Phys 321: Lecture 7 Stellar Evolu>on Prof. Bin Chen, Tiernan Hall 101, [email protected] Stellar Evoluon • Formaon of protostars (covered in Phys 320; briefly reviewed here) • Pre-main-sequence evolu>on (this lecture) • Evolu>on on the main sequence (this lecture) • Post-main-sequence evolu>on (this lecture) • Stellar death (next lecture) The Interstellar Medium and Star Formation the cloud’s internal kinetic energy, given by The Interstellar Medium and Star Formation the cloud’s internal kinetic energy, given by 3 K NkT, 3 = 2 The Interstellar MediumK andNkT, Star Formation = 2 where N is the total number2 ofTHE particles. FORMATIONwhere ButNNisis the just total OF number PROTOSTARS of particles. But N is just M Mc N c , Our understandingN , of stellar evolution has= µm developedH significantly since the 1960s, reaching = µmH the pointwhere whereµ is the much mean of molecular the life weight. history Now, of by a the star virial is theorem, well determined. the condition for This collapse success has been where µ is the mean molecular weight.due to advances Now,(2K< byU the in) becomes observational virial theorem, techniques, the condition improvements for collapse in our knowledge of the physical | | (2K< U ) becomes processes important in stars, and increases in computational2 power. In the remainder of this | | 3MckT 3 GMc chapter, we will present an overview of< the lives. of stars, leaving de(12)tailed discussions 2 µmH 5 Rc of s3oMmeckTspecia3l pGMhasces of evolution until later, specifically stellar pulsation, supernovae, The radius< may be replaced. by using the initial mass density(12) of the cloud, ρ , assumed here µmH 5 Rc 0 and comtop beac constantt objec throughoutts (stellar theco cloud,rpses). The radius may be replaced by using theForma8on initial mass density of of the cloud,protostarsρ0M, assumed1/3 here 3 c The Jeans Criterion Rc . (13) to be constant throughout the cloud, = 4πρ ! 0 " DespiteAfter many substitution successes,1/3 into Eq. important ( 12), we questions may solve for remain the minimum concerning mass necessary how to stars initiate change during • 3Mc theirR lifetimes.c Clouds of gas and dust collapse and form Young the spontaneous One area collapse. where of the the cloud. picture This condition is far from is(13) known complete as the Jeans is in criterion the earliest: stage of evo- = 4πρ lution, theStellar Objects (YSOs) ! formation0 " of pre-nuclear-burningThe objects Interstellar known Medium as protostars and Starfrom Formation interstellar Mc >MJ , After substitution into Eq. ( 12),molecular we may clouds.solve for the minimum mass necessary to initiate where the cloud’s internal kinetic energy, given by the spontaneous collapse of the cloud.If globules ThisCondi>on for spontaneous collapse to occur: condition and cores is knownin molecular as the cloudsJeans criterion are the sites: of star formation, what conditions must exist for collapse to occur? Sir James Jeans (1877–1946) first investigated this prob- 5kT 3/2 3 1/2 3 lem in 1902Mc >M by consideringJ , Where theMJ effects of small deviationsis the fromJeans mass hydrostatic(14)K equilibrium.NkT, ≃ GµmH 4πρ0 2 Although several simplifying assumptions! " are! made" in the analysis, such as= neglecting ef- where Mass of the cloud fects dueis tocalled rotation, the Jeans turbulence, mass. Using and Eq.where ( galactic 13),N theis magnetic Jeans theFor a typical dense molecular H total criterion fields,number may also it ofprovides be particles. expressed important2 cloud with in But N is insights just T = 10 K and n = 1010 m-3, M ~ 8 Msun into theterms development of the minimum of protostars. radius necessary to collapse a cloud of densityH2 ρ0: J 3/2 1/2 M The5 virialkT theorem,3 R >R , (15) c MJ This is the consequence of the virial theorem: c J (14) N , ≃ Gµm 4πρ = µm ! H " ! 0 " H whereFor a system in equilibrium: 2K U 0, where µ+is the= mean molecular weight. Now, by the virial theorem, the condition for collapse is called the Jeans mass. Using Eq. ( 13), the Jeans criterion may also15 bekT expressedInternal energy is too low -> pressure 1/2 in For a system to collapseR: J(2K< U ) becomes (16) 4 terms of the minimum radius necessarydescribes to the collapse condition a cloud of equilibrium of density≃ ρ for40π:Gµm a stable,ρ can not support against gravity gravitationally bound system. We have ! | |H 0 " already seen that the virial theorem arises naturally in the discussion of orbital motion, and2 3MckT 3 GM we haveisR also thec >RJeans invokedJ length, it. in estimating the amount of(15) gravitational energy contained< withinc . (12) a star. TheThevi Jeansrial masstheo derivationrem ma giveny a abovelso b neglectede used theto importantestima factte t thathe therecond mustiµmtio existnHs nece5ssaRryc for where an external pressure on the cloud due to the surrounding interstellar medium (such as the protosteencompassingllar collaps GMCe. in the case ofThe an embedded radius maydense becore). replaced Although by we using will not the derive initial mass density of the cloud, ρ , assumed here If twice the total internal kinetic energy of a molecular cloud (2K) exceeds the absolute 0 1/2 to be constant throughout the cloud, value of the15 gravitationalkT potential energy ( U ), the force due to the gas pressure will RJ | | (16) dominate≃ 4 theπGµm forceH ρ of0 gravity and the cloud will expand. On the other hand, if the internal1/3 ! " 3Mc kinetic energy is too low, the cloud will collapse. The boundary betweenRc these two cases. (13) = 4πρ0 is the Jeans length. describes the critical condition for stability when rotation, turbulence, and magnetic! " fields The Jeans mass derivation givenare neglected. above neglected the important factAfter that substitution there must into exist Eq. ( 12), we may solve for the minimum mass necessary to initiate an external pressure on the cloud dueAssuming to the surrounding a spherical interstellar cloud ofthe constant medium spontaneous density, (such collapse as the the gravitational of the cloud. potential This condition energy is is known as the Jeans criterion: encompassing GMC in the caseapproximately of an embedded dense core). Although we will not derive Mc >MJ , 3 GM2 U c , where∼−5 Rc where Mc and Rc are the mass and radius of the cloud, respectively. We may5kT also3/ estimate2 3 1/2 MJ (14) ≃ GµmH 4πρ0 4We have implicitly assumed that the kinetic and potential energy terms are averaged over! time. " ! " is called the Jeans mass. Using Eq. ( 13), the Jeans criterion may also be expressed in terms of the minimum radius necessary to collapse a cloud of density ρ0: Rc >RJ , (15) where 15kT 1/2 RJ (16) ≃ 4πGµm ρ ! H 0 " is the Jeans length. The Jeans mass derivation given above neglected the important fact that there must exist an external pressure on the cloud due to the surrounding interstellar medium (such as the encompassing GMC in the case of an embedded dense core). Although we will not derive The Interstellar Medium and Star Formation which leads to the differential equation dθ 1 1/2 χ 1 . (21) dt = − θ − ! " Making yet another substitution, θ cos2 ξ, (22) ≡ and after some manipulation, Eq. ( 21) becomes dξ χ cos2 ξ . (23) dt = 2 Equation ( 23) may now be integrated directly with respect to t to yield ξ 1 χ sin 2ξ t C2. (24) 2 + 4 = 2 + Lastly, the integration constant, C2, must be evaluated. Doing so requires that r r0 when t 0, which implies that θ 1, or ξ 0 at the beginning of the collapse. Therefore,= = = = C2 0. We= have finally arrived at the equation of motion for the gravitational collapse of the cloud, given in parameterized form by Forma8on1 of protostars ξ sin 2ξ χt. (25) + 2 = Our task now is to extract the behavior of the collapsing cloud from this equation. From Eq. ( 25), it is possible to• calculateOnce spontaneous collapse occurs, the >me scale is the free-fall timescale for a cloud that has satisfied the Jeans criterion. Let t tff set by the when the radius offree-fall me the collapsing sphere reaches zero (θ 0, ξ π/2).6 Then = = = The Interstellar Medium and Star Formation The Interstellar Medium and Star Formation π tff . 3 Free-fall >me, or the >me takes = 2χ However, if the cloud is somewhat centrally condensed when the collapse begins,10 M the for a par>cle to fall from any 0.7 Myr 0.5 Myr0.3 Myr 150 kyr free-fallSubstituting time will the be value shorter for forχ, we material have near the center than for material2 farther out. Thus, 60 kyr 30 kyr point of the cloud freely: 20 kyr 2 M 10 kyr as the collapse progresses, the density will increase more rapidly near the center than in 8 kyr 4 kyr 1/2 other regions. In this case the collapse is referred3π 1 to as an inside-out collapse1 . 1 M t . DM 1994 2 M (26) ff 0.5 M = 32 Gρ0 0.1 M 1.5 kyr ! " 0 0.05 M Example 2.2. Using data given in Example 2.1 for a dense core) Luminosity (L of a giant molecular 10 DM 1994 1 M cloud, weYou may should estimate notice the that amount the free-fall of time time required is actually for independent the collapse. ofLog the Assuming initial radius a density of Again for the dense molecular H –1 the sphere. Consequently,17 3 as long as the original density of2 the spherical molecularDM 1994 0.5 M cloud of ρ0 3 10− kg m− that is constant throughout the10 core,-3 Eq. ( 26) gives 1 kyr =was uniform,× all parts ofcloud with T = 10 K and n the cloud will take the sameH2 amount= 10 m of time to collapse, and the 5 –2 DM 1994 0.1 M density will increase at the same ratetff everywhere.3.8 10 Thisyr behavior.