Introduction

Self-gravitating polytropic gas spheres in astronomy are crucial to understanding star structure and evolution. The polytropic equation of state \(P = K\rho^{1 + 1/n}\) governs the relation between pressure P and density in star modeling, where n is the polytropic index. The Lane–Emden equation, a second-order nonlinear differential equation that explains the radial density distribution in a hydrostatic equilibrium star, is essential to this approach. First presented by1 and extended by2, the equation has been intensively explored for various values of n, which correlate to varied physical conditions in star interiors.

The value of the polytropic index n has a significant impact on the solutions to the Lane–Emden equation. Different physical conditions in the star’s interior are represented by different values of n. For instance: The equation simulates an isothermal gas sphere for n = 1, which is equivalent to stars with an internal temperature that is almost constant. White dwarfs, where the electron degeneracy pressure is crucial to preserving equilibrium, are modeled using the equation for n = 3. The equation roughly describes main-sequence stars with radiation pressure as their primary governing factor for n = 5. The polytropic index n has a direct impact on the star’s mass, radius, and density distribution, among other important stellar characteristics. Generally speaking, a star with a higher core density and a lower radius is more compact when n is greater. In contrast, bigger, less dense stars are linked to a lower value of n3,4.

A valuable mathematical tool for modeling and studying complex systems with memory effects, nonlocal interactions, and anomalous behavior is fractional calculus, an extension of classical integer-order calculus. Recently, there has been a lot of interest in using fractional calculus in astrophysics since it provides a fresh framework for explaining various phenomena that traditional calculus cannot sufficiently describe. Applications in astrophysics have been covered in many books and articles5,6,7,8,9,10,11,12,13,14,15,16,17,18. An example of application in astrophysics, incorporating fractional calculus offers novel approaches to tackling intricate problems in cosmology, such as cosmic acceleration and the dynamics of dark energy. El-Nabulsi and Torres19 and El-Nabulsi et al.20 formulated a fractional action principle that alters conventional variational methods in cosmology. This approach facilitates the reevaluation of the Friedmann–Robertson–Walker (FRW) cosmos by integrating fractional derivatives to investigate novel processes of cosmic expansion and structure development.

Compared to conventional integer-order derivatives, fractional derivatives provide a more versatile method for simulating physical systems. The fact that fractional derivatives often comprise an integral across a temporal or geographical domain, indicating memory effects or nonlocal interactions within the system, is one of their essential characteristics. Nevertheless, these memory effects are not relevant in static stellar models, because time dependency is not present. These models apply fractional derivatives to parameters like temperature, pressure, and mass at certain scaled radial coordinates r/R, where R is the radius of the star. By efficiently sampling the complete star, these fractional derivatives provide a more thorough explanation of its underlying structure21. Furthermore, compared to the traditional calculus of variations, the fractional calculus of variations has certain benefits, especially when representing systems with dissipation. The fractional technique enables a more precise and nuanced understanding of dissipative classical and quantum systems, while standard approaches often fail to properly capture their dynamics. Because of this, fractional calculus is particularly helpful for simulating complicated processes where dissipation, memory effects, and nonlocal interactions are significant22.

There are many investigations spread in the literature into Lane–Emden equations. The fractional action-like variational method was first presented as a subclass of the fractional calculus of variations22,23,24,25. To get the generalized Lane–Emden-type equations, El-Nabulsi22 expanded the fractional action-like variational technique for the case of non-standard power-law Lagrangians. El-Nabulsi12 also yielded the fractional Chandrasekhar or Lane–Emden nonlinear differential equation of white dwarfs. Luchko and Gorenflo26 investigated the application of Riesz fractional derivatives in the Lane–Emden equation, primarily focusing on systems with radial symmetry. The fractional isothermal and polytropic gas spheres were studied by27,28, and21 in terms of the modified Riemann–Liouville (mRL) fractional derivatives. Abdel-Salam et al.29 used a conformable Adomian decomposition approach to develop a divergent series solution for the Lane–Emden type equations. Yousif et al.30 developed general analytical formulations for the fractional isothermal gas sphere within the framework of the Newtonian hydrostatic equilibrium. Aboueisha et al.31 examined the fractional parameter’s effect on the mass–radius relation of white dwarfs and the fractional relativistic polytropic gas sphere.

The truncated M-fractional calculus (TMD) has received considerable attention for its capacity to integrate non-local effects and memory-dependent phenomena into physical models. This method, initially proposed by32, presents a more adaptable framework than conventional fractional calculus, enabling the specification of fractional derivatives of orders α and β, thereby serving as a robust tool for modeling systems characterized by complex interactions and long-range effects. The TMD method has been effectively utilized to modify and improve various physical models, particularly in fluid dynamics and nonlinear wave equations (e.g., 33,34). This research marks a significant advancement through the application of TMD to the Lane–Emden equation, a topic that, to our knowledge, has not been previously investigated. The Lane–Emden equation serves as a fundamental component in astrophysical modeling, detailing the structure of self-gravitating polytropic gas spheres, such as white dwarfs and various stellar entities. The incorporation of TMD into this framework provides a distinct opportunity to analyze stellar structures, integrating fractional derivatives that consider memory effects and nonlocal interactions.

In this paper, we formulate the Lane–Emden equation within the context of M-fractional derivatives (LETMD), a version of fractional calculus that bridges both integer and fractional orders of differentiation. The LETMD formulation enables us to examine the dynamics of polytropic gas spheres under fractional effects. To solve the resultant problem, we apply the accelerated power series approach, which offers an efficient way of deriving solutions in the presence of fractional components. Additionally, we investigate the TMD polytropic models for the Sun and white dwarfs, studying how modifying the fractional orders α and β affects important stellar properties such as mass, radius, and density distribution. These findings show that the incorporation of TMD in modeling stellar structures gives fresh insights into the behavior of self-gravitating systems, especially in cases where long-range interactions and memory effects are substantial. Thus, truncated M-fractional derivatives in the context of the Lane–Emden equation constitute a unique and promising method in astrophysical modeling, with potential applications in understanding the behavior of stars, stellar remnants, and other complicated astronomical phenomena.

The rest of the paper is organized as follows. In Sect. “The truncated M-fractional calculus (TMD)”, we review the principles of TMD. In Sect. “Fractional Lane–Emden equation (LETMD)”, we introduce the fractional extension of the Lane–Emden equation and discuss the implications of the TMD. Section “Analytical solution to LETMD” deals with the analytical solution of the LETMD. In Sect. “Physical characteristics of the TFM polytrope”, we formulate the physical characteristics of the TMD sphere. The numerical results and comparisons are presented in Sect. “Results”, followed by conclusions in Sect. “Conclusion”.

The truncated M-fractional calculus (TMD)

Definition 2.1

The truncated Mittag–Leffler function (MLF) is defined as follows32:

$${}_{l}{\rm E}_{\beta } (\varepsilon y^{\alpha } ) = \sum\limits_{j = 0}^{l} {\frac{{(\varepsilon y^{\alpha } )^{j} }}{\Gamma (\beta j + 1)}} ,\,\,\,\,\,\beta > 0,\,\,\,\,\,y \in {\mathbb{C}}.$$
(1)

Definition 2.2

Let \(\psi :[0,\,\infty ) \to \Re\) be a function, the local truncated M-fractional derivative (TMD) of \(\psi\) of order α ∈ (0, 1) with respect to y is given:

$${}_{l}D_{M,y}^{\alpha ,\beta } \psi (y) = \mathop {\lim }\limits_{\varepsilon \to 0} \,\,\frac{{\psi (y{}_{l}{\rm E}_{\beta } (\varepsilon y^{ - \alpha } )) - \psi (y)}}{\varepsilon },\,\,\,\,\,\,\forall \,\beta ,\,y > 0,\,\,\,\,\,\alpha \in (0,1).$$
(2)

The TMD adheres to the following hypotheses:

$${}_{l}D_{M}^{\alpha ,\beta } \,y^{m} \, = \frac{m\,}{{\Gamma (\beta + 1)}}y^{m - \alpha } ,\,\,\,\,m \in \,R,\,\,\,\,\,\,\,\,\,{}_{l}D_{M}^{\alpha ,\beta } c = 0,\,\,\,\,\,\,\,\,\forall \,\psi (y) = c,$$
(3)
$${}_{l}D_{M}^{\alpha ,\beta } (c_{1} \,\psi + c_{2} \,\varphi ) = c_{1} \,{}_{l}D_{M}^{\alpha ,\beta } \psi \, + c_{2} \,{}_{l}D_{M}^{\alpha ,\beta } \varphi ,\,\,\,\,\,\,\,\,\,\,\,\,\forall \,\,c_{1} \,,c_{2} \,\, \in \Re ,$$
(4)
$${}_{l}D_{M}^{\alpha ,\beta } (\psi \varphi ) = \psi {}_{l}D_{M}^{\alpha ,\beta } \varphi + \varphi {}_{l}D_{M}^{\alpha ,\beta } \psi ,$$
(5)
$${}_{l}D_{M}^{\alpha ,\beta } \left( {\frac{\psi }{\varphi }} \right) = \frac{{\varphi {}_{l}D_{M}^{\alpha ,\beta } \psi - \psi {}_{l}D_{M}^{\alpha ,\beta } \varphi }}{{\varphi^{2} }},$$
(6)
$${}_{l}D_{M}^{\alpha ,\beta } \varphi (\psi ) = \frac{d\varphi }{{d\psi }}{}_{l}D_{M}^{\alpha ,\beta } \psi ,$$
(7)
$${}_{l}D_{M}^{\alpha ,\beta } \varphi (y) = \frac{{y^{1 - \alpha } }}{\Gamma (\beta + 1)}\frac{d\varphi }{{dy}}.$$
(8)

With \(\varphi ,\,\,\psi\) represents two \(\,\alpha -\) differentiable functions of a dependent variable, the above relations are proved in30.

Choosing \(\beta = 1\) and \(l = 1\) on both sides of Eq. (2), we have

$${}_{1}D_{M,y}^{\alpha ,1} \psi (y) = \mathop {\lim }\limits_{\varepsilon \to 0} \,\,\frac{{\psi (y{}_{1}{\rm E}_{1} (\varepsilon y^{ - \alpha } )) - \psi (y)}}{\varepsilon },\,\,\,\,\,\,\forall \,y > 0,\,\,\,\,\,\alpha \in (0,1).$$
(9)

But, it is known that

$${}_{1}{\rm E}_{1} (\varepsilon y^{ - \alpha } ) = \sum\limits_{j = 0}^{1} {\frac{{(\varepsilon y^{ - \alpha } )^{j} }}{\Gamma (j + 1)}} = 1 + \varepsilon y^{ - \alpha } .$$
(10)

Thus, we conclude that

$${}_{1}D_{M,y}^{\alpha ,1} \psi (y) = \mathop {\lim }\limits_{\varepsilon \to 0} \frac{{\psi (y + \varepsilon y^{1 - \alpha } ) - \psi (y)}}{\varepsilon } = D_{{}}^{\alpha } \psi (y),\,\,\,\,\,\,\forall \,y > 0,\,\,\,\,\,\alpha \in (0,1],$$
(11)

which is exactly the conformable fractional derivative. Simply, we write \({}_{1}D_{M}^{\alpha ,\beta }\) as \(D_{M}^{\alpha ,\beta }\). The MFD of some functions

$$D_{M}^{\alpha ,\beta } e^{c\,t} = \frac{{ct^{1 - \alpha } }}{\Gamma (\beta + 1)}e^{c\,t} ,$$
(12)
$$D_{M}^{\alpha ,\beta } \sin (ct) = \frac{{ct^{1 - \alpha } }}{\Gamma (\beta + 1)}\cos (ct),$$
(13)
$$D_{M}^{\alpha ,\beta } \cos (ct) = - \frac{{c\,t^{1 - \alpha } }}{\Gamma (\beta + 1)}\sin (ct),$$
(14)
$$D_{M}^{\alpha ,\beta } e^{{c\,t^{\alpha } }} = \frac{c\,\alpha }{{\Gamma (\beta + 1)}}e^{{c\,t^{\alpha } }} ,$$
(15)
$$D_{M}^{\alpha ,\beta } \sin (ct^{\alpha } ) = \frac{c\,\alpha }{{\Gamma (\beta + 1)}}\cos (ct^{\alpha } ),$$
(16)
$$D_{M}^{\alpha ,\beta } \cos (ct^{\alpha } ) = - \frac{c\alpha }{{\Gamma (\beta + 1)}}\sin (ct^{\alpha } ).$$
(17)

Fractional Lane–Emden equation (LETMD)

The polytropic equation of state has the form

$$p = K\rho^{\gamma } ,$$
(18)

where

$$\gamma = 1 + \frac{1}{n}.$$
(19)

where \(n\) is the polytropic index and \(K\) is called the pressure constant. The equilibrium configuration of a self-gravitating gas sphere is obtained from the hydrostatic equilibrium equations. The most straightforward scenario is a spherical, non-rotating, static setup, whereby a single quantity, such as the central density, for a particular equation of state characterizes all macroscopic attributes.

The TMD form of equations of mass conservation and hydrostatic equilibrium is given by

$$M(r^{\alpha } ) = \frac{{4\Gamma (\beta + 1)^{3} }}{{3\alpha^{3} }}\pi r^{3\alpha } \rho ,$$
(20)

where M is the mass contained in a radius r. By applying the TMD derivative, Eq. (3), to Eqs. (18) and (20), we obtain

$$D_{M,r}^{\alpha ,\beta } M(r^{\alpha } ) = \frac{{4\Gamma (\beta + 1)^{2} }}{{\alpha^{2} }}\pi r^{2\alpha } \rho ,$$
(21)

and

$$D_{M,r}^{\alpha ,\beta } \,P(r^{\alpha } ) = - \frac{{\alpha^{2} }}{{\Gamma (\beta + 1)^{2} }}\frac{{G\,M(r^{\alpha } )}}{{r^{2\alpha } }}\,\rho .$$
(22)

Rearrange Eq. (22) we get

$$\frac{{\Gamma (\beta + 1)^{2} }}{{\alpha^{2} }}\frac{{r^{2\alpha } }}{\rho }\,D_{M,r}^{\alpha ,\beta } \,P(r^{\alpha } ) = \, - \,G\,M(r^{\alpha } ).$$
(23)

By performing the first fractional derivative of Eq. (23), we get

$$D_{M,r}^{\alpha ,\beta } \left( {\frac{{\Gamma (\beta + 1)^{2} }}{{\alpha^{2} }}\frac{{r^{2\alpha } }}{\rho }\,D_{M,r}^{\alpha ,\beta } P(r^{\alpha } )} \right) = \, - \,G\,D_{M,r}^{\alpha ,\beta } \,M(r^{\alpha } ).$$
(24)

Combining Eqs. (18) and (24) we get

$$D_{M,r}^{\alpha ,\beta } \left( {\frac{{\Gamma (\beta + 1)^{2} }}{{\alpha^{2} }}\frac{{r^{2\alpha } }}{\rho }\,D_{M,r}^{\alpha ,\beta } P(r^{\alpha } )} \right) = \, - \,\frac{{4\Gamma (\beta + 1)^{2} }}{{\alpha^{2} }}\pi G\,r^{2\alpha } \rho ,$$
(25)

rearrange terms, we get

$$\frac{{\alpha^{2} }}{{\Gamma (\beta + 1)^{2} }}\frac{1}{{r^{2\alpha } }}D_{M,r}^{\alpha ,\beta } \left( {\frac{{\Gamma (\beta + 1)^{2} }}{{\alpha^{2} }}\frac{{r^{2\alpha } }}{\rho }\,D_{M,r}^{\alpha ,\beta } p(r^{\alpha } )} \right) = \, - \,4\pi G\,\rho .$$
(26)

Now, by defining the dimensionless function u (Emden function) as

$$\rho = \rho_{c} u^{n} ,$$
(27)

where \(\rho\) and \(\rho_{c}\) are the density and central density respectively. The dimensionless radius of the sphere, x, could be written as

$$x = \frac{r\,\Gamma (\beta + 1)}{{a\alpha }},$$
(28)

where a is an arbitrary constant determined later by Eq. (33). Inserting Eqs. (18) and (27) in Eq. (26) we get

$$\frac{K}{{(a\,x^{\alpha } )^{2} }}D_{{M,\,\frac{a\alpha }{{\Gamma (\beta + 1)}}x}}^{\alpha ,\beta } \left( {\frac{{(a\,x^{\alpha } )^{2} }}{{\rho_{c} u^{n} }}\,D_{{M,\,\frac{a\alpha }{{\Gamma (\beta + 1)}}x}}^{\alpha ,\beta } \,(\rho_{c} u^{n} )^{{1 + \frac{1}{n}}} } \right) = - 4\pi G\rho_{c} u^{n} .$$
(29)

The fractional derivative of the Emden function u could be written as

$$D_{{M,\,\frac{a\alpha }{{\Gamma (\beta + 1)}}x}}^{\alpha ,\beta } u^{n + 1} = (n + 1) \, u^{n} D_{{M,\,\frac{a\alpha }{{\Gamma (\beta + 1)}}x}}^{\alpha ,\beta } u.$$
(30)

Inserting Eq. (30) in Eq. (29) we get

$$\frac{{K\Gamma (\beta + 1)^{2} }}{{a^{2} \alpha^{2} x^{2\alpha } }}D_{M,\,x}^{\alpha ,\beta } \left( {\frac{{\,(n + 1)x^{2\alpha } \rho_{c}^{{1 + \frac{1}{n}}} u^{n} }}{{\rho_{c} u^{n} }}\,D_{M,\,x}^{\alpha ,\beta } u} \right) = \, - \,4\pi G\,\rho_{c} u^{n} ,$$
(31)

rearrange terms, we have

$$\frac{{K(n + 1)\rho_{c}^{{\frac{1}{n} - 1}} \Gamma (\beta + 1)^{2} }}{{4\pi Ga^{2} \alpha^{2} }}\,\frac{1}{{x^{2\alpha } }}D_{M,\,x}^{\alpha ,\beta } \left( {x^{2\alpha } \,D_{M,\,x}^{\alpha ,\beta } u} \right) = \, - \,u^{n} .$$
(32)

Now by taking

$$a^{2} = \frac{{\Gamma (\beta + 1)^{2} K(n + 1)\rho_{c}^{{\frac{1}{n} - 1}} }}{{\,4\pi G\alpha^{2} }},$$
(33)

then the Lane–Emden equation in its fractional form is given by

$$\,\frac{1}{{x^{2\alpha } }}D_{M,\,x}^{\alpha ,\beta } \left( {x^{2\alpha } \,D_{M,\,x}^{\alpha ,\beta } u} \right) = \, - \,u^{n} .$$
(34)

Analytical solution to LETMD

Lane–Emden equation in its fractional form (Eq. (34)) could be written as

$$x^{ - 2\alpha } D_{M,x}^{\alpha ,\beta } \left( {x^{2\alpha } D_{M,x}^{\alpha ,\beta } } \right)u + u^{n} = 0,$$
(35)

with the initial conditions

$$\,u(0) = 1,\,\,\,\,\,\,\,\,\,\,\,\,\,D_{M,x}^{\alpha ,\beta } u(0) = 0.$$
(36)

where \(\,u = u(x)\), is the Emden function and \(0{ < }\alpha \, \le { 1,}\, \, \beta { > 0}\).

Assume the transform \(\,X = x^{\alpha }\), the Emden function takes the form

$$u(X) = \sum\limits_{m = 0}^{\infty } {A_{m} X^{m} } .$$
(37)

Inserting the initial conditions (Eq. (36)) into Eq. (38), we get \(A_{0} = 1\). Applying Eqs. (6) and (7) to Eq. (37), we get

$$D_{M,x}^{\alpha ,\beta } u = \frac{\alpha }{\Gamma (\beta + 1)}(A_{1} + 2A_{2} \, x^{\alpha } + 3A_{3} \, x^{2\alpha } + 4A_{4} \, x^{3\alpha } + ...),$$
(38)

and

$$D_{x}^{\alpha } u(0) = \frac{{\alpha A_{1} }}{\Gamma (\beta + 1)},$$
(39)

then, \(A_{1} = 0\) and Eq. (37) could be written as

$$u(X) = 1 + \sum\limits_{m = 2}^{\infty } {A_{m} X^{m} } .$$
(40)

Differentiate the Emden function \(u\) once more, and we get

$$D_{M,x}^{\alpha \alpha ,\beta } u = D_{M,x}^{\alpha ,\beta } D_{M,x}^{\alpha ,\beta } u = \left( {\frac{\alpha }{\Gamma (\beta + 1)}} \right)^{2} \left( {2A_{2} + 6A_{3} \, x^{\alpha } + 12A_{4} \, x^{2\alpha } + 20A_{5} \, x^{3\alpha } ...} \right).$$
(41)

At \(X = 0\) we have

$$D_{x}^{\alpha \alpha } u(0) = \frac{{2\alpha^{2} A_{2} }}{{\Gamma (\beta + 1)^{2} }}.$$
(42)

Differentiate the last equation j times

$$D_{M,x}^{{{\overbrace {\alpha \,\,\,\,\,\,\,...\alpha }^{j\,{\text{times}}}},\beta }} u(0) = \overbrace {{D_{M,x}^{\alpha ,\beta } ...D_{M,x}^{\alpha ,\beta } }}^{{j\,{\text{times}}}}u(0) = \frac{{j!\alpha^{j} }}{{\Gamma (\beta + 1)^{j} }}A_{j} ,\,$$
(43)

where \(A_{j}\) are constants to be determined. Now suppose that

$$u^{n} = G(X) = \sum\limits_{m = 0}^{\infty } {Q_{m} X^{m} } .$$
(44)

At \(X = 0\) we have after j times derivatives

$$D_{M,x}^{{{\overbrace {\alpha \,\,\,\,\,\,\,...\alpha }^{j\,{\text{times}}}},\beta }} G(0) = \overbrace {{D_{M,x}^{\alpha ,\beta } ...D_{M,x}^{\alpha ,\beta } }}^{{j\,{\text{times}}}}G(0) = \frac{{j!\alpha^{j} }}{{\Gamma (\beta + 1)^{j} }}Q_{j} .$$
(45)

Differentiate both sides of Eq. (40), we get

$$D_{M,x}^{\alpha ,\beta } u^{n} = D_{M,x}^{\alpha ,\beta } G,$$
(46)

that is

$$n\,u^{n - 1} D_{M,x}^{\alpha ,\beta } u = D_{M,x}^{\alpha ,\beta } G,$$
(47)

or

$$n\,\,u^{n} D_{M,x}^{\alpha ,\beta } u = u\,D_{M,x}^{\alpha ,\beta } G\,.$$
(48)

Differentiating both sides of Eq. (44) \(k\) times we have

$$D_{M,x}^{{{\overbrace {\alpha \,\,\,\,\,\,\,...\alpha }^{k\,{\text{times}}}},\beta }} [n\,G\,D_{x}^{\alpha } u] = D_{M,x}^{{{\overbrace {\alpha \,\,\,\,\,\,\,...\alpha }^{k\,{\text{times}}}},\beta }} \,(u\,D_{x}^{\alpha } G)\,,$$
(49)

then we have

$$n\sum\limits_{j = 0}^{k} {\,\left( {_{j}^{k} } \right)D_{M,x}^{{{\overbrace {\alpha \,\,\,\,\,\,\,...\alpha }^{j + 1\,{\text{times}}}},\beta }} u\,D_{M,x}^{{{\overbrace {\alpha \,\,\,\,\,\,\,...\alpha }^{k - j\,{\text{times}}}},\beta }} G} = \sum\limits_{j = 0}^{k} {\,\left( {_{j}^{k} } \right)D_{M,x}^{{{\overbrace {\alpha \,\,\,\,\,\,\,...\alpha }^{j + 1\,{\text{times}}}},\beta }} G\,\,D_{M,x}^{{{\overbrace {\alpha \,\,\,\,\,\,\,...\alpha }^{k - j\,{\text{times}}}},\beta }} u} ,\,$$
(50)

at \(X = 0\), we have

$$n\sum\limits_{j = 0}^{k} {\,\left( {_{j}^{k} } \right)D_{M,x}^{{{\overbrace {\alpha \,\,\,\,\,\,\,...\alpha }^{j + 1\,{\text{times}}}},\beta }} \,u\,(0)D_{M,x}^{{{\overbrace {\alpha \,\,\,\,\,\,\,...\alpha }^{k - j\,{\text{times}}}},\beta }} G} (0) = \sum\limits_{j = 0}^{k} {\,\left( {_{j}^{k} } \right)D_{M,x}^{{{\overbrace {\alpha \,\,\,\,\,\,\,...\alpha }^{j + 1\,{\text{times}}}},\beta }} G(0)\,\,D_{M,x}^{{{\overbrace {\alpha \,\,\,\,\,\,\,...\alpha }^{k - j\,{\text{times}}}},\beta }} u(0)} ,\,$$
(51)

or

$$n\sum\limits_{j = 0}^{k} {\left( {_{j}^{k} } \right)\frac{{(j + 1)!\alpha^{j + 1} }}{{\Gamma (\beta + 1)^{j + 1} }}A_{j + 1} \frac{{(k - j)!\alpha^{k - j} }}{{\Gamma (\beta + 1)^{k - j} }}Q_{k - j} } = \sum\limits_{j = 0}^{k} {\left( {_{j}^{k} } \right)\frac{{(j + 1)!\alpha^{j + 1} }}{{\Gamma (\beta + 1)^{j + 1} }}Q_{j + 1} \frac{{(k - j)!\alpha^{k - j} }}{{\Gamma (\beta + 1)^{k - j} }}A_{k - j} } ,$$
(52)
$$n\sum\limits_{j = 0}^{k} {\,\left( {_{j}^{k} } \right)(j + 1)!(k - j)!A_{j + 1} Q_{k - j} } = \sum\limits_{j = 0}^{k} {\,\left( {_{j}^{k} } \right)(j + 1)!(k - j)Q_{j + 1} A_{k - j} } ,$$
(53)

so, we get the following equations

$$n\sum\limits_{j = 0}^{k} {\,\frac{k!(j + 1)!(k - j)!}{{j!(k - j)!}}A_{j + 1} Q_{k - j} } = \sum\limits_{j = 0}^{k} {\frac{k!(j + 1)!(k - j)!}{{j!(k - j)!}}A_{k - j} Q_{j + 1} } ,$$
(54)
$$n\sum\limits_{j = 0}^{k} {\,k!(j + 1)A_{j + 1} Q_{k - j} } = \sum\limits_{j = 0}^{k} {k!(j + 1)A_{k - j} Q_{j + 1} } ,$$
(55)
$$n\sum\limits_{j = 0}^{k} {\,k!(j + 1)A_{j + 1} Q_{k - j} } = \sum\limits_{j = 0}^{k - 1} {k!(j + 1)A_{k - j} Q_{j + 1} } + k!(k + 1)A_{0} Q_{k + 1} ,$$
(56)

and

$$k!(k + 1)A_{0} Q_{k + 1} = n\sum\limits_{j = 0}^{k} {\,k!(j + 1)A_{j + 1} Q_{k - j} } - \sum\limits_{j = 0}^{k - 1} {k!(j + 1)A_{k - j} Q_{j + 1} } .$$
(57)

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

$$k!(k + 1)A_{0} Q_{k + 1} = n\sum\limits_{i = 1}^{k + 1} {\,k!(i)A_{i} Q_{k + 1 - i} } - \sum\limits_{i = 1}^{k} {k!(k + 1 - i)A_{i} Q_{k + 1 - i} } ,$$
(58)

if \(m = k + 1\), then

$$m!A_{0} Q_{m} = n\sum\limits_{i = 1}^{m} {\,(m - 1)!(i)A_{i} Q_{m - i} } - \sum\limits_{i = 1}^{m - 1} {(m - 1)!(m - i)A_{i} Q_{m - i} } .$$
(59)

By adding the zero value \(\,\left\{ { - (m - 1)!(m - m)A_{m} Q_{0} } \right\}\) to the second summation, we get

$$m!A_{0} Q_{m} = n\sum\limits_{i = 1}^{m} {\,(m - 1)!(i)A_{i} Q_{m - i} } - \sum\limits_{i = 1}^{m - 1} {(m - 1)!(m - i)A_{i} Q_{m - i} } - (m - 1)!(m - m)A_{m} Q_{0} ,$$
$$m!A_{0} Q_{m} = n\sum\limits_{i = 1}^{m} {\,(m - 1)!(i)A_{i} Q_{m - i} } - \sum\limits_{i = 1}^{m} {(m - 1)!(m - i)A_{i} Q_{m - i} } ,$$
(60)

then the coefficients \(Q_{m}\) could be written as

$$Q_{m} = \frac{1}{{m\,A_{0} }}\sum\limits_{i = 1}^{m} {(in - m + i)A_{i} Q_{m - i} } ,\,\,\,\forall \,\,m \ge 1,\,\,\,\,\,$$
(61)

where

\(A_{0} = 1,\,\,\,A_{1} = 0,\,\,\,\,\,\,\,Q_{0} = A_{0}^{n} = 1,\,\,\,Q_{1} = \frac{n}{{A_{0} }}\,A_{1} Q_{0} = 0.\)

Using \(u = 1 + \sum\limits_{m = 2}^{\infty } {A_{m} X^{m} }\) we obtain

$$D_{M,x}^{\alpha ,\beta } u = \,\sum\limits_{m = 2}^{\infty } {\frac{{\alpha mA_{m} }}{\Gamma (\beta + 1)}X^{m - 1} x^{\alpha - \alpha } } = \sum\limits_{m = 2}^{\infty } {\frac{{\alpha mA_{m} }}{\Gamma (\beta + 1)}X^{m - 1} } .$$
(62)

The second derivative of the Emden function \(u\) could be given by

$$D_{M,x}^{\alpha ,\beta } D_{M,x}^{\alpha ,\beta } u = \sum\limits_{m = 2}^{\infty } {\frac{{\alpha^{2} m(m - 1)A_{m} }}{{\Gamma (\beta + 1)^{2} }}X^{m - 2} } x^{\alpha - \alpha } = \sum\limits_{m = 2}^{\infty } {\frac{{\alpha^{2} m(m - 1)A_{m} }}{{\Gamma (\beta + 1)^{2} }}X^{m - 2} } .$$
(63)

Substituting Eqs. (62) and (63) in Eq. (35) we have

$$\begin{gathered} x^{ - 2\alpha } D_{M,x}^{\alpha ,\beta } \left( {x^{2\alpha } D_{M,x}^{\alpha ,\beta } } \right)u + u^{n} = 0,\, \Rightarrow x^{ - 2\alpha } \left[ {x^{2\alpha } D_{M,x}^{\alpha \alpha ,\beta } u + \frac{{2\alpha x^{\alpha } }}{\Gamma (\beta + 1)}D_{M,x}^{\alpha ,\beta } u} \right] + u^{n} = 0, \hfill \\ D_{M,x}^{\alpha \alpha ,\beta } u + \frac{{2\alpha x^{ - \alpha } }}{\Gamma (\beta + 1)}D_{M,x}^{\alpha ,\beta } u + u^{n} = 0, \hfill \\ x^{2\alpha } \sum\limits_{m = 2}^{\infty } {\frac{{\alpha^{2} m(m - 1)A_{m} }}{{\Gamma (\beta + 1)^{2} }}X^{m - 2} } + \frac{{2\alpha x^{\alpha } }}{\Gamma (\beta + 1)}\sum\limits_{m = 2}^{\infty } {\frac{{\alpha mA_{m} }}{\Gamma (\beta + 1)}X^{m - 1} } + x^{2\alpha } \left[ {1 + \sum\limits_{m = 2}^{\infty } {Q_{m} X^{m} } } \right] = 0, \hfill \\ \end{gathered}$$
(64)

or

$$\begin{gathered} \sum\limits_{m = 2}^{\infty } {\frac{{\alpha^{2} A_{m} m(m - 1)}}{{\Gamma (\beta + 1)^{2} }}X^{m} } + \sum\limits_{m = 2}^{\infty } {\frac{{2\alpha^{2} A_{m} \,m}}{{\Gamma (\beta + 1)^{2} }}X^{m} } + X^{2} + \sum\limits_{m = 2}^{\infty } {Q_{m} X^{m + 2} } = 0, \hfill \\ \sum\limits_{m = 2}^{\infty } {\alpha^{2} \frac{{A_{m} m(m - 1 + 2)}}{{\Gamma (\beta + 1)^{2} }}X^{m} } + X^{2} + \sum\limits_{m = 2}^{\infty } {Q_{m} X^{m + 2} } = 0, \hfill \\ \sum\limits_{m = 2}^{\infty } {\frac{{\alpha^{2} A_{m} m(m + 1)}}{{\Gamma (\beta + 1)^{2} }}X^{m} } + X^{2} + \sum\limits_{m = 2}^{\infty } {Q_{m} X^{m + 2} } = 0. \hfill \\ \end{gathered}$$
(65)

By putting \(m = k + 2\) in the first part and \(m = k\) in the third part of Eq. (63), we get

$$\sum\limits_{k = 0}^{\infty } {\frac{{\alpha^{2} A_{k + 2} (k + 2)(k + 3)}}{{\Gamma (\beta + 1)^{2} }}X^{k + 2} } + X^{2} + \sum\limits_{k = 2}^{\infty } {Q_{k} X^{k + 2} } = 0.$$

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

$$\frac{{\alpha^{2} (k + 2)(k + 3)}}{{\Gamma (\beta + 1)^{2} }}A_{k + 2} + Q_{k} = 0,\,\,\,\,\,\,\,\forall \,\,k \ge 2.$$

The coefficients of the series expansion could be obtained from

$$A_{k + 2} = - \frac{{Q_{k} \Gamma (\beta + 1)^{2} }}{{\alpha^{2} (k + 2)(k + 3)}}{, }\forall \,\,k \ge 2,$$
(66)

and

$$Q_{m} = \frac{1}{{mA_{0} }}\sum\limits_{i = 1}^{m} {\,(in - m + i)A_{i} Q_{m - i} } ,\,\,\,\forall \,\,m \ge 1.$$
(67)

If we put \(\alpha = 1\) in Eqs. (66) and (67), we get the series coefficients of the integer LEE. If we insert \(k = 0,{ 1, 2, 3}\) in Eqs. (66) and (67), we get

$$A_{0} = 1,\,\,\,A_{1} = 0,\,\,\,\,\,\,\,Q_{0} = 1,\,\,\,Q_{1} = 0, \, A_{2} = - \frac{{\Gamma (\beta + 1)^{2} }}{{6\,\alpha^{2} }}, \, A_{3} = 0, \, A_{4} = \frac{{n\Gamma (\beta + 1)^{4} \,}}{{120\,\alpha^{4} }}, \, A_{5} = 0.$$

Then, the series solution at \(\beta = 1\) is reduced to

$$u_{n} (x) = 1 - \frac{1}{{6\alpha^{2} }}x + \frac{n}{{120\alpha^{4} }}x^{2} - ............$$
(68)

Then, the series solution at \(\alpha = \beta = 1\) is reduced to the integer version of LEE as

$$u_{n} (x) = 1 - \frac{1}{6}x + \frac{n}{120}x^{2} - ............$$
(69)

Physical characteristics of the TFM polytrope

The mass contained in a radius \(r\) is given by

$$M(r^{\alpha } ) = \int_{0}^{r} {\frac{{4\Gamma (\beta + 1)^{2} \,}}{{\alpha^{2} }}\pi \,r^{2\alpha } \rho \,dr^{\alpha } } .$$
(70)

Inserting Eqs. (27) and (28) for \(\rho\) and \(r^{\alpha }\) we found

$$M(x^{\alpha } ) = \frac{{4\Gamma (\beta + 1)^{2} \pi \,a^{3} \rho_{c} }}{{\alpha^{2} }}\int_{0}^{x} {x^{2\alpha } u^{n} dx^{\alpha } } ,$$
(71)

by substituting Eq. (34) for the Emden function \(u^{n}\) we get

$$\begin{aligned} M(x^{\alpha } ) =& \,\frac{{4\Gamma (\beta + 1)^{2} \pi \,a^{3} \rho_{c} \,}}{{\alpha^{2} }}\int_{0}^{x} {x^{2\alpha } \left[ { - \frac{1}{{x^{2\alpha } }}\frac{{d^{\alpha } \,}}{{d\,x^{\alpha } }}\left( {x^{2\alpha } \,D_{M,x}^{\alpha ,\beta } u} \right)} \right]\,dx^{\alpha } } , \\ =&\, \frac{{4\Gamma (\beta + 1)^{2} \pi \,a^{3} \rho_{c} \,}}{{\alpha^{2} }}\int_{0}^{x} {\left[ { - \frac{{d^{\alpha } \,}}{{d\,x^{\alpha } }}\left( {x^{2\alpha } \,D_{M,x}^{\alpha ,\beta } u} \right)} \right]\,dx^{\alpha } } , \\ =& \,\frac{{4\Gamma (\beta + 1)^{2} \pi \,a^{3} \rho_{c} \,}}{{\alpha^{2} }}\left[ { - \left( {x^{2\alpha } \,D_{M,x}^{\alpha ,\beta } u} \right)} \right], \end{aligned}$$
(72)

with \(a\) could be followed by Eq. (33), then the mass is given by

$$M(x^{\alpha } ) = \frac{{4\Gamma (\beta + 1)^{2} \pi \,}}{{\alpha^{2} }}\left[ {\frac{K(n + 1)}{{4\pi G}}} \right]^{\frac{3}{2}} \,\,\rho_{c}^{{\frac{3 - n}{{2n}}}} \left[ { - \left( {x^{2\alpha } D_{M,x}^{\alpha ,\beta } u} \right)} \right]_{{x = x_{1} }} .$$
(73)

The radius of the polytrope is given by

$$R^{\alpha } = \frac{a\alpha }{{\Gamma (\beta + 1)}}\,x_{1}^{\alpha } ,$$
(74)

where \(x_{1}^{\alpha }\) is the first zero of the Lane–Emden function. Inserting \(a\) in Eq. (74) we get

$$R^{\alpha } = \frac{\alpha \,}{{\Gamma (\beta + 1)}}\left[ {\frac{K(n + 1)}{{4\pi G}}} \right]^{\frac{1}{2}} \,\,\rho_{c}^{{\frac{1 - n}{{2n}}}} \,x_{1}^{\alpha } .$$
(75)

The pressure and temperature of the polytrope are given by

$$P = P_{c} \, u^{n + 1} ,$$
(76)

and

$$T = T_{c} \, u^{n} .$$
(77)

The central density is computed from the Eq. (3)

$$\frac{{\rho_{c} }}{{\overline{\rho }}} = { - }\frac{{x_{1}^{2\alpha } }}{{3\left[ {\left( {D_{M,x}^{\alpha ,\beta } u} \right)} \right]}}_{{x = x_{1} }} .$$
(78)

where the mean density \(\overline{\rho }\) is given in terms of the mass M and radius R of the sphere as

$$\overline{\rho } = \frac{3M}{{4\pi R^{3} }},$$
(79)

then, the central density could be computed from the formula

$$\rho_{c} = { - }\frac{{x^{2\alpha } M}}{{4\,\pi \,R_{{}}^{3} \left[ {\left( {\,D_{M,x}^{\alpha ,\beta } u} \right)} \right]}}_{{x = x_{1} }} .$$
(80)

Results

The Lane–Emden equation is intended to be solved using the Mathematica code shown for a range of polytropic indices, 0 ≤ n ≤ 5. The structure of stars, namely the distributions of density, temperature, and pressure, is modeled using the Lane–Emden equation. There are two main sections of the code: The code’s initial section is dedicated to figuring out the radius of convergence of the series. This is significant because the radius of convergence enables us to determine the range in which the series expansion is valid, and the series solution to the Lane–Emden equation only converges within that range. Important polytropic physical characteristics, including mass, radius, temperature, density, and pressure, are computed in the second section of the algorithm. These characteristics are crucial for characterizing a star’s structure and are obtained using the Lane–Emden equation. In particular, the code calculates these parameters using Eqs. (75) and (77)–(81). Usually, the ideal gas law, hydrostatic equilibrium, and the polytropic connections between density and pressure serve as the foundation for these equations.

The distance from the polytrope’s center at which the series solution starts to diverge and necessitates the radius of convergence indicates the use of specialist methods for further analysis. To address this divergence, the code is initially run using starting values α = β = 1, representing the integer version of the LE equation. Two acceleration techniques, the Euler-Abel transformation and the Pade′ approximation35, are used for values of n > 1.9 (we discussed in Appendix A these two acceleration techniques). The Euler-Abel technique speeds up convergence by averaging the series’ higher-order elements, while the Pade′ approximation36,37 enhances convergence by describing the series as a ratio of polynomials. Even when the original series fails beyond a certain radius, these methods extend the solution to the polytrope’s surface, where the density becomes zero. Figure 1 displays the Emden function (u) for the polytropic index n = 2.5 and the fractional parameters α = β = 0.95; as is clear, the original series diverges (the red line) before reaching the surface of the polytrope (at about x = 2.9), the surface of the polytrope (or the first zero of the Emden function) occurs where u (x = x1) = 0, x1 = 5.667 for the present model. As the blue line shows, the accelerated series converges well to the desired value.

Fig. 1
figure 1

The Emden function of the TMD polytrope with n = 2.5, α = β = 0.95. The red line represents the function before series acceleration, while the blue line represents the function after series acceleration.

Full size image

The physical properties of the gas sphere (i.e., density, pressure temperature, and mass) are calculated for different values of the polytropic index n after the series has been accelerated to the polytrope’s surface. The radius (r) must be established first (using Eq. (28)), representing the distance at which the density falls to zero and denotes the polytrope’s border. The polytrope’s mass, temperature, and density distributions are calculated once the radius has been established. These values are obtained using the solutions of the Lane–Emden equation at various locations inside the polytrope and integrating the density across the volume. We computed models with the polytropic indexes n = 1 and n = 3; the fractional parameters take the values (α = β = 1), (α = 1, β = 0.95), (α = β = 0.95), (α = β = 0.9), and (α = β = 0.8).

In Table 1, we listed the zero of the Emden function (x1) and the ratio of the central density to the mean density (\(\rho_{c} /\overline{\rho }\)) of the TMD polytropes with indexes n = 1 and n = 3. The radius at which the density of the polytrope becomes zero is represented by the value of x1, which is the first zero of the Lane–Emden function. This is known and fixed for a particular polytropic index for classical polytropes. In M-fractional polytropes, ξ1 rises as the fractional orders α and β fall. This indicates that the system is being stretched by the fractional calculus, which enables it to attain a zero-density state at a greater radius. This may be a physical scenario in which the gas sphere expands more than it would in the classical scenario due to pressure support or gravitational attraction. The density of mass in the polytrope’s core relative to the rest of the system is shown by the ratio \(\rho_{c} /\overline{\rho }\). The polytropic index n determines this ratio for classical polytropes; greater values of n result in more peaked density profiles (higher \(\rho_{c} /\overline{\rho }\)). In the M-fractional framework, \(\rho_{c} /\overline{\rho }\) greatly rises when α and β decrease, particularly for larger values of n. This implies that the polytrope may concentrate its mass closer to the center thanks to fractional calculus. There is a significant density drop-off in the outer layers as the system gets more centrally concentrated. For instance, the ratio \(\rho_{c} /\overline{\rho }\) = 4.035 for n = 1 with α = 0.9 indicates a moderate concentration. The ratio increases to 78.657 for n = 3 with α = 0.9, suggesting an exceptionally dense core concerning the mean density.

Table 1 The zeroth of the Emden function (ξ1) and the ratio of the central density to the mean density (\(\rho_{c} /\overline{\rho }\)) of the TMD polytropes with polytopic indexes n = 1 and n = 3.
Full size table

Figures 2 and 3 show the behavior of the Emden function (u) and its first derivative (u′) for a polytropic index n = 1 (the upper panel) and n = 3 (the lower panel). The behavior of u varies with α and β, influencing how severely the density decreases as we travel radially outward. The curves show that the rate of change of density is more important in the outer layers, as demonstrated by the separation of the curves at increasing x. The effects are more noticeable at bigger x for n = 3, owing to the longer tail of the density distribution at higher polytropes. The impact of the two fractional parameters α and β on u′ will, in turn, affect the mass of the polytrope compared to the traditional Emden solution (which would have α = β = 1). The profile varies somewhat when α and β diverge from 1 (e.g., α = 0.9 or β = 0.95), indicating that these factors add a fractional adjustment to the polytropic structure. The effects of α and β on the first derivative u′ are most noticeable in the polytrope’s outer regions, where the curves divide. For n = 3, u′ is more complicated than n = 1, with more dramatic variances depending on fractional parameters.

Fig. 2
figure 2

The Emden function and its first derivative for the M-truncated fractional polytrope with n = 1. The calculation goes for different values of the fractional parameters.

Full size image
Fig. 3
figure 3

The Emden function and its first derivative for the M-truncated fractional polytrope with n = 3. The calculation goes for different values of the fractional parameters.

Full size image

The sun is commonly modeled using classical polytropic models with a polytropic index n = 3 to n = 438,39, which corresponds to stars supported primarily by radiation pressure, Fig. 4 compares various values of the fractional parameters α and β and shows the mass–radius relation of fractional polytropes for polytropic indices n = 3 and n = 3.25; through the calculations, the central density is fixed at ρc = 151 g cm−3. It is evident that from the figure, more mass is distributed to the outer layers as the mass accumulates more slowly at a smaller radius when both α and β decline from 1 to 0.8 (α = 0.95, β = 0.95; α = 0.9, β = 0.9; α = 0.8, β = 0.8). This points to a more gradual mass accumulation of the star toward the outer regions and a softer equation of state. The traditional polytropic mass–radius relation of the sun (n = 3, α = 1, β = 1) is contrasted with that of fractional polytropes, in which the mass accumulation inside the star is modified by fractional parameters α and β. A wider variety of stellar structures is possible than with the sun’s classical model thanks to the addition of fractional impacts, which describe stars with more gradual mass distributions and less clear surface boundaries.

Fig. 4
figure 4

The mass function for the polytropic model of the sun. The left panel is for the polytrope with n = 3, while the right panel is for the polytrope with n = 3.25.

Full size image

Using Eqs. (66), (68), and (73), we provide in Tables 3 and 4 the physical parameters ρc, M, and R for TMD polytropes with polytropic indices n = 3 and n = 3.25 across various values of fractional parameters α and β. MESA code (Modules for Experiments in Stellar Astrophysics) conducts intricate numerical models of stellar evolution that include increasingly sophisticated processes, including energy transmission, nuclear reactions, and opacity40,41,42,43.

The values of the masses in Tables 2 and 3 for n = 3 and n = 3.25 are approximately varying between 0.8499 and 1.084. MESA simulations of the sun provide a very accurate estimate of M = 1.00 Mʘ, indicating that polytropic approximations with fractional orders are somewhat close in some places, however not as exact as MESA. The radius values in the tables are slightly less for n = 3 (range from 0.738 (α = β = 0.8) to 1.0418 (α = 1, β = 0.95)) and substantially greater for n = 3.25 (from 0.7381 (α = β = 0.8) to 1.0378 (α = 1, β = 0.95)). This indicates that the TMD polytropic models with fractional parameters may marginally underestimate or overestimate the sun’s radius, contingent upon the values of α and β. For n = 3 and n = 3.25, smaller fractional parameters α = 0.8 and β = 0.8 often provide a smaller radius.

Table 2 Physical parameters for the polytropes with n = 3.
Full size table
Table 3 Physical parameters for the polytropes with n = 3.25.
Full size table

The central density in the table rises with lowering α and β, which is predicted as the fractional parameters affecting the polytropic pressure support. For example, for n = 3, ρc grows from 73.171 g cm−3 at α = β = 1(is much smaller than the MESA value ρc = 151 g cm−3) to 170.187 g cm−3 at α = 0.8, β = 0.8 (is closer to MESA). MESA’s comprehensive solar models estimate a core density for the sun of roughly 150 g cm−3. The polytropic models with n = 3.25 tend to generate greater values for central density, mass, and radius compared to the n = 3 models. This is consistent with the assumption that the star becomes more centrally concentrated as the polytropic index grows, with greater core densities and larger radii. The model for n = 3 (with α = 0.8, β = 0.8) provides a central density ρc = 170.187 g cm−3, which is the closest value to MESA’s estimate of roughly 150 g cm−3. The similar model for n = 3.25 also provides a mass M = 0.8499 Mʘ, and a radius R = 0.7386 Rʘ, both exceptionally far from the MESA solar estimates, but the model with α = 0.95, β = 0.95 produces ρc = 156.803 g cm−3, nearly equal to the MESA value for the sun’s core density. This model also provides M = 0.9901 Mʘ, which is close to MESA’s estimate, and a radius R = 0.9417 Rʘ, likewise reasonably near to the solar radius.

White dwarfs are often represented as polytropes with an index of n = 3 (in more massive white dwarfs) since they are mainly sustained against gravitational collapse by electron degeneracy pressure. When electrons are non-relativistic at low masses, the white dwarf functions as a polytrope with n = 3/2. The Lane–Emden equation facilitates the calculation of the critical mass for white dwarfs (around 1.44 Mʘ), referred to as the Chandrasekhar limit44. Beyond this mass, electron degeneracy pressure can no longer maintain the star, leading to collapse and the production of a neutron star or black hole.

We computed TMD polytropic models for white dwarfs with n = 3/2. The fractional parameters are (α = 1, β = 1), (α = 1, β = 0.95), (α = 0.95, β = 1), (α = 0.95, β = 0.95), (α = 0.9, β = 0.9), (α = 0.85, β = 0.85), and (α = 0.8, β = 0.8). Assuming a white dwarf with a mass of Mwd = 1.4 Mʘ and radius of Rwd = 0.001 Rʘ, we get the results in Table 4; column 2 is the first root of the Emden function (x1), column 3 is the central density (ρc), column 4 is the mean density \((\overline{\rho})\), column 5 is the radius of the star R*/Rwd, and column 6 is the mass of the star M*/Mwd (R* and M* are the radius and mass of the star computed from the polytropic model).

Table 4 Structure of white dwarf star computed under a TMD polytrope with n = 1.5.
Full size table

In Table 4, for the three fractional parameters (α = 1, β = 0.95), (α = β = 0.95), and (α = β = 0.9), both the mass and the radius drop when we lower α and β. This suggests that the fractional parameters produce a white dwarf with a smaller radius and less mass, as α and β fall, the central density ρc and mean density  \((\overline{\rho})\) rise, suggesting that the white dwarf becomes denser generally. This is consistent with the hypothesis that more compact stars may result from fractional polytropic parameters; for example, Luchko and Gorenflo26 for the modified Riemann Liouville derivatives fractional polytrope and45 for the conformable fractional polytrope. In the other three cases in the table; (α = 0.95, β = 1), (α = β = 0.85), and (α = β = 0.9), the central density increases, but the ratio M/R decreases. This suggests that the relationship between central density and compactness (M/R) isn’t always direct. It depends on how both α and β are adjusted. When both are lowered together (like 0.95,0.95; 0.9,0.9), M/R remains near 1 or slightly below, even as ρc increases. But when only one parameter is changed (like 0.95,1), ρc increases but M/R decreases. Possible reasons could involve how α and β affect the equation of state. If α and β control different aspects of the pressure-density relation, changing them asymmetrically might lead to structural changes where the radius increases more than mass when only one parameter is adjusted, thus lowering M/R even with higher central density. When both are adjusted, maybe the mass and radius scale in a way that keeps M/R roughly stable or slightly decreasing.

With several curves representing various combinations of α and β, Fig. 5 displays the mass–radius relation for white dwarf stars. The integer version of the polytropic model of the white dwarf is represented by the curve with α = β = 1. The white dwarf has the reference mass and radius in this instance, as the mass-to-radius ratios are equal to 1. As α and β fall, the curves move to the left and downward. This supports the pattern in Table 4 by demonstrating that both mass and radius drop for some fractional values, which may indicate more compact and dense white dwarfs. Some reported white dwarfs deviate from the traditional mass–radius relation, appearing either too small for their mass or very compact; an example of these extreme white dwarfs is the star RE J0317-853 with 1.35 M⊙ and 0.0034 R⊙46.

Fig. 5
figure 5

The mass function for white dwarfs with polytropic index n = 1.5. The central density and other physical parameters of each model are listed in Table 3.

Full size image

According to the equation of state of non-relativistic degenerate matter, a polytrope with index n = 1.5 is a reasonable model for low-mass white dwarfs40,41,42,43. Table 5 lists the mass and radius of 24 low-mass white dwarfs computed based on a detailed spectroscopic and photometric analysis of 219 DA and DB white dwarfs for which trigonometric parallax measurements are available47. To compare the observed mass and radius of white dwarf stars with the mass–radius relation calculated from fractional turncated M-derivatives polytropes with index n = 1.5, we computed models with mean densities \((\overline{\rho})\) = 105–106 g cm−3, and α = β = 1, α = β = 0.95, α = β = 0.8. Figure 6 compares the empirical mass–radius relation (M–R) of white dwarfs with the theoretical forecasts derived from fractional truncated M-derivative polytropes, using a polytropic index of n = 1.5. The theoretical curves represent three distinct sets of fractional parameters: the traditional case (α = 1, β = 1) and three fractional models (α = β = 0.95, α = β = 0.8, α = 0.9, β = 1.1). These models seek to investigate the impact of fractional calculus alterations on the structure of low-mass white dwarfs. The integer polytropic model (α = 1, β = 1) strongly corresponds with the observational data, especially within the intermediate mass range (0.4–0.6 M⊙). Nevertheless, discrepancies emerge for lower and higher masses, indicating that further physical factors may need to be included for a more accurate match. The variability in the data points may be ascribed to several astrophysical phenomena. White dwarfs may have various compositions, predominantly carbon–oxygen (C/O) or helium-dominated cores, which alter their mass–radius relation. Additionally, rotation and magnetic fields may impact the equilibrium structure, resulting in departures from the usual theoretical curves. The influence of the fractional parameters α and β is obvious in the shift of the M–R curves. As α and β drop, the radius at a given mass rises, reflecting a shift in the pressure balance maintaining the star against gravitational collapse. This shows that fractional calculus gives a new degree of freedom in modeling white dwarfs, perhaps accounting for physical factors beyond integer polytropes.

Table 5 Mass and radius of 24 low-mass white dwarfs48.
Full size table
Fig. 6
figure 6

Comparison between the observed mass–radius relation of 20 white dwarfs48 with the theoretical predictions obtained from fractional truncated M-derivative polytropes with a polytropic index n = 1.5.

Full size image

Conclusion

In the present paper, we formulated the Lane–Emden equation of the polytropic gas sphere in the frame of the truncated M-fractional derivatives. We solved the LETMD using a power series; then, this series is accelerated using Euler-Abel-Pade transformations to reach the surface of the gas sphere. We elaborated a MATHEMATICA code to construct models with the polytropic index n = 0, n = 1, n = 1.5, n = 3, n = 3.25, and n = 4; the fractional parameters (α and β) range from 0.8 to 1. For a given polytropic index, the value of x1 denotes the scaled radial point at which the density first hits zero, indicating the theoretical border of the polytropic sphere. As the fractional parameters α and β fall, x1 rises, the TMD polytropic structure gets “stretched” in the fractional framework. This expansion indicates a more expansive outer zone, where density approaches 0 at a larger scaled radial distance than in the classical model.

Notwithstanding the rise in x1 when α and β diminish, the physical radius R of the polytrope (such as the sun or a white dwarf) decreases. This decrease arises from the influence of fractional factors on the radius scaling factor, which includes α and β, hence impacts the actual physical dimensions of the model. The reduction in R signifies that while the theoretical limit of the density profile (represented by x1 extends outward in dimensionless terms, the tangible physical structure is more condensed. Fractional derivatives result in a greater mass concentration in the center, diminishing the physical radius while elongating the mathematical radius in its dimensionless representation. This result underscores the capacity of TMD polytropes to accommodate intricate density distributions, such as those seen in compact stellar entities, by including a degree of flexibility in mass distribution that traditional polytropic models lack.