Past exam of the mathematics course of the University of Cambridge 2008 iii Paper 72 4 ii Solution Created 2026-10-03 Updated 2026-10-14
Interpret the model comparison at the exam's 2008 epoch. A simulation's failure depends on its treatment of baryons, resolution, and adopted stellar feedback; it is not a theorem that the Lambda-CDM model cannot produce the observation under any physical implementation. The following were important tensions for then-standard realizations, with different implications for monolithic galaxy formation.
Old, alpha-enhanced massive spheroids and downsizing. The most massive quiescent elliptical galaxies tend to contain older stellar populations and elevated alpha element abundances relative to iron, indicating that much of their star formation occurred early and rapidly. Prompt core-collapse enrichment followed by shutdown before substantial delayed iron enrichment accounts for the abundance pattern. Models that spread a massive galaxy's star formation over many late progenitor-merger episodes can make its stars too young or insufficiently alpha enhanced. A rapid early dissipative collapse naturally gives a short formation interval and is therefore more immediately consistent with this particular observation. However, downsizing in galaxy formation is about stellar formation, not necessarily final assembly: late dry galaxy mergers can assemble already old alpha-enhanced stars. Early quenching and an appropriate chemical-enrichment model can consequently reconcile hierarchical assembly with the fossil record.
Thin, high-angular-momentum stellar discs. The existence of extended thin galactic disks, including systems with small galactic bulges, challenged simulations that produced compact central stellar concentrations or overly thick discs. Premature gas cooling into small progenitors, transfer of specific angular momentum to the halo, and repeated dynamical perturbations could cause the galactic angular-momentum catastrophe; satellite encounters also increase galactic disk heating. A comparatively quiet dissipative collapse with approximate conservation of baryonic specific angular momentum more readily produces an extended rotating disc, and avoiding subsequent strong perturbations helps retain thinness. Nevertheless, rapid collapse alone does not explain a disc's prolonged star formation and gas replenishment. Hierarchical histories with quieter late accretion, better feedback, delayed cooling, and dissipative disc regrowth can reduce these tensions; a problem in early implementations is not an unavoidable property of the cosmological framework.
Cores inferred in low-mass galaxies. Resolved inner galaxy rotation curves in a number of low-surface-brightness and dwarf systems favoured shallower central mass densities than the cusps of collisionless cold-dark-matter simulations. For a roughly constant-density dark-matter core, and ; an inner Navarro--Frenk--White profile has , hence and . The differing slopes explain the cusp--core problem. Beam smearing, inclination, and noncircular motions complicate inference, but do not make the issue merely a normalization mismatch. A simple early dissipative collapse does not automatically solve it: baryonic contraction can deepen the central concentration. Potential fluctuations from gas loss and stellar feedback can instead redistribute dark matter and modify a cusp. Thus this observation tests the joint halo-and-baryon model rather than preferentially vindicating the old collapse picture.
Past exam of the mathematics course of the University of Cambridge 2009 iii Paper 65 4 vii Solution Created 2026-10-03 Updated 2026-10-14
Massive stars rapidly return oxygen, magnesium, neon and other alpha elements through core-collapse supernovae, whereas Type Ia supernovae supply a substantial delayed iron contribution. If the galactic bulge and stellar halo formed much of their stellar mass rapidly, their stars locked in enhanced alpha-element to iron ratios before much delayed iron enrichment, while prolonged star formation in the local galactic disk included more iron-rich gas. The abundance contrast chiefly records different enrichment timescales, with the initial mass function and gas exchange also affecting the ratios.