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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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AlanFox86
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  • Related: Please clarify how entropy increases when matter gravitationally coalesces – Sten Jul 05 '23 at 00:50
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    Don't get so lost in the sauce of thermodynamics that you end up thinking nothing can ever compress under gravity. Do you normally consider the gravitational potential of an ideal gas? Of course not. Thus this is not just an ideal gas - there are interactions between the particles that are not normally included in the ideal gas law. Thus new phenomena will occur that would not occur on an ideal gas by itself. Indeed an ideal gas alone will not spontaneously compress itself. – AXensen Jul 05 '23 at 08:25

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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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  • Side note: in dissipationless systems (e.g. dark matter), since no energy can be lost to radiation, some of the matter must be ejected from the system to carry off enough kinetic energy to fulfill the virial theorem. (This is for the case of an isolated system; it's not typical behavior for realistic dark matter systems since they continue to accrete.) – Sten Jul 05 '23 at 18:44
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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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