Einasto Profile

The Einasto profile (or Einasto model) is a three-parameter, spherically symmetric mass-density law introduced by Estonian astrophysicist Jaan Einasto in 1965 to model the three-dimensional spatial distribution of stellar populations in galaxies—including the Milky Way and the Andromeda Galaxy (M31)—and subsequently established as a high-precision analytical description of cold dark matter halos in ΛCDM cosmological simulations. Mathematically, the Einasto profile is the three-dimensional spatial analog of José Luis Sérsic's 1963 R^1/n law for two-dimensional projected surface brightness: instead of transitioning between two fixed asymptotic power-law slopes like the Navarro–Frenk–White (NFW), Hernquist profile, or Dehnen profile models, the Einasto profile defines a density distribution whose logarithmic slope γ(r) = −d ln ρ / d ln r steepens continuously with radius as a power law, (r/r₋₂)^α. Because of this curved logarithmic slope, the Einasto model possesses both a finite central density ρ(0) and a finite total mass M_tot for all positive shape parameters α > 0.

An Einasto sphere is defined by the differential requirement that the logarithmic slope of its three-dimensional space density ρ(r) be a power law of galactocentric radius r with Einasto shape parameter α > 0 (often expressed via the Einasto index n = 1/α):

d ln ρ(r) / d ln r = −2 (r / r₋₂)^α

where r₋₂ (also denoted r_s) is the characteristic scale radius at which the logarithmic density slope equals the isothermal value −2 (so that r²ρ(r) reaches its global maximum at r = r₋₂). Integrating this differential equation with respect to ln r and normalizing by the density ρ₋₂ = ρ(r₋₂) at the scale radius yields the canonical form of the Einasto density profile:

ρ(r) = ρ₋₂ exp{ −(2/α) [(r/r₋₂)^α − 1] }

Evaluating ρ(r) at the origin (r = 0) gives a finite central density ρ₀ = ρ(0) = ρ₋₂ exp(2/α), allowing the profile to be written in terms of its central density as ρ(r) = ρ₀ exp[−(r/h)^α], where h = (α/2)^1/αr₋₂. Small values of α (α ≈ 0.14–0.20, n ≈ 5–7) produce a gradual logarithmic steepening over several decades in radius typical of cold dark matter halos, whereas α = 1 recovers an exp(−2r/r₋₂) exponential decay and α = 2 recovers a Gaussian distribution exp(−r²/r₋₂²).

Unlike the NFW profile—whose outer ρ ∝ r⁻³ slope causes the cumulative mass integral to diverge logarithmically at large radii—the super-exponential decay exp[−(2/α)(r/r₋₂)^α] of the Einasto profile ensures a finite total mass M_tot for any α > 0. Substituting the dimensionless transformed variable x(r) = (2/α)(r/r₋₂)^α into the spherical mass integral M(r) = ∫₀^r 4πr′²ρ(r′) dr′ expresses the cumulative enclosed mass in terms of the regularized lower incomplete gamma function P(k, x) = γ(k, x)/Γ(k):

M(r) = M_tot P(3/α, (2/α)(r/r₋₂)^α)

where Γ(3/α) is the complete Euler gamma function and the finite total mass of the Einasto sphere is:

M_tot = 4π ρ₋₂ r₋₂³ (e^2/α/α) (α/2)^3/α Γ(3/α)

Applying Newton's shell theorem yields the closed-form gravitational potential Φ(r) in terms of the lower and upper regularized incomplete gamma functions P(k, x) and Q(k, x) = 1 − P(k, x):

Φ(r) = −(GM_tot/r) { P(3/α, x) + [x^1/α Γ(2/α)/Γ(3/α)] Q(2/α, x) }

At the center (r → 0), the potential approaches a finite depth Φ(0) = −(GM_tot/r₋₂)(2/α)^1/α Γ(2/α)/Γ(3/α). The circular velocity curve v_c²(r) = GM(r)/r reaches its peak value v_max at radius r_max ≈ 2.163*r*₋₂ for α = 0.17.

Following the introduction of the two-parameter NFW profile by Julio Navarro, Carlos Frenk, and Simon White (1996, 1997), high-resolution cosmological N-body experiments—including Navarro et al. (2004), Merritt et al. (2006), and the Aquarius Project (Springel et al., 2008)—demonstrated that dark-matter halos formed via hierarchical clustering and violent relaxation do not converge to a fixed inner power-law cusp. Instead, the logarithmic slope γ(r) = −d ln ρ / d ln r decreases monotonically toward the center in agreement with the Einasto power law γ(r) = 2(r/r₋₂)^α down to the innermost resolved radii (r < 0.001*r*₂₀₀). Across cosmological simulations, α increases systematically with primordial density peak height ν = δ_c/σ(M, z) as α(ν) ≈ 0.155 + 0.0095ν². While the Einasto profile lacks the kiloparsec-scale constant-density cores modeled by the Burkert profile in baryon-dominated dwarf galaxies, its finite central density eliminates singular cusps and provides accurate predictions for dark-matter annihilation J-factors and gravitational lensing convergence maps.

Projecting the three-dimensional Einasto density ρ(r) along the line of sight yields a surface mass density Σ(R) expressible in terms of Fox's H-function or Meijer's G-function (Retana-Montenegro et al., 2012), which is closely approximated across all radii by a two-dimensional Sérsic profile with Sérsic index N ≈ 0.90(1/α) − 0.35. This correspondence explains why elliptical galaxies whose projected starlight obeys a de Vaucouleurs R^1/4 law (N = 4) possess three-dimensional stellar mass distributions following an Einasto profile with α ≈ 0.20–0.22, whose kinematics are modeled via the Jeans equations, collisionless Boltzmann equation, and Eddington inversion formula.

How does the Einasto profile differ from the Sérsic profile?

Both profiles share the functional form exp[−b(r/r₀)^1/n]. However, the Sérsic profile describes the two-dimensional sky-projected surface brightness I(R) on the plane of the sky, whereas the Einasto profile describes the three-dimensional spatial mass density ρ(r) as a function of spherical radius r.

Why is the scale radius written as r_(-2) in the Einasto profile?

The scale radius r₋₂ is defined as the unique radius where the logarithmic density slope d ln ρ / d ln r equals −2, which matches the slope of a singular isothermal sphere and marks the radius where r²ρ(r) reaches its peak. Using (r₋₂, ρ₋₂) makes the Einasto parameters directly comparable to the scale radius r_s of an NFW profile.

Does an Einasto halo have a central cusp or a finite central density?

At r = 0, the Einasto density evaluates to a finite value ρ(0) = ρ₋₂ exp(2/α) and its radial derivative vanishes (dρ/dr → 0 for α > 0). For cold dark matter values α ≈ 0.16–0.17, the central density ratio exp(2/α) ≈ 10⁵ is large and the logarithmic slope remains γ(r) ≈ 0.8–1.0 at r = 0.01*r*₋₂, so the profile rises steeply across observable galactocentric radii without diverging at the origin.

1. Navarro, J. F., Hayashi, E., Power, C., Jenkins, A. R., Frenk, C. S., White, S. D. M., Springel, V., Stadel, J., & Quinn, T. R. (2004). "The inner structure of ΛCDM halos — III. Universality and asymptotic slopes." Monthly Notices of the Royal Astronomical Society, 349(3), 1039–1051. https://doi.org/10.1111/j.1365-2966.2004.07586.x
2. Merritt, D., Graham, A. W., Moore, B., Diemand, J., & Terzić, B. (2006). "Empirical models for dark matter halos. I. Nonparametric construction of density profiles and comparison with parametric models." The Astronomical Journal, 132(6), 2685–2700. https://doi.org/10.1086/508988

3.Springel, V., Wang, J., Vogelsberger, M., Ludlow, A., Jenkins, A., Helmi, A., Navarro, J. F., Frenk, C. S., & White, S. D. M. (2008). "The Aquarius Project: the subhaloes of galactic haloes." Monthly Notices of the Royal Astronomical Society, 391(4), 1685–1711. https://doi.org/10.1111/j.1365-2966.2008.14066.x

1. Dutton, A. A., & Macciò, A. V. (2014). "Cold dark matter haloes in the Planck era: evolution of structural parameters for Einasto and NFW profiles." Monthly Notices of the Royal Astronomical Society, 441(4), 3359–3374. https://doi.org/10.1093/mnras/stu742
2. Retana-Montenegro, E., Van Hese, E., Gentile, G., Baes, M., & Frutos-Alfaro, F. (2012). "Analytical properties of Einasto dark matter haloes." Astronomy & Astrophysics, 540, A70. https://doi.org/10.1051/0004-6361/201118543