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# Relaxation time

Relaxation time is a general concept in physics for the characteristic time in which a system changes to an equilibrium condition from a non-equilibrium condition. It can measure the time-dependent response of a system to well-defined external stimuli.

## Examples

### Amorphous solids

An amorphous solid, such as amorphous indomethacin displays a temperature dependence of molecular motion, which can be quantified as the average relaxation time for the solid in a metastable supercooled liquid or glass to approach the molecular motion characteristic of a crystal. Differential scanning calorimetry can be used to quantify enthalpy change due to molecular structural relaxation.

### Mathematical example: Damped unforced oscillator

Let the homogenous differential equation: $mfrac\left\{d^2 y\right\}\left\{d t^2\right\}+gammafrac\left\{d y\right\}\left\{d t\right\}+ky=0$ model damped unforced oscillations of a weight on a spring.

The displacement will then be of the form $y\left(t\right) = A e^\left\{-t/T\right\} cos\left(mu t - delta\right)$. The constant T is called the relaxation time of the system and the constant μ is the quasi-frequency.

### Astronomy

In astronomy, relaxation time relates to clusters of gravitationally-interacting bodies (star clusters, galaxy clusters, globular clusters). The relaxation time is a measure of the time it takes for one object in a system to be significantly perturbed by other objects in the system. In the case of stars in a galaxy, the relaxation time measures the time for one star to undergo a strong encounter with another star. Various events occur on timescales relating to the relaxation time, including core collapse and energy exchange between stars (minimization of the total energy in a cluster).

The relaxation time is related to the velocity of a body (typically a star) and the perturbation rate. In the example of a star cluster, a particular star will have an orbit with a velocity v. As the star passes by other stars, the orbit will be perturbed by the gravitational field of nearby stars. The relaxation time is similar to the ratio of the velocity to the time derivative of the perturbation.

## Further reading

• Sparke, L. & Gallagher, J. (2000). Galaxies in the Universe: An Introduction, 1st ed., Sec. 3.2. Cambridge University Press. ISBN 0-521-59241-0

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