= Kepler–isochrone radial transformation
{c}
{title2=$r=\sqrt{\bar E/(\lambda E)}(\sqrt{\bar r^2+b^2}-b)$}
With $a=(1-\lambda)GM$, $h=\lambda E>0$ and $\bar E>0$, set $\bar E q^2=hr^2+ar$. If $a\geq0$, its positive-radius root is $r=\sqrt{\bar E/h}(\sqrt{q^2+b^2}-b)$, with $b=a/(2\sqrt{h\bar E})$. The transformed <Newtonian gravitational potential> is an additive constant plus $-G\bar M/(b+\sqrt{q^2+b^2})$, the <spherical isochrone model>. The <mass> sign and branch conditions must be checked.
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