Dynamical time scale

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The dynamical time scale, sometimes known as the freefall time scale, is in general, the length of time over which changes in one part of a body can be communicated to the rest of that body. This is often related to the time taken for a system to move from one equilibrium state to another after a sudden change.

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[edit] Astronomical use

[edit] Stellar astrophysics

For a star, the dynamical time scale is defined as the time that would be taken for a test particle released at the surface to fall under the star's potential to the centre point, if pressure forces were negligible. In other words, the dynamical time scale measures the amount of time it would take a certain star to collapse in the absence of any internal pressure. By appropriate manipulation of the equations of stellar structure this can be found to be

\tau_{dynamical} \simeq \frac{R}{v} = \sqrt{\frac{R^3}{2GM}}

where R is the radius of the star, G is the gravitational constant, M is the mass of the star and v is the escape velocity. As an example, the Sun dynamical time scale is approximately 1133 seconds. Note that the actual time it would take a star like the Sun to collapse is greater because internal pressure is present.

The 'fundamental' oscillatory mode of a star will be at approximately the dynamical time scale. Oscillations at this frequency are seen in Cepheid variables.

[edit] Time measurement

In the late nineteenth century it was found that the rotation of the Earth (i.e. the length of the day) was both irregular on short time scales, and was slowing down on longer time scales. In fact, observing the position of the Moon, Sun and planets and comparing this with their ephemerides was a better way to determine the time.

Using the ephemerides based on the theory of the apparent motion of the Sun by Simon Newcomb (1898), the SI second was defined in 1960 as:

the fraction 1/31,556,925.9747 of the tropical year for 1900 January 0 at 12 hours ephemeris time.

Caesium atomic clocks became operational in 1955, and quickly made it evident that the rotation of the earth fluctuated randomly. This confirmed the utter unsuitability of the mean solar second of Universal Time as a measure of time interval. After three years of comparisons with lunar observations it was determined that the ephemeris second corresponded to 9,192,631,770 cycles of the caesium resonance. Between 1960 and 1984 the length of the SI second was defined to be equal to the ephemeris second.

In 1976, however, the IAU resolved that the theoretical basis for Ephemeris Time is wholly non-relativistic, and therefore, beginning in 1984 Ephemeris Time would be replaced by the two relativistic timescales based on Dynamical Time, the Barycentric Dynamical Time (TDB) and Terrestrial Dynamical Time (TDT). For practical purposes the length of the ephemeris second can be taken as equal to the length of the TDB or TDT second.

[edit] References

  • P.K.Seidelmann (ed.), Explanatory Supplement to the Astronomical Almanac. University Science Books, CA, 1992 ; ISBN 0-935702-68-7



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