Canadian Journal of Physics. Estimation of the physical parameters of planets and stars in the gravitational equilibrium model

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1 Canadian Journal of Physis Estimation of the physial parameters of planets and stars in the gravitational equilibrium model Journal: Canadian Journal of Physis Manusript ID jp r1 Manusript Type: Artile Date Submitted by the Author: 15-De-15 Complete List of Authors: Fedosin, Sergey; N/A Keyword: field theory, aeleration field, pressure field, gravitational field, gravitational equilibrium model

2 Page 1 of 9 Canadian Journal of Physis Estimation of the physial parameters of planets and stars in the gravitational equilibrium model Sergey G. Fedosin Sviazeva Str. -79, Perm, 61488, Perm region, Russian Federation intelli@list.ru Abstrat: The motion equations of matter in the gravitational field, aeleration field, pressure field and other fields are onsidered based on the field theory. This enables us to derive simple formulas in the framework of the gravitational equilibrium model, whih allow us to estimate the physial parameters of osmi bodies. The aeleration field oeffiient and the pressure field oeffiient are a funtion of the state of matter, and their sum is lose in magnitude to the gravitational onstant G. In the presented model the dependene is found of the internal temperature and pressure on the urrent radius. The entral temperatures and pressures are alulated for the Earth and the Sun, for a typial neutron star and a white dwarf. The heat flux and the thermal ondutivity oeffiient of the matter of these objets are found, and the formula for estimating the entropy is provided. All the quantities are ompared with the alulation results in different models of osmi bodies. The disovered good agreement with these data proves the effetiveness and universality of the proposed model for estimating the parameters of planets and stars and for more preise alulation of physial quantities. Keywords: field theory; aeleration field; pressure field; gravitational field; gravitational equilibrium model. PACS:..+p,.5.-s,.65.pm, 4.4.-b, 95..Sf 1

3 Canadian Journal of Physis Page of 9 Résumé: Sur la base de la théorie des hamps les équations du mouvement de la matière dans le hamp gravitationnel, le hamp de l'aélération, le hamp de pression et dans autres hamps sont onsidérées. Cela permet de dériver dans le domaine du modèle d'équilibre gravitationnel des formules simples qui permettent de faire des estimations des paramètres physiques des orps spatiaux. Le oeffiient du hamp de l'aélération et le oeffiient du hamp de pression sont une fontion de l'état de la matière, et leur somme est prohe en amplitude à la onstante gravitationnelle G. Dans le présent modèle, les dépendanes de temperature et pression intérieures du rayon atuel sont trouvées. Les températures entrales et les pressions sont alulées pour la Terre et le Soleil, pour une étoile à neutrons typique et une naine blanhe. Le flux thermique et le oeffiient de ondutivité thermique de la matière de es objets sont trouvés et une formule d'estimation de l'entropie est présentée. Toutes les quantités sont omparées ave les résultats des aluls en les différents modèles de orps spatiaux. Une bonne onformité trouvée ave les données onfirme l'effiaité et l'universalité du modèle proposé pour estimer les paramètres des planètes et des étoiles, et pour les aluls plus préis de quantités physiques. Les mots-lés: la théorie du hamp, le hamp d aélération, le hamp de pression, le hamp gravitationnel, modèle d équilibre gravitationnel. 1. Introdution The most aurate models of osmi objets inlude detailed numerial alulations of ertain internal strutures (the solid or liquid ore, shell, onvetive zone) with the use of equations of the state of different phases of matter and ating fields. For ompat objets it is neessary to take into aount the quantum and relativisti effets. However, physis has in store the models that allow us to quikly estimate the harateristi parameters of planets and stars based only on the observable data, suh as radius, luminosity, spetrum, surfae temperature, gravitational redshift, asteroseismology

4 Page of 9 Canadian Journal of Physis data, et. A well-known example is a polytropi model, in whih the gas pressure is related to the mass density by a polytrope at the onstant thermal apaity of the matter [1-]. With polytropi index n and n / the model gives the orret order of suh quantities as the entral density, temperature, pressure, potential gravitational energy and a number of other quantities. White dwarfs are objets, in whih the eletron gas degenerates and makes the major ontribution to the pressure in the matter. For neutron stars, the same is true for the neutron gas. There is a well-known simple alulation of the state of matter in white dwarfs and neutron stars, based on the equality of the gravitational energy and quantum mehanial energy and providing the typial values of the masses and radii of these objets. A more detailed analysis leads to the Chandrasekhar limit [4-5] as the greatest mass of a white dwarf, beyond whih it beomes a neutron star. In [6] the struture of ompat stars is modeled by solving the equation of hydrostati equilibrium by parameterization of the mass density dependene on the radius. This leads to the dependene of the masses and radii of the objets on the entral density and the dependene of the pressure on the radius, expressed in terms of the gamma-funtion and the hypergeometri funtion. The natural drawbaks of the above-mentioned approahes are the limited range of appliation or the low auray of preditions of physial quantities and the internal struture of objets. Next, we will present the model of gravitational equilibrium, whih is based on the field theory. In order to illustrate the possibilities of this model, we will alulate some physial parameters of a number of objets and will ompare them with the results of alulations made by other authors. The positive aspet of the proposed approah is its universality, whih allows applying it to any osmi objets. In addition, this model provides very simple

5 Canadian Journal of Physis Page 4 of 9 formulas to estimate the parameters of planets and stars at minimum of neessary assumptions.. The model desription In the model of gravitational equilibrium it is assumed that the orresponding objet (planet, star) is in a state when the proesses of energy exhange between the gravitational field and other fields have finished in it. From a theoretial point of view, all the fields ating in osmi objets an be viewed as the omponents of a single general field [7]. In addition to the gravitational field, whih is the main omponent, ontribution to the general field an also be made by the eletromagneti field, pressure field, aeleration field, dissipation field, strong interation field, weak interation field, as well as other fields in the matter of the objets under onsideration. At equilibrium, all the fields are relatively independent, beause the energy fluxes between the fields and the matter on the average tend to zero. The expression for the general field equations follows from the priniple of least ation: 4 s J, s, (1) where s is the general field tensor, is the general field oeffiient, is the Levi-Civita symbol or ompletely antisymmetri unit tensor, J u is the mass four-urrent, is the mass density in the referene frame assoiated with the partile, dx u is the four-veloity of a point partile, is the speed of light. ds 4

6 Page 5 of 9 Canadian Journal of Physis Sine the general field tensor is the sum of tensors of partiular fields, then the equations of any field an be represented in the form of (1) after the respetive substitution of the field tensor, the onstant of this field and the four-urrent. A harateristi feature of (1) is the fat that the equations of the general field and of eah partiular field have the form of Maxwell equations for the eletromagneti field, written in a ovariant form in the urved spae for non-inertial referene frames.. The aeleration field and the temperature The equations of the aeleration field aording to (1) have the following form [8]: 4 u J, u, () where u is the aeleration field tensor, is the aeleration field oeffiient. The tensor u inludes the vetor omponents S and N, whih an be found by the rule: 1 ui ui iu S i, u i j iu j j ui N k, () where the indies i, j, k form a triples of non-reurring numbers of the form 1,, or,1, or,,1; the three-vetors S and N an be expanded into the omponents: S S i ( S1, S, S) ( Sx, S y, Sz ) ; N i N1 N N N x N y N z N (,, ) (,, ). 5

7 Canadian Journal of Physis Page 6 of 9 For simpliity, we will onsider equations () in the flat Minkowski spae, that is, within the framework of the speial theory of relativity. In this ase, equations () are written as the equations for strength S and for the solenoidal vetor N of the aeleration field: 1 S 4 v S 4, N, N, t N S. (4) t With the help of the vetors S and N we an form an aeleration four-vetor, harateristi of the body partiles moving at the veloity v and having the Lorentz fator : du a u J d S v, dui ai ui J [ ] d S vn. (5) A gravitationally bound body usually has a spherial shape, so that the four-aeleration a with a ovariant index will be a ertain oordinate funtion. The vetors S and N also allow us to alulate the stress-energy tensor of the aeleration field B and the vetor i K B [ SN ], whih is the vetor of the energy-momentum flux density of the 4 aeleration field. The main reason that the aeleration field of the matter partiles has its own energy density, energy flux density and field strength is the gravitation fore. Under the ation of this fore the gradients of pressure, temperature, mass density and other quantities are formed in osmi bodies. The loser to the enter of the body we approah, the higher the temperature and, onsequently, the average veloity of the partiles and the value of the four-veloity 6

8 Page 7 of 9 Canadian Journal of Physis beome. For a single partile, the four-potential of the aeleration field is the ovariant fourveloity u dx dx g of this partile. However, in ase of a set of losely ds ds interating partiles it is not so in the total four-potential of the system s aeleration field u, U the salar potential and the vetor potential U of the aeleration field beome independent quantities as a onsequene of different rules of summation of ontributions from the salar and vetor quantities of different partiles. In [9], we alulated the energy and the vetor of the energy-momentum flux density K of the aeleration field for a set of similarly harged partiles that form a gravitationally bound system in the form of some liquid and filling a spherial volume. The same was done for other fields, inluding the gravitational and eletromagneti fields, as well as the pressure field. It allowed estimating the aeleration field oeffiient in a first approximation with the help of the gravitational onstant G, the vauum permittivity and the relation q/ m for the partiles in question: q G. (6) 4 m In artile [1], the onept of aeleration field allowed us to alulate the relativisti energy of the system of partiles and the gravitational mass of the system; and in [11] it allowed us to derive the relativisti Navier-Stokes equations for visous harged matter, taking into aount the pressure field and dissipation field. Sine in the definition u u u u u the aeleration tensor is expressed in terms of the four-potential of the aeleration field, in equations () we an pass on from the aeleration tensor to the four-potential u. This leads to the wave equations for the potentials and U of the aeleration field [1]: 7

9 Canadian Journal of Physis Page 8 of 9 4 ( s s g u g s s s u su s u u s ) J. (7) In (7) u g u is the four-potential of the aeleration field, expressed with a ontravariant index using the metri tensor g, and are the Christoffel symbols, whih are the funtion of g. We solved equation (7) in [9] for the ase of a set of randomly moving partiles without general rotation, whih are onneted with eah other by means of gravitation and the eletromagneti field inluding the pressure field. In Minkowski spae the salar omponent (7) is redued to the equation for the Lorentz fator, thus we obtain the following: gu u, 4,, (8) u u where the Lorentz fator 1 1 v is the funtion of the urrent radius r inside the sphere and the average value for the set of partiles, v is the average veloity of the partiles in the referene frame K, whih is assoiated with the enter of inertia of the referene frame. The solution of (8) in ase of uniform density is the following expression: sin, (9) r 4 r r 4 8

10 Page 9 of 9 Canadian Journal of Physis where 1 1 v is the Lorentz fator for the veloities v of the partile in the enter of the sphere. From (9) by raising to the square we obtain approximately the following: 4 r v v, (1) so that as the urrent radius r inside the sphere inreases while moving from the enter to the periphery of the sphere, the veloity v of the partiles dereases. Assuming that the veloity v is the root mean square veloity of the partiles, and taking into aount its relation to the kineti temperature in the form of 1 mu v kt, where k is the Boltzmann onstant, expression (1) is transformed into the temperature-radius dependene: 4 mu r mum ( r) T T T. (11) 9k kr where T is the temperature at the enter of the sphere, M() r is the mass within the urrent radius r. m u is the mass of one gas partile, From (11) it follows that the temperature inside the osmi bodies in the first approximation dereases parabolially, depending on the square of the radius of the observation point. Assuming in (11) that at the body radius r 9 R the body mass is equal to

11 Canadian Journal of Physis Page 1 of 9 M( R) M, and negleting the surfae temperature T( R ), we find the formula for the temperature in the enter of the body: T mm u. (1) kr 4. The pressure field The four-potential of the pressure field for one partile is found by multiplying the funtion depending on the pressure and density by the ovariant four-veloity [8-9], [1]: p, Π, (1) u where p and denote the pressure and density in the referene frame K p of the p partile, the dimensionless ratio is proportional to the pressure energy of the partile per unit mass of the partile, and Π are the salar and vetor potentials of the pressure field. For a system of partiles, (1) an also be onsidered valid, but u should be onsidered not as the four-veloity of an individual partile, but as the four-veloity averaged with respet to some ensemble of partiles near the observation point. The equations of the pressure field aording to (1) are as follows: 4 f J, f, (14) 1

12 Page 11 of 9 Canadian Journal of Physis where f is the pressure field tensor, is the pressure field oeffiient. The tensor f is the result of applying the four-url to the four-potential : f, (15) f C C C x y z C C x y C I z I y I I x z I z y I x. (16) here the tensor omponents are the omponents of the strength vetor C and the solenoidal vetor I of the pressure field. In Minkowski spae, equations of the pressure field (14) are onsiderably simplified: 1 C 4 v C 4, I, I, t I C. (17) t It is suffiient to know the vetors C and I in order to determine the stress-energy tensor P, the vetor of the energy flux density F of the pressure field and the pressure fore density in the matter. 11

13 Canadian Journal of Physis Page 1 of 9 field: Substituting (15) into (14) gives the wave equation for the four-potential of the pressure 4 ( s s g g s s s s s s ) J. (18) In ase of a self-gravitating system of partiles without rotation, whih oupies a spherial volume, equation (18) and its solution for the salar potential in view of (9) is redued to the following: 4. (19) r r sin 4. r 4 p Sine there is a relation, at a onstant density the solution of (19) is transformed into the dependene of the pressure inside the system: r M ( r) p p p. () 4 8 r In large osmi bodies the pressure on the surfae at r R is rather low and we an neglet it in (). Then, to estimate the pressure in the enter of the body we obtain a simple formula: 1

14 Page 1 of 9 Canadian Journal of Physis p M. (1) 4 8 R 5. The gravitational field and the equation of motion of matter The equations of the gravitational field in the ovariant theory of gravitation [1-16] orrespond in their form to (1): 4 G Φ J, Φ, () where Φ is the gravitational field tensor, whih inludes the omponents of the gravitational strength vetor Γ and the solenoidal vetor Ω of the torsion field: Φ x y z x y z z y. z x y x The starting point of the ovariant theory of gravitation is the four-potential of the gravitational field D, D, whih is desribed by the field s salar potential and vetor potential D. The four-potential is part of the Lagrangian and it allows us to derive the equations of the gravitational field from the priniple of least ation, while the field tensor is assoiated with the four-potential: Φ D D D D. This equality an be written in vetor notation as follows: 1

15 Canadian Journal of Physis Page 14 of 9 D Γ, Ω D. () t In Minkowski spae, the mass density, the mass urrent density J v, the mass four-urrent J (, ) (, ) v J, and equations () have the following form: 1 Γ 4 GJ Γ 4G, Ω, Ω, t Ω Γ. t If in () we turn from the tensor Φ to the four-potential D, we will obtain the wave equation: 4 ( s s G g D g s s s D s D s D D s ) J. In Minkowski spae, this equation falls into two equations for the potentials of the gravitational field: 1 4 G, t 1 D 4 GJ D. (4) t If the system of partiles does not move in spae as a whole and has no general rotation, then it has the vetor potential D and the torsion field Ω. And if the potential does not depend on time, then the gravitational field beomes stati. In this ase, aording to [9], 14

16 Page 15 of 9 Canadian Journal of Physis the salar potential inside the body at a onstant mass density in view of (9) is defined by the formula: G a G r G ( r a ) i os 4 sin 4. r 4 (5) In the onept of the general field [7] the equation of motion of matter is as follows: du u J a uu Φ J F j f J h J J w J. d (6) where a is the four-aeleration in the urved spae, F is the eletromagneti tensor, j is the eletromagneti four-urrent, h is the dissipation field tensor, is the strong interation field tensor, w is the weak interation field tensor. The tensors and w are important in the ases, when the equilibrium inside osmi bodies is supported by the additional pressure from thermonulear reations or radioative deay. The vetor omponent (6) in view of (5) is redued to the following: dui S [ v N] d Γ vω E vb C vi X v Y L vμ Q v π q q [ ] [ ] [ ] [ ] [ ] [ ], (7) 15

17 Canadian Journal of Physis Page 16 of 9 where the vetors X, L and Q denote the strengths of the dissipation field, strong interation field and weak interation field, respetively, and the vetors Y, μ and π are the solenoidal vetors of these fields. If we onsider the equation of motion in Minkowski spae and take into aount only the gravitation, the aeleration field and the pressure field, then in the stati ase we an assume N, Ω, and I. In view of () and (5) we have the following: 4 G r G M ( r) Γ i r. (8) r The pressure field strength is given by the formula: Π C, and if the vetor t potential is Π, then using (19) we find the following: 4 r Mr ( ) r C r. (9) Aording to () and the definition of the aeleration field four-potential in the form of u, U, the aeleration field strength equals: U S. For a non-rotating t body both the vetor potential U and the solenoidal vetor this ase, using (8-9) inside the body we find the following: N U are equal to zero. In 4 r Mr ( ) r S r. () 16

18 Page 17 of 9 Canadian Journal of Physis Substituting Γ, C and S in (7) in the absene of other fields, we obtain the equality for the field strengths and the relation for the field oeffiients: d( v) S Γ C, G. (1) dt Outside the system, near its border, we an assume that M () r M, r R, and there is a gravitational fore ating on a ertain test partile, but the pressure tends to zero due to the low mass density. Then in (1) we should assume C, and the gravitational field strength Γ will determine the entripetal aeleration of individual partiles, rotating around the system. In the non-relativisti ase, this an be written as: v R GM R. In view of the equality 1 mu v kt for the kineti temperature of the partiles near the surfae of the system we must obtain: T s Gmu M. () kr The kineti temperature T s refers to the kineti energy of the partiles during their rotation around the system and it should exeed the average temperature T of the gas of these partiles, whih is the measure of the gas thermal energy near the system. The best onditions for () to hold must be in gas louds, for example, in Bok globules, small dark osmi louds of gas and dust. In [17] it was found that the radius of a typial globule is.5 parses, the mass is 11 Solar masses, and the reorded temperature of dust in some globules an reah 6 K. 17

19 Canadian Journal of Physis Page 18 of 9 Assuming in () that m u is equal to an atomi mass unit, we find the temperature of partiles on the surfae for a typial globule: Ts 5,5 K. If in (11) we assume M () r M, r R, T ( R) Ts, and taking into aount equation (6) for, the temperature in the enter of the globule is of the order of K, whih is lose enough to the observations. Using () for the Earth gives the kineti temperature T s of about 5 K, and for the Sun about 7.7 million degrees. Suh temperatures are atually observed at the Earth's ionosphere the thermal temperature T is over K, and at the Solar orona the average temperature is about 5 million degrees. Due to the ation of the Sun s gravitation the partiles are orbiting around the star for a long time, almost without losing their energy. This solves the well-known oronal heating problem, aording to whih a fast-moving heated gas should evaporate quikly and the orona should rapidly ool down due the insuffiient rate of its heating from the photosphere. It is obvious that in this analysis of the problem it is not taken into aount that the partiles an be fully retained by the gravitational field of the Sun and rotate around it. In addition, if the partiles are always lose to the Sun, they an be heated for a long time due to solar flares and similar phenomena, finally reahing the observed temperature of millions of degrees. However, we must limit the use of formulas (1) and () due to their inompleteness, beause along with the motion of partiles in solid bodies and stellar plasma, the main ontribution into the pressure is made by the interatomi fores, inluding the eletromagneti fores of eletri harges and the strong gravitation as the omponent of strong interation (in the gravitational model of strong interation [14]). These fores an signifiantly hange the partile aeleration in the equation of motion (7). For an ideal solid body there is no motion of partiles inside the body, S, and from p (1), in view of (8), and from the relation it follows: 18

20 Page 19 of 9 Canadian Journal of Physis p G M () r C Γ i r. () r Meanwhile, modelling of the osmi objets is usually based on the so-alled hydrostati equilibrium equation, whih has the following form without orretions of the general theory of relativity: GM () r r p r. (4) We see that (4) orresponds to () with the differene that in () the mass density is under the gradient sign together with the pressure. At onstant density both expressions are equivalent, but sine the density is usually a funtion of the radius, expressions () and (4) do not fully oinide. In the event of notieable rotation of the objet, in (7) we should take into aount the non-zero torsion field vetor Ω and the solenoidal vetor I of the pressure field, then hydrostati equation (4) beomes even more inaurate. 6. The ase of non-uniform density In [7] and [9], we made estimates of the temperature and pressure in the enter of various osmi objets by formulas (1) and (1), and we obtained quite a good agreement with the models of stars, planets and gas louds. In this setion, we plan to inrease the auray of our alulations. In (11) and () the density was assumed to be a onstant, although the temperature and pressure vary within wide ranges approximately quadratially. In general, when solving the 19

21 Canadian Journal of Physis Page of 9 wave equations (8) and (19) we should take into aount that the density is also a ertain funtion of the radius. We will ontinue to use the following approximation: Ar Br. (5) Substituting (5) in (8), where we introdue an auxiliary funtion Z Z() r in the form of Zr, we express the Laplaian as the funtion of the urrent radius: 1 d d 1 d Z 4 Z r dr dr r dr r r Ar Br. (6) It is onvenient to seek the solution of this differential equation in the form of a series with onstant oeffiients: Z k r k r k r k r n 1... n. Limiting the series by the value n 5, after substituting Z in (6) and anelling the similar terms we an alulate the oeffiients k n. All of these oeffiients appear proportional to the oeffiient k 1, and k. Speifying k1, taking into aount the relation Zr, we find the approximate dependene of the Lorentz fator on the urrent radius: 4 r A r r B. (7) 5

22 Page 1 of 9 Canadian Journal of Physis In (7) we an neglet the term in the brakets, whih is small even for neutron stars. In the first approximation, we have: 1 1 v v 1, 1 1 v v 1. Taking into aount the relationship between the partile veloity v and the temperature 1 mu v kt, and passing on in (7) from the Lorentz fator to the veloities and then to the temperature, we an now speify (11) for the temperature dependene: 4 4 mu r Amu r Bmu r T T. (8) 9k 9k 15k In order to speify the pressure dependene, in (19) we will replae substitute (5) and (7) into it: Yr and will 1 d d 1 d Y r r dr dr r dr 4 r A r B r 4 Ar Br. 5 (9) We will seek the solution of equation (9) in the form of a quinti polynomial with onstant oeffiients: Y r r r r r

23 Canadian Journal of Physis Page of 9 After substituting Y in (9) and anelling the similar terms, we see that, 1. Determining other oeffiients, we find the funtion Y, and then the salar potential of the pressure field: 4 r A r r B. (4) 5 The term in omparison with B in (4) an be negleted. As we an see, the oeffiients A and B in the density-radius dependene (5) make additional ontribution into the dependenes of the temperature (8) and the pressure field potential (4) on the urrent radius. 7. The temperature and pressure estimates We will alulate the volume-averaged mass density by integrating (5) with respet to the radius from zero to the body s radius R : R R Ar Br dv Ar Br r dr R R R M ( ) ( ). 4 AR BR (41) In (8) and (4) we will assume r R and substitute there, expressed from (41) in terms of the average density. We will also use the expression for the body mass

24 Page of 9 Canadian Journal of Physis M 4R. This allows us to estimate the surfae temperature Ts T( R) and the pressure field potential at the surfae ( R) : s T s 4 mu M Amu R Bmu R T. (4) kr 9k 15k 4 M A R B R s. (4) R 6 5 In (4) we an assume that the salar potential osmi bodies is lose to zero as ompared to the potential alulate, as well as the pressure at the enter: s of the pressure field on the surfae of at the enter. This allows us to p 4 M A R B R. R 6 5 In this relation, the density at the enter an be expressed using (41) in terms of the average density and we an use the equation M 4R : p M M A 9 M BR A R ABR B R 8 R (44) The number of terms in (44) is greater than in (1), whih inreases the auray of alulations. Let us now onsider the possible values of oeffiients of the pressure field and

25 Canadian Journal of Physis Page 4 of 9 aeleration field. Based on the equation of motion, in (1) we found the equality G. If we ompare the pressure at the enter (1) and the temperature at the enter (1) at onstant density, we obtain: p kt m. (45) u On the other hand, the standard expression for the pressure in view of the radiation pressure has the following form: p kt m 1 4 at, (46) u where 4 k a is the radiation density onstant, is the number of nuleons per 15 ionized gas partile, so that the gas partile an be an atom, ion, eletron or a single nuleon, depending on its state. By definition: 1 i m u x i (1 j ) y i j, where x i is the mass fration of the element i mi j with atomi number i, mi Ai mu is the nulear mass of the atom with atomi number i and atomi mass Ai 4, y ij is the degree of j -multiple ionization of i -th element, so that i j y ij 1. In fully ionized gas, onsisting of hydrogen, helium, and other elements with Ai 1 i 1, the expression for beomes as follows: x x x 4 H He A 1, where m H m, mhe 4m u u x, A x. i6 i 4

26 Page 5 of 9 Canadian Journal of Physis If we do not take into aount the radiation pressure, then omparison of (45) and (46) implies the equation:. Assuming G, we find: G, G. (47) G For the gas of nuleons or hydrogen atoms 1 and, 5 G 5 ; for the gas of fully ionized hydrogen 1 and G, 7 4G ; for the gas of fully ionized helium 7 4 and G, G ; for the fully ionized gas of heavier hemial elements G G and,. 4 4 Aording to the Earth's model, the inner ore temperature reahes 6 K, and the pressure is 11 p.6 1 Pa [18]. Sine the Earth is substantially inhomogeneous, we will use the data for the Earth's outer ore: the radius of 48 km, the mass kg, the temperature on the ore surfae of the order of Ts 4 K, and the pressure p 1. 1 s 11 Pa. From the analysis of the density-radius dependene of the form of (5), for the model of Earth's ore in view of (41) we an estimate the oeffiients A 1. 1 kg/m 4 and B kg/m 5 with the entral density kg/m. Based on these data, G given that, 1, and using 4 m u as the atomi mass unit, from (4) it follows that the temperature at the enter of the Earth's ore is of the order of T 5475 K. It is lear that the matter at the enter of the Earth is not a fully ionized gas, but rather solid rystalline matter. If 5

27 Canadian Journal of Physis Page 6 of 9 we assume the temperature at the enter equal to 6 K, from (4) we an estimate the effetive value of the aeleration field oeffiient: 1.8G. G For the pressure aording to (44), provided, we obtain the value 4 p Pa, and taking into aount the additional pressure of the rust p s the pressure at the enter of the Earth must be p p p 11 s Pa. This pressure is 1.9 times less than the pressure in the standard model. We an explain this, among other things, by the fat that the equations of motion (-4) for an ideal solid body are not ompletely aurate, sine they do not take into aount the ontribution of the aeleration field expliitly. These equations imply the equality of the gravitational fore and the pressure fore, whih leads to the equality of the aeleration inside the body to zero and to the relation G. But in fat, the aeleration inside the body is different from zero and is alulated in (7) and (1) with the use of the aeleration field, as a result, aording to (47) G. For the model of the Earth as a whole the oeffiients in (5) are equal to A 1. 1 kg/m 4 and B kg/m 5. Substituting these oeffiients in (44), we find the pressure at the enter of the Earth: 11 p Pa. This pressure is even less than the above pressure estimate made with the help of the oeffiients for the ore, whih illustrates the effet of inhomogeneity inside the Earth. Let us now onsider a neutron star with the radius of 1 km and the mass 1.5M, where M S denotes the mass of the Sun. We will take as an estimate of the entral stellar density the S value kg/m, with the equation of the state of matter aording to the potentials AV18 + UIX in [19]. Using the density-radius dependene from [], we an estimate the oeffiients in (5): 1 A.8 1 kg/m 4, 9 B kg/m 5. Negleting the surfae G temperature T s, with ondition in (4) we obtain the estimate of the temperature at 5 6

28 Page 7 of 9 Canadian Journal of Physis the enter of a neutron star of the order of the temperature K. Let us note that aording to [1] up to 11 1 K stable atomi nulei an exist in the matter of neutron stars. G For the pressure at the enter of a neutron star from (44) with ondition we obtain 5 the value. 1 Pa. This an be ompared with the pressure of the nulear matter. 1 Pa at the density kg/m aording to [] and the pressure of the order of. 1 Pa in [], while in different models of neutron stars aording to [] the pressure 4 does not exeed Pa. The helium white dwarf with the mass.6m must have the radius S m and the entral density 9 1 kg/m, aording to [4]. In view of (41) from the density-radius dependene in [5] the oeffiients follow in relation (5): A.6 1 kg/m 4, B kg/m 5. With this in mind, from (4) with G we find the temperature at 4 the enter of the white dwarf: K, and the pressure at the enter, aording to (44), is equal to Pa. In the NASA model, at the enter of the Sun the supposed mass density is kg/m, the pressure is about 16 p Pa and the temperature is K [6]. Sine the main sequene stars are muh larger in size than white dwarfs and neutron stars, the density variation in dependene (5) moving from the ore to the stellar surfae is very large, and two terms are not enough for aeptable auray. Therefore, we will turn from the model of the Sun as a whole to the model of its ore, whih is muh more uniform. The ore radius is estimated at the value, whih is five times less than the radius of the Sun, the ore mass is equal to.4m, the pressure on the ore surfae is not less than S 15 ps 4. 1 Pa [7], and temperature about 6 Ts K. So the oeffiients for the ore in (5) are as follows: A kg/m 4, B kg/m 5. Aording to [8], for the Solar ore 7

29 Canadian Journal of Physis Page 8 of 9.7, then from (47) we obtain.5g,.48g. From (4) we estimate the temperature at the enter of the ore: K. From (44) for the pressure we obtain 16 p Pa. These values are somewhat lower than in the NASA model, but it should be noted that the Sun s rust also exerts pressure on the Solar ore. If we sum up the pressures p and p s, the result will be muh loser to p. 8. Thermal ondutivity Aording to the model of stellar evolution, all of the main sequene stars over time turn into white dwarfs and neutron stars. It is assumed that in ompat stars there are no observable soures of internal energy, assoiated with nulear transformations. As a result, after formation white dwarfs and neutron stars have to ool down slowly over many billions of years. The surfae temperatures of some of the observed hot white dwarfs reah 15 K, and the surfae of ooled white dwarfs has a temperature below 4 K. The respetive luminosities orresponding to these temperatures are.1l S and 5 1 LS for a dwarf with the typial mass of.6m, and ooling down up to 4 K takes about 1 billion years [9]. To S haraterize the heat propagation inside a star we will onsider a phenomenologial differential heat-flux equation, desribing the Fourier s law of thermal ondution: h T. (48) In (48), the energy flux density vetor h is proportional to the thermal ondution oeffiient and the temperature gradient. In the onsidered model of gravitational equilibrium, a star or planet annot ool down infinitely. Indeed, the temperature distribution (8) was derived by us based on the fat that 8

30 Page 9 of 9 Canadian Journal of Physis the gravitational field was ounterated by the aeleration field and pressure field. The equilibrium state must be maintained at any time, as well after the star ools down. Suppose that the observed ooled white dwarfs are in suh a state that the temperature distribution in them is lose to the equilibrium distribution (8). Substituting the temperature-radius dependene (8) into (48) we find the vetor h : m 4 4Br k 5 u h Ar r. (49) Substituting in (49) r R, G, 1, kg/m and the oeffiients A and B for the white dwarf from the previous setion, we obtain the absolute value of the vetor of the energy flux density at the surfae: h 56. The integral of the vetor h aross s the entire surfae of the white dwarf must be equal to the stellar luminosity. Sine the vetor h is perpendiular to the surfae of the star, then the luminosity is equal to: L 4 R hs. On the other hand, the luminosity of a white dwarf in the steady state after long-term ooling is presumably equal to L L. From the equation L Lmin we find the estimate of the 5 min 1 S thermal ondutivity of the stellar matter: W/(m K). This value mainly haraterizes the thermal ondutivity of the upper rust and the atmosphere of the white dwarf, while the thermal ondutivity of the interior, rust and atmosphere of the star may differ many times due to the differene in temperatures and the state of matter. For omparison, the thermal ondution oeffiient of a diamond with impurities at room temperature is 1 W/(m K), while the purified diamond s oeffiient reahes W/(m K) [], whih is one of the highest experimental values for the known substanes. 9

31 Canadian Journal of Physis Page of 9 about The temperature of the surfae of neutron stars an be years of ooling [1]. If we use the stellar radius of K or less, whih requires 4 R 1. 1 m, then the orresponding minimal luminosity will be Lmin 4 R T L 4 SB s S with SB as the Stefan Boltzmann onstant, whih is lose to the observed luminosities of ertain stars []. Substituting in (49) the data for a neutron star at r G R,, 1, the entral 5 density of kg/m and the oeffiients A and B from setion 7, we obtain the absolute value of the vetor of the energy flux density at the surfae: hs Comparison of the luminosities L 4 R hs and min L allows us to estimate the thermal ondution oeffiient of the stellar matter: W/(m K). This estimate is onsistent with the results of alulation of the ioni thermal ondutivity in the shell of a neutron star []. We will use (49) with r R for the Earth as a whole, taking into aount G 4, the entral density kg/m and the orresponding oeffiients A 1. 1 kg/m 4 and B kg/m 5 from setion 7. The energy flux density at the surfae is equal to hs The measurements of the heat flux on land and in the oeans give the average value of h s W/m [4], whih leads to the thermal ondution oeffiient of 155 W/(m K). Aording to [5], the thermal ondution oeffiient in the ore of the Earth must be 7 W/(m K), orresponding to the thermal ondutivity of iron, on the ore-mantle boundary is redued to 16.4 W/(m K) [6], and the thermal ondutivity of the shell should be almost an order of magnitude less. The value we found 155W/(m K) is greater than the expeted value for the shell of the Earth, whih an be attributed to inauray of measuring the density using two oeffiients. In addition, we did not take into aount in the alulation that a signifiant part

32 Page 1 of 9 Canadian Journal of Physis of the thermal energy inside the Earth was generated due to the radioative deay of ertain isotopes. If we repeat all the alulations only for the outer ore of the Earth, with its radius of 48 km, mass of kg and the supposed total ore s heat flux of 1.6 TW, aording to [6], then with the oeffiients A 1. 1 kg/m 4 and B kg/m 5 we will obtain a more aeptable value for the upper ore: 19 W/(m K). To improve the results, the matter density should be speified in (5) with muh greater auray and all the soures of thermal energy should be taken into aount. This is even more important for modelling the main sequene stars and the Sun. 9. Entropy Planets and stars onsist of moleules, atoms, ions, eletrons, and as for the white dwarfs and neutron stars the list of basi matter partiles also inludes the atomi nulei and individual nuleons. For multiomponent systems the speifi entropy per one matter nuleon is alulated by summing up for eah type of partiles, and taking into aount the radiation entropy aording to [7] it looks as follows: i k 5 mi kt g ij kn e 5 me kt 4aT s nij ln ln n i j ni j n ne n, (5) where n is the onentration of nuleons, m u is the atomi mass unit, mi Ai mu is m u the mass of the atomi nuleus with the harge number i and the mass number Ai 4, g ij is the statistial weight of the ion of i -th hemial element in j -th ionization state, n ij xi yi j is the onentration of ions in the element i in j -th ionization state, x i is the m i 1

33 Canadian Journal of Physis Page of 9 mass fration of the element with the harge number i, y ij is the degree j -fold ionization of the i -th element, so that i yij 1, is Dira onstant, j n e i jn is the onentration i j1 i j of eletrons under eletroneutrality onditions. We will apply relation (5) to the ase of a monatomi neutral gas without taking into aount the ontribution of the eletrons: 5 mu kt 1 4aT s k ln n n. (51) Expression (51) an be used to estimate the speifi entropy in the newly formed neutron star, onsisting mostly of neutrons with admixture of protons, eletrons and atomi nulei in the shell of the star. If the star s mass is M 1.5M S and its radius is 4 R 1. 1 m, then n M m -1. Substituting in (51) the temperature mu 4 mur K at the enter of the neutron star, found in Setion 7, we obtain the speifi entropy per nuleon: sm 1.k. But at the initial time point the entropy is onsiderably higher, sine at the same temperature the radius of a hot star exeeds R and the density n is less. At this temperature, the ontribution from the entropy of radiation is 1.5 times less than the ontribution from the entropy of nuleons. The speifi entropy value in the equilibrium state should be less than s m, as due to ooling the temperature in the main volume of the star would be less than the temperature at the enter K. The estimation of the speifi entropy an be done in a different way. As it was shown in [1], the energy of the aeleration field as part of the energy of random motion of partiles an be expressed as follows:

34 Page of 9 Canadian Journal of Physis W a M. 1R To this energy we must add the energy of the partiles due to their interation with the four-potential of the aeleration field. For exat alulation of the kineti energy of partiles we an use the virial theorem or the kineti energy definition as the differene between the relativisti energy of the moving partiles and their rest energy. In both ases we obtain for the kineti energy the following: W k 411M.67M. 11R R By definition, the inrement of the speifi entropy is given by the formula: S Q s, where S is the inrement of the total entropy, N T N M N is the number of m u nuleons, Q is the inrement of the thermal energy at the temperature T. In the neutron star formation we an assume in a first approximation that Q W, as well s s. Then for the k speifi entropy we obtain the following relation:.67 Mmu s. (5) RT G Substituting here at 1 from (47) and the entral temperature 5 find s.77k K, we, whih is less than s m. However, this result is subjet to further orretion in view of the fat that the star during transition to the equilibrium radius loses its energy with the neutrinos that fly away and is getting rapidly ooled, reduing its entropy.

35 Canadian Journal of Physis Page 4 of 9 For omparison, there was found in [8] that a hot protoneutron star with the mass M 1.M S, the radius of km and the temperature K has the speifi entropy equal to s 1.k. When the star reahes its equilibrium radius of km, its speifi entropy at the effetive temperature of matter of the order of 9 1 K beomes equal to s.54k and then dereases ontinuously due to the subsequent ooling. Calulating the entropy of planets and stars with the help of (5) requires knowledge of the quantitative hemial omposition of matter. If we use (5) for initial estimation of the entropy, it is neessary to know the mass, radius and average temperature. For Jupiter, the G speifi entropy, aording to (5), at will be of the order of s 1.65k, if as T we 4 use the temperature value T 57 K expeted at the enter of Jupiter within the framework of the modern models of gas giants. This value of the speifi entropy should be inreased beause as T we should use a less temperature value, averaged over the entire volume. In partiular, alulations by a standard method give s 9k for a planet like Jupiter [9]. 1. Conlusion In the presented model of gravitational equilibrium, an important role is played by the aeleration field and the pressure field. The equations of these fields () and (14) allow us to alulate the strengths and the solenoidal vetors of the fields and to determine the distribution of temperature and pressure inside the osmi bodies, aused by gravitation, using simple formulas. Equations () allow us to find the gravitational field strength and the solenoidal vetor of the torsion field in the ovariant theory of gravitation. The knowledge of field strengths and solenoidal vetors is suffiient for the analysis of the equations of motion of matter (6) and for onstrution of the model basis. In partiular, the equation of motion 4

36 Page 5 of 9 Canadian Journal of Physis leads to orrelation (1) between the oeffiients of the aeleration field and the pressure field, and in (47) these oeffiients are expressed in terms of the thermodynami parameter. Using the values of these oeffiients, we determine the temperature and pressure at the enter of various objets in Bok globules, inside the Earth, the Sun, in a white dwarf and a neutron star. In addition, we obtain the estimates of the heat flux and the thermal ondutivity oeffiient, whih haraterizes this or that objet, and we also estimate the entropy of a neutron star and a gas planet. The analysis of the results shows that they agree well with the data provided by other authors. Based on the signifiantly higher thermal ondutivity of the ore and rust as ompared to the shell, some authors in a number of studies suggest the existene in a typial neutron star (and a white dwarf) of an almost isothermal ore with a small temperature gradient. In our approah, the temperature distribution inside a star is assoiated with dependene (7) for the Lorentz fator, while the high value of thermal ondutivity does not play an essential role and annot lead to the isothermal ore. In turn, relation (7) follows from the wave equation of the aeleration field (7) and is determined only by the matter density distribution and the matter motion. From the physial standpoint, (7) and the similar dependene for the pressure field potential (4) are assoiated with the energy distribution between the gravitational field, aeleration field and pressure field. The result of this energy distribution is a ertain state of equilibrium and ertain distribution of physial parameters, depending on the urrent radius, inside of the objet under onsideration. If we proeed from the Le Sage's theory of gravitation, then the atual agents, performing the gravitational ontration and equilibrium heating of matter of osmi objets, are the fluxes of gravitons. The energy density, the rosssetion of interation between the graviton field and the nuleon matter, as well as other parameters an be alulated based on the fat that the ation of graviton fluxes should lead to the Newton's law of universal gravitation [4-41]. 5

37 Canadian Journal of Physis Page 6 of 9 The universality of the gravitational equilibrium model is ensured by the fat that the field equations an be applied to any objets, regardless of their state of matter. A ertain restrition is the need for a detailed representation of the mass density distribution in order to obtain aurate results, however the same is required in any other approahes, inluding the ommonly used polytropi model. As a rule, to estimate the mass density distribution the measurement of the veloities of longitudinal and transverse seismi waves is used, as well as the orrespondene between the radial distribution of mass density and the total body mass and its moment of inertia. Another restrition is the use in formula (4) of the atomi mass unit as m u, whih is permissible for the hydrogen gas and the uniform nulear matter of nuleons. Meanwhile, in previously derived orrelations (11) and (8) m u has the meaning of the mass of one gas partile, whih an be different from the atomi mass unit. Thus, in white dwarfs a signifiant ontribution into the state of matter is made by eletrons, the relativisti masses of whih depend on the temperature. In stars and planets there are both neutral and ionized atoms with the masses of the order of Am u, where A is the mass number of the orresponding hemial isotope. Due to the omplex omposition of partiles in the matter, formulas (4) for the temperature and (5) for the speifi entropy need to be speified for eah type of osmi objets. Referenes 1. J.H. Lane. Am. J. Si. 5, 57 (187). doi:1.475/ajs.s G.P. Horedt. Polytropes appliations in astrophysis and related fields. Kluwer Aademi Publishers, Dordreht. 4.. L. Herrera and W. Barreto. Phys. Rev. D. 87, 87 (1). doi:1.11/physrevd