- #1
etotheipi
- Homework Statement
- A spherical cloud of interstellar gas (which can be considered to be ideal) collapses under its own gravity. During much of the collapse, the star can be considered to be at the same temperature as its surroundings, ##T##. Calculate the heat radiated away during collapse from a radius ##r_0## to ##r##.
- Relevant Equations
- N/A
When I did this the first time I didn't really think too much about it, so I just wrote$$p = \frac{nRT}{V} \implies W = \Delta U - Q = - Q = -\int_{\frac{4}{3}\pi r_0^3}^{\frac{4}{3}\pi r^3} \frac{nRT}{V} dV$$That turned out to be correct, but when I thought about it I didn't understand why this expression would be meaningful. The external pressure to the protostar is presumably zero, since the thing is surrounded by vacuum, so the PV-work on the protostar will be zero. What is really responsible for the generated heat is the work done by gravity/decrease in gravitational self energy, ##\Delta U_g = \frac{3GM}{5}\Delta \frac{1}{r}##.
So I wondered how do we actually interpret the thermodynamic result at the top? Furthermore in an actual gravitational collapse the pressure is dependent on the radial coordinate, but in the top it's assumed to be uniform. The resultant pressure force on any spherical shell also won't equal the gravitational force on that shell. Thanks
So I wondered how do we actually interpret the thermodynamic result at the top? Furthermore in an actual gravitational collapse the pressure is dependent on the radial coordinate, but in the top it's assumed to be uniform. The resultant pressure force on any spherical shell also won't equal the gravitational force on that shell. Thanks
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