Wednesday, April 1, 2015

Darwin-term in the Kepler-problem

planets are not pointlike objects and the total force in a gravitational field would need to be integrated over the planet's volume, i.e. the acceleration of the centre of gravity would need to be corrected by tidal forces. can you estimate how large that correction would be?

1 comment:

  1. the Darwin-term can be estimated by perturbation theory on the effective potential
    \begin{equation}
    V = -\frac{GM}{r} + \frac{L^2}{2mr^2}
    \end{equation}
    from which one derives first of all the equilibrium condition
    \begin{equation}
    r = \frac{L^2}{GMm}
    \end{equation}
    the first non-vanishing term would be
    \begin{equation}
    \int_{r-\delta}^{r+\delta}\mathrm{d}r\:\frac{\partial^3V}{\partial r^3} = 2\frac{GM\delta}{r^3}
    \end{equation}
    with the planet's radius $\delta$, the Sun's mass $M$ and the planet's distance $r$. The effect is therefore a fraction $2\delta/r$ of the total gravitational acceleration $GM/r^2$, which for the Earth would be about $10^{-4}$.

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