Dynamical friction
Encyclopedia
Dynamical friction is a term in astrophysics
related to loss of momentum
and kinetic energy
of moving bodies through a gravitation
al 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.
Another equivalent way of thinking about this process is that the light bodies are attracted by gravity toward the larger body moving through the cloud, and therefore the density at that location increases and is referred to as a gravitational wake. In the meantime, object under consideration has moved forward. Therefore, the gravitational attraction of the wake pulls it 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 the most probable outcome for an object moving through a cloud is a loss of momentum and energy, as described intuitively above, in the general case it might be either loss or gain. When the body under consideration is gaining momentum and energy the same physical mechanism is called slingshot effect, or gravity assist. This technique is sometimes used by interplanetary probes to obtain a boost in velocity by passing close by a planet.
density of the field of matter and is far from transparent.
A commonly used special case is where there is a uniform density in the field of matter, with matter particles significantly lighter than the major particle under consideration and with a Maxwellian
distribution for the velocity of matter particles. In this case, the dynamical friction force is as follows:
where
In general, a simplified equation for the force from dynamical friction
has the form
where the dimensionless numerical factor
depends on how 
compares to the velocity dispersion of the surrounding matter.
and the protoplanetary disk
causes energy to be transferred from the protoplanet to the disk. This results in the inward migration of the protoplanet.
.
However the observed velocity dispersion of galaxies within a galaxy cluster does not depend on the mass of the galaxies. The explanation is that a galaxy cluster relaxes by violent relaxation, which sets the velocity dispersion to a value independent of the galaxy's mass.
proposed in 1929 that a gravitational drag effect on photons could be used to explain cosmological redshift as a form of tired light
. However, his analysis had a mathematical error, and his approximation to the magnitude of the effect should actually have been zero, as pointed out in the same year by Arthur Stanley Eddington
. Zwicky promptly acknowledged the correction, although he continued to hope that a full treatment would be able to show the effect.
It is now known that the effect of dynamical friction on photons or other particles moving at relativistic speeds is negligible, since the magnitude of the drag is inversely proportional to the square of velocity. Cosmological redshift is conventionally understood to be a consequence of the metric expansion of space
.
Astrophysics
Astrophysics is the branch of astronomy that deals with the physics of the universe, including the physical properties of celestial objects, as well as their interactions and behavior...
related to loss of momentum
Momentum
In classical mechanics, linear momentum or translational momentum is the product of the mass and velocity of an object...
and kinetic energy
Kinetic energy
The kinetic energy of an object is the energy which it possesses due to its motion.It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its acceleration, the body maintains this kinetic energy unless its speed changes...
of moving bodies through a gravitation
Gravitation
Gravitation, or gravity, is a natural phenomenon by which physical bodies attract with a force proportional to their mass. Gravitation is most familiar as the agent that gives weight to objects with mass and causes them to fall to the ground when dropped...
al interaction with surrounding matter
Matter
Matter is a general term for the substance of which all physical objects consist. Typically, matter includes atoms and other particles which have mass. A common way of defining matter is as anything that has mass and occupies volume...
in space. It is sometimes referred to as gravitational drag, and was first discussed in detail by Subrahmanyan Chandrasekhar
Subrahmanyan Chandrasekhar
Subrahmanyan Chandrasekhar, FRS ) was an Indian origin American astrophysicist who, with William A. Fowler, won the 1983 Nobel Prize for Physics for key discoveries that led to the currently accepted theory on the later evolutionary stages of massive stars...
in 1943.
Intuitive account
An intuition for the effect can be obtained by thinking of a massive object moving through a cloud of smaller lighter bodies. The effect of gravity causes the light bodies to accelerate and gain momentum and kinetic energy (see slingshot effect). By conservation of energy and momentum, we may conclude that the heavier body will be slowed by an amount to compensate. Since there is a loss of momentum and kinetic energy for the body under consideration, the effect is called dynamical friction.Another equivalent way of thinking about this process is that the light bodies are attracted by gravity toward the larger body moving through the cloud, and therefore the density at that location increases and is referred to as a gravitational wake. In the meantime, object under consideration has moved forward. Therefore, the gravitational attraction of the wake pulls it 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 the most probable outcome for an object moving through a cloud is a loss of momentum and energy, as described intuitively above, in the general case it might be either loss or gain. When the body under consideration is gaining momentum and energy the same physical mechanism is called slingshot effect, or gravity assist. This technique is sometimes used by interplanetary probes to obtain a boost in velocity by passing close by a planet.
Chandrasekhar dynamical friction formula
The full Chandrasekhar dynamical friction formula for the change in velocity of the object involves integrating over the phase spacePhase space
In mathematics and physics, a phase space, introduced by Willard Gibbs in 1901, is a space in which all possible states of a system are represented, with each possible state of the system corresponding to one unique point in the phase space...
density of the field of matter and is far from transparent.
A commonly used special case is where there is a uniform density in the field of matter, with matter particles significantly lighter than the major particle under consideration and with a Maxwellian
Maxwellian
* Maxwell's equations * Maxwell–Boltzmann distribution* Maxwell's electromagnetic wave equation...
distribution for the velocity of matter particles. In this case, the dynamical friction force is as follows:
where
- G is the gravitational constantGravitational constantThe gravitational constant, denoted G, is an empirical physical constant involved in the calculation of the gravitational attraction between objects with mass. It appears in Newton's law of universal gravitation and in Einstein's theory of general relativity. It is also known as the universal...
. - M is the mass under consideration.
-
is the velocity of the object under consideration, in a frame where the center of gravity of the matter field is initially at rest. -
is the ratio of the velocity of the object under consideration to the modal velocity of the Maxwellian distribution. (
is the velocity dispersion). -
is the "error functionError functionIn mathematics, the error function is a special function of sigmoid shape which occurs in probability, statistics and partial differential equations...
" obtained by integrating the normal distribution. -
is the density of the matter field. -
is the "Coulomb logarithmCoulomb collisionA Coulomb collision is a binary elastic collision between two charged particles interacting through their own Electric Field. As with any inverse-square law, the resulting trajectories of the colliding particles is a hyperbolic Keplerian orbit...
".
In general, a simplified equation for the force from dynamical friction
has the formwhere the dimensionless numerical factor
depends on how compares to the velocity dispersion of the surrounding matter.Density of the surrounding medium
The greater the density of the surrounding medium, 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.Applications
Dynamical friction is particularly important in the formation of planetary systems and interactions between galaxies.Protoplanets
During the formation of planetary systems, dynamical friction between the protoplanetProtoplanet
Protoplanets are large planetary embryos that originate within protoplanetary discs and have undergone internal melting to produce differentiated interiors. They are believed to form out of kilometer-sized planetesimals that attract each other gravitationally and collide...
and the protoplanetary disk
Protoplanetary disk
A protoplanetary disk is a rotating circumstellar disk of dense gas surrounding a young newly formed star, a T Tauri star, or Herbig Ae/Be star...
causes energy to be transferred from the protoplanet to the disk. This results in the inward migration of the protoplanet.
Galaxies
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 galaxyElliptical galaxy
An elliptical galaxy is a galaxy having an approximately ellipsoidal shape and a smooth, nearly featureless brightness profile. They range in shape from nearly spherical to highly flat and in size from hundreds of millions to over one trillion stars...
.
Galaxy Clusters
The effect of dynamical friction explains why the brightest (more massive) galaxy is found at the center of a galaxy cluster. The effect of the two body collisions slows down the galaxy, and the drag effect is greater the larger the galaxy mass. When the galaxy loses kinetic energy, it moves towards the center of the cluster.However the observed velocity dispersion of galaxies within a galaxy cluster does not depend on the mass of the galaxies. The explanation is that a galaxy cluster relaxes by violent relaxation, which sets the velocity dispersion to a value independent of the galaxy's mass.
Photons
Fritz ZwickyFritz Zwicky
Fritz Zwicky was a Swiss astronomer. He worked most of his life at the California Institute of Technology in the United States of America, where he made many important contributions in theoretical and observational astronomy.- Biography :Fritz Zwicky was born in Varna, Bulgaria to a Swiss father....
proposed in 1929 that a gravitational drag effect on photons could be used to explain cosmological redshift as a form of tired light
Tired light
Tired light is a class of hypothetical redshift mechanisms that was proposed as an alternative explanation for the redshift-distance relationship. These models have been proposed as alternatives to the metric expansion of space of which the Big Bang and the Steady State cosmologies are the most...
. However, his analysis had a mathematical error, and his approximation to the magnitude of the effect should actually have been zero, as pointed out in the same year by Arthur Stanley Eddington
Arthur Stanley Eddington
Sir Arthur Stanley Eddington, OM, FRS was a British astrophysicist of the early 20th century. He was also a philosopher of science and a popularizer of science...
. Zwicky promptly acknowledged the correction, although he continued to hope that a full treatment would be able to show the effect.
It is now known that the effect of dynamical friction on photons or other particles moving at relativistic speeds is negligible, since the magnitude of the drag is inversely proportional to the square of velocity. Cosmological redshift is conventionally understood to be a consequence of the metric expansion of space
Metric expansion of space
The metric expansion of space is the increase of distance between distant parts of the universe with time. It is an intrinsic expansion—that is, it is defined by the relative separation of parts of the universe and not by motion "outward" into preexisting space...
.