4. Orbits in spherical mass distributions¶
“Orbits” are the trajectories that bodies travel on under the influence of gravity in a gravitational field. As discussed in the previous chapters, these trajectories can be computed using Newton’s second law \begin{equation}\label{eq-newton-eom} m\, \ddot{\vec{x}} = m\,\vec{g}(\vec{x})\,, \end{equation} where \(\vec{g}(\vec{x})\) is the gravitational field, \(m\) is the mass of the body, and \(\ddot{\vec{x}}\) is the acceleration vector. As discussed in Section 2.1, because we can cancel the mass on both sides of this equation, the acceleration \(\ddot{\vec{x}}\) in a gravitational field is independent of the mass of the body. If, furthermore, the mass of the body is so small that it does not affect the evolution of the gravitational field, the entire orbit is independent of mass. This is typically the case for orbiting bodies in galaxies (stars, dark matter particles, but not for heavy bodies such as massive black holes and large satellite galaxies orbiting within a bigger galaxy) and such bodies are known as test particles. Because of this, we can typically ignore the actual mass \(m\) of bodies orbiting in a galactic gravitational field and consider familiar quantities such as the energy, momentum, angular momentum, and the Lagrangian or the Hamiltonian as that per unit mass. For example, the momentum per unit mass—or the specific momentum—is \(\dot{\vec{x}}\) and this is often referred to simply as the momentum.
Because the orbital trajectory is determined by the second-order differential equation (4.1), an orbit is fully determined by its initial phase-space coordinate \(\vec{w}_0 = (\vec{x}_0,\vec{v}_0)\) and phase-space is filled by a dense set of non-crossing orbits. Again, this is only the case when the mass is so small that it can be ignored, because otherwise the trajectory also depends on other properties of the body, such as its mass (for a massive black hole) or its mass profile (for a satellite galaxy). Thus, we can discuss the properties of orbits without specifying whether these are the orbits of low- or high-mass stars, neutron stars, black holes, dark-matter particles, or of any other body with mass \(\ll\) the mass of the gravitational system.
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