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The Jeans mass, given by $M_J=\sqrt{\left(\frac{-5k_BT}{Gm}\right)^3\cdot\left(\frac{3}{4\pi\rho}\right)}$, is the threshold mass a dust cloud must have in order to begin gravitationally collapsing onto itself. However, wouldn't this violate the second law of thermodynamics? Since the volume is decreasing ($V_f<V_0$) and attending to the definition of the entropy increase for an ideal gas (according to Eddington, ordinary stars behave like ideal gases), $\Delta S=Nk_Blog(\frac{V_f}{V_0}) \Longrightarrow \Delta S < 0$.

Therefore, does it make sense for the dust cloud to collapse onto itself even if its mass surpasses a given threshold value? According to the argument I've given, a cloud could never collapse onto itself as long as it's considered an isolated system because it would violate the second principle.

Sten
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Lagrangiano
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2 Answers2

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The equation for the change in entropy you have given does not account for (i) the increase in internal kinetic energy of particles in the gas - i.e. the temperature increases; and (ii) the cloud will radiate away its internal energy as it collapses

The virial theorem tells us that both of these are important - half of the gravitational potential released in the collapse goes into heating the cloud and half is radiated away.

ProfRob
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$\Delta S=Nk_Blog(\frac{V_f}{V_0})$

That expression applies only if the temperature is held fixed as the volume is changed. That is not the case during gravitational collapse. As a cloud gravitationally collapses, its particles gain speed -- gravitational potential energy is converted into kinetic energy.

Note that these considerations aren't specific to gravitational systems. Adiabatic compression of an ideal gas conserves entropy; the gas heats as it is compressed.

Sten
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