Compton scattering in a converging fluid flow - III Spherical supercritical accretion

Radiative transfer in spherical, supercritical accretion on to a massive black hole is considered. Particular emphasis is placed on the case of non-adiabatic flow in which electron scattering in the converging flow is the dominant source of opacity and photon heating. In escaping diffusively, the photons, which are produced mainly at the trapping radius, undergo ∼(c/u_(tr))^2 scatterings (where utr is the velocity of the flow at the trapping radius), each one giving a secular fractional energy increase ∼(u_(tr)/c)^2 and a total average increase of order unity. The emitted radiation spectrum will be a superposition of the locally produced spectra at low frequencies and a power-law at high frequencies. For gas accreting radially with the free-fall speed, the spectral index observed in the outflowing radiation is α~2. Both thermal and non-thermal emission processes are discussed. The conditions under which bulk heating of the photons is important are specified

Hydromagnetic flows from accretion discs and the production of radio jets

We examine the possibility that energy and angular momentum are removed magnetically from accretion discs, by field lines that leave the disc surface and extend to large distances. We illustrate this mechanism by solving the equations of magnetohydrodynamics, assuming infinite conductivity, for axially symmetric, self-similar, cold magnetospheric flow from a Keplerian accretion disc in which the field strength B scales with radius r as B ∝r^(-5/4). We show that a centrifugally driven outflow of matter from the disc is possible, if the poloidal component of the magnetic field makes an angle of less than 60° with the disc surface. At large distances from the disc, the toroidal component of the magnetic field becomes important and collimates the outflow into a pair of anti-parallel jets moving perpendicular to the disc. Close to the disc, the flow is probably driven by gas pressure in a hot magnetically dominated corona. In this way, magnetic stresses can extract the angular momentum from a thin accretion disc and thus enable matter to be accreted, independently of the presence of viscosity. These jet solutions have the property that most of the power is concentrated within a central core, while most of the angular momentum and magnetic flux is carried near the jet walls. The relevance of this mechanism for the evolution of accretion discs around massive black holes in galactic nuclei and the production of jets in extragalactic radio sources is described

The gender and access to health services study: final report

Men and women frequently think and behave differently. To observe this is not to suggest anything so absurdly simple as that there are only male and female ways of being; behaviours and thought processes vary according to numerous other factors besides gender. That this is very generally the case however, does mean that there are broad - and often broadly predictable - differences in the way men and women engage with the world. Most commercial organisations understand this very well and plan accordingly. Many public authorities recognise it too and take these differences into account when developing and providing services. For historical reasons however the NHS has rarely done so. It is widely known that there are differences between men and women in the incidence and prevalence of most health conditions. Sometimes there are clear biological reasons for these differences but often there are not. Where biology offers little or no enlightenment, other questions need to be asked: · Do men and women behave in ways that predispose them to particular health conditions to different degrees? · Do men and women use health services with different degrees of effectiveness? · Do men and women receive differerent kinds of service from the NHS? The answer is – yes, these things happen frequently. This is sometimes to the disadvantage of one sex and sometimes to the disadvantage of the other. Sometimes it is to the disadvantage of both. And when these things happen, health outcomes are often affected. This report looks at the reasons why gender is such an important and fundamental determinant of health status and considers the ways in which gender inequalities can be tackled within the present legislative and policy framework. It also brings together the knowledge and evidence in relation to six specific areas of health concerns

X-ray Spectral Formation in a Converging Fluid Flow: Spherical Accretion into Black Holes

We study Compton upscattering of low-frequency photons in a converging flow of thermal plasma. The photons escape diffusively and electron scattering is the dominant source of opacity. We solve numerically and approximately analytically the equation of radiative transfer in the case of spherical, steady state accretion into black holes. Unlike previous work on this subject, we consider the inner boundary at a finite radius and this has a significant effect on the emergent spectrum. It is shown that the bulk motion of the converging flow is more efficient in upscattering photons than thermal Comptonization, provided that the electron temperature in the flow is of order a few keV or less. In this case, the spectrum observed at infinity consists of a soft component coming from those input photons which escaped after a few scatterings without any significant energy change and of a power law which extends to high energies and is made of those photons which underwent significant upscattering. The luminosity of the power law is relatively small compared to that of the soft component. The more reflective the inner boundary is, the flatter the power-law spectrum becomes. The spectral energy power-law index for black-hole accretion is always higher than 1 and it is approximately 1.5 for high accretion rates. This result tempts us to say that bulk motion Comptonization might be the mechanism behind the power-law spectra seen in black-hole X-ray sources.Comment: 37 pages, LaTex, AAS Macros, 8 ps figures, to appear in Ap

Compton scattering in a converging fluid flow - I. The transfer equation

The equations describing Compton scattering in an optically thick fluid flow are derived under the diffusion approximation. The relative importance of the bulk and random motions of the scattering electrons is discussed. In a converging fluid flow of speed u, bulk acceleration of photons dominates thermal Comptonization when u ≳ (12kT_e/m_e)^(1/2)⁠

A second eigenvalue bound for the Dirichlet Schroedinger operator

Let $\lambda_i(\Omega,V)$ be the $i$th eigenvalue of the Schr\"odinger operator with Dirichlet boundary conditions on a bounded domain $\Omega \subset \R^n$ and with the positive potential $V$. Following the spirit of the Payne-P\'olya-Weinberger conjecture and under some convexity assumptions on the spherically rearranged potential $V_\star$, we prove that $\lambda_2(\Omega,V) \le \lambda_2(S_1,V_\star)$. Here $S_1$ denotes the ball, centered at the origin, that satisfies the condition $\lambda_1(\Omega,V) = \lambda_1(S_1,V_\star)$. Further we prove under the same convexity assumptions on a spherically symmetric potential $V$, that $\lambda_2(B_R, V) / \lambda_1(B_R, V)$ decreases when the radius $R$ of the ball $B_R$ increases. We conclude with several results about the first two eigenvalues of the Laplace operator with respect to a measure of Gaussian or inverted Gaussian density

Compton scattering in a converging fluid flow - II. Radiation-dominated shock

The problem of Compton scattering in an optically thick fluid flow in which bulk motion is the dominant source of photon heating is illustrated by analysing a radiation-dominated, plane-parallel shock of speed u with photon to electron ratio greatly exceeding ∼(m_p/m_e)⁠. In traversing the shock (of thickness ∼(c/u) Thomson optical depths), a typical photon experiences (c/u)2 scatterings, each one giving a secular fractional energy increase ∼(u/c)^2 and a total average increase of order unity. In a converging fluid flow, an exponentially small number of photons are accelerated to an exponentially large energy. Thus, a power-law spectrum will be transmitted at high frequencies. For a shock of Mach number M, bulk acceleration produces a spectral index α = (M^2 − ½)(M^2 + 6)(M^2 – 1)^(−2), which tends to unity for a strong shock. The applicability of these results to quasars and the microwave background is briefly discussed

Shell-model calculations for the three-nucleon system

We use Faddeev's decomposition to solve the shell-model problem for three nucleons. The dependence on harmonic-oscillator excitations allowed in the model space, up to $32 \hbar\Omega$ in the present calculations, and on the harmonic-oscillator frequency is studied. Effective interactions derived from Nijmegen II and Reid93 potentials are used in the calculations. The binding energies obtained are close to those calculated by other methods. The structure of the Faddeev equations is discussed and a simple formula for matrix elements of the permutation operators in a harmonic-oscillator basis is given. The Pauli principle is properly treated in the calculations.Comment: 11 pages. REVTEX. 6 PostScript figure

Some remarks on the isoperimetric problem for the higher eigenvalues of the Robin and Wentzell Laplacians

We consider the problem of minimising the $k$th eigenvalue, $k \geq 2$, of the ($p$-)Laplacian with Robin boundary conditions with respect to all domains in $\mathbb{R}^N$ of given volume $M$. When $k=2$, we prove that the second eigenvalue of the $p$-Laplacian is minimised by the domain consisting of the disjoint union of two balls of equal volume, and that this is the unique domain with this property. For $p=2$ and $k \geq 3$, we prove that in many cases a minimiser cannot be independent of the value of the constant $\alpha$ in the boundary condition, or equivalently of the volume $M$. We obtain similar results for the Laplacian with generalised Wentzell boundary conditions $\Delta u + \beta \frac{\partial u}{\partial \nu} + \gamma u = 0$.Comment: 16 page