Canonical units

The Canonical Distance Unit ${\displaystyle \;{\text{[CDU]}}\;}$ is defined to be the mean radius of the reference orbit.

The Canonical Time Unit ${\displaystyle \;{\text{[CTU]}}\;}$ is defined by the gravitational parameter ${\displaystyle \;\mu \;}$:

${\displaystyle \mu \equiv G\,M~}$

where

${\displaystyle \;G\;}$ is the gravitational constant

${\displaystyle \;M\equiv {\text{[CMU]}}\;}$ is the mass of the central reference body

In canonical units, the gravitational parameter is given by:

${\displaystyle \mu =1\,{\frac {\;{\text{[CDU]}}^{3}\,}{\;{\text{[CTU]}}^{2}~}}}$

Any triplet of numbers, ${\displaystyle \,M\,,}$ ${\displaystyle \,{\text{[CDU]}}\,,}$ and ${\displaystyle \,{\text{[CTU]}}\,,}$ that satisfy the equation above is a “canonical” set.

The quantity of the time unit [CTU] can be solved in another unit system (e.g. the metric system) if the mass and radius of the central body have been determined. Using the above equation and applying dimensional analysis, set the two equations expressing ${\displaystyle \;\mu \;}$ equal to each other:

${\displaystyle \mu \equiv G\times M=1\,{\frac {\;{\text{[CDU]}}^{3}\,}{\;{\text{[CTU]}}^{2}\,}}~}$

The time unit ([CTU]) can be converted to another unit system for a more useful qualitative solution using the following equation:

${\displaystyle {\text{[CTU]}}={\sqrt {\frac {{\text{[CDU]}}^{3}}{G\,M}}}~}$

For Earth-orbiting satellites, approximate unit conversions are as follows:

• 1 [CDU] = 6378.1 km = 20,925,524.97 ft
• 1 [CDU]/[CTU] = 7.90538 km/s = 25,936.29 ft/sec
• 1 [CTU] = 806.80415 s

The astronomical unit (AU) is the canonical distance unit for the orbit around the Sun of the combined Earth-Moon system (based on the formerly best-known value). The corresponding time unit is the (sidereal) year)), and the mass is the total mass of the Sun (M_☉).[lower-alpha 1]