Confinement

Gravity

The gravitational force on an atmospheric particle of mass $m$ is $${\bf F}_G = m {\bf g}$$where ${\bf g}$ is a vector representing the acceleration that gravity induces.

Spherical Environment

The magnitude of the gravitational acceleration at a radius $r$ from the center of a spherical environment is $$g = \frac {GM_r} {r^2}$$where $G$ is the gravitational constant and $M_r$ is the mass within radius $r$.

Circular Velocity

It is often useful to express the gravitational acceleration in the environment of a galaxy in terms of the speed $$v_c = (gr)^{1/2} = \sqrt{\frac {G M_r} {r}}$$of a circular orbit around the galaxy’s center. The quantity $v_c$ is known as a circular velocity and is typically 100-300 km/s in the vicinity of a galaxy. For comparison, the gravitational acceleration at Earth’s surface corresponds to a circular velocity of about 8 km/s.

Gravitational Potential

More generally, gravitational acceleration can be expressed in terms of a gravitational potential function $\varphi({\bf r})$ defined so that $${\bf g} ({\bf r}) = - \nabla \varphi$$at a location given by the vector ${\bf r}$. The change in gravitational potential energy of a particle of mass $m$ that moves from location ${\bf r}_1$ to a location ${\bf r}_2$ is then $m \varphi ({\bf r}_2) - m \varphi ({\bf r}_1)$.

Isothermal Potential

In the vicinity of a typical galaxy, the circular velocity remains approximately constant as radius increases, meaning that $g = v_c^2 / r \propto 1/r$. Consequently, the approximate gravitational potential is $$\varphi = v_c^2 \ln \left( \frac {r} {r_0} \right)$$where $r_0$ is a reference radius at which $\varphi$ is set equal to zero. This type of gravitational potential is called isothermal because the orbital speeds of particles confined within it tend to be similar at all radii.