aa r X i v : . [ phy s i c s . g e n - ph ] D ec Structure of the Star with Ideal Gases
Ying-Qiu Gu ∗ School of Mathematical Science, Fudan University, Shanghai 200433, China (Dated: November 10, 2018)In this paper, we provide a simplified stellar structure model for ideal gases, inwhich the particles are only driven by gravity. According to the model, the structuralinformation of the star can be roughly solved by the total mass and radius of a star.To get more accurate results, the model should be modified by introducing otherinteraction among particles and rotation of the star.
Key Words: equation of state, thermodynamics, stellar structure, neutron star
PACS numbers: 95.30.Tg, 97.10.Cv, 97.60.Lf, 96.60.Jw
I. SOME PHENOMENA INSIDE A STAR
The spherical symmetric metric for a static star is described by Schwarzschild metric g µν = diag (cid:0) b ( r ) , − a ( r ) , − r , − r sin θ (cid:1) . (1.1)For the energy momentum tensor of perfect fluid T µν = ( ρ + P ) U µ U ν − P g µν = diag (cid:0) bρ, aP, r P, r sin θP (cid:1) , (1.2)where ρ ( r ) , P ( r ) are proper mass energy density and pressure, U µ = ( √ b, , , (cid:16) ra (cid:17) ′ = 1 − πGρr , ( G = − πGT ) , (1.3) b ′ b = a − r + 8 πGP ar, ( G = − πGT ) , (1.4) P ′ = − ( ρ + P ) b ′ b , ( T ν ; ν = 0) . (1.5)To determine the stellar structure of an irrotational star, we solve these equations. How-ever (1.3)-(1.4) is not a closed system, the solution depends on an equation of state(EOS) ∗ Electronic address: [email protected] P = P ( ρ ). For polytropes, we take P = P ρ γ [1]. For the compact stars, there are a lot ofEOS derived from particle models[2]-[7], which provide the structural information and pa-rameters such as the maximum mass for neutron stars. However, in these EOS the boostingeffects of the gravity on particles seem to be overlooked or adopted unconsciously. As shownin [8], the static equilibrium equation (1.5) is insufficient to describe such dynamical effectson fluid, and a Gibbs’ type law should be included. In this paper, according to this Gibbs’type law, we derive the stellar structure model, which shows that the boosting effects ofgravity play an important role to the stellar structure.Before expanding the model, we make a few simple calculations and examine the behaviorof the metric and particles inside a star to get some intuition. The first phenomenon is that,the temporal singularity and spatial singularity occur at different time and place if thespacetime becomes singular, and the temporal one seems to occur firstly.Denoting the mass distribution by M ( r ) = 4 πG Z r ρr dr, R = 2 M ( r ) . (1.6)Then by (1.3) we have solution a = (cid:16) − R ( r ) r (cid:17) − , if r < R, (cid:0) − R s r (cid:1) − , if r ≥ R, (1.7)where R is the radius of the star, and the Schwarzschild radius becomes R s = 2 M ( R ) = R ( R ) = 8 πG Z R ρr dr. (1.8)For any normal star with R > R s . From the above solution we learn a ( r ) is a continuousfunction and a ≥ , a (0) = lim r →∞ a = 1 , a max = a ( r m ) , (0 < r m ≤ R ) . (1.9)So the spatial singularity a → ∞ does not appear at the center of the star when thesingularity begins to form.On the other hand, by (1.4) and P ≥
0, we find b ′ ( r ) is a continuous function satisfying b ′ (0) = 0 , b ′ ( r ) > , ( ∀ r > , b = 1 − R s r , ( r ≥ R ) . (1.10)(1.9) shows b ( r ) is a monotone increasing function of r with smoothness at least C ([0 , ∞ )).Consequently, the temporal singularity b → M e t r i c f un c t i on s a (r) & b (r) ← R ← R ← R R s =1 ↓↓ Temporalsingularitya(r)>1 b(r)<1 b(0) → → infinity FIG. 1: The trends of ( a, b ) as R → R s , which show the spatial singularity does not occur at thecenter, but temporal singularity may occur at the center before a → ∞ The trends of a ( r ) , b ( r ) are shown in FIG.1, where we take the Schwarzschild radius R s = 1 as length unit. From FIG.1 and some simplified calculations, we find b (0) → a → ∞ .The second phenomenon is that, the particles near the center of the star are unbalanced,and violent explosion takes place inside the star before the temporal singularity occurs.When b (0) →
0, by (1.10) we have b → b r α , ( α > , if r ≪ R. (1.11)Substituting it into (1.5), we find − P ′ → ( ρ + P ) α r → + ∞ , ( r → . (1.12)According to fluid mechanics, − ∂ r P corresponds to the radial boosting force, so (1.12) meansviolent explosion.More clearly, we examine the motion of a particle inside the star. Solving the geodesicin the orthogonal subspace ( t, r, θ ), we get[9, 10]˙ t = 1 C b , ˙ θ = C r , ˙ ϕ = 0 , ˙ r = 1 r (cid:18) C b − C r − (cid:19) , (1.13)where C , C are constants. The normal velocity of the particle is given by v r = adr bdt = 1 − C b (cid:18) C r (cid:19) , v θ = r dθ bdt = C C br , v ϕ = 0 . (1.14)The sum of the speeds provides an equality similar to the energy conservation law v = 1 − C b ( r ) , with v ≡ v r + v θ + v ϕ . (1.15)(1.15) holds for all particles with v ϕ = 0 due to the symmetry of the spacetime.From (1.15) we learn v → b →
0, this means all particles escape at light velocitywhen the temporal singularity occurs. So instead of a final collapse, the fate of a star withheavy mass may be explosion and disintegration. The gravity of a star drives the insideparticles to move rapidly and leads to high temperature. How the particles to react to thecollapse of a star needs further research with dynamical models. A heuristic computationfor axisymmetrical collapse is presented in [11], which reveals that the fate of a collapsingstar sensitively depends on the parameters in the EOS.
II. THE EQUATIONS FOR STELLAR STRUCTURE
In this paper, we simply take the star as a ball of ideal gases, which satisfies the followingassumptions:(A1) All particles are classical ones only driven by the gravity, namely, they are charac-terized by 4-vector momentum p µk and move along geodesic.(A2) The collisions among particles are elastic, and then they can be ignored in statisticalsense[9, 10].(A3) The nuclear reaction and radiation are stable and slowly varying process in a normalstar, in contrast with the mass energy, the energy related to this process is small noise, sowe treat all photons as particles and omit the process of its generation and radiation.These are some usual assumptions suitable for fluid stars. However, together with (1.3)-(1.4), they are enough to give us a simplified self consistent stellar structure theory. In [8],we derived the EOS for such system as follows N = N [ J ( J + 2 σ )] , J ≡ kT ¯ mc , (2.1) ρ = N (cid:18) ¯ mc + 32 kT (cid:19) = ̺ [ J ( J + 2 σ )] (cid:18) J (cid:19) c , (2.2) P = N kT σ ¯ mc + kT σ ¯ mc + kT ) = ̺ [ J ( J + 2 σ )] c σ + J ) , (2.3)where N is the number density of particles, N = N ( ¯ m, σ ) is related to property of theparticles but independent of J , c = 2 . × m/s the light velocity, σ ˙= a factor reflectingthe energy distribution function, ¯ m the mean static mass of all particles, J dimensionlesstemperature, which is used as independent variable, ( ρ, P ) are usual energy density andpressure, ̺ a mass density defined by ̺ ≡ N ¯ m. (2.4)By (2.2) and (2.3), we get the polytropic index γ is not a constant for large range of T satisfying 1 < γ < , and the velocity of sound C sound ≡ c s dPdρ = √ (cid:18) c J (2 σ + J )(5 σ + 8 σJ + 4 J )( σ + J ) [2 σ + (2 + 5 σ ) J + 4 J ] (cid:19) < √ c, (2.5)which shows that the EOS is regular and the causality condition holds.Equation (1.5) can be rewritten as dbdJ = − bρ + P dPdJ . (2.6)Substituting (2.2) and (2.3) into (2.6), we solve b = 4( R − R s )( σ + J ) R [2 σ + (2 + 5 σ ) J + 4 J ] , (2.7)where R is the radius of the star, and R s = 2 M tot the Schwarzschild radius. Substituting(1.7) and (2.2)-(2.7) into (1.3) and (1.4), instead of the Tolman-Oppenheimer-Volkoff equa-tion and Lane-Emden equation[1], we get the following dimensionless equations for stellarstructure, R ′ ( r ) = (cid:18) rχ (cid:19) (2 + 3 J )[ J ( J + 2 σ )] , (2.8) J ′ ( r ) = − ξ ( J )2( r − R ) (cid:18) rχ (cid:19) [ J ( J + 2 σ )] + R r ( σ + J ) ! , (2.9)where χ is a constant length scale, χ = c (4 πG̺ ) − = c (4 πG N ¯ m ) − , (2.10) ξ ≡ σ + (2 + 5 σ ) J + 4 J σ + 8 σJ + 4 J , ( ξ ≈ . (2.11)The exact solution to (2.8) and (2.9) seems not easy to be obtained. However, they aredimensionless equations convenient for numeric resolution. If we take χ = 1 as the unit oflength, the solution can be uniquely determined by the following boundary conditions R (0) = 0 , J (0) = J > , J ( r ) = 0 , ( ∀ r ≥ R ) . (2.12) −1 −0.5 0 0.5 1 1.5 2 2.5−0.4−0.3−0.2−0.100.10.20.3 Dimensionless Mass vs. TemperatureDimensionless Central Temperature log(1/ J ) D i m en s i on l e ss S c h w a r zsc h il d R ad i u s l og R s ↑ Rs=1.9, J =0.13 ↑ Rs=0.97, J =1.2 ↓ Rs=1.2, J =3.0(Length unit: χ ) ← T increasing direction J Y BJ =0.3 N FIG. 2: Relation between mass and cen-tral temperature similar to H-R dia-gram, where it is concentrated by thescale χ and J . D i m en s i on l e ss S c h w a r zsc h il d R ad i u s l og R s ↑ Rs=1.9, R=8.4 ↑ Rs=0.97, R=5.5 ↓ Rs=1.2, R=6.2(Length unit: χ ) Y=X+log(Rs/R), → X AY MRs=1.59 R=5.77
FIG. 3: Relation between mass and Ra-dius. All structural information of astar are determined by a given radii pair( R s , R ). Adjusting J , we get different solution. Some solutions are displayed in FIG.2-FIG.5. Usuallywe can easily measure the radius R and the total mass or equivalent Schwarzschild radius R s of a star. Then the other structural parameters can be determined by this radii pair( R s , R ). In what follows, we show how to use FIG.2 and FIG.3 to solve practical problems.For the sun, we have the radii pair as R s = 2 . × m , R ⊙ = 6 . × m . (2.13)By R s /R ⊙ = 4 . × − , according to the relation shown in FIG.3, we can solve theintersection A and get the dimensionless radii pair X = log( R/χ ) = 2 . , Y = log( R s /χ ) = − . . (2.14)Consequently, we have χ = 10 − . R = 3 . × m . (2.15)By Y we get intersection B in FIG.2, and then get the central dimensionless temperature J = 1 . × − for the sun. Taking it as initial value we can solve (2.8) and (2.9), andthen get detailed structural information for the sun.By (2.10) and (2.15), we get ̺ = c πGχ = 9 . × (kg / m ) . (2.16) −1 −0.5 0 0.5 1 1.5 2 2.5−0.5−0.4−0.3−0.2−0.100.10.20.30.4 Influence of σ on Mass MDimensionless Central Temperature log(1/ J ) D i m en s i on l e ss S c h w a r zsc h il d R ad i u s l og R s M=R s /2 δ M ~ 15 % M σ =0.75 σσ =2/5 σ =1.25 σ FIG. 4: The influence of energy distri-bution on solutions. The results are notsensitive to σ . M e t r i c ( a , b ) , M a ss & T e m pe r a t u r e ( R , J ) r=R RsR(r)=2M(r) J(r)a(r) b(r)(Length unit: χ ) FIG. 5: Structural functions for thecompactest stars. The trends are typi-cal for all stars.
Then by (2.2) and (2.3) we solve the mass density and pressure at the center ρ (0) = ̺ [ J ( J + 2 σ )] (cid:18) J (cid:19) = 8 . × (kg / m ) , (2.17) P (0) = 12 ̺ [ J ( J + 2 σ )] ( σ + J ) − c = 8 . × (MPa) . (2.18)The temperature depends on the mean mass ¯ m . By (2.1), we have T c = ¯ mc J /k = n p ( m p c J /k ) = 1 . × n p (K) , (2.19)where m p = 1 . × − kg is the static mass of proton, n p is the equivalent proton numberfor the particles.If the ionization in the sun is about H + + N + + 2 e − , then we have n p = (70% × × / . , (2.20)¯ m = n p m p = 2 . × − (kg) , (2.21) T c = 1 . × n p = 1 . × (K) , (2.22) N = c / (4 πG ¯ mχ ) = 4 . × (m − ) . (2.23)The compactest stars(with the maximum R s /R ) correspond to the points M, N in FIG.3and FIG.2. The radii pair is R s : R = 1 .
59 : 5 .
77 and the central temperature J = 0 . M ⊙ , we get the length unit and radius as χ = R s / .
59 = 1 .
86 km , R = 5 . χ = 10 . . (2.24)Along the above procedure, we solve ρ (0) = 8 . × kg / m , (2.25) P (0) = 1 . × MPa , (2.26) T (0) = 3 . × n p K . (2.27)These are typical data for a neutron star[1]-[7]. The metric functions ( a, b ) and the mass,temperature distributions ( R , J ) for this star are displayed in FIG.5. III. DISCUSSION AND CONCLUSION
The above numerical results show that, although the orbits of the particles are not simpleellipses due to collisions, the influence of the gravity still exists and leads to high temperatureinside the star. The main part of the EOS may be related to gravity.In contrast with (2.17) and (2.18), we find the central density and pressure in the sun areabout one order of magnitude less than the current data ρ (0) = 1 . × kg / m , P (0) = 2 . × MPa . (3.1)This difference is an unsettled serious problem, which seems to be caused by the dynamicaleffect of gravity.FIG.4 shows that the solutions are not very sensitive to the concrete energy distribu-tion or σ , so one needs not to solve specific problems via calculating complex distributionfunctions[8].For photons, by the Stefan-Boltzmann’s law ρ = 8 π ( kT ) hc ) , (3.2)we get N → π (cid:16) ¯ mch (cid:17) = 2 π (cid:16) ¯ mc ~ (cid:17) , ( ¯ m → . (3.3)How to determine the concrete function N ( ¯ m, σ ) is an interesting problem.The dimensionless equations (2.8) and (2.9) simplify the relations between parameters.This function is similar to that of the similarity theory [12]. However in these equations theinformation of the interaction among particles is ignored, so it can not provide the criticaldata such as the maximum density, the largest mass. To get such data the potentials andinteractive fields should be introduced to the energy momentum tensor[8]. Acknowledgments
The author is grateful to his supervisor Prof. Ta-Tsien Li for his encouragement andguidance. Thanks to Prof. Jia-Xing Hong for kind help. [1] S. L. Weinberg,
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