Evolution Off the Main Sequence: Supernovae

Evolution oﬀ the main sequence: Supernovae

March 16, 2020

1 Supernova observations

Only a three supernovae have been observed within the Milky way galaxy and ﬁve anywhere before the use of telescopes: 1. HB9 was marked on star charts from India about 5000 years ago. 2. SN 185 was viewed by Chinese astronomers in 185 CE 3. On about April 30, 1006 a magnitude −9 star suddenly appeared in the constellation Lupus , bright enough to be seen in daytime and to read by at night. It was reported by astrologers in Europe, the Middle East, and Asia. 4. July 4, 1054, Yang Wei-T’e documented a “guest star” in Taurus, which gradually became invisible after a year. It was also observed in Japan, Korea and Arabia. It was also visible in daylight. This supernova is the source of the Crab nebula (Crab supernova remnant). (Milky Way) 5. Tycho Brahe recorded a supernova in 1572. (Tycho’s supernova) (Milky Way) 6. Johannes Kepler (Tycho Brahe’s student) recorded a supernova in 1604. (Kepler’s supernova) (Milky Way) The closest supernova since Kepler’s supernova occured in the Large Magellenic Cloud in 1987. (SN 1987A)

2 Types of supernovae

Supernovae fall into two (or four) distince classes: • Thermal runaway – Thermal runaway happens either when a white dwarf star accumulates enough matter to reignite, or two white dwarf stars collide. In either case, the reignition disrupts the star, leading to a supernova explosion and a collapsed core. • Core collapse – Direct core collapse occurs as the end state of large stars.

NASA clip on supernova formation Supernovae are classiﬁed by certain distinct spectral characteristics,

• Type I: No Hydrogen lines, indicating that the outer layers of the star were stripped before the supernova

1 – Type Ia: Strong Si II line at 615 nm: found in all galaxies, including those with little new star formation (thermal runaway) – Type Ib: Strong Helium lines: seen only in spiral galaxies, near star forming regions (core collapse) – Type Ic: No strong Helium lines: seen only in spiral galaxies, near star forming regions (core collapse)

• Type II: Strong Hydrogen lines (The Crab Nebula and SN 1987A were Type II.) (core collapse) – Type II-P (plateau) (90% of Type II) – Type II-L (linear) (10% of Type II)

but a moType Ia supernovae are produced by runaway fusion ignited on degenerate white dwarf progenitors, while the spectrally similar Type Ib/c are produced from massive Wolf–Rayet progenitors by core collapse. Brightness of a Type Ia supernova: MB = −18.4. For comparison, the apparent magnitude of the sun is m = −26.7 and the moon is mmoon = −12.6. (The absolute magnitude of the sun, i.e., its brightness if it were 10 parsecs away, is M = +4.7.) Other Types are1.5 to 2 times dimmer.

2.1 Can the energy release of a supernova be produced by nuclear fusion? The question here is similar to whether the sun’s energy could be provided (for billions of years) by chemical burning, but the power released by a supernova is much greater. Suppose free neutrons and protons were somehow suddenly compressed into iron nucleii. This gives the most energy that could be produced from nuclear fusion. To match the light curve of a supernova, this fusion would have to occur in a matter of days. The energy released by a Type II supernova is on the order of 1046J The release of energy in forming one iron nucleus from free nucleons is the binding energy of iron, 492.26 MeV = 7.9 × 10−11J so the number of iron atoms required is 1046 N = = 1.27 × 1056 7.9 × 10−11 Since one iron nucleus weighs

mF e = Z × mp = 56u × 1.66 × 10−27kg/u = 9.3 × 10−26kg the mass of this much iron is 56 −26 Miron = 1.27 × 10 × 9.3 × 10 = 1.18 × 1031

= 5.9 M Forcing six solar masses of iron to fuse in a matter of days seems unlikely; moreover the majority of the star’s mass is already comprised of heavier elements, not free nucleons. We conclude that nuclear fusion is unlikely to be the sole source of the energy of a supernova.

2 2.2 Can the energy of a supernova be produced by gravitational collapse? Consider a diﬀerent possibility. As the core of the star reaches nuclear density, the electrons, protons and neutrons of the atoms are in close proximity. If the electrons and protons were to combine into neutrons, the volume of the star would decrease substantially, releasing gravitational energy. Can this happen energetically? Can it provide enough energy (on the order of 1046J) to drive a supernova? Consider a 20 M star. There are three things to examine: 1. Free neutrons decay, giving oﬀ .782 MeV . Reversing this requires neutrons to form from electrons and protons, and the reaction must provide this much energy for each neutron formed. A 20 M star contains about 1057 protons so collapse will require about .782 × 1057 = 7.82 × 1056MeV J = 7.82 × 1056MeV × 1.6 × 10−13 MeV = 1.25 × 1044J to be lost into the neutrons. 2. The volume change in going from a white dwarf or the core of a large star (say, 5000 km, roughly the size of Earth) to a degenerate neutron gas. The degenerate neutron gas will have at least nuclear density, 2.3 × 1017kg/m3, so the volume of a 4 × 1031kg star will be

Mstar Vstar = ρnuclear 4 × 1031 = m3 2.3 × 1017 = 1.74 × 1014m3

4πR3 and therefore, setting V = 3 , the radius after collapse will be

3 1/3 R = V 4π 3 1/3 = × 1.74 × 1014 4π = 3.46 × 104m = 34.6 km or less. 3. Now we find the gravitational energy released. The total contribution to gravitational potential energy of a constant density shell of mass dm = 4πρr2dr dropped from infinity onto a sphere of the same material of radius r is GM dm dE = − r r G 4πρr3 = − 4πρr2dr r 3 16π2Gρ2 = − r4dr 3 Integrating as r grows from zero to the final radius R of the star, 16π2Gρ2R5 E = − 15

3 4πρR3 4πρR3 9G = − 3 3 15R 3GM 2 = − 5R This is the binding energy of a star of mass M and radius R. The energy change of a star of mass 20 M as the radius changes from 5000 km to 34.6 km is 3GM 2 3GM 2 ∆E = − − − 5R5000 5R50 3GM 2 1 1 = − − 5 R5000 R50 3 × 6.67 × 10−11 × 400 × 4 × 1060 1 1 = − − 5 5000 36.4 = 1.75 × 1051

If only a small fraction of this energy ultimately drives the supernova explosion, there is more than enough gravitational energy.

2.3 Fusion processes in massive stars

For stars above eight solar masses 8 M , the early evolution is similar to that of low mass stars. Looking at the evolution on the Hertzsprung-Russell diagram (see tiff) the main difference is the amount of time spent burning helium. The difference begins at the centers of large stars, which become hot and dense enough to enable further fusion. 4 Adding alpha particles 2He to various fusion products, we find 12 4 16 6 C + 2He ⇒ 8 O + γ 16 4 20 8 O + 2He ⇒ 10Ne + γ 20 4 24 23 + 10Ne + 2He ⇒ 12Mg + γ ⇒ 11Na + e + ν 24 1 23 + 12Mg + 0p ⇒ 12Mg + e + ν Adding another alpha particle to magnesium, we reach silicon,

24 4 28 12Mg + +2He ⇒ 14Si + γ At this point, the core temperature reaches about 3 × 109K (yes, billion!) and the reactions take on a diﬀerent character because the ambient energy is so high. While it is energetically favorable for Si to continue to absorb further alpha particles to yield sulfur, argon and other elements up to iron, the abundance of high energy photons makes it possible for these reactions to go in the other direction as well (⇔), creating an equilibrium mixture of many elements,

28 4 32 14Si + +2He ⇐⇒ 16S + γ 32 4 36 16S + +2He ⇐⇒ 18Ar + γ . . 52 4 56 24Cr + +2He ⇐⇒ 26Ni + γ

56 54 56 56 These equilibrium reactions can include elements above iron 26F e as well, with 26F e, 26F e and 28Ni most abundant. As with hydrogen, helium and carbon, the heavier elements migrate toward the core and the lighter elements at larger radii in layers. The dominant layers outside the iron core are silicon, oxygen, carbon, helium and hydrogen. The bottom of each of these layers is hot and dense enough to continue fusion.

4 2.4 Collapse In addition to the reversal of the fusion reactions, it is also possible for the high photon enegies to break down nuclei through other photodisintegration reactions,

56 4 26F e + γ ⇒ 132He + 4n 4 + 2He + γ ⇒ 2p + 2n Photodisintegration happens in a short period of time once the core becomes hot enough, undoing the long fusion process. A great deal of energy is required for the process, but as we have seen, the energy diﬀerence between iron and free nucleons is of order 1.25 × 1044J, much less than the energy released by gravitational collapse. The two processes drive one another. Energy loss though photodisintegration lowers the central pressure, allowing the core to collapse. The collapse releases gravitational energy, driving further photodisintegration. Core masses at this stage vary from one to a few solar masses. When the temperature and density (for a 15 M star) reach about

9 Tcore ∼ 8 × 10 K kg ρ ∼ 1013 core m3 electron capture by protons occurs, + − p + e → n + νe As we have seen, this also requires energy. A huge amount of energy escapes with the neutrinos (107 times the energy released by Si fusion). Now things happen fast: • Removal of electrons also removes the electron degeneracy of the core. This, together with the energy loss due to escaping neutrinos leads to core collapse. A core the size of Earth can collapse to tens of kilometers in less than one second. This collapse occurs uniformly in the inner parts of the star – matter at twice the distance from the center falls with twice the speed. • For the outer layers of the star, where the velocity exceeds the speed of sound in the star, the infall is slowed by the supersonic shock wave. The oxygen, carbon and helium shells cannot keep up and are left falling inward without participating in the catastrophic core collapse. • The core reaches three times nuclear density, where neutron degeneracy (neutrons held apart by the exclusion principle) causes a rebound. The rebound reaches the speed of sound and forms an outbound shock wave. • The shock, losing energy through photodisintegration of the remaining iron layer, stalls, turning into a stationary accretion shock. • The outer layers accrete onto the stationary accretion shock; the density becomes high enough to trap the outflowing neutrinos (omg!). • Details are murky at this point, but presumably the build-up of neutrino energy is sufficient to drive the shock back into motion toward the surface. This, finally, provides the visible explosion that is the supernova. About one percent of the total neutrino energy is conveyed to the expanding outer layers of the star. The rest escape. When the exploding outer layers reach 100 AU the material is diffuse enough to allows photons to escape. The brightness is a billion times the solar luminosity, comparable to an entire galaxy. The difference between this Type II scenario and the Type Ib and Type Ic supernovae have to do with the composition of the outer layers at the time of core collapse.

5 2.5 Remnant For stars less than about 25 solar masses, the remnant inner core – now a neutron star — remains supported by neutron degeneracy. The bulk of the energy of the gravitational collapse is carried oﬀ by neutrinos. For more massive stars, neutron and quark degeneracy is insuﬃcient to support the star, and it collapses into a black hole.

Extras: 3 An alternate derivation of the Chandrasekhar Limit

In order for gravitational collapse to occur, gravity must overcome the exclusion principle. In the dense center of a large star, the core is held up by electron degeneracy. Subrahmanyan Chandrasekhar received the Nobel prize in 1983 "for his theoretical studies of the physical processes of importance to the structure and evolution of the stars". Speciﬁcally, Chandrasekhar showed that electron degeneracy cannot support a white dwarf star of more than a certain mass, called the Chandrasekhar limit. To understand the limits of the exclusion principle, we ﬁrst need to examine the pressure associated with electron degeneracy and from gravity.

3.1 The pressure of a degenerate electron gas From the uncertainty relation, ∆p ∆x ≈ ~ x 2 where we neglect the factor of 2. As material is compressed, the electrons must have an uncertainty in momentum (and therefore a momentum) on the order of

p ≥ ∆p ≈ ~ x x 2∆x

with the same relation holding in the y and z directions. If the number of electrons per unit volume is ne, 1 −1/3 then the volume per electron is about Ve = so a small cubic volume has linear dimension ∆x ∼ ne , n3 1 p ≈ n1/3 x 2~ e √ q 2 2 2 1/3 and total pressure p = px + py + pz = 3~n . Since our approximation is coarse, we drop the small numerical factors and simply write 1/3 p ≈ ~ne If the electrons are nonrelativistic, the velocity is v = p . me Now compute the pressure. In a small cubical volume of side ∆x, the time between collisions with one wall is ∆t = 2∆x . Assuming elastic collisions with the wall, the momentum imparted to the wall by a particle vx going from momentum px to −px is the impulse, 2∆x 2px = F ∆t = F vx so the force on the wall from one collision is v p 1 pv F = x x ≈ x ∆x 3 ∆x

6 The pressure per collision is given by the force per unit perpendicular area, F 1 pv 1 P = = × A 3 ∆x ∆y∆z 1 pv = 3 ∆V

N The number density relates the number of particles in the volume ∆V by n = ∆V so the total pressure in the volume is 1 P = NP = npv total 3 1 In keeping with our approximation we ignore the factor of 3 and substitute for the momentum p and the velocity p , m3

Ptotal ∼ npv 1/3 1/3 ~ne = n × ~ne × me 2n5/3 = ~ e me

Zne and ﬁnally, replacing the number density of electrons by the mass density of the star, ρ ≈ mH n = mH A

2 5/3 ~ ρ Pdegeneracy ∼ me mH

Z where we again neglect a small factor, A ∼ 2. This result is actually pretty close.

3.1.1 The pressure due to gravity For a star of constant density ρ, the gavitational pressure may be found by integrating. Of course, the density is not constant, but this gives a crude approximation. First, the mass within a radius r is given by

r m (r) = 4π ρr2dr ˆ 0 4π = ρr3 3 At radius r, a small cylinder of height dr and cross-section A has a diﬀerence in pressure between the top and bottom dependent on the additional force exerted by the mass within the cylinder. The mass dm within the cylinder is dm = ρAdr and the gravitational force on the cylinder is

Gm (r) G 4π dm = ρr3 ρAdr r2 r2 3

The pressure diﬀerence is the change in force per unit area, so

4πGρ2 dP (r) = rdr grav 3

7 Since the pressure at the surface drops to zero, we choose the surface r = R for our reference point and integrate from rto the surface. We ﬁnd

R 4πGρ2 P (r) = rdr grav 3 ˆ r 2πGρ2 = R2 − r2 3

which correctly gives Pgrav (R) = 0. In this approximation, at the center of the star, r = 0, the pressure due to gravity is therefore 2π P = Gρ2R2 grav 3 The crucial observation indicating gravitational collapse is that gravitational pressure increases as the square 5 of the density while the degeneracy pressure supporting the star goes only as density to the 3 . Dropping the 2π 3 and other small numerical factors, these pressures become balance when

2 ρ 5/3 Gρ2R2 ≈ ~ me mH 2 Gρ1/3R2 = ~ 5/3 memH M 1/3 2 G R2 = ~ R3 5/3 memH 2 1 R = ~ 5/3 M 1/3 GmemH This means that the radius of the core decreases as its mass increases. This essential result for fermion degeneracy may be summarized by taking the cube. Since the volume is proportional to R3 we have

MdegenerateVdegenerate = constant

This same relation will hold whether the degenerate particles are electrons, neutrons, or quarks. As more iron builds up in the core of a massive star, its density increases without bound. Eventually, the degenerate electrons must be conﬁned to the volume of the nucleus. We show below that at high enough density and temperature, there is enough ambient photon energy γ to reverse the neutron decay process from + − n → p + e +ν ¯e to

+ − p + e + γ → n + νe

At this point, the core begins to collapse. The mass at which this happens is the Chandrasekhar limit.

4 Progenitors: Highly variable type O stars

Certain stars are extremely variable and have unusual spectra: • Luminous blue variables (extreme upper left in HR diagram) • Wolf-Rayet stars: unusual spectra: – WN: He and Nitrogen lines

8 – WC: He and Carbon lines – WO: Oxygen lines – Subclasses within these by degree of ionization See images. These are all now seen to be late evolutionary stages of stars of type O stars. In all of these, substantial expulsion of mass occurs when the luminosity pressure exceeds the gravitational attraction. As they expel large amounts of mass from their outer layers, they expose deeper and deeper layers of the star containing diﬀerent elements.

M > 85M O supergiant ⇒ O : strong lines LBV W R : Nitrogen W R : Carbon SN 40M < M < 85M O ⇒ strong lines W R : NWR : CSN 25M < M < 40M O ⇒ Red supergiant W R : NWR : CSN 20M < M < 25M O ⇒ Red supergiant W R : NSN 10M < M < 20M O ⇒ Red supergiant Blue supergiant SN

In each of these classes, SN stands for supernova.