Compact r  [0,R] self-gravitating ball (SGB) at the state of hydrostatic equilibrium (HSE)

1. On the impossibility of hydrostatic equilibrium of a star.Some properties of non-equilibrated star.A. V. Chigirinsky, Dnepropetrovsk, Ukrainechigirinsky@farlep.dp.ua+38(095)490-49-87 Modern astrophysics on the stellar structure– axiom of hydrostatic equilibrium versus protostellar cloud singularityOn the singularity of a compact ball. Recursive isolation of the central pointConceptual framework of Lagrange. Definitive isolation of the central pointRadially symmetric dynamics of a star. Three theoretical way of the development--------------------------------------------------------------• Rayleigh's cavitation of a void - infinite velocity• Stellar electromagnetism due to singular permanent shock wave• Relativistic limit of a collapse• Phenomenon of a pulsar App. 1. On the singularity of “solution” of the HSE-equation.

2. Compact r[0,R] self-gravitating ball (SGB) at the state of hydrostatic equilibrium (HSE) rt´(m,t)≡0; m[0,M]; t(-∞,+∞) if F∆p+ Fg=0 at dv=r2drdΩ then r(m) p=f(ρ) – equation of state p – pressure, ρ – density ρ(0)= ρ0 < ∞ a priori ρ(R)= ρ(∞)=0 Else ?..

3. radially symmetric d’Alembert’s wave u(r, t) =q(r ± st)/r; s=const for wave eq. def u(0,t)≡ δr(0,t)=∞ the Rayleigh’s cavitation of a void - the limit case of Rayleigh-Plesset equation rt´|r=0= ∞ (rt´(m,t)≠0 ) results in the central controversy …Else the model of gravitational collapse of isothermal protostellar cloud ??? ρ(0) = ∞ instability of relativistic(?) star theorem of Zel’dovich] ??? etc… A-bomb, H-bomb, SBSL [Gaitan, D.F., 1992] … So, can a process come to a stopat singular position/moment?

4. dfi Differential equation expresses the idea of equilibrium ofmaterial content of differential element Σdf = 0 dfi dfi differential element has to be • regular (single-type shape) • nontrivial dv > 0 dfi dv=0 means “nowhere” dm=0 means “nothing” ? ? dv|r=0= r2drdΩ = 0 dm|r=0= ρdv|r=0 = 0

5. differential shell, regular element compact ball - recursive object means “kernel >>shell” Statement: there is no regular differential decomposition of a compact ball (CB) hence neither radially symmetric boundary problemcan be formulated on the CB. (N-1)-th spherical shell CB as a whole CB – kernel,irregular 1-st spherical shell; thin shell approximation, linear term of the difference [0, R] ═> [0, 1] means CB is a finite object

6. dv = r2drdΩ – fragment of spherical shell- regular DE within segment (ε,R); ε > 0 however- non-applicable at r=0- trivial at r=0 - has no physical meaning drv-2/3dv dv|r=0= ⅓ ε3dΩ – fragment of compact ball in a whole skin;- non-trivial at r=0 - has physical meaning however- irregular finite element- non-applicable thereof subsequent infinite recursion- non-negligible since ε >> dr|r=ε

7. r(0) is multivalued function since each CB [0,y] [0, rL] contains a void; r(M) is multivalued function since each CB [0,y] [0, R] contains M. Lagrangian definition of radially symmetric materialball (i) compact ball of variable radius [0,r(m)) contains invariable mass m; (ii) definitional domain of material ball is an open segment m(0, M); strictly monotonous increasing function r(m) maps it into hollow ball; Thus– the kernel [0, rL] is immaterial CB – evacuated L-cavity (Lagrangian); – the space [R, ∞] is the Universe. Otherwise (ifdefinitional domain [0, M]) Lagrangian void possesses its boundary whereas Lagrangian mass does not: [0, ∞] = [0, rL]  (rL, R)  [R, ∞] Note that writings (0,0), [0,0) and (0,0] are mathematical catachreses

8. r before L-perturbations r+dr after r+δr r+dr +δ(r+dr)  δ(dv) = 4π δ(r2dr); dr << r −variance of the differential  δ(dv) = 4π d(r2δr); |δr| << r −differentialof the variance but before dynamical kernel – evacuated cavity –indefinite variance having exact shape of the CB r=0 after dr δr|r=0 L-cavity has appeared! δ(dr)

9. p(0)=  p(M)=0 Radially symmetric dynamics in the L-SGB self-contained system of total energy E = K+H+U = const, where K, H, U are kinetic, internal heat, and gravitational energies. Zel’dovich, 1981:min E  δE=0 complete context of the problem Scalar product 0 ≡ – L-potential

10. Final paradigm of 0 ≡ – Case meets the requirement,however UL = f(r) is arbitrary function in this case—i.e. static state of SGB is incognizable one. – Case ≡ 0 as a whole:if there are such then corresponds to self-orthogonal L-flow,the idea of the HSE is irrelevant one. – Case :differential form of radially symmetric Bernoulli’s lawmeets the requirement.The HSE-state is inaccesible due to central singularity.

11. irrelevant► ◄incognizable HSE inaccesible► The Knight at the Crossroads by Vasnetsov V.M. 1882

12. – initial A planet [star] as an orbit p(0)=  p(M)=0 – boundary p =  f(m)– constrain function K(m´)+H(m´)+U(m´)< K(m)+H(m)+U(m) < K(M)+H(M)+U(M) < U|r=∞ : m´ < m < M – finitary orbit The SGB is a degradedNewtonian orbit—the radially symmetric Bernoulli’s flow.It is a continuous medium that to orbit at the initial conditions assumed to be arbitrary.Function rL(t)=r(0, t) describesthe internal side of the orbit. Do not ask Newton where initial conditions of an orbitcome from!He does not know.He does not care.

13. Primitive δ(dH)≡0unit pulsar. Classical dynamics.  u α ; ± ± u′τ α′τ α′′ττ ± ± u′′ττ Cavitation of a void [Lord Rayleigh, 1917] ; concept α(t)

14. Pressure HSE shockwave

15. Electromagnetic phenomenon of the SGB West-east asymmetry of Jovian «synchrotron» radio emission δ2P δE δE δH δ2P δH Base fig. from [Levin S.M.et.al., 2001] Shock wave => electric charge separation [Institute for High Energy Densities of RAS] in a strip plasma of L-cavity => Global radial charge redistribution “electron-excess core” and “electron-deficit residue” <=> self-consistent electric field of the SGB. The vortexes of all the charged components => magnetic field of the SGB. Total dynamo. Single-Global-Phase U&VLF-electromagnetic phenomenon)– δE=j(r, t)‹E(r)›; δH=h(r,φ,t)‹H(r,φ)›;=> δ2P=j(r,t)h(r,φ,t)[‹E›×‹H›] =>a) tangent at the magnetic meridian; looks like an orbit; (“Orbits” ofplasmospheric hissingsf[100 Hz – 2 kHz], ‹ ν›≈700 Hz,[Molchanov О. А., 1985. ]);b) West-east asymmetry;c) pulsatile acoustic modulation of the bright hemisphere emission.

16. – chaotic radio-pulsar … I am toldthis isa periodic pulsar… h'm!

17. Relativistic limit of the central collapse Ideally spheric self-gravitating shell of ideal dust falls into its center from the infinity where its own rest mass energy was ε∞=m∞c2.Let the energy ε = m∞c2 [1 - (v/c) 2] -1/2 gravitate as m=ε/c2.The energy conservation equation takes the form rc= r(2)—Schwarzschild’s radius—is minimal radius of the SRT-collapse; the shell may ”oscillate” within the finite energy range the shell may expand with the increasing of its energy behavior of the shell at the state r = rccannot be interpreted in terms of the continuum dynamical limit, it has to move at speed uc= ±c√3/2.

18. Chaotic instantaneous seriesof radial pulses ‘collapse-expansion’[rmax = 4 m; ‹ν›≈800 Hz] • R(t) - ‘ambient noise’ ∆R(t) ≈10-12m; • Singular  relativistic as rL→0; • Emission provides each rebound[all sorts of hard radiation, geoneutrino,γ→e-avalanches, thermal outflow]; • Repeatable ‘tiny Big Bang’[10 kt TNT energetic amplitude; 1/800 loss]; • Interior looks like ‘a star turned inside out’; • Extremely sharp shockwave nearby rL  - (rmax/r) 4 /rmax; rL(t) min ≈ 106 m/s2

19. References 1. Zel’dovich, Y.B., Blinnikov, S.I., Shakura N.I., 1981. Physical Fundamentals of Structure and Evolution of Stars. (Moscow State Univ., Moscow) (in Russian) p20-212. L. Rayleigh, 1917. On the pressure developed in a liquid during the collapse of a spherical cavity. Philos. Mag. 34, 94983. Gaitan, D.F., Crum, L.A., Roy, et al.1992. Sonoluminescence and bubble dynamics for a single, stable, cavitation bubble. J. Acoust. Soc. Am., 91(6), 3166-31834. Hammer, D., Frommhold, L., 2001. Sonoluminescence: how bubbles glow. Journal of Modern Optics, 48(2), 239-2775. Brenner, M.P., Witelsky, T.P., 1998. On Spherically Symmetric Gravitational Collapse. J. Stat. Phys., 93(3-4), 863-899, doi:10.1023/B:JOSS.0000033167.19114.b86. Simmons, W., Learned, J., Pakvasa, S., et al.1998. Sonoluminescence in neutron stars. Phys. Lett. B427, 109-1137. Molchanov, O.A., 1985. Low Frequency Waves and Induced Emission in the Near-Earth Plasma, Moscow, Nauka, (in Russian)8. Dwyer, J. R. et al., 2004, A ground level gammaray burst observed in association with rocket triggered lightning, Geophys. Res. Lett., 31, L05119, doi:10.1029/2003GL0187719. Fishman G. J. et al., 1994, Discovery of Intense Gamma-Ray Flashes of Atmospheric Origin, Science, 264, 1313 10. Simmons, W., Learned, J., Pakvasa, S., et al.1998. Sonoluminescence in neutron stars. Phys. Lett. B427, 109-113 11. Levin, S.M., Bolton, S.J, Gulkis, S.J., Klein M.J., 2001. Modeling Jupiter's synchrotron radiation Geophys. Res. Lett.,Vol. 28, No. 5, pp 903-906, March 1 12. Miloslavljevich, M., Nakar, E., Spitkovsky, A. STEADY-STATE ELECTROSTATIC LAYERS FROM WEIBEL INSTABILITY IN RELATIVISTIC COLLISIONLESS SHOCKS 13. Vazquez-Semadeni E., Shadmehri M., Ballesteros-Paredes J. CAN HYDROSTATIC CORES FORM WITHIN ISOTHERMAL MOLECULAR CLOUDS? arXiv:astro-ph/0208245 v2 20 Aug 2003. 14. Vazquez-Semadeni1 E., Gomez1 G.C., Jappsen A.K., Ballesteros-Paredes1 J., Gonzalez1 R.F., Klessen R.S.,Molecular Cloud Evolution II. From cloud formation to the early stages of star formation in decaying conditions.

20. On the singularity of “solution” of the HSE-equation The Emden’s polytropicstar. To solve the problem, Emden imposes the boundaryvalues Then, each Emden’s solutiontakes the form of the convergent series However, assuming the mass density to be arbitraryfunction of its parameter, the Taylor’s series constitutes the complete context of the density definition, i.e. the function dm(dv) must be definite before that linear approximation dm = ρdv could be used properly. Hence,the last sum must vanish or, as the same, the requirement is that Alas, the Emden’s central density is indefinitesince the function has no Taylor’sexpansion at v = 0, i.e. does not exist as a thermodynamicalfunction even after it has been admitted tobe definite a priori; this is a spurious solution.

21. The tower of Babel. Pieter Bruegel the Elder. 1563.