- Email: [email protected]
Computational method for a fractional model of the helium burning network
Accepted Manuscript
Computational Method for a Fractional Model of the Helium Burning Network Mohamed I. Nouh PII: DOI: Reference:
S1384-1076(18)30148-9 10.1016/j.newast.2018.07.006 NEASPA 1212
To appear in:
New Astronomy
Received date: Revised date: Accepted date:
21 May 2018 4 July 2018 18 July 2018
Please cite this article as: Mohamed I. Nouh , Computational Method for a Fractional Model of the Helium Burning Network, New Astronomy (2018), doi: 10.1016/j.newast.2018.07.006
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
ACCEPTED MANUSCRIPT
Highlights We considered the fractional model of the helium burning phase. The system of differential equations is solved via series expansion. Comparison with the numerical solution is performed. The effects of the fractional parameter on the product abundances is investigated.
AC
CE
PT
ED
M
AN US
CR IP T
1
ACCEPTED MANUSCRIPT
Computational Method for a Fractional Model of the Helium Burning Network Mohamed I. Nouh1,2 1
Department of Physics, College of Science, Northern Border University, Arar, Saudi Arabia. Email: [email protected], [email protected]
Department of Astronomy, National research Institute of Astronomy and Geophysics, 11421 Helwan, Cairo, Egypt
CR IP T
2
Abstract: Stellar cores may be considered as a nuclear reactor that play important role in injecting new synthesized elements in the interstellar medium. Helium burning is an important stage that contribute to the synthesis of key elements such as carbon, through the triple- α
AN US
process, and oxygen. In the present paper, we introduce a computational method for the fractional model of the nuclear helium burning in stellar cores. The system of fractional differential equations is solved simultaneously using series expansion method. The calculations are performed in the sense of modified Riemann-Liouville fractional derivative. Analytic expressions are obtained for the abundance of each element as a function of time. Comparing the
M
abundances calculated at the fractional parameter 1 , which represents the integer solution, with the numerical solution revealed a good agreement with maximum error 0.003 . The
ED
product abundances are calculated at 0.25, 0.5, 0.75 to declare the effects of changing the fractional parameters on the calculations.
PT
Keywords: System of fractional differential equations; Methods: Series expansion; Stellar
CE
burning stages: Helium burning; Nuclear reaction rate.
1. Introduction
AC
Nowadays, generalizations of some of the basic differential equations in physics led to new and more deep insight to the macroscopic phenomena in a wide range of areas like anomalous diffusion, signal processing and quantum mechanics (Podlubny 1999; Sokolov et al. 2002; Kilbas et al. 2006; Laskin 2000). There are many authors present applications of fractional calculus in astrophysics of them are: Stanislavsky(2007), El-Nabulsi (2011, 2012, 2013, 2015, 2016, 2017a, 2017b); Golmankhaneh et al. (2015), Bayian and Krisch (2015), Abdel-Salam and
2
ACCEPTED MANUSCRIPT
Nouh (2016), Nouh and Abdel-Salam (2017, 2018), Calcagni et al. (2016) and Calcagni et al. (2017).
In most stars, energies are derived from the conversion of hydrogen into helium. As a result, the temperature goes high (about 100 million degrees) and the hydrogen fusion starts, the stellar core contracts gravitationally until the temperature is high enough to start helium burning. This
CR IP T
set of reactions is called the triple alpha process. Helium brining occurs come into two categories, the first is the hydrostatic burning and the second is the explosive burning.
The nucleosynthesis of the elements in stars is described by kinetic equations governing the change of the number density Ni of species i over time, that is (Kourganoff, 1973)
v
j
where
v
mn
ij
N k Nl k ,l i
v
kl
AN US
dNi / dt Ni N j
represents the reaction cross section for an interaction involving species m and
n, and the summation is taken over all reactions which either produce or destroy the species i (Haubold and Mathai, 1998). For a gas of mass density , the number density Ni of the species i
M
is expressed in terms of its abundance X i , by the relation Ni N A X i / Ai , where N A is
ED
Avogadro’s number and Ai is the mass of i in mass units. Haubold & Mathai (2000) introduced a solution of the fractional generalized kinetic equation in terms of H-functions. Their equation is suitable for incorporating changes in the
PT
Maxwell-Boltzmann distribution function. The derived solution of a fractional kinetic equation contains the particle reaction rate (or thermonuclear function) as a time constant, and provided
CE
the analytic technique to further investigate possible modifications of the reaction rate through a kinetic equation. They concluded that the Riemann-Liouville operator in the fractional kinetic
AC
equation introduces a convolution integral with slowly-decaying power-law kernel, which is typical for memory effects referred to in Kaniadakis et al. (1997) and Coraddu et al. (1998). Saxena, Mathai, and Haubold (2002) derived solutions for the generalized fractional
kinetic equations and concluded that the ordinary and generalized Mittag-Leffler functions interpolate between a purely exponential law and power-like behavior of phenomena governed by the integer kinetic equations and their fractional counterparts, respectively (Lang, 1999; Hilfer, 2000). 3
ACCEPTED MANUSCRIPT
Solutions to some fractional kinetic equations in a series form of the Lorenzo-Hartley function have been introduced by Chaurasia and Pandey (2010). Lorenzo-Hartley function could be considered as a compact and elegant expression used for the computation and has a close relationship with the R-function, the generalized Mittag-Leffler functions, the Mittag-Leffler
CR IP T
function and the Robotnov & Hartley function.
One of the most important problems in astrophysics is the nucleosynthesis in the stellar cores. Hydrogen burning, helium burning, CNO cycle are the main stages that the stars produce in their nuclear reactors, Clayton (1983). The basic equations of the burning networks are differential equations that may be solved simultaneously by numerical or analytical methods,
AN US
Duroah and Kushwaha (1963), Clayton (1983), Hix and Thielemann (1999) and Nouh et al. (2003). This system of equations may be called the integer version of the burning models, and the general form could be called the fractional models.
In the present paper we are going to introduce a computational method to fractional model the helium burning network in stellar core. The system of the fractional differential
M
equations will be solved by power series expansion. We derive general recurrence relation for the series coefficients in terms of the fractional parameter and what is called the fractal index.
ED
The calculations will be performed in the sense of modified Riemann-Liouville fractional Derivative. Comparison with the solution of the integer version is presented to declare the effects of the fractional parameter on the calculations as well as the synthesis process. The paper is
PT
organized as follows. In section 2 we introduce the helium burning model. Section 3 is devoted to the series solution of the system of the differential equations. In section 4 we introduce series
CE
solution to the helium burning network.
AC
2. Helium Burning Network
In stellar helium core, a triple alpha collision occurs at temperature of the order of 100 million degrees and the product is C12. Then C12 captures He4 to form O16 which in turn captures He4 to form Ne20 and so on, Nouh et al. (2003). These reactions could be listed as
4
ACCEPTED MANUSCRIPT
3He4 C12 7.281Mev C12 He4 O16 7.150Mev O16 He4 Ne20 4.750Mev
By considering the above reactions, Clayton (1983) set up the following model for the helium
CR IP T
burning phase. If x, y, z and r are the helium, carbon, oxygen and neon number of atoms per unit mass of stellar material. The next four equations govern the time dependent change of the abundance as
dx / dt 3ax3 bxy cxz, dz / dt bxy cxz , dr / dt cxz. where a, b, and c are the reaction rates.
AN US
dy / dt ax3 bxy,
(1)
The factor 3 appears in the equation because three helium nuclei join to form one carbon
M
nucleus.
The system of differential equations (Equation (1)) could be solved by many analytical or
ED
numerical methods. The accuracy of the methods of numerical integrations depend mainly on the interval step used through the solution (time step in our case). Bad choice of this time step may lead to bad accuracy as well as numerical instability. This difficulty could be made quantitative if
PT
we use analytical methods. In the present article we will implement power series expansion to
CE
solve the fractional helium burning network.
AC
3. Series Solution of the System of the Fractional Differential Equations
Principles of the fractional calculus implemented in the present paper could be found in Nouh and Abdel-Salam (2018).
4.1. Series Expressions for the Unknowns The system of differential equations (Equation (1)) could be written in the fractional form as
5
ACCEPTED MANUSCRIPT
Dt x 3a x3 b x y c x z , Dt y a x 3 b x y, Dt z b x y c x z ,
(2)
Dt r c x z ,
Assuming the transform T t , the solution can be expressed in a series form as
CR IP T
x X mT m X 0 X 1T X 2T 2 X 3T 3 ... m 0
X 0 X 1t X 2t 2 X 3t 3 ...,
y YmT m Y0 Y1T Y2T 2 Y3T 3 ... m 0
Y0 Y1t Y2t 2 Y3t 3 ...,
(3)
z Z mT Z 0 Z1T Z 2T Z 3T ... m
2
3
m 0
AN US
Z 0 Z1t Z 2t 2 Z 3t 3 ...,
r RmT m R0 R1T R2T 2 R3T 3 ... m 0
M
R0 R1t R2t 2 R3t 3 ... .
ED
where X m , Ym , Zm , Rm are constants to be determined.
4.2. Fractional derivative of the function raised to powers
CE
PT
The left hand side of the system in Equation (2) represents the abundances of the elements where the helium abundance (x) is raised to power 3. To obtain the fractional derivative of u n we apply the fractional derivative of the product two functions. Taking the fractional derivative of both sides of Equation (2), we have
AC
n u n Dx u u Dx G
(4)
Differentiating both sides of Equation (4) k times -derivatives we obtain k
n j 0
D k j
k
x ...Dx u Dx ...Dx G
j 1times
k j times
j 0
D k j
x
...Dx G Dx ...Dx u .
j 1times
k j times
At x 0 , we have
6
ACCEPTED MANUSCRIPT
k
n j 0
k j
k
Dx ...Dx u (0) Dx ...Dx G (0) j 1times
j 0
k j times
D k j
x
...Dx G (0) Dx ...Dx u (0)
j 1times
(5)
k j times
where Dx ...Dx u (0) X j 1(( j 1) 1), j 1 times
CR IP T
Dx ...Dx G (0) Qk j ( (k j ) 1), k j times
Dx ...Dx G (0) Q j 1(( j 1) 1), j 1times
Dx ...Dx u (0) X k j ( (k j ).
4.2.
AN US
k j times
(6)
Recurrence Relations
Substituting Equation (5) in Equation (6) we have k
n
k
X j 1(( j 1) 1)Qk j ( (k j ) 1) j 0
Q k j
(( j 1) 1) X k j ( ( k j ) 1),
j 1
M
j 0
k j
after some algebraic manipulation we obtain k
k !( (k j ) 1)(( j 1) 1) X j 1Qk j j !(k j )!
ED
((k 1) 1) X 0Qk 1 n j 0
k !(( j 1) 1) ( (k j ) 1) X k j Q j 1 j !(k j )! j 0
PT
k 1
CE
Let i j 1 in the first sum and i k - j in the second sum, then we get
k 1
AC
((k 1) 1) X 0Qk 1 n i 1
k !( (k 1 i ) 1)(i 1) X i Qk 1i (i 1)!(k 1 i )! k !((k 1 i ) 1) (i 1) X i Qk 1i (k i )!i ! i 1 k
,
put m k 1 and adding the zero value (m m) (m 1) X mQ0 / m to the second sum of the last equation, we get 7
ACCEPTED MANUSCRIPT
m
(m 1) X 0Qm n i 1
(m 1)!( (m i) 1)(i 1) X i Qm i (i 1)!(m i )! (m 1)!((m i) 1) (i 1) X i Qm i (m 1 i )!i ! i 1 m
1 (m 1) X 0
m
i 1
(m 1)!( (m i) 1)(i 1) in m i X iQmi , m 1 i !(m i )!
at m 0 , Q0 X 0n , Q1
( 1)n X1Q0 . ( 1) A0
Putting n 3 in the last equation gives
1 (m 1) X 0
and Q0 X 03 , Q1
m
i 1
, (m 1)!( (m i) 1)(i 1) 4i m X iQmi , m 1 i !(m i )!
3 X 1Q0 , .... X0
(7)
M
Qm
.
AN US
Qm
CR IP T
then the coefficients Qm could be determined by
,
ED
Now by differentiating Equation (3) -times and after some manipulations we have (n 1) X nT n 1 , n 1 ( n 1 )
PT
Dt x
(n 1) YnT n 1 , n 1 ( n 1 )
Dt y
(8)
(n 1) Dt z Z nT n 1 , ( n 1 ) n 1
CE
(n 1) RnT n 1 , n 1 ( n 1 )
AC
Dt r
substituting Equations (3) and (8) in Equation (2) for x, y, z and r respectively, we could determine the coefficients of the power series X n1, Yn1, Zn1 and Rn1 from
X n1
n n (n 1) 3 aQ b X Y c X j Z n j , n j n j ((n 1) 1) j 0 j 0
8
ACCEPTED MANUSCRIPT
Yn1 Z n1
n (n 1) aQ b n X jYn j , ((n 1) 1) j 0 n (n 1) n b X jYn j c X j Z n j , ((n 1) 1) j 0 j 0
and c (n 1) n X j Z n j . ((n 1) 1) j 0
CR IP T
Rn 1
with the initial values X 0 ;Y0 ; Z0 ; R0 . 4. Results
First, we run the MATHEMATICA code to calculate the abundance of each element at 1 .
AN US
We use six series terms and the time 200. The calculations are performed for the pure helium gas with X 0 1;Y0 0; Z0 0; R0 0 , the temperature and density are
109 K and 104 gcm-3
respectively. Comparison with the numerical solution gives good agreement with maximum relative errors are 0.003, 0.002, 0.00012 and 0.00025 for the x, y,z and r respectively. To check
M
the accuracy of the computations, we investigate the conservation formula x y z r 1 . If the sum is not 1, so there is a trouble through the computations, Table 1 lists the absolute errors
ED
( ) at different times. As we see from the table, the absolute error of the sum increases with increasing time and the maximum error is 0.0004 . To investigate the effects of changing the fraction parameter on the abundance of the elements,
we
have
performed
PT
four
the
calculations
at
the
fractional
parameters
0.25, 0.5, 0.75 and 1 respectively. Figures 1 to 3 illustrate the relation between helium
CE
concentration and the product abundances of C12, O16 and Ne20 respectively. When the helium concentration is near 0.7 the difference between the product abundance of C12 computed at
AC
different values of is very small. This differences become larger when the helium abundance goes to zero.
The behavior of the product of the O16 is different than that of C12, the product abundance
computed at 1 is smaller than of smaller than one. The situation is reversed as the helium concentration goes from 0.3 to zero, the product of Ne20 have the same behavior when the helium concentration less than 0.5, after that the difference become remarkable. To build an interior model one need to estimate the energy generation rates and the central temperature of the star. 9
ACCEPTED MANUSCRIPT
So, possible application of the fractional burning network is to model the stellar interior with two or more layers having different fractional parameters, i.e. the fractional parameter could be used to adjust the product abundance of the syntheses elements.
Table 1: The absolute errors of the sum of product abundances.
CR IP T
Time (s)
0.75
0.5
0.25
0
1.2 107
1.0 107
2.0 107
2.0 107
500
4.5 105
5.6 105
4.7 105
4.9 105
1000
9.4 105
1.3 104
1.7 104
1.9 104
1500
1.4 104
1.9 104
2.5 104
2.9 104
2000
1.9 104
2.6 104
3.3 104
3.8 104
AC
CE
PT
ED
M
AN US
1
10
ED
M
AN US
CR IP T
ACCEPTED MANUSCRIPT
Figure 1: The helium concentration as a function of the product abundance of C12. The
AC
CE
PT
different line shapes indicate different values of the fractional parameter .
11
CE
PT
ED
M
AN US
CR IP T
ACCEPTED MANUSCRIPT
Figure 2: The helium concentration as a function of the product abundance of O16. The
AC
different line shapes indicate different values of the fractional parameter .
12
CE
PT
ED
M
AN US
CR IP T
ACCEPTED MANUSCRIPT
Figure 3: The helium concentration as a function of the product abundance of
20
Ne. The
AC
different line shapes indicate different values of the fractional parameter .
5. Conclusions
In summarizing the present paper, analytical solution to the abundances differential equations of the helium burning network in its fractional form has been performed. We constructed recurrence relations for the power series coefficients of product abundance. Symbolic as well as numerical computations at 1 (the integer version of the network) are computed and compared. The 13
ACCEPTED MANUSCRIPT
maximum relative error is 0.003 . The effects of the fractional parameter on the product abundances has been investigated, we found that the product abundance is affected by changing the fractional parameter. The numerical results showed different behaviors when compared with integer model solutions. The most important output from the helium burning model is the energy generation rate
CR IP T
that is related to the luminosity of the star which in turns is the first physical parameter required to build interior model. The variation of the final product abundance will lead to a change in the energy generation rate and consequently to the calculated luminosity of the star.
AN US
References
Abdel-Salam, E. A-B., and Nouh, M. I., 2016, Astrophysics, 59, 398. Bayin, S. S. and Krisch, J. P., 2015, Ap&SS, 359, 58.
Calcagni, G., Kuroyanagi, S., Tsujikawa, S., 2016, JCAP 8, 39.
M
Calcagni, G., Rodríguez Fernández, D., Ronco, M., 2017, Europ. Phys. J. C 77, 335
Press, Chicago.
ED
Clayton, D. D., 1983, Principles of Stellar Evolution and Nucleosynthesis, University of Chicago
Chaurasia, V. and Pandey, S., 2010, Research in Astron. Astrophys, 10, 22.
PT
Coraddu, M., Kaniadakis, G., Lavagno, A., Lissia, M., Mezzorani, G., and Quarati, P.: 1998, Thermal distributions in stellar plasma, nuclear reactions and solar neutrinos,
CE
http://xxx.lanl.gov/abs/nucl-th/981108.
Duorah, H. L. and Kushwaha, R. S., 1963, Helium-Burning Reaction Products and the Rate of
AC
Energy Generation, ApJ, 137, 566.
El-Nabulsi, R.A., 2011, Appl. Math. Comput. 218, 2837. El-Nabulsi, R. A., 2012, Int. J. Theor. Phys. 51, 3978. El-Nabulsi, R. A., 2013, Canadian J. Phys. 91, 618. El-Nabulsi, R. A., 2015, Eur. Phys. J. Plus 130, 102.
14
ACCEPTED MANUSCRIPT
El-Nabulsi, R. A., 2016, Revista Mexicana de Fısica 62, 240. El-Nabulsi, R. A., 2017a, Int. J. Theor. Phys. 56, 1159. El-Nabulsi, R. A., 2017b, Canadian J. Phys. 95, 605.
CR IP T
Golmankhaneh, K., Yang, X. J. , Baleanu, D., 2015, Rom. J. Phys. 60, 22.
Hilfer, R. (ed.): 2000, Applications of Fractional Calculus in Physics, World Scientific, Singapore.
Hix, W. R.; Thielemann, F.-K., 1999, Computational methods for nucleosynthesis and nuclear energy generation, J. Comput. Appl. Math., 109, 321.
Haubold, H.J. and Mathai, A.M.: 1998, On thermonuclear reaction rates, Astrophys. Space Sci.
AN US
258, 185-199.
Haubold, H.J. and Mathai, A.M.: 2000, The fractional kinetic equation and thermonuclear functions, Astrophysics and Space Science, 273, 53-63.\
Kaniadakis, G., Lavagno, A., Lissia, M., and Quarati, P.: 1998, Anomalous diffusion modifies solar neutrino fluxes, Physica A261, 359-373, http://xxx.lanl.gov/abs/astro-ph/9710173.
M
Kilbas, A.A., Srivastava, H.M., Trujillo, J.J., 2006, Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam.
ED
Kourganoff, V.: 1973, Introduction to the Physics of Stellar Interiors, D. Reidel Publishing Company, Dordrecht.
PT
Lang, K.R.: 1999, Astrophysical Formulae Vol. I (Radiation, Gas Processes and High Energy Astrophysics) and Vol. II (Space, Time, Matter and Cosmology), Springer-Verlag,
CE
Berlin-Heidelberg.
Laskin, N., 2000, Phys. Rev. E 62, 3135. Nouh, M. I., Sharaf, M. A. and Saad, A. S., 2003, Astron. Nachr., 324, 432.
AC
Nouh, M. I and Abdel-Salam, E. A., 2017, Iranian Journal of Science and Technology, Transactions A: Science, in press.
Nouh, M. I and Abdel-Salam, E. A., 2018, European Physical Journal Plus, 133, 149. Podlubny, I., 1999, Fractional Differential Equations, Acad. Press, London. Saxena, R.K., Mathai, A.M., and Haubold, H.J., 2002, On fractional kinetic equations, Astrophysics and Space Science, 282, 281-287.
15
ACCEPTED MANUSCRIPT
Stanislavsky, A.A., 2007, Astron. Astrophys. Transactions 26(6), 655.
Sokolov, I.M., Klafter, J., Blumen, A., 2002, Phys. Today 55, 48.
AC
CE
PT
ED
M
AN US
CR IP T
Zaslavsky, G. M., 1994, Chaos 4, 25.
16
Computational Method for a Fractional Model of the Helium Burning Network Mohamed I. Nouh PII: DOI: Reference:
S1384-1076(18)30148-9 10.1016/j.newast.2018.07.006 NEASPA 1212
To appear in:
New Astronomy
Received date: Revised date: Accepted date:
21 May 2018 4 July 2018 18 July 2018
Please cite this article as: Mohamed I. Nouh , Computational Method for a Fractional Model of the Helium Burning Network, New Astronomy (2018), doi: 10.1016/j.newast.2018.07.006
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
ACCEPTED MANUSCRIPT
Highlights We considered the fractional model of the helium burning phase. The system of differential equations is solved via series expansion. Comparison with the numerical solution is performed. The effects of the fractional parameter on the product abundances is investigated.
AC
CE
PT
ED
M
AN US
CR IP T
1
ACCEPTED MANUSCRIPT
Computational Method for a Fractional Model of the Helium Burning Network Mohamed I. Nouh1,2 1
Department of Physics, College of Science, Northern Border University, Arar, Saudi Arabia. Email: [email protected], [email protected]
Department of Astronomy, National research Institute of Astronomy and Geophysics, 11421 Helwan, Cairo, Egypt
CR IP T
2
Abstract: Stellar cores may be considered as a nuclear reactor that play important role in injecting new synthesized elements in the interstellar medium. Helium burning is an important stage that contribute to the synthesis of key elements such as carbon, through the triple- α
AN US
process, and oxygen. In the present paper, we introduce a computational method for the fractional model of the nuclear helium burning in stellar cores. The system of fractional differential equations is solved simultaneously using series expansion method. The calculations are performed in the sense of modified Riemann-Liouville fractional derivative. Analytic expressions are obtained for the abundance of each element as a function of time. Comparing the
M
abundances calculated at the fractional parameter 1 , which represents the integer solution, with the numerical solution revealed a good agreement with maximum error 0.003 . The
ED
product abundances are calculated at 0.25, 0.5, 0.75 to declare the effects of changing the fractional parameters on the calculations.
PT
Keywords: System of fractional differential equations; Methods: Series expansion; Stellar
CE
burning stages: Helium burning; Nuclear reaction rate.
1. Introduction
AC
Nowadays, generalizations of some of the basic differential equations in physics led to new and more deep insight to the macroscopic phenomena in a wide range of areas like anomalous diffusion, signal processing and quantum mechanics (Podlubny 1999; Sokolov et al. 2002; Kilbas et al. 2006; Laskin 2000). There are many authors present applications of fractional calculus in astrophysics of them are: Stanislavsky(2007), El-Nabulsi (2011, 2012, 2013, 2015, 2016, 2017a, 2017b); Golmankhaneh et al. (2015), Bayian and Krisch (2015), Abdel-Salam and
2
ACCEPTED MANUSCRIPT
Nouh (2016), Nouh and Abdel-Salam (2017, 2018), Calcagni et al. (2016) and Calcagni et al. (2017).
In most stars, energies are derived from the conversion of hydrogen into helium. As a result, the temperature goes high (about 100 million degrees) and the hydrogen fusion starts, the stellar core contracts gravitationally until the temperature is high enough to start helium burning. This
CR IP T
set of reactions is called the triple alpha process. Helium brining occurs come into two categories, the first is the hydrostatic burning and the second is the explosive burning.
The nucleosynthesis of the elements in stars is described by kinetic equations governing the change of the number density Ni of species i over time, that is (Kourganoff, 1973)
v
j
where
v
mn
ij
N k Nl k ,l i
v
kl
AN US
dNi / dt Ni N j
represents the reaction cross section for an interaction involving species m and
n, and the summation is taken over all reactions which either produce or destroy the species i (Haubold and Mathai, 1998). For a gas of mass density , the number density Ni of the species i
M
is expressed in terms of its abundance X i , by the relation Ni N A X i / Ai , where N A is
ED
Avogadro’s number and Ai is the mass of i in mass units. Haubold & Mathai (2000) introduced a solution of the fractional generalized kinetic equation in terms of H-functions. Their equation is suitable for incorporating changes in the
PT
Maxwell-Boltzmann distribution function. The derived solution of a fractional kinetic equation contains the particle reaction rate (or thermonuclear function) as a time constant, and provided
CE
the analytic technique to further investigate possible modifications of the reaction rate through a kinetic equation. They concluded that the Riemann-Liouville operator in the fractional kinetic
AC
equation introduces a convolution integral with slowly-decaying power-law kernel, which is typical for memory effects referred to in Kaniadakis et al. (1997) and Coraddu et al. (1998). Saxena, Mathai, and Haubold (2002) derived solutions for the generalized fractional
kinetic equations and concluded that the ordinary and generalized Mittag-Leffler functions interpolate between a purely exponential law and power-like behavior of phenomena governed by the integer kinetic equations and their fractional counterparts, respectively (Lang, 1999; Hilfer, 2000). 3
ACCEPTED MANUSCRIPT
Solutions to some fractional kinetic equations in a series form of the Lorenzo-Hartley function have been introduced by Chaurasia and Pandey (2010). Lorenzo-Hartley function could be considered as a compact and elegant expression used for the computation and has a close relationship with the R-function, the generalized Mittag-Leffler functions, the Mittag-Leffler
CR IP T
function and the Robotnov & Hartley function.
One of the most important problems in astrophysics is the nucleosynthesis in the stellar cores. Hydrogen burning, helium burning, CNO cycle are the main stages that the stars produce in their nuclear reactors, Clayton (1983). The basic equations of the burning networks are differential equations that may be solved simultaneously by numerical or analytical methods,
AN US
Duroah and Kushwaha (1963), Clayton (1983), Hix and Thielemann (1999) and Nouh et al. (2003). This system of equations may be called the integer version of the burning models, and the general form could be called the fractional models.
In the present paper we are going to introduce a computational method to fractional model the helium burning network in stellar core. The system of the fractional differential
M
equations will be solved by power series expansion. We derive general recurrence relation for the series coefficients in terms of the fractional parameter and what is called the fractal index.
ED
The calculations will be performed in the sense of modified Riemann-Liouville fractional Derivative. Comparison with the solution of the integer version is presented to declare the effects of the fractional parameter on the calculations as well as the synthesis process. The paper is
PT
organized as follows. In section 2 we introduce the helium burning model. Section 3 is devoted to the series solution of the system of the differential equations. In section 4 we introduce series
CE
solution to the helium burning network.
AC
2. Helium Burning Network
In stellar helium core, a triple alpha collision occurs at temperature of the order of 100 million degrees and the product is C12. Then C12 captures He4 to form O16 which in turn captures He4 to form Ne20 and so on, Nouh et al. (2003). These reactions could be listed as
4
ACCEPTED MANUSCRIPT
3He4 C12 7.281Mev C12 He4 O16 7.150Mev O16 He4 Ne20 4.750Mev
By considering the above reactions, Clayton (1983) set up the following model for the helium
CR IP T
burning phase. If x, y, z and r are the helium, carbon, oxygen and neon number of atoms per unit mass of stellar material. The next four equations govern the time dependent change of the abundance as
dx / dt 3ax3 bxy cxz, dz / dt bxy cxz , dr / dt cxz. where a, b, and c are the reaction rates.
AN US
dy / dt ax3 bxy,
(1)
The factor 3 appears in the equation because three helium nuclei join to form one carbon
M
nucleus.
The system of differential equations (Equation (1)) could be solved by many analytical or
ED
numerical methods. The accuracy of the methods of numerical integrations depend mainly on the interval step used through the solution (time step in our case). Bad choice of this time step may lead to bad accuracy as well as numerical instability. This difficulty could be made quantitative if
PT
we use analytical methods. In the present article we will implement power series expansion to
CE
solve the fractional helium burning network.
AC
3. Series Solution of the System of the Fractional Differential Equations
Principles of the fractional calculus implemented in the present paper could be found in Nouh and Abdel-Salam (2018).
4.1. Series Expressions for the Unknowns The system of differential equations (Equation (1)) could be written in the fractional form as
5
ACCEPTED MANUSCRIPT
Dt x 3a x3 b x y c x z , Dt y a x 3 b x y, Dt z b x y c x z ,
(2)
Dt r c x z ,
Assuming the transform T t , the solution can be expressed in a series form as
CR IP T
x X mT m X 0 X 1T X 2T 2 X 3T 3 ... m 0
X 0 X 1t X 2t 2 X 3t 3 ...,
y YmT m Y0 Y1T Y2T 2 Y3T 3 ... m 0
Y0 Y1t Y2t 2 Y3t 3 ...,
(3)
z Z mT Z 0 Z1T Z 2T Z 3T ... m
2
3
m 0
AN US
Z 0 Z1t Z 2t 2 Z 3t 3 ...,
r RmT m R0 R1T R2T 2 R3T 3 ... m 0
M
R0 R1t R2t 2 R3t 3 ... .
ED
where X m , Ym , Zm , Rm are constants to be determined.
4.2. Fractional derivative of the function raised to powers
CE
PT
The left hand side of the system in Equation (2) represents the abundances of the elements where the helium abundance (x) is raised to power 3. To obtain the fractional derivative of u n we apply the fractional derivative of the product two functions. Taking the fractional derivative of both sides of Equation (2), we have
AC
n u n Dx u u Dx G
(4)
Differentiating both sides of Equation (4) k times -derivatives we obtain k
n j 0
D k j
k
x ...Dx u Dx ...Dx G
j 1times
k j times
j 0
D k j
x
...Dx G Dx ...Dx u .
j 1times
k j times
At x 0 , we have
6
ACCEPTED MANUSCRIPT
k
n j 0
k j
k
Dx ...Dx u (0) Dx ...Dx G (0) j 1times
j 0
k j times
D k j
x
...Dx G (0) Dx ...Dx u (0)
j 1times
(5)
k j times
where Dx ...Dx u (0) X j 1(( j 1) 1), j 1 times
CR IP T
Dx ...Dx G (0) Qk j ( (k j ) 1), k j times
Dx ...Dx G (0) Q j 1(( j 1) 1), j 1times
Dx ...Dx u (0) X k j ( (k j ).
4.2.
AN US
k j times
(6)
Recurrence Relations
Substituting Equation (5) in Equation (6) we have k
n
k
X j 1(( j 1) 1)Qk j ( (k j ) 1) j 0
Q k j
(( j 1) 1) X k j ( ( k j ) 1),
j 1
M
j 0
k j
after some algebraic manipulation we obtain k
k !( (k j ) 1)(( j 1) 1) X j 1Qk j j !(k j )!
ED
((k 1) 1) X 0Qk 1 n j 0
k !(( j 1) 1) ( (k j ) 1) X k j Q j 1 j !(k j )! j 0
PT
k 1
CE
Let i j 1 in the first sum and i k - j in the second sum, then we get
k 1
AC
((k 1) 1) X 0Qk 1 n i 1
k !( (k 1 i ) 1)(i 1) X i Qk 1i (i 1)!(k 1 i )! k !((k 1 i ) 1) (i 1) X i Qk 1i (k i )!i ! i 1 k
,
put m k 1 and adding the zero value (m m) (m 1) X mQ0 / m to the second sum of the last equation, we get 7
ACCEPTED MANUSCRIPT
m
(m 1) X 0Qm n i 1
(m 1)!( (m i) 1)(i 1) X i Qm i (i 1)!(m i )! (m 1)!((m i) 1) (i 1) X i Qm i (m 1 i )!i ! i 1 m
1 (m 1) X 0
m
i 1
(m 1)!( (m i) 1)(i 1) in m i X iQmi , m 1 i !(m i )!
at m 0 , Q0 X 0n , Q1
( 1)n X1Q0 . ( 1) A0
Putting n 3 in the last equation gives
1 (m 1) X 0
and Q0 X 03 , Q1
m
i 1
, (m 1)!( (m i) 1)(i 1) 4i m X iQmi , m 1 i !(m i )!
3 X 1Q0 , .... X0
(7)
M
Qm
.
AN US
Qm
CR IP T
then the coefficients Qm could be determined by
,
ED
Now by differentiating Equation (3) -times and after some manipulations we have (n 1) X nT n 1 , n 1 ( n 1 )
PT
Dt x
(n 1) YnT n 1 , n 1 ( n 1 )
Dt y
(8)
(n 1) Dt z Z nT n 1 , ( n 1 ) n 1
CE
(n 1) RnT n 1 , n 1 ( n 1 )
AC
Dt r
substituting Equations (3) and (8) in Equation (2) for x, y, z and r respectively, we could determine the coefficients of the power series X n1, Yn1, Zn1 and Rn1 from
X n1
n n (n 1) 3 aQ b X Y c X j Z n j , n j n j ((n 1) 1) j 0 j 0
8
ACCEPTED MANUSCRIPT
Yn1 Z n1
n (n 1) aQ b n X jYn j , ((n 1) 1) j 0 n (n 1) n b X jYn j c X j Z n j , ((n 1) 1) j 0 j 0
and c (n 1) n X j Z n j . ((n 1) 1) j 0
CR IP T
Rn 1
with the initial values X 0 ;Y0 ; Z0 ; R0 . 4. Results
First, we run the MATHEMATICA code to calculate the abundance of each element at 1 .
AN US
We use six series terms and the time 200. The calculations are performed for the pure helium gas with X 0 1;Y0 0; Z0 0; R0 0 , the temperature and density are
109 K and 104 gcm-3
respectively. Comparison with the numerical solution gives good agreement with maximum relative errors are 0.003, 0.002, 0.00012 and 0.00025 for the x, y,z and r respectively. To check
M
the accuracy of the computations, we investigate the conservation formula x y z r 1 . If the sum is not 1, so there is a trouble through the computations, Table 1 lists the absolute errors
ED
( ) at different times. As we see from the table, the absolute error of the sum increases with increasing time and the maximum error is 0.0004 . To investigate the effects of changing the fraction parameter on the abundance of the elements,
we
have
performed
PT
four
the
calculations
at
the
fractional
parameters
0.25, 0.5, 0.75 and 1 respectively. Figures 1 to 3 illustrate the relation between helium
CE
concentration and the product abundances of C12, O16 and Ne20 respectively. When the helium concentration is near 0.7 the difference between the product abundance of C12 computed at
AC
different values of is very small. This differences become larger when the helium abundance goes to zero.
The behavior of the product of the O16 is different than that of C12, the product abundance
computed at 1 is smaller than of smaller than one. The situation is reversed as the helium concentration goes from 0.3 to zero, the product of Ne20 have the same behavior when the helium concentration less than 0.5, after that the difference become remarkable. To build an interior model one need to estimate the energy generation rates and the central temperature of the star. 9
ACCEPTED MANUSCRIPT
So, possible application of the fractional burning network is to model the stellar interior with two or more layers having different fractional parameters, i.e. the fractional parameter could be used to adjust the product abundance of the syntheses elements.
Table 1: The absolute errors of the sum of product abundances.
CR IP T
Time (s)
0.75
0.5
0.25
0
1.2 107
1.0 107
2.0 107
2.0 107
500
4.5 105
5.6 105
4.7 105
4.9 105
1000
9.4 105
1.3 104
1.7 104
1.9 104
1500
1.4 104
1.9 104
2.5 104
2.9 104
2000
1.9 104
2.6 104
3.3 104
3.8 104
AC
CE
PT
ED
M
AN US
1
10
ED
M
AN US
CR IP T
ACCEPTED MANUSCRIPT
Figure 1: The helium concentration as a function of the product abundance of C12. The
AC
CE
PT
different line shapes indicate different values of the fractional parameter .
11
CE
PT
ED
M
AN US
CR IP T
ACCEPTED MANUSCRIPT
Figure 2: The helium concentration as a function of the product abundance of O16. The
AC
different line shapes indicate different values of the fractional parameter .
12
CE
PT
ED
M
AN US
CR IP T
ACCEPTED MANUSCRIPT
Figure 3: The helium concentration as a function of the product abundance of
20
Ne. The
AC
different line shapes indicate different values of the fractional parameter .
5. Conclusions
In summarizing the present paper, analytical solution to the abundances differential equations of the helium burning network in its fractional form has been performed. We constructed recurrence relations for the power series coefficients of product abundance. Symbolic as well as numerical computations at 1 (the integer version of the network) are computed and compared. The 13
ACCEPTED MANUSCRIPT
maximum relative error is 0.003 . The effects of the fractional parameter on the product abundances has been investigated, we found that the product abundance is affected by changing the fractional parameter. The numerical results showed different behaviors when compared with integer model solutions. The most important output from the helium burning model is the energy generation rate
CR IP T
that is related to the luminosity of the star which in turns is the first physical parameter required to build interior model. The variation of the final product abundance will lead to a change in the energy generation rate and consequently to the calculated luminosity of the star.
AN US
References
Abdel-Salam, E. A-B., and Nouh, M. I., 2016, Astrophysics, 59, 398. Bayin, S. S. and Krisch, J. P., 2015, Ap&SS, 359, 58.
Calcagni, G., Kuroyanagi, S., Tsujikawa, S., 2016, JCAP 8, 39.
M
Calcagni, G., Rodríguez Fernández, D., Ronco, M., 2017, Europ. Phys. J. C 77, 335
Press, Chicago.
ED
Clayton, D. D., 1983, Principles of Stellar Evolution and Nucleosynthesis, University of Chicago
Chaurasia, V. and Pandey, S., 2010, Research in Astron. Astrophys, 10, 22.
PT
Coraddu, M., Kaniadakis, G., Lavagno, A., Lissia, M., Mezzorani, G., and Quarati, P.: 1998, Thermal distributions in stellar plasma, nuclear reactions and solar neutrinos,
CE
http://xxx.lanl.gov/abs/nucl-th/981108.
Duorah, H. L. and Kushwaha, R. S., 1963, Helium-Burning Reaction Products and the Rate of
AC
Energy Generation, ApJ, 137, 566.
El-Nabulsi, R.A., 2011, Appl. Math. Comput. 218, 2837. El-Nabulsi, R. A., 2012, Int. J. Theor. Phys. 51, 3978. El-Nabulsi, R. A., 2013, Canadian J. Phys. 91, 618. El-Nabulsi, R. A., 2015, Eur. Phys. J. Plus 130, 102.
14
ACCEPTED MANUSCRIPT
El-Nabulsi, R. A., 2016, Revista Mexicana de Fısica 62, 240. El-Nabulsi, R. A., 2017a, Int. J. Theor. Phys. 56, 1159. El-Nabulsi, R. A., 2017b, Canadian J. Phys. 95, 605.
CR IP T
Golmankhaneh, K., Yang, X. J. , Baleanu, D., 2015, Rom. J. Phys. 60, 22.
Hilfer, R. (ed.): 2000, Applications of Fractional Calculus in Physics, World Scientific, Singapore.
Hix, W. R.; Thielemann, F.-K., 1999, Computational methods for nucleosynthesis and nuclear energy generation, J. Comput. Appl. Math., 109, 321.
Haubold, H.J. and Mathai, A.M.: 1998, On thermonuclear reaction rates, Astrophys. Space Sci.
AN US
258, 185-199.
Haubold, H.J. and Mathai, A.M.: 2000, The fractional kinetic equation and thermonuclear functions, Astrophysics and Space Science, 273, 53-63.\
Kaniadakis, G., Lavagno, A., Lissia, M., and Quarati, P.: 1998, Anomalous diffusion modifies solar neutrino fluxes, Physica A261, 359-373, http://xxx.lanl.gov/abs/astro-ph/9710173.
M
Kilbas, A.A., Srivastava, H.M., Trujillo, J.J., 2006, Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam.
ED
Kourganoff, V.: 1973, Introduction to the Physics of Stellar Interiors, D. Reidel Publishing Company, Dordrecht.
PT
Lang, K.R.: 1999, Astrophysical Formulae Vol. I (Radiation, Gas Processes and High Energy Astrophysics) and Vol. II (Space, Time, Matter and Cosmology), Springer-Verlag,
CE
Berlin-Heidelberg.
Laskin, N., 2000, Phys. Rev. E 62, 3135. Nouh, M. I., Sharaf, M. A. and Saad, A. S., 2003, Astron. Nachr., 324, 432.
AC
Nouh, M. I and Abdel-Salam, E. A., 2017, Iranian Journal of Science and Technology, Transactions A: Science, in press.
Nouh, M. I and Abdel-Salam, E. A., 2018, European Physical Journal Plus, 133, 149. Podlubny, I., 1999, Fractional Differential Equations, Acad. Press, London. Saxena, R.K., Mathai, A.M., and Haubold, H.J., 2002, On fractional kinetic equations, Astrophysics and Space Science, 282, 281-287.
15
ACCEPTED MANUSCRIPT
Stanislavsky, A.A., 2007, Astron. Astrophys. Transactions 26(6), 655.
Sokolov, I.M., Klafter, J., Blumen, A., 2002, Phys. Today 55, 48.
AC
CE
PT
ED
M
AN US
CR IP T
Zaslavsky, G. M., 1994, Chaos 4, 25.
16