---
title: The Sun and the Life of Stars
draft: false
module: Orientation
moduleNumber: 1
lessonNumber: 1
order: 101
summary: >
  The Sun is the one star close enough to study in detail: its luminosity fixes
  a surface temperature of 5780 K, and the proton-proton fusion cycle in its
  1.5-million-kelvin core supplies its power. Measuring other stars needs the
  magnitude scale, parallax, and the distance ladder; plotting luminosity
  against temperature builds the Hertzsprung-Russell diagram, on which a star's
  mass sets its lifetime and its evolutionary track off the main sequence.
topics: [The Sun, Stellar magnitudes, The H-R diagram]
sources:
  - book: Tipler & Llewellyn
    ref: "Ch. 13 — Astrophysics and Cosmology; §13-1 The Sun, §13-2 The Stars, §13-3 The Evolution of Stars"
  - book: Tipler & Mosca
    ref: "Ch. 34 — Wave-Particle Duality and Quantum Physics (blackbody context)"
---

Astrophysics applies the physics of relativity, quanta, atoms, and nuclei to
objects that cannot be brought into a laboratory. All the information arrives as
electromagnetic radiation and the occasional particle, emitted in the past and
happening to reach Earth. The working assumption is that the laws found on Earth
hold everywhere. The scale jumps from the femtometer of a nucleus to the parsec
of interstellar space, more than forty orders of magnitude, and the Sun is the
only star near enough to resolve as more than a point.

## The Sun's luminosity and surface temperature

The visible surface of the Sun is the **photosphere**, a thin layer that emits
most of the light. The energy per second per square meter reaching the top of
Earth's atmosphere is the **solar constant** (or solar irradiance),

$$
f = 1.365 \times 10^3 \ \text{W/m}^2 .
$$

Energy conservation converts this into the total power radiated. A sphere of
radius equal to the Earth-Sun distance, one astronomical unit
$1\,\text{AU} = 1.496 \times 10^{11}\ \text{m}$, has area $A = 4\pi r^2$, and
every square meter of it receives energy at the rate $f$. The **luminosity**
$L_\odot$, the total power radiated, is therefore

$$
L_\odot = A f = 4\pi (1.496 \times 10^{11}\ \text{m})^2 (1.365 \times 10^3\ \text{W/m}^2)
= 3.84 \times 10^{26}\ \text{W}.
$$

Treating the Sun as a blackbody ties this power to a surface temperature through
the [Stefan-Boltzmann law](/quantum-mechanics/old-quantum-theory/blackbody-radiation-and-the-planck-quantum),
$R = \sigma T^4$, with $\sigma = 5.67 \times 10^{-8}\ \text{W/m}^2\text{K}^4$. The
intensity radiated at the solar surface, of radius
$R_\odot = 6.96 \times 10^8\ \text{m}$, is $R = L_\odot / 4\pi R_\odot^2$. The
**effective temperature** $T_e$ is the blackbody temperature that produces this
intensity:

$$
T_e = \left( \frac{L_\odot}{4\pi\sigma R_\odot^2} \right)^{1/4} = 5780\ \text{K}.
$$

The measured solar spectrum matches a Planck curve at about $5800\ \text{K}$
across the range that carries 99% of the emitted power, peaking in the yellow
part of the visible band. Deviations appear at short wavelengths, where extra
x-rays come from the much hotter corona.

$$
% caption: Measured solar spectral radiance follows a 5800 K blackbody across
% the visible band; the excess at short wavelengths comes from the hot corona.
\begin{tikzpicture}[scale=1.0, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
% axes
\draw[->, black] (0,0) -- (8.4,0) node[right, black!70] {wavelength};
\draw[->, black] (0,0) -- (0,4.4) node[above, black!70] {radiance};
% blackbody-ish curve
\draw[acc, very thick] (0.4,0.3) .. controls (1.6,0.6) and (2.0,3.6) .. (2.9,3.7)
  .. controls (4.4,3.8) and (5.4,1.4) .. (8.0,0.5);
% visible band shading marker
\draw[black, dashed] (2.9,0) -- (2.9,3.7);
\node[black, anchor=north] at (2.9,-0.05) {visible peak};
% corona x-ray hump at short wavelength
\draw[acc, thick, dashed] (0.35,1.5) .. controls (0.7,1.0) and (1.0,0.9) .. (1.4,0.7);
\node[acc, anchor=west] at (0.9,1.9) {corona x-rays};
\node[black!70, anchor=south west] at (3.6,3.4) {5800 K blackbody};
\end{tikzpicture}
$$

## Layers of the Sun

Above the photosphere lie two atmospheric layers, normally hidden by its glare.
The **chromosphere**, visible for seconds during a total eclipse, has a
temperature rising with height to about $15{,}000\ \text{K}$. Beyond it the
**corona**, seen at totality as faint streamers, reaches roughly
$2 \times 10^6\ \text{K}$; its gas is so rarefied that its total emission is
tiny, though it supplies the Sun's x-rays and drives the **solar wind** of
protons and electrons that fills the solar system.

$$
% caption: The Sun in concentric zones, from the fusion core out through the
% radiative and convective interior to the photosphere and hot outer atmosphere.
\begin{tikzpicture}[scale=1.0, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[thick] (0,0) circle (3.6);
\draw[thick] (0,0) circle (2.7);
\draw[thick] (0,0) circle (1.7);
\draw[very thick] (0,0) circle (0.9);
\node at (0,0) {core};
\node at (0,1.25) {radiative};
\node at (0,2.2) {convective};
\node[anchor=south] at (0,3.05) {photosphere};
% outer atmosphere marker
\draw[dashed] (0,0) circle (3.95);
\node[anchor=west] at (2.9,3.6) {corona};
% temperature annotations to the right
\node[black, anchor=west] at (4.2,0.0) {core: 15 million K};
\node[black, anchor=west] at (4.2,-0.7) {surface: 5780 K};
\node[black, anchor=west] at (4.2,-1.4) {corona: 2 million K};
\end{tikzpicture}
$$

The interior cannot be seen through the photosphere; models treat the Sun as a
nonrotating sphere in **hydrostatic equilibrium**, with outward pressure from
energy generation balancing inward gravity at every point. The Sun's mass,
found from Newtonian gravitation and orbital motion, is
$M_\odot = 1.99 \times 10^{30}\ \text{kg}$. The pressure at the center is of
order $10^{15}\ \text{N/m}^2$, far larger than the Coulomb pressure binding an
electron to a proton, so the core matter is fully ionized **plasma**. The
ideal-gas law then gives a core temperature near $1.5 \times 10^7\ \text{K}$.

## The proton-proton cycle

Kelvin noted that the Sun's thermal and gravitational energy would be radiated
away in about $3 \times 10^7$ years, yet Earth has held life a hundred times
longer. The Sun's power must come from a far larger reservoir: **nuclear
fusion**. As the young Sun contracted under gravity, released potential energy
raised the core temperature until protons could fuse. The chain that burns
hydrogen to helium is the **proton-proton cycle**, whose first step is

$$
{}^1\text{H} + {}^1\text{H} \longrightarrow {}^2\text{H} + e^+ + \nu_e + 0.42\ \text{MeV}.
$$

The Coulomb barrier makes this step improbable except in the high-energy tail of
the Maxwell-Boltzmann distribution, and even there it proceeds only by
[quantum tunneling](/quantum-mechanics/wave-mechanics-1d/barrier-penetration-and-quantum-tunneling). That
low probability is the bottleneck that limits the fusion rate and guarantees the
Sun a long life. Once deuterium forms, the remaining steps run readily:

$$
{}^2\text{H} + {}^1\text{H} \longrightarrow {}^3\text{He} + \gamma + 5.49\ \text{MeV},
\qquad
{}^3\text{He} + {}^3\text{He} \longrightarrow {}^4\text{He} + 2\,{}^1\text{H} + 12.86\ \text{MeV}.
$$

The net conversion of four protons into one $ {}^4\text{He}$ nucleus, including
the positron annihilation and electron binding, releases about
$26.7\ \text{MeV}$, the [binding energy](/nuclear-physics/nuclear-properties/nuclear-masses-binding-energy)
appearing as radiation and neutrino energy.

$$
% caption: The proton-proton cycle fuses four protons into one helium-4 nucleus,
% releasing about 26.7 MeV per helium formed as photons and neutrinos.
\begin{tikzpicture}[scale=1.0, font=\footnotesize,
  nn/.style={circle, draw, minimum size=6mm, inner sep=0pt, font=\scriptsize}]
\definecolor{acc}{HTML}{4A6FA5}
\node[nn] (p1) at (0,1.4) {p};
\node[nn] (p2) at (0,0)   {p};
\node[nn] (d1) at (2.4,0.7) {d};
\draw[->, thick] (p1) -- (d1);
\draw[->, thick] (p2) -- (d1);
\node[nn] (p3) at (2.4,2.1) {p};
\node[nn] (he3) at (4.8,1.4) {He 3};
\draw[->, thick] (d1) -- (he3);
\draw[->, thick] (p3) -- (he3);
% second He-3 from mirror branch
\node[nn] (he3b) at (4.8,-0.6) {He 3};
\node[nn, draw=acc, very thick, fill=acc!18] (he4) at (7.4,0.4) {He 4};
\draw[->, thick] (he3) -- (he4);
\draw[->, thick] (he3b) -- (he4);
% returned protons
\node[nn] (r1) at (7.4,2.0) {p};
\node[nn] (r2) at (7.4,-1.4) {p};
\draw[->, black, dashed] (he4) -- (r1);
\draw[->, black, dashed] (he4) -- (r2);
\node[black, anchor=north] at (3.7,-1.9) {net: 4 p give He 4 plus 26.7 MeV};
\end{tikzpicture}
$$

A carbon-nitrogen-oxygen (CNO) cycle produces the same net result and supplies
about 1.5% of solar luminosity; it dominates in stars slightly more massive than
the Sun. Neutrinos from the proton-proton cycle escape the core directly and are
the only particles that reach Earth from the interior. Ray Davis measured only
about 32% of the predicted electron-neutrino flux, the **solar-neutrino
problem**, resolved by
[neutrino oscillations](/particle-physics/beyond-standard-model/beyond-standard-model):
electron neutrinos change flavor en route, and Davis's detector saw only one
flavor. The resolution requires that neutrinos have nonzero mass.[^tl-sun]

The active Sun adds transient magnetic phenomena. **Sunspots** are cooler
regions, near $3800\ \text{K}$, where bundles of field lines pierce the surface;
their number cycles over about 11 years, in step with a reversal of the Sun's
general field of about $10^{-4}\ \text{T}$. **Solar flares** eject particles and
radiation from x-ray to radio wavelengths.

## Measuring the stars

The **radiant flux** of a star is its version of the solar constant,

$$
F = \frac{L}{4\pi R^2},
$$

with $R$ the distance from Earth. Two properties classify a star: its luminosity
$L$ and its effective temperature $T_e$, inferred from its spectrum. Historically
brightness came first, through the **apparent magnitude** $m$: a difference of 5
in magnitude corresponds to a factor of 100 in brightness, so one step is
$100^{1/5} = 2.512$, and smaller $m$ means brighter. The Sun has
$m = -26.81$; the faintest detectable objects reach $m \approx +29$. Because two
stars of equal luminosity at different distances have different $m$, the
**absolute magnitude** $M$ is defined as the apparent magnitude the star would
have at a distance of 10 parsecs:

$$
100^{(m - M)/5} = \left( \frac{R}{10\ \text{pc}} \right)^2 .
$$

Stars are grouped into spectral types along a temperature sequence,
**O B A F G K M**, from hottest blue to coolest red, each subdivided 0-9; the Sun
is a G2 star. Composition splits stars into **population I** (metal-rich, about
2-3% heavier than helium, like the Sun) and older **population II** (metal-poor,
0.01-0.1%), a record of successive generations of fusion enriching the
interstellar medium.[^tl-stars]

| Type | Surface $T$ (K) | $L/L_\odot$ | $R/R_\odot$ | $M/M_\odot$ |
| --- | --- | --- | --- | --- |
| O5 | 44,500 | 790,000 | 15 | 60 |
| B0 | 30,000 | 52,000 | 8 | 18 |
| A0 | 9,520 | 54 | 3 | 3 |
| F0 | 7,200 | 6 | 2 | 2 |
| G2 (Sun) | 5,800 | 1.0 | 1.0 | 1.0 |
| K0 | 5,300 | 0.4 | 0.8 | 0.8 |
| M0 | 3,900 | 0.08 | 0.6 | 0.5 |

### Distance by parallax

Over one orbit of Earth a nearby star traces a small ellipse against the distant
background. Half the angular width of that ellipse is the **parallax angle**
$\theta$, related to the distance $r$ by

$$
\theta = \frac{1\ \text{AU}}{r} \qquad (\theta \text{ in radians}).
$$

One **parsec** is the distance at which $1\ \text{AU}$ subtends 1 arc second:

$$
1\ \text{pc} = \frac{1\ \text{AU}}{1'' } = 3.086 \times 10^{16}\ \text{m} = 3.26\ \text{ly}.
$$

$$
% caption: Parallax geometry: a baseline of 1 AU subtends the angle theta at the
% star, so a 1 arc-second angle defines a distance of one parsec.
\begin{tikzpicture}[scale=1.0, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\coordinate (star) at (0,1.5);
\coordinate (sun) at (7.5,1.5);
\coordinate (e1) at (7.5,2.7);
\coordinate (e2) at (7.5,0.3);
\fill[acc] (star) circle (3pt);
\node[acc, anchor=south] at (star) {star};
\fill[black!70] (sun) circle (2pt);
\node[black!70, anchor=west] at (sun) {Sun};
\draw[black] (7.5,1.5) circle (1.2);
\fill[black] (e1) circle (1.6pt);
\fill[black] (e2) circle (1.6pt);
\draw[acc, thick] (star) -- (e1);
\draw[acc, thick] (star) -- (e2);
\draw[black] (star) -- (sun);
% baseline label
\draw[black, <->] (7.7,1.5) -- (7.7,2.7);
\node[black, anchor=west] at (7.7,2.1) {1 AU};
\node[acc, anchor=south] at (1.35,1.75) {angle};
\node[black, anchor=north] at (4.6,1.45) {distance $r$};
\end{tikzpicture}
$$

Parallax works out to about $1\ \text{kpc}$, covering only nearby stars. Beyond
that the **distance ladder** takes over: the periods of Cepheid variables give
distances to about $29\ \text{Mpc}$, and Type Ia supernovae, with nearly
identical peak luminosities, serve as standard candles to much greater range and
provided the evidence for accelerating expansion.

## The Hertzsprung-Russell diagram

Plotting luminosity against effective temperature, with temperature increasing
to the left, produces the **Hertzsprung-Russell (H-R) diagram**. Between 80% and
90% of stars fall on a diagonal band, the **main sequence**, where they fuse
hydrogen to helium. Cool dim stars sit at the lower right, hot bright ones at the
upper left. Off the main sequence lie red giants and supergiants (cool but
enormous, hence luminous) at the upper right and white dwarfs (hot but tiny,
hence faint) at the lower left.

$$
% caption: The Hertzsprung-Russell diagram: most stars lie on the main sequence,
% with red giants above and to the right and white dwarfs below and to the left.
\begin{tikzpicture}[scale=1.0, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->, black] (8.3,0) -- (0,0) node[left, black!70] {hotter};
\draw[->, black] (0,0) -- (0,5.2) node[above, black!70] {luminosity};
% spectral type labels along top
\foreach \x/\lab in {0.6/O, 1.8/B, 3.0/A, 4.2/F, 5.4/G, 6.6/K, 7.8/M}
  \node[black, font=\scriptsize] at (\x,5.0) {\lab};
% main sequence band
\draw[acc, very thick] (0.8,4.7) -- (7.6,0.5);
\node[acc, rotate=-27, anchor=south] at (4.2,2.8) {main sequence};
% Sun marker
\fill[acc] (5.4,2.35) circle (2.6pt);
\node[acc, anchor=west, font=\scriptsize] at (5.5,2.35) {Sun};
% red giants cloud
\draw[black] (6.6,3.9) ellipse (0.95 and 0.55);
\node[black!70, font=\scriptsize] at (6.6,3.9) {red giants};
\draw[black] (7.4,4.7) ellipse (0.7 and 0.35);
\node[black!70, font=\scriptsize] at (7.4,4.7) {supergiants};
% white dwarfs
\draw[black] (2.4,1.0) ellipse (0.9 and 0.45);
\node[black!70, font=\scriptsize] at (2.4,1.0) {white dwarfs};
\end{tikzpicture}
$$

Binary-star masses show that luminosity rises steeply with mass:

$$
L \propto M^4 .
$$

A star's lifetime is the available fuel (proportional to mass, since
$E = Mc^2$ per the [mass-energy relation](/relativity/foundations/relativistic-momentum-energy))
divided by the rate of consumption (the luminosity):

$$
t_L \propto \frac{M}{L} \propto \frac{M}{M^4} = M^{-3}.
$$

More massive stars burn far faster: a star of twice the Sun's mass lasts only
about $1/8$ as long. Energy balance on the main sequence also gives $R \propto M$
and, combined with the effective-temperature relation, $T_e \propto M^{1/2}$, so
heavier stars are both hotter and brighter, fixing their place along the band.[^tl-evol]

## Evolution off the main sequence

When the core hydrogen runs out, what follows depends on the initial mass. In a
low-mass star like the Sun, the core contracts and heats until, at about
$10^8\ \text{K}$, helium ignites and fuses toward carbon. The outer layers swell:
the radius grows while luminosity stays nearly constant, so the surface cools and
reddens into a **red giant**. Helium ignition moves the star to the horizontal
branch; when core helium is spent, carbon fusion drives it up the giant branch
again as a **red supergiant**, such as Betelgeuse. Eventually the star may shed
its outer layers as a **planetary nebula**, leaving a **white dwarf** that cools
toward equilibrium.

$$
% caption: A low-mass star's evolutionary track leaves the main sequence, climbs
% to the red-giant branch, and ends by shedding a planetary nebula.
\begin{tikzpicture}[scale=1.0, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->, black] (8.3,0) -- (0,0) node[left, black!70] {hotter};
\draw[->, black] (0,0) -- (0,5.2) node[above, black!70] {luminosity};
\draw[black, thick] (0.8,4.7) -- (7.6,0.5);
\node[black, rotate=-27, anchor=south, font=\scriptsize] at (3.0,3.5) {main sequence};
% track points
\coordinate (a) at (5.4,2.35);
\coordinate (b) at (5.9,2.0);
\coordinate (c) at (6.6,3.0);
\coordinate (d) at (7.2,4.1);
\coordinate (e) at (6.4,4.4);
\coordinate (f) at (2.6,1.1);
\draw[acc, very thick, ->] (a) -- (b);
\draw[acc, very thick, ->] (b) .. controls (6.3,2.2) .. (c);
\draw[acc, very thick, ->] (c) -- (d);
\draw[acc, very thick, ->] (d) .. controls (6.9,4.5) .. (e);
\draw[acc, very thick, dashed, ->] (e) .. controls (4.0,4.6) and (2.6,3.0) .. (f);
\fill[acc] (a) circle (2pt) node[anchor=north west, font=\scriptsize, black!70] {Sun now};
\node[black!70, font=\scriptsize, anchor=south] at (7.0,4.2) {red giant};
\node[black!70, font=\scriptsize, anchor=east] at (2.5,1.1) {white dwarf};
\node[acc, font=\scriptsize, anchor=south] at (4.3,4.5) {planetary nebula};
\end{tikzpicture}
$$

High-mass stars, above about $6M_\odot$, evolve far faster, as $t_L \propto
M^{-3}$ predicts. Their gravity generates the pressures and temperatures needed
to ignite oxygen, neon, and silicon, fusing all the way to iron. Because iron has
the highest binding energy per nucleon, fusing it absorbs rather than releases
energy, so the core can go no further by fusion. That endpoint leads to the
catastrophic events and compact remnants covered in
[the next lesson](/astrophysics-cosmology/orientation/stellar-death-final-states).

[^tl-sun]: Tipler & Llewellyn, §13-1 — The Sun: solar constant, luminosity, effective temperature, the proton-proton cycle, solar neutrinos, and the active Sun.
[^tl-stars]: Tipler & Llewellyn, §13-2 — The Stars: magnitude scales, spectral classification, stellar populations, and parallax distances.
[^tl-evol]: Tipler & Llewellyn, §13-3 — The Evolution of Stars: the H-R diagram, the mass-luminosity and mass-lifetime relations, and post-main-sequence tracks.