Space: stars, distances and cosmology

Black-body radiation, stellar luminosity and distances, the Hertzsprung-Russell diagram, redshift and the fate of the universe.

Understand what is meant by a black body radiator and be able to interpret radiation curves for such a radiator

Be able to use the Stefan-Boltzmann law equation $L = \sigma AT^4$ for black body radiators

Be able to use Wien's law equation for black body radiators

Be able to use the equation intensity $I = \frac{L}{4\pi d^2}$ , where $L$ is luminosity and $d$ is distance from the source

Understand how astronomical distances can be determined using trigonometric parallax

Understand how astronomical distances can be determined using measurements of intensity received from standard candles (objects of known luminosity)

Be able to sketch and interpret a simple Hertzsprung-Russell diagram that relates stellar luminosity to surface temperature

Understand how to relate the Hertzsprung-Russell diagram to the life cycle of stars

Understand how the movement of a source of waves relative to an observer/detector gives rise to a shift in frequency (Doppler effect)

Be able to use the equations for redshift $z = \frac{\Delta\lambda}{\lambda} \approx \frac{\Delta f}{f} \approx \frac{v}{c}$ for a source of electromagnetic radiation moving relative to an observer, and $v = H_0 d$ for objects at cosmological distances

Understand the controversy over the age and ultimate fate of the universe associated with the value of the Hubble constant and the possible existence of dark matter

A star has a luminosity of $5.0 \times 10^{28}\,\text{W}$ and a surface temperature of $9000\,\text{K}$ .

Calculate the radius of the star.

Uses Stefan–Boltzmann law $L = 4\pi r^2\sigma T^4$ (1)
$r^2 = \dfrac{5.0 \times 10^{28}}{4\pi \times 5.67 \times 10^{-8} \times 9000^4}$ (1)
$r = 3.3 \times 10^{9}\,\text{m}$ (1)

The hydrogen-alpha line has a wavelength of $656.3\,\text{nm}$ in the laboratory. In the spectrum of a distant galaxy it is observed at $669.4\,\text{nm}$ . Take the Hubble constant $H_0 = 2.3 \times 10^{-18}\,\text{s}^{-1}$ .

Astronomers use trigonometric parallax for nearby stars and standard candles for more distant objects.

Explain how each method is used to determine the distance to a star or galaxy, and why standard candles are needed for very distant objects.

Indicative content:

Parallax: the apparent position of a nearby star is measured against distant background stars at two times six months apart (1)
The parallax angle and the radius of the Earth's orbit (1 AU) give the distance using trigonometry, $d = r/\tan\theta$ (or $d \approx r/\theta$ ) (1)

$z = \Delta\lambda/\lambda = 13.1/656.3 = 0.020$ (1)
$v = zc = 0.020 \times 3.00 \times 10^{8} = 6.0 \times 10^{6}\,\text{m s}^{-1}$ (1)
$d = v/H_0 = 6.0 \times 10^{6}/2.3 \times 10^{-18} = 2.6 \times 10^{24}\,\text{m}$ (1)
Age $\approx 1/H_0 = 4.3 \times 10^{17}\,\text{s} = 1.4 \times 10^{10}\,\text{years}$ (1)

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