The Sun and the Life of Stars

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),

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

Treating the Sun as a blackbody ties this power to a surface temperature through the Stefan-Boltzmann law, $R=\sigma T^4$, 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:

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.

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.

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

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. 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:

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 appearing as radiation and neutrino energy.

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: electron neutrinos change flavor en route, and Davis's detector saw only one flavor. The resolution requires that neutrinos have nonzero mass.¹

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,

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:

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.²

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

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

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.

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

A star's lifetime is the available fuel (proportional to mass, since $E=M{c}^2$ per the mass-energy relation) divided by the rate of consumption (the luminosity):

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.³

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.

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.