Gravity and the Formation of Stars

[daisey]

Earlier today I took a break from following the thread here on the Fukushima Nuc Plant problems (very interesting, by the way) and watched some TV, specifically a show I recorded on the formation of Stars. It reminded me of something that has been bothering me for some time - Gravity and how it contributes to the formation of stars.

This TV show made it sound as if a star begins life as a tremendously large cloud of hydrogen atoms, floating merrily along in space. It explains that gravity works its magic and pulls each of these little atoms together over millions of years until a star is formed. What I fail to understand is HOW the force of gravity, which is kinda weak, can force all of these atoms so closely together with such force that atomic fission begins. Furthermore, I also fail to understand how one atom of hydrogen floating along in space 100 meters from another can be pulled together by gravity, I don't care how long gravity tries. I'm aware there is no distance limit to the force of gravity, but this weakling of a force decreases with distance.

I am sure someone can explain mathematically that this is in fact possible. But can anyone explain how this is possible in terms that a lay person like myself can understand? Sorry, but it just sounds too hard to believe, although I guess it must be or we would not be here.

 

[JesseM]

Each atom is being pulled toward the system's center of mass by the combined gravity of every other atom (and there's something like 10^57 atoms in a star), so I don't understand why you bring up the example of two individual atoms pulling on each other (even in that case they would be pulled together if they started out with zero velocity in the center-of-mass frame, but the escape velocity would be very small so random differences in initial velocity could prevent them converging).

 

[qqkitty]

Hello daisey,

You bring up an excellent point! Normally, we are accustomed to thinking of gas as a crazy concoction of fast-moving molecules that are weakly interacting (i.e. an ideal gas). In the case of stellar formation, there are actual a couple of extremes we must consider. I will go ahead and write up some math to aid the explanation, because I think that the math contributes to our understanding of stellar formation.

First, as you probably already know, stars are hypothesized (and there is a lot of evidence to suggest this) to be born at the centers of dense regions of cold HI (neutral hydrogen) gas - called Giant Molecular Clouds (GMCs). When I say cold, I mean pretty cold. Around 10-100K! Typical GMCs have roughly 10$$^{4}$$ solar masses of gas and are on the order of 10pc in diameter! If we assume a spherical distribution of gas, we can get an estimate for the gas density, but that is unimportant for this explanation. Why do I mention this? Because I want you to get an idea of the extremes under which these so-called protostars develop. There are a number of contributing factors to protostar development such as their overall angular momentum, turbulence, and magnetic fields. To simplify our discussion, we shall assume a GMC is as boring as possible: it is spherically symmetric and that only gravitational effects are considered.

The virial theorem, which is a statement about the time average of the kinetic energy and potential energy of a system, is given as:

2K + U = 0. (1)

In other words, a gravitationally bound system in equilibrium (that is, not expanding or contracting. It is in equilibrium after all.), will obey the above relation. We want our cloud to collapse; therefore, we need the molecules in the GMC to be moving with energy less than that required to unbind the system, so that

2K < |U|. (2)

From classical statistical mechanics, the kinetic energy of an ideal gas consisting of N particles at temperature T is given by

K = $$\frac{3NkT}{2}$$. (3)

The gravitational potential energy of the entire cloud is given (approximately) by

U = $$\frac{-3GM^{2}}{5R}$$ (4),

where M is the mass of the cloud, and R is the radius of the cloud.

Finally (2) implies that

2K = $$\frac{6NkT}{2}$$ < $$\frac{-3GM^{2}}{5R}$$ = U (5)

If the cloud is made entirely of hydrogen, we can write

N = $$\frac{M}{m_{H}}$$, where we know the mass of hydrogen.

Then, if we know the initial density of the cloud $$\rho_{0}$$, we can find the initial radius of the cloud in terms of its mass and density and substitude into (5). We then isolate for M and receive something called the Jeans mass:

M = $$\left(\frac{5kT}{Gm_{H}}\right)^{3/2}\left(\frac{3}{4\pi\rho_{0}}\right)^{1/2}$$

Okay, so what have we discovered? This mass is the minimum mass of a cloud required before it undergoes an unstable collapse at the temperature T with density $$\rho_{0}$$. Cool, eh? Basically this means that if we play with the temperature a little bit, we get a different mass requirement. If the cloud is less dense, then then the mass required for spontaneous collapse of the cloud is greater. If the temperature of the cloud is greater, then we need yet more mass.

Basically the answer to your question lies in the initial condition of the gas cloud! Its density and its temperature. In more complicated scenarios, things such as external magnetic fields slow down the collapse of the cloud, and turbulence plays a mysterious role as well. The temperature and density extremes that permit the construction of protostars are certainly not everyday conditions. To be certain that you understand, I will mention that just because there exists some cloud of gas somewhere in empty space does not mean that it will collapse. In the ideal case of only gravitational interaction, the cloud will collapse only if it has sufficient mass as to create a sufficiently large gravitational potential well that is larger than the average kinetic energy of the gas itself. If the gas is too energetic, the cloud will just fly apart, as you would expect. It all depends on its initial density and initial temperature. I hope that answers your question a little bit.

P.S. It's my first time using Latex on these forums, so bear with the terrible notation. I can try and fix it up, but the key ideas are there. Sorry in advance!

 

[daisey]

That is interesting. I guess I could understand the gravitational effects if we assume the starting point is an already massive dense cloud of HI. But how does it get to be a dense cloud in the first place? Is this a remnant of the Big Bang?

I gave the example of two HI atoms because I assumed the ultimate starting point was a large but widely dispersed and non-dense cloud (lots of examples of two atoms pulling on each other).

 

[qqkitty]

This is yet another good question.

In my last post, I determined that whether or not a gas cloud will collapse has to do with its initial temperature and density. Keeping this in mind, we can explain why some stars are born larger than others. The larger stars were born in larger gas clouds! Alright, so now you ask, from where do these gas clouds come?

You're right in thinking that it is a remnant from the big bang. Hydrogen is a relatively simple atom, and it is not surprising that it is the most abundant element in the universe. After the big bang, there was a period of intense heat and a lot of radiation. There were protons and electrons flying around, as well as a sea of other stuff including photons of course. We'll focus on the protons and electrons. Technically speaking, the proton is an ionized hydrogen atom, but we want cold neutral hydrogen - that is - a hydrogen atom with a proton-electron pair.

For a relatively short time (I think on the order of hundreds of thousands of years), electrons and protons were unable to recombine because as soon as an electron was bound to a proton, it would immediately be ionized by the surrounding radiation because it was too energetic. As the universe expanded, the radiation that existed throughout the universe (and still exists today as the cosmic background radiation, though it is much colder now: 2.725 K) cooled and allowed the electrons and protons to recombine without being immediately ionized.

This period of time in the universe is known as the recombination period. In fact, from what I have said, you can calculate yourself what the temperature of the background radiation must have been in order for it to not immediately ionize a hydrogen atom if you know the ionization energy of hydrogen.

As the universe cooled further, the neutral hydrogen became less energetic and clouds of it began to collapse according to a mechanism similar to that which I presented in my previous post.

Hope that helps.

Edit: Another thing worth mentioning is that the universe did not expand the same in all directions, so there were regions were hydrogen was denser than in others. Analysis of the CMB demonstrates this fact.

 

[daisey]

Is it true that all stars were formed from H1 clouds left flating around from the Big Bang? Or does this H1 get recycled over and over? If so, that again means stars can be formed from widely dispersed H1 atoms, because a Supernova widely disperses its remnants, no? That would mean gravity is somehow pulling these tiny atoms together that were blasted widely apart. But then again, why would H1 be found in a s-nova? Should all be cunsumed before star dies.

 

[qqkitty]

All stars are formed from the material that was available to them before they collapsed. Not all stars go supernova, but those that do will populate the interstellar medium with metals that were produced in the star during its lifetime. There are actually many different kinds of supernova and each are a consequence of the conditions under which the star had expired its primary fuel source. As a result, not all supernova are thermonuclear explosions.

If a star can supernova, long before it does, it undergoes a giant phase where it actually burns many different elements at once. The pressure in the core changes with each new element that is undergoing fusion and the star rapidly expands as the pressure in the core changes. Burning will occur in shells: in the outer shells, helium fusion occurs and in the inner shells, carbon and oxygen fuse. The star is like a giant onion! The mass of the star determines the fusion that occurs in shells, and is the primary determinant of whether a star will supernova. As that star expands, it will become so large that the gas at the 'surface' of the star is bound very loosely by gravity. This outermost gas is usually hydrogen and so a lot of it just floats off into space and escapes the star. This process recycles some hydrogen back into the ISM.

Once the star explodes, it dumps out most of what it produced back into the ISM. In fact, in very short periods of time, a lot of the iron that was produced in the core just before the star exploded decays back into smaller elements (even back into individual protons) and releases a lot of energy in the form of neutrinos that will give the star's layers enough kinetic energy to be pushed out into space and that is what we see as the supernova.

I have left out a lot of details about the process of the supernova because I do not want you to forget what the overall point is. When this kind of star explodes, it will populate the interstellar medium with clouds containing hydrogen, helium and other metals. The next generation of stars will be formed when these clouds cool sufficiently so that they can collapse again and produce new stars. These new stars will have a relatively higher concentration of metals in them than the stars in the previous generation.

Thus we have learned something from the above discussion: to an extent the hydrogen gets recycled from one generation to the next. What will happen far far far into the future? In my very first post I discussed a scheme under which a protostar can be formed from clouds of cold hydrogen. Why did I pick cold hydrogen instead of any other element? I did so because the temperature requirement for hydrogen fusion is the easiest attained and that hydrogen is so populous in the ISM. In principle, ithe universe could have enormous clouds of helium that can collapse and form stars. But the mass requirement of such clouds is much greater than that of hydrogen because the temperature required for helium fusion is greater than that for hydrogen. Similar for carbon and oxygen stars. We don't see such stars being formed out of the ISM because there isn't enough of these elements in clumps in the universe. Remember, the reason a star supports itself under gravity is because of the energy released during fusion. The star doesn't 'care' what it has to fuse in order to be supported, just that it can no longer fuse atoms when the temperature is not high enough for the atoms left over from the previous fusion process to themselves fuse.

At the present time, hydrogen is the primary constituent in the baryonic ism, so that you don't need to worry about there being enough hydrogen for new stars. Also, not all the hydrogen gets consumed. It's not that simple. Fusion happens when the temperature is high enough. Fusion doesn't happen everywhere in the star!

Finally, it is indeed true that the remnants of supernova get blasted into space and it is probably difficult for you to imagine that the remnants can form stars right away. It is difficult for me to imagine too because it does NOT happen right away. The stuff that gets blasted off has too much kinetic energy. Over time it will collide with gas clouds in the medium and slow down and eventually, after long periods of time, will that material be 'calm' enough to produce new stars. Again, I refer you to my original post for the conditions under which a star can form. It seems to me that you have a hard time imagining this process because you haven't considered how cold the gas is, how much of it there is, and how long it takes for this process to occur. I encourage you to think carefully about these extremes and how they affect these small atoms individually.

 

[Q-reeus]

Nothing against the OP's query, but why has this thread persisted in this section despite having nothing to do with QM?