New Astronomy 26 (2014) 33–39

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New Astronomy

journal homepage: www.elsevier .com/locate /newast

The relativistic equation of state in accretion and wind flows

1384-1076/$ - see front matter � 2013 Elsevier B.V. All rights reserved.http://dx.doi.org/10.1016/j.newast.2013.04.010

⇑ Corresponding author. Present address: Ramakrishna Mission ResidentialCollege, Narendrapur, Kolkata 700 103, India. Fax: +91 033 24773597.

E-mail address: crabhorizon@gmail.com (S. Mondal).

Prasad Basu a, Soumen Mondal b,c,⇑a National Institute of Technology, Sikkim, Indiab Ramakrishna Mission Residential College, Narendrapur, Kolkata 700 103, Indiac Korea Astronomy and Space Science Institute, Daejeon 305-348, Republic of Korea

h i g h l i g h t s

�We derive a 4-velocity distribution function to relativistic ideal gas and find parametric form of the relativistic EOS.� The new form of our relativistic EOS well fits with the correct EOS calculated from canonical ensemble theory.� In the non relativistic and extreme ultra-relativistic limits our EOS perfectly reduces to the correct EOS.� The major advantages is that the probability distribution function in this case can be factorized.� The new EOS is very helpful in probing the complex astrophysical systems e.g. accretion flows.

a r t i c l e i n f o

Article history:Received 4 August 2012Received in revised form 9 April 2013Accepted 24 April 2013Available online 18 May 2013

Communicated by M. van der Klis

Keywords:Equations of stateHydrodynamicsRelativityGravitationThermodynamic properties

a b s t r a c t

In the present study we derive a 4-velocity distribution function for the relativistic ideal gas following theoriginal approach of Maxwell–Boltzmann (MB). Using this distribution function, the relativistic equation

of state (EOS): q� q0 ¼ ðc� 1Þ�1p, is expressed in the parametric form: q ¼ q0f ðkÞ, and p ¼ q0gðkÞ,where k is a parameter related to the kinetic energy, and hence, to the temperature of the gas. In the non-relativistic limit, this distribution function perfectly reduces to original MB distribution and the EOSreduces to q� q0 ¼ 3

2 p, whereas in the extreme ultra-relativistic limit, the EOS becomes q ¼ 3p correctly.

Using these parametric equations the adiabatic index c ¼ cp

cv

� �and the sound speed as are calculated as a

function of k. They also satisfy the inequalities: 43 6 c 6 5

3 and as 61ffiffi3p perfectly. The computed distribution

function, adiabatic index c, and the sound speed as are compared with the results obtained from thecanonical ensemble theory which nicely match with the standard results (Synge, 1957 and Chandrase-khar, 1939). The main advantage in using the EOS is that the probability distribution function can be fac-torized and therefore, may be helpful to solve complex dynamics of the astrophysical system.Interestingly, in one of the astrophysical application revels that shocks in accretion flows become unlikelyand except for the region very nearby the compact object, the EOS remains non-relativistic (Mondal andBasu, 2011). We therefore, conclude that the new form of EOS will be helpful to verify many conventionalideas in many astrophysical problems.

� 2013 Elsevier B.V. All rights reserved.

1. Introduction

The thermodynamics of matter plays an important roll in thestudy of its fluid dynamics. The solution of hydrodynamic equa-tions, both in the relativistic and non-relativistic formalism, re-quire a knowledge about the equation of state (EOS) of thesystem. An EOS is essentially a relation between the macroscopicquantities e.g. the pressure p, the total mass-energy density q,and rest mass energy density q0. One could arrive into such a rela-tion from the knowledge of the probability distribution of various

microscopic quantities (e.g molecular velocity and energy). For therelativistic gas, such a study has been done using the formalism ofstatistical mechanics in a Lorentz covariant framework (see Synge,1957 and Ref. therein; Chandrasekhar, 1939; Tolman, 1934; Taub,1948). Distribution function of co-ordinate velocity dx

dt

� �and the

EOS can be derived using this formalism. Several authors tried touse relativistic EOS to solve the relativistic hydrodynamics (RHD)equation in various astrophysical problems (Shen et al., 1998).But a direct application (Falle and Komissarov, 1996) of the exactrelativistic EOS turns out to be a less feasible method for numericalcomputing because of its complexity. Thus in the popular studiesof numerical RHD in astrophysical problems (see review Wilsonand Mathews, 2003), people prefer to use various alternative mod-els of EOS which can reproduce various features of exact EOS with

34 P. Basu, S. Mondal / New Astronomy 26 (2014) 33–39

a good approximation (Ryu et al., 2006; Mignone et al., 2005). Allthese models (Sokolov et al., 2001) are proposed in an ad hoc man-ner and are not derived consistently from the first principle.

The present status of the EOS in the RHD study motivates us toanalyze this problem in a model independent way in particular, wewish to find a simpler form of the EOS which not only matches atthe boundaries (relativistic and non-relativistic), but also can bederived from very first principle of kinetic theory. At this point, itwould be interesting to see whether one could derive a probabilitydistribution function of 4-velocity following the basic methods ofprobability theory and assumptions of isotropy as originally hadbeen done by MB in the case of non-relativistic gas. This approach,to the best of our knowledge, is yet unexplored and one couldexamine it at least for the sake of conceptual completeness. AnEOS could be derived then using this distribution function. In thisarticle, we follow the original line of arguments of MB to find a dis-tribution function of 4-velocity ui ¼ dxi

ds

� �, instead of co-ordinate

velocity dxi

dt , for a relativistic gas. We then use these Maxwell like4-velocity distribution function to derive a parametric form ofthe EOS. Using this parametric form of the EOS, we furthercompute the thermodynamical functions such as the adiabatic in-dex c, and the sound speed as and specific enthalpy. The behaviorof the distribution function, adiabatic index c, and the sound speedas are once again compared with the standard results (Synge, 1957and Chandrasekhar, 1939) obtained from the canonical ensembletheory and find that they nicely match with it.

This paper is organized as follows. In Section 2, we derive Max-well like 4-velocity distribution function for relativistic gas. In Sec-tion 3, we apply it to find the EOS for relativistic gas. In Section 4,we investigate the behavior of some physical quantities (e.g. c,sound speed, etc.) and compared with the standard results. The ex-treme limits: ultra-relativistic and non-relativistic are also dis-cussed. Finally conclusion are drawn in Section 5.

2. Four velocity distribution function using Maxwell–Boltzmann’s original approach of a relativistic ideal gas

We consider a gas consisting of one species of non-interactingparticles. The temperature of the gas is so high that the average ki-netic energy of a constituent particle is comparable to its rest massenergy. In other words, the thermal energy of the gas is compara-ble with its rest mass energy. The co-ordinate velocity and the 4-velocity of a gas particle are defined respectively as v i ¼ dxi

dt andul ¼ c dxl

ds where ds2 ¼ c2dt2 � ðd~xÞ2 is the length element in theMinkowski space–time. ul’s are related to the co-ordinate velocityas

ul ¼ cdxl

ds¼ vl

� ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� v2

c2

r; ð1Þ

where vl ¼ ð1;~vÞ. The molecules of the gas interacts only throughelastic collision so that all of their energy is kinetic energy. In a Lor-entz frame, where the center of mass of the container is at rest, thedistribution of molecular speed is isotropic. The choice of such aframe is unique up to a three dimensional rotation. In differentframes in which the container is at rest, u0 takes same value anduis are transformed by a 3-D Euclidean rotation i.e they transformlike a components of an ordinary three dimensional vector innon-relativistic case. According to special relativity

v2x þ v2

y þ v2z 6 c2; ð2Þ

where v i ¼ dxi

dt , is the co-ordinate velocity. In the non-relativisticcase (where vx; vy; vz are not constraint by the above inequality),we write the probability of finding the particle simultaneously inthe velocity range vx to vx þ dvx; vy to vy þ dvy, and, vz tovz þ dvz as the product of the individual probabilities:

pðvx;vy;vzÞdvx dvy dvz ¼ f ðvxÞf ðvyÞf ðvzÞdvx dvy dvz ð3Þ

where f ðv iÞdv i represents the probability of finding the particle inthe velocity component range v i to v i þ dv i. The functional formof ‘f’ is same for all the components due to isotropy. This assump-tions (3) is no longer valid in relativistic case as the co-ordinatevelocity component are no longer independent due the aboveinequality and can only vary from �c to c. This problem can be cir-cumvented if we seek a formula for the probability distribution of4-velocity instead of coordinate velocity. The 4-velocity compo-nents are related through the equation

X3

i¼1

ðuiÞ2 ¼ c2½ðu0Þ2 � 1� ¼ v2=ð1� v2

c2 Þ: ð4Þ

But, as v ! c; u0 !1 and therefore, each of the three spacialcomponents ðu1;u2;u3Þ of the 4-velocity can take values from�1 to 1 independently, irrespective of the values of other twocomponents. Therefore, just as the 3-velocity components in thenon-relativistic case, the three spacial components of 4-velocitieshave no constraint among themselves. In the chosen Lorentz frame,the center of mass of the gas system is at rest. The average 4-momentum of the total gas system is zero in all directions. This im-plies that the probability distribution function of the three spacialcomponents of the 4-velocity is isotropic just as the probabilitydistribution function for velocities in non-relativistic case. Let theprobability of finding a gas molecule in between ui to ui þ dui be

piðuiÞdui ¼ FiðuiÞdui; ði ¼ 1;2;3Þ: ð5Þ

The assumption of isotropy demands that the functional from of Fi

is same for all values of i, i.e. F1 ¼ F2 ¼ F3. As already explainedabove the three spatial components of 4-velocity u1; u2; u3 arenot constrained, hence the probability that a gas particle is foundsimultaneously within the 4-velocity range u1 to u1 þ du1

; u2 tou2 þ du2, and u3 to u3 þ du3 is

pðu1;u2;u3Þdu1 du2 du3 ¼ Fðu1ÞFðu2ÞFðu3Þdu1 du2 du3: ð6Þ

But, on account of isotropy, pðu1; u2;u3Þ is invariant under a threedimensional rotation and must be a function ofðu1Þ2 þ ðu2Þ2 þ ðu3Þ2 ¼ ðuÞ2. This gives

Fðu1ÞFðu2ÞFðu3Þ ¼ wððuÞ2Þ: ð7Þ

Differentiating Eq. 7 partially, w.r.t u1; u2, and u3, and dividing byFðu1ÞFðu2ÞFðu3Þ, one arrives at the equation

F 0ðu1Þu1Fðu1Þ ¼

F 0ðu2Þu2Fðu2Þ ¼

F 0ðu3Þu3Fðu3Þ ¼

w0ðuÞuw

¼ �2k ð8Þ

where k is a constant which is positive because of the convergencerequirement of F. The Eq. (8) is solved to give: FðuiÞ ¼ Ae�kðuiÞ2 andtherefore,

Fðu1ÞFðu2ÞFðu3Þ ¼ A3e�kðuÞ2 : ð9Þ

Integrating this probability over spherical shell of radius u anduþ du, one finds the total probability of finding the particle be-tween the speed range u and uþ du, as

FðuÞdu ¼ 4pA3u2e�ku2du: ð10Þ

The normalization condition:R1

0 FðuÞdu ¼ 1 relates A with k asA ¼

ffiffiffikp

q. The constant k is related to the average energy of the gas

particle as

hu0i ¼Z 1

0u0FðuÞdu ¼ 4pA3

Z 1

0u2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ u2

c2

� se�ku2

du:

As in the case of MB, the parameter k in this case, is also related tothe average kinetic energy (hEki) of the gas molecule, however not in

P. Basu, S. Mondal / New Astronomy 26 (2014) 33–39 35

a simple manner. If m is the rest mass of the gas molecule thenhEkðkÞi ¼ mc2½hu0i � 1�

¼ 4pmc2A3Z 1

0u2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ u2

c2

� se�ku2

du

" #�mc2; ð11Þ

where hu0imc2 represents average energy hEi (including rest massenergy) of the gas molecule. The above equation gives the physicalmeaning of the constant k as it relates this constant to the physi-cally measurable quantity hEki, i.e. k is a measure of the temperatureT of an ideal gas. hEki, and T are monotonically decreasing functionof k. Therefore, the non-relativistic limit is achieved in large k limit.The function FðuÞ has non-negligible value only in the region where

ku2 � 1 or, u2 � 1k. For large k, this implies u2 ¼ v2= 1� v2

c2

� �� 1, i.e.

v2

c2 � 1. This gives ui � v i, therefore, FðuÞdu ¼ 4pA3v2e�kv2dv , is the

velocity distribution function in the non-relativistic case.However, note that if we apply general results from the canon-

ical ensemble to the ideal gas we obtain a four velocity distributionfunction (CE-distribution), instead of Eq. (10), as

Fuc

� �d

uc

� �¼ h

K2ðhÞuc

� �2exp �h

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ u

c

� �2r( )

duc

� �: ð12Þ

where h equals to ðmc2

kBT Þ and K2 is the modified Bessel function ofsecond kind. Clearly, the u dependence is different and therefore,the distribution function obtained form canonical ensemble (CE)theory to the relativistic ideal gas does not match with the one ob-tained form MB-probabilistic approach. As to compare both the dis-tribution functions we first need to find a relation between theparameters k and h ¼ mc2

kBT

� �. We have already seen (in Eq. (11) that

the parameters k and T are related, and that the larger is one thesmaller is the other. In order to find a unique relation between themsome additional assumption have to be done. One possible assump-tion is to consider any physically measurable quantity and to equatethem at a given temperature in the both distributions. Here wechoose the average energy hEi (including rest mass energy) or theaverage kinetic energy (excluding rest mass energy) given in Eq.(11) as one of the physically measurable quantity. Now equating,hEMBðq0; kÞi ¼ hECEðq0; hÞi, a relation between k and h ¼ mc2

kBT

� �is ob-

tained. Employing that relation, we plot both the dimensionless dis-tribution functions Fðu=cÞwith respect to parameter u=c in Fig. 1 for

0 2 4 6 8 10 12 14u/c

0

0.5

1

1.5

f(u/

c)

Fig. 1. Three pairs of the dimensionless distribution functions have shown w.r.t u=cin the figure. One pair for non-relativistic case ðmc2 > kBTÞ, shown in the extremeleft. Another pair of curves has drawn for relativistic case ðmc2 < kBTÞ, shown in theextreme right. In the middle, the pair of curves has drawn at mc2 � kBT. The solidcurves represent standard CE-distribution and the dashed curves represent MB-distribution.

three different values of h and for the corresponding k values. In thefigure, solid curves corresponds to CE-distribution whereas dashedcurves corresponds to MB-distribution. A pair of curves has drawnfor non-relativistic case (shown in the extreme left) i.e., whenmc2 > kBT and another pair of curves has drawn for relativistic case(shown in the extreme right) i.e., when mc2 < kBT. In the middle,one more pair of curves has drawn at mc2 � kBT. We see that asthe temperature increases kBT > mc2, i.e, in the relativistic regimediscrepancy increases however in the non-relativistic limit both thecurves merges to each.

Even though there exist some discrepancies in MB-distribution,however, it is to be noted, the most striking difference is that theindividual probabilities do not become independent in CE-distribu-tion. We would again like to emphasis that in case of MB distribu-tion the distribution function get factorized because of thestatistical independence of the vx; vy; vz, i.e., because each ofvx; vy, and vz can vary from �1 to 1 irrespective of the valuesof others, the individual probabilities become independent andhence get factorized. Although, this assumption no longer holdsfor coordinate velocity components in case of a relativistic gas asthe velocity components are now constrained by the inequalityv2

x þ v2y þ v2

z < c2, the same assumption holds for four velocitycomponents as each of them can vary from �1 to 1 irrespectiveof the value of each other and hence can be regarded as statisticallyindependent variable. If we now apply the same arguments fol-lowed by Maxwell for nonrelativistic gas, we would arrive our fourvelocity distribution formula. It would be quite interesting toinvestigate both the theoretical reason and practical implicationof this discrepancy at least for the sake of conceptual complete-ness. This is one of the major motivation of the present study.

In this aspect, note that even though the distribution functionobtained through on probabilistic approach is different from thatobtained from canonical ensemble theory and the later one cannotbe factorized, as an consequence of that one would expect that theresults obtained using them will be quite different but we find (dis-cuss later) the variation of the adiabatic index c matches closely tothe exact theoretical result. Not only that, at the both extreme lim-its (ultra relativistic and non-relativistic) the EOS, the adiabatic in-dex c, and the sound speed as matches exactly with the theoreticalresults (end of Section 4). It is one of the basic requirement of theEOS while studying the accretion flow into a black hole since theouter boundary is non-relativistic while the inner boundary is ultrarelativistic in this case.

Further, it is always advantageous in any kind of theoretical/experimental study if the total probability get factorized per de-gree of freedom. Quite often we reduce higher dimensional prob-lems into lower dimension exploiting or using the method offactorizing the variables. In such cases our factorized probabilitydistribution function will be ease to use because its functional formwith respect to one coordinate is only require. Any small discrep-ancy arising due to this can be neglected while studying more com-plex dynamics of astronomical system such as accretion processes.

So far, three dimensional accretion disk has been mostly studiedanalytically considering azimuthal symmetry and using geometri-cal thin disk approximation (Chakrabarti, 1989). Three dimensional(3-D) problem immediately reduces to one or one and half dimen-sion problem which then possible to investigate analytically (Chak-rabarti, 1996a; Chakrabarti, 1996b). However, if one includes theeffects of various the physical processes involved in such systemsuch as ionization, comptonization, and pair production, same can-not be solved analytically. In such cases, if the probability distribu-tion function is known along a particular co-ordinate, it may bethen helpful to find the fraction of molecule taking part of a phys-ical processes and also may be simpler to establish relations be-tween the thermodynamic variables to find the inter-dependencies of accretion flow parameters.

0 1 2 3 4 5λ

0

3

6

9

12

15

Sp. Internal Energy( )Sp. Enthalpy( )

ε,h

hε

Fig. 2. The variation of specific internal energy (e) and specific enthalpy (h) areshown as a function of k.

36 P. Basu, S. Mondal / New Astronomy 26 (2014) 33–39

Our current interest is to include the effects of these physicalprocesses and study the accretion flow in details which will be re-ported in near future.

3. The relativistic equation of state in accretion and wind flows

In the general case of a relativistic gas, one can relate p;q;q0

and the adiabatic index c ¼ cp

cv

� �as

p ¼ ðc� 1Þðq� q0Þ: ð13Þ

It is easy to show that for a cool non-relativistic gas c becomes 53,

and for an extreme relativistic gas of photon c ¼ 43 (Weinberg,

1972). Therefore, the value of c in general depends on the relativis-tic nature of the gas, i.e. the dominance of the kinetic energy densityover the rest mass density q0. To calculate the dependence of c on q,we first expressed p, and q, in terms of q0. This can be done by con-sidering that the pressure p is the momentum flux through unit areaaveraged over all the molecules of the fluid-blob in the co-movingframe. We take a surface element of the fluid blob oriented to any

one axis (ith axis say,) the pressure p is p ¼ nhpiv ii. Here n is thenumber density of the molecule, pi is the ith component of the 4-

momentum pl ¼ mul. Using the relation v i ¼ ui

u0, and takingðu1Þ2

u0

D E¼ ðu2Þ2

u0

D E¼ ðu3Þ2

u0

D E¼ 1

3ðuÞ2u0

D Edue to the isotropy, we write

the pressure as

p ¼ 13q0 hu0i � 1

u0

�� ¼ q0gðkÞ: ð14Þ

Here and henceforth we work with c ¼ 1. The total energy density qis related to q0 as

q ¼ nmhu0ðkÞi ¼ q0hu0ðkÞi ¼ q0f ðkÞ: ð15Þ

Note that the above expressions of p and q are independent ofthe distribution function used. Once a particular distributionfunction is given then the functions f ðkÞ and gðkÞ can be deter-mined. In our present interest of the MB-distribution, we findthey are related to the modified Bessel functions of second kindKnðkÞ through the

hu0ðkÞi ¼Z 1

0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ u2

pFðuÞdu ¼ pA3e

k2

kK1

k2

� ; ð16Þ

1u0ðkÞ

�¼Z 1

0

FðuÞduffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ u2p ¼ pA3e

k2 K1

k2

� � K0

k2

� � : ð17Þ

The properties of KnðkÞ functions and its derivatives (require later)are well known in mathematics (Abramowitz and Stegun, 1972).The value of 4pA3 is given before using the normalization condition.

However, the standard results obtained using Eq. 12 in the sta-tistical ensemble theory the above expressions of p and q becomes

p ¼ 13q0 hu0i � 1

u0

�� ¼ q0

h¼ q0gCEðhÞ: ð18Þ

q ¼ q0hu0ðkÞi ¼ q03h2 �

1h

K1ðhÞK2ðhÞ

� ¼ q0fCEðhÞ; ð19Þ

and h ¼ mc2

kBT .

3.1. The flow variables: the specific heat ratio and the sound speed

Using the EOS (14, 15), one could to find the variation of adia-batic index c w.r.t k for a relativistic gas in many astrophysicalapplications. Therefore, our next job is to express the thermody-namic variables in terms of k alone. Using thermodynamic defini-tions, we obtain the specific internal energy

e ¼ q� q0

q0¼ f � 1 ð20Þ

and the specific enthalpy

h ¼ pþ qq0

¼ f þ g ð21Þ

of the gas. The variation of e, and h are plotted in Fig. 2 as a functionof k. Further, simultaneous solutions of three equations:p ¼ ðc� 1Þðq� q0Þ, q ¼ q0f ðkÞ, and p ¼ q0gðkÞ, provide the specificheat ratio

c ¼ ðf � 1Þ�1ðh� 1Þ; ð22Þ

as function of k only. To find the expression for the sound speed, one

needs to start from the definition a2s ¼

@p@q

� �s, where s be the specific

entropy. Now Tds ¼ d q�q0q0

� �� p

q20

dq0, we have, @q0@q

� �s¼ q0

pþq ¼ 1h. Now

since, p ¼ pðq;q0Þ we have,

dp ¼ @p@q

� q0

dqþ @p@q0

� q

dq0: ð23Þ

Therefore, @p@q

� �s¼ @p

@q

� �q0

þ @p@q0

� �q

@q0@q

� �s¼ a2

s . Again putting the va-

lue of dq from equation q ¼ qðq0; kÞ in the Eq. (23) we get,

dp ¼ @p@q

� q0

@q@q0

� k

dq0 þ@q@k

� q0

dk

" #þ @p

@q0

� qdq0: ð24Þ

Therefore, @p@k

� �q0¼ @p

@q

� �q0

@q@k

� �q0

and @p@q0

� �k¼ @p

@q

� �q0

@q@q0

� �kþ @p

@q0

� �q.

Using the equations p ¼ q0gðkÞ and q ¼ q0f ðkÞ, we get @p@q

� �q0

¼ g0

f 0

and @p@q0

� �q¼ g � g0

f 0 f . Putting these values in the expression of sound

speed and using Eq. 21, we finally get

a2s ¼

@p@q

� q0

1� @q0

@q

� k

� þ 1

h@p@q0

� k

¼ 1þ g0

f 0

� g

f þ g

� ; ð25Þ

in terms of k completely. Here the prime (‘0’) denotes derivativesw.r.t k.

Instead of using the EOS (14, 15), if we use the EOS (18, 19) thenthe above analysis would be equally valid and therefore, theexpressions of the adiabatic index c, the specific enthalpy h, andthe sound speed as would be same as before but the functions f

and g would have to replace with fCEmc2

kBT

� �and gCE

mc2

kBT

� �. Taking

these functions we again compute the standard results of CE-distri-bution namely, the adiabatic index, the sound speed as to compare

0 5 10 15 20 25 30 35 40 45 50

1.35

1.4

1.45

1.5

1.55

1.6

1.65

γ

θ( )mck T=−−

2

B

Fig. 3. The solid (dashed) curve shows the variations of the adiabatic index ccalculated using the theory of statistical mechanics (following MB-distribution)

w.r.t h ¼ mc2

kB T

� �. Both the adiabatic index perfectly matches at the boundaries

43 6 c 6 5

3

� �.

0 5 10 15 20 25 30 35 40 45 500

0.1

0.2

0.3

0.4

0.5

θ

as

Fig. 4. A plot of the sound speed as vs. h is shown in figure. The inequality,as 6 1=

ffiffiffi3p

satisfies when h! 0. The solid (CE-distribution) and dashed (MB-distribution) curves are almost coincide each other.

1.35 1.4 1.45 1.5 1.55 1.6 1.65

0.2

0.3

0.4

0.5

γ

as

Fig. 5. A plot of the sound speed as vs. the adiabatic index c is shown in the figure.The solid line obtained in statistical mechanical theory while the dashedcorresponds the result in MB probabilistic theory. We see that the overall variationis roughly same and both the curves exactly converges at the boundaries.

P. Basu, S. Mondal / New Astronomy 26 (2014) 33–39 37

with the MB-distribution results. To do that, we first obtained a

relation between the parameters k and h ¼ mc2

kBT

� �by equating aver-

age energy per particle i.e., hEMBðq0; kÞi ¼ hECEðq0; hÞi as discussedbefore.

To check the variation of the adiabatic index c w.r.t h which in-deed depends on k, we plot Fig. 3. In the figure, the solid curve cor-responds to CE-distribution whereas the dashed curve correspondsto MB-distribution. Both the curves perfectly match at the extremeboundaries i.e., 4

3 6 c 6 53 which implies that the extreme EOSs are

consistent with that of the theoretical results. We see that solidand dashed curves are close to each other indicating that the dis-crepancies are small. We can therefore, say that the overall varia-tions of c in both the cases are quite similar. In the figure, wenote that there is no appreciable variation of c for large range ofh values i.e., the EOS is mostly non-relativistic in this region andthere is a sharp decay of c values when h close to zero i.e., whenit reaches relativistic to ultra-relativistic value. This indicates thatstarted with a cool non-relativistic gas at the outer boundary theEOS becomes relativistic to ultra-relativistic at the inner boundaryof the black holes.

In a similar manner, two plots of the sound speeds as vs. hðkÞ arealso shown in Fig. 4 as to make a comparative study of their vari-ations of both the cases. We also give an attention in choosing thecorresponding values of k for each values of h in CE-distributionwhile computing the sound speed in MB-distribution. Here alsothe solid curve is plotted for CE-distribution and the dashed curvefor MB-distribution. We see that both curves almost coincide eachother once again indicating that the deviations are small. This im-plies that a similar variations of the EOS in both the cases. We alsosee that at the extreme limit, the sound speed as 6 1=

ffiffiffi3p

satisfiesas h or k! 0. Note that unlike c, the change of the sound speedas in the non-relativistic domain is prominent even in the relativelyhigher h=k values.

So far we have studied the behavior of the thermodynamicsfunctions in parametric way. Now we would like to re-check theirinter-dependencies in a non-parametric way (without k or h). To dothat we consider the thermodynamical variables such as c and as

and compute the variation of as with respect to c eliminating theparameters k and h. To compare the results obtained in the MBprobabilistic approach (dashed curve) and in the statisticalmechanical approach (solid curve), we plot as as a function of cin Fig. 5. We see that both the curve varies approximately in a sim-

ilar way and exactly matches at the boundaries reminding the con-sistency in the earlier results.

Accumulating all the results of above three figures, we now con-clude that the overall variations of the thermodynamics functionsand therefore, the EOS in MB approach is almost similar to that ofthe CE approach. We also think that a small deviations from thestandard values can be neglected while studying a complexdynamical system such as accretion processes.

4. Extreme limits of EOS: the ultra-relativistic and the non-relativistic

It is important to show that this expression for p gives correctresults for non-relativistic and ultra-relativistic limit. In the non-

relativistic limit v � c. Therefore, u0 � 1þ 12

v2

c2

� �and

1u0 � 1� 1

2v2

c2

� �. Therefore, p ¼ 1

3 q0hv2i. Now, the average kinetic

energy density is q0hEki ¼ q0ðhu0i � 1Þ ¼ 12 q0hv2i. Therefore,

q� q0 ¼ 12 q0hv2i. Using above results, we find

q� q0 ¼32

p; ð26Þ

0.5 1 1.5 2 2.5Log(r)

1.35

1.4

1.45

1.5

1.55

1.6

1.65

γ

0.5 1 1.5 2 2.5

Log(r)0

0.5

1

1.5

2

Mac

h N

o.

O(1)

Fig. 7. The variation of the adiabatic index c as a function of radius is plotted. The cvaries 4

3 6 c 6 53

� �w.r.t radius for different flow topologies solution is also shown in

the sub-panel. Notice that with this variable c only one saddle type (‘X’) criticalpoints present in the solutions.

38 P. Basu, S. Mondal / New Astronomy 26 (2014) 33–39

an exact EOS in the non-relativistic limit in which c ¼ 53. For an ul-

tra-relativistic gas of photon v ! c and m! 0. However,c2mu0 ¼ Ephoton remains finite. Therefore, for a photon gas u0 !1and 1

u0 ! 0. This gives q0 ¼ 0; q ¼ nEphoton and p ¼ 13 nEphoton.

Therefore, in the ultra-relativistic limit, EOS and c respectivelybecomes

p ¼ 13q; c ¼ 4

3: ð27Þ

For a gas of massive particle, the values of the c and as in the ex-treme relativistic regime can be obtain from Eq. (22) and Eq. (25) inthe limit k! 0. The integral in Eq. (16) and (17) and their deriva-tives w.r.t k, take the value for k! 0 (by putting

cu2 ¼ yÞ : hu0i ¼ 2k�1

2ffiffiffipp ; hu0i0 ¼ � k

�32ffiffiffipp ; h 1

u0i ¼ 2k12ffiffiffipp ; h 1

u0 i0 ¼ k�1

2ffiffiffipp . Substituting

above values in Eq. (22), and (25), we find

limk!0

c ¼ 43

and limk!0

a2s ¼

13: ð28Þ

Similarly, it can be shown that in the large k limit, c ¼ 53. We empha-

sis these points in Fig. 3, where y-axis are set at y ¼ 43 and 5

3. We see,43 6 c 6 5

3 perfectly holds in our case. Also, in Fig. 4, we find thesound speed as 6

1ffiffi3p satisfies.

5. The accretion and the wind flows around black holes

An immediate application of our EOS can be made in accretionphysics. The accreting matter remains cool (non-relativistic) at theouter boundary and becomes very hot (extreme relativistic) at theinner edge while the boundaries are reverse in the wind flows.Therefore, the variation of c from 5

3! 43 with temperature needs

to be taken into account while solving the hydrodynamic equa-tions. So far, in the conventional study of accretion/wind flows(Chakrabarti, 1989; Chakrabarti, 1996a; Chakrabarti, 1996b), peo-ple use constant ultra-relativistic (UR) value of c ¼ 4

3. This particu-lar choice of c, although takes into account some of the UR-effects,misses an essential feature namely, the variation of c with temper-ature. As a artifact of c ¼ 4

3, the existence of two saddle (‘X’)-typecritical point becomes probable for a wide range of parameters val-ues. Therefore, the formation of shocks in the flow which requiresthe presence of two saddle type points becomes likely. In Fig. 6, anexample of black hole accretion solutions (shown by arrow marks)where the inner boundary (r ! horizon) v ! c, and at the outerboundary (r !1) v ! 0, is presented. The two saddle (‘X’)-typecritical point inner (1), and outer (O or 3), respectively are formed

0.5 1 1.5 2 2.5 3Log(r)

0

0.5

1

1.5

2

2.5

Mac

h no

.

O(3)(1) (2).

Fig. 6. An example of accretion solutions having shocks is plotted. The solutionsgenerally have three critical points among them (1) and (3) are saddle type (‘X’)critical points when the ultra-relativistic values of the adiabatic index c ¼ 4=3 ischosen.

in the either side of middle critical point at (2) when c ¼ 43 is cho-

sen. In the traditional approach of black hole accretion, the sub-sonic matter crossing the outer sonic point becomes super sonicand may reach to horizon or flow may jump to subsonic branchthrough a shock transition (a vertical arrow line) if the flow satis-fies shock conditions in between two ‘X’-type sonic points (i.e,within (1 and 3)).

However, in our recent work (Mondal and Basu, 2011), employ-ing this EOS in which c varies from 4

3 6 c 6 53, we review the tradi-

tional hydrodynamic solutions for accretion/wind flows onceagain. In comparison to the previous solution with c ¼ 4

3, we findthat for a wide range of parameters the flow mostly has one saddle(‘X’)-type critical point (instead of two saddle points) and the twosaddle type critical point becomes a rare to exist. In the sub-panelof Fig. 7, the solution with variable c is shown. The black holeaccretion solution is indicated by the arrow marks. The corre-sponding variation of kðrÞ w.r.t the radial distance ðrÞ from outerto inner edge is also plotted in Fig. 7. Most interestingly, we seethat in both the accretion/wind flows, except for the region verynearby the compact object, the c remains mostly close to its non-relativistic value and not close to 4

3 (which was previously chosen).As a natural consequence of that the flow has only one ‘X’-typecritical point which implies the formation of shocks becomesunlikely.

6. Conclusion

In this communication we express the relativistic EOS in a newform through two parametric equations: q ¼ q0f ðkÞ, andp ¼ q0gðkÞ, where k is a parameter related to kinetic energy ofthe gas. The EOS is obtained by using a 4-velocity distribution func-tion which we derived by applying the basic principles of probabil-ity theory with the assumptions of isotropy as originally developedby Maxwell–Boltzmann (MB). In the ultra-relativistic regime, thesenew equations perfectly reproduces well-known results, namely,the EOS: q ¼ 3p, the value of c ¼ 4

3, and sound speed 6 1ffiffi3p have

an exact match, whereas, in the non-relativistic regime, EOS cor-rectly reduces to q� q0 ¼ 3

2 p, which implies c ¼ 53. With this cor-

rect choice of EOS, c varies continuously from 53 P c P 4

3 from theouter edge to inner edge.

We further compute the distribution function and thermody-namical functions such as pressure, total energy density (or relativ-istic EOS), the adiabatic index c, and the sound speed as from the

P. Basu, S. Mondal / New Astronomy 26 (2014) 33–39 39

canonical ensemble theory and compare our MB-distribution re-sults with the standard results. We find the overall variations ofthe thermodynamics functions in MB-distribution are almost closeto that of the CE approach. Therefore, if we neglect the small devi-ations from the standard values, the behavior of the EOS in MB ap-proach is quite similar to that of the CE approach.

Interestingly, as a consequence variation of c in the range53 P c P 4

3 in EOS, it revels various interesting results which breaksmany conventional ideas that people usually have in the study ofhydrodynamic flows. First of all, astrophysicists use constant ul-tra-relativistic value of c ¼ 4

3, however, we find that the EOS (andtherefore, c) evolves abruptly at the end phase of ultra-relativisticregime and except for that regime very nearby the compact object,the EOS of the matter mostly remains non-relativistic in nature.Taking account the fact, we have studied one such important astro-physical application. We review the accretion and wind solutionswith variable adiabatic index and verify the real possibilities ofshock in accretion flows. We first express the variation of c as afunction of the radius where it varies 5

3! 43

� �from the outer edge

to inner edge. As an immediate effect of that it results that the cand therefore, the EOS mostly remains non-relativistic very closeto inner edge of the accretion disk. More interestingly, it alsoclearly revels that the existence of two saddle type critical pointin the flow becomes a rare possibility indicating that this is an arti-fact of the choice of the constant ultra-relativistic c ¼ 4

3

� �. Therefore,

the formation of shocks in the flow (which requires the presence oftwo saddle type points) becomes very unlikely (Mondal and Basu,2011).

Another advantage is that the total probability distributionfunction get factorized per degree of freedom. It may be then easeto solve the dynamics of the system separately along the individualcoordinates. Exploiting this or using the method of factorizing the

variables any higher dimensional problems may get reduce intothe lower dimensions. Small deviations can be neglected whilestudying more complex dynamics of astronomical system such asaccretion processes.

Acknowledgments

Author SM acknowledges discussions with Prof. Sandip K. Chak-rabarti, of ICSP Kolkata, KASI (Korea) PDF support and thanks to IU-CAA, Pune for helping through the visiting associate program. Wewould also like to thank the anonymous referee for the suggestionsin improving this work.

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