NBPhO 2025

Difficulty hard

Prerequisites

• Newtonian gravity and gravitational potential
• Gauss's law / Poisson equation $\nabla^2\Phi = 4\pi G\rho$
• Simple harmonic motion and conservation of energy
• Phase space $(z, v_z)$ and orbital phase angle
• Linear interpolation of tabulated data

Learning objectives

• Apply Gauss's law to a horizontally infinite, mirror-symmetric slab to obtain $a_z = -4\pi G\rho_0 z$
• Derive the amplitude-dependent period $T(z_m) = \pi z_m\sqrt{2/\Phi(z_m)}$ from a matching-parabola approximation
• Reconstruct $\Phi(z)$ from a phase-space spiral by interpolating $E = \tfrac12 v_z^2 + \Phi(z)$ between adjacent axis crossings
• Invert Poisson's equation $d^2\Phi/dz^2 = 4\pi G\rho$ on numerical $\Phi(z)$ data to extract a local density
• Separate the dark-matter contribution by exploiting that the halo is uniform on the disc scale
• Date a galactic perturbation from the differential phase advance of stars at different amplitudes

Watch out for

• Factor-of-two trap in Poisson: the slab field is $a_z = -4\pi G\rho_0 z$, not $-2\pi G\rho_0 z$. The $4\pi$ comes from contributions on both sides of the test point; using $2\pi$ (the value for a single sheet) gives a period off by $\sqrt 2$.
• Matching-parabola conversion: with $k = \Phi(z_m)/z_m^2$, the SHO has potential $kz^2$ (not $\tfrac12 kz^2$), so $\omega^2 = 2k$ and $T = 2\pi/\sqrt{2k}$. Forgetting the factor of $2$ gives the period off by $\sqrt 2$.
• Each consecutive $v_z=0$ crossing along the spiral advances the orbital phase by $\pi$ (a half-period), not $2\pi$. Counting full turns instead of half-turns underestimates $t$ by a factor of two.
• Linear interpolation of $E$ along the spiral works in the crossing index, not in $z$ or $v_z$. Interpolating Φ(z) directly between $v_z=0$ crossings would miss the energy information carried by the $z=0$ crossings, which is the entire point of the method.
• The matching-parabola period systematically under-estimates the true period for non-parabolic $\Phi$ — by up to $\sim 27\%$ in the constant-force limit. So part-(vi) estimates that include the largest $z_m$ tend to come out high; cross-check with multiple inner/outer pairs.
• Part (v) needs two outer points (or one outer point plus the slope at $z = 0.7\,\text{kpc}$): a single outer datum is not enough to disentangle $g(0.7)$ from $\rho_\text{DM}$.