Burkert Profile

The Burkert profile (or Burkert halo model) is an empirical, spherically symmetric dark matter density law introduced by German astrophysicist Andreas Burkert in 1995 to model the H I 21-centimeter and Hα rotation curves of dark-matter-dominated dwarf irregular galaxies, dwarf spheroidal satellites, and low-surface-brightness (LSB) spiral galaxies. While dark-matter-only cosmological simulations in the standard ΛCDM paradigm predict that collisionless halos relaxing via violent relaxation develop steep inner density cusps—such as the ρ ∝ r⁻¹ inner cusp of the Navarro–Frenk–White (NFW) and Hernquist profile models or the curved power-law slope of the Einasto profile—high-resolution kinematic observations of gas-rich dwarf galaxies consistently reveal linearly rising inner rotation curves (v_c(r) ∝ r) indicative of a constant-density dark-matter core. Burkert formulated a two-parameter analytical density law that transitions smoothly from a flat central core (ρ ≈ ρ₀) inside scale radius r₀ to an NFW-like ρ ∝ r⁻³ outer decline at large galactocentric radii, establishing the empirical foundation for the core–cusp problem in galaxy formation.

A spherical Burkert dark-matter halo is parameterized by its central core density ρ₀ and its core radius r₀ (within which the space density drops to one-fourth of its central value, ρ(r₀) = ¼ρ₀):

ρ(r) = (ρ₀r₀³) / [(r + r₀)(r² + r₀²)]

In terms of the dimensionless radius x = r/r₀, the logarithmic density slope γ(x) = −d ln ρ / d ln x = x/(1 + x) + 2x²/(1 + x²) transitions across three radial zones:

• Inner Constant-Density Core (r ≪ r₀, x ≪ 1): Near the galactic center, ρ(r) ≈ ρ₀(1 − r/r₀) approaches a constant ρ₀ with γ(0) = 0, resembling the flat inner core of a non-singular isothermal King profile or Plummer model.
• Intermediate Isothermal Zone (r ≈ r₀ to 3r₀): Across intermediate radii, the factor (r² + r₀²)⁻¹ mimics a pseudo-isothermal halo (ρ ∝ r⁻²) with γ(1) = 3/2 and γ(1.52) = 2, producing a nearly flat rotation curve over the optical disk radius.
• Outer Cosmological Halo (r ≫ r₀, x ≫ 1): At large radii compared to the core radius, ρ(r) ≈ ρ₀(r₀/r)³ falls off as r⁻³ (γ → 3), matching the outer asymptotic behavior of cosmological NFW halos rather than the r⁻² divergence of a singular isothermal sphere or the r⁻⁴ falloff of the Dehnen profile family.

Integrating 4πr′²ρ(r′) from the center to radius r = xr₀ via partial fractions yields the closed-form cumulative enclosed mass profile M(r):

M(r) = πρ₀r₀³ { ln[(1 + x)²(1 + x²)] − 2 arctan(x) } where x = r/r₀

Evaluating M(r) at the core radius (r = r₀, x = 1) gives the core mass M₀ = M(r₀) = πρ₀r₀³(ln 8 − π/2) ≈ 1.6005ρ₀r₀³. At small radii (x ≪ 1), M(r) ≈ (4π/3)ρ₀r³, so the circular orbital velocity v_c(r) = [GM(r)/r]^1/2 rises linearly with radius (solid-body rotation, v_c(r) ≈ (4πGρ₀/3)^1/2r). The circular velocity curve reaches its global maximum v_max at radius r_max ≈ 3.2446*r*₀ with v_max ≈ 0.8636(4πGρ₀r₀²)^1/2. Beyond r_max, the enclosed mass diverges logarithmically as M(r) ≈ 4πρ₀r₀³ ln(r/r₀) and is truncated at the virial radius r₂₀₀.

Applying Newton's shell theorem to integrate the inward gravitational field g(r) = −GM(r)/r² yields the analytical gravitational potential Φ(r) of the Burkert sphere (with Φ(∞) = 0):

Φ(r) = −πGρ₀r₀² { π − 2(1 + 1/x) arctan(x) + 2(1 + 1/x) ln(1 + x) − (1 − 1/x) ln(1 + x²) }

At the halo center (x → 0), the potential converges to a finite depth Φ(0) = −π²Gρ₀r₀², corresponding to a central escape speed v_esc(0) = (2^1/2π)(Gρ₀)^1/2r₀. Solving the spherical Jeans equations for a tracer stellar population embedded inside a Burkert halo (such as red-giant stars in Milky Way dwarf spheroidal satellites) allows dynamicists to infer ρ₀ and r₀ from line-of-sight radial velocity dispersions while accounting for orbital velocity anisotropy via the collisionless Boltzmann equation and Eddington inversion formula. Projecting the Burkert density along the line of sight for gravitational lensing gives a central surface mass density Σ(0) = 2 ∫₀^∞ ρ(r) dr = (π/2 + ln 2)ρ₀r₀ ≈ 2.2639ρ₀r₀.

When Burkert (1995) fitted his profile to the H I rotation curves of dwarf irregular galaxies, he discovered that lower-luminosity galaxies with smaller core radii r₀ systematically possess higher central densities ρ₀. Subsequent analyses by Paolo Salucci and Andreas Burkert (2000) and Donato et al. (2009) across more than 1,000 galaxies spanning 14 magnitudes in luminosity (M_B = −8 to −22) demonstrated that the central dark-matter surface density μ_0D ≡ ρ₀r₀ is nearly constant across galaxy masses and Hubble types:

log₁₀(ρ₀r₀ / [M_☉ pc⁻²]) = 2.15 ± 0.20 (or ρ₀r₀ ≈ 140⁺⁸⁰₋₅₀ M_☉ pc⁻²)

To reconcile the constant-density cores of the Burkert profile with the cuspy inner halos predicted by dark-matter-only ΛCDM simulations, hydrodynamical models invoke bursty star formation and repeated supernova-driven gas outflows that fluctuate the central gravitational potential (∂Φ/∂t ≠ 0) and pump orbital energy into central dark-matter particles (Pontzen & Governato, 2012), alongside particle-physics models of self-interacting dark matter (SIDM) and ultra-light bosonic dark matter.

How does the Burkert profile differ from the pseudo-isothermal sphere?

Both models feature a flat central core ρ(0) = ρ₀ and produce a linearly rising inner rotation curve. However, at large radii (r ≫ r₀) the pseudo-isothermal sphere ρ(r) = ρ₀/(1 + r²/r_c²) falls off as r⁻², whereas the Burkert profile falls off as r⁻³, matching the outer density slope of cosmological NFW halos.

What is the universal central surface density relation for Burkert halos?

Rotation-curve decompositions and dwarf-spheroidal Jeans equations analyses show that the product of the Burkert central density ρ₀ and core radius r₀ is approximately constant across galaxies spanning six orders of magnitude in luminosity: ρ₀r₀ ≈ 140 M_☉ pc⁻² (≈ 0.03 g cm⁻²).

At what radius does the Burkert circular velocity reach its peak?

The circular velocity v_c(r) = [GM(r)/r]^1/2 of a Burkert halo reaches its maximum value v_max ≈ 0.864(4πGρ₀r₀²)^1/2 at r_max ≈ 3.245*r*₀, beyond which v_c(r) declines slowly as [ln(r/r₀)/(r/r₀)]^1/2.

1. Burkert, A. (1995). "The structure of dark matter halos in dwarf galaxies." The Astrophysical Journal Letters, 447(1), L25–L28. https://doi.org/10.1086/309560
2. Salucci, P., & Burkert, A. (2000). "Dark Matter Scaling Relations." The Astrophysical Journal Letters, 537(1), L9–L12. https://doi.org/10.1086/312747
3. Donato, F., Gentile, G., Salucci, P., Frigerio Martins, C., Wilkinson, M. I., Gilmore, G., Grebel, E. K., Koch, A., & Wyse, R. (2009). "A constant dark matter halo surface density in galaxies." Monthly Notices of the Royal Astronomical Society, 397(3), 1169–1176. https://doi.org/10.1111/j.1365-2966.2009.15004.x
4. Pontzen, A., & Governato, F. (2012). "How supernova feedback turns dark matter cusps into cores." Monthly Notices of the Royal Astronomical Society, 421(4), 3464–3471. https://doi.org/10.1111/j.1365-2966.2012.20571.x
5. Bullock, J. S., & Boylan-Kolchin, M. (2017). "Small-Scale Challenges to the ΛCDM Paradigm." Annual Review of Astronomy and Astrophysics, 55, 343–387. https://doi.org/10.1146/annurev-astro-091916-055313