Astrophysics.

HookThe photographic plate that doubled the Universe

ModelThe refractor — two lenses in normal adjustment

An astronomical refracting telescope is two converging lenses sharing an axis. The objective, with a long focal length \(f_o\), forms a real, inverted image of the effectively-at-infinity object in its focal plane. The eyepiece, focal length \(f_e\), is positioned so that this image sits in its focal plane too: parallel rays enter the objective, parallel rays leave the eyepiece, and the relaxed eye views a final image at infinity. That arrangement is normal adjustment, and it makes the lens separation \(f_o + f_e\) — the ray diagram AQA asks for is exactly this geometry, drawn with three construction rays and both focal planes marked.

The payoff is angular: the image subtends a larger angle at your eye than the object does unaided, and the angular magnification is \(M = \frac{\theta_{image}}{\theta_{object}} = \frac{f_o}{f_e}\). But refractors hit a ceiling, and it is worth knowing why: the Yerkes Observatory telescope of 1897, with a 1.02 m lens, is still the largest refractor ever built. A big lens can only be held by its rim, so it sags under its own weight; the glass absorbs light on the way through; and two surfaces per element must be figured precisely. Every serious telescope since has been a mirror instrument.

ModelReflectors, the Cassegrain, and the two aberrations

The Cassegrain reflector folds a long focal length into a short tube: a concave parabolic primary mirror gathers the light, a small convex secondary intercepts it before focus and reflects it back through a central hole in the primary, to an eyepiece or detector behind. Mirrors win at scale for the mirror-image of every refractor weakness: one optical surface instead of two, no light lost in glass, support available across the entire back face — and segments can be tiled into apertures no single casting could achieve (the JWST primary is 18 hexagons making 6.5 m).

Two named faults organise the comparison marks. Chromatic aberration afflicts lenses only: glass refracts blue more strongly than red, so each colour focuses at a different point and bright objects wear coloured fringes; an achromatic doublet (crown and flint glass cemented together) reduces but never removes it. Mirrors reflect all wavelengths identically and are immune. Spherical aberration afflicts spherical surfaces of either kind: rays striking the edge focus short of rays near the axis, blurring everything; the cure is a parabolic figure. How precisely the parabola must be made was demonstrated at enormous public expense in 1990, when the Hubble Space Telescope reached orbit with its 2.4 m primary ground about 2 μm too flat at the edge — a fiftieth of a hair's width, enough to blur every image until corrective optics were fitted by astronauts in 1993.

CaseOpening the spectrum — radio, infrared, UV and X-ray telescopes

The atmosphere transmits two clean bands — visible light and radio — and blocks or blurs almost everything else, which dictates where each telescope must live. A single-dish radio telescope like the 76 m Lovell Telescope at Jodrell Bank (finished in 1957, in time to track Sputnik's rocket) is structurally a Cassegrain writ huge: a parabolic reflector with the receiver at the focus. Because surface bumps only matter if they approach \(\lambda/20\), a dish observing 21 cm hydrogen emission can be built of mesh panels to almost any size, work through cloud and daylight, and steer to track — but it records one point of sky at a time rather than an image, and its resolution is poor for reasons the next block quantifies.

The rest of the spectrum forces you upwards. Infrared telescopes fight water vapour and their own warmth — a telescope at room temperature glows brightly at exactly the wavelengths it is trying to observe — so they sit on high, dry summits or in space, cooled hard: JWST operates beyond the Moon's orbit behind a tennis-court sunshield, its mirror below 50 K. Ultraviolet is absorbed by ozone: space only. X-rays would pass straight through any normal mirror, so X-ray telescopes focus by grazing incidence — nested cylindrical shells that deflect photons at glancing angles, a completely different structure from the near-normal reflection of optical instruments. Each waveband buys a different Universe: cold dust and newborn stars in the infrared, million-kelvin gas and black-hole accretion in X-rays, neutral hydrogen mapped across galaxies at 21 cm.

DataWhy aperture wins — collecting power, the Rayleigh criterion, and the CCD

Two numbers, both set by diameter. Collecting power scales as \(D^2\) — the mirror is a photon funnel, so doubling the diameter gathers four times the light and reaches objects four times fainter. Resolving power follows the Rayleigh criterion: two point sources are just distinguishable when their angular separation is about \(\theta \approx \frac{\lambda}{D}\) radians — diffraction smears every star into a disc, and bigger apertures (or shorter wavelengths) make smaller discs.

Run the numbers and radio astronomy's predicament appears. Lovell at \(\lambda = 0.21\ \text{m}\): \(\theta = 0.21/76 = 2.8 \times 10^{-3}\ \text{rad}\) — about a sixth of a degree, a third of the full Moon's width, from a 3,200-tonne instrument. A 150 mm amateur reflector at 550 nm: \(\theta = 5.5 \times 10^{-7}/0.15 = 3.7 \times 10^{-6}\ \text{rad}\) — roughly 750 times finer. The wavelength penalty is why radio observatories link dishes into arrays hundreds of kilometres across, synthesising one enormous \(D\).

At the focus of a modern telescope sits a CCD, and AQA loves it compared with the eye. Quantum efficiency — the fraction of arriving photons actually detected — is above 80% for a CCD against roughly 1% for the retina. A CCD integrates for hours while the eye refreshes in a twentieth of a second; its response is linear, so twice the photons means twice the signal and honest photometry; its images are digital, storable and subtractable; and it sees beyond the visible into infrared and ultraviolet. The eye's advantages are shorter: no equipment, no power supply, and a spectacular dynamic range. In any comparison question, quote quantum efficiency with numbers — it is the discriminator examiners reward first.

ModelMagnitudes — Hipparchus' scale with decimals on it

Astronomy's brightness bookkeeping is 2,100 years old and runs backwards. Hipparchus ranked naked-eye stars from 'first magnitude' (brightest) to 'sixth' (faintest), and the modern Hipparcos scale keeps his direction while fixing the steps: a difference of 5 magnitudes is defined as a factor of 100 in received intensity, so one magnitude is a factor of \(100^{1/5} = 2.51\). Lower means brighter, and the scale happily goes negative: the Sun sits at −26.7, Sirius at −1.46, Vega near 0, the naked-eye limit around +6, and the deepest space-telescope fields reach past +30 — each magnitude another division by 2.51.

Apparent magnitude \(m\) records only what arrives at Earth, hopelessly entangling luminosity with distance: a candle nearby outshines a searchlight far away. The disentangling tool is absolute magnitude \(M\) — the apparent magnitude an object would have at a standard 10 parsecs. The parsec is the surveyor's unit: the distance at which 1 AU subtends one arcsecond, equal to \(3.08 \times 10^{16}\ \text{m}\) or 3.26 light years (the light year itself being the distance light covers in a year, \(9.46 \times 10^{15}\ \text{m}\)). Since intensity dilutes with the inverse square, every factor of 10 in distance costs exactly 5 magnitudes, and the whole relationship compresses to \(m - M = 5\log\left(\frac{d}{10}\right)\) with \(d\) in parsecs. Beyond 10 pc, \(m > M\); inside 10 pc, \(m < M\) — a sanity check that catches most sign errors.

DataStars as black bodies — Wien's thermometer and Stefan's tape measure

A star's surface radiates close enough to an ideal black body — a perfect absorber and emitter — that two laws unlock its vital statistics. Wien's displacement law says the peak of the emission curve slides with temperature: \(\lambda_{max}T = 2.9 \times 10^{-3}\ \text{m K}\). Hotter stars peak bluer; measure where a star's spectrum crests and you have read its surface temperature off a curve. Stefan's law prices the total output: \(P = \sigma A T^4\), with \(\sigma = 5.67 \times 10^{-8}\ \text{W m}^{-2}\text{K}^{-4}\) and \(A = 4\pi R^2\) for a sphere.

Together they are a remote-sensing kit: Wien gives \(T\); absolute magnitude gives luminosity \(P\); Stefan then has only one unknown left, the radius. The \(T^4\) makes the law vicious in both directions — doubling temperature multiplies output sixteenfold — and it also makes error analysis pointed: a 5% slip in \(T\) becomes a 20% error in \(P\), an evaluation line worth a mark whenever the question hands you uncertain data. State the assumption too: all of this treats the star as a black body of a single surface temperature, which real, limb-darkened, spotted stars only approximate.

ModelOBAFGKM — and why hydrogen lines choose class A

Starlight arrives stamped with an absorption spectrum: the cooler gas of the star's atmosphere subtracts photons at element-specific wavelengths from the black-body glow beneath. The line pattern sorts stars into spectral classes, which run O B A F G K M from hot to cool. O stars (25,000–50,000 K) are blue, their spectra marked by ionised helium — only brutal temperatures strip helium. B (11,000–25,000 K): neutral helium and strengthening hydrogen. A (7,500–11,000 K): blue-white, with the strongest hydrogen Balmer lines. F (6,000–7,500 K) and G (5,000–6,000 K, the Sun's class, yellow-white): metal lines and ionised calcium take over. K (3,500–5,000 K): orange, neutral metals. M (below 3,500 K): red, and cool enough for molecules — titanium oxide bands are the fingerprint, chemistry that hotter atmospheres would tear apart.

The Balmer detail is a beautiful piece of physics and a guaranteed mark. Balmer absorption lines come from hydrogen electrons being excited out of the \(n = 2\) level, so they need a healthy population parked at \(n = 2\). Too cool a star, and nearly every electron sits in the ground state — no Balmer absorption. Too hot, and the hydrogen is ionised — no bound electrons at all. The lines therefore peak at a Goldilocks temperature around 10,000 K, class A, and fade in both directions. Line strength measures temperature, not hydrogen abundance: an M star is still mostly hydrogen; it is simply too cold to say so in the Balmer series. (For the sequence, coin your own mnemonic — 'Only Bold Astronomers Find Giant Kraken Monsters' works, and inventing one beats borrowing.)

CaseThe Hertzsprung–Russell diagram — a census, not a route

Plot absolute magnitude (or luminosity) vertically against temperature — with temperature increasing to the left, the axis convention that ambushes candidates — and stars refuse to scatter randomly. About 90% lie along the main sequence, the diagonal band from hot-bright (top left) to cool-dim (bottom right): these are stars fusing hydrogen in their cores, and their position is fixed by mass alone. Massive stars sit top-left and squander their fuel in a few million years; red dwarfs sip theirs for longer than the Universe has existed. Top right sit the giants and supergiants — cool but enormous, luminous by Stefan's \(R^2\) — and bottom left the white dwarfs, hot but merely Earth-sized, so faint.

The diagram is a snapshot of residences, not a track that stars slide along — but a single star does relocate as it ages, and AQA asks specifically for the Sun's itinerary. Now about 4.6 billion years into a roughly 10-billion-year main-sequence tenancy, the Sun will exhaust its core hydrogen, and the core will contract and heat while hydrogen burns in a shell around it: the envelope swells and cools, and the Sun moves up and right to become a red giant, engulfing Mercury and Venus and reaching towards Earth's orbit. Core helium then fuses to carbon and oxygen; when that too is spent, the outer layers drift away as a planetary nebula, and the exposed core — no fusion, just stored heat — settles bottom-left as a white dwarf, cooling towards invisibility for the rest of time. On the diagram: right and up off the main sequence, then a long slide down and left. Sketch it with labelled axes and arrows and the marks are yours.

CaseViolent endings — supernovae, neutron stars and black holes

Stars above roughly eight solar masses fuse their way up the periodic table until the core is iron — the peak of the binding-energy curve, beyond which fusion consumes energy rather than releasing it. The furnace fails, the core collapses in under a second, and the rebound blows the star apart: a type II supernova, briefly outshining its host galaxy, its spectrum carrying hydrogen lines from the shredded envelope and its light curve showing a fast rise then a slow, humped decline.

A type Ia supernova is a different machine with a priceless property. A white dwarf in a binary system siphons matter from its companion until it nudges a critical mass near 1.4 solar masses — the same trigger every time — and detonates completely: no hydrogen lines, and a peak absolute magnitude close to \(M \approx -19.3\), every time. That makes type Ia supernovae standard candles: read the peak apparent magnitude, insert the known \(M\) into \(m - M = 5\log(d/10)\), and out comes the distance to a galaxy billions of light years away. In 1998, two rival teams measured distant type Ia events and found them consistently fainter — more distant — than a steadily expanding Universe allowed: the expansion is accelerating, driven by something now labelled dark energy, and the 2011 Nobel Prize followed.

What the explosion leaves depends on the leftover core. Between about 1.4 and 3 solar masses: a neutron star, the mass of a star packed into a city-sized sphere ~12 km across, at around \(4 \times 10^{17}\ \text{kg m}^{-3}\) — nuclear density, a sugar-cube of it weighing several hundred million tonnes, often announcing itself as a lighthouse-spinning pulsar. Heavier cores keep collapsing past every known resistance into a black hole: an object whose escape velocity exceeds \(c\) inside the event horizon, of radius given by the Schwarzschild formula \(R_s \approx \frac{2GM}{c^2}\). Nothing that crosses it — light included — reports back.

DataDoppler, Hubble, and the age of everything

Motion along the line of sight shifts every spectral line: for speeds well below \(c\), \(\frac{\Delta\lambda}{\lambda} \approx \frac{\Delta f}{f} = \frac{v}{c}\). Recession stretches wavelengths — red shift, quantified as \(z = \Delta\lambda/\lambda\) — and approach compresses them. The same tool resolves spectroscopic binaries, star pairs too close to separate visually: each absorption line periodically splits into two (one star approaching as the other recedes) and merges again (both crossing the line of sight), and the splitting gives the orbital speeds while the repeat time gives the period — circular-motion analysis then hands you masses and separation. It is the section's favourite synoptic ambush.

Applied to galaxies, Doppler measurements built the biggest result in science. Hubble's 1929 data showed recession speed proportional to distance: \(v = Hd\). The AQA data booklet pins the Hubble constant at \(H = 65\ \text{km s}^{-1}\text{Mpc}^{-1}\) (modern measurements cluster between 67 and 73, and their stubborn disagreement — the 'Hubble tension' — is a live research problem). Crucially, the law implies no centre: every observer in every galaxy sees the same proportionality, because space itself is expanding, carrying galaxies apart like raisins in rising dough. Run the film backwards and everything converges: the Big Bang, a hot dense origin whose supporting evidence includes the red shifts themselves and the cosmic microwave background, the 2.7 K afterglow that fills every direction of the sky.

CaseQuasars and exoplanets — the two edges of the map

In 1963 Maarten Schmidt finally decoded the spectrum of 3C 273, a 13th-magnitude 'star' bright enough for a good amateur telescope: its lines were ordinary hydrogen, red-shifted by 16%. At \(v \approx 0.16c\), Hubble's law puts it around two billion light years away — and to appear that bright from that distance it must radiate like several trillion Suns. Its absolute magnitude works out near −26.7: seen from 10 parsecs, 3C 273 would blaze in the sky the way the Sun does from Earth. Yet its brightness flickers over weeks, so the emitting engine can be at most light-weeks across — a region smaller than the Solar System outshining an entire galaxy. Quasars are active galactic nuclei: matter spiralling through a superheated accretion disc onto a supermassive black hole, converting gravitational energy to radiation with brutal efficiency. Large red shift, colossal luminosity, tiny variable core: that triplet is the exam definition.

Exoplanets sit at the opposite edge: not too far to see, but too faint — a star outshines its planet by a factor of order a billion in visible light, at angular separations no current telescope resolves, so direct imaging almost always fails. Detection is inference. The radial-velocity method watches the star itself wobble around the system's centre of mass, its spectral lines Doppler-swinging with the orbital period: 51 Pegasi b, the first planet found around a Sun-like star (1995, a Nobel in 2019), announced itself as a 50 m s⁻¹ stellar wobble every 4.2 days — a fractional wavelength shift of order \(10^{-7}\), which is why the method needed heroic spectrograph stability. The transit method watches for the periodic dip as a planet crosses its star's disc: the fractional drop in brightness equals \(\left(\frac{R_p}{R_{star}}\right)^2\) — about 1% for a Jupiter crossing a Sun, a punishing 0.008% for an Earth — which is why transit hunting moved to space photometers like Kepler, and why the census it returned (thousands of confirmed planets) skews towards big planets huddled close to their stars. Combine the methods and the physics multiplies: transits give radius, wobbles give mass, together they give density — rock or gas.

VocabularyKey terms the mark scheme pays for

TrapsMisconceptions that cost marks

ExamWhat examiners want