# Dynamical time scale

Dynamical time scale has two distinct meanings and usages, both related to astronomy:

1. In one use, which occurs in stellar physics, the dynamical time scale is alternatively known as the freefall time scale, and 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.
2. In another use, in connection with time standards, dynamical time is the time-like argument of a dynamical theory; and a dynamical time scale in this sense is the realization of a time-like argument based on a dynamical theory: that is, the time and time scale are defined implicitly, inferred from the observed position of an astronomical object via a theory of its motion. A first application of this concept of dynamical time was the definition of the ephemeris time scale (ET).[1]

## Stellar physics

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

$\tau_{dynamical} \simeq \frac{R}{v} = \sqrt{\frac{R^3}{2GM}} \sim 1/\sqrt{G\rho}$

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.

## Time standards

In the late nineteenth century it was suspected, and in the early twentieth century it was established, 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. The suggestion was made, that observation of the position of the Moon, Sun and planets and comparison of the observations with their gravitational ephemerides would be a better way to determine a uniform time scale. A detailed proposal of this kind was published in 1948 and adopted by the IAU in 1952 (see Ephemeris time - history).

Using data from Newcomb's Tables of the Sun (based on the theory of the apparent motion of the Sun by Simon Newcomb, 1895, as retrospectively used in the definition of ephemeris time), 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 their use provided further confirmation that the rotation of the earth fluctuated randomly.[2] This confirmed the unsuitability of the mean solar second of Universal Time as a precision 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 +/- 20 cycles of the caesium resonance. In 1967/8 the length of the SI second was redefined to be 9,192,631,770 cycles of the caesium resonance, equal to the previous measurement result for the ephemeris second (see Ephemeris time - redefinition of the second).

In 1976, however, the IAU resolved that the theoretical basis for ephemeris time was wholly non-relativistic, and therefore, beginning in 1984 ephemeris time would be replaced by two further time scales with allowance for relativistic corrections. Their names, assigned in 1979,[3] emphasized their dynamical nature or origin, Barycentric Dynamical Time (TDB) and Terrestrial Dynamical Time (TDT). Both were defined for continuity with ET and were based on what had become the standard SI second, which in turn had been derived from the measured second of ET.

During the period 1991-2006, the TDB and TDT time scales were both redefined and replaced, owing to difficulties or inconsistencies in their original definitions. The current fundamental relativistic time scales are Geocentric Coordinate Time (TCG) and Barycentric Coordinate Time (TCB); both of these have rates that are based on the SI second in respective reference frames (and hypothetically outside the relevant gravity well), but on account of relativistic effects, their rates would appear slightly faster when observed at the Earth's surface, and therefore diverge from local earth-based time scales based on the SI second at the Earth's surface.[4] Therefore the currently defined IAU time scales also include Terrestrial Time (TT) (replacing TDT, and now defined as a re-scaling of TCG, chosen to give TT a rate that matches the SI second when observed at the Earth's surface),[5] and a redefined Barycentric Dynamical Time (TDB), a re-scaling of TCB to give TDB a rate that matches the SI second at the Earth's surface.

## References

• P.K.Seidelmann (ed.), Explanatory Supplement to the Astronomical Almanac. University Science Books, CA, 1992 ; ISBN 0-935702-68-7
1. ^ B Guinot and P K Seidelmann (1988), "Time scales - Their history, definition and interpretation", Astronomy and Astrophysics vol. 194, no.1-2, April 1988, p.304-308; at p.304. See also P K Seidelmann (ed.), "Explanatory Supplement to the Astronomical Almanac", University Science Books, CA, 1992,at page 41.
2. ^ W Markowitz, 'Variations in the Rotation of the Earth, Results Obtained with the Dual-Rate Moon Camera and Photograhic Zenith Tubes', Astron J v64 (1959) 106-113.
3. ^ Guinot & Seidelmann, 1988, op.cit.
4. ^ See S Klioner et al, "Units of relativistic time scales and associated quantities", IAU Symposium 261 (2009).
5. ^ IAU 2000 resolutions, at Resolution B1.9.

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