Help Deriving Hydrostatic Equilibrium and Virial Theorem

In summary, the conversation discusses the process of deriving hydrostatic equilibrium and the virial theorem for a research paper on star formation and the life cycle of stars. The derivation involves using the Taylor expansion to approximate the pressure at a nearby point and applying Newton's second law of motion to calculate the forces acting on an infinitesimal element. The final equation for hydrostatic equilibrium is a result of balancing the gravitational force with the pressure force. The virial theorem can also be derived from this equation.
  • #1
jumi
28
0
Hey all,

Long story short, for my Modern Physics course, we have to do a research paper on a physics topic we didn't cover in class. Since I've always been interested in astronomy and the cosmos, I figured I'd do star formation / life cycle of stars. The paper has to have mathematical and physical reasoning for everything we present.

Anyway, I found some cool books that have helped me out thus far, but I'm having trouble following an explanation for hydrostatic equilibrium (which more or less directly leads to the virial theorem).

The book in question "Evolution of Stars and Stellar Populations" by Maurizio Salaris and Santi Cassisi.

They start the derivation by "finding the equation of motion of a generic infinitesimal cylindrical volume element with axis along the radial direction, located between radii r and r+dr", with a base (perpendicular to the radial direction) of area dA and density ρ.

They then obtain the mass, dm, contained in the element: dm = ρdrdA.

"[They] neglect rotation and consider self-gravity and internal pressure as the only forces in action. The mass enclosed within the radius r acts as a gravitational mass located at the center of the star; this generates an inward gravitational acceleration: g(r) = G[itex]m_{r}[/itex] / [itex]r^{2}[/itex]."

"Due to spherical symmetry, the pressure forces acting on both sides perpendicular to the radial direction are balanced, and only the pressure acting along the radial direction has to be determined. The force acting on the top of the cylinder is P(r+dr)dA, whereas the force acting on the base of the element is P(r)dA. By writing: P(r+dr) = P(r) + [itex]\frac{dP}{dr}[/itex]dr and remembering that drdA = dm/ρ, the equation of motion for the volume element can be written as: [itex]\frac{d^{2}r}{dt^{2}}[/itex]dm = -g(r)dm - [itex]\frac{dP}{dr}\frac{dm}{ρ}[/itex]."

---

They go further to get the final equation, but I get lost at this last paragraph. I don't understand where the last term of this equation comes from: P(r+dr) = P(r) + [itex]\frac{dP}{dr}[/itex]dr. Why wouldn't it just be P(r+dr) = P(r)?

And then once they get that equation, how do they get to this equation: [itex]\frac{d^{2}r}{dt^{2}}[/itex]dm = -g(r)dm - [itex]\frac{dP}{dr}\frac{dm}{ρ}[/itex]? I know they skipped a bunch of steps... Maybe I'm just not thinking right.

Any help would be appreciated. (BTW, I'm assuming once I understand this, I'll be able to work out the virial theorem on my own, but if not, I'll ask it in a response.)

Thanks in advance.
 
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  • #2


I understand that the derivation of hydrostatic equilibrium and the virial theorem can be confusing and overwhelming. Let me try to break down the steps for you and hopefully clarify some of the concepts.

Firstly, the equation P(r+dr) = P(r) + \frac{dP}{dr}dr is known as the Taylor expansion. It is a mathematical tool used to approximate a function at a nearby point by using information about its value and derivatives at a known point. In this case, the pressure at a point (r+dr) is approximated by the pressure at a known point (r) plus the change in pressure over a small distance (dr). This is necessary in the derivation because we are dealing with infinitesimal elements and need to express the forces acting on them.

Moving on to the final equation, \frac{d^{2}r}{dt^{2}}dm = -g(r)dm - \frac{dP}{dr}\frac{dm}{ρ}, it is essentially an application of Newton's second law of motion (F=ma). The left hand side represents the acceleration of the element, which is equal to the sum of the forces acting on it (mass times acceleration). The first term on the right hand side represents the gravitational force, which is proportional to the mass enclosed within the radius r. The second term represents the pressure force, which is proportional to the change in pressure over a small distance (as we saw in the Taylor expansion) and the mass of the element (since pressure acts on the entire surface area of the element).

I hope this helps you understand the derivation better. As for the virial theorem, it is a consequence of the hydrostatic equilibrium equation and can be derived from it. Good luck with your research paper! Feel free to ask any further questions if needed.
 

1. What is hydrostatic equilibrium?

Hydrostatic equilibrium is a state in which the forces acting on a fluid are balanced, resulting in a stable and stationary system. This means that the pressure gradient within the fluid is equal to the gravitational force acting on the fluid.

2. How is hydrostatic equilibrium related to the Virial Theorem?

The Virial Theorem is a mathematical relationship between the total kinetic and potential energy of a system. In the case of hydrostatic equilibrium, the Virial Theorem can be used to show that the total gravitational potential energy of a star is related to its temperature and density.

3. What is the significance of hydrostatic equilibrium in stars?

Hydrostatic equilibrium is essential for the stability of stars. It ensures that the gravitational force pulling inward is balanced by the outward pressure generated by the high temperatures and densities in the star's core. This balance allows stars to maintain a stable size and energy output.

4. How do scientists derive the hydrostatic equilibrium equation?

The hydrostatic equilibrium equation is derived by applying the principles of Newton's second law of motion to a small parcel of fluid within a larger body. This results in an equation that relates the pressure gradient, density, and gravitational force within the fluid.

5. Can the hydrostatic equilibrium equation be applied to other objects besides stars?

Yes, the hydrostatic equilibrium equation can be applied to any system in which a fluid or gas is in a stable state, such as planets, galaxies, or even the Earth's atmosphere. However, the specific equations and factors may vary depending on the properties of the system being studied.

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