting stars, colloiding galaxies, collapsing black holes - the cosmos is a surprisingly or-derly place. Supermassive black holes, dark matter and stars are some of the contributors to the overall entropy of the univer. The microscopic explanation of entropy has been challenged both from the experimental and theoretical point of view [11,12]. Entropy is a mathe-matical formula. Standard calculations have shown that the entropy of our univer is dominated by black holes, who entropy is of the order of their area in planck units [13]. An analysis by Chas Egan of the Australian National University in Canberra indicates that the col-lective entropy of all the supermassive black holes at the centers of galaxies is about 100 times higher than previ-ously calculated. Statistical entr
opy is logrithmic of the number of microstates consistent with the obrved macroscopic properties of a system hence a measure of uncertainty about its preci state. Statistical mechanics explains entropy as the amount of uncertainty which remains about a system after its obrvable macroscopic properties have been taken into account. For a given t of macroscopic quantities like temperature and volume, the entropy is a function of the probability that the sys-tem is in various quantumn states. The more states avail-able to the system with higher probability, the greater the
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N. Iqbal et al. / Natural Science 3 (2011) 65-68
66 disorder and thus greater the entropy [2]. In real experi-ments, it is quite difficult to measure the entropy of a system. The technique for doing so is bad on the thermodynamic definition of entropy. We discuss the applicability of statistical mechanics and thermodynam-ics for gravitating systems and explain in what n the entropy change S – S 0 shows a changing behaviour with respect to the measuring correlation parameter b = 0 – 1.
2. THERMODYNAMIC DESCRIPTION OF GALAXY CLUSTERS
A system of many point particles which interacts by Newtonian gravity is always unstable. The basic insta-bilities which may occur involve the overall contraction (or expansion) of the system, and the formation of clus-ters within the system. The rates and forms of the in-stabilities are governed by the distribution of kinetic and potential energy and the momentum among the particles. For example, a finite spherical system which approxi-mately satisfies the viral theorem, contracts slowly
compared to the crossing time ~ ()12
G ρ- due to the evaporation of high energy particles [3] and the lack of equipartition among particles of different mass [4]. We consider here a thermodynamic description for the sys-tem (univer). The univer is considered to be an infi-nite gas in which each gas molecule is treated to be a
galaxy. The gravitational force is a binary interaction and as a result a number of particles cluster together. We u the same approximation of binary interaction for our univer (system) consisting of large number of galaxies clustering together under the influence of gravitational force. It is important to mention here that the characteri-zation of this clustering is a problem of current interest. The physical validity of the application of thermody-namics in the clustering of galaxies and galaxy clusters has been discusd on the basis of N-body computer simulation results [5]. Equations of state for internal energy U and pressure P are of the form [6]:
(3122
NT
U =
-)b (1) (1NT
P V
=
-)b (2) b defines the measuring correlation parameter and is dimensionless, given by [8]
()202,23W n
b Gm n T r K T
τξ∞
=-
=⎰,rdr (3)
W is the potential energy and K the kinetic energy of
the particles in a system. n N V = is the average num-ber density of the system of particles each of mass m, T is the temperature, V the volume, G is the universal
gravitational constant. (),,n T r ξ is the two particle correlation function and r is the inter-particle distance. An overall study of (),n T r ξ has already been dis-cusd by [7]. For an ideal gas behaviour b = 0 and for non-ideal gas system b varies between 0 and 1. Previ-ously some workers [7,8] have derived b in the form of:
3
3
1nT b nT ββ--=
+ (4) Eq.4 indicates that b has a specific dependence on the combination 3nT -.
3. ENTROPY CALCULATIONS
Thermodynamics and statistical mechanics have been found to be equal tools in describing entropy of a system. Thermodynamic entropy is a non-conrved state func-tion that is of great importance in science. Historically the concept of entropy evolved in order to explain why some process are spontaneous and others are not; sys-tems tend to progress in the direction of increasing en-tropy [9]. Following statistical mechanics and the work carried out by [10], the grand canonical partition func-tion is given by
()32
1
3212,1!N N N N mkT Z T V V nT N πβ--⎛⎫⎡=
+ ⎪⎣Λ⎝⎭
⎤⎦
(5)
where N! is due to the distinguishability of particles. Λ
reprents the volume of a pha space cell. N is the number of paricles (galaxies) with point mass approxi-mation. The Helmholtz free energy is given by:
ln N A T Z =- (6)
Thermodynamic description of entropy can be calcu-lated as:
,N V
A S T ∂⎛⎫=- ⎪∂⎝⎭ (7)
The u of Eq.5 and Eq.6 in Eq.7 gives
()3120ln ln 13S S n T b b -⎛⎫-=-- ⎪ ⎪⎝⎭
- (8) where S 0 is an arbitary constant. From Eq.4 we write
()3
1b
n b T β-=
- (9)
Using Eq.9, Eq.8 becomes as
3
203ln S S b bT ⎡⎤
-=-+⎢⎣⎦
⎥ (10)
Again from Eq.4
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N. Iqbal et al. / Natural Science 3 (2011) 65-68 67
67()13
2
2
1n b T b β-⎡⎤
=⎢⎣⎦
⎥ (11)
with the help of Eq.11, Eq.10 becomes as
()01
1ln ln 1322S S n b b b ⎡-=-+-+⎡⎤⎣⎦⎢⎥⎣⎦
⎤ (12) This is the expression for entropy of a system consist-ing of point mass particles, but actually galaxies have extended structures, therefore the point mass concept is only an approximation. For extended mass structures we make u of softening parameter ε who value is taken between 0.01 and 0.05 (in the units of total radius). Following the same procedure, Eq.8 becomes as
()320ln ln 13N S S N T N b Nb V εε⎡⎤
-=---⎢⎥⎣⎦
(13)
For extended structures of galaxies, Eq.4 gets modi-fied to
()()
331nT R b nT R εβαεβαε--=+ (14)
where α is a constant, R is the radius of a cell in a pha space in which number of particles (galaxies) is N and volume is V . The relation between b and b ε is given by: ()
11b b b εα
α=
+- (15) b ε reprents the correlation energy for extended mass particles clustering gravitationally in an expanding uni-ver. The above Eq.10 and Eq.12 take the form respec-tively as;
()()3
203ln 111bT b S S b b ααα⎡⎤
⎢⎥-=-+⎢⎥+-+-⎢⎥⎣⎦
1 (16) ()()()120113ln ln 2111b b b S S n b b ααα⎡⎤-⎡⎤⎢⎥⎣⎦-=-++⎢⎥+-+-⎢⎥⎣
⎦
1 (17)
where
2
R R ε
εεα⎛⎫⎛⎫=
⎪ ⎪⎝⎭⎝⎭
(18)
If ε = 0, α = 1 the entropy equations for extended mass galaxies are exactly same with that of a system of point mass galaxies approximation. Eq.10, Eq.12, Eq.16
and Eq.17 are ud here to study the entropy changes in
the cosmological many body problem. Various entropy change results S – S 0 for both the point mass approxima-tion and of extended mass approximation of particles (galaxies) are shown in (
Figures 1
and
2). The results
have been calculated analytically for different values of
Figure 1. (Color online) Comparison of isothermal entropy changes for non-point and point mass particles (galaxies) for an infinite gravitating system as a function of average relative temperature T and the parameter b . For non-point mass ε = 0.03 and R = 0.06 (left panel), ε = 0.04 and R = 0.04 (right panel).
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N. Iqbal et al. / Natural Science 3 (2011) 65-68 68
Figure 2. (Color online) Comparison of equi-density entropy changes for non-point and point mass particles (galaxies) for an infinite gravitating system as a function of average relative density n and th
e parameter b. For non-point mass ε= 0.03 and R = 0.04.
R (cell size) corresponding to different values of soften-ing parameter ε. We study the variations of entropy changes S – S0with the changing parameter b for differ-ent values of n and T. Some graphical variations for S – S0with b for different values of n = 0, 1, 100 and aver-age temperature T = 1, 10 and 100 and by fixing value of cell size R = 0.04 and 0.06 are shown. The graphical analysis can be repeated for different values of R and by fixing values of εfor different ts like 0.04 and 0.05. From both the figures shown in 1 and 2, the dashed line reprents variation for point mass particles and the solid line reprents variation for extended (non-point mass) particles (galaxies) clustering together. It has been ob-rved that the nature of the variation remains more or less same except with some minor difference.
4. RESULTS
The formula for entropy calculated in this paper has provided a convenient way to study the entropy changes in gravitational galaxy clusters in an expanding univer. Gravity changes things that we have witnesd in this rearch. Clustering of galaxies in an expanding univer, which is like that of a lf gravitating gas increas the gas volume which increas the entropy, but it also increas t
he potential energy and thus decreas the kinetic energy as particles must work against the attrac-tive gravitational field. So we expect expanding gas to cool down, and therefore there is a probability that the entropy has to decrea which gets confirmed from our theoretical calculations as shown in Figures 1 and 2. Entropy has remained an important contributor to our understanding in cosmology. Everything from gravita-tional clustering to supernova are contributors to entropy budget of the univer. A new calculation and study of entropy results given by Eqs.10, 12, 16 and 17 shows that the entropy of the univer decreas first with the clustering rate of the particles and then gradually in-creas as the system attains viral equilibrium. The gravitational entropy in this paper furthermore suggests that the univer is different than scientists had thought.
5. ACKNOWLEDGEMENTS
We are thankful to Interuniversity centre for Astronomy and Astro-physics Pune India for providing a warm hospitality and facilities during the cour of this work.
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