Modification to the Jeans criterion by external tides: Anisotropic fragmentation and formation of filaments

The Jeans criterion sets the foundation of our understanding of gravitational collapse. Jog studied the fragmentation of gas under external tides and derived a dispersion relation $$l^{\mathrm{\prime }}=l_{\mathrm{J}\mathrm{e}\mathrm{a}\mathrm{n}\mathrm{s}}\frac{1}{(1+{\lambda }_0^{\mathrm{\prime }}\mathrm{/}4\pi G{\rho }_0{)}^{1\mathrm{/}2}}\mathrm{.}$$ She further concludes that the Jeans mass is $m_{\mathrm{i}\mathrm{n}\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{r}\mathrm{e}\mathrm{c}\mathrm{t}}^{\mathrm{\prime }}=m_{\mathrm{J}\mathrm{e}\mathrm{a}\mathrm{n}\mathrm{s}}(1\mathrm{/}(1+{\lambda }_0^{\mathrm{\prime }}\mathrm{/}4\pi G{\rho }_0{)}^{3\mathrm{/}2})$. We clarify that due to the inhomogeneous nature of tides, this characteristic mass is incorrect. Under weak tides, the mass is $m\approx \rho l_1{l}_2{l}_3$, where the modifications to Jeans lengths along all three dimensions need to be considered; when the tide is strong enough, collapse can only occur once 1 or 2 dimensions. In the latter case, tides can stretch the gas, leading to the formation of filaments.