# The Virial theorem and Cosmology

1. May 24, 2006

### oldman

Whenever a diffuse gravitating system condenses into a stable and more compact object, energy must be removed from it.

This is a consequence of the Virial theorem, which mandates that in a stable system of gravitating particles there must be a proportional balance between the magnitudes of their kinetic and potential gravitational energies. The former must be equal to half the latter.

For example, as a stable, hot, compact proto-star forms from a cold, diffuse cloud of gas and dust, energy conservation ensures that gravitational potential energy is converted into an equal amount of other forms of energy. The condensing gas cloud heats up and radiates energy. In this process the virial theorem mandates that the internal kinetic energy added to the gas be only half the converted potential energy, if the proto-star is to form quasi-statically and not to oscillate. The balance of half the converted potential energy must be dissipated from the condensing star as radiant energy during the normal process of star formation.

In short, the virial theorem tells the star to shine, as it were; shine out into interstellar or ultimately intergalactic space, where plenty of room for emitted photons has been cleared by earlier condensations.

The relevance of the virial theorem to cosmology is the following. The real universe is lumpy. It is composed of a hierarchy of stable (on human time scales) compact astronomical structures, ranging from gas clouds, planets and stars through globular clusters and galaxies to clusters of galaxies. All these structures are thought to have formed by the gravitational condensation of more diffuse arrangements of matter.

Ultimately, all the radiation emitted by condensing matter over the estimated 13.8-billion-year life of the universe has been derived gravitational potential energy. Its emission has been mandated by the virial theorem. This radiation cannot have escaped from the universe in the way starlight escapes from a galaxy; after all by definition the universe includes all that now exists. This sea of trapped radiation may have been red-shifted as the universe has evolved, but it cannot have vanished.

My question is: where is this radiation? And does it contribute to the background radiation?

2. May 24, 2006

### kmarinas86

In space. Yes. A very small fraction of the photons from stars and galaxies reach the earth. When resolved, they appear as stars and galaxies in telescope images.

3. May 24, 2006

### marcus

interesting to think that by means of infrared astronony we ought to be able to and probably can OBSERVE the event of a cloud contracting and condensing

that is, if the condensation event is in our lightcone we can SEE a little warm patch that is a few kelvin warmer than background and even if there are no optical-visible stars there yet, and no hotter stuff, we can see it contract (because of the radiated infrared warmth)

so this huge history that we witness of gravitational potential being converted to light covers a wide spectrum

In a way, oldman's question sounds like a question about the power spectrum---like do the numbers to show why all the light coming to us from stars and other condensed/condensing objects is NEGLIGIBLE power compared with the CMB

also perhaps intuitively there is a kind of DARK NIGHT SKY question lurking here. why does not the light from past event stick around and accumulate and eventually cook us? what happens to the visible or invisible light from a star or condensing cloud after it has whizzed on by us? where does it go? Shouldnt it always be part of an accumulating background of old light?

the actual physical expansion of space (with its finite history and finite horizon) is an elegant solution to the embarrassment of an otherwise hot sky---it almost could have been INVENTED as a kind of airconditioning to keep us comfortable (naaaah! boo anthropy booooooo! )

kidding aside===I like kmarinas clear answer. the invisible infra from contracting clouds is JUST LIKE ANY other kind of starlight (and just as negligible compared with CMB)

Last edited: May 24, 2006
4. May 24, 2006

### SpaceTiger

Staff Emeritus
For those who aren't familiar, the virial theorem for a gravitating system says

$$2K+U=0$$

where K is the kinetic energy and U is the potential energy. For a spherically symmetric system (like a cloud or a cluster of stars), this can be rewritten as

$$E=K+U=\frac{U}{2}\propto-\frac{M^2}{R}$$

As the radius gets smaller, the energy goes down. Since energy must be conserved overall, the excess must go somewhere. In the case of a cloud, it usually goes into electromagnetic radiation. In the case of a star cluster, it's usually in the form of expelled stars.

In order for this to hold, however, the object must be virialized -- the relation doesn't hold for all objects. A galaxy cluster usually is virialized soon after it collapses, but not before or during. Beforehand, it is just an overdensity in a nearly uniform field of matter. At that stage, it is not self-gravitating and it's influenced by the Hubble flow, so the virial theorem does not apply.

Last edited: May 24, 2006
5. May 24, 2006

### Gokul43201

Staff Emeritus
Just to clarify, those are time averages (of K and U above, over all time), and not instantaneous values.

6. May 24, 2006

### SpaceTiger

Staff Emeritus
Technically, we'd want to average over all time in an ideal steady-state system with the same globally averaged properties (not the actual system), but that's far more precision than is needed for astronomical purposes. For this analysis, I'd say we can treat them as instantaneous because the statistical fluctuations over systems with so many particles/stars are tiny. Besides, astronomical observations can only sample the system at an instant in its dynamical history, so this is also how it's done in practice (in astronomy). You won't get an exact result, but then there are a lot of other more significant factors that contribute to the inexactness of such a computation (for example, the assumed density profile used to compute the potential energy).

7. May 24, 2006

### Gokul43201

Staff Emeritus
I know precious little about Cosmology, so I probably shouldn't have stepped in here in the first place. But the virial theorem is more general than just its application to bound gravitating systems.

And often one finds theoretical contructs applied outside their regimes of applicability. I thought the question in the OP was eventually quite independent of the virial theorem and so, could have done without it, but I may just be grossly wrong there. In this particular case, I defer to your judgement on how good an approximation this is. It seems not at all implausible that for a system with a large number of bodies, a time average over a small period will work well.

(Now to let my cosmological ignorance out in full view) I see no reason why the OP's question can not simply be asked as "where does starlight go ?". But it appears that the OP specifically differentiates between starlight and radiation resulting from gravitational collapse.

...or maybe I'm just misinterpreting that statement.

8. May 24, 2006

### SpaceTiger

Staff Emeritus
Quite the contrary, I'd love to see more of you (and other experts from outside astronomy). It never hurts to have an outside perspective.

Yeah, that's why I went out of my way to specify only a gravitating system. In general, there will be other terms in the equation that account for the various energy sinks in the system in question. For example, protostars may have dynamically important magnetic fields.

I'm sure there are other ways of approaching the problem, but the OP's approach looks alright to me. The virial theorem is very frequently applied to problems such as this. I'm curious, how would you approach the problem?

The reason I didn't interpret it that way is that, in a galaxy cluster, emitted starlight is unrelated to the collapse of the cluster. A star is its own self-gravitating system, so has its own virial eqilibrium. There is a large quantity of intracluster gas that is emitting X-rays and is dynamically coupled to the rest of the cluster, but I'm pretty sure this radiation has little impact on the size of the cluster because the majority of the cluster's mass is dark and not radiating.

Of course, that depends somewhat on how you define the size of the cluster...

9. May 25, 2006

### oldman

Well .... in space! I'm not surprised.

And when they're not resolved, and originated from the surplus kinetic energy of diffuse matter condensing into galaxies 13 or 14 billion years ago? I agree that we don't receive many such photons, but then the CBR looks pretty dim as well.

10. May 25, 2006

### oldman

.

Yes, it is such a question, but I'd like to have an estimate of those numbers --- I have no idea of them myself.

Yes. You are clearly familiar with Olbers' paradox and its resolution by the red-shifting of the remote universe. But I'd still like to know about the numbers!

11. May 25, 2006

### marcus

I like it that you ask it this way. One can do a back-of-envelope (if one is not too lazy) and also I have seen a detailed energy budget somewhere that gives the density of energy in space broken down into categories.

I estimted the energy density of the CMB a few years back when I was not so lazy, maybe I can do it again. One just uses the Stef-Boltz fourth power law. I remember being amazed (impressed anyway) at how much energy there is in the CMB compared with other radiation.

It is an honorable question to ask. If Nick or hellfire or someone does not give you those numbers momentarily, I will try to estimate at least one of them.

12. May 25, 2006

### marcus

Oh, for lazy people there is an INTUITION about the energy density of the CMB

Imagine that you are inside a large tungsten sphere which is at 3000 kelvin (this is the CMB at the time of recombination)

since last scatter, since recombination time, the spherical volume has expanded linear by a factor of 1100 (this is the CMB redshift), but still with the same number of photons inside

so picture the volume increasing by 11003 and the same number of photons, but each now has only 1/1100 as much energy now because of stretching

Now the energy density is less by a factor of 11004, than at the time of last scattering.

So, since the CMB began, its energy density has only decreased BY A MERE FACTOR OF A TRILLION. (I mean 1012)

Now think about the radiation from the ordinary stars and galaxies. They are only tiny isolated points of light, so it is intuitively hard for them to compete with this solid glowing wall of hot tungsten, even if you throw in a factor of a trillion handicap. There is a qualitative difference.

I sympathise with your wanting actual numbers, however. If no one else volunteers I will try to supply you with part of what you want.

Last edited: May 25, 2006
13. May 25, 2006

### marcus

before doing back of envelope it is good to cheat and make sure of answer ahead of time so I looked at Wikipedia
http://en.wikipedia.org/wiki/Cosmic_microwave_background

and it says that most of the RADIATION energy in the universe is the CMB

and if you average out the density of energy in all forms including matter, then the CMB (says Wiki) represents the fraction 5E-5 of the total energy density. 5x10-5

and the same Wiki article also gives the Stef-Boltz formula for energy density (different from the power density on a surface formula).

Comically enough it looks like i will have to go buy a battery for my calculator before I get around to this (the calculator is my envelope).
Maybe i will just leave it for more macho people to do, like hellfire

I remember one time i calculated that the energy density the universe was 0.8 joules per cubic kilometer-----or 0.83, but roughly 0.8. (just to be extra clear and all, that includes dark and bright matter and also dark energy and starlight etc. the whole schmeer)

so multiply by 5E-5 and you get 40 microjoules per cubic klick.
that would be the CMB by itself

So my guess is that if anybody does get their calculator out and does a Stef-Boltz with the temperature 2.725 kelvin, that they will come up with something roughly equivalent to

40 microjoules per cubic kilometer

prove me wrong anybody?

Last edited: May 25, 2006
14. May 25, 2006

### SpaceTiger

Staff Emeritus
In the Milky Way, the CMB and starlight have comparable energy densities (several tenths of an eV per cubic centimeter), but on average in the universe and well outside a galaxy, the CMB will dominate over starlight by many orders of magnitude. I'm not sure exactly how many off the top of my head.

15. May 25, 2006

### SpaceTiger

Staff Emeritus
Just as with galaxy clusters, galaxies did not begin as virialized objects, so the same arguments do not necessarily apply. One could, in theory, have an overdensity collapse into a galaxy without forming stars -- their energy loss is unrelated to the collapse of the larger system.

Stars themselves are in virial equilibrium, but most of their light is drawn from nuclear fusion at the core, not gravitational collapse (except in the pre-protostar phase). Your arguments for a collapsing virialized object are correct under certain conditions, but most of the light in the universe is drawn from other sources.

My apologies if I'm misinterpreting your question, but are we clear now that the "radiation arising from condensing matter" is not the same as starlight? In other words, the question "where has the radiation from condensing matter gone?" is not the same as the question, "where has all the starlight gone?"

16. May 26, 2006

### oldman

Curdling the universe

Yes indeed. I’m afraid that in my original post I confused folk by introducing an example of the virial theorem in action, namely the emission of radiation from a proto-star that allows it to condense. I wish I hadn’t done this. I was also stupid to tie my argument so closely to the
virial theorem. It would have been better to rely only on the opening general statement I made:

“ Whenever a diffuse gravitating system condenses into a stable and more compact object, energy must be removed from it.”

which is true for all sorts of situations where the virial theorem doesn't apply that I now won’t dare to give examples of.

It is this energy which I was asking about, not starlight, which of course is mostly derived from nuclear reactions. Some of the surplus energy dissipated to make the universe lumpy must have ended up as radiation, and I was asking if this contributes perceptibly to the CMB. I
accept that some of it must be the kinetic energy of matter, such as jets emitted from collapsing objects or of stars expelled from a cluster, as you mentioned.

But the end product of dissipation is so often radiant energy.

Marcus in his posts #11,#12 and #13 above has kindly estimated the energy in the CMB for me, and I have followed the link he gave to the Wikipedia... Thanks, Marcus. I hope you found batteries for the calculator.

Would it be possible to make a similarly simple estimate of the energy dissipated to curdle the universe, as it were? The latent heat of curdling?

17. May 26, 2006

### marcus

compliments on excellent use of the verb "to curdle"
sort of thing that keeps English alive as language

18. May 26, 2006

### SpaceTiger

Staff Emeritus
Let's just be clear -- you weren't stupid. Your questions (particularly this and the cluster question) are among the most sophisticated that I've seen in a while. Your understanding of the virial theorem was sound, you just didn't always know where to apply it. I'm glad you brought it up, because it gives an explanation for why you would expect radiation from a collapsing object.

Not necessarily. If the system is not in virial equilibrium, then the decrease in potential energy due to collapse can be translated into kinetic energy of its constituents. In fact, the early stages of star, galaxy and galaxy cluster formation have been hypothesized to proceed in just this way -- free fall of an initially unbound cloud.

Refer to my comment about X-ray emission from galaxy clusters. Although this isn't the dominant driver of the collapse, energy losses from hot gas have contributed to condensing galaxies and clusters of galaxies. In galaxies, the overall mass distribution is not changed a great deal by radiation, but the distribution of the baryonic component (that is, the stars and gas that we see) is much more centrally concentrated than the dark matter, partly due to this radiation. The radiation comes out in many forms, including emission lines, synchrotron radiation, free-free radiation...

19. May 27, 2006

### Chronos

The frightfully difficult part of addressing questions like this is how you manage to conserve energy. I am absolutely convinced this is necessary to preserve causality [a thermodynamics thing]. I cringe every time I see a new study that 'relaxes' this constraint.

20. May 29, 2006

### hellfire

My contribution to the requested numbers.

The number density of photons of a black body radiation:

$$N_{\gamma} = 1.2 \cdot 16 \pi \left( \frac{k_B T}{hc} \right)^3$$

With $T = 2.7 K$ for the cosmic microwave background, this is:

$$N_{\gamma}^{CMB} = 400 cm^{-3}$$

Assume that all stars are equal to the sun and they emit about $3.8 \cdot 10^{26} W$. Assuming a temperature of 5700 K, and according to Wien's law, the wavelength of maximum intensity is:

$$\frac{0.3} {5700} = 5.26 \cdot 10^{-5} cm$$

Assume that this wavelenght contributes to 100% of the luminosity. The energy of such a single photon is:

$$E = \frac{h c} {\lambda} = 1.6 \cdot 10^{-18} J$$

The number of photons emitted per second is then:

$$N_{\gamma}^{\odot} = \frac{3.8 \cdot 10^{26} W} {1.6 \cdot 10^{-18} J} = 2.3 \cdot 10^{44} s^{-1}$$

The average number of stars per cc in the universe (don't ask me where I got that number...):

$$N_{stars} = 10^{-63} cm^{-3}$$

Assume that these stars emit light during the whole history of the universe (13.7 Gy). The total number of photons per cc is:

$$N_{\gamma}^{stars} = 4.25 \cdot 10^{17} s \cdot 10^{-63} cm^{-3} \cdot 2.3 \cdot 10^{44} s^{-1} = 0.1 cm^{-3}$$

We see that the number of photons provided by the stars is negligible compared to the number of photons in the CMB:

$$\frac{N_{\gamma}^{stars}}{N_{\gamma}^{CMB}} = 0.00025$$

I don't know how to estimate the radiation from gas such as the intra-cluster gas or emission nebulae and I have no idea whether this would change the result.

Last edited: May 29, 2006