Reaching Equilibrium: Estimating Time to Balance

[Ely Rajo]

When can I say that the system is in equilibrium?
How can I estimate how quickly it takes to the system to reach equilibrium?
Thanks.

 

[Astronuc]

That depends on how one defines equilibrium, which often infers no change, i.e the system is more or less "steady-state".

Equilibrium for a star would usually infer a relatively constant rate of energy output, which then also infers a constant rate of fuel consumption (fusion).

Also, if there were not significant fluctuations in size or luminosity of a star, this would also be a form of equilibrium.

 

[Soul Surfer]

The title of your topic and its context suggests that you are i nterested in the carbon nitrogen oxygen cycle of helium nucleosysnthesis in stars. I am not an expert in this area but I think that I can give some general pointers taken from firs principles.

In any situation for an equilibrium to exist there must be an element of negative feedback to exist. The main areas where this has been studied is control theory and I suggest that you look this up to get some general ideas.

The operation of the nucleosynthesis chain generates heat in the cetre of a star and the reaction goes faster and generates more heat as it gets hotter but this will also increase the pressure and cause the star to expand and cool down a bit and so reduce the speed of the reaction thus causing a negative feedback.

If I remember right the reaction speed varies extremely quickly once the temperature threshold for its initiation has been reached. so the extra heat could possibly run away before the negative feedback and cooling happens.

Your interest was in how long it takes to reach an equilibrium again you will have to look up textbooks on this but radiative transfer of energy in stars is quite slow and convective transfer by motion of the less dense heated plasma can be much quicker.

The fact that the outputs of stars on the main sequence are quite stable suggests that a stable equilibrium has been achived but if I remember right there has been a lot of arguments about whether this is truly stable or pusing itself in a limit cycle too quickly for the changes to be observed on the outside of the star

lot of the time when a star is in the main sequence necleosynthesis energy generation is going on in the core

 

[SpaceTiger]

If I remember right the reaction speed varies extremely quickly once the temperature threshold for its initiation has been reached.

Yeah, in the regimes of interest, the CNO cycle goes approximately as T¹⁶.

so the extra heat could possibly run away before the negative feedback and cooling happens.

To my knowledge, this never happens unless the core is supported by degeneracy pressure, which does not respond (for the most part) to rises in temperature. Changes in pressure will be communicated at the sound speed, which should be much quicker than the fusion timescale in the core. This means that, if the pressure responds to changes in temperature, equilibrium will be quickly reached.

Your interest was in how long it takes to reach an equilibrium again you will have to look up textbooks on this but radiative transfer of energy in stars is quite slow and convective transfer by motion of the less dense heated plasma can be much quicker.

Sometimes, yes. Our sun, for example, has both a convective and a radiative zone. There is more discussion here:

https://www.physicsforums.com/showthread.php?p=745266"

lot of the time when a star is in the main sequence necleosynthesis energy generation is going on in the core

Pretty much by definition, the main sequence is usually reserved for actively hydrogen-burning stars. There is also a "helium main sequence", but generic references to the main sequence are usually to the hydrogen-burning one.

 

[SpaceTiger]

When can I say that the system is in equilibrium?
How can I estimate how quickly it takes to the system to reach equilibrium?

Which equilibrium are you referring to, exactly? Local thermodynamic equilibrium in the core? Pressure equilibrium?

 

[Labguy]

When can I say that the system is in equilibrium?
How can I estimate how quickly it takes to the system to reach equilibrium?
Thanks.

and:

Which equilibrium are you referring to, exactly? Local thermodynamic equilibrium in the core? Pressure equilibrium?

Actually, I believe the answer to both above is that there is never an "equilibrium". the OP mentioned CNO cycle (Stars greater than ~1.1 M_s) but for H fusion the Proton-Proton chain would be the same in as far as H fusion/equilibrium is concerned so we can use our Sun as an example. There will always be changes in both the core and envelope layers as evidenced by the observed photosphere pulsation and magnetic and acoustic variations throughout the whole star. A star using predominately the P-P chain might be less active as far as the oscillations are concerned because of the lower T=P=lifespans, but oscillations are oscillations. Predominant because there is always at least a little bit of both P-P and CNO in any star, they aren't mutually exclusive.

See http://solarphysics.livingreviews.org/open?pubNo=lrsp-2005-1&page=articlese1.html and its sub-links for decent explanations on how much the Sun, and likely most all stars, fluctuate even when on an apparent stable main sequence. The helioseismology sections are particularly interesting. After He fusion starts, nothing is even close to equilibrium/stable.

 

[neurocomp2003]

i eblieve the equilbrium he's looking for is across one of the radial cores ...that maintains the pressure equilbrium that spacetiger mentioned. Either its from the core expanding or before changing to another burning.

 

[Labguy]

i eblieve the equilbrium he's looking for is across one of the radial cores ...that maintains the pressure equilbrium that spacetiger mentioned. Either its from the core expanding or before changing to another burning.

That may be, but no core ever actually reaches an actual equilibrium, it changes continuously. ST's mention was pressure equilibrium with a question mark, not as an answer.

 

[SpaceTiger]

and:Actually, I believe the answer to both above is that there is never an "equilibrium".

Well, that's certainly the most literal answer, but it may or may not be the most useful. If we want to pick nits, we could say that nothing in the universe is in true equilibrium because there are no closed systems.

Yes, it's certainly true that there are oscillations, even in stars like the sun, but they are usually of very low amplitude. Think about how sound waves are constantly moving through the air outside my apartment, yet I still find it useful to hang a thermometer by my window. Clearly, we find local thermodynamic equilibrium to be a useful approximation. What about pressure equilibrium? Should scuba divers avoid diving at all for fear that the disturbances in the water will cause the pressure to fluctuate wildly? Certainly not.

If we agree that equilibrium can be a useful concept, it's then a matter of determining how much precision the OP really wanted and over what length and time scales. Depending on their response, it may turn out that they shouldn't approximate it as being in equilibrium, in which case your answer could be the best.

 

[Labguy]

If we agree that equilibrium can be a useful concept, it's then a matter of determining how much precision the OP really wanted and over what length and time scales. Depending on their response, it may turn out that they shouldn't approximate it as being in equilibrium, in which case your answer could be the best.

In this case a star's imbalances are on a much higher scale than water pressure "fluctuating wildly" from minor disturbances. Too many people are given information that leads them to think that stars/evolution is an orderly process, when the variables would actually take volumns of books to describe.

On some posts I give a "generic", good-enough answer and someone comes along and pounces in with great detail. Next post I put up some detail and then it is criticized as not generic enough. Someone should poop or get off the pot.

A perfect example, ST, was your statement to someone a few weeks ago that:(approximate quote from memory)
"When a white dwarf accretes enough mass and passes the Chandrasekhar limit, it becomes a Type Ia supernova." I hope you know that this description was so broad and innacurate as to be essentially meaningless, much less misleading to any newcomer who might then go around quoting that as gospel because Space Tiger said so.

NOT!

If anyone would like, I'll post for about the fifth time what is really necessary for a Type Ia supernova, but that (1) can be found somewhere in a PF search and (2) is nit-picking detail...
Do the search.

 

[SpaceTiger]

In this case a star's imbalances are on a much higher scale than water pressure "fluctuating wildly" from minor disturbances. Too many people are given information that leads them to think that stars/evolution is an orderly process, when the variables would actually take volumns of books to describe.

I'm sorry, but I don't agree that one needs to read many volumes of books to have a grasp of stellar evolution. In fact, I think the fundamentals are very easily communicated in a few chapters. Helioseismology is, I believe, correctly treated as a side subject, good for probing the interiors of stars, but not particularly important for getting a qualitative understanding of stars like the sun.

Furthermore, the fluctuations we see on the surface of the sun are of very low amplitude and I've never seen them incorporated into a solar model. Could you please cite some examples of this?

On some posts I give a "generic", good-enough answer and someone comes along and pounces in with great detail. Next post I put up some detail and then it is criticized as not generic enough.

I haven't had any problem with your posts. Perhaps you're just being oversensitive. Just because someone decides to give more information doesn't mean that they're criticizing or correcting the information you provided.

A perfect example, ST, was your statement to someone a few weeks ago that:(approximate quote from memory)...

The proper quote is (please do the search in the future):

The best way to destroy a degenerate object is to give it mass until it passes the Chandrasekhar Limit. When you do this to a white dwarf, you get a Type Ia supernova.

I hope you know that this description was so broad and innacurate as to be essentially meaningless, much less misleading to any newcomer who might then go around quoting that as gospel because Space Tiger said so.

Again, I disagree that what I said was inaccurate. I think you'd be hard-pressed to find an astronomer who didn't think that accreting past the Chandrasekhar limit of a white dwarf would lead to a Type Ia supernova. There's serious debate about whether this is the only source of Type Ia's, but for the purpose of the post (found https://www.physicsforums.com/showpost.php?p=843864&postcount=24"), I hardly think it was necessary to get into that.

NOT!

Let's try to be civil here.

If anyone would like, I'll post for about the fifth time what is really necessary for a Type Ia supernova

If you know, you better alert the astronomical community, cause the rest of us are still pretty unsure about it.

 

[Labguy]

I'm sorry, but I don't agree that one needs to read many volumes of books to have a grasp of stellar evolution. In fact, I think the fundamentals are very easily communicated in a few chapters and I think this is the most important thing for people to understand. Helioseismology is, I believe, correctly treated as a side subject, good for probing the interiors of stars, but not particularly important for getting a qualitative understanding of stars like the sun.

Fundamentals maybe, but is that enough to correctly answer questions on PF? To me it isn't; someone always asks for more and then the "fundamental" guy has to search for a bunch of links. Bummer.

Furthermore, the fluctuations we see on the surface of the sun are of very low amplitude and I've never seen them incorporated into a solar model. Could you please cite some examples of this?.

Examples of what? How different cores act/react based on mass and chemical composition? Stellar pulsations? Main sequence and off-main sequence movement? End-life based on original masses? Radiative and convective zones? Why some stars have radiative cores and others have convective cores? Why PP chain and CNO cycle can, and do, occur at the same time? Stellar acoustics? What were the two different "limits" calculated by Chandrasekhar? Difference between Chandra's mass and Chandra's limit(s)? Significance of the 1.39 mass limit? Dozens of other differences? This doesn't lead me to believe that the "fundamentals are very easily communicated in a few chapters."

I haven't had any problem with your posts. Perhaps you're just being oversensitive. Just because someone decides to give more information doesn't mean that they're criticizing or correcting the information you provided.

More information doesn't usually bother me because any reader can just peruse the parts needed for his satisfaction. It is the oversimplifications that do, as in your statement about Type Ia supernovae below.

The proper quote is (please do the search in the future):

I already qualified that in my statement that it was approximate based on memory, no search. Let your fingers do the walking.

Again, I disagree that what I said was inaccurate. I think you'd be hard-pressed to find an astronomer who didn't think that accreting past the Chandrasekhar limit of a white dwarf would lead to a Type Ia supernova. There's serious debate about whether this is the only source of Type Ia's, but for the purpose of the post (found https://www.physicsforums.com/showpost.php?p=843864&postcount=24"), I hardly think it was necessary to get into that.)

Actually it was very inaccurate. Does 1.39 M_s ring a bell? Chandra's "limit" of 1.44 M_s doesn't apply. Carbon deflagration and detonation? Assymetric detonation? Specific chemical composition? S.E Woosley is considered about the foremost stellar physicist working today, and he (and friends) does a very nice job in explaining why so very few accreting white dwarf stars ever result in a Type Ia supernova. Check out some of his books/papers.

Let's try to be civil here.

I'm always civil; just sometimes hostile.

If you know, you better alert the astronomical community, cause the rest of us are still pretty unsure about it.

Again I would point out the work of Woosley and friends, in books and some on the internet.

If "the fundamentals are very easily communicated in a few chapters" I wouldn't have 26 folders with 224 files saved in "favorites" just on stellar evolution and another 9 folders and 80 files on stellar composition (chemical) and H-R diagrams. I can't post them all here. And after reading them all, I'm still not even close to being considered an expert on stellar evolution but am probably more versed on that particular subject than your above-average PF reader. I will often type a long post without having to run to internet links; just based on what little I can remember from what I have already read, yesterday and 10 years ago.

 

[SpaceTiger]

Fundamentals maybe, but is that enough to correctly answer questions on PF? To me it isn't; someone always asks for more and then the "fundamental" guy has to search for a bunch of links. Bummer.

You can answer things your way and I'll answer them mine. I disagree with you in almost every possible way on this issue, so it's unlikely that we'll reach a common ground.

Examples of what?

Examples of what you were claiming -- the pulsations in the sun causing problems for the approximation of hydrostatic equilibrium.

...This doesn't lead me to believe that the "fundamentals are very easily communicated in a few chapters."

One of the challenges in astronomy (or any science, for that matter), is developing an intuition for the fundamental concepts from the details. The list of detailed concepts in stellar astrophysics is surely much larger than you give there, but what are the ideas that drive them? When are they important? It's important to have an understanding for when something isn't relevant for the problem at hand.

Do you suppose that the OPs come here to be told to read a textbook? No, they almost always come here to get an understanding for the fundamental concepts that drive stellar astrophysics. Giving them a lecture on helioseismology in response to a question about the CNO cycle is probably not very helpful. It's not enough just to know a lot -- if you want to teach people, you need to condense it into bite-sized chunks that communicate the important ideas. If they want to learn more, they can look into it further. Providing links is an excellent thing to do and I certainly wouldn't discourage it. I would, however, discourage the layman from doing a blind Google search. There's an awful lot of nonsense on the internet.

Actually it was very inaccurate. Does 1.39 M_s ring a bell? Chandra's "limit" of 1.44 M_s doesn't apply.

It's still the Chandrasekhar mass/limit. The name of the limit doesn't change when the number changes (think about Hubble's constant).

Carbon deflagration and detonation? Assymetric detonation? Specific chemical composition?

All details that are interesting to the advanced reader, but incomprehensible to the novice. I'm not going to bother writing a textbook in every post, but you certainly shouldn't hesitate to add this information if you want. If it's correct, I certainly won't argue with you.

S.E Woosley is considered about the foremost stellar physicist working today, and he (and friends) does a very nice job in explaining why so very few accreting white dwarf stars ever result in a Type Ia supernova. Check out some of his books/papers.

If taken literally, your statement isn't in contradiction to what I said. If you mean that few that accrete past the Chandrasekhar limit explode in a Type Ia, then that's news to me. I saw no mention of it in the review article I read (found http://adsabs.harvard.edu/cgi-bin/bib_query?2000ARA%26A..38..191H"). Perhaps you can provide me with the link you're referring to.

I'm always civil; just sometimes hostile.

If you wish to confront me about this sort of thing, you should do so privately. Also, if you have corrections or additions, please place them in the thread in which they're appropriate and preferably at the time they were posted.

Again I would point out the work of Woosley and friends, in books and some on the internet.

Woosley is a theorist and, like all theorists, he has his own pet theories and models. The precise mechanisms for Type Ia supernovae are still being debated, as is explained in the review article I linked. Woosley is a very smart guy, no doubt, and I wouldn't be surprised if he has a lot of things right, but the theoretical picture is far from settled.

If "the fundamentals are very easily communicated in a few chapters" I wouldn't have 26 folders with 224 files saved in "favorites" just on stellar evolution and another 9 folders and 80 files on stellar composition (chemical) and H-R diagrams.

You and I seem to have very different ideas about what is "fundamental".

And after reading them all, I'm still not even close to being considered an expert on stellar evolution but am probably more versed on that particular subject than your above-average PF reader.

I figured this went without saying, but what I wonder is why you demand that all PFers obtain the level of knowledge that you have?

 

[Labguy]

That's too long for me to answer at the speed I type (read ss-lll-oooo-www). All I can add is that, yes, we disagree on this issue; the issue being equilibrium. The OP didn't state thermal, pressure, radiative or any specific. Other posters asked about that starting with Astronuc and your first post. My first post was straightforward and mainly pointed out that CNO or PP, either method causes the H2 fusion and that stars are always fluctuating/pulsating/changing. Nothing too detailed or textbook there. I never mentioned "the approximation of hydrostatic equilibrium."

I totally agree that readers don't come here to read textbooks and that bite-sized comments are sufficient unless innacurate. Providing links is also excellent and many of my posts are simply that; easy enough to find just by looking back at a few of them.

However, when someone mentions the "Chandrasekhar Limit" almost everyone thinks about the famous 1.44 M_solar and if the number needed for an answer is something different I think that should be noted. In 1932 Chandra calculated two different limits. 1.44 M_solar for stars with a (primarily) He core and 1.76 M_solar for stars with an iron core. (From Supernovae by Paul and Leslie Murdin, page 120, Cambridge University Press). Anytiing between 1.76 and 1.44 depends on the chemical composition of the core and is usually referred to as a Chandrasekhar Mass, not limit. Even your link provided above states 1.39 mass, not limit. These differences are necessary, to me, when posting about specific differences such as the Type Ia supernova. On PF, I don't think that this is too complicated to post and certainly not "incomprehensible to the novice". We should note the differences for any reader and I don't see that as too complicated. Also, I consider it basic when referenced on that particular subject. Otherwise, they go away with a misconception.

I looked through a bunch of my bookmarks a while ago but not all X-hundred of them. Didn't find the specific I wanted from Woosley but I did notice that almost every one of those on supernovae referenced Woosley several times. I can look more if necessary but would rather not. Woosley provides numerous details in chapter 8 of the book Supernovae, edited by A.G. Petschek, pages 182-211, Springer-Verlag Publishing. Sorry to reference books and not links, but that is all I have at hand at the moment. There are many parameters that must be met for a white dwarf to go Type Ia bang. The most critical are that the remaining core is of an ~specific combination of H and O and that a smaller Fe core is needed for the carbon burning (deflagration) to become a stellar detonation, i.e. Type Ia crowd pleaser. Other white dwarfs that pass the M_Chand mass without becoming a supernova are those without the necessary chemical composition where (1) they accrete matter and flare as repeating Nova (not super) and the well known collapse to a neutron star without any bang. There are others.

I don't consider this as "confronting" you; its just a way of saying that when a question can be answered by (generic) basics, that's Ok. But, when specifics are needed for a decent explanation they should be provided. Sending that to you in private wouldn't ever let the reader see what specifics are, or aren't, being discussed. And because of that, I really don't think that I "demand that all PFers obtain the level of knowledge" that is anything beyond enough to correctly answer the question at hand. Give them less and they'll go away confused; give too much and they'll go away confused again.

That's all I'll do on this subject.

EDIT: You have no idea how long that took.

 

[SpaceTiger]

I totally agree that readers don't come here to read textbooks and that bite-sized comments are sufficient unless innacurate.

If I say something that's completely inaccurate, you should of course speak up, because I assure you I didn't intend to. If I say something that's inaccurate only in very rare circumstances or to very high precision, then it probably was intentional. I don't think those details are important to the casual reader unless they ask for them. If you want to add them, however, I certainly won't complain. On the things we're discussing, you haven't yet convinced me that I said anything inaccurate.

However, when someone mentions the "Chandrasekhar Limit" almost everyone thinks about the famous 1.44 M_solar and if the number needed for an answer is something different I think that should be noted.

As far as I know, there's no distinction between the terms "Chandrasekhar Mass" and "Chandrasekhar Limit". The number associated with it depends, as you've said, on the circumstances, but it still means the same thing -- the maximum mass for that degenerate object. The fact that I didn't specify a number makes what I said more true, not less.

On PF, I don't think that this is too complicated to post and certainly not "incomprehensible to the novice". We should note the differences for any reader and I don't see that as too complicated.

It depends a great deal on the sophistication of the questioner and astrophysical importance of the complication. There are no general rules for this sort of thing, you just have to make a judgement for each circumstance. You can make yours and I'll make mine, nothing you've said here has convinced me to alter my posting methods.

EDIT: You have no idea how long that took.

I don't feel sorry for you at all. If you didn't want to spend a lot of time on this, you shouldn't have made an issue of it to start with. I did a lot of reading on Type Ia supernovae last night and, although I learned a number of new things, nothing suggested that what I said was inaccurate.

 

[Labguy]

I did a lot of reading on Type Ia supernovae last night and, although I learned a number of new things, nothing suggested that what I said was inaccurate.

On this last point only, and my last point only, is that "a white dwarf star that passes the Chandrasekhar limit will become a Type Ia supernova" (approximation quote) is not correct and that Ia's are rare. This is only because so many specific conditions/properties of the WD have to be so precise that only a very few >mass_Chandra go Type Ia.

One easy link, of many, describes the specifics of DDT and Type Ia conditions. It only mentions Woosley 69 times in 44 pages:

http://astronomy.sussex.ac.uk/~romer/Distant/annualreviews/2000/-annurev.astro.38.1.191.pdf

Boring for PF readers and "too specific" but it contains:

Only the prompt detonation mechanism is agreed to be inconsistent with SN Ia spectra, as it fails to produce sufficient amounts of intermediate mass elements (Arnett 1969, Arnett et al 1971). This apparently slow progress is essentially a consequence of the overwhelming complexity of turbulent flame physics and deflagration-detonation transitions (DDTs) (Williams 1985, Zeldovich et al 1985) that makes first-principle predictions based on Mchan explosion models nearly impossible. The close analogy with thin chemical premixed flames has been exploited to develop a conceptual framework that covers all scales from the white dwarf radius to the microscopic flame thickness and dissipation scales (Khokhlov 1995, Niemeyer & Woosley 1997).

Laminar thermonuclear carbon and oxygen flames at high to intermediate densities were investigated by Buchler et al (1980), Ivanova et al (1982), and Woosley & Weaver (1986b), and, using a variety of different techniques and nuclear networks, by Timmes & Woosley (1992).

The ratio of the laminar flame speed and the turbulent velocity on the
scale of the flame thickness, K D Sl=v._/, plays an important role: If K _ 1, the laminar flame structure is nearly unaffected by turbulent fluctuations. Inserting the results of Timmes & Woosley (1992) for Sl and _ as functions of density, and using a typical turbulence velocity v.106cm/_107 cm s−1, the transition from flamelet to distributed burning can be shown to occur at a density of _dis _ 107 g cm−3 (Niemeyer & Kerstein 1997).

Given no time to expand prior to being burned, the CCO material in this scenario is transformed almost completely into iron-peak nuclei and thus fails to produce significant amounts of intermediate mass elements, in contradiction to observations (Filippenko 1997a,b). It is for this reason that prompt detonations are generally considered ruled out as viable candidates for the SN Ia explosion mechanism. In addition to the empirical evidence, the ignition of a detonation in the highdensity medium of the white dwarf core was argued to be an unlikely event. In spite of the smallness of the critical mass for detonation at _ _ 2 _ 109 g cm−3 (Niemeyer & Woosley 1997, Khokhlov et al 1997) and the correspondingly large number of critical volumes in the core (_1018), the stringent uniformity condition for the temperature gradient of the runaway region (Blinnikov & Khokhlov 1986, 1987) was shown to be violated even by the minute amounts of heat dissipated by convective motions (Niemeyer & Woosley 1997).

A different argument against the occurrence of a prompt detonation in CCO white dwarf cores was given by Kriminski et al (1998), who found that CCO detonations may be subject to selfquenching at high material densities ( _ > 2_107 g cm−3) (see also Imshennik et al 1999).

One of the most successful examples, model W7 of Nomoto et al (1984), clearly demonstrates the excellent agreement of “fast” deflagration models
with SN Ia spectra and light curves. St has been parameterized differently
by different authors, for instance as a constant fraction of the local sound speed (Hoeflich & Khokhlov 1996, Iwamoto et al 1999), using time-dependent convection theory (Nomoto et al 1976, 1984; Buchler & Mazurek 1975; Woosley et al 1984), or with a phenomenological fractal model describing the multiscale character of the wrinkled flame surface (Woosley 1990, 1997b). All these studies essentially agree that very good agreement with the observations is obtained if St accelerates up to roughly 30% of the sound speed.

I added the italics and bold. The point is that so many specifics have to occur that most white dwarfs, regardless of high mass, don't make it to a Type Ia.

What the authors didn't do was to stop and state that: "Gee, guys, most white dwarf stars need not apply". I think that they assume we know that from their other material. This would suggest that you were inaccurate.

 

[SpaceTiger]

Boring for PF readers and "too specific" but it contains:
I added the italics and bold. The point is that so many specifics have to occur that most white dwarfs, regardless of high mass, don't make it to a Type Ia.

My interpretation of that was that the detonation model was an incorrect physical description of an exploding white dwarf, not that it happened frequently without producing a Type Ia. When they said:

the ignition of a detonation in the highdensity medium of the white dwarf core was argued to be an unlikely event.

it sounded to me like they were just ruling out one of the explanations for Type Ia's, not hypothesizing another type of supernova.

Clearly, a white dwarf cannot survive beyond the mass limit, so there must be some kind of explosion. It's hard for me to imagine that a significant fraction of white dwarfs explode without producing an observable amount of light.