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Consider a star of Radius R and mass M, with a pressure gradient given by

[itex]\frac{dP}{dr}[/itex] = [itex]\frac{4\pi}{3}[/itex]G[itex]\rho[/itex]

where [itex]\rho[/itex] is the central density. calculate the gravitational energy, using the Virial theorem. Show that in the limit [itex]\lambda[/itex] « R this energy is given by

E = [itex]\frac{RGMM}{3R\lambda}[/itex]for tecnical reasons:

MM = M

rr = r2

[itex]\lambda\lambda[/itex] = [itex]\lambda[/itex]

[itex]\frac{dP}{dr}[/itex] = [itex]\frac{4\pi}{3}[/itex]G[itex]\rho[/itex]

^{2}r exp(-[itex]\frac{rr}{\lambda\lambda}[/itex])where [itex]\rho[/itex] is the central density. calculate the gravitational energy, using the Virial theorem. Show that in the limit [itex]\lambda[/itex] « R this energy is given by

E = [itex]\frac{RGMM}{3R\lambda}[/itex]for tecnical reasons:

MM = M

^{2}rr = r2

[itex]\lambda\lambda[/itex] = [itex]\lambda[/itex]

^{2}
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