Conformable fractional polytropic gas spheres

New Astronomy 76 (2020) 101322

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Conformable fractional polytropic gas spheres a

T

b,⁎

Emad.A.-B. Abdel-Salam , Mohamed I. Nouh a b

Department of Mathematics, Faculty of Science, New Valley University, El-Kharja 72511, Egypt Department of Astronomy, National Research Institute of Astronomy and Geophysics, Helwan, Cairo 17211, Egypt

ARTICLE INFO

ABSTRACT

Keywords: Stellar structure Polytropic gas sphere Fractional Lane-Emden equation Conformable fractional derivatives

Lane –Emden differential equation of the polytropic gas sphere could be used to construct simple models of stellar structures, star clusters and many configurations in astrophysics. This differential equation suffers from the singularity at the center and has an exact solution only for the polytropic index n = 0, 1and 5. In the present paper, we present an analytical solution to the fractional polytropic gas sphere via accelerated series expansion. The solution is performed in the frame of conformable fractional derivatives. The calculated models recover the well-known series of solutions when = 1. Physical parameters such as mass-radius relation, density ratio, pressure ratio and temperature ratio for different fractional models have been calculated and investigated. We found that the present models of the conformable fractional stars have smaller volume and mass than that of both the integer star and fractional models performed in the frame of modified Rienmann Liouville derivatives.

1. Introduction In the last decades, fractional calculus has important applications in physics and engineering, such as particle physics, wave mechanics, electrical systems, fractal wave propagations, and various real-life problems are described exactly based on the fractional differential equations, Stanislavsky (2010) and Herrmann (2014). Mathieu et al. (2003) and Pu et al. (2008) used fractional derivatives to improve the criterion of thin detection arose in signal processing. In the framework of fractional action cosmology, Debnath et al. (2012) generalized second law of thermodynamics for the Friedmann Universe enclosed by a boundary and Debnath et al. (2013) reconstructed the scalar potentials and scalar fields. Momeni and Rashid (2012) introduced a model of dark energy by choosing a power-law weight function in the fractional action cosmology and obtained relevant cosmological parameters. El-Nabulsi (2012) introduced a new theory of massive gravity and concluded that the fractional graviton masses which for a very low cosmic fluid density are different of zero. Shchigolev (2011, 2016) obtained exact solutions to fractional cosmological models with dynamical equations containing fractional derivatives or derived from the Einstein-Hilbert action. El-Nabulsi (2013) generalized the Einstein's field equations based on fractional derivatives and obtained non-local fractional Einstein's field equations. El-Nabulsi (2017) introduced a generalized derivative operator and obtained a family of Emden–Fowler differential equations. El-Nabulsi (2016) used Ornstein-Uhlenbeck-like fractional differential

⁎

equation and introduced a generalized fractional scale factor and a time-dependent Hubble parameter and described the accelerated expansion of a non-singular universe with and without the presence of scalar fields. Lane–Emden equation has a long history in modeling several phenomena in mathematical physics and astrophysics such as the theory of stellar structure, the thermal behavior of a spherical cloud of gas, isothermal gas spheres, and the theory of thermionic currents. The main difficulty of solving this Lane–Emden equation comes from the singular behavior that occurs at the center of the polytrope. Many authors introduced several approaches to solving the integer version of the LaneEmden equation, Chowdhury (2009), Ibrahim and Dares (2008), Momani and Ibrahim (2008), Nouh (2004), Hunter (2001). The fractional version of the Newtonian stellar polytrope has been investigated by El-Nabulsi (2011) for the fractional white dwarf model, Bayian and Krisch (2015) for the incompressible gas sphere, AbdelSalam and Nouh (2016) for the fractional isothermal gas sphere and Nouh and Abdel-Salam (2018) for the polytropic gas sphere. The polytropic Lane-Emden equation in its fractional form is given by (Nouh and Abdel-Salam, 2018)

1 d x2 d x

x2

d u dx

=

un

Where the dimensionless function u (Emden function) is given by

=

n cu ,

Corresponding author. E-mail address: [email protected] (M.I. Nouh).

https://doi.org/10.1016/j.newast.2019.101322 Received 14 August 2019; Received in revised form 2 October 2019; Accepted 18 October 2019 Available online 18 October 2019 1384-1076/ © 2019 Published by Elsevier B.V.

(1)

New Astronomy 76 (2020) 101322

E.A.-B. Abdel-Salam and M.I. Nouh

ρ and ρc are the density and central density respectively, and x is a dimensionless variable. In the present article, we introduce an approximate solution for the conformable fractional polytropic gas sphere. We construct a recurrence relation for the coefficient of the power series and compare the numerical results of the present algorithm with those of Nouh and Abdel-Salam (2018). The physical parameters of the polytrope are deduced and fractional models for the solar-type stars and white dwarfs are calculated to declare the effects of the fractional parameter on the structure of the stars. These calculations are motivated by the property of the conformable derivatives; where it could be applied to the continuous and discontinuous media and consequently may be physically relevant to the polytropic and isothermal gas spheres. The structure of the paper is as follows. In Section 2, principles of the conformable fractional derivatives are introduced. The series solution to the fractional polytropic gas sphere is described in Section 3. The present results are presented in Section 4 and Section 5 is devoted to the conclusion reached.

Assume the transform X = x , the Emden function takes the form

Inserting the initial conditions Eq. (10)) to Eq. (11) we get A0 = 1 and applying Eqs. (4) and ((5) to Eq. (11) we get

f (t + t 1

)

f (t )

and

Dx u (0) = A1 Then, A1 = 0 and Eq. (11) could be written as

t > 0,

(0, 1],

0

, p

,

D c = 0,

D (a f + b g ) = a D f + b D g ,

f (t ) = c,

a, b

Differentiate the Emden functionu once more, we get

D f ( t) =

t1

df , dg

Dx u (0) = 2 2A2 , Differentiate the last equation j times

D cos(ct ) =

c

D sin(ct ) = c t 1 t1

D e c t = c ec t , D cos(ct ) =

(3)

un = G (X ) =

Dx . ..Dx G (0) = j! jQj , .

(4)

Differentiate both sides of Eq. (14), we get

(5)

Dx un = Dx G, that is

n un 1Dx u = Dx G .

(7)

or

x

Dx

Dx ) u +

Differentiating both sides of Eq. (15) k times we have ...

D

cos(ct ),

= 0,

Dx u (0) = 0

k times

k times

...

[n G Dx u] = D

(u Dx G ) ,

Then we have

(8)

k

( kj ) D

n

...

j + 1 times

uD

...

k j times

G

j=0 k

( kj ) D

=

...

j + 1 times

...

G D

k j times

u,

j=0

at X = 0 , we have k

( kj ) D

n

(9)

...

j + 1 times

u (0) D

...

k j times

G (0)

j=0 k

with the initial conditions

u (0) = 1,

(16)

n unDx u = u Dx G

Lane-Emden equation in its fractional form (Eq. (1)) could be written as

un

(15)

At X = 0 we have after j times derivatives

3. Analytical Solution to conformable fractional LEE

(x 2

Qm X m , m =0

The CFD could be applied for differentiable and non-differentiable functions. So, CFD could be applied for continuous and discontinuous media.

2

u (0) = Dx . ..Dx u (0) = j! jAj ,

whereAjare constants to be determined. Now suppose that

cos(ct ),

c sin(ct ).

j times

(2)

sin(ct ),

D sin(ct ) = c

...

D

where f, g are two differentiable functions and c is an arbitrary constant. Eqs. (5) and (6) are proved by Khalil et al. (2014). The conformable fractional derivative of some functions

D e c t = c t1 e c t ,

(14)

At X = 0 we have

(6)

D ( f g ) = f D g + f D g, df D g, D f ( g) = dg

(13)

m=2

Here f(α)(0) is not defined. When = 1 this fractional derivative reduces to the ordinary derivative. The conformable fractional derivative has the following properties:

D tp = p tp

Am X m

u (X ) = 1 +

Dx u = Dx Dx u = 2 2A2 + 6 2A3 x + 12 2A3 x 2 + ...

f ( ) (0) = lim+ f ( ) (t ). t

(12)

Dx u = A1 + 2 A1 x + 3 A2 x 2 + ...

There are various definitions of fractional derivatives. Examples include Riemann–Liouville, Caputo, modified Riemann–Liouville, Kolwankar–Gangal, Cresson's and Chen's fractal derivatives, Mainardi (2010) and Herrmann (2014). Khalil et al. (2014) introduced the conformable fractional derivative (CFD) by using the limits in the form 0

(11)

m =0

2. Conformable fractional derivatives

D f (t ) = lim

Am X m ,

u (X ) =

( kj ) D

=

(10)

j=0

where u = u (x ), is the Emden function and 0 < α ≤ 1.

or 2

...

j + 1 times

G (0) D

...

k j times

u (0),

New Astronomy 76 (2020) 101322

E.A.-B. Abdel-Salam and M.I. Nouh k j=0

n

k j =0

= k j=0

n

( kj )(j + 1) !

( kj )(j + 1) !

(k j ) Q k j

j) !

(j + 1) Q j + 1 (k

( kj )(j + 1) !(k k j =0

=

(j + 1) A j + 1 (k

j) !

( kj )(j + 1) ! (k

(k + 1) A j + 1 Qk j (k + 1) Q

j) !

j + 1 Ak j ,

k k ! (j + 1) ! (k j=0 j ! (k

j) ! (k + 1) Aj + 1 Qk j j) !

n

k j=0

k! (j + 1)

(k + 1) A j + 1 Qk j

n

k j=0

k! (j + 1) Aj + 1 Qk

n

k j=0

k! (j + 1) Aj + 1 Qk

j j

k j=0

=

=

k j =0

=

k 1 k !(j j =0

j) ! (k + 1) Ak j Qj + 1 j) !

2A m (m m

1 + 2) X m + X 2 +

m=2

2A m (m m

2A k + 2 (k

(k + 1) A k j Qj + 1,

k! (j + 1)

+ 1) X m + X 2 +

m=2

m=2

m=2

Qm X m + 2

Qm X m + 2 = 0,

Qm X m + 2 = 0.

+ 2)(k + 3) X k + 2 + X 2 +

k= 0

Qk X k + 2 = 0 k=2

After some manipulations, we get the recurrence relation of the coefficients as

k! (j + 1) Ak j Qj + 1, + 1) Ak j Qj + 1 + k! (k + 1) A 0 Qk + 1,

2 (k

and

+ 2)(k + 3) Ak + 2 + Qk = 0,

k

2

The coefficients of the series expansion could be obtained from k

k 1

k! (k + 1) A0 Qk + 1 = n

k! (j + 1) Aj + 1 Qk

Ak + 2 =

k! (j + 1) Ak j Qj + 1

j

j=0

j =0

In the last equation, let i = j + 1in the first summation and i = k in the second summation, we get k+1

k !(i ) Ai Qk + 1

k ! (k + 1

i

i=1

j

Qm =

i=1

m 1

(m

1) ! (i ) Ai Qm

(m

i

i=1

By adding the zero value { (m summation, we get

(m

(m

1) ! (i) Ai Qm

1) ! (m

m i=1

m ! A0 Qm = n

1) ! (m

n = 120

i ) Ai Qm i ,

i=1

m i=1

(m

m 1 i=1

i

(m

1) ! (m

i) Ai Qm

u n (x ) = 1

i

m ) Am Q0,

1) ! (i) Ai Qm

m i=1

i

(m

1) !(m

1 m !A 0

1) !(in

m + i ) Ai Qm i ,

m

1,

(17)

where

Q0 = A0n = 1, Q1 =

A0 = 1, A1 = 0, Using u = 1 +

m=2

n A1 Q0 = 0. A0

Am X m we obtain

Am mX m 1x

Dx u =

Am mX m 1 ,

=

m=2

(18)

m =2

The second derivative of the Emden function u could be given by 2A m (m m

Dx Dx u =

1) X m 2 x

m=2

2A m (m m

=

1) X m

2,

m =2

Substituting Eqs. (18) and (19) in Eq. (9) we have

m=2

+

x2

2A m (m m

[1 +

m

(m

m =2

1) X m 2 Qm

X m]

2

[x 2 Dx u + 2 x Dx u] + un = 0,

+2 x

m=2

Am

1) !(in

m + i ) Ai Qm i ,

m

1,

(22)

i=1

4

Q0 = 1, Q1 = 0, A2 =

1 6

2

, A3 = 0, A 4

, A5 = 0.

1 n 2 x+ x 6 120

= 1is reduced to the integer version of

............

Table 1 Radius of convergence of the n = =3 fractional polytrope without series acceleration.

(19)

x 2 Dx (x 2 Dx ) u + un = 0, x Dx u + 2 x Dx u + un = 0,

(21)

The analytical solution of Eq. (9) with the initial conditions in Eq. (10) determines the polytropic structure of the configuration. This solution is represented by Eq. (13) together with Eqs. (21) and (22). First, we compare the present solution with the solution presented by Nouh and Abdel-Salam (2018), where they have simulated the fractional polytropic gas sphere via accelerated series expansion; the results indicated that the fractional polytropic sphere is smaller than the integer one. We updated our code to the conformable formulations, so the modified code contains the two series expansions. We run the code for the polytrope with n = =3 at various values ofα. The results are listed in Table 1, where column 1 represents the results from the solution by Nouh and Abdel-Salam (2018); hereafter we shall call it FP; and column 2 represents the results from the present algorithm (hereafter we shall call it CFP), we fixed the number of series terms for all calculations. The radius of convergence x1 of the power series solution without applying acceleration techniques is limited.

m

(m

2

4. Results

i) Ai Qm i ,

i=1

k

Then the series solution at LEE (Nouh, 2004) as

m ) Am Q0} to the second

1) ! (m

then the coefficients Qmcould be written as

Qm =

1 m !A 0

A0 = 1, A1 = 0,

m

m ! A0 Qm = n

Qk , + 2)(k + 3)

If we put = 1 in Eqs. (21) and (22), we get the series coefficients of the integer LEE. If we insert k = 0, 1, 2, 3in Eqs. (21) and (22) we get

i ) Ai Qk + 1 i ,

if m = k + 1, then

m !A 0 Qm = n

2 (k

and

k

k! (k + 1) A0 Qk + 1 = n

x2

2 2Am mX m + X 2 +

(20)

k k ! (j + 1) ! (k j =0 j ! (k

=

m=2

m =2

By putting m = k + 2 in the first part and m = k in the third part of Eq. (19), we get

So, we get the following equations

n

1) X m +

= 0,

(k j ) A k j,

j) !

2A m (m m

m =2

mX m 1

= 0,

or 3

α

X1

1 0.99 0.98 0.97 0.96 0.95

2.46 2.37 2.32 2.19 2.1 2

FP

X1

CFP

2.46 2.45 2.44 2.435 2.43 2.42

New Astronomy 76 (2020) 101322

E.A.-B. Abdel-Salam and M.I. Nouh

Table 2 Accelerated fractional polytropic gas spheres with n = =3. α

1 X1 FP X1 CFP (R*/R0) FP

(R*/R0)

CFP

(M*/M0)

FP

(M*/M0)

0.99

0.98

0.97

0.96

0.95

6.89 6.89 1

6.53 6.04 0.981

6.3 5.6 0.963

6.15 5.29 0.946

6.01 5.06 0.930

5.92 4.89 0.918

1

0.969

0.951

0.933

0.915

0.897

1

0.963

0.928

0.896

0.866

0.842

1

0.950

0.909

0.874

0.840

0.809

CFP

To improve the radius of convergence of the divergent series, we implemented the scheme proposed by Nouh (2004) to enable the series to reach the surface of the polytrope, the results are listed in the second row of Table 2. The situation is different than that appeared in Table 1; the CFP has a radius less than that of the FP, which consequently means that the CFP polytrope has a lower volume than FP polytrope and the integer one. The mass contained in a radiusr, radius, pressure, and temperature of the polytrope are given by (Nouh and Abd-Elsalam, 2018)

K (n + 1) 4 G

M (x ) = 4

R =

1 2

K (n + 1) 4 G

1 n 2n

c

3 2

3 n 2n

c

x2

d u dx

Fig. 1. Emden function of the conformable fractional polytrope with n = =3.

x= x1

x1 (47)

P = Pc un + 1 and

(48)

T = Tc un The central density is computed from the equation c

Fig 2. Distribution of the pressure ratio of the conformable fractional polytrope with n = =3.

x 2 M0

= 4

R 03

( ) d u dx

x= x1

(49)

We first calculate a fractional model with a polytropic index n = =3 suitable for the sun and solar-type stars. To calculate the solar physical parameters, we take the mass, radius and central temperature of the sun R 0 = 6.9598 × 108m M0 = 1.989 × 1033gm , Tc = as, and 1.570 × 107K respectively. The results are tabulated in Table 2, where row 1 represents the fractional parameter α, row 2 is the first zero X1 FP calculated by Nouh and Abdel-Salam (2018), row 3 is the first zero X1 CFP calculated by the present algorithm, row 4 is the radius of the polytrope calculated by Nouh and Abdel-Salam (2018), row 5 is the radius of the polytrope calculated by the present algorithm, row 6 is the mass calculated by Nouh and Abdel-Salam (2018) and row 7 is the mass calculated by the present algorithm. The results indicate that the CFP star has a smaller volume and mass than the FP and integer stars. Figs. 1, 2 and 3 illustrate the distribution of the Emden function, pressure distribution and mass-radius relation for the polytrope with n = =3. The effects of the fractional parameter on those distributions are remarkable, especially for the mass-radius relation. We note that the CFP spheres have a radius smaller than that of the FP spheres. The second application may be to the white dwarf stars, the interior of the compact stars is nearly isothermal and the core temperature approximately equals the temperature at the core envelope-boundary. Because of the pressure of the degenerate matter is nearly independent of the temperature, so polytropic models may be used. White dwarf models use values greater than one. To declare the effects of the fractional parameters on the structure of the white dwarfs, we constructed

Fig 3. Mass-Radius relation of the conformable fractional polytrope with n = =3.

fractional polytropic models with n = =1.5. We assume the ratio of the white dwarf radius to the solar radiusRwd / R0 = 0.00188 and the mass ratio is Mwd/ M0 = 1.41096. The central density is given by Eq. (49). Table 3 lists the results, where column 2 is the first zero of the fractional polytrope, column 3 represents the white dwarf radius ratio, column 4

4

New Astronomy 76 (2020) 101322

E.A.-B. Abdel-Salam and M.I. Nouh

Table 3 White dwarf polytropic Mwd / M0 = 1.41096 . α

X1

1 0.99 0.98 0.97 0.96 0.95

3.653 3.602 3.552 3.504 3.458 3.412

CFP

models

with

n = =1.5,

mass-radius relation for the white dwarf model.

Rwd / R 0 = 0.00188 ,

(R*/Rwd)

(M*/Mwd)

ρc(109) (g/cm3)

1 0.986 0.974 0.962 0.949 0.939

1 0.974 0.950 0.927 0.904 0.884

1.792 1.659 1.544 1.444 1.356 1.277

5. Conclusion To model physical systems via fractional derivatives one need derivative involves an integral over a spatial or time region. In static stellar models, there is no time dependence, then there are no memory effects, but the fractional derivatives of mass, pressure, temperature are given at a particular scaled radius, r/R, with R the radius of the star, they do spatially sample the entire star. In the present paper, we solved the fractional Lane-Emden equation using accelerated series expansion. The solution is performed in the frame of conformable fractional derivatives. The calculated models recover the well-known series of solutions when = 1. Physical parameters such as mass-radius relation, density ratio, pressure ratio and temperature ratio for different fractional models have been calculated and investigated. Fractional massradius relations, pressure distributions and temperature distributions for polytropic indices suitable for the sun and white dwarfs structures were calculated and investigated. We found that the present models of the conformable fractional stars have smaller volume and mass than that of both the integer models and the fractional models by Nouh and Abdel-Salam (2018), which consequently means that the fractional models are denser, more stressed and hotter. Applications of these models to different stellar configurations could be done by using models with a unique polytropic index but with different fractional parameters. Nouh and Abdel-Salam (2018) introduced a possible application of the fractional model by modeling the physical structure of the sun with n = =3 but with two different fractional parameters. Murphy and Fiedler (1985) and Murphy (1983) calculated integer models with two polytropic indices and explained how these models would be useful to interpret surface radial pulsations. Similar to two zones polytropic models, fractional polytropic models presented here are a possible explanation for some phenomena arises in stellar structure theory.

Fig 4. Mass-Radius relation of the conformable fractional polytrope with n = =1.5.

Declaration of Competing Interest

is the white dwarf mass ratio and column 5 is the central density. The effect of the fractional parameter is clear. Fig. 4 shows the fractional

None.

Appendix I. Derivation of Lane-Emden Equation via Lagrangian Calculus of Variation In this appendix, we introduce alternative methods to derive Lane-Emden equation using Lagrangian Calculus of Variation, López E et al. (2012). The Lagrangian of the fractional version of the L E equation could be written in the form

L = x2

{

1 (D x y ) 2 2

1 yn+1 n+1

}

(A1)

and the Euler-Lagrange equation for the fractional LE equation has the form

L y

Dx

L =0 Dx y

(A2)

Eq. (A1) satisfies Eq. (A2), so by taking the partial derivatives with respect to y we have

L = y

x 2 yn

(A3)

Also, by taking the partial derivatives with respect to Dx y , we have

L = x2 D xy Dx y This gives

Dx

L = Dx (x 2 D x y ) Dx y

(A4)

Substituting (A3) and (A4) into (A2), we have

Dx (x 2 D x y ) + x 2 y n = 0 5

New Astronomy 76 (2020) 101322

E.A.-B. Abdel-Salam and M.I. Nouh

or

1 [Dx (x 2 D x y )] + y n = 0, x2 Which is the fractional LE equation.

Mainardi, F., 2010. Fractional Calculus and Waves in Linear Viscoelasticity. An Introduction to Mathematical Models. Imperial College Press, London. Mathieu, B., Melchior, P., Oustaloup, A., Ceyral, Ch, 2003. Signal Process. 83, 2421. Momeni, J.M., Rashid, D., 2012. J. Phys. Conf. Ser. 354, 012008. Momani, S.M., Ibrahim, R.W., 2008. J. Math. Anal. Appl. 339, 1210. Fiedler, R., Murphy, J.O., 1985. PASA 6, 219. Murphy, J.O., 1983. Aust. J. Phys. 36, 453. Nouh, M.I., 2004. New Astron 9, 467. Nouh, M.I., Abdel-Salam, E.A.-B., 2018. EPJP 133, 149. Pu, Y., Wang, W.X., Zhou, J.L., Wand, Y.Y., Jia, H.D., 2008. Sci. China Ser. F Inf. Sci. 51 (9), 1319. Shchigolev, V.K., 2011. Commun. Theor. Phys. 56, 389. Shchigolev, V.K., 2016. Eur. Phys. J. Plus 131, 256. Stanislavsky, A.A., 2010. Astrophysical applications of fractional calculus. In: Haubold, H., Mathai, A. (Eds.), Proceedings of the Third UN/ESA/NASA Workshop on the International Heliophysical Year 2007 and Basic Space Science. Astrophysics and Space Science Proceedings. Springer, Berlin, Heidelberg.

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