Galactic Rotation Curves

Galactic Rotation Curves

Stars and gas orbit the center of a galaxy under gravity. The circular velocity $v_c(r)$ at radius $r$ depends on the total mass $M(r)$ enclosed within that radius:

$v_c(r) = \sqrt{\frac{G\,M(r)}{r}}$

Three Regimes

The Dark Matter Problem

Observations of spiral galaxies show that $v_c(r) \approx$ const far beyond the visible disk — a flat rotation curve. This cannot be explained by visible matter alone. For a flat curve, the enclosed mass must grow as:

$M(r) = \frac{v_c^2\,r}{G}$

This implies vast amounts of unseen dark matter in an extended halo.

The Milky Way

Our Galaxy has a roughly flat rotation curve with $v_c \approx 220$ km/s at the Sun's position $r \approx 8.5$ kpc. Using $1\text{ kpc} = 3.086 \times 10^{19}$ m, the enclosed mass within 8.5 kpc is roughly $10^{11}\,M_\odot$.

Your Task

Implement three functions. All constants must be defined inside each function.

• circular_velocity_m_s(M_enclosed_kg, r_m) — returns $v_c = \sqrt{G M / r}$ in m/s
• enclosed_mass_flat_rotation(v_c_m_s, r_m) — returns $M = v_c^2 r / G$ in kg (mass implied by a flat rotation curve)
• kpc_to_m(kpc) — returns distance in metres ($1\text{ kpc} = 3.086 \times 10^{19}$ m)

Use $G = 6.674 \times 10^{-11}$ N m² kg⁻².