Jeans instability can also give rise to fragmentation in certain conditions. To derive the condition for fragmentation an adiabatic process is assumed in an ideal gas and also a polytropic equation of state is taken. The derivation is showed below through a dimensional analysis:
- For adiabatic processes,
- For an ideal gas,
- Polytropic equation of state,
- Jeans mass,
If adiabatic index, Jeans mass increases with increasing density while if Jeans mass decreases with increasing density. During gravitational collapse density always increases, thus in the second case Jeans mass will decrease during collapse allowing smaller overdense regions to collapse leading to fragmentation of the giant molecular cloud. For an ideal monoatomic gas, the adiabatic index is 5/3 but in astrophysical objects this value is usually even lower than 1. So the second case is the rule rather than an exception in stars. This is the reason why stars usually form in clusters.
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