Hydrostatic Equilibrium, Energy Principles, and the Virial Theorem

Hydrostatic Equilibrium, Energy Principles, and the Virial Theorem#

Friday, Jan. 17, 2025

astrophysics of stars and planets - spring 2025 - university of arizona, steward observatory

Today’s Agenda#

  1. Announcements / Recap + Survey! (5m)

  2. PCA Discussion (10m)

  3. Reading Overview/Key Points (10m)

  4. In-Class Assignment (Groups of 2) (20m)

  5. Debrief on ICA (5m)

Hydrostatic Equilibrium#

Consider the following assumptions of an object:

  • spherically symmetric

  • nonrotating

  • nonmagnetic

  • single star on which there are

  • no net forces acting and, hence, no accelerations

Then, we ask the question, how can we describe this “star” in equations?

First, some convections:

  • radius: \(r\) is the radial distance measured from the stellar center (cm)

  • density: \(p(r)\) is the mass density at \(r\) (g cm\(^{-3}\))

  • temperature: \(T(r)\) is the temperature at \(r\) (K)

  • pressure: \(P(r)\) is the pressure at \(r\) (dyne cm\(^{-2}\) = erg cm\(^{-3}\))

  • mass: \(m(r)r\) is the mass contained within a sphere of radius \(r\) (g)

  • luminosity: \(L(r)\), the rate of energy How through a sphere at \(r\) (erg s\(^{-1}\))

  • local gravity: \(g(r)\) local acceleration due to gravity (cm s\(^{-2}\))

Following Eqns. (1.1) to (1.4), we arrive at the

Definition 1

equation of hydrostatic (or mechanical) equilibrium (HSE)

\[ \frac{dP}{dr} = -\frac{Gm}{r^2}\rho \equiv -\rho g \]

Observation 1

Since \(g,\rho \geq 0\), it follows that \(\frac{dP}{dr} \leq 0\), and the pressure must decrease from the center outwards, everywhere.

If this equation is violated, local acceleration must occur, the “star” is said be out of HSE.

An Energy Principle#

We can start by defining the total gravitational potential energy.

Definition 2

the total gravitational potential energy, \(\Omega\), of a self-gravitating body is defined as the negative of the total amount of energy required to disperse all mass elements of the body to infinity.

\[ \Omega = -q \frac{GM}{R} (\textup{ergs}) \]

where here, we have performed the integral over \(r\) and \(M\) and \(R\) represent the total mass and radius of the star, respectively. For a uniform density sphere \(q=3/5\).

Next, consider the total internal energy

Definition 3

total internal energy arising from microscopic processes

\[ U = \int_{m}E \textup{d}m \]

After some math and application of the first and second laws of thermodynamics, we arrive at a similar conclusion regarding the assumptions we first made when we set out.

The equation of HSE written in Lagrangian form (the independent variable is \(dm\)),

\[ \frac{dP}{dm} = -\frac{Gm}{4 \pi r^4}~. \]

In fact, we can use Eqn. 1.1 to recover the first definition for HSE - (Definition 1).

The Virial Theorem#

Following the discussion in HKT 1.3, we arrived at a form of the Virial Theorem:

Definition 4

\[ \frac{1}{2}\frac{dI^2}{dt^2} = 2K + \Omega \]

where

\(K\) is twice the total of the kinetic energy and \(\Omega\) is the same as defined above (Definition 2).

Assumption 1

We note that the above definition assumes that the sums are over the entire star.

In-Class Activity#

Head over to ICA1, work with the person next to you on 1.a-d.

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