Coaching Centre
The H–R diagram and stellar evolution
Using the Hertzsprung–Russell diagram at HSC level: luminosity, temperature, colour and radius; luminosity classes; what a star's position says about its mass and evolutionary stage; the main-sequence mass–luminosity–lifetime relation; and the evolutionary tracks of low-mass and high-mass stars to white dwarfs, neutron stars and black holes.
Every star you can see is at some stage in a life story that lasts millions or billions of years, far too slow to watch. Astronomers worked out the story the way you might work out how people age from a single photo of a crowded beach: by looking at huge numbers of stars, each caught at a different stage, and asking which combinations are common, which are rare, and which never happen.
Reading the H–R diagram
You met the H–R diagram in Year 10. At HSC level you need to read off a star's properties precisely and explain what its position means.
- Vertical axis: luminosity $L$, the total power the star radiates, usually in multiples of the Sun's ($L_\odot = 3.83 \times 10^{26}$ W) on a logarithmic scale. Some versions use absolute magnitude, which runs the other way: more negative is brighter.
- Horizontal axis: surface temperature $T$, decreasing to the right, also on a log scale. Equivalent labels are spectral class (O on the left, M on the right) or colour (blue to red).
Because luminosity depends on both surface area and temperature, radius can be read off too. The Stefan–Boltzmann law (not on the data sheet, but worth knowing for reasoning) gives
$$L = 4\pi R^2 \sigma T^4$$
so stars of the same temperature but higher luminosity must be bigger. Lines of constant radius run diagonally from top left to bottom right: the largest stars are at the top right, the smallest at the bottom left.
Aldebaran has a surface temperature of about 3900 K and a luminosity of about $440\,L_\odot$. The Sun's surface is 5800 K. How many times larger than the Sun is Aldebaran's radius?
From $L = 4\pi R^2\sigma T^4$: $\dfrac{R}{R_\odot} = \sqrt{\dfrac{L}{L_\odot}} \times \left(\dfrac{T_\odot}{T}\right)^2 = \sqrt{440} \times \left(\dfrac{5800}{3900}\right)^2$
$= 21.0 \times 2.21 = 46$. Aldebaran is about 46 times the Sun's radius: cooler per square metre, but with over 2000 times the surface area.
Regions and luminosity classes
| Region | Luminosity class | What the stars are doing |
|---|---|---|
| Main sequence | V | Fusing hydrogen to helium in the core. About 90% of stars. Position set mainly by mass: massive stars hot and luminous at top left, red dwarfs at bottom right. |
| Subgiants | IV | Core hydrogen just used up; starting to expand. |
| Giants | III | Hydrogen fusing in a shell around the core; later helium fusing in the core. Cool and very large. |
| Bright giants and supergiants | II and I | Evolved massive stars, up to 1000 times the Sun's radius, fusing heavier elements. |
| White dwarfs | D | Exposed cores of dead low-mass stars. No fusion; hot but Earth-sized, so dim. Slowly cooling. |
Mass on the main sequence
For main-sequence stars, mass decides everything. A more massive star has stronger gravity squeezing its core, so the core is hotter and denser and fusion runs much faster. Roughly,
$$L \propto M^{3.5}$$
(a rough rule for stars from about 0.5 to 10 solar masses, again not on the data sheet). The lifetime on the main sequence is fuel divided by rate of use, so $t \propto M/L \propto M^{-2.5}$: massive stars burn out far faster.
| Mass ($M_\odot$) | Class | Surface temperature (K) | Luminosity ($L_\odot$) | Main-sequence lifetime |
|---|---|---|---|---|
| 0.2 | M4 | 3200 | 0.005 | over a trillion years |
| 1 | G2 | 5800 | 1 | 10 billion years |
| 5 | B5 | 15 000 | about 600 | about 100 million years |
| 15 | B0 | 30 000 | about 25 000 | about 12 million years |
The main sequence is not a path stars travel along. A star arrives at the point set by its mass and stays near it for about 90% of its life, brightening slowly as helium builds up in its core.
The life of a Sun-like star
- Protostar. A collapsing cloud of gas heats up as gravitational potential energy is converted to thermal energy. It glows but has no fusion yet.
- Main sequence. The core reaches about 10 to 15 million K and hydrogen fuses to helium, mostly by the proton–proton chain. Outward gas and radiation pressure balances gravity. The Sun is here, about halfway through.
- Subgiant and red giant. Core hydrogen runs out. The inert helium core contracts and heats, hydrogen fuses in a shell around it, and the outer layers swell and cool. The star moves up and right to the red giant branch, reaching hundreds of times the Sun's radius.
- Helium flash and horizontal branch. At about 100 million K the core ignites helium fusion (the triple-alpha process, page 8E). The star shrinks a little and settles to steady helium burning.
- Asymptotic giant branch. Core helium runs out, leaving carbon and oxygen. Helium and hydrogen burn in shells and the star swells again. It loses mass in strong winds. Slow neutron capture (the s-process) in these stars makes about half the elements heavier than iron.
- Planetary nebula and white dwarf. The outer layers drift off as a glowing shell; the hot carbon–oxygen core is left as a white dwarf, held up by electron degeneracy pressure. It moves to the bottom left of the diagram and cools for billions of years.
A white dwarf can't be more massive than about 1.4 solar masses (the Chandrasekhar limit). Stars up to about 8 solar masses lose enough mass on the way to stay under it.
The life of a massive star
A star of more than about 8 solar masses runs through its fuel in a few million years. After hydrogen and helium, its core is hot enough to fuse carbon, neon, oxygen and silicon, each stage faster than the last (silicon burning lasts about a day). The star becomes a red supergiant with an onion-like structure of burning shells around an iron core.
Iron cannot release energy by fusion (page 8K explains why). When the iron core passes about 1.4 solar masses, it collapses in under a second. Electrons are forced into protons to make neutrons and neutrinos, the collapse stops suddenly, and the outer layers are blasted out in a core-collapse supernova, briefly as bright as a whole galaxy. What's left is either
- a neutron star, about 1.4 to 2 solar masses in a ball about 20 km across, held up by neutron degeneracy pressure, or
- a black hole, if the remaining core is more than about 2 to 3 solar masses. This usually happens for stars that began with more than about 20 to 25 solar masses, but the dividing line is not sharp and depends on how much mass the star lost.
In 1987 a supernova in the Large Magellanic Cloud, SN 1987A, was the nearest seen since 1604, and was studied closely from southern-hemisphere observatories including the Anglo-Australian Telescope. Neutrino detectors in Japan and the USA caught a burst of 25 neutrinos from the core collapse, about three hours before the light arrived.
- With the 1 M☉ star, step from the protostar to "Main sequence (today)". Read the age: 4.6 billion years. What is fusing in the core?
- Step on to "Red giant tip". By what factor has the luminosity increased (about 2500)? The temperature has dropped to 3100 K, so what must have happened to the radius?
- Continue to "White dwarf". Where on the diagram does it end up, and why is it so dim despite being 15 000 K?
- Switch to 15 M☉. How long is its main-sequence lifetime compared with the Sun's? Step to "Red supergiant" and list the shells in the core cross-section from outside in.
- Compare the 5 M☉ star's ending with the 15 M☉ star's. What decides which path a star takes?
Using the H–R diagram to age a cluster
All the stars in a star cluster formed at about the same time from the same cloud. Plot them on an H–R diagram and the most massive stars, which age fastest, will already have left the main sequence. The point where the cluster's main sequence bends away, the turn-off point, shows which mass of star is just now running out of hydrogen. Its main-sequence lifetime is the cluster's age. Young clusters like the Jewel Box near the Southern Cross still have hot blue stars at the top of the main sequence; old globular clusters have turned off just below the Sun's position and are over 12 billion years old.
Test yourself
Star X has the same surface temperature as the Sun but is 100 times more luminous. Its radius is
Same T, so $L \propto R^2$: $R = \sqrt{100} = 10$ times.
A star is classified as M2 III. It is
M means cool (red); luminosity class III means giant.
Which property mainly determines where a star sits on the main sequence?
Mass sets the core temperature, fusion rate, luminosity and surface temperature.
Why does a massive star's core collapse once it is made of iron?
Iron (and nickel) are near the peak of the binding-energy curve, so fusion beyond them needs energy input.
A cluster's main-sequence turn-off is at stars of 5 solar masses. The cluster's age is closest to
5 M☉ stars last about 100 million years on the main sequence; the turn-off mass is just finishing now.
Label the evolutionary tracks.
Put the stages of a Sun-like star's life in order.
Which kind of star does each describe?
Stars of the same temperature but greater luminosity have a larger . On the main sequence, the most massive stars have the lifetimes. A white dwarf cannot be more than about solar masses. The remnant of a supernova may be a neutron star or a .
Explain how a 1 solar-mass star's position on the Hertzsprung–Russell diagram changes from the main sequence to the end of its life. Relate each change in position to the processes occurring in the star.
On the main sequence the star fuses hydrogen into helium in its core and stays at roughly the same point (about 5800 K, 1 L☉) for about 10 billion years. When core hydrogen runs out, the helium core contracts and heats up, hydrogen fusion continues in a shell around it, and the outer layers expand and cool. The star moves to the right (cooler) and then up (much more luminous) as a subgiant and then a red giant, up to a few thousand L☉ and over 100 times its original radius. When the core reaches about 100 million K, helium fuses into carbon and oxygen; the star shrinks and becomes somewhat hotter and less luminous, moving down and left to the horizontal branch. When core helium is used up, helium and hydrogen burn in shells around a carbon–oxygen core and the star climbs up and right again on the asymptotic giant branch. It then sheds its outer layers as a planetary nebula, exposing the very hot core, which moves rapidly left across the top of the diagram. With no fusion, it becomes a white dwarf: hot (about 10 000 to 100 000 K) but only about Earth-sized, so it drops to the bottom left of the diagram and slowly cools, moving down and to the right.