Dehnen Profile

The Dehnen profile—also known as the γ-model or Tremaine–Dehnen model—is a unified one-parameter family of analytical, spherically symmetric gravitational potential–density pairs introduced independently by German astrophysicist Walter Dehnen in 1993 and by Scott Tremaine and collaborators in 1994 (who termed them η-models, with η = 3 − γ) to describe the three-dimensional mass distribution, gravitational potential, and orbital kinematics of elliptical galaxies, galactic bulges, and dark matter halos. Motivated by Hubble Space Telescope Planetary Camera observations in the early 1990s showing that the central regions of early-type galaxies do not have flat isothermal cores like the Plummer model or King profile, but instead exhibit power-law inner surface-brightness cusps whose logarithmic slopes vary systematically with galaxy luminosity, Dehnen generalized the Hernquist profile (γ = 1) and Jaffe profile (γ = 2) into a continuous sequence indexed by the inner logarithmic cusp slope γ ∈ [0, 3).

A spherical Dehnen galaxy of total mass M, characteristic scale radius a, and inner cusp slope parameter γ (with 0 ≤ γ < 3; or η ≡ 3 − γ ∈ (0, 3]) is defined by the three-dimensional mass density law:

ρ(r) = [(3 − γ)M / 4π] a / [r^γ(r + a)^4−γ]

In terms of the Zhao (1996) double power-law density profile ρ(r) ∝ (r/a)^−γ[1 + (r/a)^1/α]^{−(β−γ)α}, the Dehnen family forms the (α, β, γ) = (1, 4, γ) sequence:

• Inner Central Cusp (r ≪ a): Near the origin, (r + a)^4−γ ≈ a^4−γ and the density scales as a power law ρ(r) ∝ r^−γ. The condition γ < 3 ensures that the mass enclosed near the origin (M(r) ∝ r^3−γ) remains finite and vanishes as r → 0.
• Outer Violent-Relaxation Halo (r ≫ a): At radii large compared to the scale radius a, r^γ(r + a)^4−γ ≈ r⁴ for all values of γ, so every Dehnen model shares the ρ(r) ∝ r⁻⁴ outer halo falloff characteristic of incomplete violent relaxation and de Vaucouleurs's R^1/4 surface-brightness law.

Integrating 4πr′²ρ(r′) from the center to radius r yields a single elementary power law for the cumulative enclosed mass profile M(r) across the γ-family:

M(r) = M [r / (r + a)]^3−γ

Setting M(r_h) = ½M determines the three-dimensional half-mass radius r_h = a/(2^1/(3−γ) − 1). Integrating the radial gravitational field g(r) = −GM(r)/r² from r to infinity yields the Newtonian gravitational potential Φ(r):

Φ(r) = −[GM / ((2 − γ)a)] { 1 − [r / (r + a)]^2−γ } (for γ ≠ 2)

and in the singular limit γ = 2 (Jaffe profile), L'Hôpital's rule recovers the logarithmic potential Φ(r) = (GM/a) ln[r/(r + a)]. For 0 ≤ γ < 2, the central potential at r = 0 is finite and equals Φ(0) = −GM/[(2 − γ)a], whereas for 2 ≤ γ < 3 the potential diverges to −∞ at the origin. The circular orbital velocity v_c²(r) = GMr^2−γ/(r + a)^3−γ peaks at radius r_max = (2 − γ)a for 0 ≤ γ ≤ 2.

The parameter γ unifies several models of galactic dynamics and mirrors the observational dichotomy in early-type galaxy nuclei revealed by Hubble Space Telescope imaging:

1. Cored Models (γ = 0) and Weak Cusps (0 < γ < 1): Setting γ = 0 yields ρ(r) = (3*Ma*)/[4π(r + a)⁴], which possesses a finite central density ρ(0) = 3M/(4πa³). On the sky, models with 0 ≤ γ < 1 project to flat central surface-brightness cores (I(R) → I(0) = const as R → 0), matching luminous, slowly rotating boxy elliptical galaxies (M_V < −21.5, such as Messier 87 and Messier 60) where binary supermassive black hole scouring ejects stars from the nucleus.
2. The Hernquist Profile (γ = 1): At the transition boundary γ = 1, the three-dimensional density has an r⁻¹ cusp and the projected surface density diverges logarithmically at the origin as Σ(R) ∝ ln(a/R).
3. Power-Law Cusps (1 < γ ≤ 2) and the Jaffe Profile (γ = 2): For γ > 1, the sky-projected surface brightness diverges at small radii as a power law, I(R) ∝ R^1−γ, matching intermediate- and low-luminosity disky elliptical galaxies. At γ = 2 (Jaffe profile), the inner density matches a singular isothermal sphere (ρ ∝ r⁻², I(R) ∝ R⁻¹). Across 0 ≤ γ ≤ 2, the projected effective radius satisfies R_e/r_h ≈ 0.754.

Solving the isotropic spherical Jeans equations d(ρσ_r²)/dr = −ρ*GM*(r)/r² yields the intrinsic radial velocity dispersion profile σ_r²(r) for any γ ∈ [0, 3):

σ_r²(r) = GMr^γ(r + a)^4−γ ∫_r^∞ [r′^1−2γ / (r′ + a)^7−2γ] dr′

In self-gravitating Dehnen spheres without a central black hole, σ_r²(0) = GM/(30*a*) is finite for γ = 0, vanishes at the origin (σ_r(0) = 0) for all 0 < γ < 2, jumps to the isothermal value GM/(2a) at γ = 2, and diverges at the origin for 2 < γ < 3. Adding a central supermassive black hole of mass M_BH forces the central velocity dispersion to rise Keplerianly for all γ > 0 as σ_r²(r) ≈ GM_BH/[(1 + γ)r].

Inverting the potential Ψ(r) = −Φ(r) expresses the space density ρ(Ψ) as an algebraic function of Ψ, and substituting ρ(Ψ) into the Eddington inversion formula shows that the ergodic phase-space distribution function f(ℰ) solving the collisionless Boltzmann equation is positive everywhere (f(ℰ) > 0 for all 0 ≤ γ < 3). Finally, the total self-gravitational potential energy W is finite for 0 ≤ γ < 5/2, where W = −GM²/[2(5 − 2γ)a].

How are the Dehnen γ parameter and the Tremaine η parameter related?

Walter Dehnen (1993) parameterized the family by the inner logarithmic cusp slope γ (where ρ ∝ r^−γ as r → 0, with 0 ≤ γ < 3), whereas Scott Tremaine et al. (1994) parameterized the same models by η = 3 − γ (where the enclosed mass scales as M(r) ∝ r^η as r → 0, with 0 < η ≤ 3). Thus, the Hernquist profile is γ = 1 (η = 2) and the Jaffe profile is γ = 2 (η = 1).

Why does the projected surface brightness have a flat core for γ < 1 even though the 3D density diverges?

When projecting a three-dimensional density cusp ρ(r) ∝ r^−γ along the line of sight via the Abel projection integral Σ(R) = 2 ∫_R^∞ ρ(r)r dr/(r² − R²)^1/2, the line-of-sight integration smooths the cusp by one power of radius. If 0 < γ < 1, the integral converges at R = 0 to a finite central surface density Σ(0).

Why does the total potential energy diverge for γ ≥ 5/2?

The gravitational potential energy inside a small sphere of radius r scales as ΔW(r) ~ −G[M(r)]²/r. Because M(r) ∝ r^3−γ near the center of a Dehnen profile, [M(r)]²/r scales as r^5−2γ. When γ ≥ 5/2, the exponent 5 − 2γ ≤ 0, so the self-binding energy of the central cusp diverges at r → 0.

1. 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
2. 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
3. Zhao, H. (1996). "Analytical models for galactic nuclei." Monthly Notices of the Royal Astronomical Society, 278(2), 488–496. https://doi.org/10.1093/mnras/278.2.488
4. Carollo, C. M., de Zeeuw, P. T., & van der Marel, R. P. (1995). "The intrinsic shape of the velocity distribution of stars in galaxies: the γ-models." Monthly Notices of the Royal Astronomical Society, 276(4), 1131–1140. https://doi.org/10.1093/mnras/276.4.1131
5. Binney, J., & Tremaine, S. (2008). Galactic Dynamics (2nd ed.). Princeton University Press. https://doi.org/10.1515/9781400828722