Darwin–Radau equation

In astrophysics, the Darwin–Radau equation gives a relation between the third principal moment of inertia of the Earth and its rotational speed and shape. It is assumed that the rotating Earth is in hydrodynamic equilibrium and is an ellipsoid of revolution. The Darwin–Radau equation states^[1]

$\frac{C}{MR_{e}^{2}} = \frac{2}{3\lambda} = \frac{2}{3} \left( 1 - \frac{2}{5} \sqrt{1 + \eta} \right)$

where M and R_e represent the mass and mean equatorial radius of the Earth. Here λ is the d'Alembert parameter and the Radau parameter η is defined as

$\eta = \frac{5q}{2\epsilon} - 2$

where q is the geodynamical constant

$q = \frac{\omega^{2} R_{e}^{3}}{GM} \approx 3.461391 \times 10^{-3}$

and ε is the geometrical flattening

$\epsilon = \frac{R_{p} - R_{e}}{R_{e}}$

where R_p is the mean polar radius of the Earth.

References

1. ^ Bourda, G; Capitaine N (2004). "Precession, nutation, and space geodetic determination of the Earth's variable gravity field". Astronomy and Astrophysics 428: 691–702. Bibcode 2004A&A...428..691B. doi:10.1051/0004-6361:20041533.