Dynamical friction

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Dynamical friction is a term in astrophysics related to loss of momentum and kinetic energy of moving bodies through a gravitational interaction with surrounding matter in space. It is sometimes referred to as gravitational drag, and was first discussed in detail by Subrahmanyan Chandrasekhar in 1943.

The effect must exist if the principle of conservation of energy and momentum is valid since any gravitational interaction between two or more bodies corresponds to elastic collisions between those bodies.

E.g. when a heavy body B moves through a cloud of lighter bodies, the gravitational interaction between B and the light bodies causes the light bodies to accelerate and gain momentum and kinetic energy (see sling effect). Since energy and momentum are conserved, B has to lose a part of its momentum and energy equal to the sums of all momenta and energies gained by the light bodies. Because of the loss of momentum and kinetic energy of the body under consideration the effect is called dynamical friction.

Another equivalent way of thinking about this process is that the light bodies near B are attracted by its gravity toward its position and therefore the density at that location increases and is referred to as a gravitational wake. In the meantime, B has moved forward. Therefore, the gravitational attraction of the wake pulls B backward and slows it down.

Of course the mechanism works the same for all masses of interacting bodies and for any relative velocities between them. However, while in the above case the most probable outcome is the loss of momentum and energy by the body under consideration, in the general case it might be either loss or gain (when one body loses momentum and energy in an elastic collision the other one gains them). In a case when the body under consideration is gaining momentum and energy the same physical mechanism is called sling effect.

The full Chandrasekhar dynamical friction formula for the change in velocity of the object involves integrating over the phase space density of the field of matter and is far from transparent. By assuming a constant density though, a simplified equation for the force from dynamical friction, $f d$ , may be found to be

$f_d \approx C \frac{G^2 M^2 \rho}{v^2_M}$

where $G$ is the gravitational constant, $M$ is the mass of the moving object, $ρ$ is the density, and $v M$ is the velocity of the object in the frame in which the surrounding matter was initially at rest. In this equation $C$ is not a constant but depends on how $v M$ compares to the velocity dispersion of the surrounding matter (Carroll and Ostlie 1996).

The greater the density of the surrounding media, the stronger the force from dynamical friction. Similarly, the force is proportional to the square of the mass of the object. One of these terms is from the gravitational force between the object and the wake. The second term is because the more massive the object, the more matter will be pulled into the wake. The force is also proportional to the inverse square of the velocity. This means the fractional rate of energy loss drops rapidly at high velocities. Dynamical friction is, therefore, unimportant for objects that move relativistically, such as photons. This can be rationalized by realizing that the faster the object moves though the media, the less time there is for a wake to build up behind it.

Dynamical friction is particularly important in the formation of planetary systems and interactions between galaxies. During the formation of planetary systems, dynamical friction between the protoplanet and the protoplanetary disk causes energy to be transferred from the protoplanet to the disk. This results in the inward migration of the protoplanet. When galaxies interact through collisions, dynamical friction between stars causes matter to sink toward the center of the galaxy and for the orbits of stars to be randomized. This process is called violent relaxation and can change two spiral galaxies into one larger elliptical galaxy.