The Direction of Pressure Gradient in Hydrostatic Equilibrium

Question

The equation for Hydrostatic Equlibrium in galaxies and galaxy clusters is quoted as: $$\frac{dP}{dr}\times\frac{1}{\rho} = -\frac{GM}{r^2}$$The minus sign on the right hand side indicates that the direction of the force of gravity is towards the center of the object (as it should be). However, the left hand side of the term also resolves to a negative acceleration. Is the physical interpretation that the pressure gradient as well is directed inward? I've read that the force of pressure is directed along a vector from the highest density to the lowest density, but that is with an unbalanced force (such as wind). What is the right way to interpret the vectors in this equation?

Answer 1

The pressure gradient is actually a vector, written $\nabla P$. Its direction is that in which the pressure gradient is steepest. In a spherically symmetric situation, the gradient in pressure is radially inwards and can be written as $(dP/dr) \hat{r}$. i.e. $dP/dr$ written as a scalar, is a negative number. The force per unit area on an object due to a pressure gradient is $-\nabla P$.

In a spherically symmetric "ball of gas", hydrostatic equilibrium can be written $$\frac{dP}{dr}\, \hat{r} = -\rho(r) g(r) \hat{r}\, $$ where $g(r)$ is the magnitude of the gravitational acceleration at radius $r$. The right hand side is negative because the force due to gravity is inwards.

Perhaps it's clearer if you write it as $$\frac{dP}{dr}\, \hat{r} + \rho(r) g(r) \hat{r} = 0 $$

The first term has negative magnitude and the second term has positive magnitude.