A star is born (Part 3: The Virial Theorem, Jeans mass and Bok globules)

In this article I will try to show by skipping calculations and focus on the process on how to get to the result, that from a theorem called the Virial theorem, given a molecular cloud composition (mostly hydrogen), known density e.g. n = 4 10^3 cm^-3 and initial temperature. e.g. 50K, it is possible to find out the mass of this collapsing cloud, its radius, and the average luminosity of the protostar as it collapses.

Let’s start with the Virial theorem. Let us consider a system of N particles with masses m_i and position vectors r_i with i = 1,…,N. We define the virial of the system as:

V = \displaystyle\sum_{i=1}^{N} m_i \vec{r_i} \cdot \dot{\vec r_i}

Taking the time derivative in dot notation, we obtain:

\dot V = \displaystyle\sum_{i=1}^{N} m_i \dot{\vec r_i} \cdot \dot{\vec r_i} + \displaystyle\sum_{i=1}^{N} m_i \vec{r_i} \cdot \ddot{\vec r_i} = 2K + \displaystyle\sum_{i=1}^{N} \vec{r_i} \cdot \vec F_i = 2K + U

Using the facts that Fij=-Fji = m r_dot_dot and that F can be expressed in terms of the gravitational attraction between each particle of the system and that F_ii is zero because particles do not exert any force on itself. We get derivate of V equals 2K + U, two times the kinetic energy plus the potential energy of the system. Now taking the time average of that derivative of V and assuming that it is a periodic system of period T, in that case let’s assume that V(T) is approx V(0) and the time average of the derivative of V is then zero.

\langle V \rangle = \frac{1}{T} \int\limits_0^T V \mathrm{d}t = \frac{1}{T} [V(T)-V(0)] = 0

If we skip the time average symbolism and use scalar values for K and U, the Virial theorem is: 2K + U = 0. This means that in equilibrium, the kinetic energy and the gravitational energy balance each other.

An important application of this theorem arises when analyzing the evolution of large gravitationally bound systems, such as clouds of gas and dust in spiral galaxies whose collapse leads to the formation of stars. These clouds normally maintain equilibrium, in which case 2K + U = 0. In a system bound by gravitation, the potential energy of the system is negative (because it is in a “bound state” due to gravitation, an attractive force) and U = – cst. GM^2/R, therefore in 2K = -U, both the lefthand side and the righthand side are positive values.

At some stage they might start collapsing due to some perturbation, such as a nearby supernova explosion. The cloud starts collapsing and the Virial theorem is no longer applicable. In this case, gravitity wins over kinetic energy and 2K is lower than −U. Eventually the heat generated due to collapse leads to an increase in kinetic energy K of the cloud of gas and again returns to equilibrium beacuse If 2K is greater than -U, kinetic energy wins over gravity and the cloud expands.

Given the fact that the gravitational potential energy of a uniform sphere of mass M and radius R is: U = – cstGM^2/R

That the temperature and mean energy of gas particles is (The total energy K+U): cst kT

Jeans Radius for cloud collapse (2K < -U). A cloud with radius R, mass M, and temperature T:

cst k T = cst GM^2/R = cst G rho R^2

Because the mass M of the cloud of gas can be expressed in terms of it radius R knowing the cloud’s density rho (M = cst rho R^3).

R_{jeans} = \sqrt{\frac{cst kT}{G} \rho}

M_{jeans} = (\left(\frac{5kT}{Gm}\right)^{3/2} \sqrt{\frac{3}{4\pi\rho}}) \propto T^{3/2}

Assuming that the dust/molecular cloud is mostly hydrogen, given the density (rho) and temperature stated above, one can find that the Jean’s radius is approx. between 0,4 and 1 parsec. This means that a cloud will start to collapse if it’s radius is greater than the critical radius or the Jean’s radius, the mass range of the cloud can be calculated too and expressed in terms of solar masses. Using the time it takes for particle to freefall onto the surface of the star (ignoring star’s rotation and other factors), and that gravitational potential energy is converted into radiation. One can find that the luminosity of the cloud can be quite small, which means that the collapsing cloud looks very dark. These regions are sometimes called Bok globules.

References:

Pankaj Jain (2015). An Introduction to ASTRONOMY AND ASTROPHYSICS.

Jeff Hester et al. (2010). 21th Century Astronomy.

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