Approximate Author Title Comments price

Good overall introduction to Stellar Structure Free O. R. stellar evolution at the upper and Evolution (pdf available) ASTRONOMY 220C Pols division undergraduate level.

Stellar Structure Free ADVANCED STAGES OF STELLAR EVOLUTION Collins Web based graduate level text AND NUCLEOSYNTHESIS and Evolution (on line)

Kippenhahn and Stellar Structure $88.67 Great “introductory” textbook Winter, 2019 Weigert and Evolution (hardcover) (more for 220A though)

Excellent recent (2017) book on just Branch and Supernova the supernova par of the course. http://www.ucolick.org/~woosley $109.81 Wheeler Explosions Expensive. Can be “rented” for half price.

A classic. Great on nuclear physics Principles of Stellar Don $<40.85 and basic stellar physics. Good on Evolution and Clayton (paperback) the s-process, but quite dated Nucleosynthesis otherwise. Used $10.00. Buy it.

I. Preliminaries

Stars are gravitationally confined thermonuclear reactors.

The life of any star is a continual struggle between Lecture 1 the force of gravity, seeking to reduce the star to a point, and pressure, which holds it up. A balance is maintained.

So long as they remain non-degenerate and have not Overview encountered the any instabilities, overheating leads to expansion and cooling. Cooling, on the other hand, leads to contraction and heating. Hence stars are stable. Time Scales, Temperature-density The Virial Theorem works. Scalings, Critical Masses But, since ideal gas pressure depends on temperature, stars must remain hot. By being hot, they are compelled to radiate. In order to replenish the energy lost to radiation, stars must either contract or obtain energy from nuclear reactions. Since nuclear reactions change their composition, stars must evolve. 18 Theory of supernovae

3.2. Final fate as a function of stellar mass

As seen from eq. (3), smaller-mass stars have higher interior densities than larger-mass stars for the same central temperature and thus tend to form electron degenerate cores at earlier stages of evolution. The Virial Theorem implies that if a star (with constant In fig. 3, the earlier evolution of the central density, Pc, and temperature, T o is shown for helium stars density in this example) is neither too degenerate nor too relativistic Central Conditions at death the (radiation or pair dominated) (or cores) of mass M = 8, 6, 3.3, 3.0, 2.8, and 2.2 M e [27]. The approximate ignition lines for carbon, iron cores of 1 3GM 2 neon, oxygen, and silicon, where the nuclear energy generation rate is equalmassive to neutrino stars energy losses, ∼2MN kT (for an ideal ionized hydrogen, constant T ) 2 5R A are shown. The line for $ = 10 approximately separates the electron degenerateare somewhat and nondegenerate 3GM degenerate T ∼ regions, where ~ is the chemical potential of an electron in units of kT. These evolutionary paths clearly 20N AkR 1/3 indicate the effect of electron degeneracy and its dependence on stellar mass. ⎛ 3M ⎞ R ∼ for constant density ⎝⎜ 4πρ ⎠⎟ Helium stars of M~ = 4-8 M e undergo nuclear burning under nondegenerate conditions. In contrast, 1/3 2/3 2/3 the 2.2 M o helium core enters the strongly degenerate region after carbon burning. Stars of M~ = ⎛ 3 ⎞ ⎛ 4π ⎞ GM 1/3 GM 1/3 So T∼ ⎜ ⎟ ⎜ ⎟ ρ = 0.093 ρ ⎝ 20⎠ ⎝ 3 ⎠ N Ak N Ak 2.8-3.3 M e are intermediate: the O + Ne + Mg core becomes semi-degenerate and whether they will This is about 5 MK for the average temperature of the sun, while the enter the degenerate or nondegenerate region is an interesting question. The final stages of evolution central temperature is about 2 to 3 times greater. are classified by the stellar mass as follows (see refs. [34, 25, 45] for reviews and references). If the central temperature has the same sort of scaling (1) Mm, <0.08 Me: The star will become a planet-like blackρ ∝ Tdwarf3 without igniting hydrogen as the average, 2/3 burning. GM 1/3 Tc ∼ ρc N Ak (2) 0.08 M O < Mms < 0.45 Mo: The star will end up as a helium white dwarf, though such a single star has not evolved off the main sequence in a Hubble time. That is as stars of ideal gas contract, they get hotter and since That is, as a star of given mass evolves, its central temperature a given fuel (H, He, C etc) burns at about the same temperature, (3) 0.45 M O < Mms < 8 M®: The star forms a C-O core of mass smaller than 1.06 M e, which then more massive stars will burn their fuels at lower density, i.e., becomes strongly degenerate. rises rMostoughly aofs ththesee cub e starsroot o fwill its ce becomentral densi tC-Oy while dwarfs by losing their −2 3 higher entropy. ρ ∝ M T hydrogen-rich envelope, but some (6-8 Me) could reach a supernova stage by increasing the C-O core mass to the Chandrasekhar mass and igniting carbon deflagration. The explosion may look like a Type II-L supernova [9, 40]. (4) 8 M@ < Mms < 10 Mo: The star undergoes nondegenerate carbon burning and forms a degener- ate O + Ne + Mg core [23, 26]. The mass of the O + Ne + Mg core (<1.37 Me) is too small to ignite More generally for helium cores of constant M ≈ 10 − 25 M Burning Processes neon. Further evolution is due to the growth of the mass of the O +Z AMNeS + Mg core⊙ toward the mass, 2.2, 2.8, 3.0, 3.3, 6 and 8 MO (Nomoto and Hashimoto 1988)

/ I I I I I I I I _ _ I photo disintegration IU.L} F 7" < 415 / I G R /z../H d/'/,/H~-LU'~ I ,IX'- / ~ =~ 95L I/,Y , Si v • "/Ill/In ...... :', ...... I o .- s.s c,." 2 I e+e-pair ,,,~Ne '8"'~"'

Fig. 3. The evolution of the central density,It turns Pc, out and that temperature, MHe =35 MTo,O iswill shown just forbrush helium the starse+e -ofpair mass instability M s = 8, 6, 3,3, 3.0, 2.8, and 2.2 M e. The ignition lines for carbon, neon, oxygen, and silicon, where the nuclear energy generation rate is equal to neutrino energy losses, are shown. The line for ¢, = 10 approximately separates the electron degenerate and nondegenerate regions, where ~b is the chemical potential of an electron in units of kT. These instabilities (cross hatched lines) have dramatic consequences for the star: Supernovae are powered by one of two sources: • Pair instability can lead to pulsations (pulsational pair- instability supernovae), explosion (pair-instability supernovae), or collapse to a black hole strong • Thermonuclear - white dwarf explosions and force • Electron capture can rob the core of pressure support pair instability and cause collapse to a neutron star – resulting in a supernova) • Gravitational collapse – aka “core collapse” - gravity the a fraction of the binding energy of the neutron star • Photodisintegration can also cause collapse to a weakest force or black hole transported by neutrinos or rotation neutron star or black hole and make a supernova and magnetic fields

There are critical masses for all these occurrences

Similarly the light curve and spectrum depend on What kind of supernova you see depends on the the properties of the star that blew up, especially whether properties of the star (and its surroundings) in which it had a hydrogen envelope or not. Obviously white dwarfs these instabilities operate and Wolf-Rayet stars will not make Type II supernovae.

• White dwarf – explosion shatters the star, but by the time the debris expand enough to let the light out the initial explosion energy has been degraded to essentially nothing. Entirely a radioactive display

• Giant star – enough energy is retained (1%) that when the supernova expands and releases it (100 AU), the supernova stays bright for months

• Wolf-Rayet star – like a white dwarf, the display is chiefly radioactive with perhaps some early activity from the explosion, but the explosion mechanism is collapse.

• Magnetar, circumstellar interaction, and pair instability for special cases Stars inherently turn light elements into heavier ones

Pair instability SNe and white dwarf explosions Massive stars and the supernovae (and the neutron stars leave behind nothing. The rest leave an interesting that they make) are responsible for the synthesis of most distribution of compact remnants of the elements heavier than helium. (Some are made by lighter stars and one or two by cosmic ray spallation)

Including envelope Helium core

Sukhbold and Woosley (2018)

78 WOOSLEY ET AL. Vol. 151

1 Primordial α−process# 1957 4 H He e-process Part of the KEPLER network 3 4 Hydrogen burning 5 6 7 8 9 10 Li Be B C N O F Ne x-process s-process Each nucleus is subject to 11 12 Helium burning r-process 13 14 15 16 17 36 up to 15 strong and weak Na Mg Al Si P S Cl Ar reactions (plus neutrinos) 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 K Ca Sc Ti V Cr Mn Fe Co Ni Cu Zn Ga Ge As Se Br Kr

37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 Rb Sr Y Zr Nb Mo Tc Ru Rh Pd Ag Cd In Sn Sb Te I Xe

1 Now 4 H Big Bang Oxygen burning He 3 4 Neutrino irradia=on Silicon burning 5 6 7 8 9 10 during supernova Li Be and the e-process B C N O F Ne Carbon and 11 12 Neon burning 13 14 15 16 17 36 Na Mg Al Si P S Cl Ar

19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 Fig. 1.—Color-codedDuring the nuclearevolution mass excesses of a taken massive from different star data its sources constituent in the region of nuclei interest for are the rp subject-process. A darkto a line vast shows array the location of of nuclei K Ca Sc Ti V Cr Mn Fe Co Ni Cu Zn Ga Ge As Se Br Kr with Z nuclearN. The actual reactions size of the network involving, used in these protons, studies varied neutrons, with time and alpha model (-particles,2.1). Red triangles photons, indicate nuclei electrons, with experimentally determined mass excesses¼ (Audi & Wapstra 1995). Nuclei with black circles surrounding a symbol indicatex that an extrapolated or interpolated mass is available from Audi & 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 Wapstra.positrons, For those circles and enclosing neutrinos. a red triangle, The these estimatedKEPLER values nuclear were used. Greendata squares deck , circled for orstellar not, show nucleosynthesis where data from Brown et studies al. (2002) and A. Brown (2002, private communication) were used. These add a calculated displacement energy to the Audi-Wapstra masses for N Z to obtain the masses of Rb Sr Y Zr Nb Mo Tc Ru Rh Pd Ag Cd In Sn Sb Te I Xe mirror nucleiIncludes with Z > 5442N. For a nuclei subset of theseand (black 105,000 squares inside reactions green squares (plus) from Browntheir et inverses). al. (2002), the errors Most in the (non calculated-r!-process) displacement energies are 100 keV.studies The solid use green squaresabout that 1/3 cover of a wider this. region are from A. Brown (2002, private communication) and have a larger error of several hundred keV in the $calculated displacement energy due to the wider extrapolation and the possible effects of deformation in the A 80 mass region. In the region of N Z from A 76 100, the error in the mass excess is dominated by the error in the Audi & Wapstra extrapolations, which¼ are on the order of 0.5 MeV. Finally, blue¼ circles indicate¼ mass excess data taken from Mo¨ller et al. (1995). These were used wherever shown (circled or uncircled).

nuclei is reasonable. Figure 2 illustrates important nuclear and dergo e+-decay to their isospin mirrors. For these nuclei, the weak flows near the waiting point nucleus 72Kr. These flows ground state of the (e+)daughteristheisobaricanalogstate are typical of those near the other waiting point nuclei in this (IAS) of the parent ground state, the first excited state of the mass region. daughter is the IAS of the first excited state in the parent, and As can be seen from Figure 2, important weak decay so on for all of the levels. Because the nuclear Hamiltonian parents fall into four categories. Nuclei with Z N 1un- commutes with the isospin raising and lowering operators, the ¼ þ Q-values and rates for all of these (parent level) (IAS in ! TABLE 1 daughter) Fermi transitions are approximately the same. GT Proton Separation Energies of Isotones near the Long-lived transitions are not expected to dominate the decay rates be- Waiting Point Nuclei 64Ge, 68Se, 72Kr, and 76Sr cause the GTstrength is typically spread out over a wide range in daughter excitation energy. In addition, electron capture a Sp Uncertainty cannot compete with these large Q-value + decays at the Nucleus (MeV) (keV) small temperatures and electron Fermi energies achieved in 65As ...... 0.43 290 X-ray bursts. Consequently, the decay rates of Z N 1 À ¼ þ 66Se...... 2.43 180 nuclei are essentially temperature and density independent 69Br...... 0.73 320 (for the range of conditions found in X-ray bursts). For nuclei À 70Kr ...... 2.14 190 with Z N 2 even, the situation is similar. Each parent 73Rb...... 0.55 320 state decays¼ þ via a¼ Fermi transition to an IAS in the daughter, À 74Sr ...... 1.69 210 and the decay rates for these nuclei are well approximated by 77Y...... 0.23 Unknown À the ground state decay rate. 78Zr...... 1.28 Unknown Nuclei with N Z odd also decay principally via Fermi ¼ ¼ a Taken from the compilation of Brown et al. 2002, except for the transitions. However, while for these nuclei it is true that every proton separation energies of 77Yand 78Zr, which were taken from the low-lying daughter state has a low-lying IAS in the parent, the unpublished calculations of A. Brown 2002 (private communication). converse is not true. Parent states in these N Z odd nuclei ¼ ¼ STELLAR PHYSICS The evolution of stars, supernovae, and explosive transients is You may know better the equations of stellar governed by gravity, thermodynamics, and hydrodynamics as structure (Q =0) in Lagrangian coordinates embodied in two equations (Landau and Lifshitz 1959) – plus many subsidiary conditions – opacity, mixing, transport rotation, etc ∂r 1 = continuity ∂m 4πr 2ρ dm= 4πr 2ρ dr − 4πr 2ρv dt Force or Euler equation momentum ∂L ∂ε P ∂ρ ∂ ρr 2v ∂ρ −2 ( ) = Snuc − + 2 ⇒ = − r ∂m ∂t ρ ∂t ∂t ∂r energy see A.Weiss notes ∂P Gm 1 ∂v = − − ∂m 4πr 4 4πr 2 ∂t 4 ∂ v / r where Q ≡ η r 4 ( ) and η is the dynamic viscosity 3 v r V ∂T GmT ∂ = ∇ and the rest of the terms have their usual meaning. ε is ∂m 4πr 4P 3κ L(m)P ∂L or the internal energy per gram and is the flux of energy ∇ = ∇rad = 4 ∂m 16πcGmaT ∇ <∇ is stable due to convection or diffusion. Rotation and magnetic ⎛ ∂ lnT ⎞ r ad ∇ = ∇ad =⎜ ⎟ ⎝ ∂ lnP ⎠ S fields are neglected but could be included.

Or neglecting time dependent terms and expressing in In addition there is an equation that describes the radial (Eulerian) coordinates mixing of composition

dm 2 2 ∂X ∂ ⎡ 2 ∂X ⎤ = 4πr ρ mass conservation i = 4πr ρ D i dr t m ⎢( ) r ⎥ ∂ ∂ ⎣ ∂ ⎦

dP Gm(r)ρ = − 2 hydrostatic equilibrium Ions are mixed by convection (time-dependent) dr r and by other processes – rotation, semiconvection, and convective overshoot – but not by diffusion. dL = 4πr 2ρS dr nuc energy generation The transport physics enters in to the calculation of D. See Podsiadlowski notes.

dT 3κρ L(r) 1 = 3 2 = 2 L(r) diffusion dr 16πacT r 4πr Kcond TIME SCALES The free fall time scale is ~R/vesc so These equations have associated with them four time scales. The 3 1/2 R ⎛ R ⎞ M 4πρ shortest is the time required to approach and maintain hydrostatic ~ τ esc ∼ = ⎜ ⎟ 3 equilibrium. Stars not in a state of dynamical implosion or explosion vesc ⎝ 2GM ⎠ R 3 maintain a balance between pressure and gravity on a few sound 1/2 ⎛ 3 ⎞ R crossing times. The sound crossing time is typically comparable ∼ = ⎜ 8πGρ ⎟ R" to the free fall time scale. ⎝ ⎠

dP GM ρ The number out front depends upon how the = − dr r2 time scale is evaluated. The e-folding time for the GM ρ density is 1/3 of this Pcent ∼ 2R and 1/2 ⎛ 1 ⎞ 446 ! 1/2 1/2 τ sec R ρ! 1 ⎛ P⎞ GM HD  ⎜ ⎟  3 = = ⎛ ⎞ ⎝ 24πGρ ⎠ ρ R cs = γ ∼ ⎜ ⎟ ρ τ HD ⎝⎜ ρ ⎠⎟ ⎝ R ⎠ 1/2 ⎛ 2GM ⎞ v ∼ esc ⎝⎜ R ⎠⎟ which is often used to describe explosions as well as collapse. So the escape speed and the sound speed in the deep interior are comparable. A shock wave can thus lead to mass ejection.

The thermal diffusion coefficient (also called the thermal The second relevant adjustment time, moving up in diffusivity) is defined as scale, is the thermal time scale given by radiative ⎛ conductivity ⎞ K diffusion, appropriately modified for convection. D = = T ⎜ heat capacity⎟ C This is the time it takes for a star to come into and ⎝ ⎠ P ρ maintain thermal steady state, e.g., for the energy generated where K appears in Fourier's equation in the interior to balance that emitted in the form of radiation heat flow = -K ∇T at the surface. (not all stars are in thermal steady state) For radiative diffusion the "conductivity", K, is given by 4acT 3 R2 K = (see Clayton 3-12) τ = 3κρ therm D therm where κ is the opacity (cm2 g−1), thus ⎛ 4a cT3 ⎞ D = where D is the diffusion coefficient. In the simplest case, T ⎜ 2 ⎟ ⎝ 3κC ρ ⎠ D is characteristically 1/3 times a length scale (e.g. the mean P −1 −1 free path) times a characteristic speed (e.g., the speed of light). where CP is the heat capacity (erg g K ) D here thus has units cm2 s−1 If most of the energy does not reside in radiation this may be T Note that multiplied by a dimensionless correction factor. ⎛ c ⎞ ⎛ aT 4 ⎞ ⎛ c ⎞ ⎛ radiation energy content ⎞ D ≈ = ⎝⎜κρ ⎠⎟ ⎜ ρC T ⎟ ⎝⎜κρ ⎠⎟ ⎝⎜ total heat content ⎠⎟ ⎝ P ⎠ If radiation energy density is a substantial fraction of the internal energy (not true in the general case), D ~ c/κρ with κ the opacity, A closely related time scale is the Kelvin Helmholtz time scale and taking advantage of the fact that in massive stars electron scattering GM 2 M 5/3ρ1/3 τ ≈ ∝ 1/3 2 -1 KH i.e. R ~ (M / ρ) dominates so that κ = 0.2 to 0.4 cm g , the thermal time scale then 2RL L scales like Except for very massive stars, L on the main sequance R2κρ ⎛ 0.2R2 ⎞ 3M M t ≈ ~ ⎛ ⎞ rad ⎜ ⎟ ⎜ 3 ⎟ ∝ c ⎝ c ⎠ ⎝ 4πR ⎠ R is proportional to M to roughly the power 2 to 4, and ρ decreases however, massive stars have convective cores so the with M so the Kelvin Helmholtz time scale is shorter for more thermal time is generally governed by the diffusion massive stars. Note that there are numerous Kelvin time in their outer layers. Since the dimensions are still Helmholtz time scales for massive stars since they (several) solar radii while the densities are less and the typically go through six stages of nuclear burning. opacity about the same, the radiative time scales are somewhat During the stages after helium burning, L in the heavy less than the sun (1.7 ×105 yr; Mitalas and Sills, ApJ, 401, 759 (1992)). element core is given by pair neutrino emission and the Kelvin Helmhotz time scale becomes quite short - e.g. a protoneutron star evolves in a few seconds.

25 M Presupernova Star (typical for 9 - 130 M )   1400 R  0.5 R  H, He He 240,000 L 

Different time scales will characterize different regions of the star C,O Si, S, Ar, Ca

Fe O,Mg,Ne

He, C e.g., the sun is in thermal 0.1 R 0.003 R steady state. A presupernova   star is not. Examples of Time Scales Examples of Time Scales

• In stellar explosions, the relevant time scale is the hydrodynamic • In the pulsational pair instability supernova the time between one. pulses is the Kelvin Helmholtz time.

• Explosive nucleosynthesis happens when τ ~ τ • In Type Ia supernovae, the runaway occurs when the convection HD nuc (i.e., thermal) time scale equals the nuclear time scale. In between stages of nuclear burning, < . The evolution • τKH τnuc • Rotation and accretion can add additional time scales. E.g. Eddington occurs on a Kelvin Helmholtz time Sweet vs nuclear. Accretion vs nuclear. Convection also has its own time scale. • In a massive presupernova star the nuclear time scale in the inner core is less than the thermal time scale in the envelope. Summary τnuc < τthermal. Thus the outer layers are not in thermal equilibrium with the interior. The core evolves like a separate star. Time scale Value in sun Scaling ! − HD 30 min (R3/GM)1/2 • In a supernova of Type I maximum light occurs when the age ! − diffusion 2 x 105 y , M/(Rc) is equal to the diffusion time ! − KH 3 x 107 y GM2/(RL) ! − nuc 6 x 109 y qM/L

Relevant Stellar Physics Stellar Physics

Mass Loss – M! (L,R,X ), binary mass exchange • i • Equation of state – P(ρ,T,X i), ε(ρ,T,X i) Perhaps the greatest uncertainty affecting modern Perhaps the best understood part except at super studies of stellar evolution. Moderately well understood nuclear density. Can be complicated when the for line driven mass loss in hot stars. Very poorly electrons are semidegenerate and semirelativistic understood for giant stars and for mass exchanging and in the presence of partial ionization but modern binaries. Very uncertain for massive stars (e.g. Eta computers handle it easily. Carina). Scaling with metallicity uncertain. • Convection • Opacity – κ(ρ,T,X i) Moderately well understood in slowly evolving stars Easy for electron scattering – though again far from convective boundaries. Poorly understood problematic for partial ionization. Complex otherwise at convective boundaries (overshoot, undershoot), but there are tables. Uncertain at low temperature in semiconvective regions, in rotating stars, and where dust and molecules form. Uncertain for in stars where the convective time scale is close to heavy element hydrogen-free compositions the evolutionary time scale Stellar Physics Stellar Physics

• Nuclear Physics – Snuc(T,ρ,Xi) • Explosion Physics

Becoming well understood. Still a few critical An area of great activity and uncertainty for decades. reaction rates poorly determined. A variety of mechanisms operate involving thermonuclear • Rotation and Magnetic Fields instability, neutrino transport, rotation, and magnetic fields. There are basic physics problems afflicting Rotation affects compositional mixing which affects each of them - except for pair-instability supernovae. the overall evolution. Processes understood qualitatively E.g., flame physics and detonation, 3D neutrino transport, but not very quantitative. Angular momentum transport and magnetic instabilities. by magnetic torques during the evolution is a source of great uncertainty which affects our understanding of the explosion process and compact remnant properties.

Stellar Physics

• Abundances The initial star has a composition given by all the activity that went on before it was born. A common assumption is that modern stars are born with a composition like the sun. Other “metallicities” require Some Critical Masses some assumptions about how different elemental ratios scale.

• Chaos A recent realization is that even for well defined physics and initial conditions the final outcome of the evolution of a given massive star may be at some level indeterminate. Central Conditions for Polytropes Eq. 2-313

since 1/3 1 ⎛ 4πρ ⎞ = c R ⎜ 3M ⎟ ⎝ ⎠

for constant density

= 24

most stars have 1.5 < n < 3

Actually the dimensions of Y are Mole/gm and NA has dimensions particles per Mole.

N k P = Σ n kT = A ρT i µ Xi= mass fraction of species “i” −1 ⎛ ⎞ Y = ⎜ ∑ i ⎟ ⎝ ⎠

N Y n Z n ρ A e = e = ∑ i i = ZY ∑ i i N ZY = ρ A ∑ i i

0.5 < µ < 2 −1 = (1+ Z )Y ( i i )

(The limit µ=2 is achieved as A goes to infinity and Z = A/2, i.e. electrons dominate)

For advanced stages of evolution where A > 1, most of the pressure is due to the electrons(and radiation)

This would suggest that the Tc P ∝ρ ⎛ Pideal ⎞ c c µ ratio would increase as the =⎜ ⎟ star evolved and µ became greater. ⎝ Ptotal ⎠ P T P ∝ ideal ∝ ρ c tot β c µβ Decrease in β as star evolves acts to lower T3/ρ.

T ∝ µρ1/3 c c As before but now the relation for Tc is more correctly derived. CRITICAL MASSES

The decline of T3/ρ mostly reflects the fact that beyond H burning the star becomes a red giant and no longer is a single polytrope.

Mcore is essentially reduced. This more than compensates for the increase in µ. A decreasing β also decreases T3/ρ. Ιn the final stages degeneracy becomes important.

Ideal gas plus radiation

ρ ∝ T 3

note cancellation of ρ c

= 1.45 M  if Ye = 0.50 neglecting Coulomb corrections

and relativistic corrections. 1.39 M if they are included. The Chandrasekhar Mass For lower densities and hence degenerate cores Specifying a core mass gives a maximum temperature significantly less than the Chandrasekhar mass achieved before degeneracy supports the star. If that maximum non-relativistic degeneracy pressure gives another temperature exceeds a critical value, burning ignites solution (γ = 5/3)

M1 > M 2 M

This is the well known central density mass relation for (non-relativistic) white dwarfs

Critical Temperatures He core 100% He Given by “balanced power” for post-helium burning phases CO Core 50%C- 50% O and stellar models or polytropes for H and He. Ne-O Core 75% O- 25% Ne

Ne-O cores

CO-cores

He-cores The calculations shown give critical masses: Main Sequence Critical Masses

0.08 M⊙ Lower limit for hydrogen ignition C ~1.0 0.45 M⊙ helium ignition Ne ~1.25 All stars with main sequence mass O 1.39 above the Chandrasekhar 7.25 M⊙ carbon ignition mass could in principle go on to Si 1.39 burn Si. In fact, that never 9.00 M⊙ neon, oxygen, silicon ignition (off center) happens below ~8 solar masses. 10.5 M ignite all stages at the stellar center Stars develop a red ⊙ giant structure with a low density ~ 70 M First encounter the pair instability (neglecting mass loss) surrounding a compact core. ⊙ The convective envelope ~35 M⊙ Lose envelope if solar metallicity star dredges up helium core material and causes it to shrink. Only for stars above about 8 These are for single stars calculated with the KEPLER code or 9 solar masses does the He including semiconvection and convective overshoot mixing core stay greater than the Chandrasekhar mass after but ignoring rotation. With rotation the numbers may be helium burning. shifted to slightly lower values. Low metallicity may raise the numbers slightly since less initial He means a smaller helium core. Other codes give different results typically These are core masses. The corresponding to within 1 solar mass. Results for binaries will differ. main sequence masses are larger.

Mass (solar End point Remnant masses) Between 8 and 10.5 solar masses the evolution < 7 to ~8 planetary nebula CO white dwarf can be quite complicated and code dependent degenerate core Ne-O WD below 9? owing to the combined effects of degeneracy and ~8 to ~11 neutrino-powreed neutron star above 9 neutrino losses. Off-center ignition is the norm for low energy SN the post-carbon burning stages. neutrino-powered normal supernova; SN Ibc in binary. neutron stars and ~11 - ~20 Mass loss introduces additional uncertainty, especially Islands of explosion at black holes higher mass with regard to final outcome. Does a 8.5 solar mass without mass loss main sequence star produce a NeO white dwarf black hole probably no SN (unless or an electron-capture superno. 20 - 70 rotationally powered); if enough mass loss with mass loss SN Ibc neutron star Above 9 solar masses an iron core eventually pulsational pair SN 70 – 150 black hole forms – on up to the pair instability limit. if low mass loss pair instability SN 150 - 260 none if low mass loss pair induced collapse if > 260 low black hole mass loss Minimum mass supernova Presupernova stars – Type IIp and II-L

Smartt 2009 ARAA Smartt, 2009 Fig. 6b ARAA

Based on the previous figure. Solid lines use The solid line is for a Salpeter IMF with a maximum mass of 16.5 solar masses. The dashed line is a Salpeter IMF with a maximum of 35 observed preSN only. Dashed lines include upper solar masses limits.