spherical dp dr The gravitational force acting inwards on the same element of gas is,

Transcription

1 Rik s Cosmology Tutoial: Chapte Stella Stutue Pat Chapte Stella Stutue Pat : Pessue, Density and Tempeatue Distibutions in Spheially Symmeti, Main Sequene Stas The Clayton Model Last Update: July 006. Intodution This is the fist of a sequene of Chaptes onsideing the stutue of stas. Stas ae not simple beasts. In thei lives they undego seveal qualitatively diffeent phases. The fomation of a sta fom a diffuse gas loud is onsideably moe poblematial to undestand than might be supposed. The idea that the gas loud meely ollapses unde gavity until the ente is hot and dense enough to initiate nulea eations, whilst essentially oet, athe misses the point. Paadoxially, in ode to get hot enough by ollapse, the gas loud must have an effetive ooling mehanism. Without a ooling mehanism, the gas loud is suppoted against ollapse by its pessue, o by adiation pessue. If a ooling mehanism is available, the ollapse of a vey lage gas loud may be subjet to instabilities leading to fagmentation into seveal egions eah with thei own distint ollapse ente. One nulea buning has initiated, the sta goes though a tansient peiod whilst steady onditions ae set up, the so-alled Hayashi tak. Typial stas then spend the bulk of thei lives buning hydogen into helium. Stas buning hydogen in a steady state ae alled main sequene stas. Obsevationally, main sequene stas ae identified by the fat that they lie on a haateisti uve on a plot of luminosity against sufae tempeatue (i.e. olou), alled a Hetzspung-Russell diagam. Whilst buning thei ental stok of hydogen, stas do not hange thei position on the H-R diagam vey muh. Stas lying at diffeent points on the H-R main sequene uve have diffeent mass. Hotte, blue, stas have geate luminosity and ae moe massive. When hydogen beomes exhausted in the ente of a sta, things get moe ompliated. A sta suppots itself against gavitational ollapse by vitue of the pessue of the (ionised) gas of whih it is omposed. The pessue is theefoe geatest at the ente. Beause the sta is adiating its heat away into spae, if the nulea eations eased its tempeatue would dop. As a onsequene, its pessue would also dop. The esult of a sta unning out of hydogen fuel in its ental oe egion is, theefoe, that it an no longe suppot itself against gavitational ollapse. Consequently gavitational ollapse ous until the ental tempeatue is high enough to tigge the nulea buning of helium, at about 00 MK. Cudely speaking, a new equilibium ondition an then be set up in whih the ate of nulea heat podution balanes against the adiant loss. Howeve, some omplex physis is going on. Fo one thing, hydogen buning has not eased. In a shell aound the oe, hydogen is still being bunt. Hydogen was only exhausted in the oe egion. The futhe gavitational ollapse has eated tempeatues suffiient to bun hydogen (>0 MK) to aise in egions whih wee peviously too ool and hene whih still ontain hydogen. Between the helium buning oe and the hydogen buning shell thee will be a egion whih is not poduing nulea heating, being without hydogen but not hot enough to bun helium. Page of

2 Rik s Cosmology Tutoial: Chapte Stella Stutue Pat The tak of a helium buning sta on the H-R diagam is inteesting. Suh stas leave the main sequene. Unlike main sequene stas, fo whih luminous means blue, helium buning stas initially beome moe luminous whilst also beoming edde. Sine edde means a lowe sufae tempeatue, and beause this implies a edued adiant powe pe unit aea, the sta an only be moe luminous if it is fa bigge in size. In shot, the sta beomes a ed giant. Duing the helium buning peiod, the sta moves in a wide zigzag on the H-R diagam, whih we will not attempt to desibe futhe hee. The eventual exhaustion of the oe helium fuel leads to futhe gavitational ollapse. The omposition of the oe will now be pedominantly abon and oxygen. The next steady phase is that of abon buning in the oe, aound whih thee is an annula shell buning helium, and aound that anothe shell buning hydogen. The hydogen buning shell feeds helium fuel into the shell below, whilst the helium buning shell in tun feeds abon fuel into the oe. I suppose this ould be alled a fusion-beede eato. Suffiiently massive stas ultimately have five onenti egions buning, fom outside inwads: hydogen, helium, abon, oxygen and silion. By this time the oe is ion. Beause ion has the highest nulea binding enegy pe nuleon of all the nulides, it is the stable end point of the hain of nulea eations. Eah phase of buning ous at an aeleating ate. Whilst hydogen buning in a sola mass sta may last fo 0 Bys, the buning of silion into ion in massive stas ous in a timesale of days. It is athe as if the Ceato has deided to get a move on. The impession is onfimed by the subsequent, athe ass, display of powe. The supenova whih ous in suffiiently massive stas eates all the elements beyond ion in a split seond. How is the above pitue of the evolution of stas known? The answe is a ombination of alulation and obsevation. What annot be obseved, of ouse, is the evolution of a single sta as it moves aound the H-R diagam. That would be ideal, but unfotunately the human lifespan is too pathetially shot. Howeve, the whole population of stas an be used as an indiation of the tajetoies of individual stas. Thei elative spasity in some egions of the H-R diagam implies elatively bief peiods of time spent by individual stas in these egions. As egads alulations, the undelying physis of stella evolution is geneally quite well undestood and ompute models ae sophistiated. Whilst diffeent eseah goups will diffe on the details, the moe signifiant obsevation is the boad ageement egading stella evolution fom the stat of the main sequene to the supenova pogenito ondition. Howeve, onsideing stella evolution in this level of detail is well beyond the sope of these notes. Take, fo example, just one issue that of heat tanspot. A sta must tanspot heat as fast as it is podued if it is to emain in a steady state. Thee ae two mehanisms of heat tanspot available: adiation and onvetion. The effetiveness of the fome depends upon the degee of opaity of the ionised gas to the adiation. We shall say moe about this in late haptes. If the adiation mehanism is not suffiiently effetive, the esulting tempeatue gadients will dive a onvetive mehanism. As well as tanspoting heat, onvetive mehanisms also tanspot matte. Thus, egions of onvetion will mix the stella omposition. The poduts of nulea fusion eations do not, theefoe, stay put. They an be tanspoted to oute Page of

3 Rik s Cosmology Tutoial: Chapte Stella Stutue Pat egions whee onditions ae too ool fo them to have been podued dietly. In patie most stas, at most times, have some egions whih ae adiative and some whih ae onvetive. Moeove, the boundaies between these egions hanges. At key times, alled dedge-ups, the onvetive egion tansiently penetates deep into the sta towads its oe. As the name implies, this has the effet of dedging up nulea fusion poduts the hemial elements into the oute egions of the sta. Modelling onvetion, patiulaly with apidly moving onvetion-adiative boundaies, and hene the esulting hanges in the hemial omposition of the stella medium at any position, equies sophistiated ompute odes. To give one futhe example, in the ed giant phase at least fo some stella masses stas exhibit themal instabilities. Thus, we ae not eally dealing with a steady state but a dynamial ondition of osillating powe densities. One again we see that models apable of epoduing all the main featues of stella evolution ae beyond the eah of amateu dabbles. In any ase, the attempt to podue a detailed stella model would obsue the essential featues whih we wish to emphasise. In this Chapte, and up to Chapte 9, we shall ty to bing out the essential physis without undue omplexity. We shall onentate upon main sequene, hydogen-buning stas. In this hapte we shall stat by pesenting a simple, if inomplete, model of the pessue, density and tempeatue distibution within a sta. Rathe emakably, the poblem of the pessue, density and tempeatue distibutions within stas in a quasi-stati state an be deoupled fom the issues of nulea enegy podution and heat tanspot. O, athe, suh deoupling is possible povided that we ae able to speify a bounday ondition value, suh as a ental density, whih is not detemined by the model itself. This would not be neessay, and the omplete solution ould be found fo a sta of a given mass, if ompatibility with nulea heat geneation and heat tanspot wee imposed. This Chapte onsides stella models using this simplified, but estited, appoah. Chapte 8 will deive a slightly moe omplete model inopoating the nulea heating ingedient. We shall not, though, develop a omplete model whih also addesses heat tanspot thooughly. Thee will be some disussion of this in Chapte 8, i.e. how the omplete poblem is fomulated and simple losed-fom solutions (the homologous models) unde highly idealised assumptions. Thus, the poblem onsideed hee is a body of gas whih is in stati equilibium unde its own gavity. We will not onside hee the fomation of the sta, i.e. the peiod of gavitational ollapse. We wish to detemine the distibutions of pessue, density and tempeatue when the gas is in equilibium. We shall undestand by the tem gas eithe a gas of atoms o moleules o a gas of ionised plasma. Henefoth we shall dop the inveted ommas, but in most ases the gas in question will be a plasma. In this Chapte we shall initially deive a non-linea diffeential equation fo the density whih aises fom just two ingedients: hydostati equilibium and an assumed polytopi equation of state fo the ionised gas ompising the sta. This nonlinea diffeential equation, the stutue equation, will be used in Chapte 8. Howeve, the simplified appoah of this Chapte postulates an analytial fom fo the pessue distibution and imposes only the equiements of hydostati equilibium Page of

4 Rik s Cosmology Tutoial: Chapte Stella Stutue Pat (i.e. the polytopi equation of state is not used). The advantage of this appoah is that simple expessions fo the pessue, density and tempeatue distibutions esult, and thei oigin is lea fom the algebai deivation. The method is easonably auate (pehaps within ~5%?) suffiiently nea the ente of a sta. Its peditions ae ompaed with those fom moe omplete stella stutue models. Howeve, not all the elevant physial onstaints ae imposed by the model, e.g. the speifi equation of state fo the gas, ompatibility with nulea heating ates, and ompatibility with heat tanspot ates. As a onsequene, thee ae some undetemined paametes in this simplified appoah whih have to be supplied as givens. In addition, the model beomes poo away fom the ente of the sta, and annot be used even appoximately nea the sta s sufae.. Hydostati Equilibium Only two foes at within the body of ionised gas envisaged: gavity and pessue. In this Setion we onside what distibutions of pessue will maintain the gas in stati equilibium against the gavitational foes whih tend to ause it to ollapse. We onside only the spheially symmeti poblem (hene a sta whih is not otating an unealisti simplifiation in tuth). Call the pessue P() at any adius. Conside an element of gas of adial thikness and subtending a solid angle about the ente. Consideing fistly only the pessue foes ating on the spheial sufaes of this element, the net outwad foe is, F P() P( )( ) spheial pessue () The net adial foe due to pessue ating on the sides of the element (i.e. the faes fo whih one of the pola angles o is onstant) an easily be dedued. Imagine gavity being swithed off, so that the pessue was also unifom. The /d tem in () is then zeo, and, sine thee is no gavitational foe to balane, the sides must ontibute a foe whih anels with the tem -P in (). Hene, (tuning gavity on again) the net pessue foe ating outwads is, F d d P pessue () The gavitational foe ating inwads on the same element of gas is, F Gm() () gavity () The seond fato in () is simply the mass of the element of gas, whee () is its density at adius. The tem m() means the mass of gas within adius, i.e., m () ( ) d () 0 The equation fo hydostati equilibium follows fom equating () and (), i.e., Page of

5 Rik s Cosmology Tutoial: Chapte Stella Stutue Pat Gm() () d (5) It follows fom (5) that the pessue must fall monotonially away fom the ente (sine the LHS is positive) whih is not supising. Equ.(5) may also be witten in a fom whih eliminates m(). Fistly we diffeentiate (5), and then use () to substitute fo dm/d, giving, dm G d d d. d G (6) Anothe fom of the hydostati equation whih an be useful is, Gmdm d G. G d (7) This fom is useful beause the LHS is an exat diffeential and hene an be integated expliitly to give, Gm () 8 (8) o. The Stutue Equation In Setion, the equations deived fo hydostati equilibium, i..e. Equs.(5-8), apply whateve the soue of the pessue. In geneal this will be a ombination of gas pessue and adiation pessue. In this Setion we shall assume that the gas pessue dominates. The hydostati equation [say, Equ.(6)] ontains two independent funtions of adius: the density and the pessue. To ahieve a soluble system we theefoe equie an additional equation. This is povided by the equation of state of the ionised gas. In some iumstanes this an be appoximated as a polytopi equation of the fom, P K (9) whee K and ae onstants, i.e. the same at all positions in the sta. This equation of state is analogous to the adiabati equation fo pefet gases. We may expet to be something like the atio of speifis heats and hene lie between / o 5/. It tuns out that stability does indeed equie / < 5/ though we will not justify this hee. Howeve, the oeffiients in the polytopi equation ae not igoously unifom thoughout the sta. Hene, the use of Equ.(9) is a onvenient appoximation only. Diffeentiating (9) and substituting the esult into (6) gives, d K d. G (0) d d Page 5 of

6 Rik s Cosmology Tutoial: Chapte Stella Stutue Pat This seond ode DE in density povides the basis fo ou simplified stella stutue detemination. It would be linea only if =, whih is not possible. Atually < and the equation is non-linea. The bounday onditions ae, ) ( ) 0 (ondition fo a finite sized sta) (0a) d ) 0 d 0 (0b) The seond b.. follows fom (9) and ou obsevation, following (5), that the pessue must be maximum at the ente. We have, m ( 0) () d 0 G () Thus, if we knew the polytopi paametes K and, Equ.(0) ould be solved (numeially) to find the density distibution fo the bounday onditions (0a) and (0b), togethe with some assumed ental density,. Fom the density, the pessue distibution is then found fom the polytopi equation of state, Equ.(9). Finally, the tempeatue would be found fom the ideal gas law, mass P kt () whee <mass> is the aveage mass pe patile (inluding nulei, ions and eletons). In the above desiption, the only things whih distinguish one sta fom anothe ae the values of the polytopi onstants, K and, and the assumed ental density,. Thus, we may expet that K depends on the sta s total mass, fo example. Thus, K is not only a popety of the gas. The value of K is theefoe an undetemined paamete whih must be dedued fo the patiula ase. Fo the est of this Chapte, we dop the polytopi equation and onside only hydostati equilibium.. Clayton Model The Clayton model onsists of assuming a speifi analyti fom fo the pessue distibution, P(), whih espets the equiement of hydostati equilibium, and whih is also onsistent with vitually the whole mass, M, of the sta being ontained within a speified adius, R. The latte means that, to a vey good appoximation. Using this in (5) gives, m (R) M () Page 6 of

7 Rik s Cosmology Tutoial: Chapte Stella Stutue Pat d R GM (R) R (5) The fom hosen by Clayton was, d G e / a (6) whee a is an initially unknown length sale. Equ.(6) is onsistent with Equ.() as 0. Equating (5) and (6) at = R gives, (R) R. e M R / a (7) The signifiant featue of (7) is that, if the length sale paamete, a, is small ompaed with R (say, if a < R/5), then the gas density at the sufae of the sta is vanishingly small. This is, of ouse, what is equied physially sine the sufae of a sta intefaes with intestella spae, and hene is a vitually pefet vauum (at least by teestial standads). We note that the analyti fom (6) is onvenient beause it may be integated expliitly to give the pessue, x P (x) Ga e, whee x (8) a and whee we have appealed to the bounday ondition that pessue falls to zeo at infinity. (This diffes fom the oiginal Clayton model whih imposed zeo pessue at the sufae, = R, of the sta). So fa we have noted only that the Clayton fom, Equ.(6), has sensible bounday behaviou. We now also impose hydostati equilibium. The fom Equ.(7) is onvenient. Integating it and inseting (6) gives, m G G e / a d and hene, m, whee a 5 x x 6 x e dx 6 (x x )e (9) whee we have imposed the bounday ondition m(0) = 0, i.e. that the density at the ente is finite (i.e. Equ.). The density distibution may now be found fom m(x) using the definition of m, Equ.(), whih gives, Page 7 of

8 Rik s Cosmology Tutoial: Chapte Stella Stutue Pat x x e ( x ) (0) 5 x d d 6x e whee we have used.. Noting that we also need to obey dx dx the bounday ondition Equ.(7), we equate () at = R with (7) whih simplifies to an expession fo the length sale, a, as follows, M a. () whee we have assumed that R is suffiiently lage than a (e.g. R > a) so that 6. The same expession fo the length sale a esults fom imposing the ondition Equ.() on the expliit expession Equ.(9) fo m, i.e. fom equiing that the whole of the sta s mass is ontained within = R to a vey good appoximation. In summay, the Clayton model of stella stutue gives the pessue distibution Equ.(8) and the density distibution Equ.(0), whee the paamete a is given by Equ.() and the funtion is given by Equ.(9). These distibutions of pessue and density ontain just two undetemined paametes, the total mass M of the sta and the density at its ente. Given the mass and the density at the ente, the pessue and density ae detemined eveywhee. It is useful to note that, fom (8) and (), the pessue at the ente of the sta is given by, P GM () 6 Hene, we see that, at the ente of the sta, the Clayton model would be onsistent with a polytopi equation of state, Equ.(9), with = / and K =0.GM /. As sumised above, the oeffiient K is mass dependent. [Note, howeve, that assuming an = / polytopi elation held eveywhee would atually esult in a ental density about 8% lowe than that given by Eq.()]. Howeve, at othe loations the pessue is not given in tems of the density by a powe-law like Equ.(9), i.e. Equs.(8) and (0) ae not elated by a powe law. In as fa as Equ.(9) is a good appoximation to the equation of state, this exposes the appoximation inheent in the Clayton models. It is of inteest to see how fa adift the Clayton model is fom the polytopi state. To do this we inset the density given by Equ.(0) into the following polytopi expession fo the pessue, P polytopi GM () 6 Page 8 of

9 Rik s Cosmology Tutoial: Chapte Stella Stutue Pat and ompae the esult with Equ.(8), the Clayton expession fo pessue. The two agee at the ente, as shown by Equ.(). In geneal thei atio is, P P Clayton polytopi x / x e e x () This is plotted against x below,.8.6 PClayton /Ppolytopi x = /a This would seem to imply that the Clayton model pessue might be quite seiously in eo exept nea the ente. Fo x <0.5 the eo appeas to be <%. Howeve, the Clayton model pessue is not even indiative of the ode of magnitude at the sufae of a sta. The sta sufae may be at x values appoahing ~6, fo whih the above atio exeeds ~00. Having said this, the polytopi assumption is not stitly oet eithe. Example of the Clayton Model The Sun: We note fom Equ.(9) that, at suffiiently lage x, tends to 6. Hene the whole mass of the sta is given by M 6 / a. Thus we may e-wite (9) simply as m ( x) ( x) M / 6. Hene / 6 is the fation of the total mass within the adius in question. Sine Equ.(9) gives ( ) we an intepet the length sale a as the adius within whih the fation 0.69/ 6 = 8.% of the sta s total mass esides. Page 9 of

10 Rik s Cosmology Tutoial: Chapte Stella Stutue Pat To apply the Clayton model to the sun we fist hoose a value of R/a whih we ae ontent to teat as the sufae:-.5.5 Phi x = /a Fom this gaph, appeas to have vitually eahed its asymptote ( 6) by x ~.. Close inspetions eveals that the deviation of / 6, the mass fation, fom unity is as follows, x = eo in deimal plae Consequently it s a little diffiult to be peise as egads the hoie of x to assoiate with the sta s sufae. Fo illustation we shall use both R/a =. and R/a = 5., oesponding to 99.9% of the mass and all but a fation 0-0 of the mass espetively. (The sufae is best defined as that egion of finite thikness suh that the thikness equals the mean fee path of a photon i.e. the photosphee). To detemine the two degees of feedom we impose the atual values of the mass and size of the sun, i.e. x 0 0 kg and 6.96 x 0 8 m espetively. Hene the two values fo the a paamete ae:- ) R/a =.: a =.05 x 0 8 m =,700 kg/m P =.0 x 0 5 Pa ) R/a = 5.: a =.9 x 0 8 m = 9,000 kg/m P =.9 x 0 6 Pa Page 0 of

11 Rik s Cosmology Tutoial: Chapte Stella Stutue Pat whee the ental density fom Equ.() has also been given fo these values of a and M. The lage of these densities is.6 times the density of teestial steel. Standad sola models give the ental density as ~9 x 0 kg/m, so the Clayton model with R/a = 5. is the best hoie. The ental pessue fom Equ.() is also given above (using G = 6.67 x 0 - in MKSA units). Sine atm = 0 5 Pa, the lage of the above pessues is a huge.9 x 0 atmosphees pessue. Standad sola models give the ental pessue as.65 x 0 6 Pa, so the Clayton model with R/a ~ 5. is easonably lose. To deive the tempeatue fom Equ.() we need the aveage mass of the patiles whih give ise to the pessue. We shall assume that the ontibuting patiles ae potons, eletons and helium- nulei. Othe nulei total only a small fation. We assume that neutinos, whih will also be pesent in lage numbes due to the nulea eations, inteat suffiiently weakly to make negligible ontibution to the pessue (o the tempeatue of othe speies). Helium- is ~5% of the nulei by mass. (This assumes a newly fomed sta with a omposition equal to that afte the Big Bang). Sine helium is times heavie than a poton, this means that fo evey 75 potons thee ae 6.5 helium nulei. Thee ae also 75 eletons to math the potons, plus.5 eletons to math the helium nulei. Hene, the aveage mass an be evaluated fom, speies elative numbe peent numbe mass(mev) ontibn. potons 75.% 99 7 helium % x 99 9 eletons % Total 00% Hene aveage mass = 556MeV = 0.59 x poton mass = 0.59 x.67 x 0-7 kg = 9.9 x 0-8 kg. Use of this mass in Equ.() togethe with the above esults fo the pessue and density at the ente of the sun give estimates fo the tempeatue at the ente of the sun of 9.5 x 0 6 o K (fo R/a =.) and 5 x 0 6 o K (fo R/a = 5.). Standad sola models give the ental tempeatue as.7 x 0 6 o K, so the Clayton model with R/a ~ 5. is again quite lose. The R/a =. model is poo sine the pedited tempeatue is too low fo the equied nulea eation ates. We have seen above that the Clayton model is expeted to be seiously in eo fo values of x geate than unity, and etainly gossly wong at the sufae of the sta. Nevetheless, it is of inteest to see what numeial values ae pedited at the sufae. Fo this we use R/a = 5., whih gives the density, pessue and tempeatue at the sufae to be. x 0-6 kg/m,. x 0 Pa (=0.0 atm) and,000 o K espetively. The density and pessue ae at least small - though whethe they ae small enough is not obvious without ompaison with the oet esult. The density is equivalent to about 7 x 0 0 potons pe m, and a pessue of 0.0atm is a vey poo vauum. The tempeatue is etainly wong by a lage fato, the oet value being about 6,000 o K. Hene, fo easonable peditions nea the sufae of the sta we need to solve the full stutue equation, (0), fo whih a polytopi equation of state is equied. This is the subjet of Chapte 8. Auate values of the sufae tempeatue Page of

12 Rik s Cosmology Tutoial: Chapte Stella Stutue Pat ae best obtained by balaning the luminosity against the ate of nulea heat podution in the ente of the sta. This will also be dealt with in Chapte 8. The following gaphs plot the density, pessue and tempeatue distibutions fom the Clayton model, nomalised by the value in the ente (and we must bea in mind that the model is inauate fo x > ).. Density density / ental density x = /a Page of

13 Rik s Cosmology Tutoial: Chapte Stella Stutue Pat 0.9 Pessue Pessue / Cental Pessue x = /a Tempeatue Tempeatue / Cental Tempeatue x = /a These uves have been geneated fom the expessions, x x e x, P Pe (x), T T (x) x (5) Page of

14 Rik s Cosmology Tutoial: Chapte Stella Stutue Pat whee ( x) is given by Equ.(9). Thus, we see that the density and pessue fall off in a manne that looks qualitatively like a Gaussian, being dominated by the e -x^ tem. The tempeatue, howeve, falls off muh moe slowly sine the exponential tem anels in the atio of pessue and density whih entes the expession fo tempeatue. 5. Conlusion We have deived a non-linea diffeential equation in density whih embodies the equiement fo hydostati equilibium, Equ.(0), fo a gas obeying a polytopi equation of state. We have defeed until Chapte 8 a full numeial solution of this equation. In this Chapte we have exploed a simple analytial appoximation to this solution, the Clayton model, whih is auate nea the ente of a sta but not at its sufae. The value of this model is that it gives simple losed-fom expessions fo the density, pessue and tempeatue distibutions whih have the oet qualitative fom. In patiula, they fall off apidly away fom the ente, beoming vey small fations of thei ental values at just a fation of the sta s adius. Beause the model is based on hydostati equilibium alone, it shows that this qualitative featue of the pessue, density and tempeatue distibutions is expeted fo any stas unde any onditions, i.e. P, T and will always edued apidly away fom the ente. Page of

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