5.8 Stars: radiation, distances and the HR diagram

What is a black body, and what does its radiation curve show?

Three equations and two diagrams unlock almost everything we know about stars. The colour of a star gives its temperature (Wien), temperature and size give its power output (Stefan–Boltzmann), and power and distance give how bright it looks from Earth (inverse-square law). All three rest on one idealisation.

• Every object above absolute zero emits electromagnetic radiation. A black body is the perfect emitter as well as the perfect absorber: no object at the same temperature emits more at any wavelength.
• The spectrum is continuous (all wavelengths are present), but the power is not spread evenly: the curve of intensity against wavelength rises steeply to a peak at \(\lambda_{\max}\) and falls away more gradually at long wavelengths.
• Stars are a good approximation to black bodies, so their continuous spectra can be used to find their surface temperatures.

Reading a black-body curve

• Hotter → curve higher at every wavelength. A hotter body emits more at every wavelength, so the curves never cross.
• Hotter → peak at a shorter wavelength. The peak moves from the infrared (cool stars) through the visible to the ultraviolet (very hot stars). This is why cool stars look red and hot stars look blue-white.
• Area under the curve = total power emitted per unit area, which is proportional to \(T^4\).
• The Sun's peak, near 501 nm, lies in the visible region; a filament lamp at about 3000 K peaks in the infrared, so most of its output is heat, not light.

How do you find the temperature of a star from its spectrum?

\(\lambda_{\max}\) is the wavelength at which the intensity is greatest (m) and \(T\) the surface temperature (K). Wien's law is in the formulae list. Since \(\lambda_{\max}\propto1/T\), doubling the temperature halves the peak wavelength. The temperature must be in kelvin (\(T/\text{K}=\theta/^\circ\text{C}+273\)).

• Measure the spectrum of the star, find the wavelength at the peak, and divide \(2.898\times10^{-3}\ \text{m K}\) by it.
• This gives the surface temperature: the temperature of the layer the light escapes from, not the much hotter core.

How much power does a star radiate?

\(L\) is the luminosity (W), \(\sigma\) the Stefan–Boltzmann constant \(=5.67\times10^{-8}\ \text{W m}^{-2}\,\text{K}^{-4}\), \(A\) the surface area (m²) and \(T\) the surface temperature (K). The Stefan–Boltzmann law is in the formulae list. For a spherical star of radius \(r\), \(A=4\pi r^2\), so \(L=4\pi r^2\sigma T^4\); the area formula is not in the list (No — learn it).

• \(L\propto r^2\) and \(L\propto T^4\). Temperature has the stronger effect: doubling \(T\) multiplies \(L\) by \(2^4=16\); doubling \(r\) multiplies it by 4.
• A cool star can still be very luminous if it is very large (a red giant); a hot star can be dim if it is very small (a white dwarf).
• Astronomers often quote luminosity in units of the Sun's luminosity, \(L_\odot=3.83\times10^{26}\ \text{W}\) (radius \(R_\odot=6.96\times10^8\ \text{m}\), surface temperature ≈ 5780 K). These are not on the data list: a question will give them.

How bright does a star appear from Earth?

The star's power spreads out over the surface of a sphere of radius \(d\), area \(4\pi d^2\). The equation assumes nothing absorbs the radiation on the way (no dust or gas in between).

• Measured on Earth: the intensity \(I\) (with a detector of known area) and the spectrum (so \(\lambda_{\max}\) and \(T\)).
• Then: if \(d\) is known, \(L=4\pi d^2I\); and with \(T\) from Wien, the radius follows from \(L=4\pi r^2\sigma T^4\).
• Example: the Sun's luminosity spread over a sphere of radius 1 AU = 1.50 × 10¹¹ m gives about 1.35 × 10³ W m⁻² above the Earth's atmosphere.

How is the distance to a nearby star measured?

Hold a finger at arm's length and close each eye in turn: the finger jumps against the background. Trigonometric parallax is the same effect with the Earth's orbit as the baseline. As the Earth moves round its orbit, a nearby star appears to shift its position against the background of much more distant stars (which barely move).

• Measure the star's angular position, then again six months later, when the Earth is on the other side of the Sun, 2 AU away. Half the total angular shift is the parallax angle \(p\).
• The parallax angle is the angle at the star subtended by the radius of the Earth's orbit, \(r\) = 1 AU, so \(\tan p=r/d\).

\(d\) is the distance to the star (m), \(r\) the radius of the Earth's orbit = 1 AU = 1.50 × 10¹¹ m and \(p\) the parallax angle. This is not in the formulae list (No — learn it (trigonometry)). Parallax angles are tiny, always less than 1 arcsecond (1 arcsecond = 1/3600 of a degree), so \(\tan p\approx p\) in radians.

• Because \(d\propto1/p\): \(d\) in parsecs = 1 ÷ (\(p\) in arcseconds). A parallax of 0.5 arcseconds means 2 pc.
• Limit: for distant stars the angle is too small to measure reliably against the uncertainty, so parallax works only for relatively nearby stars (in our part of the galaxy). Telescopes in space, above the blurring atmosphere, extend the range.

How do standard candles measure greater distances?

• Measure the intensity \(I\) at Earth; with \(L\) known, rearrange the inverse-square law: \(d=\sqrt{L/4\pi I}\).
• Cepheid variable stars pulsate, and their luminosity is linked to their period of pulsation: measure the period, deduce \(L\). The link was calibrated using Cepheids whose distance was known by parallax. They work out to other nearby galaxies.
• Type Ia supernovae (exploding white dwarfs) all reach about the same peak luminosity, billions of times the Sun's. They are visible across billions of light-years.
• Each method is calibrated by the one before it (parallax → Cepheids → supernovae), so uncertainties build up as distances grow.

What is the Hertzsprung–Russell diagram?

A Hertzsprung–Russell (HR) diagram is a plot of luminosity (vertical, logarithmic scale, often in units of \(L_\odot\)) against surface temperature (horizontal, logarithmic, with temperature decreasing from left to right). Stars do not fall randomly: most lie on a diagonal band, the main sequence, and the rest gather in a few distinct groups. Because \(L=4\pi r^2\sigma T^4\), every point on the diagram also has a radius: lines of constant radius run diagonally, and stars get bigger towards the top right.

How does the HR diagram show the life cycle of a star?

• A star forms when a cloud of gas and dust (a nebula) collapses under gravity. The protostar heats up until hydrogen fusion begins in its core and it joins the main sequence. Where it joins depends on its mass.
• On the main sequence the outward pressure from the hot core balances the inward pull of gravity, and the star stays in nearly the same place on the diagram for most of its life. More massive stars use their fuel much faster, so they have shorter lives.
• When the hydrogen in the core runs out, the core contracts and heats, the outer layers expand and cool, and the star moves up and to the right on the diagram.

Low-mass stars (up to about 8 solar masses, including the Sun)

• Main sequence → red giant (helium fusion begins in the core; the outer layers swell to perhaps a hundred times their former radius).
• The outer layers drift away as a planetary nebula, exposing the hot core, which is seen briefly at the top left of the diagram.
• The core becomes a white dwarf: no fusion, very dense, about Earth-sized. It cools and fades, moving down and to the right along a line of almost constant radius.

High-mass stars (more than about 8 solar masses)

• Main sequence (high and to the left) → red supergiant, fusing heavier and heavier elements in shells around the core, up to iron.
• When the core can release no more energy by fusion it collapses in a fraction of a second and the star explodes as a supernova, briefly outshining a whole galaxy.
• The remnant core becomes a neutron star (core about 1.4 to 3 solar masses: incredibly dense, made almost entirely of neutrons) or, if more than about 3 solar masses, a black hole, whose gravity is so strong that not even light can escape.

Exam tips for 5.8

Common misconceptions

How this topic connects

Photons and line spectra (Unit 2)

The dark absorption lines on a star's continuous spectrum identify the elements in its outer layers; Wien's law uses the continuous spectrum underneath them. Intensity as power per unit area was first met with waves.

Thermal physics and 5.3 Nuclear fusion

Wien's law and the Stefan–Boltzmann law need absolute temperatures in kelvin, the same scale used for the ideal gas. The energy a star radiates as luminosity comes from fusion in its core; the mass of the star decides how fast it is used up.

5.7 Gravitation and 5.9 Cosmology

Gravity drives the collapse that forms a star and ends its life; standard candles give the distances that, with redshift, reveal the expanding universe.

Key equations

Check your understanding

Specification checklist

Numbers refer to the specification points.

• (161) I can describe what a black body is and interpret black-body radiation curves at different temperatures.
• (162) I can use \(L=\sigma AT^4\) to link the luminosity, surface area and surface temperature of a star.
• (163) I can use Wien's law, \(\lambda_{\max}T=2.898\times10^{-3}\ \text{m K}\), to find the surface temperature of a star from its spectrum.
• (164) I can use \(I=L/4\pi d^2\) to link intensity at Earth, luminosity and distance.
• (165) I can explain how trigonometric parallax gives the distance to a nearby star, and work with parsecs and light-years.
• (166) I can explain what a standard candle is and how its known luminosity gives the distance to a remote object.
• (167) I can sketch a simple Hertzsprung–Russell diagram with labelled axes and identify the main sequence, red giants, supergiants and white dwarfs.
• (168) I can relate the HR diagram to the life cycle of stars of different masses, including red giants, white dwarfs, supergiants, supernovae, neutron stars and black holes.

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