A THEOREM ON CENTRAL VELOCITY DISPERSIONS

If the dark halo diverges like a singular isothermal sphere (which behaves as r⁻²; henceforth SIS), the resulting potential is logarithmically divergent and so δ = 0. Thus, given the power-law assumption, for any cusped halo diverging slower than a SIS, we can limit δ < 0. Letting p = −δ>0 (that is, the potential behaves like Ψ ≃ Ψ₀ + Br^p/p), the leading term for σ²_r as r → 0 becomes

$$\begin{equation} \sigma _r^2\approx \left\lbrace \begin{array}{@{}l@{\quad }l@{}} \dsty \frac{B}{\gamma -2\beta -p}r^p &\mbox{if $p<\gamma -2\beta $}; \\ \ms \dsty r^p(B\ln r^{-1}+C) &\mbox{if $p=\gamma -2\beta $}; \\ \ms \dsty Cr^{\gamma -2\beta } &\mbox{if $\gamma -2\beta < p$}. \end{array}\right. \end{equation} \tag{ 7 }$$

For 0 < p < γ − 2β, we have σ²_r ∼ r^p → 0 as r → 0. If p = γ − 2β>0, then σ²_r ∼ r^pln r⁻¹ and again lim_r→0σ²_r → 0. Moreover, the logarithmic slope of σ²_r still tends to p > 0 in the limit of r → 0. Finally, if γ − 2β < p, then σ²_r ∼ r^γ−2β. However, for this last case, if γ < 2β, then r^γ−2β → ∞ as r → 0. Since any finitely deep central potential well is unable to support tracer populations with divergent velocity dispersions, the physical possibilities are limited to be γ ⩾ 2β (cf., An & Evans 2006).

In conclusion, with a finite central potential well, the possibilities are either (1) σ²_r → 0 as r → 0 (here, the logarithmic slope of σ²_r tends to min [p, γ − 2β]>0) or (2) γ = 2β with finite and nonzero σ²_r at r = 0 (see also Evans et al. 2009).

Next, we consider a centrally diverging potential, for which 0 ⩽ δ ⩽ 1. Here, the δ = 0 case corresponds to a logarithmic potential and a SIS-like dark halo cusp, whereas a point mass potential is represented by δ = 1.

For these cases, the theorem of An & Evans (2006) provides us with the constraint that $\gamma \ge \mbox{$\frac{1}{2}$}\delta +\beta (2-\delta)$, and so that $\delta -2\beta +\gamma \ge \big(\mbox{$\frac{3}{2}$}-\beta\big)\delta$. Since β ⩽ 1, we finally find that δ − 2β + γ ⩾ 0. Here, this is strictly larger than zero if δ ≠ 0. Consequently, the leading term for σ²_r as r → 0 with a divergent potential is

$$\begin{equation} \sigma _r^2\simeq \left\lbrace \begin{array}{@{}l@{\quad }l@{}} \dsty \frac{B}{\gamma -2\beta }+Cr^{\gamma -2\beta } &\mbox{if $\delta =0$ and $\gamma >2\beta $}; \\ \ms B\ln r^{-1}+C &\mbox{if $\delta =\gamma -2\beta =0$}, \end{array}\right. \end{equation} \tag{ 8 }$$

or

$$\begin{equation} \sigma _r^2\approx \frac{B}{\gamma -2\beta +\delta }\frac{1}{r^\delta } =\frac{\delta }{\gamma -2\beta +\delta }|\Psi | \end{equation} \tag{ 9 }$$

if 0 < δ ⩽ 1, for which γ − 2β + δ>0. Here, note that Ψ ≃ Bln r⁻¹ if δ = 0 and Ψ ≃ −(B/δ)r^−δ for 0 < δ ⩽ 1. In addition, ρ ∝ r^−(2+δ) for 0 ⩽ δ < 1 if the potential is generated by ρ, whereas B = GM_• and δ = 1 if a black hole of mass M_• dominates the potential.

In conclusion, σ²_r in a divergent potential well traces the potential except for the case of a logarithmically divergent potential with γ>2β, for which it is nonzero and finite.

First, let us recast the Jeans equation into a more useful form. We begin by integrating Equation (3), which leads us to

$$\begin{equation} r^2\frac{d\Psi }{dr}= GM(r)=GM_\bullet +4\pi G\int _0^r d\tilde{r}\,\tilde{r}^2\rho (\tilde{r}), \end{equation} \tag{ 10 }$$

where M(r) is the enclosed mass within the radius of r, and the integration constant M_• represents a central point mass (e.g., a supermassive black hole). However, we postpone detailed consideration of the M_• ≠ 0 case until Section 5.

Equation (1) is then equivalent, with the enclosed mass, to

$$\begin{equation} \frac{GM(r)}{r}=\sigma _r^2 (\gamma -2\beta -\alpha). \end{equation} \tag{ 11 }$$

Also we have introduced the logarithmic slopes of the tracer density and the velocity dispersion, namely (note the signs)

$$\begin{equation} \gamma =-\frac{d\ln \nu }{d\ln r} =-\frac{r}{\nu }\frac{d\nu }{dr} \,;\qquad \alpha =\frac{d\ln \sigma _r^2}{d\ln r} =\frac{r}{\sigma _r^2}\frac{d\sigma _r^2}{dr}. \end{equation} \tag{ 12 }$$

In the following, we consider the behavior of the system at the center, as indicated by the limit of Equation (11) as r → 0. All the subsequent arguments operate under the assumption that every quantity considered here is well behaved, continuous and smooth.

Here, we basically repeat the argument found in Section 5 of Evans et al. (2009) with a slight refinement. The result will form a part of the theorem to be proven in Section 5, and highlights the astrophysically relevant information.

The condition for the left-hand side of Equation (11) to vanish in the limit r → 0 is given by lim_r→0M(r) = M_• = 0 and, from l'Hôpital's rule, dM/dr|_{r = 0} = 0. The last bit is equivalent to lim_r→0ρr² = 0–that is to say, lim_r→0ρ is finite (i.e., a cored profile) or ρ diverges at the center slower than a SIS. Hence, assuming M_• = 0 (i.e., no central point mass), if lim_r→0ρr² = 0, then the right-hand side of Equation (11) should also vanish as r → 0. This is possible only if (1) σ²_r,0 = 0, or (2) α₀ = γ₀ − 2β₀. Here and throughout, the subscript "0" is used to indicate the limiting value at the center.

Suppose that (2) is the case. Here, if γ₀ < 2β₀, then it would be that $\sigma _r^2\sim r^{-(2\beta _0-\gamma _0)}\rightarrow \infty$ as r → 0. However, the SIS, for which lim_r→0ρr² is nonzero and finite, can only generate a potential diverging as fast as logarithmic. Thus, the velocity dispersion that diverges like a power law cannot be supported by the potential generated by the density cusp that is shallower than that of the SIS. Consequently, the corresponding case, γ₀ < 2β₀, is unphysical and not allowed. On the other hand, if γ₀ > 2β₀, then $\sigma _r^2\sim r^{(\gamma _0-2\beta _0)}\rightarrow 0$, which reduces to the case (1). In conclusion, a spherical dark halo with a core or a milder cusp than that of a SIS (i.e., lim_r→0ρr² = 0), can only permit tracer populations satisfying the constraint β₀ = γ₀/2 or those with σ²_r,0 = 0.

The preceding discussion indicates that σ²_r,0 = 0 is necessary for lim_r→0ρr² = 0 if γ₀ ≠ 2β₀. The initial impression of Equation (11) notwithstanding, σ²_r,0 = 0 alone, however, is not sufficient for vanishing M/r, either. Formally, this is because the behavior of Equation (11) cannot be specified for tracers for which σ²_r,0 = 0 and β₀ = −∞ without reference to the speed of each approach to its limiting value.

From its definition in Equation (2), β = −∞ if σ²_r = 0 unless σ²_θ = 0 too. Here, β₀ = −∞ indicates a tracer population with purely circular orbits toward the center. Note that in principle it is always possible to construct any spherical model with purely circular orbits (although such models are subject to resonant overstabilities; see Palmer et al. 1989). The nonzero tangential velocity dispersion here is the result of the random orientations of the orbital planes while the circular speed is uniquely specified by the enclosed mass. That is to say, the local dispersion of the speed of the tracers is actually zero although the tangential velocity dispersion may be nonzero.

On the other hand, for all finite values of β₀, then matters are simpler. Since

$$\begin{equation*} \sigma ^2_{\theta,0}=\sigma ^2_{\phi,0}=(1-\beta _0)\sigma ^2_{r,0}=0, \end{equation*}$$

we easily find that σ²_r,0 = 0 indicates that the total three-dimensional velocity dispersion at the center also vanishes. That is, the system must be dynamically cold at the center.

The conclusion of Section 4.2 therefore may be rephrased as follows: in a spherical potential well generated by a halo that is cored or cusped less severe than a SIS, the only allowed populations of tracers are those either consisting of purely circular orbits toward the center or exhibiting vanishing central velocity dispersions unless the limiting values of the cusp power index γ₀ and the anisotropy parameter β₀ at the center are constrained such that γ₀ = 2β₀.

At first glance, the result of Section 4.2 may appear to be counterintuitive as though it seems to suggest no pressure support at the center whereas the Jeans equation is supposed to balance the force. This reasoning is faulty because the actual "pressure" in this case is given by ρσ², not σ². Given the density cusp, the system possesses nonvanishing kinematic pressure at the center even though it is dynamically cold. This is obvious in the power-law solutions to the Jeans equation of Section 3, which show that νσ²_r ∼ r^{min(p − γ,−2β)}. For a system that is radially biased or isotropic toward the center (i.e., β ⩾ 0, for which γ ⩾ 0 from An & Evans 2006 and so tracers with a hole at the center are not allowed), the central "pressure" is therefore always nonvanishing as r → 0 (it would be actually divergent unless β = 0 < p − γ). The "pressure" of the tangentially biased system (β < 0) on the other hand can be vanishing if p > γ. However, in this last scenario, it can be understood that the tangentially biased system is preferentially rotationally supported toward the center.

If σ²_r indeed vanishes at the center, this implies that there are no radial orbits in the model and that the distribution function has the property ${\rm lim} _{L^2\rightarrow 0}f(L^2,E)\rightarrow 0$. This seems unusual, but there are mechanisms known that can depopulate the radial orbits—for example, scattering by the central cusp (Gerhard & Binney 1985) or the radial orbit instability (Palmer & Papaloizou 1987)—albeit at the cost of driving the system away from sphericity.

The result can also be applied to a self-consistent system (or equivalently interpreted as a constraint on the central velocity dispersion of the dark halo itself). In a spherical dark halo that has a core or a cusp less severe than a SIS, the central limiting value of the anisotropy parameter must be exactly half of the numerical value of the cusp slope unless the central velocity dispersion vanishes. In particular, a cored halo must have an isotropic velocity dispersion at the center if it is not dynamically cold there.

We start by giving the binary relation "∼" its precise mathematical meaning. In the following, it is understood to be the short-hand notation such that a ∼ b as r → 0 if and only if both lim_r→0(a/b) and lim_r→0(b/a) are finite. We also define the binary relations ⋩ and ⋨ similarly. That is to say, a⋩b (or a⋨b) as r → 0 if and only if lim_r→0a/b is divergent (or zero). In addition, it is also to be understood that the limit is taken to be always r → 0 unless the explicit reference to the limit is given to override.

Next, we consider the behavior of the left-hand side of Equation (11) in relation to the mass density profile that generates the potential. From the argument of Section 4.2, we find that M/r decays to zero in the limit r → 0 if M_• = 0 and ρ⋨r⁻². The corresponding potential is either finite at the center or diverges strictly slower than logarithmic (i.e., Ψ⋨ln r). On the other hand, it attains a nonzero finite limiting value at r → 0 if and only if M(r) ∼ r. This is equivalent to ρ ∼ r⁻² and also Ψ ∼ ln r. Finally, M/r diverges as r → 0 if M_• is nonzero (implying the presence of a central point mass) or ρ⋩r⁻² in the same limit. The corresponding potential also diverges but strictly faster than logarithmic (i.e., Ψ⋩ln r) in the same limit.

The behaviors of M/r and Ψ in relation to each other and their respective logarithmic slopes may be explored in further detail. Let us first define p, the logarithmic slope of M/r, i.e.,

$$\begin{equation} p=\frac{d\ln (M/r)}{d\ln r} =\frac{1}{d\Psi /dr}\frac{d}{dr}\left(\frac{GM}{r}\right). \end{equation} \tag{ 13 }$$

Note that in the limit of r → 0, the logarithmic slope of the potential (difference) necessarily tends to the same value as p₀. In particular, if Ψ(0) = Ψ₀ is finite, it naturally follows that GM/r = r(dΨ/dr) → 0 and so

$$\begin{equation*} \lim _{r\rightarrow 0}\frac{d\ln |\Psi -\Psi _0|}{d\ln r} =\lim _{r\rightarrow 0}\frac{GM/r}{\Psi -\Psi _0}=p_0\ge 0 \end{equation*}$$

using l'Hôpital's rule. If lim_r→0Ψ = −∞ on the other hand, we also find that

$$\begin{equation*} \lim _{r\rightarrow 0}\frac{d\ln |\Psi |}{d\ln r} =\lim _{r\rightarrow 0}\frac{GM/r}{\Psi }=p_0\le 0. \end{equation*}$$

Although the usual conditions for l'Hôpital's rule for this case are only strictly met if M/r diverges, l'Hôpital's rule can in fact be proven only assuming a divergent denominator. Thus, the result holds even though M/r tends to a finite value (including zero) as r → 0. Moreover, it is obvious that p₀ = 0 if Ψ → ∞ and M/r is finite as r → 0. Next, if p₀ ≠ 0, then M/r ∼ |ΔΨ| where ΔΨ = Ψ for lim_r→0Ψ = −∞ or ΔΨ = |Ψ − Ψ₀| for Ψ(0) = Ψ₀ being finite. By contrast, that p₀ = 0 indicates that M/r⋨|ΔΨ|. If the central potential is additionally finite (e.g., Ψ − Ψ₀ ∼ [ln r⁻¹]⁻¹, for which M/r ∼ [ln r]⁻²), then M/r → 0. The behavior of M/r as r → 0 for a divergent central potential depends on how fast Ψ diverges relative to logarithmic divergence (ln r⁻¹)—e.g., for |Ψ| ∼ ln ln r⁻¹, |Ψ| ∼ ln r, and $|\Psi |\sim \mbox{$\frac{1}{2}$}[\ln r]^2$, we have that M/r ∼ [ln r⁻¹]⁻¹, M ∼ r, and M/r ∼ ln r⁻¹, respectively.

Next, we proceed to analyzing the right-hand side of Equation (11). Here, we do not consider the β = −∞ case, which represents the formal possibility of building the system with purely circular orbits. Then, since M and σ²_r must be non-negative, γ₀ − 2β₀ − α₀ ⩾ 0. Moreover, γ₀ ⩽ 3 from the constraint that the central mass concentration must be finite. Together with the assumption that β₀ is finite, we find that (γ₀ − 2β₀ − α₀) is divergent only if α₀ diverges to negative infinity. However, then σ²_r diverges faster than any power law to an essential singularity (e.g., e^1/r). This is physically impossible, because no real potential diverges faster than 1/r nor is thus able to support such steeply diverging velocity dispersions. Therefore, we limit γ₀ − 2β₀ − α₀ to be finite.

If γ₀ − 2β₀ − α₀ > 0, it is clear that σ²_r ∼ GM/r, which also indicates that α₀ = p₀. If α₀ = p₀ ≠ 0, then σ²_r ∼ |ΔΨ|, too. If α₀ = p₀ = 0 on the other hand, the behavior of σ²_r still traces that of M/r, but σ²_r ∼ M/r⋨|ΔΨ|.

If γ₀ − 2β₀ = α₀, then σ²_r⋩M/r and so α₀ ⩽ p₀. Here, if the central potential is finite, then we find that 0 ⩽ α₀ = γ₀ − 2β₀ ⩽ p₀, from the constraint of An & Evans (2006). For a divergent potential (introducing δ₀ = −p₀, for which 0 ⩽ δ₀ ⩽ 1), the constraint of An & Evans (2006), $\gamma _0\ge \mbox{$\frac{1}{2}$}\delta _0+\beta _0(2-\delta _0)$, indicates that $\mbox{$\frac{1}{2}$}\delta _0-\beta _0\delta _0\le \gamma _0-2\beta _0= \alpha _0\le p_0=-\delta _0$. Now, if δ₀ > 0, this would imply $\beta _0\ge \mbox{$\frac{3}{2}$}$. This is obviously impossible, and therefore δ₀ = p₀ = α₀ = γ₀ − 2β₀ = 0. In addition, if M⋩r, it is clear that σ²_r → ∞. Furthermore, once Equation (11) is recast to be

$$\begin{equation*} r\frac{d\Psi }{dr}=(\gamma -2\beta)\sigma _r^2 -r\frac{d\sigma _r^2}{dr}, \end{equation*}$$

we can find for sufficiently fast-decaying γ − 2β that σ²_r ∼ |Ψ|. This essentially implies that σ²_r cannot diverge faster than Ψ.

In summary, the spherical Jeans equations permit only restricted physical possibilities regarding the limiting behaviors at the center. In particular, the central limiting value of the velocity anisotropy (β₀; Equation (2)) and those of the logarithmic slopes of the luminous tracer density (γ₀; Equation (12)), the radial velocity dispersion (α₀; Equation (12)) and the potential (p₀; Equation (13)) must meet one, and only one, of the following list of choices:

• 1.  
  p₀ = α₀ < γ₀ − 2β₀ and σ²_r ∼ M/r,
• 2.  
  p₀ ⩾ α₀ = γ₀ − 2β₀ ⩾ 0 and Ψ₀ is finite,
• 3.  
  p₀ = α₀ = γ₀ − 2β₀ = 0 and lim_r→0Ψ = −∞,
• 4.  
  β₀ = −∞.

Focusing on the behavior of the velocity dispersion, the result with the proviso β> − ∞ is summarized as

$$\begin{equation} \lim _{r\rightarrow 0}\frac{d\ln \sigma _r^2}{d\ln r}= \left\lbrace \begin{array}{@{}l@{\quad }l@{}} \min (2-\Gamma _0,\gamma _0-2\beta _0)\ge 0 &(\Gamma _0<2)\\ \ms -(\Gamma _0-2)\le \gamma _0-2\beta _0 &(\Gamma _0\ge 2)\end{array}\right. \end{equation} \tag{ 14 }$$

although this does not include all the information encompassed in the above choices. Here,

$$\begin{equation*} \Gamma =-\frac{d\ln \rho }{d\ln r} \end{equation*}$$

so that p = 2 − Γ, and extending to include the central point mass by setting Γ₀ = 3.

For a prescribed behavior of M/r or Ψ, the above list returns the natural extension and generalization of our earlier results. If M/r → 0, for example, then either (a) σ²_r → 0 with σ²_r ∼ M/r or α₀ = γ₀ − 2β₀ > 0, or (b) α₀ = γ₀ − 2β₀ = 0. Consequently, we recover the conclusion of Section 4.2. The implication of the list, however, is more detailed. First, if γ₀ − 2β₀ > p₀ ⩾ 0 (p₀ ⩾ 0 is necessary for vanishing M/r), then only the case (1) is possible and so σ²_r ∼ M/r and α₀ = p₀. If p₀ > 0 additionally, then σ²_r ∼ |Ψ − Ψ₀| with a finite Ψ₀ whereas σ²_r⋨|ΔΨ| for p₀ = 0. On the other hand, with p₀ ⩾ γ₀ − 2β₀ > 0, we have α₀ = γ₀ − 2β₀ > 0 and so σ²_r → 0 (and p₀ > 0 similarly indicating that M/r ∼ |Ψ − Ψ₀|). The remaining physical possibility, p₀ ⩾ γ₀ − 2β₀ = 0, implies that α₀ = 0, which can lead to a nonzero finite limit for lim_r→0σ²_r. If Ψ₀ is finite, then σ²_r must not diverge, but if Ψ is divergent (but not faster than logarithmic) as r → 0, the exact limiting behavior of the corresponding σ²_r should be inferred from the particular solution to the Jeans equations.

By contrast, if M/r diverges (for which Ψ → −∞), then case (1) indicates that σ²_r ∼ M/r → ∞ whereas case (3) requires σ²_r⋩M/r and so σ²_r → ∞ (but α₀ = p₀ = 0). In other words, σ²_r necessarily diverges if M/r → ∞. Furthermore, σ²_r must be divergent as fast as M/r (note that if δ₀ > 0, then σ²_r ∼ M/r ∼ |Ψ|, but σ²_r ∼ M/r⋨|Ψ| for δ₀ = 0, where δ₀ is the negative logarithmic slope of M/r or the potential) unless δ₀ = γ₀ − 2β₀ = 0 for which σ²_r diverges faster than M/r but not faster than |Ψ|.

For M ∼ r (and Ψ ∼ ln r), the result is basically that of Equation (8); case (1) yielding the possibility of a finite limiting value of σ²_r whereas case (3) is consistent with σ²_r diverging at most logarithmically or slower.

In the framework of classical Newtonian mechanics upon which the Jeans equations and the collisionless Boltzmann equation are ultimately based, the divergence of σ²_r when M/r and the corresponding potential also diverge is in principle physically acceptable despite its mathematical quirk. However, it is clear that the arguments given in this paper eventually break down as σ_r approaches the speed of light. Moreover, in the corresponding halo, M/r should be divergent as r → 0, and therefore there exists a radius below which GM(r)/c² > r. Consequently, the central cusp, if it ever were present, must collapse to a singularity. In other words, one would expect that the formal infinity of the velocity dispersion can be always circumvented through the presence of a central black hole. The proper examination of physical behaviors of the tracers and the halo under these conditions would require consideration of relativistic physics, which is out of the scope of the current paper. Of course, in reality, it is more likely that other various physical complexities in the system intervene to prevent the spherical Jeans equations to be applied uncritically all the way down to the center even before any relativistic effects become important.

The preceding discussion presumes the finite central potential well, which is appropriate for the potential dominated by the halo that is cored or cusped not so steep as the SIS. If the potential, however, is dominated by the central point mass, the theorem indicates that σ²_r ∼ |Ψ| ∼ 1/r and so Iσ²_ℓ ∼ R^−γ for γ>0 (i.e., cusped tracer populations) or Iσ²_ℓ ∼ ln R⁻¹ for γ = 0 (i.e., cored tracer populations). That is to say, the line-of-sight velocity dispersion of the population tracing the Keplerian potential is necessarily divergent with its logarithmic slope, |dln σ²_ℓ/dln R| being equal to min(1, γ) where γ is the three-dimensional density (negative) logarithmic slope of the same tracers unless the orbits of tracers are completely circularized toward the center. Nevertheless, the direct application of this inference to the observational results warrants caution since an assumption of "infinite" resolution is implicit in the argument. That is to say, the result is strictly relevant only if the observation can resolve the so-called sphere of influence of the central point mass.