Astrophysics and cosmology

Revision notes

Stars emit approximately continuous thermal spectra modified by absorption and emission features. A black-body model links the spectrum to surface temperature. Wien's displacement law, λmax T = b, shows that hotter objects peak at shorter wavelengths. The Stefan-Boltzmann relationship L = 4πR²σT⁴ connects stellar luminosity with radius and surface temperature when a star is modelled as a spherical black-body emitter.

Luminosity is total power emitted, while intensity or flux received at distance d from an isotropic source follows an inverse-square relationship I = \(\dfrac{L}{4\pi d^{2}}\). Apparent brightness therefore depends on both intrinsic luminosity and distance. Spectral lines reveal chemical composition and motion as well as temperature-related information. Parallax provides a geometrical distance method for nearby stars by observing an apparent angular shift against distant background objects from separated observing positions.

For relatively small cosmological redshifts, z = \(\dfrac{\Delta \lambda }{\lambda }\) can be related approximately to recession speed by v ≈ cz. Observations show a relationship between recession speed and distance described in simple form by v = H0 d. This is evidence for cosmic expansion rather than galaxies simply flying away from one privileged central point in ordinary static space. Estimates of H0 depend on distance calibration, velocity measurement and the sample used.

Stellar measurements

Luminosity is total power emitted by a star. Surface temperature can be inferred from spectral shape or peak wavelength where Wien's law is used.

Black-body radiation

An ideal black body emits a characteristic spectrum determined by temperature. Wien's displacement law links peak wavelength to temperature: hotter objects peak at shorter wavelengths. The Stefan-Boltzmann law relates luminosity, radius and temperature, L = 4πR²σT⁴ for a spherical black-body model.

These equations allow radius, luminosity and temperature to be connected quantitatively.

Spectra and stellar composition

Absorption or emission lines identify atomic species because energy levels are quantised. Doppler or cosmological shifts move the observed wavelengths of identifiable lines. Spectra therefore reveal composition, temperature-related information and relative motion.

Redshift

Cosmological redshift is z = \(\dfrac{\Delta \lambda }{\lambda _{\text{rest}}}\). For sufficiently small z, recession speed may be approximated by v ≈ cz. Redshift is determined from displacement of known spectral features, not simply from an object's visible colour.

Hubble relationship

Galaxy observations show an approximately linear relationship v = H0d on large scales. The gradient of recession speed against distance estimates the Hubble constant. The reciprocal of H0 gives an order-of-magnitude timescale for cosmic expansion, with unit conversion required.

Scatter in observational data does not invalidate the overall trend; uncertainties in distance measurement are especially important.

Stellar evolution

Stars form from collapsing gas and dust. Main-sequence lifetime and later evolution depend strongly on initial mass. Fusion changes composition and the balance between gravitational contraction and pressure. Lower-mass and higher-mass stars follow different later pathways.

Cosmological evidence

The expanding-Universe model is supported by redshift-distance observations. The Big Bang model is also supported by additional evidence such as the cosmic microwave background radiation and the observed abundance of hydrogen and helium. Scientific confidence comes from several independent observations fitting one model.

Exam technique

Distinguish luminosity from apparent brightness and redshift from temperature-related colour. Keep astronomical distance units consistent. When interpreting evidence for cosmology, state the measured observation first and then the model inference.

Distances

The astronomical unit (AU) is the mean Earth–Sun distance, 1.50 × 10¹¹ m. A light year is the distance light travels in one year, 9.46 × 10¹⁵ m.

The age and fate of the Universe

The Hubble constant gives an estimate of the age of the Universe: t ≈ \(\dfrac{1}{H_{0}}\), about 14 billion years for H₀ ≈ 70 km s⁻¹ Mpc⁻¹ once units are converted.