stellar spectra evolution and energy sources

A star is a self-gravitating ball of gas whose emitting surface obeys the Stefan–Boltzmann law:

$$L = 4\pi R^2 \sigma T^4 .$$

A star's spectrum forms in its photosphere: absorption lines arise from atomic transitions between levels, and their strength depends on temperature and density through the Boltzmann distribution and the Saha equation.
Surface temperature therefore sets the spectral class $O\,B\,A\,F\,G\,K\,M$, while Wien's displacement law $\lambda_{\max} T = 2.9\times10^{-3}$ m·K tells where the emission peak lies.
Observed flux and luminosity are linked by the inverse-square law $F = L/(4\pi d^2)$, so the distance scale from the prerequisite topic is needed to convert apparent magnitude into absolute magnitude and recover the luminosity.
The energy source of stars is the nuclear fusion of helium from hydrogen ($4\,{}^1\mathrm{H} \to {}^4\mathrm{He} + 2e^+ + 2\nu$), in which the mass defect $\Delta m$ becomes energy through $E = \Delta m c^2$.
Luminosity and temperature together place a star on the Hertzsprung–Russell diagram, where its position and mass determine its evolutionary path — from the main sequence to a giant, supergiant, or white dwarf.

A star has radius $R = 2\,R_\odot$ and surface temperature $T = 10\,000$ K. The Sun's temperature is $T_\odot = 5800$ K. Find the star's luminosity in solar units and the wavelength of peak emission.

From its spectrum the star is classified as B8 ($T \approx 12\,000$ K). Its apparent magnitude is $m = 5.0$ and its annual parallax is $10$ milliarcseconds. Determine its absolute magnitude, luminosity, and radius, and place it on the Hertzsprung–Russell diagram.

Estimate the main-sequence lifetime of a $10\,M_\odot$ star, assuming that 10 % of its hydrogen is fused into helium and that the fusion of ${}^4\mathrm{He}$ from ${}^1\mathrm{H}$ converts 0.7 % of the mass into energy. Take the star's luminosity as $3000\,L_\odot$. Compare with the rough estimate $t \approx 10\ \text{billion years} \times (M/M_\odot)/(L/L_\odot)$.

Self-study plan (simulation only, no sky observation).

1. Open a Hertzsprung–Russell diagram simulation (for example, the NAAP “Hertzsprung–Russell Diagram” lab) or use an archival dataset of 10 stars with known temperature, luminosity, and parallax.
2. For each star, convert parallax to distance ($d = 1/p$), then compute $M$ from $M = m - 5\log_{10}(d/10)$ and $L/L_\odot = 10^{(M_\odot - M)/2.5}$.
3. Plot the points on a luminosity–temperature diagram (temperature increasing to the left) and mark which stars lie on the main sequence and which fall in the giant or white-dwarf regions.
4. For one main-sequence star, use its simulated spectrum to list the dominant lines (hydrogen Balmer lines, Ca II lines, TiO bands) and assign the spectral class.
5. Estimate that star's lifetime with $E = 0.007 \times 0.10 \times M c^2$ and $t = E/L$, and compare it with $t \approx 10\ \text{billion years} \times (M/M_\odot)/(L/L_\odot)$.

Safety. Never point unprotected eyes, binoculars, or a telescope at the Sun without a proper solar filter. All spectra and photometry come from the simulation or archives, not from direct observation.

Show the link between an observed spectrum (a set of absorption lines) and a star's position on the Hertzsprung–Russell diagram, so that temperature, luminosity, and evolutionary stage are read as one picture.

Diagram. The main panel is the Hertzsprung–Russell diagram: the vertical logarithmic axis carries luminosity $L/L_\odot$ (from $10^{-4}$ to $10^{6}$), and the horizontal axis carries temperature decreasing to the right (from $40\,000$ K to $3000$ K), labelled with spectral classes $O\,B\,A\,F\,G\,K\,M$. The main sequence, the giant branch, the supergiants, and the white dwarfs are drawn; the Sun is marked at $L = 1$, $T = 5800$ K.

Inset. Three schematic spectra with absorption lines: class A shows strong hydrogen Balmer lines, class G shows Ca II lines, and class M shows broad TiO bands. Arrows connect each spectrum to the corresponding point on the diagram.

Additional. At the bottom, an arrow marks the evolutionary track of a massive star from the main sequence to supergiant, annotated “He synthesis from H: $E = \Delta m c^2$”.

You hold a CD fragment over a smartphone camera and point it at the Sun or a fluorescent lamp. A rainbow crossed by dark lines appears on the screen. For a CD with track spacing $d = 1.6\,\mu\text{m}$, the first-order diffraction angle for green light $\lambda = 550\,\text{nm}$ is $\theta = \arcsin(\lambda/d) \approx 20^\circ$. In the solar spectrum, the sodium doublet at $589.0\,\text{nm}$ and $589.6\,\text{nm}$ appears as a pair of dark lines.

This experiment reproduces the formation of Fraunhofer lines: the continuous spectrum of the hot, dense photosphere ($T \approx 5800\,\text{K}$) is absorbed by cooler, thinner gas in the chromosphere. The wavelengths of the lines reveal the presence of sodium, while their depth and width give the temperature and density of the absorbing layer — the basis of stellar spectral classification.

On a clear day, about $1000\,\text{W}$ arrives per square metre of Earth's surface. The Sun's luminosity $L = 3.8\times10^{26}\,\text{W}$ at distance $r = 1.5\times10^{11}\,\text{m}$ gives the solar constant $S = L/(4\pi r^2) \approx 1360\,\text{W/m}^2$. This power is generated in the core at $T \approx 1.5\times10^7\,\text{K}$ and density $1.5\times10^5\,\text{kg/m}^3$: each second about $4.3\times10^9\,\text{kg}$ of mass is converted into energy via $E = mc^2$, a mass defect of $0.7\%$ in the pp-chain.

This shows how the observed solar constant links a star's energy output to its source: the pp-chain converts four protons into a helium nucleus, releasing $4.3\times10^{-12}\,\text{J}$ per reaction. The rate of these reactions maintains the luminosity and sets the star's main-sequence lifetime; the same energy drives Earth's weather and climate.

The blue flame of a gas burner glows because excited CH and C$_2$ molecules emit lines at $430\,\text{nm}$ and $516\,\text{nm}$. Sprinkling table salt (NaCl) into the flame turns it yellow: the sodium D lines at $589.0\,\text{nm}$ and $589.6\,\text{nm}$ dominate. A thermocouple reads a flame temperature of about $1300\,\text{K}$.

This is an emission-line spectrum of a hot, low-density gas — the same type of spectrum used to classify Wolf–Rayet stars and to determine the chemical composition of stellar atmospheres. The wavelengths of the lines correspond to quantised atomic energy levels, and the relative line intensities give the temperature via the Boltzmann distribution.