Stellar Evolution

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Prof. Bin Chen, Tiernan Hall 101, [email protected] Stellar Evolution

• Formation of protostars (covered in Phys 320; briefly reviewed here) • Pre-main-sequence evolution (this lecture) • Evolution on the main sequence (this lecture) • Post-main-sequence evolution (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 theFormation 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 ThisCondition 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 Formation1 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 time 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 time 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 time, or the time takes = 2χ However, if the cloud is somewhat centrally condensed when the collapse begins,10 M the for a particle 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. is known as a homologous collapse. = × 42 To investigate the actual behaviorThis is a very rapid process of the collapse in our simplified model, weLog must10 Effective temperature (K) 6 This is obviously an unphysical final condition, since it implies infinite density. If r0 rfinal, however, then first solve Eq. ( 25) for ξ, given a value for t, and then useFIGURE Eq. 9 ≫ (Theoretical 22) evolutionary to find tracksθ of the gravitational collapse of 0.05, 0.1, 0.5, 1, 2, rfinal 0 is a reasonable approximation for our purposes here. and 10 M clouds through the protostar phase (solid lines). The dashed lines show the times since ≃ ⊙ = r/r0. However, Eq. ( 25) cannot be solved explicitly, so numericalcollapse began. techniques The light dotted lines must are pre-main-sequence be evolutionary tracks of 0.1, 0.5, 1, and 2M stars from D’Antona and Mazzitelli, Ap. J. Suppl., 90, 457, 1994. Note that the horizontal axis employed. The numerical solution of the homologous collapseis plotted⊙ of withthe effective molecular temperature cloud increasing is to the left, as is characteristic of all H–R diagrams. shown in Fig. 8. Notice that the collapse is quite slow initially(Figure adapted and fromaccelerates Wuchterl and Tscharnuter, quicklyAstron. Astrophys., 398, 1081, 2003.) as tff is approached. At the same time, the density increases veryAs the rapidly overlying material during continues the final to fall onto the hydrostatic core, the temperature of the core slowly increases. Eventually the temperature becomes high enough (approximately stages of collapse. 2000 K) to cause the molecular hydrogen to dissociate into individual atoms. This process absorbs energy that would otherwise provide a pressure gradient sufficient to maintain hydrostatic equilibrium. As a result, the core becomes dynamically unstable and a second collapse occurs. After the core radius has decreased to a value about 30% larger than the The Fragmentation of Collapsing Clouds present size of the Sun, hydrostatic equilibrium is re-established. At this point, the core mass is still much less than its final value, implying that accretion is still ongoing. After the core collapse, a second shock front is established as the envelope continues Since the masses of fairly large molecular clouds could exceedto accrete theinfalling Jeans material. limit, When the from nearly flat, roughly constant luminosity part of the Eq. ( 14) our simple analysis seems to imply that stars can formevolutionary with track very is reached large in Fig.masses, 9, accretion has settled into a quasi-steady main accretion phase. At about the same time, temperatures in the deep interior of the protostar 2 possibly up to the initial mass of the cloud. However, observationshave increase showd enough thatthat thisdeuteri doesum (1H) begins to burn, producing up to 60% of the luminosity of the 1 M protostar. Note that this reaction is favored over the first step not happen. Furthermore, it appears that stars frequently (perhapsin the PPeven I chain preferentially) because it has⊙ a fairly tend large cross section, σ (E), at low temperatures. to form in groups, ranging from binary star systems to clustersWith that only contain a finite amount hundreds of mass available of from the original cloud, and with only a limited amount of deuterium available to burn, the luminosity must eventually decrease. thousands of members. When deuterium burn-out occurs, the evolutionary track bends sharply downward and the The process of fragmentation that segments a collapsing cloud is an aspect of star formation that is under significant investigation. To see that fragmentation must occur by

8.0 1.0 7.0 0.8 6.0 )

5.0 0 ␳ 0.6 / ␳ 0 ( r

/ 4.0 r 10

0.4 3.0 Log

2.0 0.2 1.0

0.0 0.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 Time (105 yr) FIGURE 8 The homologous collapse of a molecular cloud, as discussed in Example 2.2. r/r0 is shown as the solid line and log10(ρ/ρ0) is shown as the dashed line. The initial density of the 17 3 5 cloud was ρ 3 10− kg m− and the free-fall time is 3.8 10 yr. 0 = × ×

Pre-main-sequence evolution

• The free-fall process is ended when a quasi-static protostar is formed, i.e., it is now under hydrostatic equilibrium • The rate of evolution is now controlled by the rate at which the star can thermally adjust to the contraction • Gravitational potential energy is released over time and is the source of the protostar’s luminosity What is the time scale for the pre-main-sequence evolution of an one solar mass protostar?

Note this answers below are just orders of magnitude estimates

A. 1 kyr B. 0.1 Myr C. 10 Myr D. 1 Gyr E. 10 Gyr

This time scale is just the Kelvin-Helmholtz timescale discussed in the previous lecture: Time for the gravitational potential energy to liberate by the contraction The Interstellar Medium and Star Formation TABLE 1 Pre-main-sequencePre-main-sequence contraction times contraction for the classical models times presented in Fig. 11. (Data from Bernasconi and Maeder, Astron. Astrophys., 307, 829, 1996.)

Initial Mass (M ) Contraction Time (Myr) ⊙ 60 0.0282 25 0.0708 15 0.117 9 0.288 5 1.15 3 7.24 2 23.4 1.5 35.4 1 38.9 0.8 68.4 reactions have little effect on the overall collapse; they simply slow the rate of collapse slightly. As the central temperature continues to rise, increasing levels of ionization decrease the opacity in that region and a radiative core develops, progressively encompassing more and more of the star’s mass. At the point of minimum luminosity in the tracks following the descent along the Hayashi track, the existence of the radiative core allows energy to escape into the convective envelope more readily, causing the luminosity of the star to increase again. Also, the effective temperature continues to increase, since the star is still shrinking.

At about the time that the luminosity begins to increase again, the temperature near the center has become high enough for nuclear reactions to begin in earnest, although not yet at their equilibrium rates. Initially, the first two steps of the PP I chain [the conversion 1 3 12 14 of 1Hto2He] and the CNO reactions that turn 6C into 7N dominate the nuclear energy production. With time, these reactions provide an increasingly larger fraction of the luminosity, while the energy production due to gravitational collapse makes less of a contribution to L. Due to the onset of the highly temperature-dependent CNO reactions, a steep temperature gradient is established in the core, and some convection again develops in that region. At the local maximum in the luminosity on the H–R diagram near the short dashed line, the rate of nuclear energy production has become so great that the central core is forced to expand somewhat, causing the gravitational energy term to become negative [recall that ϵ ϵnuclear ϵgravity]. This effect is apparent at the surface as the total luminosity decrea=ses towar+d its main-sequence value, accompanied by a decrease in the effective temperature. 12 When the 6C is finally exhausted, the core completes its readjustment to nuclear burning, reaching a sufficiently high temperature for the remainder of the PP I chain to become important. At the same time, with the establishment of a stable energy source, the gravitational energy term becomes insignificant and the star finally settles onto the main sequence. It is worth noting that the time required fora1M star to reach the main sequence, according to the detailed numerical model just described,⊙ is not very different from the crude estimate of the Kelvin–Helmholtz timescale.

The Hayashi track

Protostar contracts The Interstellar Medium and Star Formation

Gravitational 3

potential energy 10 M 0.7 Myr 0.5 Myr0.3 Myr 150 kyr liberates 2 60 kyr 30 kyr 20 kyr 2 M 10 kyr 8 kyr Effective temperature steadily increases 4 kyr 1 1 M DM 1994 2 M 0.5 M Some electrons are 0.1 M 1.5 kyr 0 0.05 M ionized from heavier ) Luminosity (L

10 DM 1994 1 M

elements Log –1 DM 1994 0.5 M Opacity increases due to H- ion 1 kyr –2 DM 1994 0.1 M

Log10 Effective temperature (K)

Temperature gradient required for energy FIGURE 9 Theoretical evolutionary tracks of the gravitational collapse of 0.05, 0.1, 0.5, 1, 2, and 10 M clouds through the protostar phase (solid lines). The dashed lines show the times since transfer by radiation becomes too large collapse began.⊙ The light dotted lines are pre-main-sequence evolutionary tracks of 0.1, 0.5, 1, and 2M stars from D’Antona and Mazzitelli, Ap. J. Suppl., 90, 457, 1994. Note that the horizontal axis is plotted⊙ with effective temperature increasing to the left, as is characteristic of all H–R diagrams. (Figure adapted from Wuchterl and Tscharnuter, Astron. Astrophys., 398, 1081, 2003.)

A large portion of the protostar becomes convective As the overlying material continues to fall onto the hydrostatic core, the temperature of the core slowly increases.• Slightly increasing effective Eventually the temperature becomes high enough (approximately 2000 K) to cause the molecular hydrogen to dissociate into individual atoms. This process absorbs energy that wouldtemperature otherwise provide a pressure gradient sufficient to maintain hydrostatic equilibrium.• Quickly decreasing luminosity As a result, the core becomes dynamically unstable and a second Evolution follows the nearly vertical Hayashi Track collapse occurs. After the core radius has decreased to a value about 30% larger than the present size of the Sun, hydrostatic equilibrium is re-established. At this point, the core mass is still much less than its final value, implying that accretion is still ongoing. After the core collapse, a second shock front is established as the envelope continues to accrete infalling material. When the nearly flat, roughly constant luminosity part of the evolutionary track is reached in Fig. 9, accretion has settled into a quasi-steady main accretion phase. At about the same time, temperatures in the deep interior of the protostar 2 have increased enough that deuterium (1H) begins to burn, producing up to 60% of the luminosity of the 1 M protostar. Note that this reaction is favored over the first step in the PP I chain because it has⊙ a fairly large cross section, σ (E), at low temperatures. With only a finite amount of mass available from the original cloud, and with only a limited amount of deuterium available to burn, the luminosity must eventually decrease. When deuterium burn-out occurs, the evolutionary track bends sharply downward and the

Pre-main-sequence evolutionary tracks The Interstellar Medium and Star Formation

–12 Onset of deuterium burning 6 –10 Evolution tracks 60 M 5 –8 • A plot of the location of an object on an H- 25 M R diagram at different times during its 15 M –6 lifetime is called an evolutionary track 4 • Evolutionary tracks for pre-main-sequence 9 M

–4 L / L bol (PMS) objects depend on the initial mass 3 10 M 5 M –2

of the protostar Log • They reach the main sequence at different Convection in 2 envelope stops points 3 M 0

2 M 1 2 Hayashi Y = 0.300 1.5 M Track 4 Z = 0.020 0 1 M 6 0.8 M

4.8 4.6 4.4 4.2 4 3.8 3.6

Log10 Te (K) FIGURE 11 Classical pre-main-sequence evolutionary tracks computed for stars of various masses with the composition X 0.68, Y 0.30, and Z 0.02. The direction of evolution on each track is generally from low effective= temperature= to high= effective temperature (right to left). The mass of each model is indicated beside its evolutionary track. The square on each track indicates the onset of deuterium burning in these calculations. The long-dash line represents the point on each track where convection in the envelope stops and the envelope becomes purely radiative. The short-dash line marks the onset of convection in the core of the star. Contraction times for each track are given in Table 1. (Figure adapted from Bernasconi and Maeder, Astron. Astrophys., 307, 829, 1996.)

computed with state-of-the-art physics are shown in Fig. 11, and the total time for each evolutionary track is given in Table 1. Consider the pre-main-sequence evolution ofa1M star, beginning on the Hayashi ⊙ track. With the high H− opacity near the surface, the star is completely convective during approximately the ﬁrst one million years of the collapse. In these models, deuterium burning also occurs during this early period of collapse, beginning at the square indicated on the 11 2 evolutionary tracks in Fig. 11. However, since 1H is not very abundant, the nuclear

11Note that since these calculations did not include the formation of the protostar from the direct collapse of the cloud as was done for the tracks in Fig. 9, there is a fundamental inconsistency between when deuterium burning occurs in the two sets of calculations.

Pre-main-sequence Evolution The Interstellar Medium and Star Formation

–12 Evolution stages for solar mass stars Onset of deuterium burning 6 • Protostar collapse with a large convection –10 envelope: Hayashi track 60 M 5 –8 • Deuterium burning starts 25 M • Rising temperature increases ionization 15 M –6 and decreases opacity -> radiation core 4 9 M

develops and gradually encompasses more –4 L / L bol of the star’s mass 3 10 M 5 M –2 • More energy is released into the Log Convection in convective envelop, increases the 2 envelope stops 3 M 0 luminosity • 2 M Effective temperature continue to increase 1 2 Hayashi as the star is still contracting Y = 0.300 1.5 M Track 4 • Temperature at the center becomes high Z = 0.020 0 enough for nuclear reactions to begin -> 1 M 6 enters the main sequence 0.8 M 4.8 4.6 4.4 4.2 4 3.8 3.6

Zero-age main-sequence stars (ZAMS) Log10 Te (K) FIGURE 11 Classical pre-main-sequence evolutionary tracks computed for stars of various masses with the composition X 0.68, Y 0.30, and Z 0.02. The direction of evolution on each track is generally from low effective= temperature= to high= effective temperature (right to left). The mass of each model is indicated beside its evolutionary track. The square on each track indicates the onset of deuterium burning in these calculations. The long-dash line represents the point on each track where convection in the envelope stops and the envelope becomes purely radiative. The short-dash line marks the onset of convection in the core of the star. Contraction times for each track are given in Table 1. (Figure adapted from Bernasconi and Maeder, Astron. Astrophys., 307, 829, 1996.)

H-R diagram of stars in the solar neighborhood

• Timescale governed by nuclear reactions • ~1010 years for the Sun • Stars −2.3 tstar / tsun ≈ (M star / M sun ) • 80–90% of all stars in the solar neighborhood are main-sequence stars • Width of the main sequence • Small, but nonzero • Possible causes: • Observational errors • Differing chemical compositions • Varying stages of evolution The Interiors of Stars sufﬁciently low densities and velocities. In these limits both distribution functions become indistinguishable from the classical Maxwell–Boltzmann distribution function.

The Contribution Due to Radiation Pressure

Because photons possess momentum pγ = hv /c, they are capable of delivering an impulse to other particles during absorption or reﬂection. Consequently, electro- magnetic radiation results in another form of pressure. It is instructive to rederive the expression for radiation pressure by making use of the pressure integral. Substituting the speed of light for the velocity v, using the expression for photon momentum, and using an identity for the distribution function, np dp nν dν, the general pressure integral, Eq. ( 8), now describes the effect of radiation, giving=

1 ∞ Prad hνnν dν. = 3 !0

At this point, the problem again reduces to ﬁnding an appropriate expression for nν dν. Since photons are bosons, the Bose–Einstein distribution function would apply. However, the problem may also be solved by realizing that nν dν represents the number density of photons having frequencies lying in the range between ν and ν dν. Multiplying by the energy of each photon in that range would then give the energy density+ over the frequency interval, or

1 ∞ Prad uν dν, (18) = 3 !0 where uν dν hνnν dν. But the energy density distribution function is found from the Planck functi=on for blackbody radiation. Substituting into Eq. ( 18) and performing the integration lead to

1 4 Main-sequencePrad aT , evolution(19) = 3 Main Sequence and Post-Main-Sequence Stellar Evolution where a is the radiation con•s tanHydrogen burns into helium t. In many astrophysical situations• Mean molecular weight μ increases the pressure due to photons can actually exceed by a signiﬁcant amount the pressure produced by the gas. In fact it is possible that the magnitude Post-AGB PN formation of the force due to radiation pressurein the core can become sufﬁciently great that it surpasses the Superwind gravitational force, resulting• inIf nothing else changes, what does an overall expansion of the system. First He shell flash

Combining both the ideal gasthe pressure do? and radiation pressure terms, the total pressure becomes TP-AGB Second dredge-up ρkT 1 4 He core Pt aT . Decrease! (20) flash

= µmH + 3 ) Pre-white dwarf L /

L E-AGB ( Example 2.1. Using the• resultsInsufficient pressure to support of Example 1.1, we can estimate the central tem-10 He core exhausted perature of the Sun. Neglectingoverlying layers the radiation pressure term, the central temperature is foundLog RGB He core burning • First Core gets slowly compressed dredge-up • Virial theorem: half of the gravitational continued H shell burning SGB energy goes into heat Core contraction • Core temperature increases H core exhausted ZAMS • More efficient nuclear reaction • Both luminosity and effective temperature increase To white dwarf phase 1 M Log10 (Te) FIGURE 4 A schematic diagram of the evolution of a low-mass star of 1 M from the zero-age main sequence to the formation of a white dwarf star. The dotted ph⊙ase of evolution represents rapid evolution following the helium core ﬂash. The various phases of evolution are labeled as follows: Zero-Age-Main-Sequence (ZAMS), Sub-Giant Branch (SGB), Red Giant Branch (RGB), Early Asymptotic Giant Branch (E-AGB), Thermal Pulse Asymptotic Giant Branch (TP-AGB), Post- Asymptotic Giant Branch (Post-AGB), Planetary Nebula formation (PN formation), and Pre-white dwarf phase leading to white dwarf phase.

and becomes nearly isothermal. At points 4 in Fig. 1, the Schönberg–Chandrasekhar limit is reached and the core begins to contract rapidly, causing the evolution to proceed on the much faster Kelvin–Helmholtz timescale. The gravitational energy released by the rapidly contracting core again causes the envelope of the star to expand and the effective temperature cools, resulting in redward evolution on the H–R diagram. This phase of evolution is known as the subgiant branch (SGB). As the core contracts, a nonzero temperature gradient is soon re-established because of the release of gravitational potential energy. At the same time, the temperature and density of the hydrogen-burning shell increase, and, although the shell begins to narrow signiﬁcantly, the rate at which energy is generated by the shell increases rapidly. Once again the stellar envelope expands, absorbing some of the energy produced by the shell

Evolution off the main sequence I Main Sequence and Post-Main-Sequence Stellar Evolution

• Star uses up its initial supply of H in the core • Core contracts, temperature and density in Post-AGB the region around the core increases and PN formation Superwind ignites H burning in a shell First He shell flash

TP-AGB

Hydrogen shell burning Second dredge-up He core • Helium core steadily increases in mass but flash )

Pre-white dwarf L the temperature is not high enough to ignite / L E-AGB (

helium nuclear reaction 10 He core exhausted

• Eventually the helium core cannot support Log RGB He core burning the outer layers (Schönberg-Chandrasekhar First limit) and contract rapidly on the Kelvin- dredge-up H shell burning Helmholtz timescale SGB Core contraction • Gravitational energy released, envelop of H core exhausted the star expands, effective temperature ZAMS decreases Subgiant branch (SGB) To white dwarf phase 1 M

Log10 (Te) FIGURE 4 A schematic diagram of the evolution of a low-mass star of 1 M from the zero-age main sequence to the formation of a white dwarf star. The dotted ph⊙ase of evolution represents rapid evolution following the helium core ﬂash. The various phases of evolution are labeled as follows: Zero-Age-Main-Sequence (ZAMS), Sub-Giant Branch (SGB), Red Giant Branch (RGB), Early Asymptotic Giant Branch (E-AGB), Thermal Pulse Asymptotic Giant Branch (TP-AGB), Post- Asymptotic Giant Branch (Post-AGB), Planetary Nebula formation (PN formation), and Pre-white dwarf phase leading to white dwarf phase.

Evolution off the main sequence II Main Sequence and Post-Main-Sequence Stellar Evolution

• Stellar envelope further expands, effective temperature further decreases Post-AGB • Opacity increases due to the additional PN formation Superwind contribution of the H– ion First He shell flash −3κ(r)ρ(r) L(r) TP-AGB Second dredge-up dT / dr = − 3 2 He core 4acT 4πr flash )

Pre-white dwarf L • Temperature gradient required for energy / L E-AGB (

transport by radiation is large enough for 10 He core exhausted

convection to kick in Log RGB He core burning • Convection zone develops throughout the First stellar interior dredge-up H shell burning • Evolution follows the Hayashi track of the SGB Core contraction pre-main-sequence star, but upwards! H core exhausted • Star rapidly expands and in the meantime, ZAMS cools down

Red Giant Branch (SGB) To white dwarf phase 1 M

Main Sequence and Post-Main-Sequence Stellar Evolution where µenv and µic are the mean molecular weights of the overlying envelope and the isothermal core, respectively. The Schönberg–Chandrasekhar limit is another consequence of the virial theorem. Based on the physical tools we have developed so far, an approximate form of this result can be obtained. The analysis is presented beginning on the facing page. The maximum fraction of the mass of a star that can be contained in an isothermal core and still maintain hydrostatic equilibrium is a function of the mean molecular weights of the core and the envelope. When the mass of the isothermal helium core exceeds this limit, the core collapses on a Kelvin–Helmholtz timescale, and the star evolves very rapidly relative to the nuclear timescale of main-sequence evolution. This occurs at the points labeled 4 in Fig. 1. For stars below about 1.2M , this deﬁnes the end of the main-sequence phase. What happens next is the subject of Section⊙ 2.

Example 1.1. If a star is formed with the initial composition X 0.68, Y 0.30, and Z 0.02, and if complete ionization is assumed at the core–=envelope=boundary, = we ﬁnd that µenv 0.63. Assuming that all of the hydrogen has been converted ≃ into helium in the isothermal core, µic 1.34. Therefore, from Eq. ( 1), the Schönberg– Chandrasekhar limit is ≃ M ic 0.08. M ≃ ! "SC The isothermal core will collapseEvolution if its mass exceeds off 8% of thethe star’s total main mass. sequence III Main Sequence and Post-Main-Sequence Stellar Evolution The Degenerate Electron Gas • For low-mass stars (<~ 1.8 Msun), the helium core The mass of an isothermalcontinues to collapse during evolution to the tip of core may exceed the Schönberg–Chandrasekhar limit if an additional source of pressurethe RGB branch can be found to supplement the ideal gas pressure. This can Post-AGB PN formation occur if the electrons in• theTemperature and density continue to increase in the gas start to become degenerate. When the density of a gas Superwind becomes sufﬁciently high,helium core the electrons in the gas are forced to occupy the lowest avail- First He shell flash able energy levels. Since• Electrons are forced to occupy the lowest energy electrons are fermions and obey the Pauli exclusion princi- TP-AGB ple, they cannot all occupy the same quantum state. Consequently, the electrons are levels possible. Second dredge-up stacked into progressively higher energy states, beginning with the ground state. In the case He core • Electrons are fermions, and they cannot all flash )

Pre-white dwarf of complete degeneracy, the pressure of the gas is due entirely to the resultant nonthermalL /

occupy the same quantum state L E-AGB motions of the electrons, and therefore it becomes independent of the temperature of ( the

10 He core exhausted gas. • They stacked into progressive higher energy

Log RGB If the electrons are nonrelativistic,states, and the core becomes the pressure of a completelyelectron- degenerate electron gas He core burning is given by degenerate First dredge-up • Pressure of a completely degenerate electron H shell burning 5/3 gas is Pe Kρ , (2) SGB = Core contraction • Ultimately the temperature/density become high H core exhausted 3 where K is a constant. If the degeneracy is only partial, some8 temperature dependence enough to ignite helium burning (10 K) ZAMS remains. The isothermal• coreTemperature rapidly increases, yet the pressure of ofa1M star between points 3 and 4 in Fig. 1 is partially degenerate; consequently, the core mass⊙ can reach approximately 13% of the entire mass the degenerate electron gas does not increase -> no To white dwarf phase 1 M 3Note that Eq. ( 2) is a polytropicexpansion to dissipate the energy -> more efficient equation of state with a polytropic index of n 1.5. = helium burning -> explosive energy release! Log10 (Te) FIGURE 4 A schematic diagram of the evolution of a low-mass star of 1 M from the zero-age Helium Core Flash main sequence to the formation of a white dwarf star. The dotted ph⊙ase of evolution represents rapid evolution following the helium core ﬂash. The various phases of evolution are labeled as follows: Zero-Age-Main-Sequence (ZAMS), Sub-Giant Branch (SGB), Red Giant Branch (RGB), Early Asymptotic Giant Branch (E-AGB), Thermal Pulse Asymptotic Giant Branch (TP-AGB), Post- Asymptotic Giant Branch (Post-AGB), Planetary Nebula formation (PN formation), and Pre-white dwarf phase leading to white dwarf phase.

Evolution off the main sequence IV Main Sequence and Post-Main-Sequence Stellar Evolution • Helium core flash “lifts” the degenerate electron gas in the core • Dramatic changes in the stellar interior Post-AGB PN formation • Helium core burning stabilizes Superwind First He shell flash

Helium-core-burning “main sequence” TP-AGB

Second dredge-up But the lifetime is much shorter He core flash )

Pre-white dwarf L

• When helium is exhausted in the core, the star /

L E-AGB (

begins to burn He in a shell around the now carbon 10 He core exhausted

core (resulted from the triple alpha process) -> Log RGB evolution is very similar to the hydrogen burning He core burning First SGB and RGB branch dredge-up H shell burning Asymptotic Giant Branch SGB Core contraction H core exhausted • He shell burning during the AGB phase is highly ZAMS temperature dependent and unstable, and intermittent helium shell flashes occur

• AGB stars have extremely low surface gravity To white dwarf phase 1 M and suffer high mass loss Log10 (Te) Superwind and formation of planetary nebulaeFIGURE 4 A schematic diagram of the evolution of a low-mass star of 1 M from the zero-age main sequence to the formation of a white dwarf star. The dotted ph⊙ase of evolution represents rapid evolution following the helium core ﬂash. The various phases of evolution are labeled as follows: Zero-Age-Main-Sequence (ZAMS), Sub-Giant Branch (SGB), Red Giant Branch (RGB), Early Asymptotic Giant Branch (E-AGB), Thermal Pulse Asymptotic Giant Branch (TP-AGB), Post- Asymptotic Giant Branch (Post-AGB), Planetary Nebula formation (PN formation), and Pre-white dwarf phase leading to white dwarf phase.

Planetary Nebulae

White dwarf • Planetary nebulae • Formed by the expanding gas around a white dwarf progenitor • Got the name as they look like giant gaseous planets when viewed Ring Nebula (M57) through a small telescope in the 19th century Red: Nitrogen, Green: Oxygen, Blue: Helium • Owes the appearance to the ultraviolet light emitted by the hot, “The eye of Sauron” condensed central star • UV light absorbed by the gas sounding the star, causing atoms to become excited/ionized • Excited electrons cascade back to lower energy levels, making the nebula glow in visible light Infrared Image Helix Nebula NGC 2818 Death of low-mass stars

• Crash course by Phil Plait EvolutionMain Sequence and Post-Main-Sequenceof stars with Stellar Evolutionintermediate mass

Post-AGB PN formation Superwind Third dredge-up

TP-AGB First He shell flash Second dredge-up

Pre-white dwarf He core exhausted He core HB E-AGB ) burning L

/ H shell L

( burning SGB No helium core flash 10 RGB

Log Overall contraction First dredge-up Existence of a convective core ZAMS H core exhausted

To white dwarf phase 5 M

Log10 (Te)

FIGURE 5 A schematic diagram of the evolution of an intermediate-mass star of 5 M from the zero-age main sequence to the formation of a white dwarf star. The diagram is labeled acc⊙ording to Fig. 4 with the addition of the Horizontal Branch (HB).

1.0 X 14 X 0.9 16

0.8 X4 0.7

X3 X3 0.6 X12 0.5 XH 0.4 X14

0.3 X4

0.2 X16 X12 X 0.1 14 X12 X'13 0.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Mass fraction FIGURE 6 The chemical composition as a function of interior mass fraction fora5M star during the phase of overall contraction, following the main-sequence phase of core hydrogen burning.⊙ 4 3 The maximum mass fractions of the indicated species are XH 0.708, X3 1.296 10− (2He), 4 3 12 = 3 13 = × 14 X4 0.9762 (2He), X12 3.61 10− ( 6C), X13′ 3.61 10− ( 6C), X14 0.0145 ( 7N), and X = 0.01080 (16O). (Figure= adapted× from Iben, Ap.= J., 143×, 483, 1966.) = 16 = 8

Observations of massive stars: Luminous Blue Variables (LBVs)

The Great Eruption

MV=-1

MV=8 1769 1851 1933 2015 η Carinae: an 120-solar-mass star surrounded by a large Historical visual magnitude of η Carinae nebula consisting of two lobes, each ~0.1 pc across Observations of massive stars: Wolf-Rayet Stars

• Very hot, massive stars • Effective temperature 25,000 K to 100,000 K • Extreme mass loss ~10-5 solar mass per year with powerful stellar winds • The most massive star known to date, R136a1, is a Wolf-Rayet star, estimated to be ~315x solar mass

WR 31a, surrounded by a blue bubble nebula from powerful stellar wind impacting surrounding material Evolution of massive stars

All massive stars with >~8 solar mass probably end their life in a supernova explosion Stellar clusters Main SequenceHR diagram of M3 and Post-Main-Sequence Stellar Evolution

13 M3 AGB Apparent 14 RGB • Very useful in studying stellar magnitude 15 HB 16 P-AGB

evolution, as the stars are 17 V o at the same distance 18 BS 19 TO SGB o born as the same time and 20 with the same composition Turn-off 21 MS • Can be used to find out point 22 Main Sequence– and0.4 Post-Main-Sequence 0.0 0.4 0.8Stellar Evolution 1.2 1.6 distance and age of the B – V –8 FIGURE 17 A color–magnitudeNGC 2362 diagram for M3, an old globular cluster. The major phases of cluster h + ␹ Persei 1.0 ¥ 106 stellar evolution are indicated: main sequence (MS); blueh stragglers+ ␹ Persei (BS); the6 main-sequence turn-off –6 2.0 ¥ 10 point (TO); the subgiant branch of hydrogen shell burning (SGB); the red giant branch along the

Hayashi track, prior to helium–4 core burning (RGB); the horizontal branch6.5 during¥ 106 helium core burning (HB); the asymptotic giant branch during hydrogen and helium shell burning (AGB); post-AGB Pleiades evolution proceeding to–2 the whiteM41 dwarf phase (P-AGB).M41 (Figure adapted2.8 ¥ 10 from7 Renzini and Fusi Pecci, Annu. Rev. Astron. Astrophys.,M1126, 199, 1988. ReproducedM11 with permission from the Annual

9 Clusters, and their associated4 color–magnitude diagrams, offer7.1 nearly ¥ 10 ideal point (yr) Cluster age at turn-off tests of many Absolute aspects of stellar evolution theory. BySun computing the evolutionary tracks of stars of various masses, all having the6 same composition as the cluster, it is possible2.9 ¥ 10 to10 plot the position of Globular cluster M3 magnitude each evolving model on the H–R diagram when the model reaches the age of the cluster. 8 (The curve connecting these–0.4 positions 0 0.4 is known 0.8 1.2 as an 1.6isochrone 2.0 .) The relative number of stars at each location on the isochrone dependsB – V on the number of stars in each mass range withinFIGURE the cluster 19 (theA composite initial mass color–magnitude function), diagram combined for a setwith of Populationthe different I galactic rates clusters. of evolution duringThe absolute each visual phase. magnitude Therefore, is indicated star oncounts the left-hand in a color–magnitude vertical axis, and the agediagram of the cluster, can shed light onbased the on timescales the location of involved its turn-off in point, stellar is labeled evolution. on the right-hand side. (Figure adapted from an Asoriginal the cluster diagram ages, by A. beginning Sandage.) with the initial collapse of the molecular cloud, the most massive and least abundant stars will arrive on the main sequence ﬁrst, evolving rapidly. Beforecontraction the lowest-mass phase related stars to the have Schönberg–Chandrasekhar even reached the main limit sequence, is much less the pronounced. most massive onesAs have a result, already color–magnitude evolved into diagrams the red giantof old region,globular perhaps clusters with even turn-off undergoing points near supernova or explosions.less than 1 M have continuous distributions of stars leading to the red giant region. ⊙ Relatively Few AGB and Post-AGB Stars SinceClose core inspection hydrogen-burning of Fig. 17 lifetimes also shows are that inversely a relatively related small to number mass, continued of stars exist evolution on of thethe cluster asymptotic means giant that branchthe main-sequence and only a fewturn-off stars are point to be, found deﬁned in the as the region point labeled where P- stars AGB (post-asymptotic giant branch). This is just a consequence of the very rapid pace of evolution during this phase of heavy mass loss that leads directly to the formation of white dwarfs.

Blue Stragglers It should be pointed out that a group of stars, known as blue stragglers, can be found above the turn-off point of M3. Although our understanding of these stars is incomplete, it appears that their tardiness in leaving the main sequence is due to some unusual aspect of their evolution. The most likely scenarios appear to be mass exchange with a binary star companion, or collisions between two stars, extending the star’s main-sequence lifetime.

Fate of massive stars Fate of massive stars: Supernovae Transient, but titanic explosions in the Universe

NGC 4526

SN 1994D

The Crab Nebula Reported by Chinese Astronomers in 1054 AD as a “Guest Star” The Fate of Massive Stars

Classes of Supernovae Type I Type II Type Ia Type Ib Type Ic The classification of supernovae

The Fate of Massive Stars ␤ ␣ ␥ ␭ f

FIGURE 6 • Type II, Type Ib , and Type Ic supernovae are core-collapse supernovae associated with the death of massive stars FIGURE 9

3 CORE-COLLAPSE SUPERNOVAE

Example 3.1. E mc =

m E/c /c . . = = = × = ⊙

N × . , = . / . = × ! "! × − " continued

The Fate of Massive Stars

FIGURE 10

iron core Core-collapse supernovae mechanism • Recall Fe is located at the peak of the binding energy iron peak The Fate of Massive Stars per nucleon curve, nuclear fusion stops to release ⊙ energy at iron • The core has very high T, supported by thermal electron degeneracy pressure, and electrons in the The Fate of Massive Stars core have extremely high energies – but no additional photodisintegration energy input Tc ρc − ∼ × ∼ • Under this extreme conditions, two processes occur ⊙ γ n Photodisintegration + → + γ p+ n. + → + Neutron formation p+ e− n νe. + → + • Photodisintegration removes thermal pressure; FIGURE 10 The onion-like interior of an Neutron formation removes electron degeneracy . ⊙ evolved massive star through core pressure – nothing supports the entire envelop of the . × × silicon burning star! . • Core goes through homologous free-fall collapse in a . ⊙ ⊙ ⊙ ⊙ very short time! iron core iron peak ⊙ − photodisintegration Example 3.2. R ⊕ γ n + → + γ p+ n. + → + GM U . R = − . E . ⊙ ⊙ = ⊙ ⊙ Rf R = ≪ ⊕

E R M f . . ≃! G ≃ × ≃ ⊙

Core-collapse supernovae: Energy release • During the collapse, speeds can reach 70,000 km/s (~0.2 c) • Within <1 s, a volume of the size of Earth can been compressed down to ~50 km, ~size of Rhode Island • Releases energy of ~1046 J (homework problem) in days/months, equivalent to the total output of the 100 Sun-like stars for its entire lifetime, 10 billion years! Supernova Explosion

• Catastrophic core collapse results in a blast wave sweeping outward as a shock • Shock encounters the infalling outer material and becomes an accretion shock • The accretion shock continues to march through the star’s envelope, drive the materials outward, and heats them up • When the materials become optically-thin, a tremendous optical display results, with a peak (visible) luminosity of 109x solar luminosity, capable of competing with the brightness of the entire galaxy! Supernova lightcurves Supernova Remnants

Crab Nebula (SN 1054A)

SN 1987A (in LMC) Cassiopeia A (SN 1671A?) Supernova Remnants

Shocked ISM Shocked Ejecta

Reverse Shock

Forward Shock

Cassiopeia A