Velocity-height relation for antimatter meteors

A general velocity-height relation for both antimatter and ordinary matter meteor is derived. This relation can be expressed as % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq% aHfpqDdaWgaaWcbaGaamOEaaqabaaakeaacqaHfpqDdaWgaaWcbaGa% eyOhIukabeaaaaGccqGH9aqpcaqGLbGaaeiEaiaabchacaqGGaWaam% WaaeaacqGHsisldaWcaaqaaiaadkeaaeaacaWGHbaaaiaabwgacaqG% 4bGaaeiCaiaabIcacaqGTaGaamyyaiaadQhacaGGPaaacaGLBbGaay% zxaaGaeyOeI0YaaSaaaeaacaWGdbaabaGaamOqaiabew8a1naaBaaa% leaacqGHEisPaeqaaaaakmaacmaabaGaaGymaiabgkHiTiaabwgaca% qG4bGaaeiCamaadmaabaGaeyOeI0YaaSaaaeaacaWGcbaabaGaamyy% aaaacaqGLbGaaeiEaiaabchacaqGOaGaaeylaiaadggacaWG6bGaai% ykaaGaay5waiaaw2faaaGaay5Eaiaaw2haaiaacYcaaaa!64FD!\[\frac{{\upsilon _z }}{{\upsilon _\infty }} = {\text{exp }}\left[ { - \frac{B}{a}{\text{exp( - }}az)} \right] - \frac{C}{{B\upsilon _\infty }}\left\{ {1 - {\text{exp}}\left[ { - \frac{B}{a}{\text{exp( - }}az)} \right]} \right\},\]where _z is the velocity of the meteoroid at height z, its velocity before entrance into the Earth's atmosphere, is the scale-height, and C parameter proportional to the atom-antiatom annihilation cross- section, which is experimentally unknown. The parameter B (B = DA₀/m) is the well known parameter for koinomatter (ordinary matter) meteors, D is the drag factor, ₀ is the air density at sea level, A is the cross sectional area of the meteoroid and m its mass.When the annihilation cross-section is zero — in the case of ordinary meteors — the parameter C is also zero and the above derived equation becomes % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq% aHfpqDdaWgaaWcbaGaamOEaaqabaaakeaacqaHfpqDdaWgaaWcbaGa% eyOhIukabeaaaaGccqGH9aqpcaqGLbGaaeiEaiaabchacaqGGaWaam% WaaeaacqGHsisldaWcaaqaaiaadkeaaeaacaWGHbaaaiaabwgacaqG% 4bGaaeiCaiaabIcacaqGTaGaamyyaiaadQhacaGGPaaacaGLBbGaay% zxaaGaaiilaaaa!4CF5!\[\frac{{\upsilon _z }}{{\upsilon _\infty }} = {\text{exp }}\left[ { - \frac{B}{a}{\text{exp( - }}az)} \right],\]which is the well known velocity-height relation for koinomatter meteors.In the case in which the Universe contains antimatter in compact solid structure, the velocity-height relation can be found useful.Work performed mainly at the Nuclear Physics Laboratory of the National University of Athens, Greece.     

On the periodic solutions of a particular Lame equation

We consider the Hill's equation: % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGGipm0dc9vqaqpepu0xbbG8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca% WGKbWaaWbaaSqabeaacaaIYaaaaOGaeqOVdGhabaGaamizaiaadsha% daahaaWcbeqaaiaaikdaaaaaaOGaey4kaSYaaSaaaeaacaWGTbGaai% ikaiaad2gacqGHRaWkcaaIXaGaaiykaaqaaiaaikdaaaGaam4qamaa% CaaaleqabaGaaGOmaaaakiaacIcacaWG0bGaaiykaiabe67a4jabg2% da9iaaicdaaaa!4973!\[\frac{{d^2 \xi }}{{dt^2 }} + \frac{{m(m + 1)}}{2}C^2 (t)\xi = 0\]Where C(t) = Cn (t, {frbuilt|1/2}) is the elliptic function of Jacobi and m a given real number. It is a particular case of theame equation. By the change of variable from t to defined by: % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGGipm0dc9vqaqpepu0xbbG8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaqcaawaaOWaaiqaaq% aabeqaamaalaaajaaybaGaamizaGGaaiab-z6agbqaaiaadsgacaWG% 0baaaiabg2da9OWaaOaaaKaaGfaacaGGOaqcKbaG-laaigdajaaycq% GHsislkmaaleaajeaybaGaaGymaaqaaiaaikdaaaqcaaMaaeiiaiaa% bohacaqGPbGaaeOBaOWaaWbaaKqaGfqabaGaaeOmaaaajaaycqWFMo% GrcqWFPaqkaKqaGfqaaaqcaawaaiab-z6agjab-HcaOiab-bdaWiab% -LcaPiab-1da9iab-bdaWaaakiaawUhaaaaa!51F5!\[\left\{ \begin{array}{l}\frac{{d\Phi }}{{dt}} = \sqrt {(1 - {\textstyle{1 \over 2}}{\rm{ sin}}^{\rm{2}} \Phi )} \\\Phi (0) = 0 \\\end{array} \right.\]it is transformed to the Ince equation: (1 + · cos(2)) y + b · sin(2) · y + (c + d · cos(2)) y = 0 where % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGGipm0dc9vqaqpepu0xbbG8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaqcaawaaiaadggacq% GH9aqpcqGHsislcaWGIbGaeyypa0JcdaWcgaqaaiaaigdaaeaacaaI% ZaGaaiilaiaabccacaWGJbGaeyypa0Jaamizaiabg2da9aaacaqGGa% WaaSaaaKaaGfaacaWGTbGaaiikaiaad2gacqGHRaWkcaaIXaGaaiyk% aaqaaiaaiodaaaaaaa!4777!\[a = - b = {1 \mathord{\left/{\vphantom {1 {3,{\rm{ }}c = d = }}} \right.\kern-\nulldelimiterspace} {3,{\rm{ }}c = d = }}{\rm{ }}\frac{{m(m + 1)}}{3}\]In the neighbourhood of the poles, we give the expression of the solutions.The periodic solutions of the Equation (1) correspond to the periodic solutions of the Equation (3). Magnus and Winkler give us a theory of their existence. By comparing these results to those of our study in the case of the Hill's equation, we can find the development in Fourier series of periodic solutions in function of the variable and deduce the development of solutions of (1) in function of C(t).     

Mass-height relation for antimatter meteors

A formula to compute the mass-height relation for the case of possible antimatter meteor entrance is derived.It is governed by the annihilation cross section for the atom-antiatom interactions which experimentally is unknown,and by various mechanisms which are possibly reducing its value. For the special case of thermal energies,the annihilation cross-section _an may be connected with the elastic cross-section_el by the relation % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaS% baaSqaaiaabggacaqGUbaabeaakiabg2da9iabeo8aZnaaBaaaleaa% caqGLbGaaeiBaaqabaGccqGHpis1caWGMbWaaSbaaSqaaiaadMgaae% qaaaaa!4227!\[\sigma _{{\rm{an}}} = \sigma _{{\rm{el}}} \prod f_i \],where the factors % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaaBa% aaleaacaWGPbaabeaaaaa!37F1!\[f_i \]are all less or equal to unity. Among them, the most significant is the barrier factor % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaaBa% aaleaacaWGIbaabeaaaaa!37EA!\[f_b \] _b described by many scientists, which may possibly reduce the annihilation cross-section down to lower than 10^–11 times than that of a simple elastic collision. The above formula could also be found useful, for some applications, which are currently in progress.     

Systems of colliding bodies in a gravitational field: Evolution and role of the internal rotations

To emphasize the rotational effects of a simple friction between colliding bodies in a keplerian field we investigate numerically the evolution of the rotational energies in a three dimensional system of spherical particles interacting through inelastic collisions in a deterministic model. All the particles are made of the same material but they possibly have different sizes. Each collision reduces the relative surface velocity and there are exchanges between orbital energy and rotational energy. Our results are compared with some previous papers and our aim is to supply other probabilists models with simple basic references about mean dynamical properties.The rotational energy of the colliding bodies tends to reach an equilibrium state that depends only on the rate of energy loss in the collision process. Internal rotations prevent the complete flattening of the system. With this model, light and small particles spin faster than the massive and big ones. We observe an excess of prograde rotations on counterclockwise orbits. The ratio of rotational and orbital energies is % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyramaaBa% aaleaacaWGYbaabeaakiaac+cacaWGfbWaaSbaaSqaaiaadUgaaeqa% aOGaeyisISRaaGymaiaaicdadaahaaWcbeqaaiabgkHiTiaaiodaaa% aaaa!3F83!\[E_r /E_k \approx 10^{ - 3} \] while the ratio of corresponding mean angular velocities is % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaaWaaeaacq% aHjpWDaiaawMYicaGLQmcacaGGVaWaaaWaaeaacqGHPoWvaiaawMYi% caGLQmcacqGHijYUcaaIYaaaaa!4008!\[\left\langle \omega \right\rangle /\left\langle \Omega \right\rangle \approx 2\] These values depends strongly on the dimensional scale of the model.     

Free convection effects on the flow past a moving vertical infinite porous plate

An analysis of the effects of free convection currents on the flow field of an incompressible viscous fluid past an infinite porous plate, which is uniformly accelerated upwards in its own plane, is presented, when the fluid is subjected to a variable suction (or injection) velocity. It is assumed that this normal velocity at the porous plate varies att^–1/2, wheret denotes time. The equations governing the flow are solved numerically, using two-point boundary value shooting techniques.     

Similarity solution for free and forced convection hydromagnetic flow over a horizontal semi-infinite plate through a non-homogeneous porous medium

A similarity analysis for the free and forced convection hydromagnetic flow over a horizontal semi-infinite flat plate through a non-homogeneous porous medium is presented, taking into account the hydrostatic pressure variation normal to the flat plate. The similarity solution of the problem under consideration is obtained under certain valid simplifying assumptions when, (i) the plate temperature is inversely proportional to the square root of the distance from the leading edge, (ii) the intensity of the applied magnetic field, normal to the plate, changes with the inverse square root of the distance from the leading edge, and (iii) the permeability of the porous medium, occupying a semi-infinite region of the space bounded by the flat plate, is proportional to the distance measured in the direction of the flow. A numerical solution of the resulting system of ordinary differential equations of motion and energy is obtained, depending on the Prandtl number Pr, the magnetic parameterM _n ,the bouyancy parameter , and the permeability parameterP _m .The variations of the fundamental quantities of the problem are shown graphically followed by a quantitative discussion.     

Mean values of particular functions in the elliptic motion

The mean values % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaamaalaaabaGaaGymaaqaaiaaikdacqaHapaCaaWaa8qCaeaacaGG% OaacbaGaa8NKbiabgkHiTiaadYgacaGGPaGaa8hiaiGacogacaGGVb% Gaai4Caiaa-bcacaWGRbGaa8NKbiaa-bcacaWGKbGaamiBaaWcbaGa% aGimaaqaaiaaikdacqaHapaCa0Gaey4kIipakiaa-bcacaqGHbGaae% OBaiaabsgacaWFGaWaaSaaaeaacaaIXaaabaGaaGOmaiabec8aWbaa% daWdXbqaaiaacIcacaWFsgGaeyOeI0IaamiBaiaacMcacaWFGaGaci% 4CaiaacMgacaGGUbGaa8hiaiaadUgacaWFsgGaa8hiaiaadsgacaWG% SbaaleaacaaIWaaabaGaaGOmaiabec8aWbqdcqGHRiI8aaaa!6BC2!\[\frac{1}{{2\pi }}\int\limits_0^{2\pi } {(f - l) \cos kf dl} {\rm{and}} \frac{1}{{2\pi }}\int\limits_0^{2\pi } {(f - l) \sin kf dl}\] (where f and l are respectively the true anomaly and the mean anomaly in the elliptic motion and k is an integer) are given in closed form.     

On hamiltonian averaging theories and resonance

In this article, we review the construction of Hamiltonian perturbation theories with emphasis on Hori's theory and its extension to the case of dynamical systems with several degrees of freedom and one resonant critical angle. The essential modification is the comparison of the series terms according to the degree of homogeneity in both % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGGipm0dc9vqaqpepu0xbbG8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOaaaeaacq% aH1oqzaSqabaaaaa!3699!\[\sqrt \varepsilon \]and a parameter which measures the distance from the exact resonance, instead of just % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGGipm0dc9vqaqpepu0xbbG8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOaaaeaacq% aH1oqzaSqabaaaaa!3699!\[\sqrt \varepsilon \].     

Formation of the saturnian system: A modern Laplacian theory

A theory for the formation of Saturn and its family of satellites, which is based on ideas of supersonic turbulent convection applied to the original Laplacian hypothesis, is presented. It is shown that if the primitive rotating cloud which gravitationally contracted to form Saturn possessed the same level of turbulent kinetic energy as the clouds which formed Jupiter and the Sun, given by % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSqaaSqaai% aaigdaaeaacaaIYaaaaOGaaiikaiabeg8aYnaaBaaajea4baGaamiD% aaWcbeaakiaadAhadaqhaaqcKfaGaeaadaWgaaqcKjaGaeaacaWG0b% aabeaaaSqaaiaaikdaaaGccaGGPaGaeyypa0ZaaSqaaSqaaiaaigda% aeaacaaIYaaaaOGaeqOSdiMaeqyWdiNaam4raiaad2eacaGGOaGaam% OCaiaacMcacaGGVaGaamOCaaaa!4D3D!\[\tfrac{1}{2}(\rho _t v_{_t }^2 ) = \tfrac{1}{2}\beta \rho GM(r)/r\] where =0.1065 ± 0.0015, then it would shed a concentric system of orbiting gas rings each of about the same mass: namely, 1.0 × 10^–3 M _S. The orbital radii R _n (n = 0, 1, 2, ...) of these gas rings form a geometric sequence similar to the observed distances of the regular satellites. It is proposed that the satellites condensed from the gas rings one at a time, commencing with Iapetus which originally occupied a circular orbit at radius 11.4 R _S. As the temperatures of the gas rings T _n increase with decreasing orbital size according as T _n 1/R _n , a uniform gradient should be evident amongst the satellite compositions: Mimas is expected to be the rockiest and Iapetus the least rocky satellite. The densities predicted by the model coincide with the Voyager-determined values. Iapetus contains some 8% by weight solid CH₄. Titan is believed to be a captured satellite. It was probably responsible for driving Iapetus to its present distant orbit. Accretional time-scales and the post-accretional evolution of the satellites are briefly discussed.     

Effects of shape and spin on the tidal disruption of P/Shoemaker-Levy 9

I derive an approximate criterion for the tidal disruption of a “rubble pile” body as it passes close to a planet (or the sun): % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyWdi3aaS% baaSqaaiaacogaaeqaaOGaeyisIS7aamWaaeaacaaIYaGaeqyWdihd% caWGWbGccaGGDbWaaeWaaeaadaWcaaqaaiaadkfamiaadchaaOqaai% aadkhaaaaacaGLOaGaayzkaaWaaWbaaSqabeaacaaIZaaaaOGaey4k% aSYaaeWaaeaadaWcaaqaaiabeM8a3bqaaiabeM8a3XGaaGimaaaaaO% GaayjkaiaawMcaamaaCaaaleqabaGaaGOmaaaaaOGaay5waiaaw2fa% amaabmaabaWaaSaaaeaacaWGHbaabaGaamOyaaaaaiaawIcacaGLPa% aacaGGSaaaaa!5229!\[\rho _c \approx \left[ {2\rho p]\left( {\frac{{Rp}}{r}} \right)^3 + \left( {\frac{\omega }{{\omega 0}}} \right)^2 } \right]\left( {\frac{a}{b}} \right),\] where ? _c is the critical density below which the body will be disrupted, ? _p is the density of the planet (or sun), R _p is the radius of the planet, r is the periapse distance, Ω is the rotation frequency of the body, Ω₀ is the surface orbit frequency about a body of unit density, and a/b is the axis ratio of the body, considered as a prolate ellipsoid. For P/Shoemaker Levy 9, in its passage close to Jupiter in 1992, this expression suggests that the critical density is ~1.2 for a spherical, non-spinning nucleus, but could be >2.5 for a 2:1 elongate body with a typical rotation period of ~10 hours.     

Spectra of stretching numbers and helicity angles in dynamical systems

We define a stretching number (or Lyapunov characteristic number for one period) (or stretching number) a = In % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGGipm0dc9vqaqpepu0xbbG8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaqWaaeaada% Wcaaqaaiabe67a4jaadshacqGHRaWkcaaIXaaabaGaeqOVdGNaamiD% aaaaaiaawEa7caGLiWoaaaa!3F1E!\[\left| {\frac{{\xi t + 1}}{{\xi t}}} \right|\]as the logarithm of the ratio of deviations from a given orbit at times t and t + 1. Similarly we define a helicity angle as the angle between the deviation % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGGipm0dc9vqaqpepu0xbbG8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOVdGNaam% iDaaaa!3793!\[\xi t\]and a fixed direction. The distributions of the stretching numbers and helicity angles (spectra) are invariant with respect to initial conditions in a connected chaotic domain. We study such spectra in conservative and dissipative mappings of 2 degrees of freedom and in conservative mappings of 3-degrees of freedom. In 2-D conservative systems we found that the lines of constant stretching number have a fractal form.     

Intrinsic orbital coordinates on a Hamiltonian surface

A reversible dynamical system with two degrees-of-freedom is reduced to a second-order, Hamiltonian system under a change of independent variable. In certain circumstances, the reduced order system may be integrated following an orthogonal curvilinear transformation from Cartesian x,y to intrinsic orbital coordinates , . Solutions for the orbit position and true time variables are expressed by: % MathType!MTEF!2!1!+- % feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGGipm0dc9vqaqpepu0xbbG8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabg2 % da9iaadAgacaGGOaGaeqOVdGNaaiilaiabeE7aOjaacMcacaGGSaGa % aeiiaiaadMhacqGH9aqpcaWGNbGaaiikaiabe67a4jaacYcacqaH3o % aAcaGGPaGaaiilaiaabccacaWGKbGaamiDaiabg2da9iabgglaXoaa % dmaabaWaaSaaaeaacaWGibWaa0baaKqaahaacqaH+oaEaeaacaqGYa % aaaOGaam4raiabgUcaRiaadIeadaqhaaqcbaCaaiabeE7aObqaaiaa % ikdaaaGccaWGfbaabaGaaGOmaiaacIcacaWGibGaey4kaSIaamyvai % aacMcaaaaacaGLBbGaayzxaaWaaWbaaSqabKqaGhaacaaIXaGaai4l % aiaaikdaaaGccaWGKbGaeqiXdqhaaa!6498! \[ x = f(\xi ,\eta ),{\rm{ }}y = g(\xi ,\eta ),{\rm{ }}dt = \pm \left[ {\frac{{_\xi ^{\rm{2}} {\ie} + _\eta ^2 }}{{2( + U)}}} \righ \]1446 1040 where U is the potential function, and z is the new independent variable. The functions f, g may be expressed by quadratures when the metric coefficients {\er},{\ie} are specified. Two second-order, partial differential equations specify {\er}, {\ie} and Hamiltonian {\tH}. Auxiliary conditions are needed because the solutions are underdetermined. For example, both sets of curvilinear coordinate lines are orbits when certain dynamical compatibility conditions between U and {\ie} (or {\er}) are satisfied. Alternatively, when orbits cross the parametric curves, the auxiliary condition {\er} = {\ie} specifies a conformal transformation, and the partial differential equation for {\tH} may be reduced to an ordinary differential equation for the orbit curve. In either case, integrability is guaranteed for Lionville dynamical systems. Specific applications are presented to illustrate direct solution for the orbit (e.g., two fixed centers) and inverse solution for the potential.     

Computer simulations in the general three-body problem. The theoretical bases of the studies

In this article we present a theoretical method for the study of the general three-body problem by computer simulation developed in the Leningrad State University Astronomical Observatory (LSU AO). This method permits statistical methods to be used for studying the behaviour of triple systems. This is achieved by selecting a representative sample of initial conditions which then reveal general features of the evolution.The main results of numerical experiments on the three-body problem carried out at the LSU AO during the past 25 years have been summarized in the reviews by Anosova (1985), Anosova and Orlov (1985), and Anosova (1986).Systematic studies of about 3 × 104 triple systems with negative total energy (E < 0) have yielded the following main results. Most (93.4%) of the systems decay; the decay always occurs after a close triple approach of the components. In a system with unequal masses, the escaping body usually has the smallest mass. A small fraction (4.3%) of stable systems is formed if the angular momentum is non-zero. The qualitative evolution in three-dimensional cases is the same as for planar systems. Small changes in initial conditions sometimes lead to substantial differences in the final outcome. The decay of triple systems is a stochastic process similar to radioactive decay. The estimated mean lifetime is equal to % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiiYdd9qrFfea0dXdf9vqai-hEir8Ve% ea0de9qq-hbrpepeea0db9q8as0-LqLs-Jirpepeea0-as0Fb9pgea% 0lrP0xe9Fve9Fve9qapdbaqaaeGacaGaaiaabeqaamaabaabcaGcba% Waa0aaaeaacaWGubaaaaaa!3C56!\[\overline T \] = (107.1 ± 1.8) crossing times for equal-mass components. Thus, for solar mass components and a typical dimension d = 0.01 pc, % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiiYdd9qrFfea0dXdf9vqai-hEir8Ve% ea0de9qq-hbrpepeea0db9q8as0-LqLs-Jirpepeea0-as0Fb9pgea% 0lrP0xe9Fve9Fve9qapdbaqaaeGacaGaaiaabeqaamaabaabcaGcba% Waa0aaaeaacaWGubaaaaaa!3C56!\[\overline T \] = (1.6 ± 1.5) × 10⁶ y, and for triple galaxies with M = 101° M ₀ and d = 50 kpc, % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiiYdd9qrFfea0dXdf9vqai-hEir8Ve% ea0de9qq-hbrpepeea0db9q8as0-LqLs-Jirpepeea0-as0Fb9pgea% 0lrP0xe9Fve9Fve9qapdbaqaaeGacaGaaiaabeqaamaabaabcaGcba% Waa0aaaeaacaWGubaaaaaa!3C56!\[\overline T \] = (1.8 ± 1.7) × 1011 y. The value % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiiYdd9qrFfea0dXdf9vqai-hEir8Ve% ea0de9qq-hbrpepeea0db9q8as0-LqLs-Jirpepeea0-as0Fb9pgea% 0lrP0xe9Fve9Fve9qapdbaqaaeGacaGaaiaabeqaamaabaabcaGcba% Waa0aaaeaacaWGubaaaaaa!3C56!\[\overline T \] decreases with increasing mass dispersion.In this article we also carry out a theoretical analysis of the changes of the integrals of motion in the general three-body problem used as the controls on the calculations. The following basic results have been found: (1) analytical functions of the changes of the integrals of motion during the integration time have been obtained; (2) changes in the integrals of the mass-centre of a triple system do not correlate with the cumulative integration errors; (3) the cumulative changes of the integral of energy are proportional to the sum of squares of the cumulative errors in the coordinates and the velocities of the bodies; (4) the cumulative changes of the square of the total angular momentum are proportional to the product of the square of these cumulative errors.The analysis of the accuracy of computer simulations conducted in LSU AO for the 3 × 10⁴ triple systems with E < 0 is summarized by the following basic qualitative results: (1) the unstable triple systems decay after a mean lifetime % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiiYdd9qrFfea0dXdf9vqai-hEir8Ve% ea0de9qq-hbrpepeea0db9q8as0-LqLs-Jirpepeea0-as0Fb9pgea% 0lrP0xe9Fve9Fve9qapdbaqaaeGacaGaaiaabeqaamaabaabcaGcba% Waa0aaaeaacaWGubaaaaaa!3C56!\[\overline T \] 100 or % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiiYdd9qrFfea0dXdf9vqai-hEir8Ve% ea0de9qq-hbrpepeea0db9q8as0-LqLs-Jirpepeea0-as0Fb9pgea% 0lrP0xe9Fve9Fve9qapdbaqaaeGacaGaaiaabeqaamaabaabcaGcba% Waa0aaaeaacaWGubaaaaaa!3C56!\[\overline T \] 104 % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiiYdd9qrFfea0dXdf9vqai-hEir8Ve% ea0de9qq-hbrpepeea0db9q8as0-LqLs-Jirpepeea0-as0Fb9pgea% 0lrP0xe9Fve9Fve9qapdbaqaaeGacaGaaiaabeqaamaabaabcaGcba% Waa0aaaeaacaWGObaaaaaa!3C6A!\[\overline h \]_t where is a crossing time, and % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiiYdd9qrFfea0dXdf9vqai-hEir8Ve% ea0de9qq-hbrpepeea0db9q8as0-LqLs-Jirpepeea0-as0Fb9pgea% 0lrP0xe9Fve9Fve9qapdbaqaaeGacaGaaiaabeqaamaabaabcaGcba% Waa0aaaeaacaWGObaaaaaa!3C6A!\[\overline h \], is a mean integration step After this integration time % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiiYdd9qrFfea0dXdf9vqai-hEir8Ve% ea0de9qq-hbrpepeea0db9q8as0-LqLs-Jirpepeea0-as0Fb9pgea% 0lrP0xe9Fve9Fve9qapdbaqaaeGacaGaaiaabeqaamaabaabcaGcba% Waa0aaaeaacaWGubaaaaaa!3C56!\[\overline T \] the mean cumulative relative changes % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiiYdd9qrFfea0dXdf9vqai-hEir8Ve% ea0de9qq-hbrpepeea0db9q8as0-LqLs-Jirpepeea0-as0Fb9pgea% 0lrP0xe9Fve9Fve9qapdbaqaaeGacaGaaiaabeqaamaabaabcaGcba% Waa0aaaeaacaWGebGaamyraaaaaaa!3D10!\[\overline {DE} \] of the integrals of the energy of the triple systems are equal to % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiiYdd9qrFfea0dXdf9vqai-hEir8Ve% ea0de9qq-hbrpepeea0db9q8as0-LqLs-Jirpepeea0-as0Fb9pgea% 0lrP0xe9Fve9Fve9qapdbaqaaeGacaGaaiaabeqaamaabaabcaGcba% Waa0aaaeaacaWGebGaamyraaaaaaa!3D10!\[\overline {DE} \] = (0.9±0.1) × 10^–4, and the mean cumulative relative changes % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiiYdd9qrFfea0dXdf9vqai-hEir8Ve% ea0de9qq-hbrpepeea0db9q8as0-LqLs-Jirpepeea0-as0Fb9pgea% 0lrP0xe9Fve9Fve9qapdbaqaaeGacaGaaiaabeqaamaabaabcaGcba% Waa0aaaeaacaWGebGaamitaaaaaaa!3D17!\[\overline {DL} \] of the area integrals are equal to % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiiYdd9qrFfea0dXdf9vqai-hEir8Ve% ea0de9qq-hbrpepeea0db9q8as0-LqLs-Jirpepeea0-as0Fb9pgea% 0lrP0xe9Fve9Fve9qapdbaqaaeGacaGaaiaabeqaamaabaabcaGcba% Waa0aaaeaacaWGebGaamitaaaaaaa!3D17!\[\overline {DL} \] = (1.0±0.1) × 10^–6; the mean values of the cumulative errors % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiiYdd9qrFfea0dXdf9vqai-hEir8Ve% ea0de9qq-hbrpepeea0db9q8as0-LqLs-Jirpepeea0-as0Fb9pgea% 0lrP0xe9Fve9Fve9qapdbaqaaeGacaGaaiaabeqaamaabaabcaGcba% Gaamiraiaadkhaaaa!3D2C!\[{Dr}\], % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiiYdd9qrFfea0dXdf9vqai-hEir8Ve% ea0de9qq-hbrpepeea0db9q8as0-LqLs-Jirpepeea0-as0Fb9pgea% 0lrP0xe9Fve9Fve9qapdbaqaaeGacaGaaiaabeqaamaabaabcaGcba% Waa0aaaeaacaWGebGaamOvaaaaaaa!3D21!\[\overline {Dv} \] in defining the coordinates (r) and velocities (v) of the bodies (during the total integration time % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiiYdd9qrFfea0dXdf9vqai-hEir8Ve% ea0de9qq-hbrpepeea0db9q8as0-LqLs-Jirpepeea0-as0Fb9pgea% 0lrP0xe9Fve9Fve9qapdbaqaaeGacaGaaiaabeqaamaabaabcaGcba% Waa0aaaeaacaWGubaaaaaa!3C56!\[\overline T \]) are equal to % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiiYdd9qrFfea0dXdf9vqai-hEir8Ve% ea0de9qq-hbrpepeea0db9q8as0-LqLs-Jirpepeea0-as0Fb9pgea% 0lrP0xe9Fve9Fve9qapdbaqaaeGacaGaaiaabeqaamaabaabcaGcba% Waa0aaaeaacaWGebGaamOCaaaaaaa!3D3D!\[\overline {Dr} \] = 0.5 × 10^–3 d, % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiiYdd9qrFfea0dXdf9vqai-hEir8Ve% ea0de9qq-hbrpepeea0db9q8as0-LqLs-Jirpepeea0-as0Fb9pgea% 0lrP0xe9Fve9Fve9qapdbaqaaeGacaGaaiaabeqaamaabaabcaGcba% Waa0aaaeaacaWGebGaamODaaaaaaa!3D41!\[\overline {Dv} \] = 0.5 × 10^–2 v, where d is the unit of distance, and v is the unit of velocity; the mean local integration errors (of one integration step) are equal to r= 5 × 10^–8 d, 6v = 5 × 10^–7 v; (2) the process of accumulation of integration errors has a complicated character and correlates strongly with the process of dynamical evolution of the triple systems; (a) because of the strong gravitational interplays of the bodies, the process of the accumulation of the integration errors is very intensive; however, the triple systems with these interplays of the bodies have, as a rule, a small escape time % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiiYdd9qrFfea0dXdf9vqai-hEir8Ve% ea0de9qq-hbrpepeea0db9q8as0-LqLs-Jirpepeea0-as0Fb9pgea% 0lrP0xe9Fve9Fve9qapdbaqaaeGacaGaaiaabeqaamaabaabcaGcba% Waa0aaaeaacaWGubaaaaaa!3C56!\[\overline T \] t, and the cumulative calculation errors are small too; (b) in the stable triple systems the local integration errors are practically constant during the numerical study of their evolution, and the calculations can be carried out (if it is necessary) during the time % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiiYdd9qrFfea0dXdf9vqai-hEir8Ve% ea0de9qq-hbrpepeea0db9q8as0-LqLs-Jirpepeea0-as0Fb9pgea% 0lrP0xe9Fve9Fve9qapdbaqaaeGacaGaaiaabeqaamaabaabcaGcba% Waa0aaaeaacaWGubaaaaaa!3C56!\[\overline T \] = (2–3) × 10³ without disturbing the periodical motions of the bodies; (3) thus, in the general three-body problem with different initial conditions, it is not necessary to carry out the computer simulations over long times, as most of the triple systems decay and do not have very long lifetimes; (4) the mean level of the cumulative errors Dr and Dv of the definitions of the coordinates and velocities of bodies in the different triple systems is practically equal.     

Hall effects on an oscillatory MHD flow in the Stokes problem past an infinite vertical porous plate,I

Hall effects on the hydromagnetic free convection flow of an electrically conducting incompressible viscous fluid past a steadily moving vertical porous plate has been analysed when the free stream oscillates in magnitude. The flow is subjected to a constant suction, through the porous plate, and the difference between wall temperature and the free-stream is moderately large causing the free convection currents. The mathematical analysis is presented for the hydromagnetic boundary layer flow without taking into account the induced magnetic field. This is a valid assumption for small magnetic Reynolds number. Approximate solutions for the components of velocity field and temperature field and their related quantities are obtained. The influence of various parameters entering into the problem is extensively discussed with the help of graphs and tables.     

Free convection in hydromagnetic flows of a viscous heat-generating fluid with wall temperature oscillation and Hall currents

A study is made of the free convection in hydromagnetic flows through a porous medium of a heat generating fluid past an infinite vertical porous plate. A strong magnetic field is imposed in a direction which is perpendicular to the free stream and makes an alge to the vertical direction. The governing equations for the hydromagnetic fluid flow and the heat transfer are solved analytically. The influence of Hall currents, the permeabilityK and the inclination upon the velocity field are shown in figures.     

Mixed convection over a horizontal plate with vectored mass transfer in a transverse magnetic field

An exact similarity solution is presented for developing mixed convection flows of electrically conducting fluids over a semi-infinite horizontal plate with vectored mass transfer at the wall which are subjected to an applied transverse magnetic field. This solution is given for the case of a wall temperature that is inversely proportional to the square root of the distance from the leading edge. By application of appropriate coordinate transformations, the governing momentum and energy boundary-layer equations are expressed as a set of coupled ordinary differential equations that depend on a magnetic parameter, the buoyancy parameter, and the Prandtl number. The shear stress, the total heat transfer, and the displacement thickness are calculated for different values of both buoyancy and magnetic parameters.     

Effects of shape and spin on the tidal disruption of P/Shoemaker-Levy 9

I derive an approximate criterion for the tidal disruption of a rubble pile body as it passes close to a planet (or the sun): % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyWdi3aaS% baaSqaaiaacogaaeqaaOGaeyisIS7aamWaaeaacaaIYaGaeqyWdihd% caWGWbGccaGGDbWaaeWaaeaadaWcaaqaaiaadkfamiaadchaaOqaai% aadkhaaaaacaGLOaGaayzkaaWaaWbaaSqabeaacaaIZaaaaOGaey4k% aSYaaeWaaeaadaWcaaqaaiabeM8a3bqaaiabeM8a3XGaaGimaaaaaO% GaayjkaiaawMcaamaaCaaaleqabaGaaGOmaaaaaOGaay5waiaaw2fa% amaabmaabaWaaSaaaeaacaWGHbaabaGaamOyaaaaaiaawIcacaGLPa% aacaGGSaaaaa!5229!\[\rho _c \approx \left[ {2\rho p]\left( {\frac{{Rp}}{r}} \right)^3 + \left( {\frac{\omega }{{\omega 0}}} \right)^2 } \right]\left( {\frac{a}{b}} \right),\] where _c is the critical density below which the body will be disrupted, _p is the density of the planet (or sun), R _p is the radius of the planet, r is the periapse distance, is the rotation frequency of the body, ₀ is the surface orbit frequency about a body of unit density, and a/b is the axis ratio of the body, considered as a prolate ellipsoid. For P/Shoemaker Levy 9, in its passage close to Jupiter in 1992, this expression suggests that the critical density is ~1.2 for a spherical, non-spinning nucleus, but could be >2.5 for a 2:1 elongate body with a typical rotation period of ~10 hours.     

Identification of Emissions in the Spectrum of Comet C/1989 VI (Skoritchenko–George) Obtained with the 6-m SAO RAS Reflector

The emission spectrum of comet Skoritchenko–George (C/1989 VI), unusual in its information content, was obtained on February 26.7 UT, 1990, with the use of a TV scanner installed on the 6-m BTA reflector of the Special Astronomical Observatory of the Russian Academy of Sciences (SAO RAS) in Nizhni Arkhyz. Detailed identification of the emission lines of this comet was made. The observed spectrum contains 311 emission lines, including those of the molecules. Among others, the lines of the negative carbon C ₂ ^- ion and the lines corresponding to the electron transition in the neutral CO molecule are discovered. The presence of a large number of lines of the neutral CO molecule (the Asundi bands and the triplet bands) in the visible region is one of the uncommon features of the emission spectrum of this comet. The triplet lines : 15–3, 13–2, 11–2, 9–1, 8–1, 7–1, 7–0, 5–0, 4–0; : 7–0, 6–0, 5–0; and a" : 11–1 (K = 3, 4); 16–4 (K= 0, 1, 2, 4); 9-0 (K= 0, 1, 2); 8–0 (K= 0) were identified for the first time. Prior to this work, the lines of CO in the visible range were observed only in the spectrum of comet C/1979 VI (Bradfield) in 1989.     

The dynamical flattenings of Mercury and Venus

The method which is used to calculate the dynamical flattenings % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiiYdd9qrFfea0dXdf9vqai-hEir8Ve% ea0de9qq-hbrpepeea0db9q8as0-LqLs-Jirpepeea0-as0Fb9pgea% 0lrP0xe9Fve9Fve9qapdbaqaaeGacaGaaiaabeqaamaabaabcaGcba% Gaamisaiabg2da9iaacIcacaWGdbGaeyOeI0YaaSaaaeaacaaIXaaa% baGaaGOmaaaacaGGOaGaamyqaiabgUcaRiaadkeacaGGPaGaaiykai% aac+cacaWGdbaaaa!4717!\[H = (C - \frac{1}{2}(A + B))/C\] of the Earth and Moon meets with difficulties when it applies to Mercury and Venus. In this paper, after the calculation of the dimensionless moment of inertiaC/MR ² by solving the Emden equation, the effectiveness of the method deriving dynamical flattening from the observed value of the Mercury's obliquity is analysed based on the resonance rotation theory. Some suggestions are made for the future space explorations. Finally, the ranges of dynamical flattening and of the obliquity of Venus are calculated.