Hernquist Profile

The Hernquist profile (or Hernquist model) is an analytical, spherically symmetric gravitational potential–density pair introduced by American astrophysicist Lars Hernquist in 1990 to model the three-dimensional mass distribution, kinematics, and phase-space structure of elliptical galaxies, classical galactic bulges, and cosmological dark matter halos. Following Gerard de Vaucouleurs's 1948 discovery that the sky-projected surface brightness of giant elliptical galaxies obeys the empirical R^1/4 law, dynamical studies were hindered because deprojecting the R^1/4 profile into three dimensions yields an integral with no closed-form expression for the space density ρ(r) or gravitational potential Φ(r). Hernquist resolved this limitation by formulating a rational potential–density pair whose density scales as ρ(r) ∝ r⁻¹ in the inner core and transitions smoothly to ρ(r) ∝ r⁻⁴ in the outer halo—matching the asymptotic behavior produced by incomplete violent relaxation while enabling closed-form evaluation of the Jeans equations and the isotropic phase-space distribution function via the Eddington inversion formula.

A spherical Hernquist galaxy or halo of total mass M and characteristic scale length a is defined by the three-dimensional mass density distribution:

ρ(r) = (Ma) / [2πr(r + a)³]

where r is the spherical galactocentric radius. This profile is the γ = 1 member of the one-parameter Dehnen profile family (γ-models):

• Inner Cuspy Regime (r ≪ a): Near the galactic center, (r + a)³ ≈ a³ and the space density diverges with a shallow power-law cusp, ρ(r) ≈ (M/2πa²)r⁻¹. This inner r⁻¹ slope matches the inner asymptotic slope of the cosmological Navarro–Frenk–White (NFW) dark-matter profile, making the Hernquist sphere a widely used finite-mass substitute for NFW halos in numerical galaxy-merger simulations.
• Outer Halo Regime (r ≫ a): At radii large compared to the scale radius a, (r + a)³ ≈ r³ and the density falls off as ρ(r) ≈ (Ma/2π)r⁻⁴, matching the radial signature of stars tossed onto bound, high-eccentricity Keplerian halo orbits during incomplete violent relaxation.

Integrating 4πr′²ρ(r′) outward from the origin yields a rational expression for the cumulative enclosed mass M(r) inside radius r:

M(r) = Mr² / (r + a)²

Unlike the NFW profile—whose r⁻³ outer slope causes the enclosed mass to diverge logarithmically as ln(r) at large radii—the Hernquist profile converges smoothly to the finite total mass M as r → ∞. Setting M(r_h) = ½M determines the three-dimensional half-mass radius r_h = (1 + 2^1/2)a ≈ 2.4142*a*. Integrating the radial gravitational acceleration g(r) = −GM(r)/r² = −GM/(r + a)² from r to infinity yields the Newtonian gravitational potential Φ(r):

Φ(r) = −GM / (r + a)

At the galactic center (r = 0), the potential attains a finite minimum depth Φ(0) = −GM/a, corresponding to a central escape speed v_esc(0) = (2*GM*/a)^1/2. The circular orbital speed v_c(r) = (GMr)^1/2/(r + a) reaches its global maximum v_c,max = (GM/4a)^1/2 at r = a.

Projecting the three-dimensional Hernquist density along the line of sight at projected cylindrical radius R = sa on the sky plane yields an analytical surface mass density Σ(R):

Σ(R) = [M / (2πa²(s² − 1)²)] [(2 + s²)X(s) − 3]

where X(s) = (1 − s²)^−1/2 arccosh(1/s) for s < 1, X(s) = (s² − 1)^−1/2 arccos(1/s) for s > 1, and Σ(a) = (2/15π)(M/a²) at s = 1. Integrating Σ(R) over a circular aperture of projected radius R gives the enclosed projected mass M_proj(R) = Ms²[X(s) − 1]/(s² − 1). Setting M_proj(R_e) = ½M determines the sky-projected effective (half-light) radius R_e ≈ 1.8153*a* (or a ≈ 0.5509*R*_e). Over the radial range 0.06 ≤ R/R_e ≤ 14.5—which contains 94% of a galaxy's integrated light—the projected Hernquist profile matches the empirical de Vaucouleurs R^1/4 profile to within ±0.10 magnitudes per square arcsecond, while avoiding the steeper central cusp of the Jaffe profile.

For a spherical system with an isotropic velocity dispersion tensor (β = 0), integrating the spherical Jeans equations d(ρσ_r²)/dr = −ρ*GM*(r)/r² yields the closed-form radial velocity dispersion σ_r²(r):

σ_r²(r) = (GM/12*a*) { [12*r*(r + a)³/a⁴] ln[(r + a)/r] − [r/(r + a)][25 + 52(r/a) + 42(r/a)² + 12(r/a)³] }

Because the inner density cusp is ρ ∝ r⁻¹ while the inward gravitational acceleration approaches −GM/a² at the origin, the isotropic velocity dispersion vanishes logarithmically at r = 0 as σ_r²(r) ≈ (GM/a)(r/a) ln(a/r), peaks at σ_r,max ≈ 0.459(GM/a)^1/2 at r ≈ 0.33*a*, and declines Keplerianly as GM/(5r) at large radii. Adding a central supermassive black hole of mass M_BH shifts the inner velocity dispersion to a Keplerian rise σ_r²(r) ≈ GM_BH/(2r).

Inverting Ψ(r) = GM/(r + a) to write r(Ψ) = (GM/Ψ) − a enables evaluation of the Eddington inversion formula in closed analytical form. Defining q = (aℰ/GM)^1/2 ∈ [0, 1] and v_g = (GM/a)^1/2, the isotropic phase-space distribution function f(ℰ) solving the collisionless Boltzmann equation is:

f(ℰ) = [M / (8(2)^1/2π³a³v_g³)] (1 − q²)^−5/2 [3 arcsin(q) + q(1 − q²)^1/2(1 − 2q²)(8q⁴ − 8q² − 3)]

In modern extragalactic astrophysics, composite models of spiral galaxies such as the Milky Way and Andromeda Galaxy combine a central Hernquist bulge with an axisymmetric Miyamoto–Nagai potential stellar disk (satisfying the Toomre stability criterion) and an outer Einasto profile or NFW dark-matter halo. In numerical galaxy-merger simulations (Springel, Di Matteo, & Hernquist, 2005), NFW halos of virial radius r₂₀₀ and concentration c₂₀₀ are routinely mapped onto finite-mass Hernquist halos of identical inner r⁻¹ density normalization, while the analytic projected mass M_proj(R) provides closed-form deflection angles for gravitational lensing.

Why does the isotropic velocity dispersion drop to zero at the center of a Hernquist profile?

In a self-gravitating Hernquist sphere without a central black hole, the enclosed mass scales as M(r) ∝ r² at r ≪ a, so the gravitational acceleration g(r) = −GM(r)/r² approaches a constant −GM/a² while the density diverges as ρ(r) ∝ r⁻¹. Hydrostatic balance in the Jeans equations requires the radial pressure ρσ_r² to remain finite at the origin, forcing σ_r²(r) ∝ (r/a) ln(a/r) → 0 as r → 0.

How do you convert between a galaxy's effective radius R_e and the Hernquist scale length a?

Integrating the projected Hernquist surface brightness over circular apertures shows that half of the total projected light lies within R_e ≈ 1.8153*a*. Given an observed half-light radius R_e, the corresponding Hernquist scale length is a ≈ 0.5509*R*_e.

What is the total gravitational potential energy of a Hernquist sphere?

Integrating ½ ∫ ρ(r)Φ(r) 4πr² dr yields a self-gravitational potential energy of W = −GM²/(6a). By the virial theorem, the equilibrium kinetic energy is K = GM²/(12*a*) and the global one-dimensional velocity dispersion is ⟨σ_1D²⟩ = GM/(18*a*).

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2. Dehnen, W. (1993). "A family of potential-density pairs for spherical galaxies and bulges." Monthly Notices of the Royal Astronomical Society, 265(1), 250–256. https://doi.org/10.1093/mnras/265.1.250
3. Tremaine, S., Richstone, D. O., Byun, Y.-I., Dressler, A., Faber, S. M., Grillmair, C., Kormendy, J., & Lauer, T. R. (1994). "A family of models for spherical stellar systems." The Astronomical Journal, 107(2), 634–644. https://doi.org/10.1086/116883
4. Baes, M., & Dejonghe, H. (2002). "The Hernquist model revisited: explicit expressions for the phase-space distribution and photometric properties." Astronomy & Astrophysics, 393(2), 485–497. https://doi.org/10.1051/0004-6361:20021009
5. Springel, V., Di Matteo, T., & Hernquist, L. (2005). "Modelling feedback from stars and black holes in galaxy mergers." Monthly Notices of the Royal Astronomical Society, 361(3), 776–794. https://doi.org/10.1111/j.1365-2966.2005.09238.x