Influence of the Irreducible Triplets on the Velocity Distribution Of Galaxies

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1 EJTP 7, No. 4 (1) Electronic Journal of Theoretical Physics Influence of the Irreducible Triplets on the Velocity Distribution Of Galaxies Farooq Ahmad, Aasifa Nazir, and Manzoor A. Malik University of Kashmir, Hazratbal Srinagar-196, J&K,India; Inter-University Centre for Astronomy and Astrophysics, Pune-4117 Received 6 July 1, Accepted 15 August 1, Published 1 October 1 Abstract: We derive the velocity distribution function of galaxies from the partition function inclusive of higher order contribution. Our result shows that the effect of higher order contribution on the velocity distribution has an appreciable effect only for small N, while for large N the effect is negligible, thus revalidating the earlier results. We also calculate the density of energy states which gives probability for bound and virilized system of galaxies. c Electronic Journal of Theoretical Physics. All rights reserved. Keywords: Cosmology; Theory Galaxies Clusters; General Gravitation Large Scale Structure Methods; Analytical PACS (1): k; r; 95.3.Sf 1. Introduction The spatial distribution function of galaxies f(n), originally derived from the first and second laws of thermodynamics [13, 14] under quasi-equilibrium conditions, is a simple and powerful statistics which characterizes the location of galaxies in space. It is closely related to the velocity distribution function [15, 8] f(v)dv which gives the probability of finding a galaxy with peculiar velocity between v and v + dv. The two distribution functions (spatial and velocity) compliment each other and their consistent combination provides a statistical description of evolution in the complex six-dimensional phase space. The velocity distribution function f(v), was origially [15] derived from the spatial distribution by making the assumptions that the combined distribution function f(n,v) separates into f(n) f(v) in quasi-equilibrium and that on average the potential energy farphy@kashmiruniversity.ac.in sheikhaasifa@yahoo.com mmalik@kashmiruniversity.net

2 396 Electronic Journal of Theoretical Physics 7, No. 4 (1) fluctuations are proportional to the kinetic energy fluctuations in any local volume. After the development of the more general, statistical mechanical, description of the cosmological many-body problem [1], the velocity distribution function was rederived [8] giving some additional information e,g. the effect of surrounding an individual galaxy with a halo making use of the softened potential (r + ɛ ) 1/, with the softening parameter ɛ as a representative of halo surrounding each galaxy. The statistical mechanical description [1] has been refined further [] and extended in different directions [3, 1] and hence needs to relook at the earlier results for understanding the theory at different levels. In this paper we would like to specifically focus on the effect of the inclusion of the irreducible triplets on the velocity distribution and the density of states. Indeed, it is important to check the self-consistency of the earlier results in the wake of our improved theoretical understanding. In section, we redrive the velocity distribution function and the allied parameters starting with the partition function that incorporates the irreducible triplets. In section 3, we see how the density of states is modified by the fresh inclusion and finally we discuss our result in section 4.. Velocity Distribution Function for Irreducible Triplets. We consider a system having volume V, containing N number of galaxies each having mass m at a temperature T and number density n. The cosmological many body partition function for such a system inclusive of irreducible triplets is given by [], Z N (T,V )= V ( N πmt N! Λ ) 3N/ [ (1 + b nt 3 ) N 1 + ] (N 1)(N ) 4 9 (b nt 3 ) 3, (1) where N! takes the distinguishability of classical particles and Λ normalizes the phase space volume and b is given by, b = 3 (Gm ) 3. () From equation (1), all thermodynamical quantities can be derived as discussed in earlier paper []. The distribution function is then derived from the grand canonical partition function and is given as, f(n) = N(1 b)[ N(1 b)+nb] N 1 + N 3 ( N) N 3 (1 b)η e Nbt N(1 b t), (3) N![1+(1 b)η] where [ ] N +3aN b (1 b) N 3 b t = b, (4) N + a N b 3 (1 b) N 3 is the clustering parameter for N 3. Here η is given by η = a N b 3 (1 b) N 4, forn 3 =, forn < 3, (5)

3 Electronic Journal of Theoretical Physics 7, No. 4 (1) where b is defined by the relation [13], b = b nt 3. (6) 1+b nt 3 b ranges between to 1 and for no clustering b =, for condensation b = 1, and a N = (N 1)(N ). (7) 9 Equation (3), improves upon the earlier results, for the spatial spatial distribution function by taking into account the higher order (triplet) contributions. The velocity distribution can be obtained from equation (3) by assuming that the fluctuations in potential energy over a given volume are proportional to the kinetic energy fluctuations. 1 N = αv. (8) r ij This assumption has been applied earlier [15] to derive the velocity distribution function. Here, we use it to rederive the velocity distribution function from the spatial distribution function given by equation (3) that includes the irreducible triplet contributions. Here we may use of the fact that potential energy fluctuations around the spatial constant mean field having N number of galaxies have an ensemble average value, where Q N (T,V )=... φ = T T (ln Q N), (9) exp( φ(r 1,r,..., r N )T 1 )d 3N r = V N [ (1 + b nt 3 ) N 1 + (N 1)(N ) 4 9 (b nt 3 ) 3 ], (1) is the configurational integral defined earlier [] whose integration is very complicated and can be evaluated by using 3N-fold integration. By using equation (1) then equation (9) becomes, φ = 3(N 1)bT a t = (N 1)bm v a t, (11) where a t = (1 b)3 +3a N (N 1) 1 b (1 b) N 1. (1) (1 b) 3 + a N b 3 (1 b) N 1 The factor a t arises due to inclusion of triplets. Potential energy may fluctuate from one volume to another and fluctuations will be caused by fluctuation in N. We may write the average potential energy in this volume as, φ = 1 i<j N Gm = Gm N(N 1) r ij 1 r ij, (13)

4 398 Electronic Journal of Theoretical Physics 7, No. 4 (1) and 1 1 = η 1, (14) r ij r ij poission where 1/r ij poission is the average inverse position separation between the ith and jth particles. The values of r ij are smaller than their poission values and η 1 represents the form factor. From equation (11) and (13), we have N = ba t 1 Gmη 1 r ij 1 v. (15) poission This equation has exactly the same form as equation (8) with α given by, α = ba t Gmη 1 = b Gmη 1 [ (1 b) 3 +3a N (N 1) 1 b (1 b) N 1 (1 b) 3 + a N b 3 (1 b) N 1 ]. (16) If a N =, then equation (16) reduces to the earlier form, devoid of triplet contributions. Using natural units G=m=R=1 the value of α is also of order unity. Velocity distribution is obtained by first rescaling (3) from density fluctuations to the kinetic energy fluctuations [15]. Then the kinetic energy fluctuations are converted to velocity fluctuations from density distribution using the jacobian transformation. This gives f(v)dv = α β(1 b)[αβ(1 b)+αbv ] αv 1 +α(αv ) 3 (αβ) αv 3 (1 b)η Γ(αv +1)[1+(1 b)η] e αbtv αβ(1 b t) vdv, (17) where β = v. (18) Above equation incorporates the effect of the irreducible triplet terms on the velocity distribution. It is obvious that for η = (i.e if there are no triplet contribution), the earlier results [8] are reproduced. Figure 1 illustrates the influence of the irreducible triplets. One can see that for small N (Figure 1a), a noticeable effect in the form of a shift in the peak of the distribution is observed. However, as N increases (Figure 1c and 1d), the distinction begins to disappear. This implies that for a reasonable number of particles the irreducible triplets have negligible effect on the velocity distribution. For comparison with observations, we need the radial velocity distribution function f(v r ) which is related to the velocity distribution function f(v). Let the spherical coordinates (r, θ, φ) in configuration space correspond to the coordinates (v, θ, φ) in velocity space. The total velocity v is given as; v = v r + v θ + v φ = v r + v (19) where v r is the radial velocity and v is the tangential velocity. Integrating equation (17) over tangential velocities now gives the radial velocity distribution function for higher

5 Electronic Journal of Theoretical Physics 7, No. 4 (1) order. where f(v r )=α β(1 b)e αβ(1 bt) [ ] v [αβ(1 b)+αb(vr + v r+v )]α(v ) 1 v r + v Γ[α(vr + v )+1][1+(1 b)η] e αb t(vr+v ) dv [ ] + e αβ(1 bt) v [α(α(vr + v ))3 (αβ) α(v r+v ) 3 (1 b)η] v r + v Γ[α(vr + v )+1][1+(1 b)η] e αb t(vr+v ) dv, () β = v = v r + v. (1) Figure shows a comparison of the observed radial peculiar velocities for a representative sample of galaxies taken from Radchadhury and Saslaw 1996 with our theoretical results equation (). The solid line is the gravitational velocity distribution of equation () with the values given in the upper side. The dotted line represents the velocity distribution with out triplet term having same values. Figure shows that the curve (including triplets) have slightly higher peak, in better agreement with observations, than without triplets. 3. Density Of States (Inclusive Of Irreducible Triplet Contributions) The general probability density for a region with N galaxies to be in the range of energy between E and E + de is as under, e E T e Nμ T P (E,N,T,V,μ)dE = g(e) de, () Z G (T,V,μ) where g(e) is the density of states, Z G is the grand canonical partition function, T is the average temperature of an ensemble and μ is the chemical potential. In order to find the density of states, it is important to rescale grand canonical ensemble where gravitational interaction of particles (Galaxies) leads to specific form of equation of state. This gives for the internal energy U and pressure P equation of state [], U = 3 NT(1 b t). (3) P = NT V (1 b t). (4) Now, the Boltzmans fundamental postulate relates density of states to the entropy in the microcanonical ensemble [7] as, i.e. Ω(E) = g(e)de = e S(E,V,N). (5) ΔE

6 4 Electronic Journal of Theoretical Physics 7, No. 4 (1) It is an established fact that the energy E in the microcanonical ensemble is very nearly equal to the average energy U in the canonical ensemble. The detailed conditions under which S(E,V,N) S(U, V, N) have been investigated by Leong and Saslaw [8]. The grand canonical ensemble in which each cell contains N-particles forms a canonical ensemble. The grand canonical entropy, which for a sub-ensemble of fixed N is the same as the canonical entropy [13]. Now we can write specific heat in terms of clustering parameter [], C V = 3 ( 1 b t T b ) t. (6) T The energy equation of state gives relation between T and E and is given as [8], E = U. (7) 3Na1/3 T = T. (8) a1/3 where T is the local fluctuating temperature, and Using equation (3), equation (7) becomes, a = b n = 3 ( ) Gm 3 N. (9) r E = T (1 b t ). (3) From equation (6), we can write correlaion parameter b in terms of T using equation (9), 3 at b = 1+aT = T. (31) Substituting equation (4) in (3), and then use equation (31) we get, E = T T 1+T 3 [ ] N(1 + T 3 ) N +3a N T 3N 6. (3) N(1 + T 3 ) N + a N NT 3N 9 The second term in the brackets on the R.H.s of the equation (3) arises due to inclusion of triplet term. The number of energy states in terms of entropy can be written as, ( ) Ω(E) =e S = e N ln N+ 5 N 3N T (1 + T 3 ) N e 3N 1+T 3 1+ a 33 NT (1 + T 3 ) N [ ( )] 3a N (T 3 ) 3 N(T 3 ) 3(1 + T 3 ) exp (1 + T 3, (33) ) a N (T 3 ) 3 +(1+T 3 ) N

7 Electronic Journal of Theoretical Physics 7, No. 4 (1) where S is the entropy ( inclusive of irreducible triplets) which is a function of temperature and is given as [], ( ) F S = T N,V ( ) [ NT 3 = N ln + N ln(1 + βnt 3 )+ln 1+ a ] N(βnT 3 ) V (1 + βnt 3 ) N 3N βnt 3 1+βnT N + 3 ( ) πm N ln Λ [ ] (βnt 3 ) 3 N(βnT 3 ) 3(1 + βnt 3 ) +3a N. (34) (1 + βnt 3 ) a N (βnt 3 )+(1+βnT 3 ) N Differentiating equation (3) w.r.t E and equation (33) w.r.t T, and Setting, ( ) πma 1/3 V /3 3N/ =1. (35) We obtain the density of states as, [g(e (T ))] = dω = dω dt = 3N ( de dt de ) 1+ a 33 NT exp (1 + T 3 ) N Λ 5 ln N+ e N N 3N T [ 3a N (T 3 ) 3 (1 + T 3 ) (1 + T 3 ) N e 3N ( N(T 3 ) 3(1 + T 3 ) a N (T 3 1+T 3 3N T 1 ) 3 +(1+T 3 ) N )]. (36) Figure 3 shows the effect of irreducible triplet contribution on the density of states. For small N (Figure 3b), a slight deviation is observed which disappears as N increases (Figure 3c), implying no or negligibly small effect of the irreducible triplet contribution on the density of states. Discussion In an earlier paper [], the effect of irreducible triplet contribution on the spatial distribution function was investigated, with the objective of checking the self-consistency of the earlier results of spatial distribution function of galaxies [1] as well as improving upon the theoretical results and the agreement with observations. Since the velocity distribution function compliments the spatial distribution, it is logical to investigate the effect of such higher order terms on the velocity distribution function, as well. This is what we have attempted in this paper. While as we could find that the inclusion of higher order terms does improve the agreement with observations (Figure 3), but the effect is quite minimal. From this we infer that the earlier results [8] that neglected such contributions are reasonable and consistent. This is further substantiated by the observation that the density of states is also not effected by the inclusion of irreducible triplets. This, therefore, serves as a revalidation of our previous results [1, 8, ], as well. Further, it may imply that the contribution of still higher order terms (like quadruplets) may not substantially destroy the agreement between our theory and observations. However, the actual effect of such terms needs further investigation.

8 4 Electronic Journal of Theoretical Physics 7, No. 4 (1) References [1] Ahmad, F., Saslaw, W. C., & Bhat, N. I., ApJ, 571, 576 [] Ahmad, F., Saslaw, W. C., & Malik, M. A. 6a, ApJ, 645, 94 [3] Ahmad, F., Malik, M. A., Masood, S. 6b, Int. J. Mod. Phys. D, 15, 8, 167 [4] Bean, A. J. 1983, MNRAS, 5, 65 [5] Fang, F; & Zhou 1994, APJ, 41, 9 [6] Fisher, K. B. 1994, ApJ, 448, 494 [7] Haung, K. 1987, Statistical mechanics (New York; Wiley) [8] Leong, B; & Saslaw, W. C. 4, ApJ, 68, 636 [9] Marzke, R. O. 1995, ApJ, 11, 477 [1] Manzoor A. Malik, Shakeel Ahmad and Sajad Masood, 8 Science for Better Tommorow, 363 (proc. of the Science Congress held at University of Kashmir, Srinagar, 6). [11] Manzoor A. Malik, Farooq Ahmad, Shakeel Ahmad and Sajad Masood 9, Int. J. Mod. Phys. D, 18, 959 [1] Raychadhury, A., & Saslaw, W.C. 1996, ApJ, 461, 514 [13] Saslaw, W. C., & Hamilton, A. J. S. 1984, ApJ, 76, 13 [14] Saslaw, W. C., & Fang, F. 1996, ApJ, 46, 16 [15] Saslaw, W. C., Chitre, S. M., Itoh, M., & Inagaki, S. 199, ApJ, 365, 419 [16] Saslaw, W.C. 1987, Gravitational physics of Stellar and Galactic System (Cambridge: Cambridge Univ. Press) [17] Saslaw, W.C., The Distribution of The Galaxies. (New York: Cambridge Univ. Press)

9 Electronic Journal of Theoretical Physics 7, No. 4 (1) L and S 4 1 L and S 4.8 N=4 b=.91.8 N=6 b= f(v) f(v) (a) v (b) v.6 L and S 4.4 L and S 4.5 N=8 b= N=1 b= f(v).3 f(v) (c) v (d) v Fig. 1 Effect of the inclusion of irreducible triplet terms on the Velocity distribution function at different values of N.

10 44 Electronic Journal of Theoretical Physics 7, No. 4 (1) L and S 4 R and S L and S 4 R and S N=85 <v r >=.515 α= 4.9 β= 1.54 b= N=78 <v r >=.531 α= 17.7 β= 1.59 b= f(v r ).5 f(v r ) v r (1km/s) (a) v r (1km/s) (b) 1 L and S 4 R and S L and S 4 R and S N=85 <v r >=.578 α=5. β=1.45 b=.68.8 α= 11.5 N=85 β=.8 <v r >=.69 b= f(v r ) f(v r ) v r (1km/s) (c) v r (1km/s) (d) L and S 4 R and S 1996 α=.87 N=1 β=53.44 <vr r >=16.76 b= L and S 4 R and S 1996 α=.74 N=1 β=47.33 <v r >=15.3 b= f(v r ).8 f(v r ) v r (natural units) (e) v r (natural units) (f) Fig. Comparison of the observed velocity distribution function taken from Raychadhury and Saslaw 1996 with and without irreducible triplet contributions.

11 Electronic Journal of Theoretical Physics 7, No. 4 (1) L and S 4 1 L and S 4 N= N= g(t *, N) g(t *, N) T * (a) T * (b) 1 L and S 4 1 L and S 4 N=6 N= g(t *, N) g(t *, N) T * (c) T * (d) Fig. 3 Density of states for different values of N. Solid line is our result from equation (36), and the dashed line is taken from Leong and Saslaw 4. But for N= the two curves coincide.

Physical Form of the Clustering Parameter and Gravitational Galaxy Clustering

EJTP 5, No. 17 (2008) 73 90 Electronic Journal of Theoretical Physics Physical Form of the Clustering Parameter and Gravitational Galaxy Clustering Sajad Masood 1, Manzoor A Malik 1, Shakeel Ahmad 1 and