astrophysics Astrophysics is a science that employs the methods and principles of physics and chemistry in the study of astronomical objects and phenomena. As one of the founders of the discipline said, Astrophysics "seeks to ascertain the nature of the he ...

, the Darwin–Radau equation (named after

Rodolphe Radau Jean Charles Rodolphe Radau (22 January 1835 – 21 December 1911) was an astronomer and mathematician who worked in Paris at the ''Revue des deux Mondes'' for most of his life. He was the co-founder of the Bulletin Astronomique. Radau was bo ...

and

Charles Galton Darwin Sir Charles Galton Darwin (19 December 1887 – 31 December 1962) was an English physicist who served as director of the National Physical Laboratory (NPL) during the Second World War. He was a son of the mathematician George Howard Darwin ...

) gives an approximate relation between the

moment of inertia factor In planetary sciences, the moment of inertia factor or normalized polar moment of inertia is a dimensionless quantity that characterizes the radial distribution of mass inside a planet or satellite. Since a moment of inertia must have dimensions of ...

of a planetary body and its rotational speed and shape. The moment of inertia factor is directly related to the largest principal

moment of inertia The moment of inertia, otherwise known as the mass moment of inertia, angular mass, second moment of mass, or most accurately, rotational inertia, of a rigid body is a quantity that determines the torque needed for a desired angular accele ...

, ''C''. It is assumed that the rotating body is in

hydrostatic equilibrium In fluid mechanics, hydrostatic equilibrium (hydrostatic balance, hydrostasy) is the condition of a fluid or plastic solid at rest, which occurs when external forces, such as gravity, are balanced by a pressure-gradient force. In the planetar ...

and is an

ellipsoid of revolution A spheroid, also known as an ellipsoid of revolution or rotational ellipsoid, is a quadric surface obtained by rotating an ellipse about one of its principal axes; in other words, an ellipsoid with two equal semi-diameters. A spheroid has circu ...

. The Darwin–Radau equation states :

\frac = \frac = \frac \left( 1 - \frac \sqrt \right)

where ''M'' and ''R_e'' represent the mass and mean equatorial radius of the body. Here λ is the

d'Alembert Jean-Baptiste le Rond d'Alembert (; ; 16 November 1717 – 29 October 1783) was a French mathematician, mechanician, physicist, philosopher, and music theorist. Until 1759 he was, together with Denis Diderot, a co-editor of the ''Encyclopéd ...

parameter and the Radau parameter η is defined as :

\eta = \frac - 2

where ''q'' is the

geodynamical constant Geodynamics is a subfield of geophysics dealing with dynamics of the Earth. It applies physics, chemistry and mathematics to the understanding of how mantle convection leads to plate tectonics and geologic phenomena such as seafloor spreading, m ...

q = \frac

and ε is the geometrical flattening :

\epsilon = \frac

where ''R_p'' is the mean polar radius and ''R_e'' is the mean equatorial radius. For

Earth Earth is the third planet from the Sun and the only astronomical object known to harbor life. While large volumes of water can be found throughout the Solar System, only Earth sustains liquid surface water. About 71% of Earth's surf ...

q \approx 3.461391 \times 10^

and

\epsilon \approx 1/298.257

, which yields

\frac \approx 0.3313

, a good approximation to the measured value of 0.3307.

References

Astrophysics Planetary science Equations of astronomy {{planetary-science-stub