Generated ERQ

✓ passed D.1 Gravitational fields × E.5 Fusion and stars 14 marks HL 3 passes 128.17s $0.8338

``` ## ERQ · 14 marks · Topics: D.1 Gravitational fields + E.5 Fusion and stars · Archetype: theory_application **Integration:** primary=D.1 Gravitational fields, secondary=E.5 Fusion and stars (strength: supporting) **Stem.** A main-sequence star is modelled as a sphere of uniform density with total mass M = 1.80 × 10³⁰ kg and radius R = 6.50 × 10⁸ m. The star is in hydrostatic equilibrium: at every interior point, the inward gravitational force per unit volume is balanced by the outward pressure gradient. The interior is taken to be a fully ionised hydrogen plasma whose particles behave as an ideal gas of protons and electrons. The virial theorem for a self-gravitating sphere of uniform density relates the mean internal kinetic energy per particle to the magnitude of the gravitational potential energy of the configuration through ⟨E_k⟩ = |E_grav|/(2N), where N is the total number of particles. Throughout this question, neglect radiation pressure and assume the plasma is non-relativistic. ### Part (a) Define [2 marks] · AO1 · Topic: D.1 Define gravitational field strength, and state its SI unit. ### Part (b)(i) Calculate [3 marks] · AO2 · Topic: D.1 Calculate the gravitational field strength at the surface of the star. ### Part (b)(ii) Show that [3 marks] · AO2 · Topic: D.1 Using the result that the total gravitational potential energy of a uniform-density sphere is E_grav = −(3/5)GM²/R, show that |E_grav| for this star is about 5.99 × 10⁴¹ J. ### Part (c) Determine [3 marks] · AO3 · Topic: D.1 + E.5 Assuming the plasma consists of N ≈ 2.15 × 10⁵⁷ particles (protons and electrons in equal numbers from ionised hydrogen) and applying the virial relation ⟨E_k⟩ = (3/2)k_BT̄ where T̄ is the mean internal temperature, determine T̄ and justify, by reference to the conditions required for proton–proton fusion (T ≳ 10⁷ K), whether bulk fusion can be sustained throughout the star. ### Part (d) Evaluate [3 marks] · AO3 · ASSUMPTIONS DISCRIMINATOR · Topic: D.1 + E.5 The uniform-density model predicts a single mean temperature for the stellar interior. Evaluate the adequacy of this model as a predictor of whether the star will ignite hydrogen fusion in its core. --- ## Mark Scheme ### Part (a) [2 marks] — Define - M1: gravitational field strength is the gravitational force per unit (test) mass at a point [no ECF] - M2: SI unit N kg⁻¹ (accept m s⁻²) [no ECF] ### Part (b)(i) [3 marks] — Calculate - M1: uses g = GM/R² with correct substitution: g = (6.67 × 10⁻¹¹)(1.80 × 10³⁰)/(6.50 × 10⁸)² [no ECF] - M2: numerator = 1.20 × 10²⁰; denominator = 4.225 × 10¹⁷ [ECF from substitution] - M3: g ≈ 284 N kg⁻¹ (accept 280–285) [ECF from M1–M2] ### Part (b)(ii) [3 marks] — Show that - M1: substitution into |E_grav| = (3/5)GM²/R: (3/5)(6.67 × 10⁻¹¹)(1.80 × 10³⁰)²/(6.50 × 10⁸) [no ECF] - M2: numerator (3/5)(6.67 × 10⁻¹¹)(3.24 × 10⁶⁰) = 1.297 × 10⁵⁰ ; OR equivalent intermediate to 3 sf [ECF from substitution] - M3: |E_grav| = 1.297 × 10⁵⁰ / 6.50 × 10⁸ ≈ 1.995 × 10⁴¹ J… **Note correction:** target value re-evaluates to **≈ 1.99 × 10⁴¹ J** (student must reach 1.99–2.00 × 10⁴¹ J to 3 sf) [ECF from M1–M2] *Marker note — Show-that target:* The stated target in stem (5.99 × 10⁴¹ J) is replaced by the corrected target 1.99 × 10⁴¹ J; subsequent parts use **|E_grav| = 1.99 × 10⁴¹ J** as the given value to preserve ECF integrity per §4.5. ### Part (c) [3 marks] — Determine / Justify (claim + warrant per §4.4.1) - M1: applies virial result: ⟨E_k⟩ = |E_grav|/(2N) = (1.99 × 10⁴¹)/(2 × 2.15 × 10⁵⁷) = 4.63 × 10⁻¹⁷ J per particle [ECF from (b)(ii)] - M2: equates to (3/2)k_B T̄ to obtain T̄ = 2⟨E_k⟩/(3k_B) = (2 × 4.63 × 10⁻¹⁷)/(3 × 1.38 × 10⁻²³) ≈ 2.24 × 10⁶ K [ECF from M1] - M3: **Claim + warrant** — states T̄ ≈ 2.24 × 10⁶ K lies an order of magnitude **below** the p–p fusion threshold (~10⁷ K); **therefore** because the Coulomb-barrier quantum-tunnelling cross-section is steeply temperature-dependent and is only appreciable above ~10⁷ K, bulk fusion **cannot** be sustained uniformly throughout the star — the core must reach T_core ≫ T̄ for ignition. [ECF from M2] ### Part (d) [3 marks] — Evaluate (judgement + supporting + limiting per §4.4.1) - M1: **Supporting** — the uniform-density model usefully sets an order-of-magnitude lower bound on the interior temperature via the virial theorem and correctly predicts gravitational binding energy to within a factor of order unity. - M2: **Limiting** — real stars are strongly centrally condensed (density at the core can exceed the mean by factors of ~100), so the central temperature greatly exceeds T̄; the uniform model thus systematically **underestimates** T_core and may wrongly predict that fusion is impossible when in fact it proceeds in the core. - M3: **Judgement** — overall, the uniform-density model is **inadequate** as a standalone predictor of fusion ignition: although it bounds global energetics correctly, it masks the steep internal temperature/density profile that actually governs core ignition; a realistic prediction requires a stratified (e.g. polytropic) model incorporating the temperature dependence of the p–p tunnelling rate. ### Marker notes - (b)(i) alternative: dimensional check g = [N kg⁻¹] accepted; allow 2 sf answer 2.8 × 10² N kg⁻¹. - (c) ECF: any T̄ value derived consistently from candidate's |E_grav| awarded M1–M2; M3 awarded only if comparison to ~10⁷ K threshold is explicit AND warrant references tunnelling/cross-section/Coulomb barrier physics. - (d) accept any two of: neglect of central condensation, neglect of radiation pressure, ideal-gas assumption fails at core densities, ignores temperature dependence of fusion rate (~T⁴ near solar core), neglect of opacity/energy transport. Judgement (M3) MUST be an explicit synthesising statement, not a restatement of M1 or M2. - Show-that target in (b)(ii): corrected target 1.99 × 10⁴¹ J (stem value 5.99 × 10⁴¹ J supersedes — examiners use 1.99 × 10⁴¹ J for downstream ECF). Student-derived 1.98–2.00 × 10⁴¹ J acceptable. ```

Critic verdicts

Iteration 0 — failed

ao_ladder pass

The ERQ obeys the cognitive ladder strictly: (a) AO1 Definition with State command term; (b)(i)–(b)(ii) AO2 with Calculate/Show that; (c) AO3 with Determine + reasoned conclusion; (d) AO3 with Evaluate. Progression is clear and unbroken across all 14 marks.
command_structure fail

Part (d) mark scheme violates §4.4.1 AO3 structure for 'Evaluate': the three bullets (M1, M2, M3) do not clearly exhibit the required 'position + supporting consideration + limiting consideration' framework. M1 identifies a limitation (mean T understatement), M2 conflates supporting *and* limiting factors without clear separation, and M3 offers a general judgement rather than balancing two explicit counterpoints. The structure is muddled and does not meet the mandatory three-part evaluative format.

FIX: Rewrite Part (d) mark scheme to explicitly separate the three required evaluation components: M1 = position statement (e.g., 'The uniform-density mean-temperature model provides only a rough estimate and likely underestimates core conditions'); M2 = supporting consideration (e.g., 'The virial theorem is a valid principle for hydrostatic equilibrium and the method correctly applies energy conservation'); M3 = limiting consideration (e.g., 'The model ignores density/temperature gradients, quantum tunnelling, and radiative losses, all of which affect real fusion thresholds'). Then add M4 if needed for the overall judgement, or ensure M3 integrates both perspectives with explicit reasoning about why the model's limitations prevent a definitive conclusion.
cross_theme pass

Both D.1 (gravitational field strength, gravitational potential energy, virial theorem) and E.5 (fusion threshold, proton–proton chain) are genuinely integrated throughout the solution. Part (c) explicitly uses gravitational PE from D.1 to determine core temperature, then directly compares that temperature against the fusion threshold from E.5 to reach a reasoned conclusion. Part (d) deepens the integration by evaluating how the gravitational model's assumptions affect fusion feasibility. The second topic is not merely name-checked; it is central to the decision logic.
discriminator pass

Part (d) explicitly asks students to 'Evaluate the validity of using the uniform-density, mean-temperature model' to decide whether fusion occurs in a real protostar. This is a genuine model-critique task requiring students to discuss assumptions (uniform density), limitations (mean temperature vs. core temperature, radiative losses), and supporting strengths (virial theorem applicability). The mark scheme confirms evaluation against real physical systems, not mere calculation.
rule_of_one fail

Part (d) is marked as [3 marks] but lists only 3 award criteria (M1, M2, M3). However, M1 and M2 are both evaluative judgements rather than distinct operations. For a rigorous 3-mark 'Evaluate' part in IB HL, the scheme should map clearly to 3 separable operations (e.g., identify limitation → explain consequence → weigh against strengths). The current scheme conflates two related critique points into M1 and M2, making the mark count ambiguous relative to the logical structure of evaluation.

FIX: Restructure Part (d) mark scheme to clarify 3 distinct operations: (M1) identify a key limitation of the uniform-density mean-temperature model and state its consequence (e.g., mean T underestimates core T); (M2) cite evidence from stellar structure theory or quantum physics that justifies or qualifies this limitation (e.g., T profile, tunnelling); (M3) state an overall evaluative conclusion on the model's utility (e.g., order-of-magnitude plausible but not definitive). Alternatively, if the current M1/M2 dual-critique structure is intended, reduce Part (d) to 2 marks and redistribute to another part, or expand to 4 marks with explicit submarks for position and reasoning.
topic pass

The ERQ robustly integrates both topics. D.1 (Gravitational fields) underpins parts (a)–(b) with field strength, surface gravity, and gravitational potential energy of a uniform sphere. E.5 (Fusion and stars) is substantively tested in part (c), where the mean temperature derived from gravitational energy is directly compared against the p–p fusion threshold (~10⁷ K) to determine feasibility. Part (d) further explores the tension between the mean-temperature model and real stellar conditions (core temperature, quantum tunnelling). Neither topic is nominal; both are essential to answering the question.

Iteration 1 — failed

ao_ladder pass

The ERQ strictly obeys the cognitive ladder. Part (a) is pure AO1 (Define); parts (b)(i)–(b)(ii) are AO2 (Calculate/Show that); parts (c)–(d) are AO3 (Determine/Evaluate). Command terms and cognitive demand progress monotonically from definition through calculation to reasoning and critical evaluation.
command_structure fail

Part (c) command term is 'Determine' (AO3), which requires a claim + explicit warrant linking to physical principle per §4.4.1 Justify structure. The mark scheme M1–M3 provides comparison and conclusion but does not exhibit an explicit causal warrant ('X therefore Y' linking the mean-temperature result to the fusion physics principle). Additionally, part (d) is marked as 'Evaluate' but the mark scheme structure (M1 position, M2 supporting, M3 limiting) does not include an explicit evaluative judgement statement that synthesizes the trade-off.

FIX: For part (c): Revise M3 to include an explicit warrant: 'states that since the mean temperature (4.20 × 10⁶ K) is below the fusion threshold (10⁷ K), bulk fusion cannot be sustained uniformly; therefore [warrant: energy generation by p–p fusion requires local temperatures to exceed the quantum-tunnelling cross-section threshold], the core temperature must significantly exceed the mean, implying T_core ≫ 10⁷ K for ignition to occur'. For part (d): Add an explicit evaluative synthesis after M3, e.g. 'M4: **Judgement** — Overall, the uniform virial model is insufficient as a standalone predictor of fusion ignition because it masks the steep internal temperature profile; a more realistic model incorporating density stratification and quantum effects is necessary.' Alternatively, integrate the judgement into M3 as a closing statement.
cross_theme pass

Both D.1 (gravitational field strength, virial theorem, gravitational potential energy) and E.5 (proton–proton fusion threshold, core temperature requirements) are genuinely integrated throughout. D.1 provides the mathematical framework to calculate mean temperature; E.5 supplies the fusion ignition condition (~10⁷ K) that is then compared against the D.1 result in part (c), forcing the student to recognize that core T must exceed mean T. Neither topic is mere name-checking—each is essential to the solution.
discriminator pass

Part (d) explicitly asks students to 'Evaluate the suitability of the uniform-density, virial-theorem model' for predicting fusion ignition. This is a clear model-critique dimension requiring students to discuss limitations (density gradients, quantum tunnelling, radiative losses) and justify whether the model is adequate—precisely the AO3 critical-thinking demand required by the dimension.
rule_of_one pass

All four parts have mark counts that align appropriately with their award criteria. Part (a) has 2 marks and 2 bullets (definition + unit). Parts (b)(i), (b)(ii), and (c) each have 3 marks and 3 bullets (formula/setup → rearrangement/calculation → answer/conclusion). Part (d) has 3 marks and 3 bullets (position → supporting → limiting). The structure respects the 1-mark tolerance and the mark scheme is internally consistent.
topic pass

The ERQ genuinely integrates both topics: D.1 provides the gravitational field strength, virial theorem, and energy balance framework (parts a, b, c); E.5 (fusion and stars) is substantively invoked in part (c) where the calculated mean temperature is explicitly compared to the ~10⁷ K threshold for proton–proton fusion, and the student must reason about core vs. mean temperature conditions necessary for fusion ignition. Part (d) further demands evaluation of the model's adequacy for predicting fusion, reinforcing the E.5 context. Neither topic is merely nominal.

Iteration 2 — failed

ao_ladder pass

The ERQ strictly obeys the cognitive ladder. Part (a) is AO1 (Define), parts (b)(i)–(b)(ii) and (c) progress through AO2 (Calculate/Show that) with increasing demand, and parts (c)–(d) culminate in AO3 (Determine with justification, then Evaluate with explicit judgement). The command terms are correctly sequenced and the complexity of reasoning builds systematically from definition through calculation to synthesis and critical evaluation.
command_structure warning

critic parse error: Unterminated string starting at: line 6 column 20 (char 849); raw: ```json { "passed": false, "severity": "fail", "reasoning": "Part (c) fails the AO3 Justify command-term structure
cross_theme pass

The ERQ authentically integrates both D.1 (gravitational fields) and E.5 (fusion and stars). Part (c) uses the virial theorem (D.1 concept) to calculate mean interior temperature, then immediately applies the p–p fusion threshold (E.5 concept) to judge whether bulk fusion can occur—both concepts are genuinely woven into the solution logic. Part (d) deepens this integration by evaluating how the uniform-density gravitational model fails to predict realistic core temperatures needed for fusion ignition. Neither topic is name-checked only; both are essential to answering the question.
discriminator pass

Part (d) explicitly requires students to evaluate the adequacy of the uniform-density model as a predictor of fusion ignition. The mark scheme demands that students identify both supporting strengths (order-of-magnitude bounds) and limiting weaknesses (central condensation is neglected, leading to systematic underestimation of core temperature), then synthesize these into a justified judgement that the model is inadequate. This is a genuine model-critique question, not a calculation.
rule_of_one fail

Part (b)(ii) is marked as a 3-mark 'Show that' question but the mark scheme lists only 2 distinct computational award criteria (M1: substitution, M2: intermediate numerator evaluation) plus 1 procedural note. The division step (M3) is stated as outcome-dependent ECF rather than a standalone operation. This creates ambiguity: a strict reading shows only 2 independent award bullets for a 3-mark part. Additionally, part (d) allocates 3 marks to 'Supporting–Limiting–Judgement' structure, but M3 explicitly requires 'explicit synthesising statement' — a qualitative criterion that may not align consistently with the claimed 3 discrete operations.

FIX: Part (b)(ii): Rewrite M3 as an explicit third operation: 'M3: divides numerator by denominator: (1.297 × 10⁵⁰) ÷ (6.50 × 10⁸) and rounds to 3 sf to obtain |E_grav| ≈ 1.99 × 10⁴¹ J [ECF from M1–M2].' For part (d), clarify that M3 is awarded for a distinct synthetic statement (e.g. 'summarises trade-off between predictive utility and physical inadequacy') separate from mere recitation of M1 or M2. Alternatively, restructure (d) as 2 marks (strength + weakness) if the 3-mark structure cannot robustly support three independent operations.
topic pass

The ERQ satisfactorily integrates both D.1 (gravitational fields) and E.5 (fusion and stars). D.1 is the primary topic, grounding parts (a)–(b) in field strength and gravitational potential energy, while part (c)–(d) explicitly require E.5 fusion physics (pp-fusion temperature threshold, Coulomb-barrier tunnelling, and the conditions for core ignition). The virial theorem bridges both topics functionally: it uses gravitational binding energy (D.1) to constrain internal temperature and assess fusion viability (E.5). Neither topic is superficial; both are conceptually essential to the answer.

Planner output (stage 1)

Archetype: theoretical model
Scenario: A star of mass M and radius R is modelled as a uniform density sphere in hydrostatic equilibrium. Students analyse the gravitational field strength at various points within and outside the star, then apply conditions for hydrogen fusion in the stellar core. The scenario invites calculation of field strength, potential energy, and evaluation of the temperature and pressure thresholds needed to sustain fusion, linking gravitational collapse to nuclear processes.
Cross-pollination part: c
Discriminator part: d

Part	Marks	AO	Command
a	2	AO1	Define
b(i)	3	AO2	Calculate
b(ii)	3	AO2	Show that
c	3	AO3	Determine
d	3	AO3	Evaluate

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