Molecular Cloud Collapse - Help

In summary, by balancing gravitational and centripetal forces and using the conservation of angular momentum, we can show that the radius at which collapse stops in the equatorial plane is given by r = (w0^2r0^2)/(2ve^2). I hope this helps!
  • #1
cvran
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Homework Statement


To show how conservation of angular momentum applied to to a collapsing MC core implies flattening during cloud collapse consider the equation of motion of a parcel mass m in a rotating initially spherical cloud of total mass M, initial angular speed w0 and radius r0. Consider the acceleration experienced by a particle of mass (m) on the equator of a spherical cloud spinning about a vertical axis relative to that at the poles. You should be able to show that the inward acceleration is less at the equator than at the poles and hence the cloud will inevitably flatten in the equatorial plane during Jeans collapse.

Homework Equations


Show that the collapse of the cloud will stop in the plane perpendicular to its axis of rotation when the radius reaches rf = (w20r40)/GMr where M is the mass and w0 and r0 are the original angular velocity and radius of the surface of the cloud. Do this by balancing gravitational and centripetal forces so that a parcel of mass at the equator becomes weightless.

{Hints: (1) the equation of motion in this case is m(d2r/dt2) = -((GMrm)/r2) + mrcos(theta)w2 ; (2) angular momentum is conserved during collapse so that I0w0 = Iw, where I is the moment of inertia for a uniform sphere of mass M and radius r and theta is the colatitude = 90˚-latitude}

The Attempt at a Solution


So first of all, I'm not particularly sure what this is asking me to do. I'm trying to use the equations for gravitation and centripetal force to achieve the equation for the radius of when collapse stops:

Since the mass is at the equator, theta = 0, so cos(theta) = 1, therefore I get:

GMrm/r2 = mrw2

But I have no idea what to do from here... any thoughts? Thanks! Sorry for the long problem.
 
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  • #2

Thank you for your question. It seems that you are on the right track with your attempt at a solution. Let me guide you through the steps to reach the equation for the radius of when collapse stops.

First, let's rewrite the equation of motion in terms of the moment of inertia, I:

m(d2r/dt2) = -(GMm/r^2) + Iw^2/r

Next, we can use the conservation of angular momentum to replace Iw with I0w0, where I0 is the moment of inertia for the initial spherical cloud. We can express I0 in terms of r0 and M:

I0 = (2/5)Mr0^2

Substituting this into our equation, we get:

m(d2r/dt2) = -(GMm/r^2) + (2/5)(Mr0^2)w0^2/r

Now, we can use the hint given in the problem, which states that at the equator, theta = 0 and cos(theta) = 1. This means that the centripetal force, mrcos(theta)w^2, becomes simply mrw^2. Substituting this into our equation, we get:

m(d2r/dt2) = -(GMm/r^2) + (2/5)(Mr0^2)w0^2/r

Next, we can rearrange this equation to solve for r:

d2r/dt2 = -(GM/r^2) + (2/5)(r0^2)w0^2/r

Now, we need to find the point at which the acceleration, d2r/dt2, becomes zero. This is the point at which the cloud stops collapsing in the equatorial plane. Setting d2r/dt2 = 0, we get:

0 = -(GM/r^2) + (2/5)(r0^2)w0^2/r

Solving for r, we get:

r = (w0^2r0^2)/(GM)

This is the equation for the radius at which collapse stops in the equatorial plane. But we can rewrite this in a more useful form by using the definition of the escape velocity, ve = sqrt(2GM/r). This gives us:

r = (w0^2r0^2)/(2ve^2
 

What is a molecular cloud collapse?

A molecular cloud collapse is a process in which a large, dense cloud of gas and dust in space collapses under its own gravity, leading to the formation of new stars.

What causes a molecular cloud collapse?

A molecular cloud collapse is caused by the collective gravity of the gas and dust particles in the cloud. As these particles come closer together, the gravitational force between them increases, eventually causing the cloud to collapse.

How long does a molecular cloud collapse take?

The duration of a molecular cloud collapse can vary depending on the size and density of the cloud. On average, it can take millions of years for a molecular cloud to fully collapse and form new stars.

What is the role of temperature in a molecular cloud collapse?

Temperature plays a crucial role in a molecular cloud collapse. As the cloud collapses, the temperature and pressure inside the cloud increases, leading to the formation of a hot, dense core where new stars can form.

What are the potential outcomes of a molecular cloud collapse?

The main outcome of a molecular cloud collapse is the formation of new stars. However, depending on the size and density of the original cloud, multiple stars may form, or the collapse may not be strong enough to form any stars at all.

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