![]() ![]() ![]() There is no unique way to measure localization, and individual measures can reflect different aspects of the same quantum state. In Section 3.4.Measuring the degree of localization of quantum states in phase space is essential for the description of the dynamics and equilibration of quantum systems, but this topic is far from being understood. Orbits in the isochrone potential and other spherical potentials ¶Īs we discussed already above, the potentials corresponding to a point mass (Kepler) and to a constant density (homogeneous sphere) bracket the range of plausible galactic potentials, as densities typically remain constant or decrease with increasing \(r\). The semi-major axis is along \(x\) and the orbit is at \(y=0\) at positive \(x\) at \(t=0\): Let’s fix the latter two of these by orienting the ellipse such that These constants are (i+ii) the semi-major and semi-minor axes of the ellipse, \(a\) and \(b\) respectively, (iii) the orientation of the ellipse within the plane, and (iv) the position along the orbit at \(t=0\). We are working in the orbital plane, which is constant, and there are therefore only two positions and velocities that count the position and velocity perpendicular to this plane are zero at all times). This solution has four constants that need to be determined from the initial condition of the orbit (remember that Note that these are not Keplerian orbits, even though Keplerian orbits are also ellipses as we will see below, Keplerian orbits are ellipses with the origin of the mass distribution at the focus of the ellipse, not at the center of the ellipse as for the homogeneous sphere. This solution describes an ellipse, with the center at the origin of the mass distribution ( \(r=0\)). First, we re-write the potential for the homogeneous sphere from Equation ( 3.41) inside of its limiting radius as The homogeneous sphere (see Section 3.4.2) has particularly simple orbits. To illustrate the concepts introduced in the previous subsection, we consider orbital properties in a few simple spherical potentials. ![]() Orbits in specific spherical potentials ¶ Radial and azimuthal frequencies are simply equal to \(2\pi\) over the radial and azimuthal period. 2007 for methods for dealing with this situation). Some care is necessary in this numerical calculation, because the integrands integrably diverge at the end-points of the integration interval (see Press et al. All of the quantities \(T_r\), \(\Delta \psi\), and \(T_\psi\) typically need to be computed numerically. Where we need the absolute value of \(\Delta \psi\), because \(\Delta \psi\) can be negative. The general theory of relativity and galaxies Part IV: Galaxy formation and evolution (under construction) Equilibria of elliptical galaxies and dark matter halos Orbits in triaxial mass distributions and surfaces of section Gravitation in elliptical galaxies and dark matter halos The kinematics and dynamics of galactic rotation Equilibria of collisionless stellar systems Orbits in the isochrone potential and other spherical potentials General properties of orbits in spherical potentials ![]()
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