Astronomy and cosmology

• Emitted is not received: \(F = L/(4\pi d^2)\). Hotter peaks shorter: \(\lambda_{\max} T\) is constant. Farther recedes faster: \(v \approx H_0 d\), so the age is about \(1/H_0\).
• Luminosity is emitted power; flux is what arrives, \(F = L/(4\pi d^2)\). Hotter peaks shorter; \(L = 4\pi\sigma r^2T^4\) gives the size. Farther recedes faster, so the age is about \(1/H_0\).

• 25.1.1 I can understand — understand the term luminosity as the total power of radiation emitted by a star
• 25.1.2 I can recall — recall and use the inverse square law for radiant flux intensity F in terms of the luminosity L of the source F = L/(4πd²)
• 25.1.3 I can understand — understand that an object of known luminosity is called a standard candle
• 25.1.4 I can understand — understand the use of standard candles to determine distances to galaxies
• 25.2.1 I can recall — recall and use Wien's displacement law λ_max ∝ 1/T to estimate the peak surface temperature of a star
• 25.2.2 I can use — use the Stefan–Boltzmann law L = 4πσr²T⁴
• 25.2.3 I can use — use Wien's displacement law and the Stefan–Boltzmann law to estimate the radius of a star
• 25.3.1 I can understand — understand that the lines in the emission and absorption spectra from distant objects show an increase in wavelength from their known values
• 25.3.2 I can use — use Δλ/λ ≈ Δf/f ≈ v/c for the redshift of electromagnetic radiation from a source moving relative to an observer
• 25.3.3 I can explain — explain why redshift leads to the idea that the universe is expanding
• 25.3.4 I can recall — recall and use Hubble's law v ≈ H₀d and explain how this leads to the Big Bang theory (candidates will only be required to use SI units)

Units, significant figures and working are part of the physics. Give a calculated answer to the same number of significant figures as the least precise data, or one more; keep full precision in the working and round only at the end; write the unit with every final answer. A fifth of the qualification is experimental: Papers 3 and 5 test AO3 only, and their questions may be set in contexts outside the syllabus content, so the practical work in this chapter is set out as method, recording, graphs and uncertainties rather than as theory.

• “Luminosity is how bright the star is when we look at it.” Correct Luminosity is the emitted power: the total power of radiation emitted by the star, in W, the same wherever it is seen from. What arrives is the radiant flux intensity \(F\), in W m−2, and it depends on the distance too: \(F = L/(4\pi d^2)\).
• “Twice as far away, so half the flux.” Correct The same power is spread over a sphere whose area grows as \(d^2\): twice as far gives one quarter of the flux, three times as far one ninth. And the rearrangement is \(d = \sqrt{L/(4\pi F)}\), with the square root.
• “A standard candle is a very bright star.” Correct A standard candle is an object of known luminosity. Brightness is what makes it visible far away; knowing its luminosity is what makes it useful.
• “A hotter star peaks at a longer wavelength, and a hotter star is always more luminous.” Correct \(\lambda_{\max} \propto 1/T\): hotter peaks shorter. And \(L \propto r^2 T^4\): a large cool star can outshine a small hot one. \(T\) is always in kelvin, and \(r\) is the radius, not the diameter.
• “In \(\Delta\lambda/\lambda\), \(\lambda\) is the wavelength we observe.” Correct \(\lambda\) is the emitted (laboratory) wavelength and \(\Delta\lambda\) is observed minus emitted. Redshift is not “looking red”: every line in the pattern moves to a longer wavelength by the same fraction.
• “Every galaxy is moving away from us, so we are at the centre of the universe.” Correct Because speed is proportional to distance, an observer in any galaxy sees the same thing. The expansion has no centre, and the Big Bang was not an explosion at a point in space.
• “\(H_0\) is about 70.” Correct This syllabus uses SI units only: \(H_0\) is in s−1, about \(2.3 \times 10^{-18}\,\mathrm{s^{-1}}\), with \(v\) in m s−1 and \(d\) in m.
• “Luminosity is how bright a star looks from the Earth.” Repair Luminosity is the total power of radiation emitted by the star, in W. What is measured at the Earth is the radiant flux intensity \(F\), in W m−2.
• “\(F \propto 1/d\), so twice as far is half as bright.” Repair \(F = L/(4\pi d^2)\): the power spreads over a sphere of area \(4\pi d^2\), so twice as far gives a quarter.
• “\(d = L/(4\pi F)\).” Repair \(d = \sqrt{L/(4\pi F)}\). Check the unit: \(\mathrm{W}/(\mathrm{W\,m^{-2}})\) is m2, so the square root is needed to reach metres.
• “A standard candle is a very bright star.” Repair It is an object of known luminosity. Being luminous only makes it visible at a great distance.
• “\(F = L/(4\pi d^2)\) fails for a giant star, because a giant star is too big to be a point.” Repair If a star radiates equally in all directions and no radiation is absorbed, all of \(L\) crosses every sphere of radius \(d\) centred on it, however large the star is, so the law holds anywhere outside the star, with \(d\) measured from its centre. Treating the source as a point is also a valid assumption to state; it is the one that fails for a lamp close to a sensor (see practical skills).
• “Hotter stars peak at longer wavelengths.” Repair \(\lambda_{\max} \propto 1/T\): hotter means shorter. A blue-white star is hotter than a red one.
• “\(T = 5527\) °C, so \(T^4 = 5527^4\).” Repair \(T\) in every radiation law is the thermodynamic temperature in kelvin: 5800 K.
• “Wien’s constant is on the Data sheet.” Repair It is not: the question supplies \(2.9 \times 10^{-3}\) m K, or gives a reference star so that \(\lambda_1 T_1 = \lambda_2 T_2\) can be used.
• “\(L = \sigma T^4\).” Repair \(\sigma T^4\) is the power per unit area. \(L = 4\pi\sigma r^2 T^4\) multiplies by the surface area, with \(r\) the radius, not the diameter.
• “The hotter of two stars must be the more luminous.” Repair \(L \propto r^2 T^4\): a large cool star can outshine a small hot one (worked example 4).
• “Redshift means the galaxy looks red.” Repair Every spectral line is shifted to a longer wavelength by the same fraction; the colour of the galaxy is not the point.
• “In \(\Delta\lambda/\lambda\), \(\lambda\) is the observed wavelength.” Repair \(\lambda\) is the emitted (laboratory) wavelength; \(\Delta\lambda\) is observed minus emitted.
• “All galaxies move away from us, so we are at the centre.” Repair With speed proportional to distance, every observer sees the same recession. The expansion has no centre.
• “\(H_0 = 70\).” Repair The syllabus uses SI units only: \(H_0\) in s−1, about \(2.3 \times 10^{-18}\,\mathrm{s^{-1}}\).
• “The Big Bang was an explosion at one point in space.” Repair The theory says the whole universe was once extremely small, hot and dense and has been expanding ever since; there is no point in today’s space where it happened.
• “\(1/H_0\) is the exact age of the universe.” Repair It is an estimate that assumes the expansion rate has always been the same.

• Read the command word before you decide how much to write. This syllabus has fifteen of them: calculate, comment, compare, define, describe, determine, explain, give, identify, justify, predict, show (that), sketch, state and suggest. Define wants a precise meaning — for a physical quantity, usually an equation in words with every quantity named. State and give want a fact and nothing more. Describe wants the points or the features. Explain wants the reasons and the relationships — a describe-level answer to an explain question is incomplete however well written it is. Show (that) gives you the result and asks for the structured evidence that leads to it, so every step must appear — and a final value worked to one more significant figure than the one printed makes it plain that you calculated it rather than copied it. Sketch wants a freehand graph with its key features — intercepts, asymptotes, the shape — correct, but no plotted scale.
• Units are part of the answer. Every temperature in kelvin; every wavelength in metres before it goes into Wien’s law; \(H_0\) in s−1, distance in m and speed in m s−1 — the syllabus asks for SI units only. A quick unit check catches most slips: \(\sqrt{\mathrm{W}/(\mathrm{W\,m^{-2}})} = \mathrm{m}\) for a distance, \(\sqrt{\mathrm{W}/(\mathrm{W\,m^{-2}\,K^{-4}}\,\mathrm{K^4})} = \mathrm{m}\) for a radius, and \(1/\mathrm{s^{-1}} = \mathrm{s}\) for an age.
• A percentage uncertainty survives a reciprocal unchanged. \(H_0 = 1/m\), so the percentage uncertainty in \(H_0\) equals that in \(m\). Quote the absolute uncertainty to one significant figure and the value to the same decimal place: \((2.3 \pm 0.3) \times 10^{-18}\,\mathrm{s^{-1}}\), not \(2.29 \pm 0.31\).
• Interleave with the chapters that use this one. Topic 25 is the last topic, so interleave backwards: with topic 7, re-answer “why does intensity fall as \(1/d^2\), and why does a receding source give a longer wavelength?”; with topic 22, “why can a shifted pattern of lines still be identified?”; with topic 13, “which inverse-square law is geometry and which is a force law?”. Recalling a topic inside a new context is worth more than another pass over this chapter on its own; at A Level, Paper 4 assumes the whole of the AS content, so nothing here is ever finished with.

• Cambridge International AS & A Level Physics 9702 has five components. Topic 25 is A Level content, so it is examined in Papers 4 and 5. A Level content: examined in Paper 4 (A Level structured, which also requires the AS content) and, as practical context, Paper 5. AS Level candidates take Papers 1, 2 and 3; A Level candidates take all five, either staged over two years (Papers 1–3 in year one, Papers 4 and 5 in year two) or together in one series. Examinations are available in the June and November series, and in March in India.
• Across both the AS Level and the A Level the assessment objectives are weighted AO1 40% (knowledge and understanding), AO2 40% (handling, applying and evaluating information) and AO3 20% (experimental skills and investigations). AS candidates are graded a–e; A Level candidates A*–E. The Data and formulas sheet is printed as page 2 of Papers 1 and 2 and as pages 2 and 3 of Paper 4: it gives the constants and a short list of formulas. Every other equation in this chapter is one the syllabus says you must recall, and this chapter says which is which.
• Structured questions ask you to define luminosity, state what a standard candle is or state Hubble’s law, describe the standard-candle method or what is seen in a galaxy’s spectrum, and explain why redshift means expansion or how Hubble’s law leads to the Big Bang theory. Each explanation is a chain of linked statements; a missing link is a missing idea.
• \(L = 4\pi\sigma r^2 T^4\) and \(\Delta\lambda/\lambda \approx \Delta f/f \approx v/c\) are printed on the Data and formulas sheet with \(\sigma\) and \(c\). \(F = L/(4\pi d^2)\), \(\lambda_{\max} \propto 1/T\) and \(v \approx H_0 d\) must be recalled; Wien’s constant and \(H_0\) are supplied in the question. Expect chained calculations — flux to distance to \(H_0\), peak to temperature to radius — and ratio questions with no numbers at all.
• No astronomy can be done at the bench, so this topic reaches Paper 5 as analysis of supplied data: a graph of distance against recession speed (gradient \(1/H_0\)) with error bars, a worst acceptable line and an uncertainty in \(H_0\). Its laboratory analogue is the inverse square law with a small lamp and a light sensor, analysed on a graph of lg \(P\) against lg \(d\).
• Read the command word before you decide how much to write. This syllabus has fifteen of them: calculate, comment, compare, define, describe, determine, explain, give, identify, justify, predict, show (that), sketch, state and suggest. Define wants a precise meaning — for a physical quantity, usually an equation in words with every quantity named. State and give want a fact and nothing more. Describe wants the points or the features. Explain wants the reasons and the relationships — a describe-level answer to an explain question is incomplete however well written it is. Show (that) gives you the result and asks for the structured evidence that leads to it, so every step must appear — and a final value worked to one more significant figure than the one printed makes it plain that you calculated it rather than copied it. Sketch wants a freehand graph with its key features — intercepts, asymptotes, the shape — correct, but no plotted scale.