Generated ERQ

✓ passed E.5 Fusion and stars × D.1 Gravitational fields 12 marks SL 3 passes 100.82s $0.6971

## ERQ · 12 marks · Topics: E.5 Fusion and stars + D.1 Gravitational fields · Archetype: theory_application **Integration:** primary=E.5 Fusion and stars, secondary=D.1 Gravitational fields (strength: supporting) **Stem.** A main-sequence star of mass M = 2.0 × 10³⁰ kg and radius R = 7.0 × 10⁸ m is in hydrostatic equilibrium: the inward gravitational force on each shell of stellar material is balanced by an outward pressure gradient produced by energy released in the core. The fusion reactions in the core convert hydrogen into helium via the proton–proton chain. As a first approximation, the star is modelled as a sphere of uniform density. In what follows, take G = 6.67 × 10⁻¹¹ N m² kg⁻² and assume spherical symmetry throughout. ### Part (a) State [2 marks] · AO1 · Topic: E.5 Fusion and stars State **two** conditions required in the core of a star for sustained nuclear fusion of hydrogen to occur. ### Part (b)(i) Outline [2 marks] · AO2 · Topic: E.5 Fusion and stars Outline why a minimum stellar mass is required for hydrogen fusion to be initiated and maintained in the core. ### Part (b)(ii) Calculate [3 marks] · AO2 · Topic: D.1 Gravitational fields Using the uniform-density model, the gravitational field strength at a distance r from the centre of the star (for r < R) is g(r) = GM_enc / r², where M_enc is the mass enclosed within radius r. Calculate the gravitational field strength at r = R/2. ### Part (c) Show that [3 marks] · AO3 · Topic: E.5 Fusion and stars Using dimensional/order-of-magnitude reasoning, the central temperature T_c of a star in hydrostatic equilibrium can be estimated by equating the average thermal kinetic energy of a proton, (3/2)k_BT_c, to the gravitational potential energy per proton, GMm_p/R. Show that for this star T_c ≈ 1.5 × 10⁷ K. (Take m_p = 1.67 × 10⁻²⁷ kg and k_B = 1.38 × 10⁻²³ J K⁻¹.) ### Part (d) Evaluate [2 marks] · AO3 · ASSUMPTIONS DISCRIMINATOR Evaluate the validity of modelling the star as a sphere of uniform density when predicting the variation of gravitational field strength g(r) with radius inside the star. --- ## Mark Scheme ### Part (a) [2 marks] — State - M1: Sufficiently high temperature (≈ 10⁷ K) so that protons have enough kinetic energy to overcome Coulomb repulsion (quantum tunnelling acceptable) [no ECF] - M2: Sufficiently high density / particle number density in the core so that the rate of collisions between protons is large enough to sustain the reaction [no ECF] ### Part (b)(i) [2 marks] — Outline - M1: A larger mass produces a stronger gravitational compression of the core, raising the core temperature and pressure [no ECF] - M2: Only above a minimum mass is the core temperature high enough for protons to overcome Coulomb repulsion / for fusion to be self-sustaining against radiative losses [no ECF] ### Part (b)(ii) [3 marks] — Calculate - M1: Enclosed mass under uniform density: M_enc = M × (r/R)³ = M × (1/2)³ = M/8 = 2.5 × 10²⁹ kg [no ECF] - M2: Correct substitution into g = GM_enc/r² with r = R/2 = 3.5 × 10⁸ m: g = (6.67 × 10⁻¹¹)(2.5 × 10²⁹)/(3.5 × 10⁸)² [ECF from M1] - M3: g ≈ 1.4 × 10² N kg⁻¹ (accept 1.3–1.4 × 10²) [ECF from M1, M2] ### Part (c) [3 marks] — Show that - M1: Equating energies: (3/2)k_B T_c = GMm_p / R [no ECF] - M2: Rearranging: T_c = 2GMm_p / (3k_B R) and substituting values: T_c = 2(6.67 × 10⁻¹¹)(2.0 × 10³⁰)(1.67 × 10⁻²⁷) / [3(1.38 × 10⁻²³)(7.0 × 10⁸)] [no ECF] - M3: T_c ≈ 1.54 × 10⁷ K, consistent with 1.5 × 10⁷ K to 2 sf [no ECF] ### Part (d) [2 marks] — Evaluate (position + supporting + limiting per §4.4.1) - M1 [Position + supporting consideration]: The uniform-density model is a useful first approximation — it correctly predicts g = 0 at the centre, recovers g = GM/R² at the surface, and gives the right order of magnitude for the interior field - M2 [Limiting consideration]: However, real stars exhibit strong central condensation (core density orders of magnitude greater than the envelope), so g(r) does not vary linearly with r as the uniform model predicts; the actual g(r) rises much more steeply near the core and the model misrepresents the internal field profile ### Marker notes - Show-that target in (c): 1.54 × 10⁷ K given to 3 sf; student-derived values 1.5–1.6 × 10⁷ K acceptable. - (b)(ii) alternative: students who note g(r) = GMr/R³ for uniform density and substitute r = R/2 directly to get g = GM/(2R²) earn M1+M2; final numerical answer for M3. - (d) accept any of: neglect of radiation/thermal pressure, omission of temperature gradient, failure to model nuclear-burning core vs radiative/convective zones, failure to predict steep rise of g near small enclosed radii.

Critic verdicts

Iteration 0 — failed

ao_ladder pass

The ERQ obeys the cognitive ladder correctly: (a) AO1/State, (b)(i) AO2/Outline, (b)(ii) AO2/Calculate, (c) AO3/Derive, (d) AO3/Evaluate. Command terms match cognitive levels throughout, and progression from recall through application to higher-order reasoning is intact.
command_structure fail

Part (d) mark scheme violates §4.4.1 AO3 Evaluate structure. The scheme provides M1 (limitation) and M2 (supporting consideration) but lacks an explicit position statement at the start. Per the spec, Evaluate requires 'position + supporting consideration + limiting consideration' in that logical order. The current structure is disorganized: M2 conflates model utility with support, and neither bullet explicitly states an overall evaluative judgment (e.g., 'The model is unrealistic because...' or 'The model remains valid despite...').

FIX: Restructure Part (d) marks as: M1: Position — 'The uniform-density prediction (linear increase of g with r) is unrealistic for an actual star.' M2: Supporting consideration — 'In reality, stellar density is highly centrally condensed, so the core has much higher density than the envelope, causing g to deviate significantly from linearity and peak below the surface.' M3: Limiting consideration — 'However, the model provides a useful first-order approximation, correctly predicting g = 0 at the centre and giving the right order of magnitude for surface field strength, making it pedagogically valuable despite its physical limitations.' Alternatively, condense to 2 marks by combining position + limitation in M1 and position + supporting consideration in M2, ensuring both perspectives are explicit.
cross_theme pass

Both E.5 (Fusion and stars) and D.1 (Gravitational fields) are genuinely integrated throughout the question. Part (b)(ii) requires the gravitational field formula from D.1 to calculate a physically meaningful quantity (field strength at a point in the core), and this result directly feeds into part (c) where the ratio of gravitational fields is derived. Part (d) evaluates the realism of the uniform-density model using insights from both topics. Neither topic is name-checked as filler; both are essential to solving the question.
discriminator pass

Part (d) explicitly asks students to 'Evaluate whether this prediction is realistic for an actual star,' requiring a critique of the uniform-density model's assumptions. The mark scheme confirms both a limitation (density profile is non-linear in reality) and supporting evidence (model utility as approximation), fulfilling the model-critique dimension.
rule_of_one pass

Each part's mark count matches the number of award criteria. Part (a) 2 marks / 2 bullets; (b)(i) 2 marks / 2 bullets; (b)(ii) 3 marks / 3 bullets (formula, evaluation, answer); (c) 3 marks / 3 bullets (formula, simplification, numerical result); (d) 2 marks / 2 bullets (limitation, supporting). The dimensional contract is satisfied throughout.
topic pass

The ERQ genuinely integrates both topics. E.5 (Fusion and stars) dominates parts (a)–(b)(i) and (c)–(d) through fusion conditions, quantum tunneling, and stellar structure. D.1 (Gravitational fields) is substantively applied in (b)(ii) and (c), where the interior gravitational field formula g(r) = GMr/R³ is used to calculate field strength and compare it to surface gravity. The hydrostatic equilibrium concept in the stem explicitly links both topics, and part (d) evaluates the model's realism using gravitational density structure—requiring synthesis of both. Neither topic is superficial.

Iteration 1 — failed

ao_ladder pass

The ERQ follows the cognitive ladder correctly: Part (a) is AO1 (State), Part (b) is AO2 (Outline/Calculate), Part (c) is AO3 (Derive), and Part (d) is AO3 (Evaluate). Command terms are appropriate to cognitive level, progression is logical, and the final part explicitly requires two-sided evaluative reasoning per IB rubric §4.4.1.
command_structure fail

Part (d) mark scheme violates §4.4.1 Evaluate structure. M1 and M2 are labelled 'Position + supporting' and 'Position + limiting' respectively, but they describe the SAME position (uniform density is unrealistic). A valid Evaluate structure requires ONE position statement plus supporting AND limiting considerations—not two separate positions. The scheme conflates structure with content.

FIX: Restructure Part (d) as: M1 [Position + supporting consideration]: 'The uniform-density model is physically unrealistic because real stars have strong central condensation with core density orders of magnitude greater than envelope density. However, it correctly predicts g = 0 at centre and gives surface field strength of the right order of magnitude.' M2 [Limiting consideration]: 'The model fails to capture the actual density profile, so g does not increase linearly with r and does not reach maximum at the surface. It also neglects the role of thermal/radiation pressure and temperature gradients that stabilise real stars.' This structure presents one position with both supporting merit and explicit limitation, conforming to §4.4.1.
cross_theme pass

Both E.5 (Fusion and stars) and D.1 (Gravitational fields) are genuinely integrated throughout the question. Part (b)(ii) explicitly requires gravitational field calculation inside the star (D.1 core concept). Part (c) derives gravitational pressure by combining surface gravitational field strength (D.1) with fusion-driven hydrostatic equilibrium (E.5). Part (d) evaluates the uniform-density assumption in the context of real stellar structure, reinforcing the interplay. Neither topic is merely name-checked; both are essential to the physical reasoning.
discriminator pass

Part (d) explicitly requires evaluation of the uniform-density assumption, asking students to critique the model by identifying both its limitations (central condensation, unrealistic density profile) and its merits (pedagogical utility, correct boundary conditions). This satisfies the dimension's requirement for model critique in the final part.
rule_of_one fail

Part (b)(ii) is marked as 3 marks but contains only 2 distinct award criteria (M1: formula derivation; M2: substitution; M3: numerical answer). The mark scheme lists 3 bullets, but M2 and M3 together form a single 'Calculate' operation (substitution + answer), violating the requirement that a 3-mark Calculate part must have exactly 3 independent operations. Part (d) has 2 marks but lists 2 distinct evaluative criteria (position+support + position+limit), which is acceptable.

FIX: Restructure Part (b)(ii) to either: (1) split into two separate parts (b)(ii) for formula [2 marks] and (b)(iii) for calculation [3 marks], or (2) rewrite as a 2-mark part if only formula+substitution are essential, or (3) add an intermediate step (e.g., M1: formula, M2: compute enclosed mass, M3: substitute and compute answer) to genuinely justify 3 distinct operations. Alternatively, redesign the mark scheme so that M2 and M3 are truly independent (e.g., M2 = substitution of r=R/2, M3 = final numerical answer with units), but currently the separation is artificial.
topic pass

The ERQ robustly integrates both topics. E.5 (Fusion and stars) is the primary focus: parts (a) and (b)(i) address fusion conditions and the role of temperature in overcoming Coulomb repulsion. D.1 (Gravitational fields) is genuinely embedded: part (b)(ii) requires calculating gravitational field strength inside a uniform sphere; part (c) derives gravitational pressure by combining surface gravitational field (D.1) with fusion-driven hydrostatic equilibrium (E.5); part (d) evaluates the uniform-density assumption's impact on gravitational modelling. Neither topic is superficial—both are conceptually necessary to complete the question coherently.

Iteration 2 — failed

ao_ladder pass

The ERQ strictly obeys the cognitive ladder. Part (a) uses State (AO1), part (b)(i) uses Outline (AO2), part (b)(ii) uses Calculate (AO2), part (c) uses Show that (AO3), and part (d) uses Evaluate (AO3). Each command term matches its Bloom tier, with progression from knowledge to application to analysis/evaluation.
command_structure fail

Part (d) mark scheme violates §4.4.1 requirement for Evaluate command term. The scheme provides position + supporting consideration + limiting consideration (correct structure), but M1 conflates these three elements into a single bullet, making it impossible to award partial credit (1 mark) if a student provides only position + support or only position + limitation. §4.4.1 requires each element to be independently markable. Additionally, M2 is phrased as a single limiting consideration without explicit scaffolding of how a student earning only M2 (without M1) would be credited.

FIX: Restructure Part (d) mark scheme to separate the three required Evaluate elements into independently awardable bullets: M1 [Position]: 'The uniform-density model is a useful first approximation.' M2 [Supporting consideration]: 'It correctly predicts g = 0 at centre, g = GM/R² at surface, and gives correct order of magnitude for interior field / it captures the linear g(r) ∝ r behaviour expected in a uniform sphere.' M3 [Limiting consideration]: 'Real stars have central condensation (core density >> envelope density), so actual g(r) rises steeply near core; the model predicts g(r) ∝ r but reality is much steeper / model misrepresents internal field profile.' This allows students to earn 1–2 marks for partial Evaluate responses and ensures the mark scheme structure matches the command term requirement.
cross_theme pass

The ERQ genuinely integrates both E.5 (Fusion and stars) and D.1 (Gravitational fields) throughout the problem. Topic E begins the stem and dominates (a), (b)(i), (c); topic D is used substantively in (b)(ii) where gravitational field calculation is central, and in (c) where gravitational potential energy is equated to thermal kinetic energy as the core physical mechanism driving fusion conditions. Neither topic is name-checked only—both are actively deployed in the solution chain.
discriminator pass

Part (d) explicitly asks students to 'Evaluate the validity of modelling the star as a sphere of uniform density when predicting the variation of gravitational field strength g(r) with radius inside the star.' This is a clear model-critique question requiring assessment of assumptions and limitations (real stars have central condensation, g(r) does not vary linearly), not a calculation. It meets the ERQ dimension's contract.
rule_of_one fail

Part (b)(ii) is marked for 3 marks but the mark scheme lists only 2 distinct award criteria (M1: enclosed mass calculation; M2: substitution; M3: final answer). However, M3 is listed as dependent on M1 and M2 via ECF, making it technically a single outcome (numerical answer) rather than an independent operation. The issue is that M2 and M3 together collapse into a single 'substitute and calculate' step, not two separate operations. A proper 3-mark Calculate part requires: formula/setup, substitution with values, and answer-with-units as three distinct checkpoints.

FIX: Restructure Part (b)(ii) mark scheme to explicitly separate: M1 = derive/state enclosed mass formula and calculate M_enc = M/8; M2 = substitute both M_enc and r = R/2 into g = GM_enc/r² showing full working; M3 = obtain final numerical answer g ≈ 1.4 × 10² N kg⁻¹ with units. Alternatively, reduce the part to 2 marks if only formula application and final answer are intended.
topic pass

The ERQ effectively integrates both topics. E.5 (Fusion and stars) is primary and substantive: parts (a), (b)(i), and (c) directly test conditions for hydrogen fusion, minimum mass requirements, and central temperature estimation. D.1 (Gravitational fields) is secondary but genuinely required: part (b)(ii) applies the gravitational field formula to a stellar interior, and part (c) uses gravitational potential energy to estimate temperature. The uniform-density model connects both topics coherently. Neither topic is merely nominal—both are invoked through essential physics concepts.

Planner output (stage 1)

Archetype: theoretical model
Scenario: A star of mass M and radius R maintains hydrostatic equilibrium, where the outward pressure from fusion in the core balances the inward gravitational force. Students explore the conditions necessary for sustained fusion, the gravitational field at the star's surface, and the limitations of simplified models that assume uniform density. The scenario bridges stellar structure (fusion energetics) with gravitational field theory.
Cross-pollination part: b(ii)
Discriminator part: d

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

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