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Considering the ideal gas law , where pressure is always positive, i wonder :

How can gravity turn a gas into a star?

Yes gas has mass too.

But a light gas obeing the ideal gas law seems problematic to me?

Without the pressence of other matter or gravity not from the gas, I think a star can not form from the gas? Because the pressure and temperature would increase When pushed together?

The repulsion in gas > the strenght of gravity right?

How about rotation?

Im confused.

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closed as unclear what you're asking by John Rennie, stafusa, hft, Jon Custer, sammy gerbil Jun 26 '18 at 18:00

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GMCs( Giant Molecular clouds)are very unlike stars: their typical diameter is 50 pc, typical mass exceeds $10^5$ solar masses and their density is $10^{20}$ times less than a typical star. Thus, the question is: How do stars form from such molecular clouds? The simplest answer to this question is that the stars form due to gravity: a cloud of gas contracts due to self-gravity. This contraction increases density and temperature of the core, initiating generation of nuclear energy. You may ask : How do we reconcile the typical mass of a star with the mass of a GMC (~$10^5 $ solar mass)? This problem was addressed by Jeans who proposed a criterion for the mass of a contracting cloud which may evolve into a star.

Jeans Criterion:- Jeans proposed that there are two competing processes in the gravitational collapse of a molecular cloud. On the one hand, the gravitational contraction increases the internal pressure of the cloud which tends to expand the cloud. On the other hand, gravity acts on the cloud and tends to further contract it. Which of these two processes will dominate is determined by the mass of the cloud. If the internal pressure is more than the gravitational force, the cloud will break up. A clump of cloud must have a minimum mass to continue collapsing and give birth to a star. This minimum mass is called the Jeans mass. It is a function of density and temperature. To obtain an expression for the Jeans mass, we make the following simplifying assumptions:

i) the cloud is uniform and non-rotating;

ii) the cloud is non-magnetic; and

iii) the gas and dust is confined to a certain region of space by the gravitational force and is in hydrostatic equilibrium.

For such a system, we may write the relation between kinetic and potential energies as Star Formation ( virial theorem):

$$2U + \Omega = 0$$

where U is the internal kinetic energy and $\Omega$ is the gravitational potential energy of the cloud.

If M and R are the mass and radius of the cloud, respectively, the potential energy of the system can be written as:

$$\Omega=\frac{-3}{5}\frac{GM^2}{R}$$

If the number of particles in the cloud is N and its temperature is T, the internal kinetic energy of the cloud can be written as:

$$U=\frac{3}{2}NK_BT$$

where $K_B$ is the Boltzmann constant. Further, the number of particles

$$N=\frac{M}{µm}$$

where µm is the mean molecular weight and m is the mass of a hydrogen atom.

If the total internal kinetic energy is less than the gravitational potential energy, the cloud will collapse. This condition reduces viral Theorem into

$$2U < |\Omega |$$

Now expression can be written as:

$$\frac{3K_BTM}{µm}<\frac{3}{5}\frac{GM^2}{R}$$

or,

$$M>\frac{5K_BTR}{µmG}$$

On substituting $$R=\left(\frac{3M}{4πρ}\right)^\frac{1}{3}$$

we find that the minimum mass that will initiate a collapse is given by:

$$M ≈ M_J={\left[\frac{5K_BT}{µmG}\right]}^\frac{3}{2}{\left[\frac{3}{4πρ}\right]}^\frac{1}{2}$$

Here $M_J$ is called the Jeans mass and this equation is known as Jeans criterion. The Jeans mass is the minimum mass needed for a cloud to balance its internal pressure with self-gravity; clouds with greater mass will collapse.

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  • $\begingroup$ You know there's MathJax, right? ;) $\endgroup$ – pela Jun 25 '18 at 15:00
  • $\begingroup$ @pela when I post that answer, I don't know that stack exchange supports Latex. It was advised to me to use Latex rather to upload images. And, I know very well about Latex. So, after knowing that I improve my mistakes... Like 105 to $10^5$.. $\endgroup$ – Sheetal Jun 25 '18 at 15:58
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    $\begingroup$ We expect and prefer members to use Mathjax on Physics SE. That means all equations should be using Mathjax and images of equations are not acceptable. Please edit your post to use Mathjax. Most members will not upvote a post with images instead of Mathjax and many would downvote it. $\endgroup$ – StephenG Jun 25 '18 at 16:33
  • $\begingroup$ @StephenG Thank you so much for your suggestion Sir. I had edited this answer using mathjax $\endgroup$ – Sheetal Jun 26 '18 at 5:16
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Here is a brief, heuristic, description of why the ideal gas law does not work for systems where gravitation is dominant.

One way you can arrive at the notion of an ideal gas and hence the ideal gas law is from the kinetic theory of gases. This makes a number of assumptions, of which the critical one here is:

except during collisions the forces that the molecules of the gas exert on each other are negligible.

This means that the molecules interact only during collisions: there are no long-range forces.

This assumption is not true. It is true to a very good approximation for relatively small systems, but gravity is a long-range force and is always attractive (unlike EM forces which are also long-range but can be attractive and repulsive, which means that the EM forces cancel on average for uncharged gases). What this means is that, for a sufficiently large system, gravity starts to mean that the molecules don't interact only during collisions: even when not colliding with each other they are feeling the gravitational pull from the rest of the molecules in the system. This means that the ideal gas law, and thermodynamics more generally, breaks down for systems which are sufficiently large that gravity is dominant.

So it is not safe to use things like the ideal gas law for volumes of gas large enough that self-gravitation becomes important: you can't understand how stars form that way.

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Yes, it's difficult for gas to collapse gravitationally to form a star, but clearly it's not impossible. ;) A collapsing gas cloud isn't a closed system: it can lose heat by radiation, although admittedly that's a rather slow process for a cloud that's almost entirely hydrogen and helium. So in the early universe, star formation was quite difficult, and it took very large quantities of gas to have enough gravity to overcome the pressure.

Those very early stars (known as population III stars) haven't been detected yet individually (AFAIK) because they are so far away, at very high redshifts, but theory tells us that they must have been enormous. Big stars burn their fuel at a fast rate, so they tend to have relatively short lifetimes. When they died they would have caused pressure waves in the interstellar medium and seeded it a little with heavier elements, both of those things made it easier for the next generation of stars (population II) to form. Because those population III stars lived so briefly they simply didn't have enough time to produce a lot of heavier elements, they mostly just converted some hydrogen to helium, although they probably made a bit of heavier stuff when they exploded. But we expect that they made some carbon, and that made a big difference to formation rates of population II stars.

Carbon is very important in star formation: it can radiate away energy in frequencies that hydrogen and helium are transparent to. This allows a collapsing gas cloud with carbon in it to shed excess heat far more effectively than a pure hydrogen + helium cloud can. Thus during the population II phase much smaller stars were able to form. And some of those stars could live long enough to synthesize a wider range of elements, enriching the interstellar medium even further. Thus population I stars like the Sun collapse from a cloud containing sufficient carbon to speed up the collapse process as well as other elements that are handy for planet formation.

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