Razor-Thin Discs and Swing Amplification

Abstract

This chapter considers the secular dynamics of razor-thin stellar discs, and shows how a proper accounting of the discs’ self-gravitating amplification allows for a precise description of their diffusion features.

The work presented in this chapter is based on Fouvry et al. (2015).

Appendices

Appendix

A  Kalnajs 2D Basis

Let us detail the 2D basis introduced in Kalnajs (1976) and used in Sect. 4.3 to compute the diffusion flux. A similar rewriting of the basis normalisations can also be found in Earn and Sellwood (1995). This basis depends on two parameters, namely ${ k_\mathrm{Ka} \!\in \! \mathbb {N} }$ and a scale radius ${ R_\mathrm{Ka} \!>\! 0 }$. In order to shorten the notations in the upcoming expressions, let us write r for the dimensionless quantity ${ r / R_\mathrm{Ka} }$. As introduced in Eqs. (4.5) and (4.6), the 2D basis elements depend on two indices: the azimuthal number ${ \ell \!\ge \! 0 }$ and the radial number ${ n \!\ge \! 0 }$. The radial component of the potential elements is given by

$$\begin{aligned} \mathcal {U}_{n}^{\ell } (r) \!=\! - \frac{\sqrt{G}}{R_\mathrm{Ka}^{1/2}} \, \mathcal {P} (k_\mathrm{Ka} , \ell , n) \, r^{\ell } \sum _{i = 0}^{k} \sum _{j = 0}^{n} \!\alpha _\mathrm{Ka} (k_\mathrm{Ka} , \ell , n , i , j) \, r^{2 i + 2 j} \, , \end{aligned}$$

(4.51)

while the radial component of the density elements reads

$$\begin{aligned} \mathcal {D}_{n}^{\ell } (r) = \frac{(-1)^{n}}{\sqrt{G} \, R_\mathrm{Ka}^{3/2}} \, \mathcal {S} (k_\mathrm{Ka} , \ell , n) \, (1 \!-\! r^{2})^{k_\mathrm{Ka} - 1/2} \, r^{\ell } \sum _{j = 0}^{n} \beta _\mathrm{Ka} (k_\mathrm{Ka} , \ell , n , j) \, (1 \!-\! r^{2})^{j} \, . \end{aligned}$$

(4.52)

In Eqs. (4.51) and (4.52), we introduced the coefficients ${ \mathcal {P} (k , \ell , n) }$ and ${ \mathcal {S} (k , \ell , n) }$ as

$$\begin{aligned} \mathcal {P} (k ,\ell , n)&\, = \Bigg \{\! \frac{[2 k \!+\! \ell \!+\! 2 n \!+\! (1\!/2)] \Gamma [2 k \!+\! \ell \!+\! n \!+\! (1\!/2)]}{\Gamma [2 k \!+\! n \!+\! 1] \, \Gamma ^{2} [\ell \!+\! 1] \, \Gamma [n \!+\! 1] } \Gamma [\ell \!+\! n \!+\! (1\!/2)] \!\Bigg \}^{1\!/2} \! \, , \nonumber \\ \mathcal {S} (k , \ell , n)&\, = \frac{\Gamma [k \!+\! 1]}{\pi \, \Gamma [2 k \!+\! 1] \, \Gamma [k \!+\! (1/2)]} \Bigg \{ \frac{[2k \!+\! \ell \!+\! 2 n \!+\! (1\!/2)] \, \Gamma [2 k \!+\! n \!+\! 1] \, \Gamma [2 k \!+\! \ell \!+\! n \!+\! (1\!/2)]}{\Gamma [\ell \!+\! n \!+\! (1/2)] \, \Gamma [n \!+\! 1]} \Bigg \}^{1/2} \, . \end{aligned}$$

(4.53)

In Eqs. (4.51) and (4.52), we also introduced the coefficients $\alpha _\mathrm{Ka}$ and $\beta _\mathrm{Ka}$ as

$$\begin{aligned} \alpha _\mathrm{Ka} (k , \ell , n , i , j)&\, = \frac{[-k]_{i} \, [\ell \!+\! (1\!/2)]_{i} \, [2 k \!+\! \ell \!+\! n \!+\! (1/2)]_{j} [i \!+\! \ell \!+\! (1\!/2)]_{j} \, [- n]_{j}}{[\ell \!+\! 1]_{i} \, [1]_{i} \, [\ell \!+\! i \!+\! 1]_{j} [\ell \!+\! (1\!/2)]_{j} [1]_{j}} \, , \nonumber \\ \beta _\mathrm{Ka} (k , \ell , n , j)&\, = \frac{[2 k \!+\! \ell \!+\! n \!+\! (1/2)]_{j} \, [k \!+\! 1]_{j} \, [- n]_{j}}{[2 k \!+\! 1]_{j} \, [k \!+\! (1/2)]_{j} \, [1]_{j} } \, , \end{aligned}$$

(4.54)

where the two previous expressions relied on the rising Pochhammer symbol ${ [a]_{i} }$ defined as

$$\begin{aligned}{}[a]_{i} = {\left\{ \begin{array}{ll} \begin{aligned} &{}1 &{} \text {if} \;\;\; i = 0 \, , \\ &{} a \, (a \!+\! 1) \, ... \, (a \!+\! n \!-\! 1) &{} \text {if} \;\;\; i > 0 \, . \end{aligned} \end{array}\right. } \end{aligned}$$

(4.55)

B  Calculation of $\aleph$

In this Appendix, we briefly detail how the analytic function $\aleph$, introduced in Eq. (4.21) to compute the response matrix, may be estimated. In order to ease the effective numerical implementation of this calculation, let us first rewrite $\aleph$ in a dimensionless fashion so that

$$\begin{aligned} \aleph (a_{g} , b_{g} , c_{g} , a_{h} , b_{h} , c_{h} , \eta , \Delta r)&\, = \!\! \int _{- \frac{\Delta r}{2}}^{\frac{\Delta r}{2}} \!\! \int _{- \frac{\Delta r}{2}}^{\frac{\Delta r}{2}} \!\! \mathrm {d} x_{\mathrm {p}} \mathrm {d} x_{\mathrm {a}} \, \frac{a_{g} \!+\! b_{g} x_{\mathrm {p}} \!+\! c_{g} x_{\mathrm {a}}}{a_{h} \!+\! b_{h} x_{\mathrm {p}} \!+\! c_{h} x_{\mathrm {a}} \!+\! \mathrm {i}\eta } \nonumber \\&\, = \frac{a_{g}}{a_{h}} (\Delta r)^{2} \!\! \int _{- \frac{1}{2}}^{\frac{1}{2}} \!\! \int _{- \frac{1}{2}}^{\frac{1}{2}} \!\! \mathrm {d} x \mathrm {d} y \, \frac{1 \!+\! \frac{b_{g} \Delta r}{a_{g}} x \!+\! \frac{c_{g} \Delta r}{a_{g}} y }{1 \!+\! \frac{b_{h} \Delta r}{a_{h}} x \!+\! \frac{c_{h} \Delta r}{a_{h}} y \!+\! \mathrm {i}\frac{\eta }{a_{h}} } \nonumber \\&\, = \frac{a_{g}}{a_{h}} (\Delta r)^{2} \, \aleph _{\mathrm {D}} \left[ \frac{b_{g} \Delta r}{a_{g}} , \frac{c_{g} \Delta r}{a_{g}} , \frac{b_{h} \Delta r}{a_{h}} , \frac{c_{h} \Delta r}{a_{h}} , \frac{\eta }{a_{h}} \right] \, , \end{aligned}$$

(4.56)

where we assumed that ${ a_{g} , a_{h} \!\ne \! 0 }$ and used the change of variables ${ x \!=\! x_{\mathrm {p}} / \Delta r }$ and ${ y \!=\! x_{\mathrm {a}} / \Delta r }$. Finally, we introduced the dimensionless function $\aleph _{\mathrm {D}}$ as

$$\begin{aligned} \aleph _{\mathrm {D}} (b , c , e , f , \eta ) = \!\! \int _{- \frac{1}{2}}^{\frac{1}{2}} \!\! \int _{- \frac{1}{2}}^{\frac{1}{2}} \!\! \mathrm {d} x \mathrm {d} y \, \frac{1 \!+\! b x \!+\! c y}{1 \!+\! e x \!+\! f y \!+\! \mathrm {i}\eta } \, . \end{aligned}$$

(4.57)

To effectively compute this integral, it only remains to exhibit a function G(x, y) so that

$$\begin{aligned} \frac{\partial ^{2} G}{\partial x \partial y} = \frac{1 \!+\! b x \!+\! c y}{1 \!+\! e x \!+\! f y \!+\! \mathrm {i}\eta } \, . \end{aligned}$$

(4.58)

One possible choice for G is given by

$$\begin{aligned}&G (x , y) = \frac{1}{4 e^2 f^2} \log [ e^2 x^2 \!+\! 2 e (f x y\!+\!x)\!+\!f^2 y^2\!+\!2 f y\!+\!\eta ^2\!+\!1 ] \nonumber \\&\!\times \! \bigg \{ b f (e^2 x^2\!-\!(f y\!+\! \mathrm {i}\eta \!+\!1)^2 ) \!+\!2 e f (e x\!+\! \mathrm {i}\eta \!+\!1) \!-\!c e \nonumber (e x\!+\! \mathrm {i}\eta \!+\!1)^2 \bigg \} \\&\!+\!\frac{\mathrm {i}}{2 e^2 f^2} \bigg \{ \frac{\pi }{2}\!-\!\tan ^{-1} \!\bigg [ \frac{e x\!+\!f y\!+\!1}{\eta } \bigg ] \bigg \} \bigg \{ b f ( e^2 x^2\!-\!(f y\!+\!\mathrm {i}\eta \!+\!1)^2 ) \!+\! 2 e f (e x\!+\! \mathrm {i}\eta \!+\!1) \!-\!c e (e x\!+\! \mathrm {i}\eta \!+\!1)^2 \bigg \} \nonumber \\&\!+\!\frac{y}{4 e^2 f} \bigg \{ f (\!-\!4 e\!+\!b (2 e x\!+\!f y\!+\!2 \mathrm {i}\eta \!+\!2))\!+\!c e (2 e x\!-\!f y\!+\!2 \mathrm {i}\eta \!+\!2)\!+\!2 e f (c y\!+\!2) \log [e x\!+\!f y\!+\! \mathrm {i}\eta \!+\!1] \bigg \} \, . \end{aligned}$$

(4.59)

In the previous expression, one should be careful with the presence of a complex logarithm and a $\tan ^{-1}$. Fortunately, because ${ e , f , \eta \!\in \! \mathbb {R} }$, and ${ \eta \!\ne \! 0 }$, one can straightforwardly show that the arguments of both of these functions never cross the usual branch-cut of these functions chosen to be ${ \big \{ \text {Im} (z) \!=\! 0 \, ; \, \text {Re} (z) \!\le \! 0 \big \} }$. Equation (4.57) can then be computed as

$$\begin{aligned} \aleph _{\mathrm {D}} = G [ \tfrac{1}{2} , \tfrac{1}{2} ] \!-\! G [ \tfrac{1}{2} , - \tfrac{1}{2} ] \!-\! G [ - \tfrac{1}{2} , \tfrac{1}{2} ] \!+\! G [ - \tfrac{1}{2} , - \tfrac{1}{2} ] \, . \end{aligned}$$

(4.60)

C  Recovering Unstable Modes

Let us detail how the matrix code presented in Sect. 4.2.3 as well as the ${N-}$body code described in Sect. 4.4.1 can be validated by recovering known unstable modes of razor-thin discs. The direct numerical calculation of modes in a galactic disc is a complex task, which has only been made for a small number of discs models (Zang 1976; Kalnajs 1977; Vauterin and Dejonghe 1996; Pichon and Cannon 1997; Evans and Read 1998b; Jalali and Hunter 2005; Polyachenko 2005; Jalali 2007, 2010; De Rijcke and Voulis 2016). Here, we will recover the results of the pioneer work of Zang (1976), extended in Evans and Read (1998), Evans and Read (1998b), and recovered numerically in Sellwood and Evans (2001). These three works were interested in recovering the precession rate ${ \omega _{0} \!=\! m_{\mathrm {p}} \Omega _{\mathrm {p}} }$ and growth rate ${ \eta \!=\! s }$ of the unstable modes a truncated Mestel disc very similar to the stable one presented in Sect. 3.7.1. The unstable discs considered thereafter are fully active discs, so that ${ \xi \!=\! 1 }$, and their radial velocity dispersion is given by ${ q \!=\! (V_{0} / \sigma _{r})^{2} \!-\! 1 \!=\! 6 }$. Finally, we consider different models of disc by varying the power index $\nu _{\mathrm {t}}$ of the inner taper function as defined in Eq. (3.90). Here, we will look for ${ m_{\mathrm {p}} \!=\! 2 }$ modes, and will consider three different truncation indices given by ${ \nu _{\mathrm {t}} \!=\! 4, 6, 8 }$. In Sect. C.1, we first recover the associated unstable modes by computing the system’s response matrix, following Sect. 4.2.3, while in Sect. C.2, we recover these modes via direct ${N-}$body simulations using the ${N-}$body implementation presented in Sect. 4.4.1.

4.1.1 C.1 The Response Matrix Validation

In order to compute the system’s response matrix, we follow the method presented in Sects. 4.2.3 and 4.2.4. In addition, we use the same numerical parameters as the ones used in Sect. 4.3.1. The grid in the ${ (r_{\mathrm {p}} , r_{\mathrm {a}})-}$space is characterised by ${ r_{\mathrm {p}}^\mathrm{min} \!=\! 0.08 }$, ${ r_{\mathrm {a}}^\mathrm{max} \!=\! 4.92 }$ and ${ \Delta r \!=\! 0.05 }$. The sum on the resonant index $m_{1}$ is limited to ${ |m_{1}| \!\le \! m_{1}^\mathrm{max} \!=\! 7 }$. Finally, we consider basis elements given by Appendix 4.A, with the parameters ${ k_\mathrm{Ka} \!=\! 7 }$ and ${ R_\mathrm{Ka} \!=\! 5 }$, with a restriction of the radial basis elements to ${ 0 \!\le \! n \!\le \! 8 }$. One should note that despite having a disc that extends up to ${ R_\mathrm{max} \!=\! 20 }$, one can still safely consider a basis truncated at such a small radius $R_\mathrm{Ka}$, which allows us to efficiently capture the self-gravitating properties of the disc in the inner regions.

In order to search for unstable modes in a disc, one has to look for complex frequencies ${ \omega \!=\! \omega _{0} \!+\! \mathrm {i}\eta }$, such that the complex response matrix ${ \widehat{\mathbf {M}} (\omega _{0} , \eta ) }$ from Eq. (4.13) possesses an eigenvalue equal to 1. This complex frequency is then associated with an unstable mode of pattern speed ${ \omega _{0} \!=\! m_{\mathrm {p}} \Omega _{\mathrm {p}} }$ and growth rate $\eta$. In order to effectively determine the characteristics of the unstable modes, we follow an approach based on Nyquist contours, as presented in Pichon and Cannon (1997). For a fixed value of $\eta$, one studies the behaviour of the function ${ \omega _{0} \!\mapsto \! \text {det} \big [ \mathbf {I} \!-\! \widehat{\mathbf {M}} (\omega _{0} , \eta ) \big ] }$, which takes the form a continuous curve in the complex plane, called a Nyquist contour. For ${ \eta \!\rightarrow \! + \infty }$, one has ${ \widehat{\mathbf {M}} (\omega , \eta ) \!\rightarrow \! 0 }$, so that the contour will shrink around the point (1, 0). As a consequence, for a given value of $\eta$, the number of windings of the Nyquist contour around the origin of the complex plane gives a lower bound of the number of unstable modes with a growth rate superior to $\eta$. By varying the value of $\eta$, one can then determine the largest value of $\eta$ which admits an unstable mode, and this is the growth rate of the most unstable mode of the disc. Figure 4.16 illustrates these Nyquist contours for an unstable Mestel disc with the truncation index ${ \nu _{\mathrm {t}} \!=\! 6 }$. We gathered in Table 4.1 the results of the measurements for the three discs considered.

Fig. 4.16

Left panel Zoomed-in Nyquist contours in the complex plane of the function ${ \omega _{0} \!\mapsto \! \text {det} \big [ \mathbf {I} \!-\! \widehat{\mathbf {M}} (\omega _{0} , \eta ) \big ] }$ for various fixed values of $\eta$ illustrated with different colours. These contours were obtained via the matrix method for a truncated Mestel disc with ${ \nu _{\mathrm {t}} \!=\! 6 }$, $q \!=\! (V_{0}/\sigma _{r})^{2} \!-\! 1 \!=\! 6$, and looking for ${ m_{\mathrm {p}} \!=\! 2 }$ modes. One can note that for ${ \eta \!=\! 0.20 }$, the contour crosses the origin, which corresponds to the presence of an unstable mode. Right panel Illustration of the behaviour of the function ${ \omega _{0} \!\mapsto \! \log |\text {det} \big [ \mathbf {I} \!\cdot \! \widehat{\mathbf {M}} (\omega _{0} , \eta ) \big ] | }$, when considering the same truncated Mestel disc as in the left panel. Each coloured curve corresponds to a different fixed value for $\eta$. This representation allows us to determine the pattern speed ${ \omega _{0} \!=\! m_{\mathrm {p}} \Omega _{\mathrm {p}} \!\simeq \! 0.94 }$ of the system’s unstable mode

After having determined the characteristics ${ (\omega _{0} , \eta ) }$ of the unstable modes, one can then study their shapes in the physical space. To do so, one can compute ${ \widehat{\mathbf {M}} (\omega _{0} , \eta ) }$ and numerically diagonalise this matrix of size ${ n_\mathrm{max} \!\times \! n_\mathrm{max} }$, where $n_\mathrm{max}$ is the number of basis elements considered. One of the matrix eigenvalues is then almost equal to 1, and one can consider its associated eigenvector $\varvec{X}_\mathrm{mode}$. The shape of the mode is then given by

$$\begin{aligned} \Sigma _\mathrm{mode} (R , \phi ) = \text {Re} \bigg [ \sum _{p} \varvec{X}_\mathrm{mode}^{p} \, \Sigma ^{(p)} (R , \phi ) \bigg ] \, , \end{aligned}$$

(4.61)

where we wrote the considered surface density basis elements as $\Sigma ^{(p)}$. Figure 4.17 illustrates the shape of the recovered unstable mode for the truncated ${ \nu _{\mathrm {t}} \!=\! 4 }$ Mestel disc.

Fig. 4.17

Illustration of the dominant ${ m_{\mathrm {p}} \!=\! 2 }$ unstable mode for a truncated ${ \nu _{\mathrm {t}} \!=\! 4 }$ Mestel disc as recovered via the matrix method presented in Sect. 4.2.3. Only positive level contours are shown and they are spaced linearly between 10 and 90% of the maximum norm. The three resonance radii, associated with the resonance ILR, COR, and OLR have been represented, as defined by ${ \omega _{0} \!=\! m_{\mathrm {p}} \Omega _{\mathrm {p}} \!=\! \varvec{m} \!\cdot \! \varvec{\Omega } (R_{\varvec{m}}) }$, where the intrinsic frequencies ${ \varvec{\Omega } (R) \!=\! (\Omega _{\phi } (R) , \kappa (R)) }$ are computed within the epicyclic approximation, as in Eq. (3.87). See Fig. 3.9 for an illustration of the signification of these resonance radii

4.1.2 C.2 The ${N-}$Body Code Validation

Let us now investigate the same unstable modes via direct ${N-}$body simulations, in order to validate the ${N-}$body implementation on which Sect. 4.4 is based. We do not detail here the initial sampling of the particles required to setup the simulations, and details can be found in Appendix E of Fouvry et al. (2015). In order not to be significantly impacted by the initial Poisson shot noise and the lack of a quiet start sampling (Sellwood 1983), for each value of the truncation power $\nu _{\mathrm {t}}$, the simulations were performed with ${ N \!=\! 20\!\times \! 10^{6} }$ particles. As can be observed in Fig. 1 of Sellwood and Evans (2001), in order to recover correctly the characteristics of the disc’s unstable modes, an appropriate setting of the ${N-}$body code parameters is crucial. Following the description from Sect. 4.4.1, we consider a cartesian grid made of ${ N_\mathrm{mesh} \!=\! 120 }$ grid cells, while using a softening length given by ${ \varepsilon \!=\! R_{\mathrm {i}} / 60 }$. We also restrict the perturbing forces only to the harmonic sector ${ m_{\phi } \!=\! 2 }$, thanks to ${ N_\mathrm{ring} \!=\! 2400 }$ radial rings with ${ N_{\phi } \!=\! 720 }$ azimuthal points.

In order to extract the characteristics of the unstable modes from ${N-}$body realisations, one may proceed as follows. For each snapshot of the simulation, one can estimate the disc’s surface density as

$$\begin{aligned} \Sigma _\mathrm{star} (\varvec{x} , t)&\, = \mu \sum _{i = 1}^{N} \delta _{\mathrm {D}} (\varvec{x} \!-\! \varvec{x}_{i} (t)) \nonumber \\&\, = \sum _{p} b_{p} (t) \, \Sigma ^{(p)} (\varvec{x}) \, , \end{aligned}$$

(4.62)

where the sum on i in the first line is made on all particles in the simulation, and ${ \varvec{x}_{i} (t) }$ stands for the position of the $i^\mathrm{th}$ particle at time t. In the second line of Eq. (4.62), the sum on p is made on all the basis elements considered. Here, we consider the same basis elements as the ones considered previously in Sect. C.1. The basis coefficients ${ b_{p} (t) }$ are straightforward to estimate thanks to the biorthogonality property from Eq. (2.12). They read

$$\begin{aligned} b_{p} (t) = - \!\! \int \!\! \mathrm {d}\varvec{x} \, \Sigma _\mathrm{star} (\varvec{x} , t) \, \psi ^{(p) *} (\varvec{x}) = - \mu \sum _{i} \, \psi ^{(p)} (\varvec{x}_{i} (t)) \, . \end{aligned}$$

(4.63)

Because we are looking for unstable modes, we expect the coefficients ${ b_{p} (t) }$ to have a temporal dependence of the form ${ b_{p} (t) \!\propto \! \exp [ - \mathrm {i}( \omega _{0} \!+\! \mathrm {i}\eta ) t ] }$, where ${ \omega _{0} \!=\! m_{\mathrm {p}} \Omega _{\mathrm {p}} }$ is the pattern speed of the mode and $\eta$ its growth rate. If an unstable mode is present in the disc, one therefore expects the relations

$$\begin{aligned} \frac{\mathrm {d}\, \text {Re} \big [\! \log (b_{p} (t)) \!\big ] }{\mathrm {d}t} = \eta \;\;\; ; \;\;\; \frac{\mathrm {d}\, \text {Im} \big [\! \log (b_{p} (t)) \!\big ]}{\mathrm {d}t} = - \omega _{0} \, , \end{aligned}$$

(4.64)

provided one pays a careful attention to the branch-cut of the complex logarithm. These linear time dependences appear therefore as appropriate measurements to estimate the growth rate and pattern speed of unstable modes. Let us note that Eq. (4.64) does not hold anymore if more than one unstable mode of similar strength are present in the disc. Figure 4.18 illustrates such measurements for different values of the truncation index $\nu _{\mathrm {t}}$.

Fig. 4.18

Illustration of the measurements of the growth rates $\eta$ (left panel) and pattern speeds $\omega _{0}$ (right panel) of the ${ m_{\mathrm {p}} \!=\!2 }$ unstable modes of truncated Mestel discs, with a radial velocity dispersion given by ${ q \!=\! (V_{0} / \sigma _{r})^{2} \!-\! 1 \!=\! 6 }$, and truncation indices given by ${ \nu _{\mathrm {t}} \!=\! 6, 8 }$. The basis coefficient plotted corresponds to the indices ${ (\ell , n) \!=\! (2, 0) }$

Following the determination of the basis coefficients ${ b_{p} (t) }$, one can consequently study the shape of the recovered unstable mode in the physical space. Indeed, similarly to Eq. (4.61), the shape of the mode is given by

$$\begin{aligned} \Sigma _\mathrm{mode} (R , \phi , t) = \text {Re} \bigg [\! \sum _{p} b_{p} (t) \, \Sigma ^{(p)} (R , \phi ) \!\bigg ] \, . \end{aligned}$$

(4.65)

Similarly to Fig. 4.17, we illustrate in Fig. 4.19 the unstable mode of the same truncated ${ \nu _{\mathrm {t}} \!=\! 4 }$ Mestel disc, as recovered from ${N-}$body simulations.

Fig. 4.19

Illustration of the dominant ${ m_{\mathrm {p}} \!=\! 2 }$ unstable mode for a truncated ${ \nu _{\mathrm {t}} \!=\! 4 }$ Mestel disc as recovered via direct ${N-}$body simulations. Only positive level contours are shown and they are spaced linearly between 20 and 80% of the maximum norm. Similarly to Fig. 4.17, the radii associated with the resonances ILR, COR, and OLR are represented

As a conclusion, we gathered in Table 4.1 the growth rates and pattern speeds obtained either via the matrix method or via direct ${N-}$body simulations. As already noted in Sellwood and Evans (2001) when considering truncated Mestel discs, the recovery of the characteristics of unstable modes from direct ${N-}$body simulations remains a difficult task, for which the convergence to the values predicted through linear theory can be delicate.

Table 4.1 Measurements of the pattern speed ${ \omega _{0} \!=\! m_{\mathrm {p}} \Omega _{\mathrm {p}} }$ and growth rate $\eta$ for unstable ${m_{\mathrm {p}} \!=\! 2 }$ modes in truncated Mestel discs. The velocity dispersion is characterised by ${ q \!=\! (V_{0} / \sigma _{r})^{2} \!-\! 1 = 6 }$, and the power indices of the inner taper are given by ${ \nu _{\mathrm {t}} \!=\! 4, 6, 8}$. The theoretical values were obtained from a tailored linear theory in Evans and Read (1998b). Our measurements were performed either via the response matrix as in Sect. C.1, or via direct ${N-}$body simulations as in Sect. C.2

D  The Case of Self-Gravitating Spheres

In this Appendix, let us show how the previous calculations of the system’s response matrix and the associated diffusion flux presented for razor-thin discs, can straightforwardly be extended to 3D spherical systems. Analytical studies of the linear collective response of spherical self-gravitating systems have been considered by a number of authors (Tremaine and Weinberg 1984; Weinberg 1989; Seguin and Dupraz 1994; Murali and Tremaine 1998; Murali 1999; Weinberg 2001a; Pichon and Aubert 2006). Such calculations are of interest if one wants to describe the long-term evolution of spherical systems such as dark matter haloes while accounting for self-gravity. In Sect. D.1, we show how the main text calculations can straightforwardly be extended to such systems, while in Sect. D.2, we illustrate how such a formalism may be applied to the study of the cusp-core problem in the context of the long-term evolution of dark matter haloes.

4.1.1 D.1  The 3D calculation

As in the case of razor-thin axisymmetric potentials, 3D spherically symmetric potentials are also guaranteed by symmetry to be integrable. The three natural actions are given by

$$\begin{aligned} J_{1} = J_{r} = \frac{1}{\pi } \!\! \int _{r_{\mathrm {p}}}^{r_{\mathrm {a}}} \!\!\!\! \mathrm {d}r \, \sqrt{2 (E \!-\! \psi _{0} (r)) - L^{2} / r^{2}} \;\;\; ; \;\;\; J_{2} = L \;\;\; ; \;\;\; J_{3} = L_{z} \, , \end{aligned}$$

(4.66)

where the radial action $J_{r}$ was already introduced in Eq. (4.1), ${ L \!>\! 0}$ stands for the magnitude of the particle’s angular momentum, and $L_{z}$ its projection along the ${z-}$axis. Here, as previously, the first action $J_{r}$ encodes the amount of radial energy of the star, L encodes the typical distance of the star to the centre, while finally $L_{z}$ characterises the vertical orientation of the orbital plane to which the particle motion is restricted. The intrinsic frequencies of motion of the associated angles are given by ${ \varvec{\Omega } \!=\! (\Omega _{1} , \Omega _{2} , \Omega _{3}) }$. For spherical systems, one should see the third action $J_{3}$ as a mute variable, so that ${ \Omega _{3} \!=\! 0 }$. Therefore, one has an additional conserved quantity, namely ${ \theta _{3} \!=\! \text {cst.} }$, which corresponds to the longitude of the ascending node. As for razor-thin discs, the two additional frequencies of motion ${ \Omega _{1} \!=\! \kappa }$ and ${ \Omega _{2} \!=\! \Omega _{\phi } }$ are given by Eqs. (4.2) and (4.3). In this context, we may also use the pericentre and apocentre ${ (r_{\mathrm {p}} , r_{\mathrm {a}}) }$ to represent the two actions ${ (J_{1} , J_{2}) }$.

Let us now introduce the 3D spherical coordinates ${ (R , \theta , \phi ) }$. For 3D systems, the generic basis element can be decomposed as

$$\begin{aligned} \psi ^{(p)} (R, \theta , \phi ) = \psi _{n}^{\ell m} (R , \theta , \phi ) = Y_{\ell }^{m} (\theta , \phi ) \, \mathcal {U}_{n}^{\ell } (R) \, , \end{aligned}$$

(4.67)

where $Y_{\ell }^{m}$ are the usual spherical harmonics and $\mathcal {U}_{\ell }^{n}$ is a real radial function. Equation (4.67) is the direct spherical equivalent of the 2D decomposition from Eq. (4.5), and allows us to separate the angular dependences from the radial one. Similarly, the associated density elements are given by

$$\begin{aligned} \rho ^{(p)} (R , \theta , \phi ) = \rho ^{\ell m}_{n} (R , \theta , \phi ) = Y_{\ell }^{m} (\theta , \phi ) \, \mathcal {D}_{n}^{\ell } (R) \, , \end{aligned}$$

(4.68)

where $\mathcal {D}_{n}^{\ell }$ is a real radial function. Explicit 3D basis of potentials and densities elements can for example be built from spherical Bessel functions (Fridman and Poliachenko 1984; Weinberg 1989; Rahmati 2009) or from ultraspherical polynomials (Hernquist and Ostriker 1992). The spherical basis elements suggested in Weinberg (1989) are illustrated in Fig. 4.20.

Fig. 4.20

Left panel Illustration of the spherical harmonics $Y_{\ell }^{m}$ used to construct the 3D basis elements from Eq. (4.67). From top to bottom, the lines are associated with ${ \ell \!=\! 0, 1, 2 }$, and on a given line, the harmonics are represented for ${ - \ell \!\le \! m \!\le \! \ell }$. Right panel Illustration of the radial dependence of the basis function $\mathcal {U}_{n}^{\ell = 2}$, based on spherical Bessel functions and introduced in Weinberg (1989), for various values of the radial index ${ n \!\ge \! 1 }$. Here, the basis elements are defined on a finite radial range ${ R \!\le \! R_\mathrm{sys} }$

In 3D, the Fourier transformed basis elements from Eq. (4.7) become

$$\begin{aligned} \psi _{\varvec{m}} (\varvec{J}) = \frac{1}{(2 \pi )^{3}} \!\! \int \!\! \mathrm {d}\theta _{1} \mathrm {d}\theta _{2} \mathrm {d}\theta _{2} \, \psi ^{(p)} (\varvec{x} (\varvec{\theta } , \varvec{J})) \, \mathrm {e}^{- \mathrm {i}\varvec{m} \cdot \varvec{\theta }} \, , \end{aligned}$$

(4.69)

while the angle mapping from Eq. (4.8) still holds. In order to describe the orientation of the orbital plane, let us define the orbit’s inclination, ${ \beta \!=\! \beta (\varvec{J}) }$, as

$$\begin{aligned} J_{3} = J_{2} \, \cos (\beta ) \;\; \text {with} \;\; 0 \!\le \! \beta \!\le \! \pi \, . \end{aligned}$$

(4.70)

Following Tremaine and Weinberg (1984), Weinberg (1989), the Fourier transform in angles of the basis element ${ \psi ^{(p)} \!=\! \psi ^{\ell ^{p} m^{p}}_{n^{p}} }$ w.r.t. the resonance vector ${ \varvec{m} \!=\! (m_{1} , m_{2} , m_{3}) }$ takes the form

$$\begin{aligned} \psi ^{(p)}_{\varvec{m}} (\varvec{J}) = \delta _{m^{p}}^{m_{3}} \, \mathcal {V}_{\ell ^{p} m_{2} m^{p}} (\beta ) \, \mathcal {W}_{\ell ^{p} m_{2} n^{p}}^{m_{1}} (\varvec{J}) \, . \end{aligned}$$

(4.71)

In the previous equation, one should pay attention to the difference between the index $m^{p}$, which is the second index of the basis element from Eq. (4.67), and ${ \varvec{m} \!=\! (m_{1} , m_{2} , m_{3}) }$ corresponding to the three indices of the Fourier transform w.r.t. the angles. In Eq. (4.71), we introduced the coefficient ${ \mathcal {W}_{\ell ^{p} m_{2} n^{p}}^{m_{1}} (\varvec{J}) }$, whose expression was already obtained in Eq. (4.10). We also introduced the coefficient ${ \mathcal {V} _{\ell ^{p} m_{2} m^{p}} (\beta ) }$, specific to the 3D basis, which reads

$$\begin{aligned} \mathcal {V}_{\ell ^{p} m_{2} m^{p}} (\beta ) = \mathrm {i}^{m^{p} - m_{2}} \, Y_{\ell ^{p}}^{m_{2}} (\pi /2 , 0) \, \mathcal {R}_{m_{2} m^{p}}^{\ell ^{p}} (\beta ) \, , \end{aligned}$$

(4.72)

where we introduced the rotation matrix for spherical harmonics, given by

$$\begin{aligned} \mathcal {R}_{m_{2} m}^{\ell } (\beta ) = \sum _{t} (-1)^{t} \!&\, \frac{\sqrt{(\ell \!+\! m_{2})! \, (\ell \!-\! m_{2})! \, (\ell \!+\! m)! \, (\ell \!-\! m)!}}{(\ell \!-\! m \!-\! t)! \, (\ell \!+\! m_{2} \!-\! t)! \, t! \, (t \!+\! m \!-\! m_{2})! } \nonumber \\&\, \times \left[ \cos ( \beta /2 ) \right] ^{2 \ell + m_{2} - m - 2 t} \! \left[ \sin ( \beta /2 ) \right] ^{2 t + m - m_{2}} \, . \end{aligned}$$

(4.73)

In Eq. (4.73), the sum is to be made on all the "t" such that the arguments of the factorials are either zero of positive. It corresponds to ${ t_\mathrm{min} \!\le \! t \!\le \! t_\mathrm{max} }$, with ${ t_\mathrm{min} = \text {Max} [0 , m_{2} \!-\! m] }$ and ${ t_\mathrm{max} \!=\! \text {Min} [\ell \!+\! m , \ell \!+\! m_{2}] }$.

Having computed the Fourier transformed basis elements, one may then proceed to the evaluation of the system’s response matrix. As already detailed in Sect. 4.2.3, we perform the estimation of the response matrix by using ${ (r_{\mathrm {p}} , r_{\mathrm {a}}) }$ as our variables. To do so, the first step of the calculation is to go from ${ \varvec{J} \!=\! (J_{1} , J_{2} , J_{3}) }$ to ${ (E , L , \cos (\beta )) }$. Similarly to Eq. (4.12), the Jacobian of this transformation is given by

$$\begin{aligned} \frac{\partial (E , L , \cos (\beta ))}{\partial (J_{1} , J_{2} , J_{3})} = \begin{vmatrix} \displaystyle \Omega _{1}&\displaystyle \Omega _{2}&\displaystyle 0 \\ \displaystyle 0&\displaystyle 1&\displaystyle 0 \\ \displaystyle 0&\displaystyle - L_{z} / L^{2}&\displaystyle 1/L \end{vmatrix} = \frac{\Omega _{1}}{L} \, . \end{aligned}$$

(4.74)

One may now perform the integration w.r.t. the inclination $\beta$. Let us therefore assume that the system’s DF is such that ${ F \!=\! F (J_{1} , J_{2}) \!=\! F (E , L) }$. In addition, we noted previously that the system’s intrinsic frequencies ${ \varvec{\Omega } \!=\! (\Omega _{1} , \Omega _{2} , \Omega _{3}) }$ are independent of $J_{3}$, so that in the expression (2.17) of the response matrix, the only remaining dependences w.r.t. $\beta$ are in the Fourier transformed basis elements from Eq. (4.71) through the rotation matrix from Eq. (4.73). Following Edmonds (1996), the rotation matrices satisfy the orthogonality relation

$$\begin{aligned} \int _{-1}^{1} \!\!\! \mathrm {d} \cos (\beta ) \, \mathcal {R}_{m_{2} m_{3} }^{\ell ^{p} } (\beta ) \, \mathcal {R}_{m_{2} m_{3} }^{\ell ^{q} } (\beta ) = \delta _{\ell ^{p}}^{\ell ^{q}} \frac{2}{2 \ell ^{p} \!+\! 1} \, . \end{aligned}$$

(4.75)

Equation (4.72) then gives

$$\begin{aligned} \int _{-1}^{1} \!\!\!\! \mathrm {d} \cos (\beta ) \, \mathcal {V}^{*}_{\ell ^{p} m_{2} m_{3}} (\beta ) \, \mathcal {V}_{\ell ^{q} m_{2} m_{3} } (\beta ) = \delta _{\ell ^{p}}^{\ell ^{q}} \, \mathcal {C}_{\ell ^{p} m_{2}} \, , \end{aligned}$$

(4.76)

where we introduced the coefficient ${ \mathcal {C}_{\ell ^{p} m_{2}} }$ as

$$\begin{aligned} {\mathcal {C}}_{\ell ^{p} m_{2}} = \frac{2}{2 \ell ^{p} \!+\! 1} \big | Y_{\ell ^{p}}^{m_{2}} (\pi / 2 , 0) \big |^{2} \, . \end{aligned}$$

(4.77)

The expression (4.71) of the Fourier transformed basis elements allows us then to rewrite the response matrix from Eq. (2.17) as

(4.78)

where one may note that the sum on $m_{3}$ was dropped thanks to the Kronecker symbol from Eq. (4.71). In addition, expression (4.77) also imposes the additional constraints ${ |m_{2}| \!\le \! \ell ^{p} }$ and ${(\ell ^{p} \!-\! m_{2})}$ even, so that the sum on $m_{2}$ may also be reduced. Finally, we also relied on the fact that the coefficient ${ \mathcal {W}_{\ell ^{p} m_{2} n^{p}}^{m_{1}} (\varvec{J}) }$ from Eq. (4.10) is real, so that no conjugates are present in Eq. (4.78). Let us now perform the change of variables ${ (E , L) \!\mapsto \! (r_{\mathrm {p}} , r_{\mathrm {a}}) }$ to rewrite Eq. (4.78) as

(4.79)

where the functions ${ g_{m_{1} m_{2}}^{\ell ^{p} n^{p} n^{q}} (r_{\mathrm {p}} , r_{\mathrm {a}}) }$ and ${ h_{m_{1} m_{2}}^{\omega } (r_{\mathrm {p}} , r_{\mathrm {a}}) }$ are respectively given by

$$\begin{aligned} g_{m_{1} m_{2}}^{\ell ^{p} n^{p} n^{q}} (r_{\mathrm {p}} , r_{\mathrm {a}}) = (2 \pi )^{3} \, \mathcal {C}_{\ell ^{p} m_{2}} \, \left| \frac{\partial (E,L)}{\partial (r_{\mathrm {p}} , r_{\mathrm {a}})} \right| \, \frac{L}{\Omega _{1}} \, \left[ \varvec{m} \!\cdot \! \frac{\partial F}{\partial \varvec{J}} \right] \mathcal {W}_{\ell ^{p} m_{2} n^{p}}^{m_{1}} (\varvec{J}) \, \mathcal {W}_{\ell ^{p} m_{2} n^{q}}^{m_{1}} (\varvec{J}) \, , \end{aligned}$$

(4.80)

and

$$\begin{aligned} h_{m_{1} m _{2}}^{\omega } (r_{\mathrm {p}} , r_{\mathrm {a}}) = \omega \!-\! (m_{1} , m_{2}) \!\cdot \! (\Omega _{1} , \Omega _{2}) \, . \end{aligned}$$

(4.81)

In Eq. (4.80), if the system’s DF is such that ${ F \!=\! F (E ,L) }$, the gradient ${ \varvec{m} \!\cdot \! \partial F / \partial \varvec{J} }$ w.r.t. the actions may be computed following Eq. (4.17). Let us finally note the very strong analogies that exist between Eqs. (4.79) and (4.14) obtained for razor-thin discs. This allows us to apply to Eq. (4.79) the exact same method as described in Sect. 4.2.4 by truncating the ${ (r_{\mathrm {p}} , r_{\mathrm {a}})-}$space in small regions on which linear approximations may be performed. We do not repeat here these calculations. As the calculation of the response matrix can be a cumbersome numerical calculation, it is important to validate its implementation by recovering known unstable modes for 3D spherical systems, e.g., in Polyachenko and Shukhman (1981), Saha (1991), Weinberg (1991). Following this approach, one is therefore able to compute the response matrix of a 3D spherical system. In addition, the computation of the Fourier transformed basis elements in Eq. (4.71) allows us to subsequently compute the associated collisionless and collisional fluxes. Let us now illustrate in the upcoming section one possible application of such an approach to describe the secular evolution of dark matter haloes.

4.1.2 D.2 An Exemple of Application: The Cusp-Core Problem

Dark matter (DM) only simulations favour the formation of a cusp in the inner region of DM haloes (Dubinski and Carlberg 1991; Navarro et al. 1997), following what appears to be an universal profile, the NFW profile. However, observations tend to recover profiles more consistent with a shallower core profile (Moore 1994; de Blok and McGaugh 1997; de Blok et al. 2001; de Blok and Bosma 2002; Kuzio de Naray et al. 2008). This discrepancy between the cuspy profile predicted by direct DM only simulations and the core profile inferred from observations is one current important challenge in astrophysics, coined the cusp-core problem.

Various solutions have been proposed to resolve this discrepancy. A first set of solutions relies on modifying the dynamical properties of the collisionless dark matter preventing it from indeed collapsing into cuspy profile. Examples include the possibility of warm DM (Kuzio de Naray et al. 2010) or of self-interacting dark matter (Spergel and Steinhardt 2000; Rocha et al. 2013). Another set of solutions rely on the remark that accounting self-consistently for the baryonic physics and its back-reactions on the DM may also be at the origin of the discrepancy. These mechanisms can be divided into three broad categories. The first one relies on dynamical friction from infalling baryonic clumps and disc instabilities (El-Zant et al. 2001; Weinberg and Katz 2002; Tonini et al. 2006; Romano-Diz et al. 2008; Goerdt et al. 2010; Cole et al. 2011; Del Popolo and Pace 2016). A second possible mechanism is associated with AGN-driven feedback (Peirani et al. 2008; Martizzi et al. 2012; Dubois et al. 2016). Finally, a third possible mechanism relies on the long-term effects associated with supernova-driven feedback (Binney et al. 2001; Gnedin and Zhao 2002; Read and Gilmore 2005; Mashchenko et al. 2006, 2008; Governato et al. 2010; Teyssier et al. 2013; Pontzen and Governato 2012; El-Zant et al. 2016).

The collisionless diffusion equation (2.31), and its associated customisation to 3D systems presented in this Appendix is the appropriate framework to investigate in detail the role that supernova feedback may have on the secular evolution of DM haloes. Can the presence of an inner stellar disc, because of the potential fluctuations it induces, lead to the secular diffusion of a cuspy DM halo to a core one?

The first step of such an analysis is to characterise these fluctuations. To do so, we rely on hydrodynamical simulations. In order to decouple the source of perturbations, i.e. the disc, from the perturbed system, i.e. the halo, these simulations are performed while using a static and inert halo. Therefore, during the numerical simulations, feedback, while still present, cannot lead to a secular evolution of the halo profile. Similarly, any back-reaction from the halo onto the disc cannot be accounted for. Such a setup allows us to measure and characterise the statistical properties of the fluctuations induced by the disc directly from simulations. Because the DM halo is analytical, this also prevents any shot noise associated with the use of a finite number of DM particles. Once these fluctuations have been estimated, their effects on the DM halo may then be quantified using the secular collisionless diffusion equation (2.31).

In order to characterise these fluctuations, we consider an analytic NFW halo profile, and embed within it a gaseous and stellar disc, paying attention to preparing the system in a quasi-stationary state. In addition, we implement a supernova feedback recipe allowing for the release of energy from the supernovae into the interstellar medium. Figure 4.21 illustrates two successive snapshots of such a hydrodynamical simulation. In Fig. 4.21, one can note that because of supernova feedback, the gas density undergoes some fluctuations. These fluctuations in the potential due to the gas will be felt by the DM halo and may therefore be the driver of a resonant forced secular diffusion in the DM halo. Because we are interested in the ensemble average autocorrelation of the feedback fluctuations, various realisations are run for the same physical setup.

Fig. 4.21

Simulations run by Rebekka Bieri. Illustration of the gas density in a hydrodynamical simulation performed with the AMR code Ramses (Teyssier 2002). The stellar and gaseous discs are embedded in a static NFW DM halo and are seen from the top (top panel) and the edge (bottom panel). A supernova feedback recipe was implemented following Kimm et al. (2005). As can be seen in these two snapshots, this leads to fluctuations in the gas density, which resonantly couple to the DM halo and induce therein a resonant diffusion

Once these simulations are performed, we characterise their statistics by computing the autocorrelation matrix $\widehat{\mathbf {C}}$ introduced in Eq. (2.25). This first requires to compute the basis coefficients ${ b_{p} (t) }$, so that the perturbing potential ${ \delta \psi ^{\mathrm {e}} }$ may be written as ${ \delta \psi ^{\mathrm {e}} (\varvec{x} , t) \!=\! \sum _{p} b_{p} (t) \psi ^{(p)} (\varvec{x}) }$. Thanks to the biorthogonality property from Eq. (2.12), these basis coefficients are immediately given by

$$\begin{aligned} b_{p} (t) = - \!\! \int \!\! \mathrm {d}\varvec{x} \, \psi ^{(p) *} (\varvec{x}) \, \delta \rho ^{\mathrm {e}} (\varvec{x} , t) \, , \end{aligned}$$

(4.82)

where ${ \delta \rho ^{\mathrm {e}} }$ stands for the star and gas density fluctuations within the disc, as illustrated in Fig. 4.21. In Fig. 4.22, we illustrate the behaviour of the function ${ t \!\mapsto \! b_{p} (t) }$ for three different basis elements and ten different realisations. In Fig. 4.22, one can note, as expected, that the fluctuations history vary from one realisation to another. Similarly, the typical frequencies of the fluctuations also depend on the considered basis elements. Once the perturbations history ${ t \!\mapsto \! b_{p} (t) }$ is extracted from the numerical simulations, one may follow Eq. (2.25) to compute their ensemble averaged autocorrelation matrix ${ \widehat{\mathbf {C}}_{pq} (\omega ) }$. This fully characterises the stochastic external source term, which sources the collisionless diffusion coefficients ${ D_{\varvec{m}} (\varvec{J}) }$ from Eq. (2.32). When the characteristics of the initial DM halo (namely its potential and associated self-consistent DF) have been specified, one may follow Appendix 4.D.1 to compute the halo’s response matrix $\widehat{\mathbf {M}}$. This allows finally for the calculation of the diffusion coefficients $D_{\varvec{m}} (\varvec{J})$, and for the estimation of the collisionless diffusion flux $\varvec{\mathcal {F}}_\mathrm{tot}$ from Eq. (2.33). This diffusion flux characterises the initial orbital restructuration undergone by the DM halo’s DF.

Fig. 4.22

Illustration of the behaviour of the basis coefficients ${ t \!\mapsto \! b_{p} (t) }$, for three different basis elements, i.e. different values of the basis indices ${ (\ell , m , n) }$. Each colour corresponds to a different realisation. One can note the presence of potential fluctuations associated with supernovae feedback. The autocorrelation of these fluctuations, captured by the matrix $\widehat{\mathbf {C}}$, will drive a feedback-induced secular diffusion in the DM halo

In this context, a final difficulty arises from the integration forward in time of the collisionless diffusion equation (2.34). Indeed, because the diffusion is self-consistent, the diffusion coefficients are slow functions of the halo’s DF. Similarly, the halo’s potential (initially cuspy) also secularly depends on the halo’s DF. The integration of the diffusion equation therefore requires a self-consistent update of the halo’s diffusion coefficients and the halo’s potential. One possible approach to solve this self-consistency problem relies on an iterative approach (Prendergast and Tomer 1970; Weinberg 2001b) that we do not detail further here.

The method described previously is expected to allow for a detailed description of the resonant collisionless diffusion occurring in a DM halo as a result of stochastic external fluctuations induced by its inner galactic disc. The same approach could also allow us to investigate how much this diffusion mechanism depends on the strength of the feedback mechanisms, by for example changing the feedback recipes used in the hydrodynamical simulations. One could determine the typical power spectrum of the perturbations induced by the feedback, or give quantitative bounds on the feedback strengths required to induce a softening of the DM halo’s profile. Similarly, the dependence of the diffusion efficiency w.r.t. the disc and halo masses could also be investigated. Finally, the efficiency of AGN feedback to induce a secular diffusion in the DM halo could also be studied within the same framework by characterising the associated potential fluctuations.