hydrostatic equilibrium illustrated

    Image 1 Caption: Hydrostatic equilibrium illustrated for a plane-parallel system of a fluid in a gravitational field.

    Features:

    1. Hydrostatic equilibrium is when a fluid is at rest everywhere because everywhere the forces are balanced.

    2. Gravity and the pressure force are usually the forces.

    3. For the diagram, Newton's 2nd law of motion (AKA F=ma) specializes to

            ma = 0 = -mg + (p_1-p_2)*A ,

      where acceleration a = 0 because of the constraint of hydrostatic equilibrium, A is the area of planar layer, m is the mass of the planar layer, -g is the gravitational field and g is positive, p_1 is the pressure below the layer, p_2 is the pressure above the layer, and we take up as positive.

    4. Note p_1 > p_2 for hydrostatic equilibrium.

    5. Another perspective is that p_1*A must equal the gravitational force on all the mass above the lower layer.

      hydrostatic equilibrium for a sphere

    6. Image 2 Caption: The spherically-symmetric hydrostatic equilibrium case.

    7. Note in both plane-parallel and spherically-symmetric hydrostatic equilibrium cases that in the directions perpendicular to the gravitational force, the pressure must be constant or there would be motion and NOT hydrostatic equilibrium.

    8. To generalize to spherically-symmetric hydrostatic equilibrium case the result we stated above, in all spherically-symmetric hydrostatic self-gravitating pressure-supported bodies, the upward pressure force of every layer must balance the gravitational force on all the mass above that layer. For example, this is true for the Earth's atmosphere and the Earth's interior. To generalize, it is true for moons, planets, and stars.

    9. Now let Δp = p_2 - p_1 .

      Let Δr be the thickness of the layer and let its density ρ be asymptotically constant as Δr→0.

      Then in limit as Δr→0, we obtain

            lim_(Δr→0)(Δp/Δr) = lim_(Δr→0)[(-m/(Δr*A)]g = -ρg ,

      or finally

            (dp/dr) = -ρg ,

      where (dp/dr) is the derivative of pressure p with respect to height r.

    10. For a star at radius r from the center, g=Gm(r)/r**2, where m(r) is the mass enclosed within the spherical shell of radius r. This formula follows from the shell theorem originally derived by Isaac Newton (1643--1727).

      Our planar layer is asymptotically equivalent to a small part of a spherical shell, and so we can substitute for g in the formula above to get

            (dp/dr) = -Gm(r)ρ(r)/r**2 ,

      where ρ(r) is the density of a star as a function of stellar radius.

      This equation is one of the 4 equations of stellar structure: the stellar hydrostatic equilibrium equation.

      The 4 equations of stellar structure are the equations that are solved to obtain a stellar structure model, a hydrostatic that is. Stars undergoing large-scale motions require hydrodynamic modeling which is much more difficult and less certain.

    11. The stellar hydrostatic equilibrium equation, as the derivation implies, apples to all spherically-symmetric hydrostatic self-gravitating pressure-supported bodies. So it applies moons, planets, and stars. Corrections do have to be made for deviations from spherical symmetry, of course.

    Images:
    1. Credit/Permission: David Bailey (AKA User:H2g2bob), 2006 (uploaded to Wikimedia Commons by User:Stannered, 2008) / Public domain.
      Image link: Wikimedia Commons: File:Hydrostatic equilibrium.svg.
      Local file: local link: hydrostatic_equilibrium.html.
    2. Credit/Permission: © David Jeffery, 2018 / Own work.
      Image link: Itself.
    File: Fluids file: hydrostatic_equilibrium.html.