Astronomy and cosmology

A-Level Physics 9702 · Topic 25 revision notes · A2

Measuring the Universe: luminosity, flux and standard candles for distance, Wien's and Stefan-Boltzmann laws for stellar radii, and redshift with Hubble's law for an expanding Universe.

9 definitions to learn word for word, 8 equations in their own column, 6 diagrams and graphs. Read it, cover it, test yourself.

25.1 Standard candles
25.2 Stellar radii
25.3 Hubble's law and the Big Bang theory

Definitions (mark-scheme wording)EquationsExam traps

25.1Standard candles

Definitions

Luminosity (L): the total power of electromagnetic radiation emitted by a star.
Radiant flux intensity (F): the radiant power passing normally through unit area (at the observer).
Standard candle: an astronomical object of known luminosity (e.g. a Cepheid variable star or a type 1a supernova).

Key points

Radiation spreads out over a sphere of area 4πd², so flux obeys an inverse-square law.
Luminosity is intrinsic to the star; flux depends on luminosity and distance.
Distance method: know L (standard candle), measure F, then d = √(L / 4πF).
Cepheid variables: pulsation period is linked to luminosity; useful to ~10⁷ pc (nearby galaxies).
Type 1a supernovae: very bright, nearly equal peak luminosity; reach distant galaxies.
Assumes no absorption of radiation between the star and Earth.

Inverse-square law: the same power L spreads over spheres of area 4πd², so doubling d quarters F.

Flux against distance: F ∝ 1/d² for a source of fixed luminosity.

Watch out!

A dim star may be distant, not low-luminosity: F depends on both L and d.
Use 4πd² (sphere area), not πd², and d in metres.

25.2Stellar radii

Definitions

Wien's displacement law: the wavelength at which a black body emits maximum power is inversely proportional to its thermodynamic temperature: λ_max ∝ 1/T.
Stefan-Boltzmann law: the power radiated by a black body is proportional to its surface area and to the fourth power of its thermodynamic temperature.
Black body: an idealised object that absorbs all incident radiation and emits a continuous spectrum depending only on its temperature.

Key points

Treat stars as black bodies: the spectrum shape depends only on surface temperature.
Hotter star: peak at shorter λ (bluer), and more power at every wavelength.
Step 1: measure λ_max from the spectrum, use Wien's law to find T.
Step 2: find L (e.g. from F and d). Step 3: r = √(L / 4πσT⁴).
Two stars of the same T but different L: the more luminous one is larger (e.g. red giant vs red dwarf).
T must be in kelvin.

Black-body curves: the hotter star is higher at every λ and peaks at a shorter λ_max (Wien's law).

Finding a stellar radius: spectrum gives T (Wien), flux and distance give L, then Stefan-Boltzmann gives r.

Watch out!

L ∝ T⁴: doubling T multiplies L by 16 (same radius).
Temperature in kelvin; λ_max in metres (not nm) with the 2.9 × 10^-3 m K constant.

25.3Hubble's law and the Big Bang theory

Definitions

Redshift: the increase in the observed wavelength (decrease in frequency) of electromagnetic radiation from a source moving away from the observer.
Hubble's law: the recession speed of a galaxy is (approximately) proportional to its distance from the Earth.
Big Bang theory: the Universe began from a single point (very hot and dense) and has been expanding ever since.

Key points

Spectral lines from distant galaxies are shifted to longer wavelengths compared with the same lines in the lab.
For v ≪ c, the fractional shift gives the recession speed (Doppler effect).
Almost all galaxies are redshifted: they are moving away from us.
Further galaxies show greater redshift: v ∝ d (Hubble's law).
Distances come from standard candles; speeds from redshift.
Hubble's law implies the Universe is expanding; running it backwards gives the Big Bang.
Age of the Universe ≈ 1/H₀ (assumes constant expansion rate), about 1.4 × 10¹⁰ years.

Absorption lines from a receding galaxy (bottom) are shifted towards the red end compared with the laboratory lines (top).

Hubble's law: recession speed against distance is a straight line through the origin; gradient = H₀.

Watch out!

For the age 1/H₀, convert H₀ to s^-1 first (from km s^-1 Mpc^-1; 1 Mpc = 3.09 × 10²² m).
Redshift is an increase in wavelength; Δλ = observed - emitted, divided by the emitted λ.
Galaxies are not flying through space from a centre: space itself expands, and every observer sees the same law.