View Full Version : Explaining virial equilibrium to nonprofessional astronomers

weltevredenkaroo

2013-Dec-07, 08:54 PM

Virial processes are key factors in star and galaxy cluster formation, yet go virtually unmentioned in even the most knowledgeable amateur or hobbyist forums. I'm doing an article re. IC 1613 for that readership and would like to describe virial equilibrium as 'the total of all energies exerting outward pressure on a body of mass balancing the sum of all energies exerting inward pressure from it.' I know this oversimplifies, yet I also need to get around the fear-of-physics and mathophobia of many readers. Suggestions for improvement welcomed.

No nonscientific dictionary I have lists the word 'virial', and even the Online Etymology Dictionary with 20-odd resources in several languages draws a blank. Wiki moves into 'canonical partition finctions' and 'Hamiltonian energy operators' within the first 2 paragraphs. This will not do for Cloudy Nights and Astromart devotees, yet discussing an object like a starburst dwarf irregular whose activity is readily seen by hobby astronomers requires more than 'I saw it, too'.

Thanks for your suggestions, Dana in S Africa

ngc3314

2013-Dec-07, 10:12 PM

As a start, how about:

The physics of statistical processes shows that when a group of objects is held together by their own mutual gravitational attraction, they approach a specific relation between their total kinetic and potential energy, known as the virial theorem -which is to say, a specified balance between their positions and velocities. This is a standard way to estimate the masses of star clusters and galaxies - of the positions, velocities, and total mass, if we know two we can often derive the third. The positions and velocities are more or less observable, except for projection effects from not knowing where the object is along the line of sight or its direction of motion, but if there are enough objects in a system an average is useful. An early application of the virial theorem in astronomy was Fritz Zwicky's 1937 discovery (http://articles.adsabs.harvard.edu/cgi-bin/nph-iarticle_query?1937ApJ....86..217Z&data_type=PDF_H IGH&whole_paper=YES&type=PRINTER&filetype=.pdf[/url) that the Coma cluster of galaxies contains vastly more mass then we can account for with its visible contents.

(I may have overshot the target level but maybe that's useful)

chornedsnorkack

2013-Dec-07, 10:33 PM

Maybe look at it this way:

A test body on a circular orbit around a point mass has at all times kinetic energy equal to one half the potential energy;

One test body on an elliptical orbit does not have such an equality. As an end member, note that a body in periapse of a very elliptic orbit has kinetic energy almost equal to potential energy.

Is it easy to demonstrate by Kepler laws that the time average kinetic energy is exactly half of the time average of potential energy?

(That takes care of point masses only... not general mas distributions).

Jeff Root

2013-Dec-08, 10:48 PM

As a general principle, I think you can completely avoid both

jargon and math at the cost of a slightly longer, more detailed

explanation. Just introduce simple ideas one at a time, in a

logical order, and link them together with a clear narrative,

and you should be fine.

-- Jeff, in Minneapolis

Ken G

2013-Dec-08, 11:23 PM

This is how I would put it. In any bound system (gravitational or otherwise, but in astronomy we care about gravity so you could specify that if you like), there are two ways that the key concept of energy appears. One is the concept of kinetic energy, which is the energy associated with motion. Given Galileo's law of inertia, an energy of motion should be associated with an energy that is trying to make a system fly apart. The second is the concept of potential energy, which is the energy associated with the attractive forces creating the binding. Given that the force is attractive, this energy should be associated with a tendency for the system to fall into itself. So then ask, why doesn't the first make the system fly apart, and why doesn't the second make it fall into itself? Of course, because they are both trying to happen, neither does, and instead a balance between the energies is acheived. In detail, the potential energy works out to be twice the kinetic, on average, but that's for the mathematically inclined, it's probably enough to say they reach a kind of balance where they are both important. Then you can point out that for a system of a given total mass, the kinetic energy relates to the speed of the particles, whereas the potential energy relates to the size of the system, so a virialized system has a given connection between the particle speed and the size of the system, but this connection is different for different masses so that brings in what ngc3314 was saying.

As a final kicker, you can point out that the fact that the potential energy is actually a bit larger in magnitude (though negative) than the kinetic energy, this means that when the system loses total energy (via loss of heat to its environment, by shining), it has to go to a state of smaller size and larger kinetic energy. So you have the counterintuitive situation that gravitating systems shrink and go to higher temperature when they lose heat! That's how you make a star, for example.

George

2013-Dec-09, 03:48 AM

As a final kicker, you can point out that the fact that the potential energy is actually a bit larger in magnitude (though negative) than the kinetic energy, this means that when the system loses total energy (via loss of heat to its environment, by shining), it has to go to a state of smaller size and larger kinetic energy. This, I assume, would make it no longer virialized (ie not 2 to 1 in energy ratio)? Or is it just a temporary loss of "virialness" if a planet, for instance, suddenly loses PE and gains KE due to impact or something, but then finds orbital stability?

So you have the counterintuitive situation that gravitating systems shrink and go to higher temperature when they lose heat! That's how you make a star, for example. This seems to support my question. If so, and in the context of the energy ratio change, is there some virial theory basics for cloud collapses and planetary formation systems?

Ken G

2013-Dec-09, 06:16 AM

This, I assume, would make it no longer virialized (ie not 2 to 1 in energy ratio)? Or is it just a temporary loss of "virialness" if a planet, for instance, suddenly loses PE and gains KE due to impact or something, but then finds orbital stability?The latter-- if you imagine taking out a small but finite amount of heat, you get a momentary loss of virialization, because you'll lose KE with no change to the PE. But rapidly it will contract, the PE will rise (in magnitude), the KE will bump up a little, and it will reach a new equilibrium when it returns to the 2-to-1 ratio. But if you take the limit as the "bump" is smaller and smaller, eventually it is essentially continuous energy changes, and virialization is maintained throughout, which is normally how we think of it. It is a little tricky that way-- you think you need the KE to drop to get contraction, but it's not actually true, if heat is lost continuously, then KE will rise continuously as it continuously contracts.

If so, and in the context of the energy ratio change, is there some virial theory basics for cloud collapses and planetary formation systems?

They all follow the same 2-to-1 virial ratio, all you need is an inverse-square force, though the ratio gets closer to 1-to-1 when the motion gets relativistic.

George

2013-Dec-09, 04:59 PM

The latter-- if you imagine taking out a small but finite amount of heat, you get a momentary loss of virialization, because you'll lose KE with no change to the PE. But rapidly it will contract, the PE will rise (in magnitude), the KE will bump up a little, and it will reach a new equilibrium when it returns to the 2-to-1 ratio. What do you mean by the PE increasing in magnitude during contraction?

Ken G

2013-Dec-09, 05:56 PM

It gets larger negatively. The "magnitude" is the absolute value.

George

2013-Dec-10, 05:57 PM

Thanks! It is interesting, as mentioned by ngc3314 above, that the Virial theorem was used by Zwicky to open the first door to dark matter. I think the theorem was fresh out of the box when Fritz put it to good work, but I am going on faint memory. I knew it was powerful but it seems much simpler than I thought it might be, at least in principle.

StupendousMan

2013-Dec-11, 12:09 AM

I think the theorem was fresh out of the box when Fritz put it to good work, but I am going on faint memory.

The ADS system allows one to search the titles, abstracts, and even full texts of published papers in the astronomical literature for any phrase. I found the first use of the phrase "virial theorem" in a paper by Rosseland in 1924

http://labs.adsabs.harvard.edu/adsabs/abs/1924MNRAS..84..720R/

but the word "virial" itself -- either alone, "the virial", or with other words, "virial of Clausius", "virial equation", -- it appears around the turn of the twentieth century. For example,

http://labs.adsabs.harvard.edu/adsabs/abs/1898KNAB....1..138V/

http://labs.adsabs.harvard.edu/adsabs/abs/1900KNAB....3..515V/

Clausius published his work outlining the relationship between potential and kinetic energy in 1870:

Clausius, RJE (1870). "On a Mechanical Theorem Applicable to Heat". Philosophical Magazine, Ser. 4 40: 122–127.

So when Fritz Zwicky used it to suggest that something fishy was going on in galaxy clusters, in 1936, he was applying an established tool.

George

2013-Dec-14, 07:52 PM

Thanks, SM. I should someday get a little more free time to follow-up on this work. I won't be shocked if it ties, somehow, into Eddington's work.

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