The intrinsic beauty of polytropic spheres in reduced variables

The concept of reduced variables is revisited with regard to van der Waals' theory and an application is made to polytropic spheres, where the reduced radial coordinate is ${\rm red}(r)=r/R=\xi/\Xi$, $R$ radius, and the reduced density is ${\rm red}(\rho)=\rho/\lambda=\theta^n$, $\lambda$ central density. Reduced density profiles are plotted for several polytropic indexes within the range, $0\le n\le5$, disclosing two noticeable features. First, any point of coordinates, $({\rm red}(r),{\rm red}(\rho))$, $0\le{\rm red}(r)\le1$, $0\le{\rm red}(\rho)\le1$, belongs to a reduced density profile of the kind considered. Second, sufficiently steep i.e. large $n$ reduced density profiles exhibit an oblique inflection point, where the threshold is found to be located at $n=n_{\rm th}=0.888715$. Reduced pressure profiles, ${\rm red}(P)=P/\varpi=\theta^{n+1}$, $\varpi$ central pressure, Lane-Emden fucntions, $\theta=(\rho/\lambda)^{1/n}$, and polytropic curves, ${\rm red}(P)={\rm red}(P)({\rm red}(\rho))$, are also plotted. The method can be extended to nonspherical polytropes with regard to a selected direction, ${\rm red}(r)(\mu)=r(\mu)/R(\mu)=\xi(\mu)/\Xi(\mu)$. The results can be extended to polytropic spheres made of collisionless particles, for polytropic index within a more restricted range, $1/2\le n\le5$.


Introduction
Models in reduced variables are useful tools for the description of the physical world, in that a single formulation relates to a whole set of configurations. For instance, let ρ(r), 0 ≤ r ≤ R, be the density profile of an assigned mass distribution along a selected direction, and let (r, ρ) = (r s , ρ s ) be fixed nonzero scaling values. Let reduced (or scaled) variables be defined as ( / r, / ρ) = (r/r s , ρ/ρ s ). Accordingly, the reduced density profile reads / ρ( / r), 0 ≤ / r ≤ / R, which includes an infinity of density profiles, r → k r r, k r > 0, ρ → k ρ ρ, k ρ > 0, in addition to the one under consideration.
A classical example of reduced variables can be found in van der Waals' theory of real gases in connection with the critical point, lying on the critical isothermal curve. The coordinates of the critical point on the Clapeyron plane are P c ≡ (V c , p c , T c ), where V c is the largest volume along the critical isothermal curve still allowing a liquid phase, p c is the lowest pressure along the critical isothermal curve still allowing a liquid phase, and T c is the temperature along the critical isothermal curve i.e. the largest temperature still allowing a liquid phase.
Though isothermal curves on the Clapeyron plane, (OV p), are different for different gases, the contrary holds on the reduced Clapeyron plane, (O / V / p), where reduced isothermal curves coincide for all gases as / V = V /V c , / p = p/p c , / T = T /T c , with extension to ideal gases. In any case, the equation of state reads p = p(V, T ).
A special case of astrophysical interest, p = p(V ) or p = p(ρ), relates to polytropic spheres or, in general, polytropes (e.g., Jeans 1929, Chap. IX, § §235-239; Chandrasekhar 1939;Horedt 2004), which are self-gravitating systems in hydrostatic equilibrium. Related scaling radius and scaling density are usually denoted as r s = α and ρ s = λ, respectively, where α depends on the central pressure, the central density, the density profile, and λ is the central density.
In a widely investigated class of polytropic spheres, the reduced radial coordinate is denoted as ξ = r/α and the reduced density as θ n = ρ/λ, where n is the polytropic index. The cases of astrophysical interest, 0 ≤ n ≤ 5, range from null (n = 0) to infinite (n = 5) degree of concentration or, in other words, from homogeneous to Roche (e.g., Jeans 1929, Chap. IX, § §229-232) or Plummer (Plummer 1911) models, according if the central density is divergent or finite, respectively. The reduced radius, Ξ = R/α, is a monotonically increasing function of the polytropic index, n, and Ξ → +∞ as n → 5 (Chandrasekhar 1939, Chap. IV, §4; Horedt 2004, Chap. 2, §2.5). Accordingly, the reduced density cannot be represented in a finite region of the reduced (Oξθ n ) plane for 0 ≤ n ≤ 5.
To this aim, a different choice of reduced radial coordinates has to be performed, / r = r/R = ξ/Ξ, while the reduced density is left as / ρ = ρ/λ = θ n . Under the restriction of null density on the boundary, θ n = 0, the whole set of reduced density profiles on the (O / r / ρ) plane lies within a square of unit sides parallel to the coordinate axes, with a vertex on the origin. A picture of the kind considered could be useful in disclosing additional features of reduced density profiles related to polytropic spheres. To this subject, the current investigation is devoted.
The paper is organized as follows. Reduced variables are introduced in Section 2, where the special case of ideal and real gases is presented as a guidance example. The special case of polytropic spheres is considered in Section 3, where reduced density profiles are plotted on the reduced (O / r / ρ) plane for several values of the polytropic index, 0 ≤ n ≤ 5, and the occurrence of an oblique inflection point is studied in detail. The reduced pressure profiles, the Lane-Emden functions, and the polytropic curves, are similarly considered therein. The discussion and the concluding remarks are drawn in Section 4.

Reduced variables
Let the equilibrium configuration of a thermodynamical system be defined by a set of physical parameters, P 1 , P 2 , ..., P N . Let P s,1 , P s,2 , ..., P s,N , be selected reference values or scaling parameters with respect to the above mentioned ones. Let the dimensionless parameters, / P 1 , / P 2 , ..., / P N , be defined as reduced or scaled parameters, with respect to the above mentioned ones. A description in terms of reduced parameters includes all configurations where P i → k i P i , k i > 0; in particular, k i = 1, 1 ≤ i ≤ N, relates to the system of interest.
As a guidance example, ideal and real gases shall be taken into consideration. For further details and exhaustive presentation, an interested reader is addressed to articles on the subject (e.g., Caimmi 2010Caimmi , 2012 or specific textbooks (e.g., Rostagni 1957;Landau and Lifchitz 1967).
The equation of state of ideal (e.g., Landau and Lifchitz 1967, Chap. IV, §42) and real (van der Waals 1873) gases, respectively, read: where p is the pressure, V the volume, T the temperature, N the particle number, k the Boltzmann constant, A and B constants which depend on the nature of the particles. In particular, B can be conceived as the volume filled by a particle of real gas and the product, NB, as the volume filled by all particles, or covolume (e.g., Landau and Lifchitz 1967, Chap. IV, §74). Van der Waals' equation of state, Eq. (2), can be rewritten as p = p(V, T ), and the partial derivatives, (∂p/∂V ) V,T , (∂ 2 p/∂V 2 ) V,T , can explicitly be expressed. Van der Waals isothermal curves may exhibit two extremum points (maximum and minimum), allowing a liquid phase, or no extremum point, allowing no liquid phase. The threshold relates to the critical isothermal curve, which shows a single extremum (horizontal inflection) point, where a liquid phase still occurs. The above mentioned inflection point is defined as critical point and related coordinates, P c ≡ (V c , p c , T c ), are defined as critical volume, critical pressure, critical temperature, respectively, where the last relates to the critical isothermal curve.
Owing to the mathematical properties of horizontal inflection points, (∂p/∂V ) Vc,Tc = 0, (∂ 2 p/∂V 2 ) Vc,Tc = 0, which, together with p c = p(V c , T c ), make a system of three equations in the three unknowns, V c , p c , T c . The solution is (e.g., Rostagni 1957 (3) in terms of the covolume, NB, and the constants, A, B, k.

Polytropic spheres 3.1 General considerations
Polytropes are special cases of barotropes i.e. self-gravitating fluids in hydrostatic equilibrium where the equation of state reads p = p(ρ) or, restricting to polytropes (e.g., Jeans 1929, Chap. IX, § §235-239): where K is a constant, n the polytropic index and ρ b the density on the boundary, which is usually taken equal to zero (e.g., Chandrasekhar 1939 where V G is the gravitational potential and V b a normalization constant. From this point on, it shall be assumed ρ b = 0 for simplicity. Let λ be the central density and α the scaling radius, defined as: where ̟ is the central pressure and Kλ 1+1/n is dimensioned as a pressure. With regard to the reduced variables: where Ξ = R/α is the reduced radius, the substitution of Eqs. where the prime denotes derivation with respect to ξ. Density profiles of astrophysical interest (i.e. decreasing with increasing radial coordinate) correspond to the range of polytropic index, 0 ≤ n ≤ 5, where n = 0 relates to homogeneous models and n = 5 to Roche (e.g., Jeans 1929, Chap. IX, § §229-232) or Plummer (Plummer 1911) models, according if the central density, λ, is divergent or finite, respectively. Reduced radii are monotonically increasing from Ξ = √ 6 (n = 0) to Ξ → +∞ (n = 5). The divergence of the reduced radius as n → 5 makes a representation of reduced density profiles on the (Oξ / ρ) plane of little utility, in that interesting features could be lost. As 0 ≤ θ n ≤ 1, a reduced radial coordinate within a similar range would be needed. To this respect, let an additional reduced radial coordinate be defined as: and let the reduced density profile, / ρ = / ρ( / r), be considered and plotted on the (O / r / ρ) plane, which implies the knowledge of θ n (ξ/Ξ), Ξ, for selected values of n.

Source of data
Physical parameters for sequences of rigidly rotating polytropes were determined in an earlier investigation (Caimmi 1983, hereafter quoted as C83), from nonrotating to maximally rotating (i.e. up to centrifugal support on the equatorial plane) configurations. Unfortunately, computations cannot be repeated as the original computer code is still in cards and no conversion into electronic format was tried in the past. For this reason, data used in the current paper are taken as specified below.

•
For integer and half-integer polytropic indexes, with the addition of n = 4.99, seven-digit tables of Lane-Emden functions (Horedt 1986, hereafter quoted as H86) are used.
• For quarter-integer polytropic indexes, with the addition of n = 0.001, 0.010, 0.050, 0.100, 0.200, 0.808, a computer code with starting series solution followed by a fourth-order Runge-Kutta interpolation method, when an assigned tolerance is exceeded, is used.
Accordingly, reduced density profiles can be plotted on the (O / r / ρ) plane.

Results
Plotting in terms of the reduced radius, via Eq. (16) implies the knowledge of the scaled radius, Ξ. Related values from the sources mentioned above are listed in Table 1 for several polytropic indexes, n, with regard to the computer code used in the present paper (pp), results from the parent paper (C83) published later (Caimmi 1985), and results from seven-digit tables of Lane-Emden functions (H86). The computer code does not hold for n = 0 due to the occurrence of undetermined forms of the kind, 0/0 and so on, and for n = 5 due to memory overflow. An inspection of Table 1 discloses that, within θ(Ξ) < 5 · 10 −7 , Ξ agrees with its counterpart from C83 and/or H86 within a few percent for 0 < n < 0.5 and within 10 −4 or less for 0.75 ≤ n ≤ 4.75.
The limiting case, n = 0, is represented by the top and right side of the box in Fig. 2. To this respect, it is worth emphasyzing polytropic spheres with n = 0 are conceptually different from MacLaurin spheres. More specifically, the latter are incompressible while the former are compressible but with infinite pressure inside and null pressure on the boundary conformly to hydrostatic equilibrium. Then the density on the boundary of polytropic Table 1: The scaled radius, Ξ, of polytropic spheres for polytropic index, n, within the range, 0 ≤ n ≤ 5, according to the present paper (pp) and earlier investigations (C83; H86). Cases related to symbols (s) are plotted in Fig. 2 2.500 5.3552750E+00 5.35527546E+00 5.35527546E+00 ⊔ 2.750 6.0355700E+00 6.03557001E+00 3.000 6.8968500E+00 6.89684862E+00 6.89684862E+00 * 3.250 8.0189350E+00 8.01893753E+00 3.500 9.5358000E+00 9.53580534E+00 9.53580534E+00 ⋄ 3.750  Fig. 2, due to Ξ → +∞, which implies finite ξ on the origin and infinite ξ for the remaining of the domain, 0 < ξ ≤ Ξ. In other words, the massive body is "compressed" into the origin while the vanishing atmosphere extends up to ξ/Ξ = 1, similarly to Roche models in the physical space (e.g., Jeans 1929, Chap. IX, § §229-232. Concerning the remaining cases, an inspection of Fig. 2 shows reduced density profiles can be divided into two main classes, namely exhibiting one or no oblique inflection point for n ≥ 1 and n ≤ 0.808, respectively, with the threshold lying in between. The monotonic trend of the reduced density, / ρ = θ n (ξ/Ξ), implies no extremum point and, in turn, oblique inflection points related to d 2 / ρ/ d / r 2 = 0, which via Eq. (14) after little algebra can be expressed as: where the prime denotes derivation with respect to ξ. The substitution of Eq. (15) into (17) yields: which is the condition for the existence of an oblique inflection point in the case under discussion. The lowest n for which Eq. (18) is still satisfied is found numerically using the computer code described above, and the result is n = 0.888715 where Ξ = 3.0459945 and the inflection point occurs at ξ = 2.650000 within a tolerance ǫ = 5 · 10 −7 . Related reduced density profile is not plotted in Fig. 2 to avoid confusion.
Reduced pressure profiles, / p = p(r)/̟ = θ n+1 (ξ), vs. reduced radial coordinates, / r = r/R = ξ/Ξ, for integer and half-integer polytropic index within the range, 0 ≤ n ≤ 5, are plotted in Fig. 3 where symbol captions are as in Fig. 2 and data are from seven-digit tables of Lane-Emden function (H86). The limiting case, n = 5, is represented by the left and bottom side of the box in Fig. 3.
An inspection of Fig. 3 shows reduced pressure profiles can be divided into two main classes, namely exhibiting one or no oblique inflection point for n ≥ 0.5 and n < 0.5, respectively. The monotonic trend of the reduced pressure, / p = θ n+1 (ξ/Ξ), implies no extremum point and, in turn, oblique inflection points related to d 2 / p/ d / r 2 = 0, which via Eq. (14) after little algebra can be expressed as: (n + 1)Ξ 2 θ n−1 [n(θ ′ ) 2 + θθ ′′ ] = 0 ; where the prime denotes derivation with respect to ξ. The substitution of Eq. (15) into (19) yields: which is the condition for the existence of an oblique inflection point in the case under discussion.
The Lane-Emden functions, θ = [ρ(r)/λ] 1/n , vs. reduced radial coordinates, / r = r/R = ξ/Ξ, for integer and half-integer polytropic index within the range, 0 ≤ n ≤ 5, with the addition of n = 4.85, 4.99, are plotted in Fig. 4 where symbol captions are as in Fig. 2 and data are from seven-digit tables of Lane-Emden function (H86) except for the case, n = 4.85, where computations were performed as outlined above. The limiting case, n = 5, is represented by the left and bottom side of the box in Fig. 4.
An inspection of Fig. 4 shows the Lane-Emden function is characterized by the occurrence of an oblique inflection point, from the boundary (n = 0) to the centre (n = 5). In fact, the monotonic trend of the Lane-Emden function, θ(ξ/Ξ), implies no extremum point and, in turn, oblique inflection points related to d 2 θ/ d / r 2 = 0, or Ξ 2 θ ′′ = 0, which via Eq. (15) can be expressed as: that in the special case, n = 0, reduces to Eq. (21). Then a "vertical" inflection point of the Lane-Emden function takes place on the boundary. Accordingly, all Lane-Emden functions, θ = [ρ(r)/λ] 1/n , within the range, 0 ≤ n ≤ 5, exhibit an oblique inflection point. The reduced polytropic curves, / p = / p( / ρ) or θ n+1 vs. θ n , for integer polytropic index within the range, 0 ≤ n ≤ 5, with the addition of n = 0.50, 0.25, 0.10, are plotted in Fig. 5 where symbol captions are as in Fig. 2 and data are from seven-digit tables of Lane-Emden function (H86) except for the cases n = 0.25, 0.10, where computations were performed as outlined above. The centre and the boundary of the sphere correspond to (1, 1) and (0, 0), respectively. The limiting case, n = 0, is represented by the right and bottom side of the box in Fig. 5.

Discussion and conclusion
An application to polytropic spheres has shown the usefulness and the power of models in reduced variables. With regard to the plane, (O / r / ρ), reduced density profiles for polytropic indexes, 0 ≤ n ≤ 5, completely fill a square of unit side, 0 ≤ / r ≤ 1, 1 ≥ / ρ ≥ 0, where the top and the right side relate to homogeneous models (n = 0), while the left and the bottom side relate to Roche and Plummer models (n = 5). The last case can be related to extremely inhomogeneous mass distributions, where the extension is finite and the density is nonzero only on the centre for Roche models (e.g., Jeans 1929, Chap. IX, § §229-232), while the extension is infinite and the density is nonzero provided the distance from the centre remains finite for Plummer models (Plummer 1911).
The reduced slope, via Eqs. (14) and (16) takes the explicit form: where θ ′ remains finite. Accordingly, κ(Ξ) → −∞ for 0 ≤ n < 1, κ(Ξ) = 1/π for n = 1, κ(Ξ) = 0 for 1 < n ≤ 5. The occurrence of an oblique inflection point on reduced density profiles for sufficiently large n, with the threshold at n = n th = 0.888715, suggests a definition of "steep" and "mild" reduced density profiles as related to n ≥ n th and n < n th , respectively, with regard to polytropic spheres. In addition, mild density profiles could be related to isothermal curves of real gases above the critical one, where no inflection point appears, and steep density profiles could be related to isothermal curves of real gases below the critical one, where (two extremum points and then) two inflection points appear, as shown in Fig. 1.
The results of the current paper can be extended to polytropic spheres made of collisionless particles, keeping in mind collisionless polytropes in rigid rotation have an exact collisional counterpart within the range of polytropic index, 1/2 ≤ n ≤ 5 (Vandervoort 1980).
In conclusion, reduced variables appear to be not restricted to the Clapeyron plane as initially conceived (van der Waals 1873), but they can be successfully extended to other physical situations as shown for polytropic spheres, which provides additional credit to van der Waals' original work.   Table 1, where the corresponding symbol is also shown. Full curves (f) relate to n = 0 (top and right side of the box), n = 5 (left and bottom side of the box), and n = 1, for which density profiles can be expressed analytically. The dashed curve (d) relates to n = 0.1. Symbols upside with respect to the full curve (n = 1) correspond to n = 0.25 (asterisks), 0.5 (diamonds), 0.808 (crosses), starting from top right. Symbols downside with respect to the full curve correspond to n = 1.5 (triangles), 2.0 (saltires), 2.5 (squares), 3.0 (asterisks), 3.5 (diamonds), 4.0 (crosses), 4.5 (triangles), 4.99 (saltires), starting from the full curve towards bottom left. Source of data: n = 0.5 and 1 < n < 5 (H86); n = 0.1, 0.25, 0.808, (present paper). See text for further details.  Table 1, where the corresponding symbol is also shown. Full curves relate to n = 0 (top right), n = 5 (left and bottom side of the box), and n = 1, for which pressure profiles can be expressed analytically. Symbols correspond to n = 0.5 (diamonds), n = 1.5 (triangles), 2.0 (saltires), 2.5 (squares), 3.0 (asterisks), 3.5 (diamonds), 4.0 (crosses), 4.5 (triangles), starting from the top right towards bottom left. Data are from seven-digit tables of Lane-Emden functions (H86). See text for further details.  Table 1, where the corresponding symbol is also shown. Full curves relate to n = 0 (top right), n = 5 (left and bottom side of the box), and n = 1, for which the Lane-Emden function can be expressed analytically. Symbols correspond to n = 0.5 (diamonds), n = 1.5 (triangles), 2.0 (saltires), 2.5 (squares), 3.0 (asterisks), 3.5 (diamonds)   Table 1, where the corresponding symbol is also shown. Full curves relate to n = 0 (right and bottom side of the box), n = 5 (top left), and n = 1, for which the Lane-Emden function can be expressed analytically. The dashed curve relates to n = 0.1. Symbols correspond to n = 0.25 (asterisks), 0.5 (diamonds), 2.0 (saltires), 3.0 (asterisks), 4.0 (crosses), starting from bottom right towards top left. Data are from seven-digit tables of Lane-Emden functions (H86) except for n = 0.10, 0.25 (pp). The centre and the boundary of the sphere correspond to (1, 1) and (0, 0), respectively. See text for further details.