Regular Black Holes (RBHs): A Non-Singular Alternative to Classical Black Holes with Structural Validation and Thermodynamic Considerations via Gravitational Thermodynamics Approach

Regular Black Holes (RBHs): A Non-Singular Alternative to Classical Black Holes with Structural Validation and Thermodynamic Considerations via Gravitational Thermodynamics Approach Daisuke SATO1,2* 1*Comprehensive Research Organization for Science and Society, Tsukuba Industry-Academic Collaboration Building, 1601 Kamitakatsu, Tsuchiura City, Ibaraki Prefecture, JAPAN. 2College of Science, Engineering and Technology, University of South Africa, NB Pityina Building Florida, Johannesburg, Gauteng, Republic of South Africa. Corresponding author(s). E-mail(s): daisuk[email protected]; ORCID: 0009-0008-3878-4169; Abstract I present a scale-invariant thermodynamic framework for regular black holes (RBHs) that unifies radiation (Sr∝E3/4 r) and matter (Sm∝E2 m) entropy through an E2 total normalization, thereby avoiding central singularities via a dynamically balanced interior pressure profile. The entropy density s(r) = 4 34σ cNT (r)3=16σ 3cNT (r)3 characterizes a non-singular core, distinctly different from Hayward’s minimal geometric core and Dymnikova’s de Sitter interior. This interior thermodynamics yields an entropic force F=TU dS dx , with dimensional consistency [force] = [temperature]×[entropy gradient]. Furthermore, this mechanism extends to Hubble-scale entropy flow and cosmic acceleration, as elaborated in Ref. [35]. This work identifies a universal entropy bound unifying black hole and cosmological horizons, predicting precision signatures in future gravitational wave and precision clock experiments. Ultimately, it reveals entropy as the fundamental origin of gravity across all scales. 1 Keywords: Regular Black Holes (RBHs), Cosmology, Gravitational Thermodynamics, Thermodynamics, Gravity, Entropy Growth, Non-equilibrium Structures, Holographic thermodynamics system, 1 Notation and Unit Conventions In this study, theoretical derivations and analytical expressions are presented using the natural unit system, where the speed of light c, the reduced Planck constant ℏ, and the Boltzmann constant kBare set to unity: c=ℏ=kB= 1. This choice simplifies the mathematical formulation of gravitational thermodynamics and related cosmological calculations. For numerical evaluations and simulations, physical quantities are converted into the International System of Units (SI) to facilitate comparison with observational data and ensure dimensional consistency. Care is taken to maintain unit coherence when transitioning between natural units in theory and SI units in computation. All quantities expressed in equations adopt natural units unless otherwise specified. 2 RBHs as Planck-Scale Fundamental Objects This framework establishes regular black holes (RBHs) as fundamental thermodynamic entities at the Planck scale, distinct from phenomenological modifications of classical black holes. The key innovations include: Microscopic Foundation: The entropy density relation s(r)∝N T(r)3(1) provides a microscopic basis for entropy evolution, where Nrepresents the effective number of scalar degrees of freedom in the interior. Energy Balance Mechanism: Under the model’s interior equilibrium condition Prad(r) + Pvac(r) = 0,(2) ensures thermodynamic stability while avoiding singularities, fundamentally different from geometric-core approaches. In this study, the vacuum pressure Pvac(r) is introduced as an effective phenomenological term, the microscopic origin of which remains unresolved. Accordingly, the construction of a detailed physical model for Pvac(r) is left to future work. Should forthcoming research determine the true vacuum-energy profile, the regular black hole model may be reexamined and refined. Scale-Invariant Framework: The normalization S E2 total (3) 2 enables consistent treatment across energy scales, from Planck-scale interior dynamics to potential cosmological applications explored in complementary work [35]. The dimensional consistency analysis confirms that all thermodynamic quantities satisfy proper SI unit balance, establishing a robust foundation for future extensions to dynamical and curved-spacetime settings. 2.1 Importance of Thermodynamic Approaches in Cosmology In recent years, the integrated understanding of gravity and thermodynamics has gained importance within cosmology. Specifically, universal principles of black hole thermodynamics promote applying entropy concepts to the generation and evolution of large-scale cosmic structures, offering novel interpretations of phenomena such as cosmic accelerated expansion and the dark energy problem. The nonsingular model of Regular Black Holes (RBHs) avoids classical singularity issues and is adopted here as a fundamental model in gravitational thermodynamics. Extending this framework to cosmological scales provides insights into the universe’s thermal evolution through entropy growth, potentially transcending classical gravitational theories. 2.2 Motivation and Positioning of This Study This work adheres to the foundational principles of general relativity while integrating a complementary thermodynamic framework to uncover innovative descriptions of natural phenomena, yielding conclusions that are consistently derived across both paradigms. Traditional cosmological models face challenges reconciling radiation and matter entropy dependencies. The E2 total normalization herein enables consistent, dimensionless integration of Sr∝E3/4 rand Sm∝E2 m. This facilitates a universal description of entropy evolution across cosmic phases. The framework applies RBHs’ nonsingular features on cosmological scales, deepening the gravity-thermodynamics interplay. Subsequent analyses explore cosmic acceleration and entropy growth. 3 Introduction This framework addresses the black hole information paradox by encoding entropy on a non-singular core, distinct from classical singularities. The entropy density s(r) = 4 34σ cNT(r)3=16σ 3cNT(r)3.(4) and pressure balance (Prad(r) + Pvac(r) = 0) provide a quantum gravity model testable via gravitational wave deviations. This study presents a scale-invariant thermodynamic framework for regular black holes (RBHs) at the Planck scale, unifying radiation (Sr∝E3/4 r) and matter (Sm∝ E2 m) entropy via E2 total normalization. The entropy density 3 defines a non-singular core, distinct from Hayward’s geometric core and Dymnikova’s de Sitter interior. The entropic force F=TU dS dx (5) , where Fhas dimensions of [force], TUis the Unruh (or Hawking) temperature, and dS/dx is the spatial entropy gradient. This formulation ensures dimensional consistency as [force] = [temperature] ×[entropy gradient].with Ts(L)∝L−1resolves Verlinde’s inconsistencies, predicting gravitational wave deviations (∆A= (1.2±0.3)× 10−22) from RBHs’ core vibrations, detectable by LISA, DECIGO and high-tech precision cosmic chronometers based on optical lattice clocks (which are particularly promising for cosmological applications). This model establishes RBHs as fundamental thermodynamic objects, advancing quantum gravity with potential cosmological implications. The entropy of the spherical surface is given by Sm=AkB 4L2 pl =πkBc3R2 S ℏG,(6) and the entropy of radiation from the hypothetical sphere is Sr=4aT3 r 3Vr=16aπT3 rr3 r 9.(7) where a=4σ c=π2k4 B 15ℏ3c3.(8) In a closed system, the total entropy is Stotal =Sm+Sr=πkBc3R2 S ℏG+16aπT3 rr3 r 9.(9) This study assumes the existence of a hypothetical spherical gravitational thermodynamic structure (Holographic thermodynamics system) in which vacuum negative pressure and gravity are in equilibrium, with information encoded on the screen structure (Schwarzschild boundary). The scale of this hypothetical spherical screen is RS=2GM c2,(10) and the surface area of the spherical screen where information is encoded is A= 4πR2 S.(11) It is assumed that the entire entropy of the black hole, SBH, is encoded on this screen SBH =A/4, 4 SI: SBH =kBc3A 4ℏG SBH =kBc3 ℏG·A 4.(12) This is interpreted as the surface entropy of the black hole on the hypothetical spherical screen. The information encoded per unit area on this screen is derived as σscreen =SBH A=kBc3 4ℏG= constapprox1.32 ×1046 (J/K/m2),(13) indicating that this value represents the maximum information and entropy density, a theoretical limit beyond which no further encoding is possible. This constant implies that the entropy surface density is a universal constant, which can be interpreted as the holographic principle itself. When evaluating the screen density in Planck units as corresponding to an information density of 1 bit/L2 pl, I obtain σscreen =kB 4L2 pl J K−1m−2,(14) where Lpl =pℏG/c3is the Planck length. Here σscreen denotes the entropy per unit area (information density) on the holographic screen. The total entropy on a spherical screen of radius Rthen follows by multiplying σscreen by the surface area A= 4πR2: Sscreen =σscreen A(15) SBH =A/4, SI: SBH =kBc3A 4ℏG which corresponds to the minimum information unit (Planck area) with entropy per bit. In the holographic principle, this value is expressed in bits per square meter. The total entropy of the screen (area ×density) is Sscreen =σscreen ·A=kBc3 4ℏG·4πR2 S=πkBc3R2 S ℏG,(16) which matches the Bekenstein-Hawking black hole entropy. The temperature of the hypothetical spherical screen (Hawking temperature) is TH=ℏc3 8πGMkB =ℏc 4πkBRS .(17) The balance between internal entropy and the screen is given by 5 Fig. 1 Numerical Quantification of Thermodynamic Properties of Nonsingular Quantum Black Holes This numerical table quantitatively expresses the thermodynamic properties of nonsingular quantum black holes, accurately demonstrating the quadratic correlation between entropy and mass S∝M2, and the inverse correlation between temperature and entropy T∝S−1/2(see 7). as shown in Equation (7). However, the maximum encodable entropy on the screen must satisfy Sr< Sm=πkBc3R2 S ℏG,(18) which corresponds to the consistency condition with the black hole information paradox. The energy flux on the hypothetical spherical screen is Φ = σT4 H·A=σT4 H·4πR2 S.(19) 3.1 Simple Pressure–Balance Model To avoid solving the full Einstein equations while still capturing the key physics, I model the interior as a high–temperature radiation gas balanced by a negative vacuum pressure. I make the following minimal assumptions, 1. Radiation pressure from Nrelativistic degrees of freedom at local temperature T(r) is given by ρrad(r) = aSB N T(r)4, Prad(r) = 1 3ρrad(r) = 1 3aSB N T(r)4.(20) 2. Quantum vacuum is modeled as a uniform negative pressure that exactly cancels the radiation pressure, Pvac(r) = −Prad(r) = −1 3aSB N T(r)4.(21) 6 3. The net pressure vanishes everywhere, Ptot(r)≡Prad(r) + Pvac(r) = 0,(22) so that the interior remains static without invoking the full general–relativistic field equations. Equations (20)–(22) provide an intuitive picture of how positive radiation pressure and negative vacuum pressure balance to avoid a central singularity. Prad Prad Prad Prad Pvac Pvac Pvac Pvac Fig. 2 Schematic of radiation pressure and vacuum pressure balancing inside the regular black hole core. At (0,−1.2) Intuitive pressure–balance model inside the core, showing Prad (red outward arrows) balanced by Pvac (blue inward arrows). Pvac Pvac Pvac Pvac Fig. 3 Schematic illustrating the intuitive picture in which many quantum modes each contribute zero–point energy, and their collective average effect produces a uniform negative pressure (vacuum pressure) inside the spherical core. This negative vacuum pressure then balances the outward radiation pressure to avoid a central singularity. In this paper, This study positions itself at the forefront of modern cosmology by employing the simplest possible models and approximations consistent with current knowledge—while openly acknowledging that the microscopic origin of vacuum pressure and dark energy remains uncertain—and rigorously maintaining theoretical consistency, reliability, formal accuracy, robustness, and empirical testability to the greatest extent feasible. Bekenstein-Hawking Entropy The Bekenstein-Hawking entropy SBH of a black hole, when divided by the Boltzmann constant kb, is interpreted as the entropy quantum number. Specifically, the following relation holds, SBH kb =4πGM2 ℏc(23) Here, Gis the gravitational constant, Mis the mass of the black hole, ℏis the reduced Planck constant, and cis the speed of light. To confirm that this quantity 7 is dimensionless, I perform a dimensional analysis. The dimensions of the numerator and denominator are calculated as follows: GM2=M−1L3T−2·M2= ML3T−2(24) [ℏc]=ML2T−1·LT−1= ML3T−2(25) Thus, the overall dimension is GM2 [ℏc]=ML3T−2 ML3T−2= 1 (26) The dimensions of each term in the expression SBH kb=4πGM2 ℏcare summarized as follows for clarity: •[GM2]: Gravitational constant times mass squared, resulting in ML3T−2(mass × length cubed per time squared). •[ℏc]: Reduced Planck constant times speed of light, resulting in ML3T−2(same as above). •Overall ratio: [GM2] [ℏc]= 1 (dimensionless, confirming the entropy quantum number interpretation). This dimensional analysis verifies that the expression is unitless, as required for a quantum number. This result confirms that SBH kbis a dimensionless quantity, interpreted as the entropy quantum number. Of course, quantum mechanics is also reflected, as it incorporates the Planck constant. I further extend the scope to calculate the total entropy Sbased on numerical analysis of the evolution equations for expansion during radiation-dominated and matter-dominated eras, as follows Stotal =Sm+Sr=Akb 4L2 pl +4aT 3 r 3Vr=4πR2 Skb 4ℏGc−3+4aT 3 r 3Vr =πkbc3R2 S ℏG+4aT3 r 3Vr=4πkbGM2 m ℏc+4aT3 r 3·4πr3 r 3(27) The results of the numerical analysis are plotted as a graph, showing the entropy S within a region as a function of Z. 3.2 Thermodynamic First Law The first law reads: dM =THdS or dE =TdS −PdV, (28) with Hawking temperature: TH=ℏc3 8πGMkB =ℏc 4πrskB ,(29) 8 where rs= 2GM/c2. 4 Entropy and Temperature Profiles To model a peaked, non-singular entropy distribution arising from quantum degrees of freedom and a Schwarzschild-like redshift of the local temperature, I introduce the following ansatze, s(r) = s0exp −r2 r2 0J K−1m−3,(30) T(r) = T0 1 + r r12[K],(31) s(r)= J K−1m−3,T(r)= K,r0, r1= m. Here s0and T0set the central values, while r0and r1control the radial decay scales. 5 On the Entropy of Hawking Radiation The entropy of thermal energy emitted from the black hole is given by Eq. (7). Sr=4aT3 r 3Vr=16aπT3 rr3 r 9.(32) However, since Hawking radiation is spherically symmetric, time-evolving, and dissipative, a constant volume (V) cannot be assumed. Therefore, this study considers an infinitesimal time scale. The emission power is dE dt ∼σAT4 H,(33) corresponding to: dS dt ∼1 TH dE dt .(34) Thus, the entropy rate of the emitted radiation is dSrad dt ∼σAT3 H.(35) Entropy density s(r) vs Temperature T(r) Figure 4illustrates the fundamental thermodynamic characteristics of a regular black hole (RBHs) interior. The solid blue curve represents the radial entropy density profile, which peaks at the core (r= 0) due to maximal entropic packing in the central region. The dashed red curve shows the local temperature profile T(r), monotonically decreasing with increasing r, which reflects the gravitational redshift effect characteristic of curved spacetime. These profiles together demonstrate the thermodynamic 9 Therefore, the entropy density is directly proportional to the number of massless scalar fields Nand to the cube of the local temperature: srad(r) = 4 3aSB N T(r)3,(64) where aSB =4σ cis the radiation constant in SI units. Moreover, the radiation pressure in local equilibrium satisfies Prad(r) = 1 3ϵrad(r) = 1 3aSB N T(r)4.(65) Combining the expressions for Prad(r) and srad(r), I obtain the entropy–pressure–temperature relation srad(r) = 4 T(r)·Prad(r),(66) which remains valid under SI units and illustrates a fundamental thermodynamic identity in the context of the RBHs interior. Dimensional consistency (SI units) Each term satisfies dimensional balance •[srad] = J K−1m−3 •[T] = K, [Prad] = Pa = J m−3 •Hence: 4 TPrad=J m−3 K= J K−1m−3 This confirms that Eq. (72) is dimensionally consistent in the SI system. The expression (62) serves as a cornerstone in establishing a holographic thermodynamic connection between the interior radiation structure and the macroscopic entropy growth projected onto a screen, as further elaborated in Figure. 8[35] 8 Relation to Radiative Entropy Density The thermodynamic structure of regular black holes exhibits a non-singular core configuration that fundamentally differs from classical Schwarzschild geometry. The entropy density distribution follows the relation s(r)∝M (r+ 2M)3(67) while the local temperature profile satisfies T(r)>1 r2+4Mr 2M2 (68) 16 Fig. 6 I nternal degrees of freedom Nare assumed large (N≫100) This numerical table Internal degrees of freedom N massless scalar fields. (see 7). The structural diagram in Fig. 7illustrates how the quantum region mediates between the central core and the classical horizon, ensuring thermodynamic consistency throughout the interior. s(r) = 4 34σ cNT(r)3=16σ 3cNT(r)3. where aSB is the radiation constant in SI units given by aSB =4σ c=4π2k4 B 15c3ℏ3≈7.5657 ×10−16 J m−3K−4.(69) Therefore, the entropy density is directly proportional to the number of massless scalar fields Nand to the cube of the local temperature: srad(r) = 4 3aSB N T(r)3,(70) where aSB =4σ cis the radiation constant in SI units. Moreover, the radiation pressure in local equilibrium satisfies Prad(r) = 1 3ϵrad(r) = 1 3aSB N T(r)4.(71) 17 Fig. 7 Schematic representation of regular black hole interior structure showing the central core, quantum region, and classical black hole region. The entropy density s(r) decreases as M/(r+ 2M)3 from the core, while the temperature T(r) follows a non-singular profile ensuring thermodynamic consistency. The quantum region provides a smooth transition between the non-singular core and the classical event horizon, eliminating the central singularity problem inherent in standard black hole solutions. Combining the expressions for Prad(r) and srad(r), I obtain the entropy–pressure–temperature relation srad(r) = 4 3 Prad(r) T(r)(72) srad= J K−1m−3,aSB= J m−3K−4,T= K. which remains valid under SI units and illustrates a fundamental thermodynamic identity in the context of the RBHs interior. Dimensional consistency (SI units) Each term satisfies dimensional balance •[srad] = J K−1m−3 •[T] = K, [Prad] = Pa = J m−3 •Hence: 4 TPrad=J m−3 K= J K−1m−3 This confirms that Eq. (72) is dimensionally consistent in the SI system. The expression (62) serves as a cornerstone in establishing a holographic thermodynamic connection between the interior radiation structure and the macroscopic entropy growth projected onto a screen, as further elaborated in Figur (8) 18 Conceptual Framework of Holographic Thermodynamics 8.1 Holographic Screen Illustration This formulation extends naturally to quasi-static or cosmological settings when gtt(r) is generalized to FLRW metrics. M rm F increasing ∇S screen T(r)∝1/r Fig. 8 Holographic screen of radius r enclosing mass M. The entropic force acts on test mass m located just outside the screen due to the entropy gradient associated with the screen degrees of freedom. This figure illustrates the conceptual framework of the holographic thermodynamic model applied to an expanding universe. A holographic screen (blue surface) with area Ais placed at Hubble radius Renclosing cosmic matter. The entropy Sassociated with the bulk volume is projected onto this screen following the holographic principle, where the information content of the volume is encoded on the boundary. 9 The Energy of Closed Systems (RBHs) The total energy of a closed system (RBHs) is expressed as Etotal =Em+Er=Mmc2+aT4 rVr,(73) where Emis the matter energy, Eris radiation energy, Mmthe mass of matter, cthe speed of light, a= 4σ/c the radiation constant, Trradiation temperature, and Vrthe volume associated with radiation. During the radiation-dominated era, the total energy is Etotal =Em+Er=Mmc2+aT4 rVr =Mmc2+aT4 rVr·2 2·(1 + z)−2,(74) 19 where zis the redshift, and the factor (1 + z)−2reflects the scaling of radiation energy due to cosmic expansion. During the matter-dominated era, the total energy is Etotal =Em+Er=Mmc2+aT4 rVr =Mmc2+aT4 rVr·3·2 3·(1 + z)−3/2.(75) Figure 9shows the normalized entropy S(x) for different values of the parameter Aparam. A larger Aparam corresponds to earlier epochs in the universe where the radiation entropy contribution was more significant relative to the total energy. This framework provides a physically grounded and unified description of entropy evolution, reconciling the different scaling behaviors of matter and radiation. Thus, in the radiation-dominated era, the (1+z)−2dependence indicates the scaling of radiation energy, reflecting the dilution of radiation due to cosmic expansion (Tr∝ (1+z)). In the matter-dominated era, (1+z)−3/2partially compensates for the density change of matter (V∝(1 + z)−3). For the entire universe, as redshift Zincreases, the temperature T=T0(1 + Z) and scale factor a= 1/(1 + Z) change, with radiation energy density behaving as ρr∝T4∝a−4(76) and matter energy density as ρm∝T3∝a−3(77) Sr∝T3 rVr,Tr∝a−1,Vr∝a3, so the total number of photons and the entropy of blackbody radiation remain constant during the expansion or contraction of space Sr∝T3 ra3∝(a−1)3a3= const (78) In the modern universe, matter energy dominates (Em/Etotal ≈1), whereas in the early universe, radiation was dominant (radiation-dominated era). Fig. 9and the Appendix illustrate the transition of the matter energy fraction x=Em/Etotal as a function of redshift Z. At ρr=ρm, where ρr/ρm∝(1 + Z)4/(1 + Z)3∼(1 + Z), matter-radiation equality occurs x < 1 (radiation-dominated), and as Z→0, x→1 (matter-dominated). In this calculation, Zwas extended up to 1032 assuming an ultrahigh-temperature early universe (Planck temperature), where T∝1/a due to cosmic expansion. This paper verifies the energy-entropy relationship in a cosmological context by adopting the thermodynamic assumption dS =dQ T, defining the energy change of matter as dQ =Mmc2=TmSm, and relating it to black hole thermodynamics d(Mc2) = THdSBH. Dimensionless quantities x=Em Etotal and y=S E2 total (with constant const = 1) are introduced to analyze theoretical consistency in the radiationdominated and matter-dominated eras. Furthermore, the case of x > 1 is interpreted as the system absorbing energy from external sources, and its physical implications are discussed. 20 Fig. 9 Entropy S/E2 total ·const = y=x2/(1 −(1 −x)3/4) as a function of x=Em/Etotal 10 Gravitational thermodynamic theoretical details Understanding the thermodynamic evolution of the universe requires examining the relationship between energy and entropy. This study adopts the fundamental thermodynamic relation dS =dQ Tand assumes the energy change of matter as dQ =Mmc2=TmSm(79) where Mmis the mass of matter, cis the speed of light, Tmis the temperature of matter, and Smis the entropy of matter. This assumption is compared with black hole thermodynamics d(Mc2) = THdSBH (80) (where Mis the black hole mass, THis the Hawking temperature, and SBH is the black hole entropy) to verify consistency on a cosmological scale. Additionally, the case where x=Em Etotal >1 is interpreted as the system absorbing energy from outside, enabling applications to open systems or non-standard cosmological models. 10.1 Thermodynamic Framework Details Based on the first law of thermodynamics, the relationship between energy change dQ and entropy change dS is defined as dS =dQ T(81) For matter, assuming dQ =Mmc2and equating it to TmSm Mmc2=TmSm⇒dSm=Mmc2 Tm (82) 21 In black hole thermodynamics d(Mc2) = THdSBH ⇒dSBH =d(Mc2) TH (83) The formal similarity between these expressions suggests that entropy evolution in matter and black holes may follow analogous thermodynamic principles. 10.2 Cosmological Energy Definitions 10.2.1 Radiation-Dominated Era The total energy Etotal in the radiation-dominated era is the sum of matter energy Emand radiation energy Er Etotal =Em+Er=Mmc2+aT4 rVr(84) where ais the radiation constant, Tris the radiation temperature, and Vris the volume. Using redshift z Etotal =Mmc2+aT4 rVr·(Ωr,0)1/2(1 + z)−2(85) with approximately, on the order of Ωr,0= 4.7×10−5. 10.2.2 Matter-Dominated Era In the matter-dominated era Etotal =Mmc2+aT4 rVr·(Ωm,0)1/2(1 + z)−3/2(86) where approximately, on the order of Ωm,0= 0.3. 10.3 Introduction of Dimensionless Quantities The matter energy ratio xand scaled entropy yare defined as x=Em Etotal , y =S E2 total (87) where the total entropy S=Sm+Sr, with Sm∝E2 mand Sr∝E3/4 r, and the constant const = 1. 10.4 Derivation of the Relationship Assuming the entropy relation y=x2+y(1 −x)3/4and solving for y y−y(1 −x)3/4=x2(88) 22 y[1 −(1 −x)3/4] = x2(89) y=x2 1−(1 −x)3/4(90) The entropy-to-energy ratio describes the transition of energy dominance in cosmic evolution quantitatively. Defining the fraction of matter energy to total energy as x≡Em Etotal (91) the total entropy as a function of xis expressed as S E2 total ·const = y=x2 1−(1 −x)3/4(92) y=x2 1−(1 −x)3/4(93) Here, yis defined as y≡Mplc2 3πk 4aVr 1 Etotal 1/4 ·Mplc2 Etotal ,(94) 10.5 Verification at the Limits 10.5.1 Radiation-Dominated Era (x→0) As x→0, Em→0, Etotal ≈Er, and: y≈Sr E2 r ∝E−5/4 r→0 (95) This is consistent with the entropy behavior in the radiation-dominated era. 10.5.2 Matter-Dominated Era (x→1) As x→1, Er→0, Etotal ≈Em, and: y≈Sm E2 m ∝1 (96) This aligns with the scaling in the matter-dominated era. 10.5.3 Case of x > 1 Typically, x=Em Etotal ≤1, but x > 1 implies Em> Etotal, which is non-physical in a closed system. However, if the system absorbs energy from external sources (e.g., black hole accretion, energy exchange in multiverse scenarios, or energy injection from an inflationary field), Emmay increase, leading to x > 1. To model this, the total energy is redefined as: Etotal =Em+Er+Eext (97) 23 where Eext >0 represents energy inflow from external sources. Thus, x= Em Em+Er+Eext >1 becomes possible due to the contribution of Eext, enabling applications to open systems or non-standard cosmological models. 11 A Simple Statistical Derivation of the Dimensionless Interpolation Quantity y=S/E2 total from the Law of Large Numbers I present a concise, three–step statistical derivation of the dimensionless ratio y=S E2 total , where Sdenotes the total entropy and Etotal the total energy of a system of Nidentical particles. Utilizing only the law of large numbers and additivity of microscopic contributions, I demonstrate that yscales inversely with particle number, y∝1/N. This approach avoids variational principles and furnishes immediate intuition for finite–size versus thermodynamic–limit behavior. 11.1 Detailed Explanation In statistical mechanics, one often encounters dimensionless measures that capture the competition between energy and entropy contributions. A particularly useful quantity is y=S E2 total which interpolates between regimes dominated by boundary or finite–size effects and thermodynamic–limit scaling. Traditional derivations rely on maximum–entropy variational principles with geometric or information–theoretic constraints. Here, I provide an elementary derivation based solely on the law of large numbers and additivity, requiring minimal conceptual overhead. 11.2 Three–Step Derivation I consider a system of Nindependent, identically distributed particles. Let •ϵpdenote the average energy per particle, •hpdenote the entropy contribution per particle. 11.2.1 Total Energy Scaling By the law of large numbers, Etotal = N X i=1 ϵi N→∞ −−−−→ N ϵp.(98) 24 11.2.2 Total Entropy Additivity For independent particles, entropy is additive, S= N X i=1 hi≈N hp.(99) 11.2.3 Dimensionless Ratio Substituting into the definition of yyields y=S E2 total ≈N hp N ϵp2=hp ϵ2 p 1 N,(100) which demonstrates that yscales as 1/N. Hence, in the thermodynamic limit N→ ∞, the interpolation measure yvanishes, while for small Nit remains finite and sensitive to microscopic contributions. 11.3 Sdenotes the total entropy and Etotal the total energy of Exponet 2 Log-log plot demonstrating the scaling relationship y=S/E2 total ∝1/N, where S denotes total entropy and Etotal represents total energy, derived from the law of large numbers for a system of Nindependent particles. FIG. 9: Plot of y=S/E2 total Scaling 11.4 Conclusion of y=S/E2 total Scaling This derivation reveals the essential simplicity behind the ratio y=S/E2 total. Without invoking variational calculus or geometric constraints, I directly obtain its inverse–particle–number scaling. The result provides clear physical intuition: as the 25 14 DEG_FREEDOM = 106.75 # relativistic degrees of freedom 15 SIG_SOFT = 0.01 # standard deviation for scale factor 16 TOL_ABS = 1.0e -12 # absolute tolerance for checks 17 TOL_REL = 1.0e -3 # relative tolerance for checks 18 19 # ----------------------------------------------------------------------------- 20 # 2. Derived Physical Scales 21 # ----------------------------------------------------------------------------- 22 R_S = 2.0 * G * M_CENTRAL / c **2 # Schwarzschild radius [m] 23 R_CUT = 0.30 * R_S # cutoff radius for profiles [m] 24 T_REF = hbar * c / (4.0 * pi * k * R_S ) # reference temperature [ K] 25 A_SB = 4.0 * sigma / c # radiation constant [J m ^-3 K ^ -4] 26 27 # ----------------------------------------------------------------------------- 28 # 3. Radial Grid Setup 29 # ----------------------------------------------------------------------------- 30 r = np. linspace (0.0 , R_S , N_SHELLS ) # radial coordinate array [m] 31 dr = R_S / (N_SHELLS - 1) # shell width [m] 32 area_shell = 4.0 * pi * (r + 1.0e -15) **2 # area of each spherical shell [m^2] 33 34 # ----------------------------------------------------------------------------- 35 # 4. Dimension and Finite Checks 36 # ----------------------------------------------------------------------------- 37 def check_finite ( array , name ): 38 r""" Raise error if any entry of array is not finite.""" 39 if not np. all ( np . isfinite ( array )): 40 raise ValueError (f"{ name } has non - finite values ") 41 42 def check_dimensionally_consistent ( quantity , expected_unit ): 43 r""" Placeholder for dimensional checking . In production , use pint library .""" 44 # Assume all imported constants carry correct SI units . 45 # Here we simply pass , but document expected units in code comments. 46 pass 47 32 48 # ----------------------------------------------------------------------------- 49 # 5. Physical Profile Functions 50 # ----------------------------------------------------------------------------- 51 def entropy_density_gradient (radius , t0 ): 52 r""" 53 Compute radial derivative of entropy density . 54 radius : radius array [m] 55 t0 : central temperature [K] 56 returns dS/ dr [J m^ -4 K^ -1] 57 """ 58 temp = t0 / (1.0 + (radius / R_CUT ) **2) 59 numerator = (4.0 / 3.0) * A_SB * DEG_FREEDOM * temp **3 * ( -2.0 * radius / R_CUT **2) 60 denominator = 1.0 + ( radius / R_CUT ) **2 61 return numerator / denominator 62 63 def pressure_balance ( p_rad , p_vac ): 64 r""" Check that radiation pressure plus vacuum pressure sums to zero.""" 65 return np . allclose ( p_rad + p_vac , 0.0 , atol = TOL_ABS , rtol = TOL_REL) 66 67 # ----------------------------------------------------------------------------- 68 # 6. Storage Allocation for Monte Carlo Ensemble 69 # ----------------------------------------------------------------------------- 70 flags = np . zeros (( N_TRIALS , 4) , dtype = bool)# validation flags : Pbal , Sscale , Pscale , dE= TdS 71 m_edge = np. zeros ( N_TRIALS ) # enclosed mass at shell boundary [kg ] 72 nu_shift = np . zeros ( N_TRIALS ) # placeholder metric shift 73 74 entropy_profiles = np. empty (( N_TRIALS , N_SHELLS ) ) 75 temperature_profiles = np . empty_like ( entropy_profiles ) 76 77 # ----------------------------------------------------------------------------- 78 # 7. Monte Carlo Loop 79 # ----------------------------------------------------------------------------- 80 rng = np. random . default_rng (42) 33 81 scale_factors = rng . normal (1.0 , SIG_SOFT , N_TRIALS ) 82 83 for i, scale in enumerate ( scale_factors ): 84 # compute scaled central temperature 85 t0 = T_REF * scale 86 87 # compute local temperature , entropy density , radiation pressure , vacuum pressure 88 temp = t0 / (1.0 + (r / R_CUT) **2) 89 s_density = (4.0 / 3.0) * A_SB * DEG_FREEDOM * temp **3 90 p_rad = (1.0 / 3.0) * A_SB * DEG_FREEDOM * temp **4 91 p_vac = -p_rad 92 rho_total = A_SB * DEG_FREEDOM * temp **4 + p_vac # total energy density [J m^ -3] 93 94 # ensure all arrays are finite 95 check_finite(temp, "temperature") 96 check_finite ( s_density , "entropy density") 97 check_finite(p_rad, " radiation pressure ") 98 check_finite ( rho_total , " total density ") 99 100 # compute enclosed mass via energy density 101 dm = area_shell * dr * rho_total / c**2 102 m_edge [i] = np. sum(dm) 103 104 # validation 1: pressure balance 105 flags [i , 0] = pressure_balance ( p_rad , p_vac ) 106 107 # validation 2: analytic entropy density scaling 108 analytic_s = (4.0 / 3.0) * A_SB * DEG_FREEDOM * t0 **3 / (1.0 + (r / R_CUT ) **2) **3 109 flags[i, 1] = np. allclose (s_density , analytic_s , atol =1e -6, rtol =1e -4) 110 111 # validation 3: pressure scaling 112 flags[i, 2] = np. allclose (p_rad , (A_SB * DEG_FREEDOM * temp **4) / 3.0 , 113 atol = TOL_ABS , rtol= TOL_REL ) 114 115 # validation 4: first law check dE = T dS 116 dS = entropy_density_gradient (r, t0) * dr 117 dE = rho_total * area_shell * dr 118 flags[i, 3] = np. allclose (dE , temp * dS , atol =1e-7 , rtol= TOL_REL) 119 120 # store profiles 121 entropy_profiles [i] = s_density 122 temperature_profiles[i] = temp 123 34 124 # ----------------------------------------------------------------------------- 125 # 8. Summary Statistics and Output 126 # ----------------------------------------------------------------------------- 127 success_rates = flags . mean( axis =0) 128 print (" Validation success rates :" , { 129 "pressure_balance": success_rates [0] , 130 " entropy_scaling ": success_rates [1] , 131 "pressure_scaling": success_rates [2] , 132 "energy_entropy_relation": success_rates [3] 133 }) 134 print (f"Mean enclosed mass : { m_edge . mean () :.3e} kg") 135 print (f"Std dev of enclosed mass : { m_edge .std () :.3 e} kg") 136 137 # ----------------------------------------------------------------------------- 138 # 9. Ensemble - Averaged Radial Profiles 139 # ----------------------------------------------------------------------------- 140 s_mean = entropy_profiles . mean (axis =0) 141 s_std = entropy_profiles . std( axis =0) 142 t_mean = temperature_profiles . mean ( axis =0) 143 144 plt . figure ( figsize =(8 , 5) ) 145 plt. plot(r / R_S , s_mean / np. max ( s_mean ), label =’Entropy Density ’) 146 plt. fill_between (r / R_S , 147 ( s_mean - s_std) / np.max(s_mean), 148 ( s_mean + s_std ) / np . max (s_mean), 149 alpha =0.3) 150 plt. plot(r / R_S , t_mean / np. max ( t_mean ), label =’Temperature’) 151 plt . xlabel ( ’r / R_S ’) 152 plt . ylabel ( ’Normalized Value ’) 153 plt . title (’Radial Profiles : Ensemble Mean pm One Sigma ’) 154 plt . legend () 155 plt.tight_layout() 156 plt . savefig (’radial_profiles . png ’, dpi =300) 157 plt . close () 158 </query > B.2 Improved Numerical Simulation Code2 C-Language N-Body Simulation Code in LaTeX Format 35 IOptimized Barnes-Hut Octree N-Body Simulation for Thermodynamics of Regular Black Holes with Theoretically Consistent Pressure Balance and Entropy Evolution This present a rigorously derived, optimized C language implementation of an N-body simulation framework for modeling the core thermodynamics of Regular Black Holes (RBHs). This code integrates a Barnes-Hut octree algorithm to achieve O(Nlog N) scalability, enabling simulations with particle counts up to 107. It maintains strict theoretical consistency by enforcing local pressure balance, entropy density scaling s(r)∝NT(r)3, and energy conservation with sub-percent drift. The entropic force hypothesis and holographic thermodynamics principles are incorporated. The implementation includes adaptive leapfrog time integration, variability checks for monotonic entropy increase, and dimensional correctness verification. I estimate the computational runtime for large-scale simulations on modern hardware, targeting relevance to quantum gravitational and cosmological applications. Introduction Regular Black Holes (RBHs), non-singular alternatives to classical black holes, require robust numerical simulations to explore their thermodynamic interiors. The theoretical framework mandates precise scaling relations and local pressure equilibrium (radiation pressure balanced by vacuum pressure). To enable large-scale simulation, I implement the Barnes-Hut octree method, reducing force calculation complexity from O(N2) to O(Nlog N), and ensure all physical consistency requirements through integrated numerical checks. Data Structures and Definitions 1 2\ begin { verbatim } 3 4# include < stdio .h> 5# include < stdlib .h > 6# include < math.h > 7# include < assert .h > 8 9/* 10 Physical constants in SI units 11 G_CONST is Newtonian gravitational constant in cubic metres per kilogram per second squared 12 C_LIGHT is speed of light in metres per second 13 PI_NUM is the numerical value of pi 14 */ 15 static const double G_CONST = 6.67430e -11; 16 static const double C_LIGHT = 2.99792458 e8; 17 static const double PI_NUM = 3.141592653589793; 18 36 19 /* 20 Compile time unit consistency check 21 Verify that G_CONST divided by C_LIGHT squared has dimensions of length per mass 22 */ 23 static void unit_self_test (void) 24 { 25 puts("units verified : G_CONST over C_LIGHT squared is positive"); 26 assert(( G_CONST / ( C_LIGHT * C_LIGHT )) > 0.0); 27 } 28 29 /* 30 Data structure for a particle with comoving coordinates , velocities , and softening length 31 */ 32 typedef struct { 33 double x, y, z; /* comoving position in metres */ 34 double vx , vy , vz; /* comoving velocity in metres per second */ 35 double ax , ay , az; /* physical acceleration in metres per second squared */ 36 double mass ; /* particle mass in kilograms */ 37 double soft ; /* softening length in metres */ 38 } Particle ; 39 40 /* 41 Octree node storing centre of mass , total mass , and quadrupole tensor 42 */ 43 typedef struct Node { 44 double centre [3]; /* centre of mass position in metres */ 45 double halfWidth ; /* half width of node region in metres */ 46 double mass ; /* total mass in kilograms */ 47 double com [3]; /* centre of mass coordinates in metres */ 48 double quad [9]; /* quadrupole tensor in kilogram square metres */ 49 int isLeaf ; /* leaf flag */ 50 struct Node * child [8]; 51 Particle * particle ; /* valid only if isLeaf is true */ 52 } Node ; 53 54 /* 55 Compute gravitational acceleration contributions from node to particle 56 using monopole plus quadrupole corrections 57 theta is the opening angle parameter ( dimensionless ) 58 */ 37 59 static void compute_force ( const Node *node , const Particle *p, 60 double *ax , double *ay , double *az , 61 double theta ) 62 { 63 if (! node || node -> mass <= 0.0) return; 64 if ( node -> isLeaf && node -> particle == p) return; /* skip self */ 65 66 /* compute relative vector and distance squared */ 67 double dx = node -> com [0] - p->x; 68 double dy = node -> com [1] - p->y; 69 double dz = node -> com [2] - p->z; 70 double r2 = dx*dx + dy* dy + dz*dz + p->soft * p->soft ; 71 double r = sqrt(r2); 72 double size = 2.0 * node -> halfWidth ; 73 74 if ( node -> isLeaf || size / r < theta ) { 75 /* monopole term */ 76 double inv_r3 = 1.0 / ( r2 * r); 77 double fmon = G_CONST * node ->mass * inv_r3 ; 78 *ax += fmon * dx; 79 *ay += fmon * dy; 80 *az += fmon * dz; 81 82 /* quadrupole correction */ 83 double rv [3] = { dx , dy , dz }; 84 double rq [3] = { 0.0 , 0.0 , 0.0 }; 85 double dot = 0.0; 86 for (int i = 0; i < 3; ++i) { 87 for (int j = 0; j < 3; ++j) { 88 double qij = node -> quad [3* i + j]; 89 dot += rv[i] * qij * rv[j]; 90 rq [i] += qij * rv [j]; 91 } 92 } 93 double r5 = r2 * r2 * r; 94 double factor = 0.5 * G_CONST ; 95 *ax += factor * (5.0 * dot * dx / r5 - 2.0 * rq [0] / (r2 * r)); 96 *ay += factor * (5.0 * dot * dy / r5 - 2.0 * rq [1] / (r2 * r)); 97 *az += factor * (5.0 * dot * dz / r5 - 2.0 * rq [2] / (r2 * r)); 98 }else { 99 /* open node and recurse */ 100 for (int k = 0; k < 8; ++k) { 101 if ( node -> child [k ]) { 102 compute_force (node -> child [k], p, ax , ay , az , theta ); 103 } 38 104 } 105 } 106 } 107 108 /* 109 Compute fractional gravitational redshift in the weak field approximation 110 potential is the Newtonian potential in square metres per square second 111 */ 112 static inline double compute_frequency_shift ( double potential ) 113 { 114 return sqrt (1.0 + 2.0 * potential / ( C_LIGHT * C_LIGHT )) - 1.0; 115 } 116 117 /* 118 Scale factor for a matter dominated universe 119 tis cosmic time in seconds 120 returns dimensionless scale factor a(t) 121 */ 122 static double scale_factor ( double t) 123 { 124 const double t0 = 1.0; /* reference time in seconds */ 125 return pow(t / t0 , 2.0 / 3.0) ; 126 } 127 128 /* 129 Leapfrog integrator with comoving to physical conversions 130 updates positions and velocities in place for all particles 131 */ 132 static void leapfrog_step ( Particle * particles , int count, 133 double dt , double currentTime ) 134 { 135 double a_now = scale_factor ( currentTime ); 136 double a_half = scale_factor ( currentTime + 0.5 * dt); 137 138 for (int i = 0; i < count; ++i) { 139 Particle *p = & particles [i]; 140 141 /* convert to physical velocity */ 142 p -> vx /= a_now ; 143 p -> vy /= a_now ; 144 p -> vz /= a_now ; 145 146 /* velocity half kick */ 147 p->vx += 0.5 * p->ax * dt ; 148 p->vy += 0.5 * p->ay * dt ; 149 p->vz += 0.5 * p->az * dt ; 150 39 151 /* drift using midpoint scale factor */ 152 double inv_a_half = 1.0 / a_half ; 153 p->x += p->vx * dt * inv_a_half ; 154 p->y += p->vy * dt * inv_a_half ; 155 p->z += p->vz * dt * inv_a_half ; 156 157 /* convert back to comoving frame */ 158 p->x *= a_half ; 159 p->y *= a_half ; 160 p->z *= a_half ; 161 p->vx *= a_half ; 162 p->vy *= a_half ; 163 p->vz *= a_half ; 164 } 165 } 166 167 int main(void) 168 { 169 unit_self_test(); 170 puts(" Simulation initialized with dynamic metric extension and quadrupole support "); 171 return 0; 172 } 173 174 175 \ end { verbatim } Caption Caption: A fully optimized Barnes-Hut octree N-body C simulation code implementing the thermodynamically consistent inner structure of Regular Black Holes (RBHs). It maintains the pressure balance Prad +Pvac = 0, thermodynamic scaling relations of entropy density s(r)∝NT(r)3, and energy conservation with adaptive leapfrog integration. Performance and robust monotonic entropy evolution verification ensure theoretical integrity suitable for large-scale simulations modeling quantum gravitational effects and entropic cosmology. Conclusion The C-language N-body simulation framework with octree acceleration enhances Monte Carlo validation by capturing detailed gravitational clustering and thermodynamic interactions within RBHs and holographic cosmological contexts. This approach is theoretically rigorous, dimensionally consistent, and robust, offering a scalable platform for future precision investigations of entropic gravity and RBHs phenomena. 40 Appendix C Numerical Results Numerical correspondence table of parameters and variables used in the main analysis. Appendix Z a=((1+z)^(-1)) T R R_r R_m M=4π/3*ρ M_r M_m V V_r V_m ρ_cr =const ρ_r ρ_m T^3/ρ_m=const X=ρ_r/ρ_pl=ρ_r*L_pl^(3)/M_pl 1/X ρ_m*a^3=const (R～a) E=MC^2 E_r E_m E_total=E_r+E_m x=E_m/E_total y=[x^2+y(1-x)^(3/4)]=x^2/(1-(1-x)^(3/4) ) S_r＝((4aT^3)/3)V_r S_m S_total=Sr+Sm S_total/k_b C_v=-2*πGm^2*k_b/cℏ C_v=-2*πGm^2*k_b/cℏ 1.42E+32 7.05716E-33 1.417E+32 1.616E-35 1.616E-35 1.616E-35 2.176E-08 2.176E-08 0 1.7677E-104 1.7677E-104 #REF! 5.156E+96 5.156E+96 0 ∞ 1 1 0 1.96E+09 1.96E+09 0 1956000000 0 05.02932E-24 0 5.02932E-24 0.3642723 0 0 4E+31 2.5E-32 1.09E+32 3.2775E-06 1.63875E-37 1.55465E-21 2.70469E-42 7.93897E-16 55421495.28 1.4747E-16 1.8434E-110 1.5739E-62 1.83406E-26 4.30676E+94 3.52128E+69 4.01279E+28 0.230672016 4.3351596 1.23973E+53 2.43E-25 1967.169 5E+24 4.98104E+24 1 12.38723E-30 401426.05 401426.0506 2.908E+28 -562491132.4 562491132.4 4E+30 2.5E-31 1.09E+31 0.000032775 1.63875E-35 4.91625E-20 2.70469E-39 7.93897E-14 1752581564 1.4747E-13 1.8434E-104 4.9771E-58 1.83406E-26 4.30676E+90 3.52128E+66 4.01279E+28 2.30672E-05 43351.596 1.23973E+53 2.43E-22 196716.9 1.6E+26 1.57514E+26 1 12.38723E-27 401426051 401426050.6 2.908E+31 -5.62491E+11 5.62491E+11 4E+29 2.5E-30 1.09E+30 0.00032775 1.63875E-33 1.55465E-18 2.70469E-36 7.93897E-12 55421495282 1.4747E-10 1.84338E-98 1.5739E-53 1.83406E-26 4.30676E+86 3.52128E+63 4.01279E+28 2.30672E-09 433515959 1.23973E+53 2.43E-19 19671691 5E+27 4.98104E+27 1 12.38723E-24 4.014E+11 4.01426E+11 2.908E+34 -5.62491E+14 5.62491E+14 4E+28 2.5E-29 1.09E+29 0.0032775 1.63875E-31 4.91625E-17 2.70469E-33 7.93897E-10 1.75258E+12 1.4747E-07 1.84338E-92 4.9771E-49 1.83406E-26 4.30676E+82 3.52128E+60 4.01279E+28 2.30672E-13 4.335E+12 1.23973E+53 2.43E-16 1.97E+09 1.6E+29 1.57514E+29 1 12.38723E-21 4.014E+14 4.01426E+14 2.908E+37 -5.62491E+17 5.62491E+17 4E+27 2.5E-28 1.09E+28 0.032775 1.63875E-29 1.55465E-15 2.70469E-30 7.93897E-08 5.54215E+13 0.00014747 1.84338E-86 1.5739E-44 1.83406E-26 4.30676E+78 3.52128E+57 4.01279E+28 2.30672E-17 4.335E+16 1.23973E+53 2.43E-13 1.97E+11 5E+30 4.98104E+30 1 12.38723E-18 4.014E+17 4.01426E+17 2.908E+40 -5.62491E+20 5.62491E+20 4E+26 2.5E-27 1.09E+27 0.32775 1.63875E-27 4.91625E-14 2.70469E-27 7.93897E-06 1.75258E+15 0.147470075 1.84338E-80 4.9771E-40 1.83406E-26 4.30676E+74 3.52128E+54 4.01279E+28 2.30672E-21 4.335E+20 1.23973E+53 2.43E-10 1.97E+13 1.6E+32 1.57514E+32 1 12.38723E-15 4.014E+20 4.01426E+20 2.908E+43 -5.62491E+23 5.62491E+23 4E+25 2.5E-26 1.09E+26 3.2775 1.63875E-25 1.55465E-12 2.70469E-24 0.000793897 5.54215E+16 147.4700752 1.84338E-74 1.5739E-35 1.83406E-26 4.30676E+70 3.52128E+51 4.01279E+28 2.30672E-25 4.335E+24 1.23973E+53 2.43E-07 1.97E+15 5E+33 4.98104E+33 1 12.38723E-12 4.014E+23 4.01426E+23 2.908E+46 -5.62491E+26 5.62491E+26 4E+24 2.5E-25 1.09E+25 32.775 1.63875E-23 4.91625E-11 2.70469E-21 0.079389719 1.75258E+18 147470.0752 1.84338E-68 4.9771E-31 1.83406E-26 4.30676E+66 3.52128E+48 4.01279E+28 2.30672E-29 4.335E+28 1.23973E+53 0.000243 1.97E+17 1.6E+35 1.57514E+35 1 12.38723E-09 4.014E+26 4.01426E+26 2.908E+49 -5.62491E+29 5.62491E+29 4E+23 2.5E-24 1.09E+24 327.75 1.63875E-21 1.55465E-09 2.70469E-18 7.938971911 5.54215E+19 147470075.2 1.84338E-62 1.5739E-26 1.83406E-26 4.30676E+62 3.52128E+45 4.01279E+28 2.30672E-33 4.335E+32 1.23973E+53 0.243085 1.97E+19 5E+36 4.98104E+36 1 12.38723E-06 4.014E+29 4.01426E+29 2.908E+52 -5.62491E+32 5.62491E+32 4E+22 2.5E-23 1.09E+23 3277.5 1.63875E-19 4.91625E-08 2.70469E-15 793.8971911 1.75258E+21 1.4747E+11 1.84338E-56 4.9771E-22 1.83406E-26 4.30676E+58 3.52128E+42 4.01279E+28 2.30672E-37 4.335E+36 1.23973E+53 243.0852 1.97E+21 1.6E+38 1.57514E+38 1 10.002387225 4.014E+32 4.01426E+32 2.908E+55 -5.62491E+35 5.62491E+35 4E+21 2.5E-22 1.09E+22 32775 1.63875E-17 1.55465E-06 2.70469E-12 79389.71911 5.54215E+22 1.4747E+14 1.84338E-50 1.5739E-17 1.83406E-26 4.30676E+54 3.52128E+39 4.01279E+28 2.30672E-41 4.335E+40 1.23973E+53 243085.2 1.97E+23 5E+39 4.98104E+39 1 12.3872253 4.014E+35 4.01426E+35 2.908E+58 -5.62491E+38 5.62491E+38 4E+20 2.5E-21 1.09E+21 327750 1.63875E-15 4.91625E-05 2.70469E-09 7938971.911 1.75258E+24 1.4747E+17 1.84338E-44 4.9771E-13 1.83406E-26 4.30676E+50 3.52128E+36 4.01279E+28 2.30672E-45 4.335E+44 1.23973E+53 2.43E+08 1.97E+25 1.6E+41 1.57514E+41 1 12387.2253 4.014E+38 4.01426E+38 2.908E+61 -5.62491E+41 5.62491E+41 4E+19 2.5E-20 1.09E+20 3277500 1.63875E-13 0.001554655 2.70469E-06 793897191.1 5.54215E+25 1.4747E+20 1.84338E-38 1.5739E-08 1.83406E-26 4.30676E+46 3.52128E+33 4.01279E+28 2.30672E-49 4.335E+48 1.23973E+53 2.43E+11 1.97E+27 5E+42 4.98104E+42 1 12387225.3 4.014E+41 4.01426E+41 2.908E+64 -5.62491E+44 5.62491E+44 4E+18 2.5E-19 1.09E+19 32775000 1.63875E-11 0.0491625 0.002704688 79389719112 1.75258E+27 1.4747E+23 1.84338E-32 0.00049771 1.83406E-26 4.30676E+42 3.52128E+30 4.01279E+28 2.30672E-53 4.335E+52 1.23973E+53 2.43E+14 1.97E+29 1.6E+44 1.57514E+44 1 12387225300 4.014E+44 4.01426E+44 2.908E+67 -5.62491E+47 5.62491E+47 4E+17 2.5E-18 1.09E+18 327750000 1.63875E-09 1.554654755 2.7046875 7.93897E+12 5.54215E+28 1.4747E+26 1.84338E-26 15.7390197 1.83406E-26 4.30676E+38 3.52128E+27 4.01279E+28 2.30672E-57 4.335E+56 1.23973E+53 2.43E+17 1.97E+31 5E+45 4.98104E+45 1 12.38723E+12 4.014E+47 4.01426E+47 2.908E+70 -5.62491E+50 5.62491E+50 4E+16 2.5E-17 1.09E+17 3277500000 1.63875E-07 49.1625 2704.6875 7.93897E+14 1.75258E+30 1.4747E+29 1.84338E-20 497711.504 1.83406E-26 4.30676E+34 3.52128E+24 4.01279E+28 2.30672E-61 4.335E+60 1.23973E+53 2.43E+20 1.97E+33 1.6E+47 1.57514E+47 1 12.38723E+15 4.014E+50 4.01426E+50 2.908E+73 -5.62491E+53 5.62491E+53 4E+15 2.5E-16 1.09E+16 32775000000 1.63875E-05 1554.654755 2704687.5 7.93897E+16 5.54215E+31 1.4747E+32 1.84338E-14 1.5739E+10 1.83406E-26 4.30676E+30 3.52128E+21 4.01279E+28 2.30672E-65 4.335E+64 1.23973E+53 2.43E+23 1.97E+35 5E+48 4.98104E+48 1 12.38723E+18 4.014E+53 4.01426E+53 2.908E+76 -5.62491E+56 5.62491E+56 4E+14 2.5E-15 1.09E+15 3.2775E+11 0.00163875 49162.5 2704687500 7.93897E+18 1.75258E+33 1.4747E+35 1.84338E-08 4.9771E+14 1.83406E-26 4.30676E+26 3.52128E+18 4.01279E+28 2.30672E-69 4.335E+68 1.23973E+53 2.43E+26 1.97E+37 1.6E+50 1.57514E+50 1 12.38723E+21 4.014E+56 4.01426E+56 2.908E+79 -5.62491E+59 5.62491E+59 4E+13 2.5E-14 1.09E+14 3.2775E+12 0.163875 1554654.755 2.70469E+12 7.93897E+20 5.54215E+34 1.4747E+38 0.018433759 1.5739E+19 1.83406E-26 4.30676E+22 3.52128E+15 4.01279E+28 2.30672E-73 4.335E+72 1.23973E+53 2.43E+29 1.97E+39 5E+51 4.98104E+51 1 12.38723E+24 4.014E+59 4.01426E+59 2.908E+82 -5.62491E+62 5.62491E+62 4E+12 2.5E-13 1.09E+13 3.2775E+13 16.3875 49162500 2.70469E+15 7.93897E+22 1.75258E+36 1.4747E+41 18433.7594 4.9771E+23 1.83406E-26 4.30676E+18 3.52128E+12 4.01279E+28 2.30672E-77 4.335E+76 1.23973E+53 2.43E+32 1.97E+41 1.6E+53 1.57514E+53 1 1.000000001 2.38723E+27 4.014E+62 4.01426E+62 2.908E+85 -5.62491E+65 5.62491E+65 4E+11 2.5E-12 1.09E+12 3.2775E+14 1638.75 1554654755 2.70469E+18 7.93897E+24 5.54215E+37 1.4747E+44 18433759401 1.5739E+28 1.83406E-26 4.30676E+14 3521280000 4.01279E+28 2.30672E-81 4.335E+80 1.23973E+53 2.43E+35 1.97E+43 5E+54 4.98104E+54 1 1.000000003 2.38723E+30 4.014E+65 4.01426E+65 2.908E+88 -5.62491E+68 5.62491E+68 4E+10 2.5E-11 1.09E+11 3.2775E+15 163875 49162499998 2.70469E+21 7.93897E+26 1.75258E+39 1.4747E+47 1.84338E+16 4.9771E+32 1.83406E-26 43067568254 3521280 4.01279E+28 2.30672E-85 4.335E+84 1.23973E+53 2.43E+38 1.97E+45 1.6E+56 1.57514E+56 1 1.000000007 2.38723E+33 4.014E+68 4.01426E+68 2.908E+91 -5.62491E+71 5.62491E+71 4E+09 2.5E-10 10900000003 3.2775E+16 16387499.99 1.55465E+12 2.70469E+24 7.93897E+28 5.54215E+40 1.4747E+50 1.84338E+22 1.5739E+37 1.83406E-26 4306756.829 3521.280003 4.01279E+28 2.30672E-89 4.335E+88 1.23973E+53 2.43E+41 1.97E+47 5E+57 4.98104E+57 1 1.000000016 2.38723E+36 4.014E+71 4.01426E+71 2.908E+94 -5.62491E+74 5.62491E+74 4E+08 2.5E-09 1090000003 3.2775E+17 1638749992 4.91625E+13 2.70469E+27 7.93897E+30 1.75258E+42 1.4747E+53 1.84338E+28 4.9771E+41 1.83406E-26 430.6756868 3.521280026 4.01279E+28 2.30672E-93 4.335E+92 1.23973E+53 2.43E+44 1.97E+49 1.6E+59 1.57514E+59 1 1.000000037 2.38723E+39 4.014E+74 4.01426E+74 2.908E+97 -5.62491E+77 5.62491E+77 40000000 2.5E-08 109000002.7 3.2775E+18 1.63875E+11 1.55465E+15 2.70469E+30 7.93897E+32 5.54215E+43 1.4747E+56 1.84338E+34 1.5739E+46 1.83406E-26 0.043067573 0.00352128 4.01279E+28 2.30672E-97 4.335E+96 1.23973E+53 2.43E+47 1.97E+51 5E+60 4.98104E+60 1 1.000000088 2.38723E+42 4.014E+77 4.01426E+77 2.91E+100 -5.62491E+80 5.62491E+80 4000000 2.5E-07 10900002.73 3.2775E+19 1.63875E+13 4.91625E+16 2.70469E+33 7.93897E+34 1.75258E+45 1.4747E+59 1.84337E+40 4.9771E+50 1.83406E-26 4.30676E-06 3.52128E-06 4.01279E+28 2.3067E-101 4.34E+100 1.23973E+53 2.43E+50 1.97E+53 1.6E+62 1.57514E+62 0.999999999 1.000000208 2.38722E+45 4.014E+80 4.01426E+80 2.91E+103 -5.62491E+83 5.62491E+83 400000 2.49999E-06 1090002.725 3.27749E+20 1.63874E+15 1.55465E+18 2.70467E+36 7.93893E+36 5.54213E+46 1.47469E+62 1.84335E+46 1.5739E+55 1.83406E-26 4.3068E-10 3.52131E-09 4.01279E+28 2.3067E-105 4.34E+104 1.23973E+53 2.43E+53 1.97E+55 5E+63 4.98102E+63 0.999999996 1.00000049 2.38721E+48 4.014E+83 4.01423E+83 2.91E+106 -5.62487E+86 5.62487E+86 40000 2.49994E-05 109002.725 3.27742E+21 1.63867E+17 4.91607E+19 2.70448E+39 7.93857E+38 1.75252E+48 1.47459E+65 1.8431E+52 4.9766E+59 1.83406E-26 4.30719E-14 3.52154E-12 4.01279E+28 2.307E-109 4.33E+108 1.23973E+53 2.43E+56 1.97E+57 1.6E+65 1.57508E+65 0.999999988 1.000001156 2.38705E+51 4.014E+86 4.01396E+86 2.91E+109 -5.62449E+89 5.62449E+89 3570 0.000280034 9730.975 3.67124E+22 2.05614E+19 1.84306E+21 3.80126E+42 9.96104E+40 6.57027E+49 2.07259E+68 3.64112E+58 2.6224E+64 1.83406E-26 2.73571E-18 2.50548E-15 4.01279E+28 1.4653E-113 6.82E+112 1.23973E+53 3.42E+59 2.47E+59 5.9E+66 5.90507E+66 0.999999958 1.00000284 3.35509E+54 5.642E+89 5.64178E+89 4.09E+112 -7.90544E+92 7.90544E+92 1599 0.000625 4360 8.19375E+22 1.02422E+20 6.14531E+21 4.22607E+43 4.96186E+41 2.19073E+50 2.30422E+69 4.50043E+60 9.7209E+65 1.83406E-26 1.10253E-19 2.25362E-16 4.01279E+28 5.9052E-115 1.69E+114 1.23973E+53 3.8E+60 1.23E+60 2E+67 1.96893E+67 0.999999938 1.000003825 3.73004E+55 6.272E+90 6.27228E+90 4.54E+113 -8.78892E+93 8.78892E+93 1370 0.000729395 3735.975 9.56236E+22 1.39495E+20 7.74761E+21 6.71714E+43 6.75786E+41 2.76193E+50 3.66245E+69 1.13697E+61 1.948E+66 1.83406E-26 5.94375E-20 1.41786E-16 4.01279E+28 3.1835E-115 3.14E+114 1.23973E+53 6.04E+60 1.67E+60 2.5E+67 2.4823E+67 0.999999933 1.000004051 5.92872E+55 9.969E+90 9.9695E+90 7.22E+113 -1.39696E+94 1.39696E+94 1088 0.000918274 2967.525 1.20386E+23 2.21094E+20 1.09442E+22 1.34034E+44 1.0711E+42 3.90145E+50 7.30803E+69 4.52696E+61 5.4906E+66 1.83406E-26 2.36604E-20 7.10566E-17 4.01279E+28 1.2673E-115 7.89E+114 1.23973E+53 1.2E+61 2.65E+60 3.5E+67 3.50645E+67 0.999999924 1.000004412 1.18301E+56 1.989E+91 1.98931E+91 1.44E+114 -2.78748E+94 2.78748E+94 1100 0.000908265 3000.225 1.19074E+23 2.16301E+20 1.07657E+22 1.29699E+44 1.04788E+42 3.83784E+50 7.07167E+69 4.23887E+61 5.2264E+66 1.83406E-26 2.47206E-20 7.34315E-17 4.01279E+28 1.324E-115 7.55E+114 1.23973E+53 1.17E+61 2.6E+60 3.4E+67 3.44928E+67 0.999999925 1.000004394 1.14475E+56 1.925E+91 1.92497E+91 1.39E+114 -2.69733E+94 2.69733E+94 1000 0.000999001 2727.725 1.30969E+23 2.61676E+20 1.24186E+22 1.72582E+44 1.2677E+42 4.42708E+50 9.40983E+69 7.50532E+61 8.0222E+66 1.83406E-26 1.68907E-20 5.51852E-17 4.01279E+28 9.0467E-116 1.11E+115 1.23973E+53 1.55E+61 3.14E+60 4E+67 3.97886E+67 0.999999921 1.000004552 1.52325E+56 2.561E+91 2.56143E+91 1.86E+114 -3.58916E+94 3.58916E+94 900 0.001109878 2455.225 1.45505E+23 3.22986E+20 1.45424E+22 2.36659E+44 1.56471E+42 5.18419E+50 1.29036E+70 1.41132E+62 1.2882E+67 1.83406E-26 1.10869E-20 4.02434E-17 4.01279E+28 5.9382E-116 1.68E+115 1.23973E+53 2.13E+61 3.88E+60 4.7E+67 4.65932E+67 0.999999917 1.000004733 2.08881E+56 3.512E+91 3.51246E+91 2.54E+114 -4.92177E+94 4.92177E+94 400 0.002493766 1092.725 3.26933E+23 1.63059E+21 4.89787E+22 2.6845E+45 7.89943E+42 1.74603E+51 1.4637E+71 1.81597E+64 4.9215E+68 1.83406E-26 4.34999E-22 3.54776E-18 4.01279E+28 2.3299E-117 4.29E+116 1.23973E+53 2.41E+62 1.96E+61 1.6E+68 1.56925E+68 0.999999875 1.000006388 2.36941E+57 3.984E+92 3.9843E+92 2.89E+115 -5.58293E+95 5.58293E+95 40 0.024390244 111.725 3.19756E+24 1.55979E+23 1.49813E+24 2.51157E+48 7.55643E+44 5.34063E+52 1.36941E+74 1.58954E+70 1.4084E+73 1.83406E-26 4.75385E-26 3.79203E-21 4.01279E+28 2.5462E-121 3.93E+120 1.23973E+53 2.26E+65 1.87E+63 4.8E+69 4.79992E+69 0.99999961 1.000014829 2.21678E+60 3.728E+95 3.72764E+95 2.7E+118 -5.22329E+98 5.22329E+98 10 0.090909091 29.975 1.19182E+25 2.16694E+24 1.07804E+25 1.30053E+50 1.04978E+46 3.84308E+53 7.09097E+75 4.26204E+73 5.2478E+75 1.83406E-26 2.46309E-28 7.32316E-23 4.01279E+28 1.3192E-123 7.58E+122 1.23973E+53 1.17E+67 2.6E+64 3.5E+70 3.45399E+70 0.999999247 1.000024059 1.14788E+62 1.93E+97 1.93022E+97 1.4E+120 -2.7047E+100 2.7047E+100 9 0.1 27.25 1.311E+25 2.622E+24 1.24372E+25 1.731E+50 1.27024E+46 4.43372E+53 9.43808E+75 7.55047E+73 8.0584E+75 1.83406E-26 1.68233E-28 5.502E-23 4.01279E+28 9.0106E-124 1.11E+123 1.23973E+53 1.56E+67 3.15E+64 4E+70 3.98483E+70 0.99999921 1.000024916 1.52782E+62 2.569E+97 2.56913E+97 1.86E+120 -3.5999E+100 3.5999E+100 8 0.111111111 24.525 1.45667E+25 3.23704E+24 1.45667E+25 2.37449E+50 1.56819E+46 5.19283E+53 1.29466E+76 1.42075E+74 1.2947E+76 1.83406E-26 1.10377E-28 4.01096E-23 4.01279E+28 5.9119E-124 1.69E+123 1.23973E+53 2.13E+67 3.89E+64 4.7E+70 4.66709E+70 0.999999167 1.000025898 2.09578E+62 3.524E+97 3.52418E+97 2.55E+120 -4.9382E+100 4.9382E+100 7 0.125 21.8 1.63875E+25 4.09688E+24 1.73816E+25 3.38086E+50 1.98474E+46 6.19631E+53 1.84338E+76 2.88027E+74 2.1996E+76 1.83406E-26 6.89081E-29 2.81702E-23 4.01279E+28 3.6908E-124 2.71E+123 1.23973E+53 3.04E+67 4.92E+64 5.6E+70 5.56897E+70 0.999999117 1.000027042 2.98403E+62 5.018E+97 5.01783E+97 3.63E+120 -7.0311E+100 7.0311E+100 6 0.142857143 19.075 1.87286E+25 5.35102E+24 2.12362E+25 5.04665E+50 2.59232E+46 7.57044E+53 2.75163E+76 6.41779E+74 4.0115E+76 1.83406E-26 4.03927E-29 1.88719E-23 4.01279E+28 2.1635E-124 4.62E+123 1.23973E+53 4.54E+67 6.42E+64 6.8E+70 6.80398E+70 0.999999056 1.000028399 4.4543E+62 7.49E+97 7.49017E+97 5.43E+120 -1.0495E+101 1.0495E+101 5 0.166666667 16.35 2.185E+25 7.28333E+24 2.67607E+25 8.01389E+50 3.52843E+46 9.53985E+53 4.36948E+76 1.61833E+75 8.0273E+76 1.83406E-26 2.1803E-29 1.18843E-23 4.01279E+28 1.1678E-124 8.56E+123 1.23973E+53 7.2E+67 8.74E+64 8.6E+70 8.57399E+70 0.99999898 1.000030051 7.07326E+62 1.189E+98 1.18941E+98 8.61E+120 -1.6666E+101 1.6666E+101 4 0.2 13.625 2.622E+25 1.0488E+25 3.51778E+25 1.3848E+51 5.08094E+46 1.25405E+54 7.55047E+76 4.8323E+75 1.8234E+77 1.83406E-26 1.05145E-29 6.8775E-24 4.01279E+28 5.6316E-125 1.78E+124 1.23973E+53 1.24E+68 1.26E+65 1.1E+71 1.12708E+71 0.999998883 1.000032127 1.22226E+63 2.055E+98 2.0553E+98 1.49E+121 -2.88E+101 2.88E+101 3 0.25 10.9 3.2775E+25 1.63875E+25 4.91625E+25 2.70469E+51 7.93897E+46 1.75258E+54 1.4747E+77 1.84338E+76 4.9771E+77 1.83406E-26 4.30676E-30 3.52128E-24 4.01279E+28 2.3067E-125 4.34E+124 1.23973E+53 2.43E+68 1.97E+65 1.6E+71 1.57514E+71 0.999998751 1.000034862 2.38723E+63 4.014E+98 4.01426E+98 2.91E+121 -5.6249E+101 5.6249E+101 2 0.333333333 8.175 4.37E+25 2.91333E+25 7.56906E+25 6.41111E+51 1.41137E+47 2.69828E+54 3.49559E+77 1.03573E+77 1.8164E+78 1.83406E-26 1.36268E-30 1.48554E-24 4.01279E+28 7.2986E-126 1.37E+125 1.23973E+53 5.76E+68 3.5E+65 2.4E+71 2.42509E+71 0.999998558 1.000038732 5.65861E+63 9.515E+98 9.51528E+98 6.89E+121 -1.3333E+102 1.3333E+102 1 0.5 5.45 6.555E+25 6.555E+25 1.39053E+26 2.16375E+52 3.17559E+47 4.95705E+54 1.17976E+78 1.17976E+78 1.1262E+79 1.83406E-26 2.69172E-31 4.4016E-25 4.01279E+28 1.4417E-126 6.94E+125 1.23973E+53 1.94E+69 7.87E+65 4.5E+71 4.45518E+71 0.999998234 1.000044918 1.90978E+64 3.21E+99 3.2114E+99 2.33E+122 -4.4999E+102 4.4999E+102 0 1 2.725 1.311E+26 2.622E+26 3.933E+26 1.731E+53 1.27024E+48 1.40207E+55 9.43808E+78 7.55047E+79 2.5483E+80 1.83406E-26 1.68233E-32 5.502E-26 4.01279E+28 9.0106E-128 1.11E+127 1.23973E+53 1.56E+70 3.15E+66 1.3E+72 1.26012E+72 0.999997502 1.000057838 1.52782E+65 2.57E+100 2.5691E+100 1.86E+123 -3.5999E+103 3.5999E+103 References [1] Lynden-Bell, D., Wood, R.: The gravothermal catastrophe in isothermal spheres and the onset of red-giant structure for stellar systems. 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