Gravitational Collapse: How Long Does it Take?

[Ott Rovgeisha]

How long does the gravitational collapse of a giant molecular cloud take? The charged particles acquire quite a huge speed before hitting each other with an impact strong enough to cause nuclear forces take over. This implies that the process of acceleration should take a long time, since gravity is a week force.

Any insights?

 

[Bandersnatch]

In the absence of any other factors, the free-fall time of a homogeneous cloud is
$$t_{ff}=\sqrt{\frac{3\pi}{32G\rho}}$$
Notice how it's dependent only on density.
Example derivation here (page 4 onwards):
http://www.astro.uu.se/~hoefner/astro/teach/apd_files/apd_collapse.pdf
or in any good astronomy book.
A simplified derivation with approximate result can be found here:
http://en.wikipedia.org/wiki/Free-fall_time

The particles, however, are not charged in general - the collapse to stars happens in cold molecular clouds. They also don't acquire particle accelerator-range velocities in the process, if that's what you're implying.

 

[Ott Rovgeisha]

In the absence of any other factors, the free-fall time of a homogeneous cloud is
$$t_{ff}=\sqrt{\frac{3\pi}{32G\rho}}$$
Notice how it's dependent only on density.
Example derivation here (page 4 onwards):
http://www.astro.uu.se/~hoefner/astro/teach/apd_files/apd_collapse.pdf
or in any good astronomy book.
A simplified derivation with approximate result can be found here:
http://en.wikipedia.org/wiki/Free-fall_time

The particles, however, are not charged in general - the collapse to stars happens in cold molecular clouds. They also don't acquire particle accelerator-range velocities in the process, if that's what you're implying.

From this equation, if I am not mistaken, then if the take the density of hydrogen (though may be a dangerous idea, since the density may be different), a collapse take thousands of years, which in turn indicates that the particle gain a tremendous speed, which in turn would explain the stunningly high temperature that IS needed to start the nuclear fusion. Are those estimates OK?

 

[Bandersnatch]

More or less, yes. For a solar-like progenitor cloud it takes about 20 000 years to collapse to a radius where the further collapse is halted by heating of the gas (at about 1AU).

I wouldn't get too hung up on the time of collapse as a 'source' or reason for the heating, though. A particle can have an extremely long free-fall time and not gain much energy in the process - for example calculate how long it takes to fall to Earth from 1 ly away (imagine there's no Sun or other stars and planets around). I can already tell you it that no matter the result, and it will take a long time, it will have gained less than the Earth's escape velocity of ~11km/s.

Better to think in terms of potential energy being converted into kinetic energy, which translates to temperature (T being the average kinetic energy of molecules).

 

[Ott Rovgeisha]

Better to think in terms of potential energy being converted into kinetic energy, which translates to temperature (T being the average kinetic energy of molecules).

Well, on atom-atom basis, enormously high temperature means very high speeds, random directions, yes, but still...: this is what temperature is all about..
If the potential energy is converted into kinetic: that already means that the particles gain speed..
Frankly i do not understand what is the difference ?

 

[Bandersnatch]

The difference is that saying 'long free-fall time = high speed' is not generally true, as can be seen in the example with Earth.

 

[Ott Rovgeisha]

The difference is that saying 'long free-fall time = high speed' is not generally true, as can be seen in the example with Earth.

I still fail to understand: potential energy loss and gain in kinetic energy IS saying that a particle gains speed.
A high temperature DOES mean, that there are high speeds, there is no high temperature unless there is high speed of particles, yes the velocities have chaotic direction, but still.

You cannot have it both ways: saying "high temperature" means "high velocities"..
If the system's temperature rises by enormous amounts, then it is equal to saying that the speed of particles increases.
Let it be bit by bit (meaning that the speeds get chaotic), but still, huge velocities..

Of course, falling to the Earth from 1 ly away is not going to give a huge speed, because you are comparing a ridiculously weak gravitational field of the Earth with something that is not even comparable.

The fact that long fall time does not necessarily mean high velocities is true, but in case of star formation in plasma clouds, it should mean that, because high temperature means high speeds!

 

[Bandersnatch]

I'm sorry, I don't understand what you're arguing with. You've just repeated what I said in post #4.

My issue is only with your initial statement that long free-fall times translate to high velocities (so, also high KE, high T).

 

[Bandersnatch]

The fact that long fall time does not necessarily mean high velocities is true, but in case of star formation in plasma clouds, it should mean that, because high temperature means high speeds!

I missed this bit.
It doesn't mean that. Notice how decreasing (molecular! not plasma) cloud density increases the free-fall time. At the same time, for a given radius, it decreases the mass, and the potential energy. So particles in less dense clouds collapse longer, and gain less KE in the process.

 

[Ott Rovgeisha]

I missed this bit.
It doesn't mean that. Notice how decreasing (molecular! not plasma) cloud density increases the free-fall time. At the same time, for a given radius, it decreases the mass, and the potential energy. So particles in less dense clouds collapse longer, and gain less KE in the process.

The thing is that I never said that long time always means high velocities.

My initial post says this:
" The charged particles acquire quite a huge speed before hitting each other with an impact strong enough to cause nuclear forces take over. This implies that the process of acceleration should take a long time, since gravity is a weak force."

What is meant here, is this: in order for a weak force like gravity to cause high velocities, there has to be one condition met: the unbalanced force must take a longer time to last: therefore acceleration must last a longer time. How else can you get a high temperature with gravity. Of course the molecules or atoms collide during acceleration and that sort of spoils the acceleration in one certain direction, but the overall gain in speed is an obvious consequence.

 

[Bandersnatch]

in order for a weak force like gravity to cause high velocities, there has to be one condition met: the unbalanced force must take a longer time to last: therefore acceleration must last a longer time.

It does not follow. In post #2 I showed you the equation for free-fall time. It is solely dependent on density. A tiny cloud and a huge cloud of same densities will collapse equally fast. Even in the same cloud, the particles closer to the centre will gain less velocity than those farther away, despite equal free-fall times.
The condition for high velocities is not the duration of force acting, but the potential energy of the particle - so the mass of the cloud 'below' that particle.

And again, the denser the cloud, the shorter the free-fall time, and HIGHER the velocities.

The thing is that I never said that long time always means high velocities.

You are implying that it does mean that in this case, though.
There is no causative link between ##t_{ff}## and ##\Delta V##.

 

[Ott Rovgeisha]

It does not follow. In post #2 I showed you the equation for free-fall time. It is solely dependent on density. A tiny cloud and a huge cloud of same densities will collapse equally fast. Even in the same cloud, the particles closer to the centre will gain less velocity than those farther away, despite equal free-fall times.
The condition for high velocities is not the duration of force acting, but the potential energy of the particle - so the mass of the cloud 'below' that particle.

And again, the denser the cloud, the shorter the free-fall time, and HIGHER the velocities.You are implying that it does mean that in this case, though.
There is no causative link between ##t_{ff}## and ##\Delta V##.

I am referring to a very simple point:
If potential energy converts into kinetic energy, and in case there is a lot of it, that means high speed.
Time it takes to convert from one into another of course depends the gravitational field and the position of the particles...

If a cloud is dense, then it just requires less time to achieve the same kinetic energy.

Of course this is not the only factor, of course different parts of the clouds act differently, of course..of course..

I do not understand what does potential energy of a particle means. It is meaningless to say that, because the potential energy belongs to a system on interacting objects. If the gravitational field is stronger, then of course it takes less time for the molecules to gain the speed..

Potential energy cannot be converted into heat without molecules gaining speed. This must be the case if the temperature is high. How else can it happen...

If a rock is 20 m from the ground, it takes more time to fall that the one, that is 50 m from the ground, the acceleration lasts longer, the speed the of the rock falling from higher gains more speed, because the unbalanced force acted a longer time. There certainly is a connection.
Of course, if the gravitational field is stronger, the rocks gain as huge of a speed with less time..

If the cloud is more dense, then the gravitational field is stronger.. so it is strange to say that it solely depends on density and take this literary, because one should follow from another.

 

[Bandersnatch]

If a rock is 20 m from the ground, it takes more time to fall that the one, that is 50 m from the ground, the acceleration lasts longer, the speed the of the rock falling from higher gains more speed, because the unbalanced force acted a longer time. There certainly is a connection.

No. There is no connection for a collapsing molecular cloud, which is what we're talking about.

If potential energy converts into kinetic energy, and in case there is a lot of it, that means high speed.
Time it takes to convert from one into another of course depends the gravitational field and the position of the particles...

Not for a molecular cloud.

The difference is that the farther a particle is from the centre of the cloud, the more mass pulls it inwards. As a result, all factors bar the density cancel out. The wiki derivation shows that clearly.

 

[Ott Rovgeisha]

No. There is no connection for a collapsing molecular cloud, which is what we're talking about.

Not for a molecular cloud.

The difference is that the farther a particle is from the centre of the cloud, the more mass pulls it inwards. As a result, all factors bar the density cancel out. The wiki derivation shows that clearly.

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No. There is no connection for a collapsing molecular cloud, which is what we're talking about.

Not for a molecular cloud.

The difference is that the farther a particle is from the centre of the cloud, the more mass pulls it inwards. As a result, all factors bar the density cancel out. The wiki derivation shows that clearly.

Ok..

So... the particles farther away from the center, gain more speed. Right?

 

[Bandersnatch]

Yes. The velocity gained does depend on the position in the cloud (the potential energy in the field of the mass inferior to the particle). The time it takes to reach that velocity doesn't.

 

[Ott Rovgeisha]

Yes. The velocity gained does depend on the position in the cloud (the potential energy in the field of the mass inferior to the particle). The time it takes to reach that velocity doesn't.

But the more time passes, the more velocity the particles gain. Right?

 

[Bandersnatch]

Sure, that much is true.

 

[Ott Rovgeisha]

gt

Sure, that much is true.

So the longer THAT particular force acts, the higher the speed gets.

 

[Bandersnatch]

Yes, but the maximum velocity is set beforehand by the potential energy the particle has.

Think of it this way: before it starts collapsing, a particle in a cloud will have some potential energy dependent on the mass of the material inferior to its position in the cloud. If the mass is large, it'll have large potential energy and will eventually reach large velocities as the energy gets converted to KE. The mass can be large either because the cloud is huge, or because its density is high (or both).

Then the particle starts gaining speed from 0 to that pre-set speed. The time to complete the process will depend only on the density of the cloud. If it's higher it'll take less time.

 

[Ott Rovgeisha]

Yes, but the maximum velocity is set beforehand by the potential energy the particle has.

Yes, no argument there! I thought that was a given! You cannot get higher kinetic energies than the system has potential energy as the whole.

Think of it this way: before it starts collapsing, a particle in a cloud will have some potential energy dependent on the mass of the material inferior to its position in the cloud. If the mass is large, it'll have large potential energy and will eventually reach large velocities as the energy gets converted to KE. The mass can be large either because the cloud is huge, or because its density is high (or both).

Then the particle starts gaining speed from 0 to that pre-set speed. The time to complete the process will depend only on the density of the cloud. If it's higher it'll take less time.

That is well explained. But it does not contradict the fact that gravitational forces as a whole do have to act during a relatively long time period in order for the particles to have high speeds.

I am glad we are having this discussion I must admit.

 

[Bandersnatch]

I dig get the feeling that we were talking past each other more than once. :)

But it does not contradict the fact that gravitational forces as a whole do have to act during a relatively long time period in order for the particles to have high speeds.

The reason I have an issue with putting it like this is that it suggests a proportionality relationship between the time for the collapse and the maximum speed reached.
I don't dispute that a particle light-years away from the centre of a typical molecular cloud will take a long, long time to reach the centre, and typically gain large velocity in the process. This is, however, more of a correlation rather than causation.

 

[Ott Rovgeisha]

I dig get the feeling that we were talking past each other more than once. :)The reason I have an issue with putting it like this is that it suggests a proportionality relationship between the time for the collapse and the maximum speed reached.
I don't dispute that a particle light-years away from the centre of a typical molecular cloud will take a long, long time to reach the centre, and typically gain large velocity in the process. This is, however, more of a correlation rather than causation.

Very interesting.. the correlation-causation thing is intriguing. Just an example:

Two boxes of 2 kg are lifted. Their speeds are uniform. But one box has a speed, that is, say 4 times larger.
Both boxes gain potential energy (box-earth system gains).

Both boxes are acted upon a balanced force of 20 Newtons, because both of them move with uniform speeds.

Yet for no apparent reason, one of the 20N force does work more rapidly: is more powerful. Why, because the box has a higher velocity.
This higher velocity sort of seems to cause that force to do work much quicklier.

So, is that a cause or a correlation?

It seems that it depends of the history: at the very beginning the faster box was acted upon by a BRIEF but stronger unbalanced force that no longer exists. Yet still. the higher speed seems to cause this balanced force to be more powerful.

Doesn't this seem odd? And if it does not, then why?

 

[Itsme2033]

One thing that seemed to be overlooked in this discussion is that the particles are not in free fall. They are falling in an atmosphere. At first, it's a very thin atmosphere, and can be ignored. As they fall further inward, they will collide more and more. As such, their "temperature" will continue increasing, but their velocity will not. They will reach a terminal velocity and that terminal velocity will actually decrease as their pressure increases. All this stuff is fairly elementary, but I just didn't see it mentioned (except for a mention of the temperature).

 

[Ken G]

It seems that it depends of the history: at the very beginning the faster box was acted upon by a BRIEF but stronger unbalanced force that no longer exists. Yet still. the higher speed seems to cause this balanced force to be more powerful.

Remember that work done is force times distance over which it is applied. So if you fix the force, the gain in kinetic energy depends entirely on the distance over which it acts. If the particle is moving faster, it covers that distance faster, so acquires kinetic energy faster, but the total kinetic energy acquired will be the same if you fix the distance. Or, if you fix the time the force acts, then the faster particle will cover more distance, hence gain more kinetic energy from that same force. So it all depends on what you are keeping fixed. In the case of a freely falling cloud, the force is not constant, it rises like 1/r² as the radius r of the cloud contracts. But you end up with a gain in kinetic energy like 1/r anyway, regardless of how fast the contraction occurs, so that's more like the fixed force over the fixed distance. But also, in a collapsing cloud, the size of the force depends on the mass of the cloud, so you end up finding that more massive clouds, contracting to the same r, acquire faster speeds, regardless of how fast they get there (though it is true, as Bandersnatch said, that the higher mass cloud, starting from the same initial R, will get to any given r in less time). So that's why Bandersnatch discouraged you from thinking in terms of how long it takes-- the work depends on the force and the distance, not how long the distance gets covered.

 

[Ken G]

One thing that seemed to be overlooked in this discussion is that the particles are not in free fall. They are falling in an atmosphere. At first, it's a very thin atmosphere, and can be ignored. As they fall further inward, they will collide more and more.

This only affects their spread in velocities, not the magnitude of those velocities. The magnitude of the velocities comes from the kinetic energy, and that has to be given by the potential energy converted. The difference between free-fall, and a gradual contraction, is only a factor of 2 in kinetic energy-- in free fall, the kinetic energy is all the potential energy converted (and the velocities are mostly radially inward), whereas in a contraction that is near force balance, the kinetic energy is only 1/2 the potential energy converted (and the velocities are more isotropic). The missing energy has to get radiated away in the force-balance case. Any intermediate case is somewhere in between, so to within a factor of 2 in energy, you don't need to worry if it is in free fall or not, the distinction is not terribly important to the speed of the particles-- unless you care about the time it takes to play out (which is the short time Bandersnatch quoted for the free-fall phase, and much much longer times in the force balance phase).

As such, their "temperature" will continue increasing, but their velocity will not.

Here you must be referring to the radially inward bulk flow velocity, rather than random velocities of the particles. The latter will always increase with temperature, as I'm sure you aware.

 

[Ott Rovgeisha]

Remember that work done is force times distance over which it is applied. So if you fix the force, the gain in kinetic energy depends entirely on the distance over which it acts. If the particle is moving faster, it covers that distance faster, so acquires kinetic energy faster, but the total kinetic energy acquired will be the same if you fix the distance. Or, if you fix the time the force acts, then the faster particle will cover more distance, hence gain more kinetic energy from that same force. So it all depends on what you are keeping fixed. In the case of a freely falling cloud, the force is not constant, it rises like 1/r² as the radius r of the cloud contracts. But you end up with a gain in kinetic energy like 1/r anyway, regardless of how fast the contraction occurs, so that's more like the fixed force over the fixed distance. But also, in a collapsing cloud, the size of the force depends on the mass of the cloud, so you end up finding that more massive clouds, contracting to the same r, acquire faster speeds, regardless of how fast they get there (though it is true, as Bandersnatch said, that the higher mass cloud, starting from the same initial R, will get to any given r in less time). So that's why Bandersnatch discouraged you from thinking in terms of how long it takes-- the work depends on the force and the distance, not how long the distance gets covered.

True for the size of the work, not the how fast the work is done, though.. It is still bizarre that forces of same magnitude can do work with different speeds. And another point: a body can never gain potential energy without having kinetic energy. Sort of bizarre.