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On the impossibility of hydrostatic equilibrium of a star. Some properties of non-equilibrated star. A. V. Chigirinsky, Dnepropetrovsk, Ukraine chigirinsky@farlep.dp.ua +38(095)490-49-87 chigirinsky@farlep.dp.ua Modern astrophysics on the stellar structure – axiom of hydrostatic equilibrium versus protostellar cloud singularity On the singularity of a compact ball. Recursive isolation of the central point Conceptual framework of Lagrange. Definitive isolation of the central point Radially 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.

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

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

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Differential equation expresses the idea of equilibrium of material content of differential element differential element has to be regular (single-type shape) nontrivial d v > 0 d v =0 means “nowhere” dm=0 means “nothing” d v| r=0 = r 2 drdΩ = 0 dm | r=0 = ρ d v| r=0 = 0 ???? dfidfi dfidfi dfidfi dfidfi Σdf = 0

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

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dr v -2/3 d v d v = r 2 drdΩ – 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 d v| r=0 = ⅓ ε 3 dΩ – 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

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Lagrangian definition of radially symmetric material ball (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, r L ] is immaterial CB – evacuated L-cavity (Lagrangian); – the space [R, ∞] is the Universe. Otherwise ( if definitional domain [0, M] ) r(0) is multivalued function since each CB [0,y] [0, r L ] contains a void; r(M) is multivalued function since each CB [0,y] [0, R] contains M. Lagrangian void possesses its boundary whereas Lagrangian mass does not: [0, ∞] = [0, r L ] ( r L, R ) [R, ∞] Note that writings (0,0), [0,0) and (0,0] are mathematical catachreses

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L-perturbations but dynamical kernel – evacuated cavity – indefinite variance having exact shape of the CB L-cavity has appeared! r r+dr r+δr r+dr +δ(r+dr) δ(d v ) = 4π δ(r 2 dr); dr << r − variance of the differential δ(d v ) = 4π d(r 2 δr); |δr| << r − differential of the variance r=0 dr δr| r=0 δ(dr) before after before after

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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. p(0) = p(M) = 0 Zel’dovich, 1981: min E δE=0 complete context of the problem – L-potential0 ≡ Scalar product

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– 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 meets the requirement, however U L = f(r) is arbitrary function in this case— i.e. static state of SGB is incognizable one. Final paradigm of 0 ≡ – Case : differential form of radially symmetric Bernoulli’s law meets the requirement. The HSE-state is inaccesible due to central singularity.

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

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A planet [star] as an orbit p(0) = p(M) = 0 – boundary – initial 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 degraded Newtonian orbit— the radially symmetric Bernoulli’s flow. It is a continuous medium that to orbit at the initial conditions assumed to be arbitrary. Function r L (t)=r(0, t) describes the internal side of the orbit. p = f(m) – constrain function Do not ask Newton where initial conditions of an orbit come from! He does not know. He does not care.

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

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Pressure shockwave HSE

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Electromagnetic phenomenon of the SGB 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. δE=j(r, t) ‹ E(r) › ; δH=h(r, φ, t) ‹ H(r, φ) › ; => δ 2 P=j(r, t)h(r, φ, t)[ ‹ E › × ‹ H › ] => a) tangent at the magnetic meridian; looks like an orbit; (“Orbits” of plasmospheric hissings f [100 Hz – 2 kHz], ‹ ν ›≈700 Hz, [Molchanov О. А., 1985. ]); b) West-east asymmetry; c) pulsatile acoustic modulation of the bright hemisphere emission. Single-Global-Phase U&VLF- electromagnetic phenomenon)– δEδE δEδE δHδH δHδH δ2Pδ2P δ2Pδ2P West-east asymmetry of Jovian «synchrotron» radio emission Base fig. from [Levin S.M. et.al., 2001]

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– chaotic radio-pulsar … I am told this is a periodic pulsar… h'm!

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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 ∞ c 2. Let the energy ε = m ∞ c 2 [1 - (v/c) 2 ] -1/2 gravitate as m=ε/c 2. The energy conservation equation takes the form r c = 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 = r c cannot be interpreted in terms of the continuum dynamical limit, it has to move at speed u c = ± c√3/2.

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rL(t)rL(t) Chaotic instantaneous series of radial pulses ‘collapse-expansion’ [r max = 4 m; ‹ν› ≈800 Hz] R(t) - ‘ambient noise’ ∆R(t) ≈10 -12 m; Singular relativistic as r L →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 r L - (r max / r ) 4 /r max ; ≈ 10 6 m/s 2 min

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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) p20-21 2. L. Rayleigh, 1917. On the pressure developed in a liquid during the collapse of a spherical cavity. Philos. Mag. 34, 9498 3. 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), 3166-3183 4. Hammer, D., Frommhold, L., 2001. Sonoluminescence: how bubbles glow. Journal of Modern Optics, 48(2), 239-277 5. 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.b8 6. Simmons, W., Learned, J., Pakvasa, S., et al.1998. Sonoluminescence in neutron stars. Phys. Lett. B427, 109-113 7. 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:10.1029/2003GL018771 9. 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 SHOCKSA. 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.

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The Emden’s polytropic star. To solve the problem, Emden imposes the boundary values Then, each Emden’s solution takes the form of the convergent series However, assuming the mass density to be arbitrary function 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 indefinite since the function has no Taylor’s expansion at v = 0, i.e. does not exist as a thermodynamical function even after it has been admitted to be definite a priori; this is a spurious solution. On the singularity of “solution” of the HSE-equation

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

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