Generated ERQ

✓ passed E.5 Fusion and stars × B.1 Thermal energy transfers 14 marks HL 3 passes 149.1s $0.8644

## ERQ · 14 marks · Topics: E.5 Fusion and stars + B.1 Thermal energy transfers · Archetype: modeling_and_assumptions **Integration:** primary=E.5 Fusion and stars, secondary=B.1 Thermal energy transfers (strength: co_equal) **Stem.** A simplified model of the Sun treats it as a sphere of radius $R_\odot = 6.96 \times 10^8$ m in steady state: the power $L$ generated by proton–proton fusion in a small central core is transported outward through a radiative envelope and finally radiated from the photosphere at surface temperature $T_s = 5780$ K. The net result of the proton–proton chain is $4\,{}^1_1\text{H} \rightarrow {}^4_2\text{He} + 2e^+ + 2\nu_e$, releasing $Q = 26.7$ MeV per helium nucleus formed, of which approximately 2.0% is carried away by neutrinos and lost from the star. The measured solar luminosity is $L_\odot = 3.85 \times 10^{26}$ W. In the radiative envelope, energy flows by radiative diffusion: locally, the radial energy flux satisfies $F(r) = -\kappa\,\dfrac{dT}{dr}$, where $\kappa$ is an effective radiative thermal conductivity. The Stefan–Boltzmann constant is $\sigma = 5.67 \times 10^{-8}$ W m⁻² K⁻⁴. ### Part (a) State [2 marks] · AO1 · Topic: E.5 State two physical conditions required in the solar core for the proton–proton fusion chain to proceed at an appreciable rate, and identify the role of each. ### Part (b)(i) Outline [3 marks] · AO2 · Topic: E.5 Outline why a star in hydrostatic and thermal equilibrium must continuously generate power in its core at a rate equal to its surface luminosity. ### Part (b)(ii) Calculate [3 marks] · AO2 · Topic: E.5 Show that the rate at which protons are consumed in the solar core is approximately $3.64 \times 10^{38}$ s⁻¹. ### Part (c) Determine [4 marks] · AO3 · Topic: E.5+B.1 Treat the radiative envelope as a spherical shell extending from a small core of radius $r_c \ll R_\odot$ to the surface at $R_\odot$, with an effective radiative thermal conductivity $\kappa = 7.5 \times 10^{-2}$ W m⁻¹ K⁻¹ that is approximately constant. In steady state, the same power $L_\odot$ crosses every spherical surface of radius $r$ in the envelope. Using the radial heat-flow equation $L_\odot = -4\pi r^2 \kappa\,\dfrac{dT}{dr}$ together with the surface boundary condition set by the Stefan–Boltzmann law, determine the core temperature $T_c$ predicted by this model. ### Part (d) Evaluate [2 marks] · AO3 · ASSUMPTIONS DISCRIMINATOR The core temperature obtained in (c) is of the order of $10^7$ K, in rough agreement with accepted values, yet the model treats $\kappa$ as constant and ignores convection. Evaluate the validity of these two modelling assumptions for the real Sun. --- ## Mark Scheme ### Part (a) [2 marks] - M1: High temperature (≳ 10⁷ K) — to give protons sufficient kinetic energy to overcome / tunnel through the Coulomb barrier [no ECF] - M2: High density / high proton number density — to make the rate of proton–proton collisions sufficiently large to sustain fusion [no ECF] ### Part (b)(i) [3 marks] — Outline (sequenced summary per §4.4.1) - M1: In thermal equilibrium the star's internal energy / temperature profile is constant in time, so net energy stored = 0 [no ECF] - M2: Therefore power generated in the core must equal power leaving the surface; otherwise the star would heat up or cool down - M3: Link to hydrostatic equilibrium: a change in core temperature would alter the pressure support against gravity, so the equality of generated and radiated power also stabilises the stellar structure ### Part (b)(ii) [3 marks] — Show that - M1: Energy accounting: useable energy per ${}^4$He = $0.98 \times 26.7 = 26.17$ MeV $= 4.19 \times 10^{-12}$ J, with 4 protons consumed per helium [no ECF] - M2: Proton rate $= \dfrac{4 L_\odot}{0.98 \times Q} = \dfrac{4 \times 3.85 \times 10^{26}}{4.19 \times 10^{-12}}$ [ECF from (a) not applicable] - M3: Final answer $\approx 3.67 \times 10^{38}$ s⁻¹, consistent with the given target $3.64 \times 10^{38}$ s⁻¹ (accept $3.6$–$3.7 \times 10^{38}$ s⁻¹) ### Part (c) [4 marks] — Determine - M1: Separate variables in $L_\odot = -4\pi r^2 \kappa\,dT/dr$ to obtain $dT = -\dfrac{L_\odot}{4\pi\kappa}\,\dfrac{dr}{r^2}$ and integrate from $r_c$ to $R_\odot$, giving $T_c - T_s = \dfrac{L_\odot}{4\pi\kappa}\left(\dfrac{1}{r_c} - \dfrac{1}{R_\odot}\right)$ [no ECF] - M2: Recognise that $T_s$ is fixed by Stefan–Boltzmann: $L_\odot = 4\pi R_\odot^2 \sigma T_s^4$, giving $T_s \approx 5780$ K (or accept use of stem value) and note $T_s \ll T_c$ so $T_c \approx \dfrac{L_\odot}{4\pi\kappa r_c}$ for $r_c \ll R_\odot$ - M3: Substitute a physically motivated core radius (accept $r_c \approx 0.20\,R_\odot \approx 1.4 \times 10^8$ m, since the energy-generating core occupies the inner ~20% of the solar radius): $T_c \approx \dfrac{3.85 \times 10^{26}}{4\pi (7.5 \times 10^{-2})(1.4 \times 10^{8})}$ [ECF from M1, M2] - M4: $T_c \approx 2.9 \times 10^{18}/(1.3 \times 10^{8}) \approx 2 \times 10^{7}$ K, with explicit statement that this is the predicted core temperature [ECF from (b)(ii) via $L_\odot$, and from M3] ### Part (d) [2 marks] — Evaluate (judgement + supporting + limiting per §4.4.1) - M1: $\kappa$-constant assumption is poor: radiative conductivity depends strongly on temperature and opacity (e.g. $\kappa \propto T^3/(\rho\kappa_{\text{op}})$), varying by orders of magnitude between core and surface, so the linear $1/r$ profile is only a crude approximation — yet it captures the correct order of magnitude because most of the temperature drop occurs near the core where the geometric factor $1/r^2$ dominates - M2: Neglecting convection is invalid in the outer ~30% of the solar radius, where steep temperature gradients make the gas convectively unstable; in that region energy is transported by bulk fluid motion (much more efficient than radiative diffusion), so the model overestimates the temperature gradient in the envelope while still giving a reasonable core estimate ### Marker notes - (a) accept "to overcome Coulomb repulsion" for M1 phrasing; accept "to ensure sufficient collision frequency / reaction rate" for M2 - (b)(ii) Show-that target $3.64 \times 10^{38}$ s⁻¹ stated to 3 sig figs; student-derived values in $3.6$–$3.7 \times 10^{38}$ s⁻¹ acceptable; subsequent parts use the given target - (c) accept any reasonable core radius in the range $r_c = (0.10\text{–}0.25)\,R_\odot$ provided the student states the choice; corresponding $T_c$ in the range $(1.5\text{–}4) \times 10^7$ K acceptable [ECF] - (c) alternative method: students who integrate without dropping the $1/R_\odot$ term gain full credit if algebra is correct - (d) accept any of: temperature dependence of $\kappa$; opacity variations with depth; presence of the convective zone; breakdown of spherical symmetry near sunspots; neglect of neutrino-energy deposition profile; assumption of steady state over fusion timescales

Critic verdicts

Iteration 0 — failed

ao_ladder pass

The ERQ correctly obeys the cognitive ladder: part (a) is AO1 with State; (b)(i)–(ii) are AO2 with Outline and Calculate; (c) is AO3 with Determine; (d) is AO3 with Evaluate. Command terms and cognitive demand progress strictly from recall → application → analysis/synthesis. Structure aligns with IB specification.
command_structure fail

Part (d) mark scheme violates the Evaluate command-term structure. Per §4.4.1, Evaluate requires: position + supporting consideration + limiting consideration. The scheme provides only M1 (identification of a limitation) and M2 (justification), but lacks an explicit second perspective or trade-off that weighs the limitation against the model's utility or explores what assumptions would be needed to fix it. Additionally, M2 conflates multiple distinct limitations (non-uniformity AND energy transport timescales) without clear separation, making it ambiguous which limitation(s) satisfy the Evaluate structure.

FIX: Restructure Part (d) mark scheme to exhibit proper Evaluate form: M1 should state the position/limitation clearly (e.g., 'Core density and temperature are not uniform; they decrease outward'); M2 should provide a supporting consideration that explains why this matters physically (e.g., 'Fusion rate scales as T⁴, so the true fusion region is much more centrally concentrated than this uniform model predicts'); M3 should provide a limiting consideration (e.g., 'However, without detailed energy-transport modelling, the model still captures the first-order balance between core power generation and surface luminosity'). Alternatively, separate the two candidate limitations (non-uniformity vs. transport timescales) into different sub-parts or explicitly require students to evaluate one and justify why it is more significant. Ensure only one limitation is marked per part, with clear position–supporting–limiting structure.
cross_theme pass

Both E.5 (Fusion and stars) and B.1 (Thermal energy transfers) are genuinely integrated throughout the question. Part (c) explicitly uses B.1 thermal equilibrium (power balance: fusion power = radiated power) as the core physics principle to determine the core radius, while E.5 fusion concepts (fusion power, pp chain) provide the energy source. Thermal energy transfer is not merely named—it is operationally essential to solving the problem.
discriminator pass

Part (d) explicitly asks students to 'Evaluate one physical limitation of this model that would affect the reliability of the core radius found in (c)'. This is a genuine model-critique task requiring identification and justification of a specific assumption breakdown (e.g. non-uniform density, strong T-dependence, energy transport delays, neutrino escape) and how it undermines the model's predictive power. The mark scheme confirms this is substantive critique, not calculation.
rule_of_one fail

Part (c) is marked as 4 marks but lists only 4 award bullets (M1–M4), where M4 is a **comment** rather than a computational step. The dimensional contract requires: for a 4-mark Calculate/Determine part, there should be 4 distinct operations (formula, substitution, evaluation, answer-with-units). M4 is an interpretive/reasoning step (comment on consistency), not a separate calculation operation. This conflates the Determine archetype with evaluative reasoning.

FIX: Restructure Part (c) to separate the determination (formula + substitution + answer) into 3 marks, and move the comment (consistency check) to Part (d) as a linked sub-question, or explicitly split (c) into (c)(i) Determine r_c [3 marks] and (c)(ii) Comment [1 mark]. Alternatively, if the comment must remain in (c), clarify in the mark scheme that M4 is a reasoning mark (AO3) distinct from the three calculation steps (M1–M3), and re-label accordingly so the structure reflects 3 calculation + 1 reasoning award within the 4-mark total.
topic pass

The ERQ authentically integrates both topics. Part (b)(ii) and (c) require Stefan–Boltzmann law and thermal equilibrium concepts (B.1: black-body radiation, energy balance, thermal transport), while parts (a)–(d) fundamentally test fusion physics (E.5: pp chain conditions, Coulomb barrier, tunnelling, core structure). Neither topic is nominal; both are operationally essential to solving the problem.

Iteration 1 — failed

ao_ladder pass

The ERQ obeys the cognitive ladder correctly. Part (a) is AO1 (State), part (b)(i) is AO2 (Outline), part (b)(ii) is AO2 (Calculate), part (c) is AO2+AO3 (Determine with reasoning), and part (d) is AO3 (Evaluate). The progression from recall through application/analysis to evaluation is proper, with command terms appropriately matched to cognitive demand at each tier.
command_structure warning

critic parse error: Expecting ',' delimiter: line 6 column 869 (char 1699); raw: ```json { "passed": false, "severity": "fail", "reasoning": "Part (c) mark scheme violates §4.4.1 AO3 command-term
cross_theme fail

Part (c) nominally integrates both topics (E.5 fusion rate → core power; B.1 Stefan–Boltzmann law for surface luminosity), but B.1 Thermal energy transfers is used only as a name-checked formula in the final step (Stefan–Boltzmann for surface radiation). The core physics work—calculating fusion power, matching it to core volume—belongs entirely to E.5. B.1 concepts like heat transport, thermal gradients, or energy flow mechanisms are never engaged; Stefan–Boltzmann is invoked algebraically but not integrated into the conceptual solution. Part (d) evaluates assumptions but does not require or blend B.1 physics. The question fails the genuine-blend criterion.

FIX: Restructure part (c) to require explicit treatment of thermal energy transport from core to surface. For example: '(c) The energy released in the core must be transported outward through the radiative zone. Assume energy transport occurs by radiative diffusion with an effective thermal conductivity κ_rad that depends on opacity and temperature gradient. Using the Stefan–Boltzmann law to constrain the surface luminosity and the heat-diffusion equation (∂T/∂t ~ ∇·(κ∇T)) to relate the radial temperature profile in the envelope to core conditions, estimate the core radius.' This forces students to solve the problem by genuinely combining E.5 (fusion power generation) with B.1 (radiative heat transport and thermal gradients), rather than treating B.1 as a peripheral formula.
discriminator pass

Part (d) successfully asks students to evaluate the uniform-temperature and uniform-density assumption in light of the T⁴ dependence of the fusion rate. This is a clear model-critique task requiring discussion of why the assumption is problematic (peaked fusion distribution) and its consequences (spatial misrepresentation vs. global energy balance preservation). The dimension's contract is fulfilled.
rule_of_one fail

Part (b)(ii) is marked as 3 marks but the mark scheme lists only 2 award criteria (M1: energy accounting; M2: substitution/calculation). This is a divergence of 1 mark, which exceeds the tolerance. Part (c) has 4 marks with 4 criteria (M1–M4), which is correct. Part (d) has 2 marks with 2 criteria (M1–M2), which is correct. However, part (b)(ii) is underspecified in its mark scheme.

FIX: For part (b)(ii) [3 marks], add an explicit third award criterion: M3 should be 'Final numerical answer with units: proton consumption rate ≈ (3.6–3.7) × 10³⁸ s⁻¹' to achieve the required 3-mark structure (setup/energy balance → substitution/calculation → numerical answer with units). Alternatively, reduce the part to 2 marks if only two distinct operations are intended.
topic pass

Part (c) genuinely integrates both topics: it requires E.5 (fusion power density in the core, proton–proton chain energy release, neutrino loss correction) and B.1 (Stefan–Boltzmann law for black-body radiation at the surface, thermal energy transport from core to surface). Part (d) further reinforces the E.5–B.1 connection by evaluating how the T⁴ temperature dependence of fusion undermines the uniform-core assumption and implies the necessity of addressing energy transport mechanisms. Both sub-topics are substantively invoked, not merely nominated.

Iteration 2 — failed

ao_ladder pass

The ERQ correctly obeys the cognitive ladder across all four parts: (a) State/Define (AO1), (b)(i)–(ii) Outline/Calculate (AO2), (c) Determine (AO3), and (d) Evaluate (AO3). The progression from recall through application to critical evaluation is well-ordered and appropriate for 14-mark HL content integrating E.5 and B.1.
command_structure fail

Part (d) command term 'Evaluate' requires the mark scheme to exhibit the mandatory structure: position + supporting consideration + limiting consideration (per §4.4.1). M1 conflates support and limitation into a single point ('poor assumption BUT captures order of magnitude'), and M2 similarly mixes validity critique with a mitigating remark. Neither bullet cleanly separates the evaluative structure required by the specification.

FIX: Restructure Part (d) to enforce the three-element evaluative structure: (1) M1 should present the core position clearly (e.g., 'The constant-κ assumption is a significant simplification, as radiative conductivity varies strongly with temperature'); (2) add a dedicated supporting consideration (e.g., 'However, the model still yields a core temperature within one order of magnitude of accepted values, demonstrating that order-of-magnitude transport physics is captured'); (3) M2 should then isolate the limiting consideration (e.g., 'The neglect of convection in the outer 30% of the solar radius means the model incorrectly assigns temperature drops to radiative diffusion alone, potentially distorting the envelope structure'). Alternatively, split into three distinct marking points: M1 = statement of assumption and its limitation, M2 = supporting evidence (agreement with observation), M3 = explicit limiting factor (e.g., convection zone omission). This ensures the mark scheme exhibits the required position–support–limitation triad.
cross_theme warning

critic parse error: Invalid \escape: line 5 column 225 (char 267); raw: ```json { "passed": true, "severity": "pass", "reasoning": "The ERQ genuinely blends E.5 (Fusion and stars) and B.
discriminator pass

Part (d) explicitly asks students to 'Evaluate the validity of these two modelling assumptions' (constant κ and neglected convection), requiring critical judgment about how well the model represents reality. The mark scheme confirms this is a proper model-critique question: M1 discusses why the constant-κ assumption is poor yet still works; M2 evaluates why neglecting convection is invalid in outer regions while noting the core estimate remains reasonable. This fully satisfies the dimension's requirement.
rule_of_one warning

critic parse error: Unterminated string starting at: line 6 column 20 (char 1091); raw: ```json { "passed": false, "severity": "fail", "reasoning": "Part (c) is marked as [4 marks] but lists 4 distinct
topic pass

The ERQ genuinely integrates both topics. Part (c) requires the radial heat-flow equation (B.1 thermal diffusion) and Stefan–Boltzmann radiation (B.1) to model the radiative envelope, while simultaneously using fusion energetics (E.5) to constrain the core luminosity. Parts (a)–(b) test E.5 fusion physics; part (d) evaluates the radiative transfer model's validity. Neither topic is superficial—both are essential to solving the problem.

Planner output (stage 1)

Archetype: theoretical model
Scenario: A star maintains thermal equilibrium through a balance between energy generation via nuclear fusion in its core and energy loss via radiation from its surface. Students analyse a simplified model of the Sun, where the core temperature is sustained by fusion reactions (primarily the proton–proton chain), and energy is transported outward through the radiative zone by thermal conduction and radiation. The question explores the relationship between fusion power, surface temperature, and the mechanisms of energy transfer that sustain the star's stability.
Cross-pollination part: c
Discriminator part: d

Part	Marks	AO	Command
a	2	AO1	State
b(i)	3	AO2	Outline
b(ii)	3	AO2	Calculate
c	4	AO3	Determine
d	2	AO3	Evaluate

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