= Nonnegativity bound for a prolate logarithmic-potential distribution
{title2=$q^2\le(16\sqrt2-e)/(16\sqrt2-2e)$}
For the <logarithmic-potential two-integral distribution>, $q^2>1$ makes $A<0$. Accessible phase space satisfies $L_z^2e^{-2E/V^2}\le V^2/e$, attained by equatorial circular motion. Hence the <galactic distribution function> is nonnegative precisely when $B+AV^2/e\ge0$, giving the displayed upper bound. A larger prolate flattening parameter produces negative phase-space density even though the spatial <mass density> remains positive.
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