Astrophysics & Cosmology

Stellar luminosity, stellar evolution, the Doppler effect, Hubble's law, and the Big Bang.

Spec Points Covered
  • I can use Stefan's law L = $4\pi$r^$2 \sigma$\(T^{4}\) to calculate stellar luminosityThe total powerThe rate of energy transfer. Measured in watts (W). radiated by a star across all wavelengths. Measured in watts (W)..
  • I can use Wien's displacementThe distance moved in a particular direction from a starting point. A vector quantity. Measured in metres (m). $law \lambda_{max}$ T = $2.898 \times 10^{-3}$ m K to determine the surface temperature of a star.
  • I can use the inverse square law for intensityThe powerThe rate of energy transfer. Measured in watts (W). transmitted per unit area perpendicular to the wave direction. Measured in W m⁻². Proportional to amplitude squared. I = L/($4\pi$\(d^{2}\)) to relate luminosityThe total powerThe rate of energy transfer. Measured in watts (W). radiated by a star across all wavelengths. Measured in watts (W)., intensityThe power transmitted per unit area perpendicular to the wave direction. Measured in W m⁻². Proportional to amplitude squared., and distance.
  • I can explain the concept of a standard candleAn astronomical object whose absolute luminosityThe total power radiated by a star across all wavelengths. Measured in watts (W). is known or can be determined independently, allowing its distance to be calculated from its observed intensityThe power transmitted per unit area perpendicular to the wave direction. Measured in W m⁻². Proportional to amplitude squared.. and how it is used to determine astronomical distances.
  • I can describe the main features of a Hertzsprung-Russell diagram and locate different types of stars on it.
  • I can describe the life cycle of stars of different masses from main sequence to their final states.
  • I can use the Doppler effectThe change in observed wavelengthThe minimum distance between two points on a wave that are in phase (e.g. crest to crest). Measured in metres (m). (or frequencyThe number of complete oscillations passing a point per unit time. Measured in hertz (Hz).) of a wave due to relative motion between the source and the observer. $equation \Delta\lambda$ / $\lambda \approx$ v/c to calculate recessional velocities.
  • I can use Hubble's law $v = H_{0} d$ and explain what it tells us about the expansion of the universe.
  • I can describe the evidence for the Big Bang theoryThe theory that the universe began from an extremely hot, dense state approximately 13.8 billion years ago and has been expanding and cooling ever since. including the cosmic microwave backgroundNearly uniform microwave radiation from all directions with a black body spectrum corresponding to approximately 2.7 K. The cooled remnant of radiation from the early universe. and redshiftAn increase in the observed wavelength of light compared to the emitted wavelength, caused by the source moving away from the observer or by the expansion of space. of distant galaxies.
  • I can estimate the age of the universe from the Hubble constant using $t \approx 1/$H_0.
Notes
01 Luminosity (L) Luminosity (L) → 02 Wien's law states that the peak wavelength of the black body spectrum is inversely $\lambda_{max} T = 2.898 \times 10^{-3} \text{ m K}$ → 03 Intensity (I) Intensity (I) → 04 Standard candle Standard candle → 05 The HR diagram plots luminosity (y-axis → 06 All stars form from nebulae (clouds of gas and dust that collapse under gravity) → 07 Doppler effect Doppler effect → 08 Hubble's law $v = H_0 d$ → 09 If the universe has been expanding at a constant rate, then the age can be estimated as $t → 10 Big Bang theory Big Bang theory →
Σ Key Equations Full Reference →
On Data Sheet
Not on Data Sheet
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).
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.
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.