The fractional white dwarf hydrodynamical nonlinear differential equation and emergence of quark stars

Applied Mathematics and Computation 218 (2011) 2837–2849

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The fractional white dwarf hydrodynamical nonlinear differential equation and emergence of quark stars EL-Nabulsi Ahmad Rami ⇑ Department of Nuclear Engineering, Cheju National University, Ara-dong 1, Jeju 690-756, South Korea Key Laboratory of Numerical Simulation of Sichuan Province, Neijiang, Sichuan 641112, China College of Mathematics and Information Science, Neijiang Normal University, Neijiang, Sichuan 641112, China

a r t i c l e

i n f o

Keywords: Riemann–Liouville fractional integral White dwarfs Fractional equation of state Fractional white dwarf non-linear equation Quark stars Chandrasekhar mass

a b s t r a c t In recent years, considerable interest has been stimulated by many applications of fractional calculus in astrophysics. Motivated by recent advances of the statistical mechanical description of degenerate matter gas and fractional statistical physics, we discussed the fractional formulation of the white dwarf stellar dynamical problem. Our approach is based on the familiar definition of the Riemann–Liouville fractional integral operator of order 0 < a < 1. After deriving the fractional equation of state in D-dimensions, we focused on the three-dimensional case and we derive the fractional Chandrasekhar or Lane–Emden non-linear differential equation (LENDE) by discussing the hydrostatic equilibrium. It was observed that the equation of states for both the non-relativistic and relativistic degenerate gas are strongly influenced by the fractional parameter a. Besides, for the ultra-relativistic case, it was observed the non-existence of a unique mass for relativistic white dwarfs and hence the Chandrasekhar mass law which states that ‘‘there exist a unique mass for relativistic white dwarfs, above which hydrostatic equilibrium cannot be maintained and the stars starts to collapse’’ is violated. This violation may be realized by hypothetical quark stars from non-perturbative QCD. Additional consequences are discussed in some details. Ó 2011 Elsevier Inc. All rights reserved.

1. Introduction The fractional calculus is an old branch of applied mathematics which deals with derivatives and integrals of fractional order. It has found many applications in different field of sciences including statistical mechanics, kinetics, astrophysics and more particularly in studies of scaling phenomena [1–26]. In fact, most of the mathematical theories applicable to the study of derivatives and integrals of noninteger order were developed prior to the turn of the 20th century, especially that numerous applications and physical manifestations of fractional calculus have been found. At the moment, various definitions of fractional derivatives and fractional integrals have been given due to their usefulness in applied mathematics notably Riemann–Liouville, Caputo, Erdelyi–Kober, Saxena, Parasher, Kalla and Saxena, Lowndes, Hadamard and so on. That is the concept of differ-integral of fractional order can be introduced in several ways. The most widely used definition of an integral of fractional order is via an integral transform, called the Riemann–Liouville operator of fractional integration of order a [1,2]:

⇑ Address: Department of Nuclear Engineering, Cheju National University, Ara-dong 1, Jeju 690-756, South Korea. E-mail address: [email protected] 0096-3003/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2011.08.028

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a

a I x f ðxÞ

¼

1 CðaÞ

Z

x

ðx  XÞa1 f ðXÞdX;

ReðaÞ > 0;

ð1:1Þ

a

n

¼

d aþn f ðxÞ; n aI dx x

n < ReðaÞ 6 0:

ð1:2Þ

In this interpretation, the fractional derivative is left inverse of the fractional integral which is a natural generalization of the Cauchy formula for the n-fold primitive of a function f. The natural question that will arise concerns the physical meaning of fractional integration. In [23,24], the fractional integration can be considered as integration is some fractional dimension space, and in [27], it is shown that the geometric interpretation of fractional integration is ‘‘shadows on the walls’’ and its physical interpretation is ‘‘shadows of the past’’ whereas in [28], it was shown that there is a relation between stable probability distributions and the fractional integral. Fractional integrals provide the language for formulating and analyzing many laws in physics and more particularly in astrophysics (stellar dynamics) where fractional kinetic equations play a crucial role [26]. The fractional generalization of the standard kinetic equation was done making use of the Riemann–Liouville fractional integral operator [26], and further fractional generalization of the fractional kinetic equation were established in terms of the Mittag–Leffler functions [29], the R-function and the Lorenzo–Hartley function [30]. More generalization were also investigated by Saxena et al. [31–34]. The fractional formulation of stars physics is a fascinating topic since it involves many areas of fractional physics simultaneously: fractional hydrodynamics [24], fractional statistical mechanics [23], fractional quantum mechanics [25] and fractional relativity [35]. In this paper, we would like to investigate a well-known stellar dynamical problem making use of the Riemann–Liouville fractional integral. More precisely, we will discuss the fractional case of the white dwarf in D = 3 dimensions. The classical description of white dwarf in any D-dimensions was explored more recently in [36]. It was observed that quantum mechanics cannot balance gravitational collapse for D P 4 in similarity with Ehrenfest arguments [37] at the atomic level for Coulomb forces in Bohr’s atomic model and for the Kepler problem. We extend here the arguments of [36] and Chandrasekhar [38,39] and we model a white dwarf star as a degenerate gas sphere in hydrostatic equilibrium and where the equation of state for the degenerate interior of white dwarf will be derived from the Riemann–Liouville fractional integral. The paper is organized as follows: in Section 2, we derive the fractional equation of state for the white dwarf. In Section 3, we derive the fractional white dwarf equation by discussing the fractional hydrostatic equilibrium. In Section 4, we discuss a simple approximate solution of the fractional white dwarf non-linear differential equation. Finally, concluding remarks and perspectives are given in Section 5. 2. The fractional equation of state In the completely degenerate limit, electrons are fermions and obey Fermi–Dirac statistics. All of momentum states up to some critical Fermi momentum value p0 are filled while the states with momentum greater than p0 are empty. Their distribution function is n(p) = 2/hD where h is the Planck constant. The fractional number of electrons per unit volume is defined by:

Z

2SD

p0

jp0  pja1 pD1 dp; h CðaÞ 0 2SD CðDÞ a ; RðD  1Þ > 0; ¼ D pD1þ 0 h CðD þ aÞ

ne ¼

ð2:1Þ

D

RðaÞ > 0;

ð2:2Þ

where SD = 2pD/2/C(D/2) is the surface of a unit sphere in D-dimensions. The absolute value is introduced for physical convenience. The fractional pressure is the mean rate of transport of momentum across unit area and is defined in our framework by:

pe ¼

2SD

Z

D

h DCðaÞ

p0

jp0  pja1 pD

0

de dp; dp

ð2:3Þ

where

eðpÞ ¼ mc2

"rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi # p2 1þ 2 21 m c

ð2:4Þ

is the Fermi energy of an electron of mass m and momentum p for a completely degenerate state neglecting the ions contributions as they are not degenerate [37]. Here c is the celerity of light. Accordingly, Eq. (2.4) takes the form:

pe ¼

2SD D

h mDCðaÞ

Z

p0 0

pDþ1 jp0  pja1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dp: 2 1 þ mp2 c2

The fractional mean kinetic energy per electron is given by

ð2:5Þ

A.R. EL-Nabulsi / Applied Mathematics and Computation 218 (2011) 2837–2849

je ¼

Z

2SD D

h CðaÞ

p0

0

jp0  pja1 pD1 eðpÞdp:

2839

ð2:6Þ

At the end, the mass density of the star in the fractional framework is given by:

q ¼ ne lH;

ð2:7Þ

where H is the mass of the proton and l the molecular weight. Introducing the notation x = p0/mc, we may write Eq. (2.5) like:

pe ¼

2SD mDþa cDþaþ1 D

h DCðaÞ

Z 0

x

tDþ1 jx  tja1 pffiffiffiffiffiffiffiffiffiffiffiffiffi dt: 1 þ t2

ð2:8Þ

We may define at the moment:

f ðxÞ ¼

Z

x

tDþ1 jx  tja1 pffiffiffiffiffiffiffiffiffiffiffiffiffi dt  xDþaþ1 1 þ t2

0

Z

1 0

j1  Xja1 X Dþ1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dX; 1 þ x2 X 2

ð2:9Þ

where X = t/x and then Eq. (2.8) is reduced to:

pe ¼

2SD mDþa cDþaþ1 D

h DCðaÞ

xDþaþ1

Z 0

1

j1  Xja1 X Dþ1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dX; 1 þ x2 X 2

ð2:10Þ

The asymptotic behavior is as the following:

X1:

Z

1

0

X1:

Z

1

0

Z 1 j1  Xja1 X Dþ1 X Dþ1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dX  pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dX; 2 2 0 1þx X 1 þ x2 X 2 Z 1 a1 Dþ1 j1  Xj X X Dþa pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dX  pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dX: 0 1 þ x2 X 2 1 þ x2 X 2

ð2:11Þ ð2:12Þ

For D = 3, we find accordingly:

X1:

Z

1

0

X1:

Z

1

0

pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 j1  Xja1 X Dþ1 x x2 þ 1ð2x2  3Þ þ 3sinh x pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dX  ; 5 2 8x 1 þ x2 X 1 4þa 6þa  2 j1  Xja1 X Dþ1 2 F 1 2 ; 2 ; 2 ; x pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dX  ; 4þa 1 þ x2 X 2

ð2:13Þ ð2:14Þ

where 2F1(a, b; c; x) is the hypergeometric function. For D = 2, we find accordingly:

X1:

Z

1

0

X1:

Z

1

0

pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 j1  Xja1 X Dþ1 x2 þ 1ðx  2Þ þ 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dX  ; 2 3x4 1 þ x2 X   1 3þa 5þa 2 j1  Xja1 X Dþ1 2 F 1 2 ; 2 ; 2 ; x pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dX  ; 3þa 1 þ x2 X 2

ð2:15Þ ð2:16Þ

And finally for D = 1, we find accordingly:

X1:

Z

1

0

X1:

Z

1

0

! pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 x2 þ 1 sinh x ;  x2 x3 1 2þa 4þa  2 j1  Xja1 X Dþ1 2 F 1 2 ; 2 ; 2 ; x pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dX  ; 2þa 1 þ x2 X 2 j1  Xja1 X Dþ1 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dX  2 1 þ x2 X 2

ð2:17Þ ð2:18Þ

We therefore obtained the general fractional formula for the pressure at all values of the relativity parameter for the fully degenerate case. Through this work, we will focus on the three-dimensional case. Accordingly, Eq. (2.10) is written making use of Eqs. (2.13) and (2.14) like:

pe ¼

pffiffiffiffiffiffiffiffi 8 Dþa Dþaþ1 x x2 þ1ð2x2 3Þþ3sinh1 x aþ4 > < 2SD mD c x : X  1; 8x5 h DCðaÞ > : 2SD mDþa cDþaþ1 hD DCðaÞ

1 4þa 6þa 2 2 F 1 2; 2 ; 2 ;x

ð

4þa

Þ

:

ð2:19Þ

X  1:

In Figs. 1–3, we plot the behavior of the function f(x) for both the classical and the ultra-relativistic case and for two different values of a = 1/2 and a = 1/4: For X  1, we may discuss two independent limiting cases: the non-relativistic degeneracy (x ? 0) and the relativistic degeneracy (x ? 1):

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Fig. 1. Plot of

pffiffiffiffiffiffiffiffi

x

x2 þ1ð2x2 3Þþ3sinh1 x aþ4 x 8x5

Fig. 2. Plot of

1 4þa 6þa 2 2 F 1 2; 2 ; 2 ;x

Fig. 3. Plot of

pe ¼

ð

Þ

4þa

1 4þa 6þa 2 2 F 1 2; 2 ; 2 ;x

ð

4þa

for a = 1/2, a = 1/4 and a = 1.

for a = 1/2 and a = 1/4.

Þ

for a = 1/2 and a = 1.

( pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 xaþ4 : x ! 0; 1 2SD mDþa cDþaþ1 x x2 þ 1ð2x2  3Þ þ 3sinh x aþ4 x / D 5 8 8h DCðaÞ x xaþ3 : x ! 1: |fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl}

ð2:20Þ

A

Since the density q / x3, then the equation of state of non-relativistic degenerate gas is aþ4

pe ðx ! 0Þ / q

3

ð2:21Þ

;

whereas for a relativistic degenerate gas, we obtain:

pe ðx ! 1Þ / q

aþ3 3

ð2:22Þ

:

Putting the equation of state for the non-relativistic fully degenerate white dwarf interior into the equation of state of hydrostatic equilibrium yields the subsequent approximate consequence:

pe 

M2 R4

 /

M

R3

aþ4 3 ;

ð2:23Þ

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2841

which has the following solution: a2

R  M 3a :

ð2:24Þ

Thus for a = 1/2, we obtain R / 1/M whereas for a = 1/4, we obtain R = M7/3. Hence, for too small valued of the fractional parameter a, i.e. R  Mb, b < 1/3. For the ultra-relativistic case, we obtain a3

R  M3ða1Þ :

ð2:25Þ

We plot in Figs. 4 and 5, the variations of R for both the classical and the ultra-relativistic cases with 0 < a < 1. Apparently for the ultra-relativistic case, the term R did not cancel out as in the standard case and hence for 0 < a < 1, they do not exist a unique mass for relativistic white dwarfs and hence we have a violation of the Chandrasekhar mass law which states that ‘‘there exist a unique mass for relativistic white dwarfs, above which hydrostatic equilibrium cannot be maintained and the stars starts to collapse’’. As a result for a = 1/2, we obtain R / M5/3 whereas for a = 1/4, we obtain R = M11/9. Hence, for too small valued of the fractional parameter a, we obtain R  M. Thus more massive stars are expected to be larger. Gravity loses and the star grows. Normally, in the standard case, massive stars with masses greater than 1.4 solar masses must get rid of most of their mass as planetary nebula, otherwise they will become neutron stars of black holes. In other words, the former stars are gravitationally bounds, and for a degenerate fermion gas, the radius usually decreases with increasing mass of the configuration. There exist somewhat a violation of this rule which are exhibited by critical stars recognized in literature by ‘‘quark stars’’ from perturbative QCD which are self-bound and exhibit very roughly an increases in radius with increasing mass [40–47]. This is the case we found in the fractional approach. We entitle this mass by the ‘‘fractional quark star’’. We plot in Fig. 6 the variation of the mass with radius for different values of the fractional parameter a. 3. Fractional hydrostatic of fully degenerate star For a spherically symmetric distribution of matter, the equations of hydrostatic equilibrium are mainly:

rP ¼

dP MGq ¼  2 ¼ qru; dr r

ð3:1Þ

dM ¼ 4pr 2 q; dr du MG ru ¼ ¼ 2 ; r dr

ð3:2Þ ð3:3Þ

from which we derive

Du ¼

  1 d du r2 ¼ 4pGq: 2 r dr dr

ð3:4Þ

Here G is the gravitational constant, q(r) is the density at r, M(r) is the mass contained with r, P(r) is the pressure at r and u is the Poisson gravitational potential. By considering hydrostatic equilibrium:

dP du ¼ q dr dr

ð3:5Þ

Fig. 4. Plot of R  M(a  2)/3a.

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Fig. 5. Plot of R  M(a  3)/3(a  1).

Fig. 6. Plot of M = R3/5, M = R9/11 and M = R.

and performing the change of variable from radius r to dimensionless Fermi momentum x:

dP du ¼ q ; dx dx

ð3:6Þ

we obtain straightforwardly making use of Eqs. (2.8), (2.9), (2.20) and q = Bx3 where B is a parameter which depends on the Hydrogen mass, the electron mass, Planck’s constant, the celerity of light and the molecular weight:

  pffiffiffiffiffiffiffiffiffiffiffiffiffiffi dP 1 df 1 x4 1 ¼ A ¼ A pffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ ða  1Þ x x2 þ 1ð2x2  3Þ þ 3sinh x xa2 ; dx 8 dx 8 x2 þ 1 du : ¼ Bx3 dx

ð3:7Þ ð3:8Þ

Accordingly, we obtain after performing the integration:

ffi 1A 1 A pffiffiffiffiffiffiffiffiffiffiffiffiffi x2 þ 1  ða  1Þ 8B 8B   

2  1 a1 aþ1 2x 2 F 1  2 ; 2 ; 2 ; x2 32 F 1  12 ; a3 ; a1 ; x2 2 2  xa3  a1 a3 h 8  i93 a 1
uðxÞ ¼ 

ð3:9Þ

where C is an integration constant. One can choose the constant so that at the surface (x = 0), the potential has the normal value MG/R and then making use of the series expansion of the following integrals at x = 0:

A.R. EL-Nabulsi / Applied Mathematics and Computation 218 (2011) 2837–2849

Z n  o pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð2x2  3Þ 1 þ x2 xa4 þ 3 log x þ 1 þ x2 xa5 dx

  xa4 1 a3 a1 1 ; ; x2 3ða  3Þsinh x  3x2 F 1  ; ¼ 2 2 2 ða  4Þða  3Þ   1 a3 a1 

2  1 a1 aþ1 2 2x 2 F 1  2 ; 2 ; 2 ; x 32 F 1  2 ; 2 ; 2 ; x2 ; þ xa3  a1 a3  3 1 9x 15x3 x3 þ x1 þ  þ 0ðx4 Þ x¼0 2ð1  aÞ 40ða þ 1Þ 112ða þ 3Þ a3

 3 1 11 7 a 3 1 þx  x þ x þ x x3 þ 0ðx4 Þ ; 2ða  1Þ 8ða þ 1Þ 16ða þ 3Þ a3  xa

 xa

2843

ð3:10Þ





x¼0

 8x 4x3  þ 0ðx4 Þ ; 5ða þ 1Þ 7ða þ 3Þ

ð3:11Þ

ð3:12Þ

we obtain straightforwardly:

C¼

MG 1 A þ : R 8B

ð3:13Þ

Then

MG 1 A ffi 1 A pffiffiffiffiffiffiffiffiffiffiffiffiffi x2 þ 1  1   ða  1Þ 8B R 8 B  

2  1 a1 aþ1 2 32 F 1  12 ; a3 ; a1 ; x2 a3 2x 2 F 1  2 ; 2 ; 2 ; x 2 2  x  a1 a3 h 8  i93 1 a 4 1 a 3 a 1
uðxÞ ¼ 

ð3:13Þ

It is noteworthy that at very large distances (x = 1), we obtain:

  1 A x2 x4 x6 MG 1 A  ða  1Þ  þ þ 0ðx7 Þ  8B 2 R 8B 8 16 "      11 6C 2  a2 C a1 1 2 pffiffiffiffi þ 0 þ xa   2 x ða  7a þ 12Þ p (       a1   ! 3 ða2  7a þ 12ÞC a3 3 2C 2  ða2  7a þ 12ÞC a3 log 4 þ 2 log x  4C a1 log 2x 2 2 2  x4   2ða  4Þ2 ða  3Þ ða  4Þ2 ða  3ÞC a3 2       7 ) 6 ða2  9a þ 18ÞC a3 þ 4C a1 1 6 2 2   x þ0 þ ; ð3:14Þ x 4ða3  13a2 þ 54a  72ÞC a3 2

uðxÞ ¼ 

and hence differs completely from its standard counterpart by the presence of fractional terms. We may now substitute the fractional potential to the Poisson equation (3.4) with q = Bx3 as:

  1 d 2 du ¼ 4pGBx3 ; r r2 dr dr

ð3:15Þ

or more explicitly making use of Eq. (3.13) like:

  pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1A 1 d d 2 d 2 þ 1 þ ða  1Þr 2 x r ð IðxÞ þ JðxÞ Þ ¼ 4pGBx3 ; 8 B r 2 dr dr dr

ð3:16Þ

where

IðxÞ , xa3

  

2  1 a1 aþ1 2x 2 F 1  2 ; 2 ; 2 ; x2 32 F 1  12 ; a3 ; a1 ; x2 2 2  ; a1 a3

ð3:17Þ

and

JðxÞ ,

h  i 1 xa4 3ða  3Þsinh x  3x2 F 1  12 ; a3 ; a1 ; x2 2 2 ða  4Þða  3Þ

:

ð3:18Þ

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By performing the change of variable:

z¼

pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ x2 ¼ zc w;

ð3:19Þ

r ¼ r 1 n;

ð3:20Þ

we obtain:

 3=2   1 d 32pGB2 r 21 z3c 1 2 dw 2 d 2 z þ z ð IðwÞ þ JðwÞ Þ ¼  n ð a  1Þn  ; w c c dn dn z2c A n2 dn

where

  8   ða3Þ=2 <2z2 w2  1 F  1 ; a1 ; aþ1 ; z2 w2  1 2 2 1 2 c c 2 2 2 1 z z c c IðwÞ , zca4 w2  2 : zc a1   9 32 F 1  12 ; a3 ; a1 ; z2c w2  z12 = 2 2 c ;  ; a3

ð3:21Þ

ð3:22Þ

and

 ða4Þ=2 ( sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   zca5 w2  z12 1 1 c JðwÞ , z2c w2  2 3ða  3Þsinh zc ða  4Þða  3Þ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi     ) 1 1 a3 a1 1 : ; ; z2c w2  2  3 z2c w2  2 2 F 1  ; zc 2 2 2 zc

ð3:23Þ

Eq. (3.21) may be written in the following form:

!  3=2 2 2 dw d ðIðwÞ þ JðwÞÞ 2 d 32pGB2 r 21 z2c 1 2 þ ð ð IðwÞ þ JðwÞ Þ þ a  1Þ þ ¼   : w n dn z2c A dn2 n dn dn2 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} 2

d w

ð3:24Þ

fractional contribution

Eq. (3.24) is the ‘‘fractional white-dwarf non-linear equation’’. Notably the parameter zc is equivalent to the central energy density of the star. Now we define the new change of variable:

  1 /2 ¼ z2c w2  2 ; zc

ð3:25Þ

and then Eqs. (3.22)–(3.24) are reduced respectively to:

(    ) 2/2 2 F 1  12 ; a1 ; aþ1 ; /2 32 F 1  12 ; a3 ; a1 ; /2 2 2 2 2 ;  a1 a3

  a4 z1 1 a3 a1 1 c / Jð/Þ , ; ; ; /2 3ða  3Þsinh /  3/2 F 1  ; 2 2 2 ða  4Þða  3Þ

a3 Ið/Þ , z1 c /

ð3:26Þ ð3:27Þ

and

08 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffi 9 = /2 þ 1 d 1 d B< / d/ 32pGB2 r 21 3 C 1 þ ða  1Þ ðIð/Þ þ Jð/ÞÞ n2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffi / : A¼ 2 dn @: ; d/ dn A / n /2 þ 1

ð3:28Þ

Here we have normally:

 qffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffi   d ðIð/Þ þ Jð/ÞÞ ¼ 3/a5 log / þ /2 þ 1 þ 2/2  3 1 þ /2 /a4 ; d/

ð3:29Þ

and then Eq. (3.28) takes the special form:

9 08 1 >   > qffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffi = d/ 1 d B< / 32pGB2 r21 3 C qffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ ða  1Þ 3/a5 log / þ /2 þ 1 þ ð2/2  3Þ /2 þ 1/a4 n2 A ¼  / : 2 dn @> > A n : /2 þ 1 ; dn

ð3:30Þ

To recast the problem in a dimensionless form, we set

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A 1 r1 ¼ ; 32pGB2 zc

ð3:31Þ

A.R. EL-Nabulsi / Applied Mathematics and Computation 218 (2011) 2837–2849

2845

and then Eq. (3.30) is simplified to:

9 08 1 > > < = d/ 1 d B / 1 3 C 2 @ qffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ F a ð/Þ n A¼ 2/ ; > zc n2 dn > : /2 þ 1 ; dn

ð3:32Þ

or

0 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 0 !  2 2 2 2 / þ / þ 1 dF ð/Þ d/ / d / 2 d/ 1 a B C @ A ¼  2 /3 ; þ þ @qffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ F a ð/ÞA þ 2 d/ dn n dn zc 2 /2 þ 1 dn / þ1

ð3:33Þ

where

   qffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffi F a ð/Þ ¼ ða  1Þ 3/a5 log / þ /2 þ 1 þ ð2/2  3Þ /2 þ 1/a4 ;

ð3:34Þ

and

dF a ð/Þ ¼ða  1Þ d/ 3  qffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffi a1 a3 a5   2/  3/ þ 3/ 7 qffiffiffiffiffiffiffiffiffiffiffiffiffiffi  3ða  5Þ/a6 log / þ /2 þ 1 þ þ 2ða  2Þ/a3  3ða  4Þ/a5 /2 þ 15: 2 / þ1

ð3:36Þ

Eq. (3.33) needs to be solved numerically. The ‘‘fractional white-dwarf non-linear equation or the fractional Lane–Emden equation’’ is too complicated; nevertheless one can perform some simplifications for illustration purpose. This will be done in the next section. 4. Simple approximative solution of the fractional white-dwarf equation The main purpose of the previous section was to propose a fractional generalization of the white-dwarf equation. The difficulty of solving Eq. (3.24) or (3.33) is obvious. Recently, many analytical methods have been used to solve the standard Lane–Emden non-linear differential equations; the most important complicatedness arises in the singularity of the equation about the origin. At present, most analytical techniques are based on either series solutions, Lagrangian formulation [48] or perturbation techniques [49]. However, in the neighborhood of the origin and more particularly for zc  1, i.e. /  1, we have: Fa(/)  3(3  a)xa  4 and hence Eq. (3.33) is approximated by: 2

d / dn2

þ

 2 2 d/ a  4 d/ þ  0; n dn / dn

ð4:1Þ

and the solution is given by:

/ðnÞ ¼

 1=ða3Þ c1 n þ 3  a þ c2 ; n

ð4:2Þ

where c1 and c2 are integration constants. The boundary conditions are the following: /(0) = 1 and /0 (0) = 0. For real stars the solutions must be finite at the center n = 0, i.e. d//dn must vanish at the center. In other words, the surface of the star is defined by the value of n for which the density is zero and thus / = 0. Making use if the boundary conditions and Eq. (3.25), we obtain from the first constraint c2 = 1 whereas the second constraint is verified 8c1 2 R. For mathematical simplicity, we set c1 = 1 and hence Eq. (4.2) takes the form:

 1=ða3Þ 3a /ðnÞ ¼ 1 þ 1 þ : n

ð4:3Þ

Therefore, from Eq. (3.25) we find:

ffi vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u(  1=ða3Þ )2 1u 3  a wðnÞ ¼ t 1 þ 1 þ þ 1: zc n

ð4:4Þ

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A.R. EL-Nabulsi / Applied Mathematics and Computation 218 (2011) 2837–2849

The fractional mass of star is then:

 a43a

dw / 3  a dn

M¼ 4pqr 2 dr ¼ 4pBr 31 n2R

¼ 4pBr 31 z1

; c qffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ

dr dr n 2 R 0 r¼R

/ þ1 r¼R   1=ð a 3Þ 4a  1 þ 1 þ ð3  aÞ rR1 r 1 a3 2 1 1 þ ð3  aÞ ; ¼ 4pBr 1 zc rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n R  1=ða3Þ o2 þ1 1 þ 1 þ ð3  aÞ rR1 n o sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ða4Þ=ða3Þ 4 ð1 þ XÞ 1 þ ð1 þ XÞ1=ða3Þ 32 p GB ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi rn ;  4pr31 o2 A 1=ða3Þ 1 þ ð1 þ XÞ þ1 Z

1

ð4:5Þ

ð4:6Þ

ð4:7Þ

where

X ¼ ð3  aÞ

r1 : R

ð4:8Þ

For r1  R, the series expansion of Eq. (4.7) about X = 0 gives:

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   32pGB4 41  10a r 41 100a2  765a þ 1454 r 51 M  4p B þ    : 2r 31 þ þ 50 5A 5 R R2

ð4:9Þ

We plot in Figs. 7 and 8 the variations of the mass M for a = 1/2, a = 3/4 and for any 0 < a < 1(x  R): If in contrast r1  R (quarks stars), then Eq. (4.7) is approximated by:

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 8 p GB 8pGB4  r 1 ða4Þ=ða3Þ M  4pr31 : X ða4Þ=ða3Þ ¼ 4pr 31 ð3  aÞ A A R

ð4:10Þ

We plot in Figs. 9 and 10 the variations of the mass M for a = 1/2, a = 3/4 and for any 0 < a < 1: We stress that the illustration done here is nothing than an approximation and more details require numerical analysis and work in this direction is in progress. However, we have seen here an example of application of fractional polytropic solutions to stars with fractional barotropic equations of state neglecting the temperature contribution [50,51]. In normal stars, on the other hand, pressure is a function of density and temperature, so polytropic relations are not applicable in general. Though, there are a number of particular situations which enforce a relation between density and temperature that allows one to write pressure as a function of density alone, and in some of these situations polytropic equations give a good description of the star or parts of the star. We will come across such situations when we discuss in a future work the fractional description of normal stars in some details. 5. Conclusions and perspectives To the best of our knowledge, this work represents the first attempt to discuss the physics of stellar objects making use of the Riemann–Liouville fractional integral operator tool. The theory presented here exhibits many appealing consequences. We have formulated fractionally the statistical mechanics of degenerate gas for both the non-relativistic and ultra-relativistic case in D -dimensions. After that, we focused on D = 3 and we derived the fractional equation of state for white dwarfs. It was observed that for the ultra-relativistic case, a violation of the well-known Chandrasekhar mass law which states that ‘‘there exist a unique mass for relativistic white dwarfs, above which hydrostatic equilibrium cannot be maintained and the stars starts to collapse’’ [52–54]. This violation is not new as it founds its origin in non-perturbative formulation of Quantum

qffiffiffiffiffiffiffiffiffiffiffiffi 4 a Fig. 7. Plot of M  4pB 32p5AGB 2r 31 þ 4110 5

r 41 R

2 765aþ1454

þ 100a

50

r 51

R2

qffiffiffiffiffiffiffiffiffiffiffiffi 4 þ    for a = 1/2 and a = 3/4 with r1 = 1 and 4pB 32p5AGB ¼ 1 for illustration purpose.

A.R. EL-Nabulsi / Applied Mathematics and Computation 218 (2011) 2837–2849

qffiffiffiffiffiffiffiffiffiffiffiffi 4 a Fig. 8. Plot of M  4pB 32p5AGB 2r 31 þ 4110 5

Fig. 9. Plot of M  4pr 31

qffiffiffiffiffiffiffiffiffiffi 8pGB4 A

Fig. 10. Plot of M  4pr 31

r 41 R

ð3  aÞ rR1

qffiffiffiffiffiffiffiffiffiffi 8pGB4 A

2 765aþ1454

þ 100a

50

ða4Þ=ða3Þ

ð3  aÞ rR1

r 51

R2

2847

qffiffiffiffiffiffiffiffiffiffiffiffi 4 þ    for 0 < a < 1 with r1 = 1 and 4pB 32p5AGB ¼ 1 for illustration purpose.

qffiffiffiffiffiffiffiffiffiffiffiffi 4 for a = 1/2 and a = 3/4 with r1 = 1 and 4pB 32p5AGB ¼ 1 for illustration purpose.

ða4Þ=ða3Þ

qffiffiffiffiffiffiffiffiffiffiffiffi 4 for 0 < a < 1 with r1 = 1 and 4pB 32p5AGB ¼ 1 for illustration purpose.

Chromodynamics (QCD) and more precisely a violation caused by quark stars [55–59]. Besides, we discussed the fractional hydrodynamic equilibrium scenario and we derived resulted fractional Chandrasekhar or Lane–Emden non-linear differential equation. The later being too intricate to solve analytically, we discussed a simple approximate solution of the fractional white dwarf non-linear differential equation. In addition, it was observed that the equation of states for both the non-relativistic and relativistic degenerate gas are strongly influenced by the fractional parameter a. Follow-up studies may be devoted to understanding the relation between the fractional approaches explored here and nonperturbative studies of QCD. A comparison with observations and astrophysical applications would be a reasonable plan for future validations of the theory. Further generalization making use of different forms of fractional integral operators

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