Summary — What This Chapter Covers
- Luminosity L: total power of radiation emitted by a star. Radiant flux intensity F = L/(4πd2).
- A standard candle is an object of known luminosity; measuring F then gives d.
- Wien's law λmax ∝ 1/T; Stefan–Boltzmann L = 4πσr2T4; together they give stellar radii.
- Redshift Δλ/λ ≈ Δf/f ≈ v/c; distant galaxies are receding.
- Hubble's law v ≈ H0d implies an expanding Universe that began in a Big Bang.
1. Standard Candles
25.1
Known luminosity, measured flux
The luminosity L of a star is the total power of radiation it emits (W). The radiation spreads over a sphere of area 4πd2, so the radiant flux intensity F (power per unit area received, W m−2) falls with the inverse square of distance.
A standard candle is an object whose luminosity is known. Measure F from Earth, and d follows from the inverse square law. Cepheid variable stars (whose luminosity is linked to their pulsation period) work for nearby galaxies; type Ia supernovae, which all peak at about the same luminosity and are extremely bright, reach far more distant galaxies.
Inverse square law
F = L/(4πd2) d = √(L/(4πF))
Worked Example
A Cepheid in a distant galaxy has luminosity 3.0 × 1030 W and a measured radiant flux intensity of 2.5 × 10−12 W m−2. Find the distance to the galaxy.
Practice Question
The Sun’s luminosity is 3.83 × 1026 W and it is 1.50 × 1011 m away. Find the radiant flux intensity at Earth.
2. Stellar Radii
25.2
Wien and Stefan–Boltzmann
A star radiates approximately like a black body. Wien’s displacement law: the wavelength at which the emitted intensity peaks is inversely proportional to the thermodynamic surface temperature, λmax ∝ 1/T (λmaxT = constant, about 2.9 × 10−3 m K; questions give the constant or a reference star). Hotter stars peak at shorter wavelengths and look bluer.
The Stefan–Boltzmann law gives the luminosity from the surface area and temperature, L = 4πσr2T4, with σ = 5.67 × 10−8 W m−2 K−4. Find T from the spectrum (Wien), L from a standard-candle or distance measurement, then solve for r.
Black-body laws
λmax ∝ 1/T L = 4πσr2T4
Worked Example
A star’s spectrum peaks at 290 nm and its luminosity is 4.0 × 1028 W. Take λmaxT = 2.9 × 10−3 m K. Find its surface temperature and radius.
Practice Question
Star X has the same surface temperature as star Y but four times its luminosity. Compare their radii.
3. Hubble's Law and the Big Bang Theory
25.3
Redshift and expansion
longer wavelengths
redshifted
Hubble’s law
0
Big Bang
0
0
−1
Redshift and Hubble
Δλ/λ ≈ Δf/f ≈ v/c v ≈ H0d age ≈ 1/H0
Practice Question
Estimate the age of the Universe from H0 = 2.2 × 10−18 s−1.
Memorise