Orbital Period

Baby Collar, Dong people. China, Yunnan province, 20th century 39 x 17 cm. From the collection of Tan Tim Qing, Kunming. Photograph by D Dunlop.

Let P be a particle with an orbital frequency of ν in a reference frame that includes a clock Ω*. Definition: the orbital period is the number

$\mbox{\fontsize{14}{18}\selectfont $ \hat{\tau} \equiv 1 / \nu $}$

The units used for the period depend on the clock. For example; if Ω* is the daily orbit of the Earth, then the period is measured in days. Or if Ω* is a caesium clock, then the units are seconds, abreviated as (s). For any event, the period and angular frequency are related as

$\mbox{\fontsize{14}{18}\selectfont $ \omega = 2 \pi / \hat{\tau} $}$

Consider measuring the orbital period of some particle based on laboratory observations of its mechanical energy. The two quantities are related as

$\mbox{\fontsize{14}{18}\selectfont $ \hat{\tau} = h / E $}$

where h is Planck's constant. By the usual rules for assessing the propagation of experimental errors in a calculation, the uncertainty in the period is in the same proportion as the uncertainty in the energy measurement so that

$\mbox{\fontsize{20}{20}\selectfont $ \frac { \delta \! \hat{\tau} }{ \hat{\tau} } = \frac { \delta \! E }{ E } $}$