Starstruck: The Formation of Stars

Introduction

Here we consider the time scale of star formation and stability. We do this by considering the time it would take a cloud of particles to collapse into a star, and in this process derive an approximation for the Jean’s Length.

This is the system we must consider. The grey cloud is all the dust orbiting the point mass. The test particle is at the far right edge of the red line. The red line is the semimajor axis of the ellipse. The test particle must travel along the red line to reach the center. The time it takes to travel this distance is the free fall time.

Solution

a. Consider a test particle in an e=1 “orbit” around a point mass.

An orbit with an eccentricity of 1 would be a straight line. That is, it would mean that that the planet would orbit back and forth straight across the point mass:

If the test particle starts at the edge of the orbit - that is, r away from the point mass, then the distance it will have to fall is equal to r (the semimajor axis of the whole clous - see image above!)

Kepler’s third law, from memory, is: $\bg_black dM = p(r)4\pi r^2dr$

The expression for mass of a sphere, in terms of its density, is: $M = \frac{4\pi r^{3}\bar{p}}{3}$

Combining these two expressions, we can solve for the free fall time by considering the period. Since the period, p, is the time it would take the test particle to “cross” over the point mass twice and return to its initial position, one half of the period will equal the free-fall time. Also, we can substitute 2a for r (see the diagram above).

$\frac{a^{3}4\pi ^{2}}{Gp^{2}} = \frac{4\pi r^{3}\bar{p}}{3}$

Solve for p (period. Note that this is different from the symbol for density, which is p-bar):

$\sqrt{\frac{3\pi a^{3}}{G r^{3}\bar{p}}} = {p}$

We know the relationships between t_ffand p, as well as r and a:

$\frac{p}{2} = T_{ff}$ $2a = r$

$\sqrt{\frac{3\pi a^{3}}{G (2a)^{3}\bar{p}}} = {\frac{p}{2}}$

$\sqrt{\frac{3\pi a^{3}}{8G a^{3}\bar{p}}} = {p}$

$\frac{1}{2}\sqrt{\frac{3\pi}{8G\bar{p}}} = T_{ff}$

$\sqrt{\frac{3\pi}{32G\bar{p}}} = T_{ff}$

This expression describes the free fall time of a particle on the edge of the cloud, and this the time it would take the cloud of dust to collapse into a star. We are assuming that the test particle experiences exactly one half a period of motion.

b. Determine the dynamical time for a sound wave to cross this same distance.

Defining the speed of sound as c_s, we can define the time it would take a sound wave to cross the same distance as a test particle as;

$t_{s} = \frac{2a}{c_{s}}$

We can solve for the length variable – that is, a.

$t_{s} = t_{ff}$

$\frac{2a}{c_{s}} = \sqrt{\frac{3 \pi}{32G\bar{p}}}$

$a = \frac{c_{s}}2{}\sqrt{\frac{3 \pi}{32G\bar{p}}}$

Which is equivalent to:

$R_{j} = \sqrt{\frac{3 \pi c_{s}^{2}}{128G\bar{p}}}$

This length signifies the distance at which the forces of gravity and pressure are at an equilibrium. The gas cloud will neither expand nor contract at this size.

c. What is the significance of Jean’s Length?

Jean’s Length shows the distance at which a cloud of dust would not be stable, and thus susceptible to collapse. The force of gravity attempts to pull the dust into the center, while the force of pressure tries to push it back out. If a sound wave takes longer to cross the distance than the actual particle of dust, then the force of pressure won’t be strong enough to prevent the dust cloud from collapsing. So, if a dust cloud is larger than Jean’s Length, that means that it cannot sustain itself, and will collapse into its center, resulting in stars!

d. Consider a collapsing cloud with a radius equal to Jean’s Length: when the cloud reaches a radius equal to one half of its initial radius, by what fractional amount had R_Jchanged?

To see the ratio of the Jeans’ Length for a cloud of equal mass but different radii, we must consider how the density changes from one size to another. So, first, we can find the densities of the two clouds, assuming a constant mass.

$\bar{p_{o}}=\frac{M}{\frac{4}{3} \pi R_{o}^{3}}$ $\bar{p_{J}}=\frac{M}{\frac{4}{3} \pi R_{J}^{3}}$

So:

$\bar{p_{J}}\frac{4}{3} \pi R_{J}^{3} = \bar{p_{o}}\frac{4}{3} \pi R_{o}^{3}$

$\bar{p_{J}}R_{J}^{3} = \bar{p_{o}} R_{o}^{3}$

We can plug in 0.5R_O and see how R_Jchanges. As R_o shrinks, the density p must grow by a balancing amount.

$X\bar{p_{o}}(0.5R_{o})^{3} = \bar{p_{o}} R_{o}^{3}$

$X\bar{p_{o}}0.125R_{o}^{3} = \bar{p_{o}} R_{o}^{3}$

$(8)\bar{p_{o}}0.125R_{o}^{3} = \bar{p_{o}} R_{o}^{3}$

So the density increases by a factor of 8 if the radius decreases by a factor of 2.

Thus,

$R_{J} =\sqrt{\frac{\pi c_{s}^{2}}{G \bar{p}}}$

If R becomes 0.5R_o then the density becomes 8p:

$R_{J} =\sqrt{\frac{\pi c_{s}^{2}}{G \bar{8p}}}$

$R_{J} =\frac{1}{^{\sqrt{8}}}\sqrt{\frac{\pi c_{s}^{2}}{G \bar{p}}}$

So, R_jdecreases by a factor of 0.35. This is fractional change of about 65%; the new value of R_J will be around 65% of the old one.

Conclusion

Jeans’ Length, the point of balance between the forces of pressure and gravity, can help you determine the behavior of a cloud of dust. By knowing the radius and mass, or by knowing just the density, you can calculate whether the cloud will expand or contract. If the cloud contracts, then it will likely become a star. Decreasing the radius of a cloud by 50% decreasing the Jeans’ Length by 35%, so the new length is 65% of the old one.

Acknowledgements

Thanks to Iryna and Monica! We had a lot of fun with this problem :)

2 comments:

JackieNovember 6, 2011 at 11:10 PM
Nice write-up! a few questions:

-the second diagram: is the point mass at the center or the focus of the ellipse? where is the focus of an e=1 ellipse?

-the first equation: this equation is known as the mass conservation equation. what is kepler's third law? you use kepler's third law to get to your third equation. (side note: in a latex editor, \rho will get you a good symbol for density)

-what is the force of pressure that is balanced with the force of gravity at the Jeans length? if you had an infinite cloud with constant pressure throughout, would there be any net force due to pressure? what stellar structure equation encapsulates the balance of gravity and pressure? protostellar clouds collapse pretty much in free fall until they satisfy this particular stellar structure equation, at which point they start contracting much more slowly by radiating away gravitational energy until they get hot enough for nuclear burning.

-check your math on the last part - is the new jeans length Rj' = 1/sqrt(8) * Rj (original Jeans length) reduced by 35% or by 65%? Is the new Jeans length shorter or longer than the current size of the cloud (R' = 1/2 * R0 = 1/2 * Rj)? can smaller pieces of the cloud collapse separately?
John JohnsonNovember 8, 2011 at 8:13 AM
Great writeup! However, there are a few things I should point out that will help you with your understanding of the results you found:

1) If R_J shrinks as the cloud shrinks, what does this imply for fragmentation? In other words, what does this imply for the formation of several stars rather than one gigantic star?

2) The clouds that form stars are composed of 99% hydrogen and helium. The 1% is everything else, including a little bit of dust. You might have been misled by all the dark patches in the pictures Dr. Swift showed, but that's just because it's hard to see the bulk material: hydrogen.

3) I'm really glad you had a lot of fun with this problem!

Sunday, November 6, 2011

The Formation of Stars

2 comments: