The theory of figures of Clairaut with focus on the gravitational rigidity modulus: inequalities and an improvement in the Darwin-Radau equation. (arXiv:1811.07759v2 [astro-ph.EP] UPDATED)

The theory of figures of Clairaut with focus on the gravitational rigidity modulus: inequalities and an improvement in the Darwin-Radau equation. (arXiv:1811.07759v2 [astro-ph.EP] UPDATED)
<a href="http://arxiv.org/find/astro-ph/1/au:+Ragazzo_C/0/1/0/all/0/1">Clodoaldo Grotta Ragazzo</a>

This paper contains a review of Clairaut’s theory with focus on the
determination of a gravitational rigidity modulus $gamma$ defined as
$left(frac{C-I_o}{I_o}right)gamma=frac{2}{3}Omega^2$, where $C$ and $I_o$
are the polar and mean moment of inertia of the body and $Omega$ is the body
spin.The constant $gamma$ is related to the static fluid Love number $k_2=
frac{3I_o G}{R^5} frac{1}{gamma}$, where $R$ is the body radius and $G$ is
the gravitational constant. The new results are: a variational principle for
$gamma$, upper and lower bounds on the ellipticity that improve previous
bounds by Chandrasekhar (1963) and a semi-empirical procedure for estimating
$gamma$ from the knowledge of $m$, $I_o$, and $R$, where $m$ is the mass of
the body. The main conclusion is that for $0.2le I_o/(mR^2)le 0.4$ the
approximation $gammaapprox G sqrt{ frac{2^7}{5^5}frac{m^5}{I_o^3}}=
gamma_I$ is a better estimate for $gamma$ than that obtained from the
Darwin-Radau equation, denoted as $gamma_{DR}$. Moreover, within the range of
applicability of the Darwin-Radau equation $0.32le I_o/(mR^2)le 0.4$ the
relative difference between the two estimates, $|gamma_{DR}/gamma_I -1|$, is
less than $0.05%$.

This paper contains a review of Clairaut’s theory with focus on the
determination of a gravitational rigidity modulus $gamma$ defined as
$left(frac{C-I_o}{I_o}right)gamma=frac{2}{3}Omega^2$, where $C$ and $I_o$
are the polar and mean moment of inertia of the body and $Omega$ is the body
spin.The constant $gamma$ is related to the static fluid Love number $k_2=
frac{3I_o G}{R^5} frac{1}{gamma}$, where $R$ is the body radius and $G$ is
the gravitational constant. The new results are: a variational principle for
$gamma$, upper and lower bounds on the ellipticity that improve previous
bounds by Chandrasekhar (1963) and a semi-empirical procedure for estimating
$gamma$ from the knowledge of $m$, $I_o$, and $R$, where $m$ is the mass of
the body. The main conclusion is that for $0.2le I_o/(mR^2)le 0.4$ the
approximation $gammaapprox G sqrt{ frac{2^7}{5^5}frac{m^5}{I_o^3}}=
gamma_I$ is a better estimate for $gamma$ than that obtained from the
Darwin-Radau equation, denoted as $gamma_{DR}$. Moreover, within the range of
applicability of the Darwin-Radau equation $0.32le I_o/(mR^2)le 0.4$ the
relative difference between the two estimates, $|gamma_{DR}/gamma_I -1|$, is
less than $0.05%$.

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