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 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.

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

ρ(0) = ρ0 < ∞ a priori

ρ(R) = ρ(∞) = 0

r(m)

p=f(ρ) – equation of state

p – pressure,

ρ – density

rt´(m,t)≡0; m[0,M]; t(-∞,+∞)if

then F∆p+ Fg=0 at dv=r2drdΩ

Else ?..

…Else

ρ(0) = ∞

( rt´(m,t)≠0 ) results in the central controversy

the model of gravitational collapseof isothermal protostellar cloud ???

u(0,t) ≡ δr(0,t)=∞radially symmetric d’Alembert’s waveu(r, t) = q(r ± st)/r ; s=constfor wave eq.

def

rt´|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] ???

Differential equation

expresses the idea of equilibrium ofmaterial content of differential element

differential element has to be

•regular (single-type shape)

•nontrivial dv > 0

dv=0 means “nowhere”dm=0 means “nothing”

dv|r=0= r2drdΩ = 0

dm|r=0 = ρdv|r=0 = 0

?

?

dfi

dfi

dfi

dfi

Σdf = 0

Statement: there is no regular differential decomposition of a compact ball (CB) hence neither radially symmetric boundary problemcan 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

dr v-2/3dv

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

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

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, rL] 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, rL] contains a void;

r(M) is multivalued function since each CB [0,y] [0, R] contains M.Lagrangian void possesses its boundarywhereas Lagrangian mass does not:

[0, ∞] = [0, rL] (rL, R) [R, ∞]

Note that writings (0,0), [0,0) and (0,0] are mathematical catachreses

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(dv) = 4π δ(r2dr); dr << r − variance of the differential

δ(dv) = 4π d(r2δr); |δr| << r − differential of the variance

•

r=0

dr

δr|r=0

δ(dr)

before

after

before

after

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-potential 0 ≡

•Scalar product

– 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 UL = 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.

◄incognizable

inaccesible ►

irrelevant ►

HSE

The Knight at the Crossroads by Vasnetsov V.M. 1882

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 rL(t)=r(0, t) describesthe internal side of the orbit.

p =  f(m) – constrain function

Do not ask Newton where initial conditions of an orbitcome from!He does not know.He does not care.

Primitive δ(dH)≡0 unit pulsar. Classical dynamics.

;

± ±

±±

;

Cavitation of a void [Lord Rayleigh, 1917]

concept α(t)

α′′ττ

u′′ττ

u′τα′τ

α u

Pressure

shockwave

HSE

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)›;

=> δ2P=j(r, t)h(r, φ, t)[‹E›×‹H›] => a) tangent at the magnetic meridian; looks like an orbit; (“Orbits” of plasmospheric 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.

Single-Global-Phase U&VLF-electromagnetic phenomenon)–

δE δE

δHδHδ2P

δ2P

West-east asymmetry of Jovian «synchrotron» radio emission

Base fig. from [Levin S.M. et.al., 2001]

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

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 = rc cannot be interpreted

in terms of the continuum dynamical limit, it has to move at

speed uc = ± c√3/2.

rL(t)

•Chaotic instantaneous seriesof radial pulses ‘collapse-expansion’[rmax = 4 m; ‹ν›≈800 Hz]

•R(t) - ‘ambient noise’ ∆R(t) ≈10 -12 m;

•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;≈ 106 m/s2min

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, 131310. Simmons, W., Learned, J., Pakvasa, S., et al.1998. Sonoluminescence in neutron stars. Phys. Lett. B427, 109-11311. 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 112. Miloslavljevich, M., Nakar, E., Spitkovsky, A. STEADY-STATE ELECTROSTATIC LAYERS FROM WEIBEL INSTABILITY IN RELATIVISTIC COLLISIONLESS SHOCKS13. 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.

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 linearapproximation 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 functioneven after it has been admitted to be definite a priori; this is a spurious solution.

On the singularity of “solution” of the HSE-equation

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