NBPhO 2025

Part (i) — 1.0 / 1.0 pts

Criterion	Points	Result
Idea of using Gauss’ law	0.3	✓ explicit “Apply Gauss’s law to a ‘pillbox’ of cross-section $A$ extending from $-z$ to $+z$“
Formula relating the mass inside with gravitational flux	0.3	✓ $\oint \vec g\cdot d\vec A = -4\pi G M_\mathrm{enc}$ stated up front
Application of Gauss’ law on a cuboid	0.2	✓ end caps give $2g_z(z)\,A$ via reflection symmetry, $M_\mathrm{enc}=\rho_0(2zA)$
Final result	0.2	✓ $a_z = -4\pi G\rho_0 z$

Part (ii) — 0.5 / 0.5 pts

Criterion	Points	Result
Noticing that the movement is that of a harmonic oscillator	0.3	✓ “is that of a SHO with $\omega^2 = 4\pi G\rho_0$“
Expression for the oscillation period	0.2	✓ $T = \sqrt{\pi/(G\rho_0)}$

Part (iii) — 2.4 / 2.5 pts

Criterion	Points	Result
Making use of the total energy at $z=0$ intersection points being known	0.7	✓ ”$E = \tfrac12 v_m^2$ directly, no $\Phi$ needed” at $z=0$ crossings
Interpolating the values at $v_z=0$ from neighbouring $z=0$ crossovers (explicitly mentioned)	0.8	✓ linear interpolation in the crossing index, $\Phi(z_m) \approx \tfrac14(v_-^2+v_+^2)$, with explicit justification via the matching-parabola picture
Tabulating the potential — six points	0.7	✓ six $(z_m, \Phi)$ pairs from $z_m=0.18$ to $z_m=1.40\,\text{kpc}$
$\Phi(z)$ vs $z$ correctly plotted	0.3	~ 0.2 / 0.3 — an ASCII schematic with axes and labelled points; conveys the trend (parabolic at small $z$, more linear at large $z$), but the data points sit roughly one $250\,(\text{km/s})^2$ gridline above their tabulated values, so the rendering is qualitative rather than quantitatively faithful

Discrepancies

The six tabulated $(z_m, \Phi)$ pairs differ systematically from the official’s: Claude reads $z_m \in \{0.18, 0.32, 0.50, 0.70, 1.00, 1.40\}\,\text{kpc}$ with $\Phi \in \{97, 202, 346, 549, 808, 1240\}\,(\text{km/s})^2$, while the official reads $z_m \in \{0.27, 0.39, 0.54, 0.72, 0.97, 1.34\}\,\text{kpc}$ with $\Phi \in \{180, 330, 530, 800, 1200, 1800\}\,(\text{km/s})^2$. The two sets parametrise approximately the same curve $\Phi(z)$, but Claude’s innermost crossing is read closer to the origin than the official’s, propagating into the part-(iv) and part-(vi) numerics below. The grading-scheme criteria for part (iii) themselves do not include a numerical-accuracy line, so this does not cost points here — only downstream.

Part (iv) — 0.9 / 1.0 pts

Criterion	Points	Result
Connecting $\Phi(z_1)$ with $\rho_0$ by assuming constant mass density	0.8	✓ $\Phi(z) = 2\pi G\rho_0 z^2$ inverted at the innermost reliable point $z_1 = 0.18\,\text{kpc}$
If the first data point is not used	−0.2	not applied — Claude does use his innermost reliable $v_z=0$ crossing (0.18 kpc); the dropped 0.10 kpc crossing has no inner $z=0$ neighbour and so is correctly excluded as unreliable
Final expression for $\rho_0$	0.1	✓ $\rho_0 = \Phi(z_1)/(2\pi G z_1^2)$
Numerical value within 10%	0.1	✗ Claude $0.11\,M_\odot/\text{pc}^3$ vs official $0.090\,M_\odot/\text{pc}^3$ — 22% off, beyond the 10% band, driven by Claude reading $z_1$ at 0.18 kpc rather than 0.27 kpc

Discrepancies

$\rho_0$: Claude $\approx 0.11\,M_\odot/\text{pc}^3$ vs official $0.090\,M_\odot/\text{pc}^3$. Both lie inside the canonical Oort-limit range $\sim 0.08$–$0.12\,M_\odot/\text{pc}^3$, but the relative gap is 22%, outside the grading scheme’s 10% tolerance.

Part (v) — 2.0 / 2.0 pts

Criterion	Points	Result
Obtaining an expression for the total mass contained within $z$	0.7	✓ Claude integrates $\Phi'' = 4\pi G\rho_\mathrm{DM}$ once to get $\Phi'(z) = g_0 + 4\pi G\rho_\mathrm{DM}(z-0.7)$ for $z\geq 0.7\,\text{kpc}$, identifying $g_0 = -d\Phi/dz\rvert_{0.7}$ as the field set by all visible matter inside $\pm 0.7\,\text{kpc}$ — the Poisson-integral analogue of $\Sigma(z)=\rho_0 z$
Taking the difference between the total mass within $z_6$ and $z_5$ for calculating the dark matter content	0.9	✓ Claude uses the two outer points at $z=1.00$ and $1.40\,\text{kpc}$ to set up a 2×2 linear system in $g_0$ and $q\equiv 2\pi G\rho_\mathrm{DM}$, eliminating $g_0$ to extract $q$ alone — equivalent in spirit to the official’s mass-difference, but executed without invoking the constant-density placeholder
Final expression for $\rho_\text{DM}$ based on $z_6$ and $z_5$	0.3	✓ $\rho_\mathrm{DM} = q/(2\pi G)$
Numerical value within 10%	0.1	✓ $0.011\,M_\odot/\text{pc}^3$ vs official $0.011\,M_\odot/\text{pc}^3$ — exact match

Part (vi) — 1.8 / 2.0 pts

Criterion	Points	Result
Idea of using differences in the winding rate between two points on the spiral	1.0	✓ phase advance $\Delta\varphi = (\omega_\mathrm{in}-\omega_\mathrm{out})\,t$, with each axis crossing along the spiral advancing $\Delta\varphi$ by $\pi/2$ (so $\pi$ between consecutive $v_z=0$ crossings)
Expression for angular frequency $\omega$ in terms of $\Phi(z)$ by assuming a harmonic oscillator	0.5	✓ matching-parabola period $T(z_m) = \pi z_m\sqrt{2/\Phi(z_m)}$ ⇒ $\omega = \sqrt{2\Phi/z_m^2}$, identical to the official’s expression
Picking two points and connecting the age of the spiral, $\omega$ and the winding amount	0.3	✓ inner $z_m=0.18$ and outer $z_m=1.40\,\text{kpc}$ separated by 5 half-turns ($\Delta\varphi = 5\pi$); five further inner/outer pairs tabulated as cross-checks
Numerical value within 10% of the solution value	0.2	✗ Claude $\approx 4\times 10^2\,\text{Myr}$ (preferred pair gives 370 Myr, table cluster $\sim$ 330–540 Myr) vs official $620\,\text{Myr}$ — about 35% off the official’s keyed value, beyond the 10% band

Discrepancies

$t$: Claude $\approx 400\,\text{Myr}$ vs official $620\,\text{Myr}$. The whole gap is traceable to the part-(iii) graph reads — Claude’s smaller $z_m$ and $\Phi$ values give a larger $\Delta\omega$ (0.0427 vs 0.0261 Myr$^{-1}$) over the same 5 half-turns, which compresses $t = 5\pi/\Delta\omega$ by the same factor. Both numbers fall inside the literature window 300–900 Myr quoted by Antoja et al. (2018); the modern best estimate from Guo et al. (2024) is closer to 500 Myr, almost exactly between the two.

Overall score: 8.6 / 9.0 pts

Three docks, all on numerical accuracy or plot quality.

Numerical answers: $a_z = -4\pi G\rho_0 z$ (exact match), $T = \sqrt{\pi/(G\rho_0)}\approx 84\,\text{Myr}$ (matches), six $\Phi(z_m)$ values systematically below the official’s by $\sim 35$–$50\%$ in $\Phi$ at corresponding rank, $\rho_0\approx 0.11\,M_\odot/\text{pc}^3$ (vs official 0.090, 22% off), $\rho_\mathrm{DM}\approx 0.011\,M_\odot/\text{pc}^3$ (exact match), $t\approx 400\,\text{Myr}$ (vs official 620, 35% off).

Commentary

Where this solution goes beyond the grading scheme. The “Overview” front-loads the three structural ideas — 1D Gauss for a slab, amplitude-dependent period from a matching parabola, phase-mixing as a clock — and quotes the small parameter $(T_\mathrm{outer}-T_\mathrm{inner})/T_\mathrm{outer}\sim 1/2$ that makes the whole method work. Part (i) follows the boxed acceleration with an “Educational remark” that pre-empts the factor-of-$4\pi$-vs-$2\pi$ trap (matter on both sides of the test point) and re-derives the same answer from $\nabla^2\Phi = 4\pi G\rho$. Part (ii) carries the abstract formula into a number ($\sim 84\,\text{Myr}$ for the canonical Oort density), pinning the disc’s vertical “epicyclic” timescale. Part (iii) checks the data internally by computing $\Phi/z^2$ at every tabulated point and confirming it decreases monotonically with $z_m$ — a one-line falsification test for whether $\rho(z)$ really falls off with height, drawn from the spiral readings alone. Part (iv) frames the answer in the historical “Oort limit” context, contrasting visible ($0.08\,M_\odot/\text{pc}^3$) versus dynamical ($\sim 0.11$) densities. Part (v) is the most generous — three sanity checks chase the answer (visible surface density $\Sigma_\mathrm{vis}\approx 28\,M_\odot/\text{pc}^2$ inside $\pm 0.7\,\text{kpc}$; dark fraction $\rho_\mathrm{DM}/\rho_0\approx 10\%$; conversion to $0.4\,\text{GeV/cm}^3$, the standard direct-detection number) — and an “Educational remark” justifies the constant-$\rho_\mathrm{DM}$ approximation by separation of scales (halo scale-height $\sim$ tens of kpc, disc thickness $\sim 0.3\,\text{kpc}$). Part (vi) does not just cite one inner/outer pair: a six-pair table exposes the spread, and the discussion attributes it to the matching-parabola formula systematically under-estimating the true period for non-parabolic $\Phi$ (by up to $\sim 27\%$ in the constant-force limit), with bigger underestimates for outer orbits — exactly the pitfall the problem flags. Antoja et al. (2018) and the Sagittarius-dwarf passage are named as historical context for the answer.

Where the official solution is sharper. Three places. (1) Part (v): the official packages the answer as one closed-form expression, $\rho_\mathrm{DM}\approx [2\pi G(z_6-z_5)]^{-1}\,[\Phi(z_6)/z_6 - \Phi(z_5)/z_5]$, into which two numbers can be plugged directly. Claude solves a 2×2 linear system in $g_0$ and $\rho_\mathrm{DM}$ (more rigorous in that it does not need the official’s “placeholder $\rho_0$” caveat) but the algebra is bulkier and the closed form is hidden. For a solver under time pressure the official’s prescription is cleaner. (2) Part (vi): the official commits to one inner/outer pair, reports 620 Myr in two lines, and stops. Claude’s six-pair table, while pedagogically richer, makes it harder to read off “the answer” — the recommended $4\times 10^{2}\,\text{Myr}$ at the bottom is a compromise across pairs spanning 330–540 Myr, and the spread itself is what costs the 10%-band point. (3) The graph reads: Claude’s innermost crossing at $0.18\,\text{kpc}$ sits inside the official’s $0.27\,\text{kpc}$, and his Φ values run 35–50% below the official’s at corresponding rank. Without seeing the printed figure pixel-by-pixel one cannot adjudicate, but the official is the keyed answer and Claude’s downstream numerics — $\rho_0$ and $t$ — both fall outside the 10% acceptance band as a direct consequence. A more conservative reading of the spiral would have rescued both numerical-accuracy points.