Mechanism of Magnetic Flux Loss in Molecular Clouds

We investigate the detailed processes working in the drift of magnetic fields in molecular clouds. To the frictional force, whereby the magnetic force is transmitted to neutral molecules, ions contribute more than half only at cloud densities $n_{\rm H} < 10^4 {\rm cm}^{-3}$, and charged grains contribute more than 90% at $n_{\rm H} > 10^6 {\rm cm}^{-3}$. Thus grains play a decisive role in the process of magnetic flux loss. Approximating the flux loss time $t_B$ by a power law $t_B \propto B^{-\gamma}$, where $B$ is the mean field strength in the cloud, we find $\gamma \approx 2$, characteristic to ambipolar diffusion, only at $n_{\rm H} < 10^7 {\rm cm}^{-3}$. At higher densities, $\gamma$ decreases steeply with $n_{\rm H}$, and finally at $n_{\rm H} \approx n_{\rm dec} \approx {\rm a few} \times 10^{11} {\rm cm}^{-3}$, where magnetic fields effectively decouple from the gas, $\gamma << 1$ is attained, reminiscent of Ohmic dissipation, though flux loss occurs about 10 times faster than by Ohmic dissipation. Ohmic dissipation is dominant only at $n_{\rm H} > 1 \times 10^{12} {\rm cm}^{-3}$. While ions and electrons drift in the direction of magnetic force at all densities, grains of opposite charges drift in opposite directions at high densities, where grains are major contributors to the frictional force. Although magnetic flux loss occurs significantly faster than by Ohmic dissipation even at very high densities as $n_{\rm H} \approx n_{\rm dec}$, the process going on at high densities is quite different from ambipolar diffusion in which particles of opposite charges are supposed to drift as one unit.Comment: 34 pages including 9 postscript figures, LaTex, accepted by Astrophysical Journal (vol.573, No.1, July 1, 2002

Quasi-Monte Carlo methods for Choquet integrals

We propose numerical integration methods for Choquet integrals where the capacities are given by distortion functions of an underlying probability measure. It relies on the explicit representation of the integrals for step functions and can be seen as quasi-Monte Carlo methods in this framework. We give bounds on the approximation errors in terms of the modulus of continuity of the integrand and the star discrepancy.Comment: 6 page