Hydrostatic equilibirum in slowly rotating star

[Vrbic]

Hello, in article Slowly relativistic stars by James B. Hartle (http://adsabs.harvard.edu/full/1967ApJ...150.1005H) is equation of Newtonian hydrostatic equilibrium, eq. (5). $$const.=\int_0^p\frac{dp}{\rho}-1/2(\Omega \times r)^2+\Phi,$$ where $p$ is pressure, $\rho$ is desinty, $\Omega$ angular velocity of star and $\Phi$ is graviational potential.
How may I derive it? I can derive eq. for hydrostatic equilibrium of non rotating star, but here is in result only potetntial and it suggests some other start than I know.
My idea is that all forces have to be in equilibrium, so if I take some small piece of matter let's call it $dm$. Than $$Fp_b-Fp_t+Fg+Fc=0,$$ where $Fp_b$ is preassure force from the bottom, $Fp_t$ is preassure force from the top of $dm$, $Fg$ is gravitational force and $Fc$ is centrifugal force. But how to proceed further, I'm not sure.
Can anybody suggest something?

 

[haruspex]

Hello, in article Slowly relativistic stars by James B. Hartle (http://adsabs.harvard.edu/full/1967ApJ...150.1005H) is equation of Newtonian hydrostatic equilibrium, eq. (5). $$const.=\int_0^p\frac{dp}{\rho}-1/2(\Omega \times r)^2+\Phi,$$ where $p$ is pressure, $\rho$ is desinty, $\Omega$ angular velocity of star and $\Phi$ is graviational potential.
How may I derive it? I can derive eq. for hydrostatic equilibrium of non rotating star, but here is in result only potetntial and it suggests some other start than I know.
My idea is that all forces have to be in equilibrium, so if I take some small piece of matter let's call it $dm$. Than $$Fp_b-Fp_t+Fg+Fc=0,$$ where $Fp_b$ is preassure force from the bottom, $Fp_t$ is preassure force from the top of $dm$, $Fg$ is gravitational force and $Fc$ is centrifugal force. But how to proceed further, I'm not sure.
Can anybody suggest something?

Try comparing it with the Bernoulli equation.

 

[Vrbic]

Try comparing it with the Bernoulli equation.

Aha, thank you.