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Stars and Stellar Processes

Mike Guidry

Chapter 20

Supernovae - all with Video Answers

Educators


Chapter Questions

04:11

Problem 1

How much thermonuclear energy is produced if a $\mathrm{C}-\mathrm{O}$ white dwarf of $1.4 M_{\odot}$ is burned to ${ }^{56} \mathrm{Ni}$ ? Compare this energy with the gravitational binding energy of the white dwarf, assuming uniform density and a mass of $10.4 \mathrm{M}$.

Averell Hause
Averell Hause
Carnegie Mellon University
01:22

Problem 2

Estimate the dynamical timescale for collapse of the iron core of a $25 M_{\odot}$ star. Assume the density of the core to be $5 \times 10^9 \mathrm{~g} \mathrm{~cm}^{-3}$ at the beginning of collapse.

Zachary Warner
Zachary Warner
Numerade Educator
02:09

Problem 3

How much gravitational energy is released if the iron core of a massive star collapses to neutron-star size? Assume the core to be of uniform density $5 \times 10^9 \mathrm{~g} \mathrm{~cm}^{-3}$ with a radius of $500 \mathrm{~km}$, and that it collapses to a uniform sphere of radius $10 \mathrm{~km}$. Compare the energy released in this collapse with the total gravitational binding energy of the star before collapse, assuming it to be a supergiant of $15 M_{\odot}$ and radius $2 \mathrm{AU}_{\mathrm{AU}}$ before the collapse begins (you may use the uniform-density formula for the estimate).

Muhammad Haider
Muhammad Haider
Numerade Educator
04:57

Problem 4

When the ${ }^{56} \mathrm{Fe}$ core of a massive star becomes gravitationally unstable, two things greatly accelerate its collapse: (1) photodisintegration of ${ }^{56} \mathrm{Fe}$ is extremely endothermic and absorbs energy that can no longer be used for pressure support, and (2) rapid neutronization, $p+e \rightarrow n+v_{\mathrm{e}}$ undermines stability further because the major pressure support comes from the degenerate electrons that are disappearing, and the core is initially transparent to neutrinos so each emitted neutrino carries its energy out of the core at nearly the speed of light. Estimate the amount of energy absorbed by these two processes if the ${ }^{56} \mathrm{Fe}$ core is entirely photodisintegrated to $\alpha$ particles and neutrons, and in addition all of the available electrons undergo neutronization reactions with the protons. Assume that the core is electrically neutral, that $Q=-124.4 \mathrm{MeV}$ for ${ }^{56} \mathrm{Fe} \rightarrow 13 \alpha+4 n$, and that each emitted neutrino carries off average energy of $10 \mathrm{MeV}$. In a core collapse supernova the primary part of this energy release from photodisintegration and neutronization occurs in a fraction of a second. How long would the Sun need to radiate at its present luminosity to account for the energy loss estimated above for photodisintegration and neutronization of an iron core?***

Mayukh Banik
Mayukh Banik
Numerade Educator
03:30

Problem 5

Basic properties of the r-process suggest that successive neutron captures must occur on a timescale of microseconds. Estimate the neutron number density required to produce this capture rate, assuming a typical neutron capture cross section to be $100 \mathrm{mb}$. Hint: The solution to Problem 13.3 will be useful in this estimate.

Mayank Tripathi
Mayank Tripathi
Numerade Educator
09:53

Problem 6

A Type Ia supernova is powered by a thermonuclear runaway in electron-degenerate white dwarf matter. To see how such an instability could arise, use the approximate relationship between the central pressure and central density for a Lane-Emden polytropic solution given by Eq. (?14), and the generic equation of state
$$
\frac{d P}{P}=\alpha \frac{d \rho}{\rho}+\beta \frac{d T}{T}
$$
(where $\alpha$ and $\beta$ are non-negative) discussed in Problem 4.16, to show that thermonuclear burning is unstable at the center of a white dwarf described by a polytropic equation of state. Hint: Use Eq. (8.14), and that stability requires that changes in density and changes in temperature have the same sign.

Eduard Sanchez
Eduard Sanchez
Numerade Educator
00:56

Problem 7

The absolute bolometric magnitude for the progenitor of SN 1987A was estimated in Ref. [27] to be -7.8 . Spectroscopically it was a luminous B3 supergiant with an estimated surface temperature of 16 , $000 \mathrm{~K}$. What was its luminosity and radius?

Farhanul Hasan
Farhanul Hasan
Numerade Educator
05:28

Problem 8

The bright red supergiant Betelgeuse is a relatively nearby star that could undergo a core collapse at any time. Use the SIMBAD database [2] to retrieve the parallax and the ultraviolet magnitude $U$, the blue magnitude $B$, and the visual magnitude $V$ for this star. Determine its $U-B$ and $B-V$ color indices and its distance from Earth. Given that the (large) bolometric correction for Betelgeuse is -2.15 , what is its luminosity? If the effective surface temperature is $3500 \mathrm{~K}$, what is its radius? Other things being equal, how much larger would you expect the neutrino flux on Earth to be relative to that for Supernova 1987A when Betelgeuse goes supernova?

Andrew Duncan
Andrew Duncan
Numerade Educator
14:05

Problem 9

What was the likely host galaxy, supernova spectal type (see Eig. 20.3), and spectroscopic redshift for SN 1969L? Hint: Use the SIMBAD astronomical database [2].
1 Another proposed mechanism involves some aspects of both core collapse and thermonuclear runaway: massive stars of low metallicity can undergo a par-instability supemova, which is described in $\operatorname{Box} 20.2$
2 This is a miniscule fraction of the total diameter. The massive star is typically a supergiant by this time, with its tenuous outer layers spread over a volume that would encompass much of the inner Solar System if it were placed at the position of the Sun.
1. Recall from Section 9.5 .1 that a homologous collapse is "self-similar": it can be described by changing a seale factor.
4 This defines a unit of energy commonly used in supernova discussions that originally was called a foe: 1 foe $=10^{51}$ ergs, with the name deriving from the first letters of fifty-one ergs. This unit is now termed a bethe, in honor of the remarkable plysicist Hans Bethe (19062005), who won the 1967 Nobel prize in Physics for illuminating how nuclear reactions powered the stars and made seminal contributions to many other fields of science, including supernova physics. Bethe (whose name was pronounced the same as "beta") continued to do science until close to his death at age 98 , inspiring a joke that went "The nuclear physicists were wrong; there is no Bethe decay!"
5 The next generation is expected to be exascale systems, with speeds $\sim 10^{18}$ floating point operations per second ( 1 exaflop). The fastest computers available as this is written in 2017 are capable of $10-100$ petaflops $\left(10-100 \times 10^{15}\right.$ floating point operations per second). Optimistically the exaseale for scientific computing could be reached by the early 2020 s.
6 A standard argument was that, assuming the usual model that supemova remnants and lighteurves are the result of a shockwave that disrupts the outer layers of the star after core collapse, the properties of observed bright supernovae in distant galaxies required that the shockwave pass through mostly low-density matter on its way out of the star, and red supergiants are of much lower density than blue supergiants (see Table 2.4) [27]. As will be discussed further below, this was partially observational bias because core collapse in blue supergiants is expected to produce supemovac that are much less luminous than those originating in red supergiants. It should be noted that some theoretical work before SN 1987A considered the possibility that supernovae of lower luminosity might originate in blue supergiants (see [27] for references) but the lack of observational data for such events, and that most stellar evolution calculations of the time favored explosion in the red supergiant phase, caused this possibility to be discounted in favor of the red supergiant progenitor paradigm.

Eric Goldman
Eric Goldman
Numerade Educator