Pressure gradient in Jeans Lenght
All the demonstrations on Jeans instability start with:
[itex]\rho g \ge \nabla P[/itex]
Then they substitute [itex]\nabla P[/itex] with [itex]nkT/R[/itex].
But from ideal gas law: [itex]PV = nkT[/itex], so [itex]P = nkT/V = nkT/R^3[/itex]
(I'm not interested in proportionality factors, so let's not bother about [itex]4/3 \pi[/itex], ecc...)
Now, gradient of [itex]P[/itex], to me, means to derivate with respect to [itex]R[/itex].
[itex]\nabla P = nkT/R^4[/itex] (again, not interested in proportionality factors)
So, here's the problem:
On demonstrations: [itex]\nabla P = nkT/R[/itex]
To me: [itex]\nabla P = nkT/R^4[/itex]
Obviously, it's plain i'm missing something, still i can't see what it is.
(I apologize for my poor scholastic english, i'm not native speaker)