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Theme E — Nuclear and Quantum Physics

E.5.8 Determination of Stellar Radii — IB Physics HL Revision Notes & Topical Questions

5 min read 9 parts Lesson 45 of 45 1 quiz

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Lesson summary

E.5.8 Determination of Stellar Radii — IB PHYSICS SL notes: Calculate stellar radii by combining Wien's law, the inverse square law for flux and the Stefan–Boltzmann law.A star's radius is deduced — stars are too far away to measure directly.Treat the star as a black body — L = 4…

  • Wien's displacement law
  • A peak of 500 nm gives T
  • At distance d the power spreads over a sphere of area 4πd2
  • Rearranged, L
  • Distance comes from stellar parallax or other methods
  • Check with the Sun

Part 1 of 7Strategy

1A star's radius is deduced
  • A star's radius is deduced stars are too far away to measure directly.
2Treat the star as a black body
  • L = 4πR2σT4.
3Then R
√( L / (4πσT4) ) — needing luminosity L and surface temperature T.
4
Procedure combines three laws.
  • Wien's displacement law gives the surface temperature.
  • The inverse square law of flux gives the luminosity from radiant flux and distance.
  • The Stefan–Boltzmann law gives the radius.

Part 2 of 7Surface temperature

Wien's displacement law
  • λmaxT = 2.90 × 10-3 m K, so T = 2.90 × 10-3 / λmax.
Peak wavelength is the wavelength of greatest intensity in the spectrum.
A peak of 500 nm gives T
5800 K like the Sun — Betelgeuse at about 850 nm gives about 3400 K.
Temperature can also come from the spectral class found from absorption lines.

Part 3 of 7Luminosity from brightness and distance

Luminosity from brightness and distance
The star radiates power L equally in all directions.
At distance d the power spreads over a sphere of area 4πd2
  • At distance d the power spreads over a sphere of area 4πd2 so b = L / (4πd2).
  • b is apparent brightness (radiant flux) in W m-2, L is in W and d in m.
Rearranged, L
4πd2b.
Distance comes from stellar parallax or other methods
  • Distance comes from stellar parallax or other methods and b from a photometer.
Check with the Sun
  • Check with the Sun b = 1.36 × 103 W m-2 at 1.50 × 1011 m gives L = 3.8 × 1026 W.

Part 4 of 7Combined procedure

1
Measure the parallax angle p and find d = 1/p in parsecs, then convert to metres.
2
Measure apparent brightness b and find L = 4πd2b.
3
Measure the peak wavelength and find T from Wien's law.
4
Find R = √( L / (4πσT4) ).
If L is given directly, start from step 3.
Solar ratios
  • R/R☉ = √(L/L☉) × (T☉/T)2.
  • The Sun has L☉ = 3.83 × 1026 W, T☉ = 5800 K and R☉ = 6.96 × 108 m.
For Betelgeuse (L
4.49 × 1031 W, peak 850 nm) the radius is about 1000 times the Sun's.

Part 5 of 7Meaning of the results

Star typeTypical radius
White dwarfAbout 0.01 solar radii, comparable to Earth
Main-sequence starAbout 0.1 to 10 solar radii
Red giantTens to hundreds of solar radii
Red supergiantUp to about a thousand solar radii
Neutron starAbout 10 km
1
These sizes explain the regions of the HR diagram.
2The black-body assumption is an approximation
  • The black-body assumption is an approximation real spectra have absorption lines.
3Interstellar dust dims starlight
  • Interstellar dust dims starlight making the calculated luminosity too low unless corrected.
4
Close eclipsing binaries give radii more directly from eclipse duration and depth.

Part 6 of 7Ratios and checks

Ratios and checks
For two stars use L ∝ R2T4
  • For two stars use L ∝ R2T4 constants such as 4πσ cancel.
  • If star A has three times the radius and half the temperature of star B, LA/LB = 32 × (1/2)4 = 9/16.
Convert temperature to kelvin and distance to metres
  • Convert temperature to kelvin and distance to metres because T4 and d2 amplify errors.
Sanity checks
  • Sanity checks a main-sequence radius lies within a factor of ten of the Sun's, a giant is much larger, a white dwarf is near Earth-sized.

Part 7 of 7More from the specification

Stellar blackbody energy levels
  • Stefan-Boltzmann calculations combine radiant flux and temperature, governed by underlying quantum photonic energy levels.
Stefan-Boltzmann Law and Stellar Radii
  • The total power (luminosity L) emitted by a star is related to its surface radius R and absolute surface temperature T by L = 4π R2 σ T4,
  • allowing stellar radii to be calculated from observational data.
Hydrostatic Equilibrium in Stars
  • A stable star maintains hydrostatic equilibrium,
  • where inward gravitational pressure is precisely balanced by outward gas and radiation pressure generated by core nuclear fusion reactions.
Hydrogen Fusion in Stellar Cores
  • Main sequence stars fuse hydrogen nuclei into helium nuclei via the proton-proton chain or CNO cycle, with mass converted into released radiant energy via E = m c2.
Core Collapse and Post-Main Sequence Evolution
  • When core hydrogen is exhausted, radiation pressure drops and gravity causes core collapse until temperature and pressure rise sufficiently to trigger helium fusion.
Stefan-Boltzmann Law and Stellar Radii
  • The total power (luminosity L) emitted by a star is related to its surface radius R and absolute surface temperature T by L = 4π R2 σ T4,
  • allowing stellar radii to be calculated from observational data.
Hydrostatic Equilibrium in Stars
  • A stable star maintains hydrostatic equilibrium,
  • where inward gravitational pressure is precisely balanced by outward gas and radiation pressure generated by core nuclear fusion reactions.
Hydrogen Fusion in Stellar Cores
  • Main sequence stars fuse hydrogen nuclei into helium nuclei via the proton-proton chain or CNO cycle, with mass converted into released radiant energy via E = m c2.
Core Collapse and Post-Main Sequence Evolution
  • When core hydrogen is exhausted, radiation pressure drops and gravity causes core collapse until temperature and pressure rise sufficiently to trigger helium fusion.
Energy level
  • A discrete energy an electron in an atom can have
Kinetic energy
  • K = ½mv2, energy of motion.
Continuous spectrum
  • Spectrum with no gaps between the regions.
Fusion
  • Joining of light nuclei into a heavier one.
Pressure
  • Force per unit area, P = F/A.
    What the symbols mean: p = pressure (Pa = N m−2) · F = force acting at right angles to the surface (N) · A = area (m2)
Helium
  • Product of the fusion of hydrogen in stars.
Nucleus
  • Small, dense, positively charged centre of the atom.
Reaction
  • The partner force acting on the other object.
Radiation pressure
  • The pressure exerted on a surface by absorbed or reflected radiation because photons transfer momentum.
Binding energy
  • The minimum energy required to separate the nucleons of a nucleus to infinity.
Emission spectrum
  • The set of discrete wavelengths emitted by excited atoms of an element.

Key terms

TermMeaning
Apparent brightnessIntensity received from a star, b = L/(4πd2.
LuminosityTotal power output of a star.
Black bodyIdeal emitter whose spectrum depends only on temperature.
Wien's displacement lawλ_max T = 2.90 × 10-3 m K.
Stefan–Boltzmann lawL = σAT4 for a black body.
PhotosphereVisible surface layer of a star.
Solar radius6.96 × 108 m.

Avoid these mistakes

1
Using the Celsius scale for T — use kelvin.
2
Forgetting the factor 4π in L = 4πd2 b or in the Stefan–Boltzmann law.
3
Leaving the distance in parsecs or light-years in b = L/(4πd2 — use metres.
4
Forgetting to take the square root when finding R from R2.
5
Thinking luminosity and apparent brightness are the same — brightness depends on distance.

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