Dark Matter Halo Around a Galaxy — Rotation Curve (3D)

Visible Mass (disk)

Disk mass M_disk 6.0 ×10¹⁰ M☉

Disk scale radius r_d 3.0 kpc

Dark Matter Halo

Halo core density ρ₀ 1.0 ×10⁷ M☉/kpc³

Halo core radius r_c 5.0 kpc

Selected Star

Orbital radius r

— kpc

Orbital speed v

— km/s

v∞ of halo alone

0 km/s

Peak visible-only v

0 km/s

How it works

Circular orbital speed from enclosed mass: v(r) = √(GM(r)/r). The visible exponential disk contributes M_vis(r) = M_disk[1 − (1+r/r_d)e^−r/r_d], which saturates at large r — so v_vis(r) falls off as 1/√r, exactly the Keplerian decline you'd expect if starlight traced all the mass.

Observed spiral-galaxy rotation curves instead stay flat far outside the visible disk. A spherical dark-matter halo with a pseudo-isothermal density profile ρ(r) = ρ₀ / (1 + (r/r_c)²) explains this: its enclosed mass grows almost linearly at large r, so v_halo(r) approaches a constant v_∞ = √(4πGρ₀r_c²) instead of falling off. Toggle the halo off to see the curve revert to the Keplerian 1/√r falloff — the actual historical puzzle (Rubin, Freeman, Bosma, 1970s) that motivated the dark-matter halo hypothesis.

v_total(r) = √( v_vis(r)² + v_halo(r)² )   [halo on]
v_total(r) = v_vis(r)                       [halo off]

Visible disk stars Halo extent (translucent)

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