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Question Types — Astronomy and Cosmology

Question Type 1: Luminosity and radiant flux intensity​

如何识别​

题目给出恒星 luminosity LL 和距离 dd，要求计算 radiant flux intensity FF，或反推距离。

标准解题方法​

标准解题方法
  1. F=L/(4πd2)F = L / (4\pi d^2)
  2. FF 单位：W m−2^{-2}
  3. LL 单位：W（或 solar luminosity L⊙L_{\odot}）
  4. 已知 FF 和 dd 可求 LL，已知 FF 和 LL 可求 dd

评分标准​

评分标准（MS 模式）
  • B1: 定义 luminosity
  • C1: 代入 F=L/(4πd2)F = L / (4\pi d^2)
  • A1: 数值 + 单位

完整原题​

Example 1 — 9702/w22/qp/41 Q9 (8 marks):

(a) State what is meant by the luminosity of a star. (b) A star is at distance 8.14×1016 m8.14 \times 10^{16}\text{ m} from Earth and has luminosity 9.86×1027 W9.86 \times 10^{27}\text{ W}, surface temperature 9830 K. (i) Calculate the radiant flux intensity observed from Earth. (ii) Determine the radius of the star. (c) Explain how surface temperature of a distant star may be determined from its wavelength spectrum.

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MS:

  • B1: (a) total power of radiation emitted (by the star)
  • C1: (b)(i) F=L/(4πd2)=9.86×1027/[4π×(8.14×1016)2]F = L / (4\pi d^2) = 9.86 \times 10^{27} / [4\pi \times (8.14 \times 10^{16})^2]
  • A1: F=1.18×10−7 W m−2F = 1.18 \times 10^{-7}\text{ W m}^{-2}
  • C1: (b)(ii) L=4πσr2T4L = 4\pi\sigma r^2 T^4
  • A1: r=1.22×109 mr = 1.22 \times 10^9\text{ m}
  • B1: (c) wavelength of peak intensity determined from spectrum
  • B1: Wien's displacement law used (λmax⁡∝1/T\lambda_{\max} \propto 1/T)
  • B1: compare with known temperature object / calibration

Example 2 — 9702/s23/qp/41 Q10(b) (2 marks):

A star of luminosity 3.8×1031 W3.8 \times 10^{31}\text{ W} is at a distance of 1.8×1024 m1.8 \times 10^{24}\text{ m} from Earth. Calculate the radiant flux intensity at Earth.

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MS:

  • C1: F=L/(4πd2)=(3.8×1031)/[4π×(1.8×1024)2]F = L / (4\pi d^2) = (3.8 \times 10^{31}) / [4\pi \times (1.8 \times 10^{24})^2]
  • A1: F=9.3×10−19 W m−2F = 9.3 \times 10^{-19}\text{ W m}^{-2}

Example 3 — 9702/s24/qp/41 Q10 (8 marks):

(a)(i) State what is meant by luminosity of a star. (ii) Explain how a standard candle in a distant galaxy can be used to determine distance. (b) The Sun has mean distance from Earth 1.50×1011 m1.50 \times 10^{11}\text{ m} and radiant flux intensity 1.36×103 W m−21.36 \times 10^3\text{ W m}^{-2}. (i) Calculate luminosity of the Sun. (ii) The Sun's surface temperature is 5780 K. Calculate its radius.

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MS:

  • B1: (a)(i) total power of radiation emitted (by star)
  • B1: (a)(ii) standard candle has known luminosity, measure FF, use d2=L/(4πF)d^2 = L / (4\pi F)
  • C1: (b)(i) L⊙=F×4πd2=1.36×103×4π×(1.50×1011)2L_{\odot} = F \times 4\pi d^2 = 1.36 \times 10^3 \times 4\pi \times (1.50 \times 10^{11})^2
  • A1: L⊙=3.85×1026 WL_{\odot} = 3.85 \times 10^{26}\text{ W}
  • C1: (b)(ii) L=4πσr2T4L = 4\pi\sigma r^2 T^4, rearrange for rr
  • A1: r=6.96×108 mr = 6.96 \times 10^8\text{ m}

Question Type 2: Stellar radius using Wien and Stefan-Boltzmann​

如何识别​

题目给出恒星温度（或 λmax⁡\lambda_{\max}）和 luminosity（或 flux），要求估计恒星半径。

标准解题方法​

标准解题方法
  1. Wien's law: λmax⁡T=constant=2.9×10−3 m K\lambda_{\max} T = \text{constant} = 2.9 \times 10^{-3}\text{ m K}，求 TT
  2. Stefan-Boltzmann law: L=4πσr2T4L = 4\pi\sigma r^2 T^4
  3. 整理得 r=L/(4πσT4)r = \sqrt{L / (4\pi\sigma T^4)}
  4. 注意 LL 有时需通过 F=L/(4πd2)F = L / (4\pi d^2) 先求出

评分标准​

评分标准（MS 模式）
  • C1: 代入 Stefan-Boltzmann 公式
  • C1: 正确代入数值
  • A1: 半径数值 + 单位

完整原题​

Example 1 — 9702/w22/qp/41 Q9(b)(ii) (2 marks):

Star has luminosity 9.86×1027 W9.86 \times 10^{27}\text{ W}, surface temperature 9830 K. Determine its radius.

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MS:

  • C1: L=4πσr2T4L = 4\pi\sigma r^2 T^4, 9.86×1027=4π×5.67×10−8×r2×(9830)49.86 \times 10^{27} = 4\pi \times 5.67 \times 10^{-8} \times r^2 \times (9830)^4
  • A1: r=1.22×109 mr = 1.22 \times 10^9\text{ m}

Example 2 — 9702/s23/qp/42 Q9(c) (2 marks):

A star has radius 2.3×109 m2.3 \times 10^9\text{ m} and luminosity 1.4×1028 W1.4 \times 10^{28}\text{ W}. Calculate its surface temperature.

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MS:

  • C1: L=4πσr2T4L = 4\pi\sigma r^2 T^4, so T4=L/(4πσr2)T^4 = L / (4\pi\sigma r^2)
  • A1: T=5800 KT = 5800\text{ K} (approximately)

Question Type 3: Redshift and Hubble's law​

如何识别​

题目给出星系光谱线的实验室波长和观测波长，要求计算 redshift、recession speed、Hubble constant。

标准解题方法​

标准解题方法
  1. Redshift: Δλ=λobs−λemit\Delta\lambda = \lambda_{\text{obs}} - \lambda_{\text{emit}}（观测值增大）
  2. Δλ/λ≈v/c\Delta\lambda / \lambda \approx v / c，求 vv
  3. v≈H0dv \approx H_0 d，求 H0H_0 或 dd
  4. 注意：红移表明星系远离我们，宇宙在膨胀

评分标准​

评分标准（MS 模式）
  • B1: redshift 定义
  • C1: Δλ/λ=v/c\Delta\lambda / \lambda = v / c
  • C1: v=H0dv = H_0 d
  • A1: H0H_0 数值 + 单位（s−1^{-1}）

完整原题​

Example 1 — 9702/s23/qp/41 Q10 (9 marks):

(a) State Hubble's law. (b) Star luminosity 3.8×1031 W3.8 \times 10^{31}\text{ W}, distance 1.8×1024 m1.8 \times 10^{24}\text{ m}, calculate radiant flux. (c) Spectral line known wavelength 486 nm, observed 492 nm. (i) Explain why observed wavelength differs. (ii) Determine Hubble constant H0H_0.

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MS:

  • M1: (a) speed is (directly) proportional to distance
  • A1: v=H0dv = H_0 d, where vv is speed of recession, dd is distance
  • C1: (c)(i) galaxy is moving away (from Earth)
  • B1: wavelength increased by Doppler effect / redshift
  • C1: (c)(ii) Δλ/λ=v/c\Delta\lambda / \lambda = v / c, v=(492−486)×3.00×108/486=3.7×106 m s−1v = (492 - 486) \times 3.00 \times 10^8 / 486 = 3.7 \times 10^6\text{ m s}^{-1}
  • C1: H0=v/d=3.7×106/1.8×1024H_0 = v / d = 3.7 \times 10^6 / 1.8 \times 10^{24}
  • A1: H0=2.1×10−18 s−1H_0 = 2.1 \times 10^{-18}\text{ s}^{-1}

Example 2 — 9702/w24/qp/41 Q10 (9 marks):

(a) State what is meant by redshift and explain how it leads to the idea that the Universe is expanding. (b) A spectral line of known wavelength 658 nm from a galaxy is measured as 726 nm. (i) Calculate the distance to the galaxy using F=L/(4πd2)F = L / (4\pi d^2). (ii) Calculate the recessional speed of the galaxy. (c)(i) Sketch a graph of recessional speed against distance for galaxies. (ii) Identify the quantity given by the gradient of this graph.

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MS:

  • B1: (a) redshift is increase in observed wavelength / decrease in observed frequency (caused by Doppler effect)
  • B1: radiation from distant galaxies is observed to be redshifted
  • B1: redshift provides evidence that galaxies are moving apart
  • B1: galaxies moving apart means Universe must be expanding
  • C1: (b)(ii) Δλ/λ=v/c\Delta\lambda / \lambda = v / c, (726−658)/658=v/(3.00×108)(726 - 658) / 658 = v / (3.00 \times 10^8)
  • A1: v=3.1×107 m s−1v = 3.1 \times 10^7\text{ m s}^{-1}
  • B1: (c)(i) straight line with positive gradient through origin
  • B1: (c)(ii) Hubble constant

Example 3 — 9702/s22/qp/42 Q9 (7 marks):

(a)(i) State Hubble's law. (b) Explain how Hubble's law and the idea of the expanding Universe lead to the Big Bang theory.

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MS:

  • B1: (a)(i) v=H0dv = H_0 d, speed of recession proportional to distance
  • B1: (b) Universe has been expanding from a single point
  • B1: extrapolating backwards, all matter was at a single point
  • B1: this point is the Big Bang
  • B1: age of Universe ≈1/H0\approx 1 / H_0

Question Type 4: Standard candles​

如何识别​

题目出现 "standard candle"、"known luminosity"、"distance determination"。

标准解题方法​

标准解题方法
  1. Standard candle: 已知 luminosity 的天体
  2. 测量 radiant flux FF
  3. 使用 F=L/(4πd2)F = L / (4\pi d^2) 计算 dd
  4. 用于确定星系距离

完整原题​

Example 1 — 9702/s24/qp/41 Q10(a)(ii) (2 marks):

Explain how a standard candle in a distant galaxy can be used to determine the distance to the galaxy.

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MS:

  • B1: standard candle has known luminosity
  • B1: measure radiant flux intensity FF, then d2=L/(4πF)d^2 = L / (4\pi F)