Stellar Evolution and the Hertzsprung–Russell Diagram – Cosmology, Galaxies and Expansion of the Universe (Senior 6 Physics) | Ganzaa

Cosmology, Galaxies and Expansion of the Universe

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Senior 6 · Physics

Cosmology, Galaxies and Expansion of the Universe

5 lessons

Lesson 4 of 5

Stellar Evolution and the Hertzsprung–Russell Diagram

25 min

Stars are born, live and die

The Ring Nebula in the constellation Lyra looks like a smoke ring of glowing gas, with a tiny, very hot star at its centre. It is what remains of a star like the Sun at the end of its life: the outer layers drifting away, the exposed core cooling slowly. No astronomer can watch a star age over billions of years, but by plotting thousands of stars on one diagram, as one might photograph a crowd of people of all ages, the life story of the stars can be read. That diagram is the Hertzsprung–Russell diagram.

The Hertzsprung–Russell diagram

Around 1911–1913, Ejnar Hertzsprung and Henry Norris Russell independently plotted the luminosity of stars against their surface temperature (or spectral class). Instead of filling the graph randomly, the stars fall into a few distinct groups (Figure 4.1). By tradition, temperature decreases to the right, and both axes are logarithmic.

Figure 4.1: The Hertzsprung–Russell (H–R) diagram (schematic, with some real stars).

About 90% of stars, including the Sun, lie on the main sequence, a band from hot, luminous blue stars at the upper left to cool, dim red stars at the lower right. A star's mass sets its place on it.
Giants and supergiants lie above the main sequence on the right: cool, yet very luminous, so by $L = 4\pi R^2\sigma T^4$ they must be enormous.
White dwarfs lie below on the left: hot, yet faint, so they must be tiny, about the size of the Earth.

Because $L \propto R^2T^4$ , lines of constant radius cross the diagram diagonally: radius increases towards the upper right.

Definition

The Hertzsprung–Russell (H–R) diagram is a graph of stars' luminosity (or absolute magnitude) against their surface temperature (or spectral class), with temperature decreasing to the right. The main sequence is the band on it occupied by stars that are fusing hydrogen into helium in their cores. Example: the Sun, a G2 star of luminosity $1\,L_\odot$ , lies near the middle of the main sequence.

Worked example 1: sizing stars from the diagram

Use $\frac{R}{R_\odot} = \sqrt{\frac{L}{L_\odot}}\left(\frac{T_\odot}{T}\right)^2$ with $T_\odot = 5772\,\mathrm{K}$ to find the radius of (a) Sirius B (25 000 K, $0.056\,L_\odot$ ) and (b) Aldebaran (3900 K, $440\,L_\odot$ ), in solar radii and, for Sirius B, in kilometres ( $R_\odot = 6.96 \times 10^{5}\,\mathrm{km}$ ).

Solution

(a) $\sqrt{0.056} \times \left(\frac{5772}{25\,000}\right)^2 = 0.237 \times 0.0533 = 0.0126\,R_\odot \approx 8.8 \times 10^{3}\,\mathrm{km}$ : only about 1.4 times the Earth's radius (6371 km), yet with about the mass of the Sun (Lesson 3).

(b) $\sqrt{440} \times \left(\frac{5772}{3900}\right)^2 = 21.0 \times 2.19 = 46\,R_\odot$ : a giant.

Life on the main sequence

A star forms when a cold, dense cloud of gas and dust (a nebula) collapses under its own gravity. The collapsing protostar heats up as gravitational potential energy is converted to thermal energy. When the core reaches about 10 million kelvin, hydrogen nuclei fuse into helium and the star settles on the main sequence. In stars like the Sun the main route is the proton–proton chain, whose net effect is

$4\,{}^{1}_{1}\mathrm{H} \rightarrow {}^{4}_{2}\mathrm{He} + 2e^{+} + 2\nu_e + \text{energy} \qquad (Q = 26.7\,\mathrm{MeV}\ \text{including the positron annihilations})$

The star is in hydrostatic equilibrium: the inward pull of gravity on each layer is balanced by the outward pressure of the hot gas beneath it. Massive stars need higher core temperatures to support themselves, so they burn their fuel ferociously: luminosity rises roughly as $M^{3.5}$ while the fuel only rises as $M$ . The main-sequence lifetime is therefore

$t \propto \frac{M}{L} \propto M^{-2.5}, \qquad t \approx 10^{10}\,\text{years} \times \left(\frac{M}{M_\odot}\right)^{-2.5}$

Worked example 2: how long does a star live?

(a) Show that the Sun can shine for about $10^{10}$ years, given that about 10% of its mass ( $1.99 \times 10^{30}\,\mathrm{kg}$ ) is hydrogen in its core available for fusion, that fusion converts 0.71% of that mass into energy, and that $L_\odot = 3.83 \times 10^{26}\,\mathrm{W}$ . (b) Estimate the main-sequence lifetimes of stars of 10 and 0.5 solar masses.

Solution

(a) Energy available: $E = 0.0071 \times 0.10 \times 1.99 \times 10^{30} \times (3.00 \times 10^{8})^2 = 1.27 \times 10^{44}\,\mathrm{J}$ . Time: $t = \frac{1.27 \times 10^{44}}{3.83 \times 10^{26}} = 3.3 \times 10^{17}\,\mathrm{s} = 1.1 \times 10^{10}$ years.

(b) $10\,M_\odot$ : $10^{10} \times 10^{-2.5} = 3 \times 10^{7}$ years (30 million years). $0.5\,M_\odot$ : $10^{10} \times 0.5^{-2.5} = 5.7 \times 10^{10}$ years, longer than the present age of the universe, so no such star has yet left the main sequence.

After the main sequence

Figure 4.2: The life cycles of stars. The dividing mass of about 8 solar masses is approximate.

When the hydrogen in the core runs out, fusion stops there; the core contracts and heats, hydrogen burns in a shell around it, and the outer layers swell and cool. The star moves to the upper right of the H–R diagram.

Stars up to about 8 solar masses (including the Sun) become red giants. The core, now at about 100 million kelvin, fuses helium into carbon and oxygen. Eventually the outer layers drift away as a glowing planetary nebula (nothing to do with planets), and the exposed core remains as a white dwarf: about the mass of the Sun packed into the size of the Earth, supported not by heat but by the quantum pressure of tightly packed electrons. It can be no more massive than about 1.4 solar masses (the Chandrasekhar limit). With no fuel, it simply cools for billions of years. The Sun will reach this stage in about 5 billion years.
More massive stars become red supergiants and fuse ever heavier elements in shells (carbon, neon, oxygen, silicon) up to iron. Fusing iron releases no energy (Unit 5, the binding-energy curve), so when the iron core grows beyond the Chandrasekhar limit it collapses in under a second. The rebound and a flood of neutrinos blow the rest of the star apart in a supernova, which can briefly outshine a whole galaxy. The core is left as a neutron star, about 20 km across and denser than an atomic nucleus, or, if it is heavier than about 3 solar masses, a black hole.

Supernovae scatter carbon, oxygen, iron and the other heavy elements into space, where they are built into new stars and planets. The iron in our blood and the calcium in our bones were made inside stars that died before the Sun was born.

Definition

A white dwarf is the hot, dense remnant core of a low-mass star, about the size of the Earth, with no fusion, at most about 1.4 solar masses. A neutron star is the collapsed core left by a supernova, made almost entirely of neutrons, about 20 km across. A black hole is a region where matter is so compressed that not even light can escape from it. Example: Sirius B is a white dwarf; the Crab pulsar is a neutron star.

Warning

Common mistakes. (1) Putting hot stars on the right of the H–R diagram: temperature decreases to the right. (2) Thinking a star moves along the main sequence as it ages: it stays at nearly the same point during hydrogen burning, then leaves it. (3) Thinking massive stars live longer because they have more fuel: they burn it far faster, so they live much shorter lives. (4) Saying the Sun will become a black hole or a supernova: it is far too light, and will end as a white dwarf.

Tip

Read the H–R diagram with $L = 4\pi R^2\sigma T^4$ : upper right = big (giants), lower left = small (white dwarfs). For a star's fate, the mass decides everything: light stars fade, heavy stars explode.

Stars and the H-R Diagram, from the Royal Observatory Greenwich. Watch how stars are placed on the diagram and how their positions change as they evolve.

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Practice

Sketch an H–R diagram, labelling the axes correctly, and mark the main sequence, the giants, the supergiants, the white dwarfs and the Sun.
Two stars have the same surface temperature, but one is 100 times as luminous. Compare their radii.
Explain why a star stays on the main sequence for most of its life.
Estimate the main-sequence lifetime of a star of 3.0 solar masses.
Describe the stages in the life of the Sun from now until its end.
Explain why fusion in a massive star stops at iron.
A star has $T = 3200\,\mathrm{K}$ and $L = 0.010\,L_\odot$ . Where is it on the H–R diagram, and what is its radius in solar radii?
An astronomer finds a cluster in which all stars hotter than spectral class A0 have left the main sequence. Explain what this suggests about the cluster's age compared with a cluster that still contains O stars.

Show answersHide answers

Vertical axis: luminosity (logarithmic), horizontal: surface temperature decreasing to the right (logarithmic) or spectral class O to M. Main sequence: diagonal band from upper left to lower right; giants and supergiants: upper right; white dwarfs: lower left; the Sun: middle of the main sequence (5800 K, 1 L☉).
$L \propto R^2$ at the same $T$ : $\frac{R_1}{R_2} = \sqrt{100} = 10$ .
Fusion of hydrogen in the core, the longest-lasting fuel, balances gravity (hydrostatic equilibrium); the star changes very little until the core hydrogen runs out.
$10^{10} \times 3.0^{-2.5} = 10^{10} \times 0.064 = 6.4 \times 10^{8}$ years.
About 5 billion more years on the main sequence; then a red giant (helium fusing to carbon and oxygen); then the outer layers form a planetary nebula; the core becomes a white dwarf that slowly cools.
Iron-56 is near the peak of the binding-energy-per-nucleon curve. Fusing iron into heavier nuclei absorbs energy instead of releasing it, so it cannot support the star against gravity.
Lower right: a cool, faint main-sequence red dwarf (class M). $\frac{R}{R_\odot} = \sqrt{0.010} \times \left(\frac{5772}{3200}\right)^2 = 0.10 \times 3.25 = 0.33$ .
Stars all formed at about the same time in a cluster. The hottest, most massive stars have the shortest lives; if stars down to class A0 (about 3 solar masses, lifetime of order $10^{8}$ to $10^{9}$ years) have already left the main sequence, the cluster is several hundred million years old. A cluster that still has O stars (lifetimes of a few million years) must be very young.

Key points

H–R diagram: luminosity against temperature (decreasing to the right). Main sequence (hydrogen fusion), giants and supergiants (large, cool, luminous), white dwarfs (small, hot, faint); radius from $L = 4\pi R^2\sigma T^4$ .
Nebula → protostar → main sequence (p–p chain, 26.7 MeV per helium nucleus; hydrostatic equilibrium). Lifetime $t \approx 10^{10}\,\mathrm{yr} \times \left(\frac{M}{M_\odot}\right)^{-2.5}$ .
Below about 8 $M_\odot$ : red giant → planetary nebula → white dwarf (≤ 1.4 $M_\odot$ ). Above: red supergiant → iron core → supernova → neutron star or black hole.

Sources and credits

18.5: The H–R Diagram, 22.2: Evolution from the Main Sequence to Red Giants and 22.6: The Evolution of More Massive Stars, from Astronomy 2e by Andrew Fraknoi, David Morrison and Sidney C. Wolff, OpenStax, hosted by LibreTexts, licensed under CC BY 4.0. Adapted by Ganzaa.
Image: Hubble image of the Ring Nebula (Messier 57), NASA, ESA and C. Robert O'Dell, Wikimedia Commons, public domain.
Image: Crab Nebula.jpg, NASA, ESA, J. Hester and A. Loll, Wikimedia Commons, public domain.
Video: Stars and the H-R Diagram, Royal Observatory Greenwich, embedded from YouTube.

The two diagrams, the worked examples and the practice exercises are written by Ganzaa.

1Stars are born, live and die

2The Hertzsprung–Russell diagram

3Life on the main sequence

4After the main sequence

Stars are born, live and die

The Hertzsprung–Russell diagram

Life on the main sequence

After the main sequence