Unit 5, from zero

Heating, latent heat and internal energy

Block 1≈9 marks a papermcΔθ in 18 of 18latent heat in 15 of 18

Learn

• Kelvin: T (K) = θ (°C) + 273. A change of 1 °C is a change of 1 K, so either works for Δθ, but anything with pV or kT needs kelvin. Absolute zero (0 K, −273 °C) is the lowest possible temperature, where the molecules have minimum kinetic energy.
• Internal energy is the sum of the randomly distributed kinetic and potential energies of all the molecules. Temperature tracks the mean kinetic energy.
• Specific heat capacity c: the energy needed to raise 1 kg by 1 K, so ΔE = mcΔθ (J kg⁻¹ K⁻¹).
• Specific latent heat L: the energy needed to change the state of 1 kg with no change in temperature, so ΔE = LΔm. “Fusion” means melting or freezing; “vaporisation” means boiling or condensing.
• While a substance changes state the temperature stays constant. The energy changes the molecules’ potential energy (their separation changes), while their mean kinetic energy stays the same. Flat parts of a heating or cooling curve are changes of state.
• Heaters: energy supplied = Pt. Efficiency = useful energy ÷ total energy.
• Energy balance: energy lost by the hot things = energy gained by the cold things, and everything ends at the same temperature. With ice: energy to melt the ice (mL) + energy to warm the meltwater from 0 °C (mcΔθ) = energy lost by the drink (mcΔθ).
• Flowing liquids: mass per second = density × volume per second, and rate of energy transfer = (mass per second) × c × Δθ.

ΔE = mcΔθΔE = LΔmρ = m/VE = PtT = θ + 273energy lost by hot = energy gained by cold

What they ask

• Kettle or heater: how long to boil, or what mass boils away. Use E = Pt, spend some of it on mcΔθ up to 100 °C, and the rest boils water off (LΔm). Seen in January 2021, October 2021, October 2023, January 2025 and January 2026.
• Mixtures: “Deduce whether adding the ice cools the drink to the safe temperature.” Write the energy balance in words first, solve, then compare and conclude. Ice in a drink (January 2022), molten tin in copper (October 2024), water in mugs (October 2025).
• Big objects: a pool, an ice floe, an ice pack, a tub of ice cream. Get the mass from m = ρV first, with V in m³.
• Explain the flat part of a cooling curve, or how the internal energy changes (the June 2023 6-marker).
• Experimental errors (MCQs): ice melting because energy comes from the surroundings, the heater running longer than the timed period, energy lost to the room. Decide whether each makes L too big or too small.
• Why water makes a good coolant: its large specific heat capacity means a smaller temperature rise for the same energy transferred (June 2026).

Traps

• Grams to kg, and cm³ to m³ (1 cm³ = 10⁻⁶ m³, 1 litre = 10⁻³ m³).
• Subtracting energies in a balance where they should be added. In June 2025 this cost many people the final mark.
• Forgetting that the meltwater also has to warm up from 0 °C.
• Energy flows from hot to cold. An ice pack keeps food cold because energy flows from the food into the ice, which melts. Saying the ice “gives out cold” scores nothing, and that question averaged 1 out of 2.

Drill: Jan 2022 Q12, Oct 2021 Q12, Jun 2023 Q14, Jan 2024 Q15, Oct 2024 Q12

Ideal gases and kinetic theory

Block 1≈7.5 marks a paperpV = NkT in 18 of 18mean kinetic energy in 8 of 18

Learn

• pV = NkT: p in Pa, V in m³, N = number of molecules, k = 1.38 × 10⁻²³ J K⁻¹, T in kelvin.
• For a fixed mass of gas, N is constant, so p₁V₁/T₁ = p₂V₂/T₂. At constant temperature, pV is constant (Boyle’s law). At constant volume, p/T is constant. At constant pressure, V/T is constant.
• Total mass = N × the mass of one molecule, and density = total mass ÷ V.
• Mean kinetic energy of a molecule: ½m⟨c²⟩ = ³/₂kT. It depends only on temperature, so any two gases at the same temperature have the same mean kinetic energy, and the heavier molecules move more slowly: ⟨c²⟩ ∝ 1/m.
• Internal energy of an ideal gas is all kinetic, because there are no forces between the molecules, so there’s no potential energy: U = N × ½m⟨c²⟩ = ³/₂NkT = ³/₂pV.
• Ideal-gas assumptions (MCQ favourites): molecules have negligible volume, there are no forces between them except in collisions, collisions are elastic, and motion is random. The molecules do not have to be identical.
• Pressure in molecular terms: molecules hit the walls and change momentum, a rate of change of momentum is a force, and pressure = force ÷ area. The full chain is in the 6-mark bank.

pV = NkT½m⟨c²⟩ = ³/₂kTp₁V₁/T₁ = p₂V₂/T₂U = ³/₂NkT = ³/₂pVV = ⁴/₃πr³total mass = N × m_molecule

What they ask

• Straight pV = NkT for p, V, N or T, in every single paper. Often it continues as “calculate the mass of helium in the airship”, so find N, then multiply by the mass of one atom.
• Before and after: a sealed cylinder warmed up (June 2026), a weather balloon rising (January 2022), a bubble rising through a lake (October 2024). At depth, pressure = atmospheric pressure + pressure due to the water.
• Mean kinetic energy from ½m⟨c²⟩ = ³/₂kT, sometimes after finding T from pV = NkT. June 2026 also asked you to derive U = ³/₂pV: multiply the mean kinetic energy by N, then substitute NkT = pV.
• Checking Boyle’s law from a graph: work out pV for three points, compare, and conclude (October 2023).
• The 6-marker on why pressure changes with temperature (October 2020, January 2023, October 2025).

Traps

• Not converting °C to K. That loses the “conversion” mark and the answer.
• Radius vs diameter in ⁴/₃πr³, and cm to m before cubing. ⁴/₃πr³ isn’t on the sheet.
• “Pressure above atmospheric” means you add atmospheric pressure to get the actual pressure (the football in October 2020).
• Using the Boltzmann k in a spring question, or a spring constant here. They share a letter.

Drill: Jan 2021 Q14, Oct 2021 Q14, Jan 2023 Q17, Oct 2023 Q11–12, Jun 2024 Q13

Radioactivity and decay

Block 2≈12.5 marks a paperdecay calculation in 18 of 18nuclear equation in 17 of 18

Learn

• Alpha (α): a helium nucleus, ⁴₂He. Very strongly ionising, so a short range (a few cm of air, stopped by paper or skin). Beta-minus (β⁻): a fast electron, emitted with an antineutrino. Moderately ionising, stopped by a few mm of aluminium. Beta-plus (β⁺): a positron, emitted with a neutrino. Gamma (γ): an electromagnetic wave, weakly ionising and very penetrating, reduced by thick lead.
• Nuclear equations: the top numbers (nucleon number) and bottom numbers (proton number) must balance. Alpha: top −4, bottom −2. β⁻: top unchanged, bottom +1. β⁺: bottom −1. Gamma changes neither.
• Background radiation (rocks, radon gas, cosmic rays, medical sources): measure it with no source present, then subtract it from every reading.
• Random: you can’t predict which nucleus decays next or when, though each has a fixed probability of decaying per second. Spontaneous: decay isn’t affected by external conditions such as temperature or pressure.
• Decay constant λ is the probability of decay per second (s⁻¹). Activity A = λN in Bq (decays per second). λ = ln 2/t½. N = N₀e^−λt and A = A₀e^−λt.
• Number of nuclei from a mass: N = mass ÷ (nucleon number × 1.66 × 10⁻²⁷ kg). Turn it around to get the mass from N.
• Power from a source: P = A × energy per decay (in J).
• Safety: handle with tongs (distance cuts intensity by the inverse square law), keep the exposure time short, shield with lead, store in a lead box. Alpha sources are mainly dangerous if they get inside the body, for example breathing in radon gas.

A = λNλ = ln 2/t½N = N₀e^−λtA = A₀e^−λtN = m/(A_r × 1.66 × 10⁻²⁷)P = A × Et = ln(A₀/A)/λcorrected count = count − background

What they ask

• Complete the equation (2 marks: top line, bottom line), in 17 of 18 papers.
• The standard chain: “show that λ = …”, then the activity or the number of nuclei, then the mass of the sample (October 2023, June 2024, January 2023, October 2025, June 2026).
• “How long until the activity falls to …?” Take ln of both sides: t = ln(A₀/A)/λ (June 2022, June 2023, January 2025).
• Pick the radiation for a job. Paper-thickness gauge: beta, because it’s partly absorbed, so the count rate changes as the thickness changes. Sterilising: gamma, because it’s very penetrating. Tracers and PET scans: radiation that escapes the body, with a half-life long enough for the procedure but short enough to limit the dose.
• Cloud chamber tracks (June 2023, June 2025). Thickness comes from ionising ability: alpha tracks are thick because alpha is strongly ionising, beta tracks thin. Shape comes from mass: alpha is massive, so its track is straight; beta is light and easily deflected, so its track is twisted. Give each feature its own cause. This averaged only 2 out of 4.
• Gamma absorption (core practical 15): I = I₀e^−μx, so a graph of ln(corrected count rate) against thickness has gradient −μ. Half-value thickness = ln 2/μ.
• Why beta energies vary: three particles (nucleus, beta particle, antineutrino) share the energy, so the beta particle can have a range of energies.

Traps

• A half-life in years with an activity in Bq. Convert to seconds (1 year = 3.15 × 10⁷ s, usually given).
• Reaching for N = N₀e^−λt when the step you need is A = λN. Examiners flagged this in June 2025.
• Forgetting to subtract background before using ratios or ln.
• Inverse square for counts: subtract background, scale by (d₁/d₂)², then add background back. This MCQ caught most people in June 2025.

Drill: Oct 2023 Q19, Jun 2024 Q17, Jan 2023 Q15, Jun 2023 Q13, Jan 2024 Q20

Mass defect, binding energy, fission and fusion

Block 2≈9.5 marks a papermass difference → energy in 17 of 18fusion conditions in 11 of 18

Learn

• Energy released in a decay or reaction = (total mass before − total mass after) × c². With masses in u: multiply the difference by 1.66 × 10⁻²⁷ kg, use ΔE = c²Δm, then divide by 1.60 × 10⁻¹³ to get MeV.
• Mass defect = (mass of the separate protons and neutrons) − (mass of the nucleus). Binding energy = mass defect × c², the energy needed to separate a nucleus into its nucleons.
• Binding energy per nucleon = binding energy ÷ nucleon number. On the graph it rises steeply from hydrogen, peaks at iron-56 (the most stable nucleus, about 8.8 MeV), then falls slowly towards uranium.
• Fission: a large nucleus splits into two smaller nuclei plus neutrons. Fusion: small nuclei join. Both release energy because the binding energy per nucleon increases, so the total mass decreases.
• Fusion conditions: a very high temperature, so the nuclei have enough kinetic energy to overcome their electrostatic repulsion and get close enough to fuse; and a very high density, so collisions happen often enough to sustain fusion. In stars, gravity supplies the density. On Earth, no material container survives those temperatures, so it’s hard to sustain.
• In alpha decay the alpha particle gets most of the kinetic energy. Momentum is conserved, so the alpha and the daughter nucleus have equal and opposite momenta, and E_k = p²/2m gives the lighter alpha far more kinetic energy. The daughter recoils.

ΔE = c²ΔmE_k = p²/2m1 MeV = 1.60 × 10⁻¹³ Jmass defect = Zm_p + (A − Z)m_n − m_nucleusB.E. per nucleon = B.E./A

What they ask

• “Show that the energy released is about … MeV” (4–5 marks): mass difference, u → kg, ΔE = c²Δm, J → MeV, answer to one more significant figure than given.
• Binding energy per nucleon of a named nucleus (June 2021, October 2024, January 2025, October 2025).
• Energy from the graph: total binding energy after − total binding energy before, where total = (value per nucleon) × (nucleon number) for each nucleus (January 2024, June 2023).
• Explain why uranium fission releases energy but helium fission wouldn’t (October 2025), or why fission of one uranium nucleus releases far more than fusing two hydrogen nuclei (January 2026): both raise binding energy per nucleon, but uranium has far more nucleons.
• Fusion conditions, worth 4–6 marks, in stars or in a reactor on Earth (11 of 18 papers in some form).
• Minimum energy for a reaction where the products have more mass than the reactants (October 2023).

Traps

• Multiplying the given nuclear masses by nucleon numbers. The masses are already for the whole nucleus.
• Getting the subtraction backwards. It’s before − after.
• Rounding the mass difference early. Keep every digit the question gives until the end.
• “Energy needed to split helium-4 into nucleons” = 4 × its binding energy per nucleon.

Drill: Jun 2021 Q19(a), Oct 2021 Q13, Jun 2022 Q15, Oct 2024 Q21, Jan 2024 Q19 and Q21

Oscillations, resonance and damping

Block 3≈15 marks a paper, the biggest topicSHM definition in 13 of 18resonance explained in 11 of 18

Learn

• Definition of SHM (2 marks, learn it word for word): the acceleration (or resultant force) is proportional to the displacement from the equilibrium position, and is always directed towards the equilibrium position. Writing a = −ω²x with the symbols defined also scores.
• ω = 2πf = 2π/T. Released from the amplitude at t = 0: x = A cos ωt, v = −Aω sin ωt, a = −Aω² cos ωt.
• Maximum speed v_max = ωA, at the equilibrium position. Maximum acceleration a_max = ω²A, at the amplitude, where v = 0.
• Mass on a spring: T = 2π√(m/k). Find k from a static stretch first: mg = kΔx. Pendulum: T = 2π√(l/g), which doesn’t depend on the mass or (for small swings) the amplitude.
• Energy: kinetic + potential stays constant with no damping. E_k,max = ½m(ωA)², and the total energy ∝ A². Kinetic energy against x is an upside-down parabola, largest at x = 0. Kinetic energy against time oscillates at twice the frequency and is never negative.
• Graphs: velocity is the gradient of the displacement–time graph, and acceleration is the gradient of the velocity–time graph. Acceleration is exactly opposite (in antiphase) to displacement.
• Free oscillation: displaced and released, it oscillates at its natural frequency. Forced oscillation: a periodic driving force makes it oscillate at the driving frequency.
• Resonance: when the driving frequency equals the natural frequency, there’s a maximum transfer of energy from the driver, so the amplitude increases to a maximum.
• Damping: energy is transferred away from the oscillating system because work is done against resistive forces (friction, air resistance, drag in a liquid), so the amplitude decreases. More damping gives a lower, broader resonance peak. Materials that deform plastically absorb energy, so they make good dampers.

a = −ω²xω = 2πf = 2π/TT = 2π√(m/k)T = 2π√(l/g)v = −Aω sin ωtv_max = ωAa_max = ω²AE_k,max = ½m(ωA)²mg = kΔx

What they ask

• Define SHM, or “explain how the graph shows SHM”: an a–x graph that’s a straight line through the origin with a negative gradient.
• Find k, then T or f, or find a mass: the added mass (January 2025), an astronaut’s mass in orbit (January 2024), a bungee jumper (June 2025), a carriage on springs (June 2026).
• Maximum speed or kinetic energy from T and A, often read off a graph (October 2023 was worth 6 marks; also June 2023, January 2026, October 2025).
• A pendulum on another planet: get g = GM/r² first, then the period (January 2024).
• Resonance in context: tides, a car on speed bumps, a bridge, a building, a loudspeaker cone, a bee’s wings, a microwave oven, a glass.
• Damping: “work is done against resistive forces, so energy is transferred to the surroundings” (car suspension, dampers in a building).
• Sketch velocity from displacement, acceleration from velocity, or kinetic energy against displacement.

Traps

• Saying “displacement” without “from the equilibrium position”. Half the candidates in June 2025 lost a mark this way.
• Forgetting that sin ωt = 1 at maximum speed.
• Oscillations per minute (convert to Hz), and reading a period from a graph over one cycle instead of several.
• Forgetting to add the two masses when an extra mass is hung on.
• Bringing resonance into a damping answer. It doesn’t score there.

Drill: Oct 2023 Q20, Jun 2022 Q17, Jan 2024 Q17, Jun 2024 Q20, Jan 2022 Q11

Gravitational fields and orbits

Block 4≈11.5 marks a paperorbit calculation in 16 of 18ΔE_grav in 12 of 18, 16 with MCQs

Learn

• Newton’s law of gravitation: F = Gm₁m₂/r², with r measured between the centres. Field strength g = F/m = GM/r² in N kg⁻¹. Around a planet the field is radial and follows an inverse square law: double r and g drops to a quarter.
• Gravitational potential V = −GM/r (J kg⁻¹) is the work done per unit mass to bring a mass from infinity to that point. It’s zero at infinity and negative everywhere else.
• Change in gravitational potential energy: ΔE_grav = mΔV. Moving out from r₁ to r₂ gives a gain of GMm(1/r₁ − 1/r₂).
• Equipotentials are surfaces of equal potential, at right angles to the field lines. Around a planet, equal steps of V get further apart as you go out, because V ∝ 1/r.
• Circular orbits: gravity provides the centripetal force, so GMm/r² = mω²r = mv²/r. Put in ω = 2π/T and you get T² = (4π²/GM)r³, and the satellite’s mass cancels. Show this working: quoting Kepler’s law alone earns no “use of” mark.
• Orbit radius = the planet’s radius + the height above its surface.
• Geostationary orbit: a period of 24 hours, above the equator, moving the same way the Earth spins, so it stays over one point. Its radius is about 4.2 × 10⁷ m (a height of about 3.6 × 10⁷ m).
• Escape velocity: set ½mv² = GMm/r, so v = √(2GM/r). Setting v = c gives the radius of a black hole (October 2025).
• “Weightless” astronauts still have weight. They’re in free fall: gravity provides the centripetal force, and there’s no contact force on them.
• Gravitational vs electric fields: both are inverse square, with radial field lines and spherical equipotentials around a point mass or charge. But gravity only attracts, acts on mass and is far weaker; electric forces can attract or repel and act on charge.

F = Gm₁m₂/r²g = Gm/r²V = −Gm/rF = mrω² = mv²/rω = 2π/TT² = 4π²r³/GMΔE_grav = mΔVr = R + hv_esc = √(2GM/r)M = ρ × ⁴/₃πr³

What they ask

• g at a planet’s surface, or a ratio version where g ∝ M/r² (January 2021, June 2024, October 2025).
• Orbital period, orbits per day, or “was 8 days long enough to see the moon complete an orbit?” (October 2021, October 2024, January 2026).
• The mass of the Sun from the Earth’s orbit (January 2025), or another planet’s period from the Earth’s, since T² ∝ r³ (June 2026).
• ΔE_grav between two radii (January 2021, October 2021, June 2022, October 2023, June 2025, January 2026).
• Read a V–r graph, take one point and use V = −GM/r to find M (January 2025, January 2026).
• Derive T² ∝ r³ (June 2022, June 2024, and Kepler’s constant in January 2022).
• Explain: a lower orbit has a bigger force, so a shorter period (June 2021); the Sun losing mass makes the Earth’s orbit radius increase (June 2026).
• Dark matter from orbits: the period calculated from the visible mass is far longer than the real one, so there must be extra mass we can’t see (June 2024).

Traps

• Adding the Earth’s radius to an orbit radius that’s already measured from the centre. Only a third scored full marks on this in June 2025.
• Using mgh for a large change in height. It scores nothing, because g changes with height.
• Forgetting to square r, or leaving it in km.
• Mixing up G and g.

Drill: Jun 2022 Q16, Oct 2021 Q21, Oct 2023 Q17–18, Oct 2024 Q18, Jan 2021 Q17

Stars: Wien and Stefan

Block 5≈6 marks a paperWien in 17 of 18L = σAT⁴ in 16 of 18

Learn

• Stars behave roughly as black bodies: they emit a continuous spectrum whose shape depends only on their temperature.
• Wien’s law: λ_maxT = 2.898 × 10⁻³ m K. A hotter star peaks at a shorter wavelength.
• Stefan–Boltzmann law: L = σAT⁴, with A = 4πr². L is the total power output in W, and σ = 5.67 × 10⁻⁸ W m⁻² K⁻⁴ is on the data list.
• Ratios: L ∝ r²T⁴. Two stars with the same luminosity: the hotter one is smaller.
• If the graph’s axis is frequency, read f at the peak, convert with λ = c/f, then use Wien. Putting a frequency straight into Wien’s law was a common error in June 2025.
• Intensity at a distance d: I = L/4πd², the power per unit area in W m⁻².

λ_maxT = 2.898 × 10⁻³ m KL = σAT⁴I = L/4πd²A = 4πr²λ = c/fL ∝ r²T⁴

What they ask

• Read λ_max off a graph and find the surface temperature (June 2021, June 2023, June 2026), or from a frequency graph (October 2022, October 2024, June 2025).
• “Assess the claim that its radius is 1000 times the Sun’s.” Get T from Wien, then r from L = σ4πr²T⁴, then compare and conclude (January 2022, October 2023, October 2025, January 2026).
• Link it to a planet: is the intensity at the planet similar to Earth’s? (January 2023, June 2025). Or work back from the intensity at Earth to the Sun’s radius (June 2026).
• Explain why a star whose peak is in the infrared still looks red: it emits a range of wavelengths around λ_max, and some of them are visible red light (October 2021, January 2026).
• Why a lamp filament doesn’t fit perfectly: it isn’t a perfect black body, or the glass absorbs some radiation (January 2025).

Traps

• A sphere’s surface area is 4πr², not πr² or ⁴/₃πr³, and it isn’t on the sheet.
• nm to m (× 10⁻⁹) and μm to m (× 10⁻⁶), with T in kelvin.
• Radius vs diameter, again.
• In “assess” questions, compare two numbers and write a conclusion, or the last mark goes.

Drill: Oct 2023 Q21(a), Jan 2022 Q15, Oct 2022 Q16, Jun 2023 Q21(a)

HR diagram and stellar evolution

Block 5≈4.5 marks a paperthe 6-marker in 5 of 18in every paper, counting MCQs

Learn

• Axes: luminosity (as multiples of the Sun’s, from 10⁻⁴ to 10⁶, logarithmic) up the side. Surface temperature along the bottom, running backwards (hottest on the left) and roughly logarithmic: 40 000, 20 000, 10 000, 5000, 2500 K.
• The Sun sits at about 6000 K and L = 1, on the main sequence.
• Regions: the main sequence is the diagonal band from top left to bottom right, and its stars fuse hydrogen into helium in their cores. Red giants are top right (cool but very luminous, so huge). White dwarfs are bottom left (hot but dim, so tiny). Red dwarfs are small, cool, dim main-sequence stars at the bottom right.
• A Sun-like star: main sequence → red giant → white dwarf. A massive star: main sequence → red supergiant → supernova → neutron star or black hole.
• More massive main-sequence stars have a higher core temperature, so a higher rate of fusion, so they leave the main sequence sooner. Half the candidates scored 0 on this in June 2025.
• Cluster ages: all the stars in a cluster formed at the same time. A young cluster shows only a main sequence. An older one has lost the top of its main sequence and has red giants. An old one has white dwarfs too.
• White dwarfs: small surface area, very dense, high surface temperature, low luminosity, no fusion.
• Spectral classes, hottest to coolest: O B A F G K M.

What they ask

• Add the temperature scale: reversed, and logarithmic, so each equal step multiplies by the same factor (January 2022, June 2023, October 2025, June 2026).
• Mark the Sun at about 6000 K and L = 1, or label the regions (October 2024).
• Draw the Sun’s path: straight from its main-sequence spot to the red giant region, then down to the white dwarf region. Moving it along the main sequence first lost marks in June 2025.
• Explain why a cluster is old or young (January 2021, June 2023, October 2025).
• The 6-marker on the Sun’s evolution (January 2022, June 2024, June 2025, June 2026) or on how a cluster’s diagram changes as it ages (January 2021).
• Why massive stars spend less time on the main sequence (June 2025).

Traps

• It’s the core that contracts, not the whole star.
• The core temperature rises first, and that starts helium fusion. Helium fusion doesn’t cause the rise.
• “Becomes a red giant” is too thin. Say it expands into a red giant.

Drill: Jan 2022 Q19(c), Jun 2023 Q20(a)–(b), Jan 2024 Q11, Oct 2024 Q20(a)

Distances: parallax and standard candles

Block 5≈6 marks a paperstandard candles in 13 of 18parallax in 12 of 18 (both counting MCQs)

Learn

• Trigonometric parallax: observe a nearby star against the background of very distant stars from two positions six months apart, on opposite sides of the Earth’s orbit, and measure the change in its angular position. With the Earth–Sun distance r (1.5 × 10¹¹ m) known, d = r/tan θ ≈ r/θ, with θ in radians.
• Why parallax only works for nearby stars: for distant stars the parallax angle is so small that its percentage uncertainty is too large for the instrument’s resolution. “It’s hard to measure” scores nothing.
• A standard candle is an astronomical object of known luminosity, such as a Cepheid variable star or a Type 1a supernova. It’s an object, not a method: in June 2025, answers that called it “a method” scored zero.
• The method: locate a standard candle in the galaxy, measure the intensity of its radiation at Earth, then use I = L/4πd², naming I, L and d, to calculate the distance.
• How L is known: measure the intensity of a nearby standard candle whose distance comes from parallax, then L = 4πd²I. For a Cepheid, measure its period and read L off the period–luminosity graph.
• For the most distant galaxies: measure the redshift, work out v, then d = v/H₀.

I = L/4πd²d = r/θ (θ in rad)r = 1.5 × 10¹¹ m (Earth–Sun)

What they ask

• Define a standard candle (1 mark).
• Describe the standard-candle method (3–4 marks), or the whole chain including how L is found (the October 2022 6-marker).
• Describe parallax (3–4 marks) and explain its limit (2 marks).
• Distance from I and L: Sirius (October 2021), a standard candle in M81 (June 2022), a Cepheid from its period (January 2025).
• Parallax numbers: find d from θ (June 2022), or decide whether parallax could reach a given star (January 2022, June 2026).
• Dust in the way lowers the measured intensity, so the calculated distance comes out too big (January 2021).

Traps

• θ must be in radians for d = r/θ.
• Quoting I = L/4πd² in a description without saying what the symbols mean. That lost a mark for many people in June 2025.
• Write “measure the intensity”, not “observe” or “find” it. Examiners want the technical verb.

Drill: Jan 2023 Q18, Oct 2022 Q13, Jun 2022 Q11 and Q18(a), Oct 2021 Q17

Doppler, Hubble and dark matter

Block 5≈7.5 marks a paperDoppler calculation in 15 of 18Hubble or the age of the universe in 15 of 18, counting MCQs

Learn

• Doppler shift: z = Δλ/λ ≈ Δf/f ≈ v/c, for speeds much less than c. The bottom of the fraction is the wavelength (or frequency) that was emitted, which is the laboratory value.
• A longer wavelength received (lower frequency) is a redshift, so the source is moving away. Shorter is a blueshift, so it’s approaching.
• Redshift (2 marks): the fractional increase in the wavelength of the radiation received, because the source is moving away.
• Hubble’s law: v = H₀d. H₀ is about 2.2 × 10⁻¹⁸ s⁻¹ (about 70 km s⁻¹ Mpc⁻¹). To convert, multiply by 10³ (km to m) and divide by 3.09 × 10²² (Mpc to m).
• Age of the universe ≈ 1/H₀ in seconds (divide by 3.15 × 10⁷ for years). A smaller H₀ means an older universe.
• Evidence for expansion: distant galaxies are all redshifted, and the further away they are, the bigger the redshift.
• Dark matter is matter that has mass, so it exerts a gravitational force, but doesn’t emit electromagnetic radiation. Evidence: stars in the outer parts of galaxies orbit faster than the visible mass can explain.
• The fate of the universe depends on its average density compared with the critical density. Greater: the expansion eventually stops and it contracts. Less: it expands forever. It’s uncertain because the amount of dark matter, and so the density, is uncertain.
• Doppler from rotation: one edge of a rotating star moves towards us (blueshift) and the other away (redshift). A star with an orbiting planet wobbles, so its spectral lines shift back and forth.

z = Δλ/λ ≈ Δf/f ≈ v/cv = H₀dage ≈ 1/H₀v = 2πr/T (rotation)

What they ask

• Speed from a shifted spectral line, given in wavelength or frequency, plus “is it moving towards or away?” (15 of 18).
• Distance to a galaxy with Hubble’s law (January 2021, January 2023), or the age of the universe (June 2022, January 2024, and MCQs in June 2021, January 2025, June 2025 and October 2025).
• Convert H₀ from km s⁻¹ Mpc⁻¹ to s⁻¹ (June 2022).
• Define redshift (January 2021, October 2023, October 2024).
• Describe how the distances to the most distant galaxies are found: measure the redshift, calculate v, use v = H₀d (October 2024, June 2026).
• Dark matter: define it (2 marks: it has mass, and it emits no electromagnetic radiation) and link it to the fate of the universe (January 2022, January 2023, January 2026).
• Doppler in other settings: the rotating Sun (October 2021), a star wobbling because of a planet (October 2020), a speed camera (June 2026).

Traps

• Putting the observed wavelength in the denominator. Use the laboratory (emitted) value.
• “Redshift means it’s accelerating away.” It only means it’s moving away.
• Very distant galaxies are redshifted so far that their visible light arrives as infrared (October 2022).

Drill: Jun 2022 Q12, Jan 2023 Q13 and Q16, Oct 2024 Q17, Jan 2021 Q16, Oct 2021 Q16