---
title: Cataclysmic Events and the Final States of Stars
draft: false
module: Orientation
moduleNumber: 1
lessonNumber: 2
order: 102
summary: >
  A star's death is set by its mass. In close binaries, matter poured across the
  Roche lobe onto a white dwarf produces novae and, at the Chandrasekhar limit of
  1.4 solar masses, a Type Ia supernova; a massive star fusing to an iron core
  collapses into a Type II supernova. The remnant is a white dwarf held by
  electron degeneracy, a neutron star held by neutron degeneracy, or, above the
  neutron-star limit, a black hole inside its Schwarzschild radius.
topics: [Supernovae, Degenerate remnants, Black holes]
sources:
  - book: Tipler & Llewellyn
    ref: "Ch. 13 — Astrophysics and Cosmology; §13-4 Cataclysmic Events, §13-5 Final States of Stars"
---

Explosions and collapse are a normal part of a star's life cycle, and they forge
and scatter the elements from which later stars and planets form. The events near
the end of a star's life, and the compact object left behind, are governed almost
entirely by mass, chiefly the mass of the core.

## Close binaries and novae

More than half of all stars belong to binary or larger groups. In a **close
binary**, the two stars of masses $M_1$ and $M_2$ orbit their common center of
mass closely enough to interact. An observer rotating with the system feels the
sum of the two gravitational forces and the centrifugal pseudoforce. Along the
line joining the stars there is a point where these cancel, a **Lagrangian
point** $L$; the equipotential surface through it wraps each star in a teardrop
**Roche lobe**.

$$
% caption: Two stars share equipotential Roche lobes meeting at the Lagrangian
% point; matter overflowing star one spirals into an accretion disk around star two.
\begin{tikzpicture}[scale=1.0, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
% Roche lobes as two teardrop-ish shapes meeting at L
\draw[black] (2.4,1.4) .. controls (2.4,2.6) and (0.4,2.6) .. (0.4,1.4)
  .. controls (0.4,0.2) and (2.4,0.2) .. (2.4,1.4);
\draw[black] (5.0,1.4) .. controls (5.0,2.4) and (3.2,2.4) .. (3.2,1.4)
  .. controls (3.2,0.4) and (5.0,0.4) .. (5.0,1.4);
\fill[black] (1.4,1.4) circle (3.5pt);
\node[anchor=south] at (1.4,1.75) {$M_1$};
% accretion disk around M2
\draw[acc, thick] (4.1,1.4) ellipse (0.75 and 0.28);
\fill[black] (4.1,1.4) circle (2.5pt);
\node[anchor=south] at (4.1,1.75) {$M_2$};
\node[black, anchor=north, font=\scriptsize] at (4.1,1.05) {accretion disk};
% Lagrangian point
\fill[black!70] (2.8,1.4) circle (2pt);
\node[black!70, anchor=south, font=\scriptsize] at (2.8,1.55) {L};
% mass stream
\draw[->, acc, thick] (2.5,1.4) .. controls (3.0,1.4) and (3.3,1.6) .. (3.6,1.65);
\node[black, anchor=west] at (5.3,1.4) {mass transfer};
\end{tikzpicture}
$$

When a star fills its Roche lobe, its surface feels no gravity at $L$, so gas
pours through into the companion's lobe. Because the system rotates, the Coriolis
effect makes the stream spiral into an **accretion disk** rather than fall
straight in. If the companion $M_2$ is a **white dwarf**, cataclysm follows.
Stored disk material dumped onto the dwarf's surface heats it and raises the
luminosity by a factor of 10 to 100, a **nova**; recurrences range from weeks to
millennia. A **classical nova** brightens by up to a factor of a million when
enough hydrogen accumulates to trigger a surface thermonuclear explosion.

## Supernovae

A **supernova** is the destruction of an entire star, more luminous at peak than
its whole host galaxy. Supernovae are classified by spectrum and light curve.

$$
% caption: Supernova classification: Type I lacks hydrogen lines and splits by
% silicon and helium features; Type II shows hydrogen and splits by light-curve shape.
\begin{tikzpicture}[scale=1.0, font=\footnotesize,
  bx/.style={draw, black, minimum height=7mm, inner sep=3pt, align=center, font=\scriptsize}]
\definecolor{acc}{HTML}{4A6FA5}
\node[bx, draw=acc, fill=acc!10] (sn) at (0,0) {supernova};
\node[bx] (t1) at (3.0,1.4) {Type I \\ no H lines};
\node[bx] (t2) at (3.0,-1.4) {Type II \\ H lines};
\draw[->, acc] (sn) -- (t1);
\draw[->, acc] (sn) -- (t2);
\node[bx] (ia) at (6.4,2.4) {Ia: Si line};
\node[bx] (ib) at (6.4,1.4) {Ib: He lines};
\node[bx] (ic) at (6.4,0.4) {Ic: no He};
\draw[->, black] (t1) -- (ia);
\draw[->, black] (t1) -- (ib);
\draw[->, black] (t1) -- (ic);
\node[bx] (iil) at (6.4,-0.9) {II-L: linear};
\node[bx] (iip) at (6.4,-1.9) {II-P: plateau};
\draw[->, black] (t2) -- (iil);
\draw[->, black] (t2) -- (iip);
\end{tikzpicture}
$$

The two types have entirely different origins.

- **Type Ia**, a carbon detonation. In a binary, a companion pours mass onto a
  carbon-oxygen white dwarf. When the dwarf's mass reaches the Chandrasekhar
  limit, its central pressure ignites runaway carbon fusion, a thermonuclear
  "carbon bomb" that unbinds the star. The common trigger mass makes all Type Ia
  events nearly identical in peak brightness, which is why they serve as
  **standard candles**.
- **Type II**, core collapse. In a star above about $8M_\odot$, gravity keeps
  driving fusion through oxygen, neon, and silicon until the core is iron. Iron
  has the highest [binding energy per nucleon](/nuclear-physics/nuclear-properties/nuclear-masses-binding-energy),
  so further fusion absorbs energy, and the core has no way to resist gravity.

$$
% caption: Late in a massive star, fusion products settle in shells around an
% inert iron core, the layout that precedes a Type II core-collapse supernova.
\begin{tikzpicture}[scale=1.0, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[black] (0,0) circle (3.5) node[below=3.1cm, black] {hydrogen};
\draw[black] (0,0) circle (2.8) node[below=2.4cm, black] {helium};
\draw[black] (0,0) circle (2.1) node[below=1.7cm, black] {carbon, oxygen};
\draw[black] (0,0) circle (1.3) node[below=0.95cm, black] {silicon};
\draw[acc, very thick, fill=acc!18] (0,0) circle (0.65);
\node at (0,0) {iron};
\node[black, anchor=west] at (3.7,0) {inert core: no fusion energy};
\end{tikzpicture}
$$

Collapse compresses and heats the core past $10^9\ \text{K}$, where photons
photodisintegrate iron and then helium, each step absorbing energy and
accelerating the fall:

$$
{}^{56}\text{Fe} \longrightarrow 13\,{}^{4}\text{He} + 4n,
\qquad
{}^{4}\text{He} \longrightarrow 2p + 2n .
$$

In free fall the core crushes protons and electrons together by **inverse beta
decay**, converting into neutrons and a flood of neutrinos:

$$
p + e^- \longrightarrow n + \nu_e .
$$

The envelope outside the core is blown away in the explosion. The 1987 supernova
SN1987A in the Large Magellanic Cloud, $170{,}000\ \text{ly}$ away, released the
predicted neutrino burst, detected at the Kamiokande observatory, confirming the
core-collapse picture and hinting at nonzero
[neutrino mass](/nuclear-physics/beta-decay/double-beta-decay-neutrino-mass). The
supernova of 1054, recorded by Chinese astronomers, is now the Crab Nebula.

## White dwarfs

A star below about $6M_\odot$ ends by shedding its outer layers as a planetary
nebula, leaving a **white dwarf** of mass near $1M_\odot$ packed into a radius
of order $10^7\ \text{m}$, comparable to Earth. Its density reaches about
$5 \times 10^5\ \text{g/cm}^3$. With fusion ended, nothing thermal holds it up;
gravity compresses it until the exclusion principle stops the electrons, the same
[degeneracy pressure](/statistical-mechanics/bose-systems/bose-einstein-condensation-and-the-fermion-gas) that
resists compression of a Fermi gas. Balancing electron degeneracy pressure
against gravity gives a nonrelativistic mass-radius relation

$$
R = \left( 3.1 \times 10^{17}\ \text{m} \cdot \text{kg}^{1/3} \right) \frac{Z^{5/3}}{A}\, M^{-1/3},
$$

with $Z$ and $A$ the atomic number and mass number of the material. The radius
shrinks as the mass grows: a heavier white dwarf is smaller.

> **Definition (Chandrasekhar limit).** The maximum mass a white dwarf can
> support by electron degeneracy pressure. When the electrons become
> relativistic, Chandrasekhar's relativistic treatment gives a radius tending to
> zero at a mass of about $1.4M_\odot$. Every measured white dwarf lies below
> this limit.

A lone white dwarf radiates its heat away with no furnace to replace it, cooling
toward a **black dwarf** in equilibrium with the universe, a stage none has yet
reached. A white dwarf in a binary that accretes to the Chandrasekhar limit
instead implodes and detonates as a Type Ia supernova.

## Neutron stars

If the collapsed core exceeds the Chandrasekhar limit, electron degeneracy fails
and the core becomes neutrons. Treating those neutrons as an ideal Fermi gas
gives a mass-radius relation of the same form,

$$
R = \left( 1.6 \times 10^{14}\ \text{m} \cdot \text{kg}^{1/3} \right) M^{-1/3},
$$

so a $1.0M_\odot$ core has $R \approx 1.27 \times 10^4\ \text{m} = 12.7\
\text{km}$. Its density, about $1.2 \times 10^{14}\ \text{g/cm}^3$, is nearly
that of a bare neutron. Here the exclusion-principle repulsion of the strong
nuclear force between neutrons balances gravity. This support also has a ceiling:
current theory places the maximum **neutron star** mass between $1.7M_\odot$ and
$3M_\odot$.

$$
% caption: On a log-log mass-radius plot, degenerate stars shrink as mass grows;
% neutron stars lie about three orders of magnitude smaller than white dwarfs.
\begin{tikzpicture}[scale=1.0, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->, black] (0,0) -- (7.6,0) node[right, black!70] {mass};
\draw[->, black] (0,0) -- (0,4.8) node[above, black!70] {radius};
% white dwarf curve (upper, decreasing)
\draw[very thick] (0.7,4.2) .. controls (2.4,3.2) and (4.0,2.9) .. (5.6,2.6);
\node[anchor=south] at (2.6,3.7) {white dwarfs};
\draw[black, dashed] (5.6,2.6) -- (5.6,0) node[below, black, font=\scriptsize] {1.4 solar};
% neutron star curve (lower, decreasing)
\draw[very thick] (0.9,1.6) .. controls (2.4,1.1) and (4.2,0.9) .. (6.6,0.7);
\node[anchor=south] at (3.0,1.15) {neutron stars};
\draw[black, dashed] (6.6,0.7) -- (6.6,0) node[below, black, font=\scriptsize] {3 solar};
\end{tikzpicture}
$$

Rapidly rotating neutron stars appear as **pulsars**, regular radio (and
sometimes optical) sources first found in 1967 in supernova remnants such as the
Crab Nebula. The neutron star inherits much of the original star's angular
momentum and magnetic field, so it spins fast while dragging a tilted
magnetosphere. Charged particles accelerated along the magnetic axis radiate in a
narrow cone; as the tilted axis sweeps past Earth, the source pulses like a
lighthouse. The Crab pulsar radiates $3 \times 10^{31}\ \text{W}$ with a period
of $0.033\ \text{s}$.

$$
% caption: A pulsar's magnetic axis is tilted from its spin axis, so its
% radiation cones sweep the sky and pulse each time a beam crosses Earth.
\begin{tikzpicture}[scale=1.0, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
% spin axis (vertical)
\draw[black, thick] (0,-2.4) -- (0,2.6);
\node[black, anchor=south] at (0,2.6) {spin axis};
% neutron star
\draw[very thick] (0,0) circle (0.55);
% magnetic axis (tilted)
\coordinate (top) at (1.1,2.2);
\coordinate (bot) at (-1.1,-2.2);
\draw[acc, thick] (bot) -- (top);
\node[acc, anchor=west] at (1.6,0.9) {magnetic axis};
% radiation cones as triangles at each magnetic pole
\draw[acc, fill=acc!12] (top) -- ($(top)+(0.75,0.5)$) -- ($(top)+(0.05,0.95)$) -- cycle;
\draw[acc, fill=acc!12] (bot) -- ($(bot)+(-0.75,-0.5)$) -- ($(bot)+(-0.05,-0.95)$) -- cycle;
\node[black, anchor=west] at (2.0,2.5) {radiation beam};
% rotation arrow
\draw[->, black] (0.75,0.55) arc (0:150:0.75);
\end{tikzpicture}
$$

## Black holes

When the remnant core exceeds the neutron-star ceiling, no known pressure stops
the collapse. The **escape velocity** from a mass $M$ at radius $R$, from
equating kinetic and gravitational potential energy, is

$$
v_e = \left( \frac{2GM}{R} \right)^{1/2}.
$$

Setting $v_e = c$ defines the **Schwarzschild radius**.

> **Definition (Schwarzschild radius).** The radius
> $$
> R_S = \frac{2GM}{c^2}
> $$
> at which the Newtonian escape velocity equals the speed of light. A mass
> compressed within $R_S$ is a **black hole**: no material object escapes its
> surface. For $1M_\odot$ the Schwarzschild radius is about $3\ \text{km}$.

Radiation is trapped as surely as matter. Light of rest wavelength $\lambda_0$
emitted at radius $R$ suffers the
[gravitational redshift](/relativity/foundations/general-relativity)

$$
\frac{\lambda}{\lambda_0} = \left( 1 - \frac{R_S}{R} \right)^{-1/2}.
$$

As $R \to R_S$ the wavelength diverges and the photon energy $E = hc/\lambda$
falls to zero, so no energy escapes: the object neither emits nor reflects,
appearing perfectly black.

$$
% caption: Inside the Schwarzschild radius neither matter nor light escapes;
% passing light rays bend toward the hole, and those crossing the horizon are captured.
\begin{tikzpicture}[scale=1.0, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
% event horizon
\filldraw[acc, fill=acc!14] (0,0) circle (1.1);
\node[black!75] at (0,0) {black hole};
\draw[acc, dashed] (0,0) circle (1.1);
\node[acc, anchor=south, font=\scriptsize] at (0,1.15) {event horizon};
% radius marker
\draw[->, black] (0,0) -- (0.82,0.72);
\node[black, font=\scriptsize, anchor=north west] at (0.5,0.28) {$R_S$};
% incoming light rays bending
\draw[->, thick] (-4.2,1.9) .. controls (-2.0,1.5) and (-0.5,1.3) .. (0.9,0.7);
\draw[->, thick] (-4.2,0.6) .. controls (-2.2,0.4) and (-1.4,0.1) .. (-1.05,0.0);
\draw[thick] (-4.2,-0.9) .. controls (-2.2,-0.7) and (-1.6,-0.3) .. (-1.05,-0.15);
\node[black, anchor=west] at (-4.2,2.1) {light rays};
\node[black, anchor=west, font=\scriptsize] at (1.4,-0.9) {captured within horizon};
\end{tikzpicture}
$$

Black holes are inferred from their gravity. At the center of the Milky Way,
stars tracked orbiting the compact radio source Sagittarius A* imply an unseen
mass of about $3 \times 10^6\ M_\odot$ confined to a tiny volume. Unlike white
dwarfs and neutron stars, black holes do not cool toward equilibrium.

The compact remnants also power **gamma-ray bursts**, the brightest gamma-ray
flashes in the sky, lasting from under a second to minutes and distributed
uniformly across the sky, which places them at cosmological distances. Many
appear to accompany the collapse of very massive stars into neutron stars or
black holes.

| Remnant | Support against gravity | Typical mass | Typical radius |
| --- | --- | --- | --- |
| White dwarf | electron degeneracy pressure | $\sim 1M_\odot$ | $\sim 10^4\ \text{km}$ |
| Neutron star | neutron degeneracy / strong force | $1.4$-$3\,M_\odot$ | $\sim 12\ \text{km}$ |
| Black hole | none (collapse continues) | $> 3M_\odot$ | $R_S = 2GM/c^2$ |

[^tl-cat]: Tipler & Llewellyn, §13-4 — Cataclysmic Events: Roche lobes and accretion in close binaries, novae, and the classification and mechanisms of Type I and Type II supernovae.
[^tl-final]: Tipler & Llewellyn, §13-5 — Final States of Stars: electron-degenerate white dwarfs and the Chandrasekhar limit, neutron stars and pulsars, and the Schwarzschild radius of black holes.