AQA A-Level Physics: Astrophysics (7408) – Revision Notes

These notes condense section 3.9 Astrophysics of the AQA AS and A-level Physics specification (7407/7408, version 1.4, AS and A-level exams June 2016 onwards), sub-sections 3.9.1 to 3.9.3. Astrophysics is one of the five options (sections 3.9 to 3.13); you study one, assessed in Paper 3 Section B. It is A-level only. For full explanations and worked examples, read the Astrophysics study guide first.

Useful links: course hub, printable checklist, free diagnostics, and the Astrophysics practice questions.

Formula table

Quantity	Relationship	Notes
Angular magnification	M = β/α = fₒ/fₑ	Normal adjustment; lenses fₒ + fₑ apart
Rayleigh criterion	θ ≈ λ/D	θ in rad; smaller θ = better resolution
Collecting power	∝ D²	Area of objective
Magnitude and intensity	ratio = 2.51^Δm	Δm = 1 → 2.51; Δm = 5 → about 100
Distance modulus	m − M = 5 log₁₀(d/10)	d in parsecs
Wien’s law	λₘₐₓT = 2.9 × 10⁻³ m K	Peak wavelength → temperature
Stefan’s law	P = σAT⁴	A = 4πr² for a star
Inverse square law	I = P/4πd²	Isotropic source, no absorption
Schwarzschild radius	Rs ≈ 2GM/c²	Radius of event horizon
Doppler	Δf/f = v/c; z = Δλ/λ = −v/c	v ≪ c; v positive for approach
Binary orbit radius	r = vT/2π	Circular orbit, viewed in its plane
Hubble’s law	v = Hd	Age ≈ 1/H if H constant

Unit conversions: 1 pc ≈ 3.08 × 10¹⁶ m; 1 Mpc ≈ 3.08 × 10²² m; 1 ly ≈ 9.46 × 10¹⁵ m. H in s⁻¹ = (H in km s⁻¹ Mpc⁻¹ × 10³)/(3.08 × 10²²).

3.9.1 Telescopes

• Refractor: two converging lenses. Normal adjustment means the two principal foci coincide and the final image is at infinity. The ray through the objective’s centre fixes where the intermediate image forms in the shared focal plane; a construction line from there through the eyepiece centre sets the direction of the emerging parallel rays. Image is inverted.
• Cassegrain: parabolic concave primary, convex secondary, light returns through a hole in the primary to the eyepiece.
• Spherical aberration: edge and central rays focus at different points (spherical surfaces). Cured by a parabolic mirror.
• Chromatic aberration: lenses focus different colours at different points. Mirrors do not have it.
• Reflectors over refractors: no chromatic aberration; large mirrors supported from behind; one surface to polish. Refractors: lenses sag, glass must be flawless and absorbs some wavelengths. Reflector drawback: secondary blocks light and causes diffraction.
• Radio vs optical: both parabolic reflectors (radio can be mesh if gaps ≪ λ); radio works on the ground, day or night, through cloud, but suffers interference; poorer resolution (large λ); larger dishes give larger collecting power.
• IR: high dry sites or space, cooled detectors. UV and X-ray: space only (atmospheric absorption); X-ray mirrors at grazing incidence.
• Eye vs CCD: CCD has much higher quantum efficiency, resolution set by pixel spacing, long exposures and digital storage; eye is real-time and needs no equipment.

3.9.2 Classification of stars

• Apparent magnitude m: Hipparcos scale; dimmest visible stars m = 6; lower number = brighter; brightness is subjective.
• Absolute magnitude M: apparent magnitude at 10 pc.
• Black body: hotter → higher curve, peak at shorter λ.
• Spectral classes (hot to cool): O, B blue; A blue-white; F white; G yellow-white; K orange; M red. Lines: O He⁺, He, H; B He, H; A H strongest plus ionised metals; F ionised metals; G ionised and neutral metals; K neutral metals; M neutral atoms, TiO.
• Balmer lines need atoms in n = 2: too few excited in cool stars, mostly ionised in the hottest, strongest in A stars.
• HR diagram: M from +15 (bottom) to −10 (top); T from 50 000 K (left) to 2 500 K (right). Main sequence diagonal; giants top right; white dwarfs bottom left. Sun: main sequence, G, M about +5.
• Sun-like evolution: protostar → main sequence → red giant (up and right) → white dwarf (left and down).
• Supernova: rapid rise in absolute magnitude. Neutron star: neutrons, nuclear density. Black hole: escape velocity > c.
• Gamma ray bursts: supergiant collapse to neutron star or black hole.
• Type 1a: same peak M → standard candle; light curve quick rise, slow fall. Faint distant ones → accelerating Universe → dark energy (controversial).
• Supermassive black holes at galaxy centres.

Method: find a distance from magnitudes

1. Write m − M = 5 log₁₀(d/10).
2. Divide (m − M) by 5 and raise 10 to that power.
3. Multiply by 10 to get d in pc; convert to m only if asked.

Method: compare two stars with Stefan’s law

1. Write P₁/P₂ = (r₁/r₂)² × (T₁/T₂)⁴.
2. Get T from Wien if you are given λₘₐₓ.
3. Substitute ratios; check the hotter or bigger star comes out more powerful.

3.9.3 Cosmology

• Doppler: recession → red shift, z positive. Use z = Δλ/λ = v/c for the speed (v ≪ c). Applies to optical and radio frequencies.
• Binary star: maximum shift → orbital speed; time for one cycle of shifts → period.
• Hubble: v = Hd; expansion; age ≈ 1/H.
• Big Bang evidence: microwave background (about 2.7 K, nearly uniform in all directions); hydrogen and helium abundance (about 3 : 1 by mass).
• Quasars: most distant measurable objects; found as bright radio sources; large optical red shifts; huge power; active supermassive black holes.
• Exoplanets: direct detection hard (faint, tiny angular separation). Radial velocity method = periodic Doppler shift of the star. Transit method = periodic dip in intensity; dip ≈ (r_planet/r_star)².

Method: age of the Universe

1. Multiply H in km s⁻¹ Mpc⁻¹ by 10³.
2. Divide by 3.08 × 10²² to get s⁻¹.
3. Take 1/H in seconds; divide by 3.15 × 10⁷ for years.

Must-know distinctions

• Apparent vs absolute magnitude: seen from Earth vs at 10 pc.
• Resolving power vs collecting power: λ/D (detail) vs D² (faintness).
• Spherical vs chromatic aberration: shape of surface vs dispersion in glass.
• Radial velocity vs transit: Doppler shift of the star vs dip in its intensity.
• Type 1a supernova vs gamma ray burst: standard candle vs collapse of a supergiant.
• Neutron star vs black hole: escape velocity below c (matter at nuclear density) vs escape velocity above c (event horizon at Rs).
• Refractor vs reflector image defects: a refractor can show both aberrations; a parabolic reflector shows neither.

Quick self-test

1. fₒ = 2.0 m, fₑ = 50 mm. Find M in normal adjustment.
2. Find the minimum angular resolution of a 0.30 m telescope at 600 nm.
3. How many times greater is the collecting power of a 10 m mirror than a 2.5 m mirror?
4. Two stars differ by 3 magnitudes. Find the intensity ratio.
5. A star at 100 pc has m = 7.0. Find M.
6. A star’s spectrum peaks at 725 nm. Find T and its spectral class.
7. Two stars have the same temperature; one has 3 times the radius. Find the power ratio.
8. Find Rs for 10 solar masses (1.99 × 10³⁰ kg each; G = 6.67 × 10⁻¹¹ N m² kg⁻²; c = 3.00 × 10⁸ m s⁻¹).
9. A galaxy has z = 0.030. Find v and d if H = 65 km s⁻¹ Mpc⁻¹.
10. Estimate the age of the Universe if H = 65 km s⁻¹ Mpc⁻¹.
11. Which exoplanet method relies on a periodic dip in a star’s intensity?
12. Why are Balmer lines weak in O-class stars?

Answers

1. M = 2.0/0.050 = 40.
2. θ ≈ 600 × 10⁻⁹/0.30 = 2.0 × 10⁻⁶ rad.
3. (10/2.5)² = 16.
4. 2.51³ = 15.8.
5. 7.0 − M = 5 log(10) = 5, so M = 2.0.
6. T = 2.9 × 10⁻³/725 × 10⁻⁹ = 4000 K, class K.
7. P ∝ r², so 9.
8. Rs = 2 × 6.67 × 10⁻¹¹ × 1.99 × 10³¹/(9.00 × 10¹⁶) = 2.95 × 10⁴ m.
9. v = 0.030 × 3.00 × 10⁸ = 9.0 × 10⁶ m s⁻¹ = 9000 km s⁻¹; d = 9000/65 = 138 Mpc.
10. H = 65 × 10³/3.08 × 10²² = 2.11 × 10⁻¹⁸ s⁻¹; 1/H = 4.74 × 10¹⁷ s = 1.5 × 10¹⁰ years.
11. The transit method.
12. Most hydrogen is ionised, so few atoms have electrons in n = 2 to absorb.

Where marks are usually lost

• Writing M = fₑ/fₒ or giving the telescope length as fₒ alone.
• Leaving θ in degrees or arcseconds when comparing with λ/D.
• Calling a star with m = −1 fainter than one with m = 2.
• Using d in metres or light years in m − M = 5 log(d/10).
• Using T² instead of T⁴ in Stefan’s law, or forgetting to square the radius ratio.
• Converting H to s⁻¹ wrongly (forgetting the 10³ for km).
• Saying Balmer lines are strongest in the hottest stars.
• Describing the transit light curve without the repeat at the orbital period.
• Stating that black holes have infinite escape velocity instead of escape velocity greater than c.

Official syllabus

AQA AS and A-level Physics specification (7407/7408), version 1.4, July 2026, AS and A-level exams June 2016 onwards: section 3.9 Astrophysics (A-level only).