E2 Celestial Mechanics and Gravitation

Topic

The central element of this model is the self-gravitating system—whether it be a planet orbiting a star or an entire galaxy cluster held together by its own gravity.

Newtonian celestial mechanics governs the dynamics of such a system at any scale, from a single planet to a galaxy cluster, operating through the gravitational potential that underpins every Keplerian orbit: the closer a body is to the central mass, the deeper—that is, the more negative—this potential becomes. Poisson's equation extends this law of gravitation—originally formulated for point masses—to continuous mass distributions; solving it allows the gravitational field of an entire galaxy or cluster to be determined directly, without the need to treat the system as a single point mass.

The virial theorem establishes that, in any gravitationally bound system at equilibrium, the average kinetic energy of its components is exactly half the absolute value of its potential energy. Because this relationship holds true, simply measuring the observed velocities of the members of a globular cluster, galaxy, or galaxy cluster is sufficient to estimate the system's total mass, without requiring direct observation of the constituent mass itself.

The circular orbital velocity of each component in such a system is the speed that precisely balances the force of gravity against the centripetal acceleration required to maintain the orbit. The fact that this circular velocity remains constant—rather than decreasing—as one moves away from a galaxy's center serves as direct kinematic evidence that a dark matter halo envelops the galaxy, using its own additional gravity to sustain the speeds of the outermost components. When the total orbital energy is negative, it indicates a gravitationally bound state in any of these Keplerian orbits—such as the simple orbit of a planet around its star. Thus, from the self-gravitating system with which this document begins to the bound state with which it concludes—and encompassing the gravitational potential, the virial theorem, and circular orbital velocity—celestial mechanics ensures that the same set of relations applies, without changing form, to both describing the orbit of a single planet and weighing an entire galaxy cluster.

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