Lesson 15: Astrophysics Option Preview

Astrophysics applies the whole course — gravitation, thermal radiation, nuclear fusion — to stars. The IB treats it as an application strand within Theme E.5 (fusion and stars); this lesson previews the key quantifiable tools: how we measure stellar distances, temperatures, luminosities and radii.

Distance: parallax and beyond

Stellar parallax is the only direct distance measure: a nearby star appears to shift against distant stars as Earth orbits. The parsec is defined so that distance (pc) = 1/parallax (arcseconds). One parsec = 3.26 light years. Parallax from the ground reaches ~0.01 arcsec (100 pc); the Gaia satellite measures micro-arcseconds, mapping a billion stars. Beyond parallax, astronomers chain methods (the "distance ladder"): Cepheid variable stars, whose period reveals luminosity, then Type Ia supernovae as standard candles for other galaxies. Each rung is calibrated by the one below — a case study in how measurement uncertainty compounds.

Black-body radiation: Wien and Stefan–Boltzmann

Stars radiate approximately as black bodies. Wien's law: λmax T = 2.90 × 10⁻³ m K — the spectrum's peak gives surface temperature from colour alone. Stefan–Boltzmann law: total luminosity L = 4πR²σT⁴ (σ = 5.67 × 10⁻⁸ W m⁻² K⁻⁴). With L and T known, the radius follows — stars are measured without ever being resolved as discs. The Sun: T = 5780 K, L☉ = 3.83 × 10²⁶ W.

Worked problem 1 — characterising a star

A star's spectrum peaks at 420 nm and its radius is 9.6 × 10⁸ m. Find its surface temperature and luminosity, and compare with the Sun.

T = 2.90 × 10⁻³/420 × 10⁻⁹ = 6.90 × 10³ K
σT⁴ = 5.67 × 10⁻⁸ × (6900)⁴ = 5.67 × 10⁻⁸ × 2.27 × 10¹⁵ = 1.29 × 10⁸ W m⁻² (surface flux)
L = 4πR² × σT⁴ = 4π × (9.6 × 10⁸)² × 1.29 × 10⁸ = 1.49 × 10²⁷ W
L/L☉ = 1.49 × 10²⁷/3.83 × 10²⁶ = 3.9

Note the T⁴ sensitivity: a 10% temperature error becomes a 46% luminosity error — error propagation again (Lesson 17).

Worked problem 2 — parallax distance

A star shows a parallax angle of 0.35 arcsec. Find its distance in parsecs and light years, and the time its light takes to reach us.

d = 1/0.35 = 2.86 pc = 2.86 × 3.26 = 9.3 ly, so the light left 9.3 years ago.

The nearest star, Proxima Centauri, has p = 0.77 arcsec → 1.30 pc = 4.24 ly.

The Hertzsprung–Russell diagram and stellar life

Plotting luminosity against temperature (hot on the left, by convention) organises stars into the main sequence (hydrogen fusion in the core, where most stars spend most of their lives), red giants, supergiants and white dwarfs. A star's fate is set by its mass: Sun-like stars end as white dwarfs after shedding a planetary nebula; stars above ~8 solar masses end in supernovae, leaving neutron stars or black holes, and forging the heavy elements. Our own composition — carbon, oxygen, iron — is stellar ash: nucleosynthesis is why Theme E belongs in an astrophysics discussion.

Practical worked example — the inverse-square law in the lab

A light sensor measures intensity from a small lamp at distances 0.20–1.00 m. A log–log plot of intensity against distance should have gradient −2 (I ∝ 1/d²). Measured gradient: −2.1 ± 0.15 — consistent with −2 within uncertainty. But the first point (0.20 m) sits above the line: the lamp is not a point source at close range, and ambient light adds a constant offset. Subtracting a measured background (lights off reading) and starting at 0.30 m are the refinements. The identical issues — background and finite source size — dominate real astronomical photometry.

Nature of Science

Astrophysics cannot experiment on its objects; it is an observational science built on models tested against spectra and statistics. The steady-state versus Big Bang debate was settled by observation (the cosmic microwave background) — showing that even without controlled experiments, falsifiable prediction works.

Real-world application

Exoplanet transits — the dip in a star's luminosity when a planet crosses it — use exactly the L ∝ R² scaling: a 1% dip implies a planet with 1% of the star's disc area, hence a radius ratio of 0.1. Over 5000 exoplanets have been found this way, with school-accessible data from space telescopes.

Common mistakes (examiner style)

• Quoting parallax distances with d = 1/p but p in arcminutes or degrees.
• Placing hot stars on the right of the HR diagram (the axis is reversed by convention).
• Using L = σAT⁴ with A as the disc area πR² instead of the surface area 4πR².
• Confusing luminosity (total output, W) with brightness/intensity at Earth (W m⁻²), which falls as 1/d².
• Thinking main-sequence position depends mainly on age rather than mass.

IB Exam Experience

Paper 1-style questions. 1. A star's spectrum peaks in the ultraviolet. Compared with the Sun it is: A. cooler and redder B. hotter C. necessarily larger D. necessarily more luminous. Key: B — Wien's law: shorter λmax means higher T; size and luminosity need more data. 2. Two stars have the same surface temperature; star X has twice the radius of star Y. X's luminosity is: A. 2× B. 4× C. 8× D. 16×. Key: B — L ∝ R² at fixed T. 3. A star with parallax 0.25 arcsec is at a distance of: A. 0.25 pc B. 2.5 pc C. 4.0 pc D. 25 pc. Key: C — d = 1/0.25 = 4.0 pc.

Paper 2-style structured question (6 marks). A star has luminosity 25 L☉ and surface temperature 2 × 5780 K. (a) Show that its radius is 1.25 times the Sun's radius. [3] (b) Its parallax is measured as 0.050 arcsec. Calculate its distance in metres. [3] (Use 1 pc = 3.09 × 10¹⁶ m.) *Marking notes:* (a) L ∝ R²T⁴: 25 = (R/R☉)² × 2⁴ = 16(R/R☉)² → (R/R☉)² = 25/16 → R = 1.25 R☉. Award full marks for ratio method; absolute calculation also acceptable. (b) d = 1/0.050 = 20 pc = 20 × 3.09 × 10¹⁶ = 6.2 × 10¹⁷ m. The classic loss: converting with light years mid-calculation and returning an answer in mixed units.

IA idea: Using published transit data for an exoplanet system, determine the planet's radius ratio and orbital period, propagating the uncertainties quoted in the data tables.

The HR diagram read as a biography

The main sequence is not an evolutionary track but a population: stars of different masses sitting where their mass dictates — hot and luminous O/B stars at top-left, cool dim M dwarfs at bottom-right, the Sun unremarkably in between. A star's lifetime on the sequence scales as fuel/usage ≈ M/L ∝ M/M^3.5 = M^−2.5: a 10-solar-mass star lives about 3 × 10⁻³ of the Sun's ~10¹⁰ years — a few tens of millions of years — while a 0.3-solar-mass dwarf may shine for hundreds of billions of years, longer than the current age of the universe. When core hydrogen runs out, the star leaves the sequence: core contracts and heats, the envelope expands and cools (red giant), and the HR track moves right and up. Mass decides every ending: below the Chandrasekhar limit (1.4 solar masses) the remnant is a white dwarf supported by electron degeneracy pressure; above it, collapse continues to a neutron star or black hole.

Cepheids: standard candles worked

Cepheid variables brighten and dim with periods of days to weeks, and their period reveals their mean luminosity: a Cepheid with a 10-day period has L ≈ 10³ L☉. Measure its apparent brightness (flux) F at Earth, and the distance follows from F = L/(4πd²). For F = 2.6 × 10⁻¹² W m⁻² and L = 10³ × 3.83 × 10²⁶ = 3.83 × 10²⁹ W: d = √(L/(4πF)) = √(3.83 × 10²⁹/3.27 × 10⁻¹¹) = √(1.17 × 10⁴⁰) = 1.08 × 10²⁰ m ≈ 3500 pc. This is the method with which Hubble measured the Andromeda galaxy's distance in 1924 and ended the "island universes" debate — parallax's successor on the distance ladder.

Apparent and absolute magnitude (the astronomer's scale)

Magnitudes run backwards (brighter = smaller number): a factor 100 in brightness equals 5 magnitudes. Absolute magnitude M is the apparent magnitude a star would have at 10 pc, linking luminosity to an observational quantity. The distance modulus m − M = 5 log₁₀(d/10 pc) converts an observed m and a known M (from a spectrum or a Cepheid period) into a distance. Examiners use the concept qualitatively at IB level: the point is that luminosity is intrinsic, brightness is distance-diluted, and every method of measuring cosmic distance is a device for separating the two.

Colour index: temperature from two measurements

Astronomers quantify a star's colour by measuring its brightness through blue (B) and visual (V) filters; the colour index B − V is negative for hot blue stars and positive (~+1.5) for cool red ones, mapping monotonically onto surface temperature via the black-body curve's shape. It is Wien's law turned into a two-point measurement — robust against distance (both fluxes dilute equally) though not against interstellar reddening (dust scatters blue more, mimicking coolness). Much of observational astronomy is this kind of ingenuity: extracting intrinsic properties from ratios that distance cannot corrupt.

Stellar magnitudes: a quick calibration

The Sun's absolute magnitude is +4.83; a star 100 times more luminous has M lower by 5 (since 5 magnitudes = a factor 100): M ≈ −0.2. Sirius, the brightest night star, owes its fame mostly to proximity (2.6 pc) rather than power (L ≈ 25 L☉); Rigel, less conspicuous to the eye, radiates ~10⁵ L☉ from 260 pc away. Holding the brightness-versus-luminosity distinction firmly is the entire point — the sky's appearance is a conspiracy of distance, and astrophysics is the discipline of undoing it.