Key Equations

Astrophysics & Cosmology - OCR A-Level Physics

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Intensity inverse square law
$$I = \frac{L}{4\pi d^2}$$
  • Where:
    • $I$ = W \(m^{-2}\)
    • $L$ = W
    • $d$ = m
Intensity falls off as the square of the distance. Used with standard candles to find distances.
Stefan's law (luminosity)
$$L = 4\pi r^2 \sigma T^4$$
  • Where:
    • $L$ = W
    • $r$ = m
    • $\sigma$ = 5.67 \times \(10^{-8}\) W \(m^{-2}\) \(K^{-4}\)
    • $T$ = K
Links a star's luminosity to its radius and surface temperature. L depends on \(T^{4}\) so temperature dominates.
Wien's displacement law
$$\lambda_{max} T = 2.898 \times 10^{-3} \text{ m K}$$
  • Where:
    • $\lambda_{max}$ = m
    • $T$ = K
The peak wavelength is inversely proportional to temperature. Hotter stars peak at shorter wavelengths (bluer).
Doppler effect for light
$$\frac{\Delta\lambda}{\lambda} \approx \frac{\Delta f}{f} \approx \frac{v}{c}$$
  • Where:
    • $\Delta\lambda$ = m
    • $\lambda$ = m (emitted)
    • $v$ = m \(s^{-1}\)
    • $c$ = 3.0 \times \(10^{8}\) m \(s^{-1}\)
Valid for v << c. Positive \Delta\lambda means redshift (receding source). Use emitted wavelength in denominator.
Hubble's law
$$v = H_0 d$$
  • Where:
    • $v$ = km \(s^{-1}\) or m \(s^{-1}\)
    • $H_0$ = km \(s^{-1}\) Mpc^{-1} or \(s^{-1}\)
    • $d$ = Mpc or m
Recessional velocity is proportional to distance. Evidence for an expanding universe.
Age of the universe
$$t \approx \frac{1}{H_0}$$
  • Where:
    • $t$ = s
    • $H_0$ = \(s^{-1}\)
Must memorise. Only valid as an approximation assuming constant expansion rate. Convert H_0 to \(s^{-1}\) first.
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