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Register a new userA simple, heuristic derivation of the Balescu-Lenard kinetic equation for stellar systems
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Chris Hamilton
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The unshielded nature of gravity means that stellar systems are inherently inhomogeneous. As a result, stars do not move in straight lines. This obvious fact severely complicates the kinetic theory of stellar systems because position and velocity turn out to be poor coordinates with which to describe stellar orbits - instead, one must use angle-action variables. Moreover, the slow relaxation of star clusters and galaxies can be enhanced or suppressed by collective interactions (polarisation effects) involving many stars simultaneously. These collective effects are also present in plasmas; in that case, they are accounted for by the Balescu-Lenard (BL) equation, which is a kinetic equation in velocity space. Recently several authors have shown how to account for both inhomogeneity and collective effects in the kinetic theory of stellar systems by deriving an angle-action generalisation of the BL equation. Unfortunately their derivations are long and complicated, involving multiple coordinate transforms, contour integrals in the complex plane, and so on. On the other hand, Rostokers superposition principle allows one to pretend that a long-range interacting $N$-body system, such as a plasma or star cluster, consists merely of uncorrelated particles that are dressed by polarisation clouds. In this paper we use Rostokers principle to provide a simple, intuitive derivation of the BL equation for stellar systems which is much shorter than others in the literature. It also allows us to straightforwardly connect the BL picture of self-gravitating kinetics to the classical two-body relaxation theory of uncorrelated flybys pioneered by Chandrasekhar.
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The traditional Chandrasekhar picture of the slow relaxation of stellar systems assumes that stars orbits are only modified by occasional, uncorrelated, two-body flyby encounters with other stars. However, the long-range nature of gravity means that in reality large numbers of stars can behave collectively. In stable systems this collective behaviour (i) amplifies the noisy fluctuations in the systems gravitational potential, effectively dressing the two-body (star-star) encounters, and (ii) allows the system to support large-scale density waves (a.k.a. normal modes) which decay through resonant wave-star interactions. If the relaxation of the system is dominated by effect (i) then it is described by the Balescu-Lenard (BL) kinetic theory. Meanwhile if (ii) dominates, one must describe relaxation using quasilinear (QL) theory, though in the stellar-dynamical context the full set of QL equations has never been presented. Moreover, in some systems like open clusters and galactic disks, both (i) and (ii) might be important. Here we present for the first time the equations of a unified kinetic theory of stellar systems in angle-action variables that accounts for both effects (i) and (ii) simultaneously. We derive the equations in a heuristic, physically-motivated fashion and work in the simplest possible regime by accounting only for very weakly damped waves. This unified theory is effectively a superposition of BL and QL theories, both of which are recovered in appropriate limits. The theory is a first step towards a comprehensive description of those stellar systems for which neither the QL or BL theory will suffice.
In this work, we elucidate the mathematical structure of the integral that arises when computing the electron-ion temperature equilibration time for a homogeneous weakly-coupled plasma from the Lenard-Balescu equation. With some minor approximations, we derive an exact formula, requiring no input Coulomb logarithm, for the equilibration rate that is valid for moderate electron-ion temperature ratios and arbitrary electron degeneracy. For large temperature ratios, we derive the necessary correction to account for the coupled-mode effect, which can be evaluated very efficiently using ordinary Gaussian quadrature.
In the mean-field regime, the evolution of a gas of $N$ interacting particles is governed in first approximation by a Vlasov type equation with a self-induced force field. This equation is conservative and describes return to equilibrium only in the very weak sense of Landau damping. However, the first correction to this approximation is given by the Lenard-Balescu operator, which dissipates entropy on the very long timescale $O(N)$. In this paper, we show how one can derive rigorously this correction on intermediate timescales (of order $O(N^r)$ for $r<1$), close to equilibrium.
Starting from the action-angle variables and using a standard asymptotic expansion, here we present a new derivation of the Wave Kinetic Equation for resonant process of the type $2leftrightarrow 2$. Despite not offering new physical results and despite not being more rigorous than others, our procedure has the merit of being straightforward; it allows for a direct control of the random phase and random amplitude hypothesis of the initial wave field. We show that the Wave Kinetic Equation can be derived assuming only initial random phases. The random amplitude approximation has to be taken only at the end, after taking the weak nonlinearity and large box limits. This is because the $delta$-function over frequencies contains the amplitude-dependent nonlinear correction which should be dropped before the random amplitude approximation applies. If $epsilon$ is the small parameter in front of the anharmonic part of the Hamiltonian, the time scale associated with the Wave Kinetic equation is shown to be $1/epsilon^2$. We give evidence that random phase and amplitude hypotheses persist up to a time of the order $1/epsilon$.
We introduce CoSHA: a Code for Stellar properties Heuristic Assignment. In order to estimate the stellar properties, CoSHA implements a Gradient Tree Boosting algorithm to label each star across the parameter space ($T_{text{eff}}$, $log{g}$, $left[text{Fe}/text{H}right]$, and $left[alpha/text{Fe}right]$). We use CoSHA to estimate these stellar atmospheric parameters of $22,$k unique stars in the MaNGA Stellar Library (MaStar). To quantify the reliability of our approach, we run both internal tests using the Gottingen Stellar Library (GSL, a theoretical library) and the first data release of MaStar, and external tests by comparing the resulting distributions in the parameter space with the APOGEE estimates of the same properties. In summary, our parameter estimates span in the ranges: $T_{text{eff}}=[2900,12000],$K, $log{g}=[-0.5,5.6]$, $left[text{Fe}/text{H}right]=[-3.74,0.81]$, $left[alpha/text{Fe}right]=[-0.22,1.17]$. We report internal (external) uncertainties of the properties of $sigma_{T_{text{eff}}}sim48,(325),$K, $sigma_{log{g}}sim0.2,(0.4)$, $sigma_{left[text{Fe}/text{H}right]}sim0.13,(0.27)$, $sigma_{left[alpha/text{Fe}right]}sim0.09,(0.14)$. These uncertainties are comparable to those of other methods with similar objectives. Despite the fact that CoSHA is not aware of the spatial distribution of these physical properties in the Milky Way, we are able to recover the main trends known in the literature with great statistical confidence. The catalogue of physical properties can be accessed in url{http://ifs.astroscu.unam.mx/MaStar}.
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