Thickened Discs

Abstract

This chapter investigates the dynamics of thickened stellar discs, simplifies their dynamics via a new thickened WKB approximation, and investigates various possible sources of secular thickening.

The work presented in this chapter is based on Fouvry et al. (2017).

Appendices

Appendix

A Antisymmetric Basis

In Sect. 5.3, we restricted ourselves to the construction of the symmetric thick WKB basis elements. One can proceed similarly for the antisymmetric ones. Assuming ${ \psi _{z} (- z) \!=\! - \psi _{z} (z) }$, the ansatz from Eq. (5.12) immediately imposes ${ D \!=\! - A }$ and ${ C \!=\! - B }$, while the continuity conditions from Eq. (5.13) become

$$\begin{aligned} {\left\{ \begin{array}{ll} \displaystyle A \, \mathrm {e}^{- k_{r} h} = 2 \mathrm {i}B \sin (k_{z} h) \, , \\ \displaystyle k_{r} \, A \, \mathrm {e}^{- k_{r} h} = - 2 \mathrm {i}k_{z} B \cos (k_{z} h) \, . \end{array}\right. } \end{aligned}$$

(5.99)

Similarly to Eq. (5.14), we obtain the antisymmetric quantisation relation

$$\begin{aligned} \tan (k_{z} h) = - \frac{k_{z}}{k_{r}} \, , \end{aligned}$$

(5.100)

which is illustrated in Fig. 5.4. One can also note that the antisymmetric elements also follow the typical step distance ${ \Delta k_{z} }$ obtained in Eq. (5.17). Following Eq. (5.18), the full expression of the antisymmetric elements is given by

$$\begin{aligned} \psi ^{[k_{\phi } , k_{r} , R_{0} , n]} (R , \phi , z) = \mathcal {A} \, \psi _{r}^{[k_{\phi } , k_{r} , R_{0}]} (R , \phi ) \, {\left\{ \begin{array}{ll} \begin{aligned} \displaystyle &{} \sin (k_{z}^{n} z) &{} \text {if} &{} \;\; |z| \!\le \! h \, , \\ \displaystyle &{} \mathrm {e}^{kr h} \sin (k_{z}^{n} h) \, \mathrm {e}^{- k_{r} |z|} &{} \text {if} &{} \;\; z \!\ge \! h \, , \\ \displaystyle &{} - \mathrm {e}^{k_{r} h } \, \sin (k_{z}^{n} h) \, \mathrm {e}^{- k_{r} |z|} &{} \text {if} &{} \;\; |z| \!\le \! h \, , \end{aligned} \end{array}\right. } \end{aligned}$$

(5.101)

and

$$\begin{aligned} \rho ^{[k_{\phi } , k_{r} , R_{0} , n]} (R , \phi , z) = - \frac{k_{r}^{2} \!+\! (k_{z}^{n})^{2}}{4 \pi G} \, \psi ^{[k_{\phi } , k_{r} , R_{0} , n]} (R , \phi , z) \, \Theta \bigg [\! \frac{z}{h} \!\bigg ] \, . \end{aligned}$$

(5.102)

Similarly to Eq. (5.20), the amplitude of the antisymmetric elements is given by

$$\begin{aligned} \mathcal {A} = \sqrt{\frac{G}{R_{0} h (k_{r}^{2} \!+\! (k_{z}^{n})^{2})}} \, \beta _{n} \, , \end{aligned}$$

(5.103)

where, in analogy with Eq. (5.21), $\beta _{n}$ is a numerical prefactor reading

$$\begin{aligned} \beta _{n} = \sqrt{\frac{2}{1 \!-\! \sin (2 k_{z}^{n} h)/(2 k_{z}^{n} h)}} \, . \end{aligned}$$

(5.104)

As illustrated in Fig. 5.4, let us note that the antisymmetric quantisation relation (5.100) imposes for the antisymmetric vertical frequency to satisfy ${ k_{z}^{1} \!>\! \pi / (2 h) }$, and in this domain, one has ${ 1.3 \!\le \! \beta _{n} \!\le \! 1.5 }$. Similarly to Eq. (5.22), the Fourier transformed antisymmetric basis elements are given by

$$\begin{aligned} \psi _{\varvec{m}}^{[k_{\phi } , k_{r} , R_{0} , n]} (\varvec{J}) = \delta _{m_{\phi }}^{k_{\phi }} \, \delta _{m_{z}}^\mathrm{{odd}} \, \mathcal {A} \, \mathrm {e}^{\mathrm {i}k_{r} R_{\mathrm {g}}} \, \mathrm {i}^{m_{z} - 1 - m_{r}} \mathcal {B}_{R_{0}} (R_{\mathrm {g}}) \, \mathcal {J}_{m_{r}} \!\bigg [\! \sqrt{\!\tfrac{2 J_{r}}{\kappa }} \, k_{r} \!\bigg ] \, \mathcal {J}_{m_{z}} \!\bigg [\! \sqrt{\!\tfrac{2 J_{z}}{\nu }} \, k_{z}^{n} \!\bigg ] \, . \end{aligned}$$

(5.105)

B A Diagonal Response Matrix

In this Appendix, let us detail why we may assume, as in Eq. (5.23), that the disc’s response matrix is diagonal in the thickened WKB limit. Let us first note that because the symmetric (resp. antisymmetric) Fourier transformed basis elements from Eq. (5.22) (resp. (5.105)) involve a ${ \delta _{m_{z}}^\mathrm{{even}} }$ (resp. $\delta _{m_{z}}^\mathrm{{odd}}$), the response matrix coefficients from Eq. (2.17) are equal to zero as soon as the two considered basis elements do not share the same vertical symmetry. We may therefore treat separately the symmetric and antisymmetric cases.

The thickened WKB basis elements introduced in Eq. (5.6) depend on four indices ${ [k_{\phi } , k_{r} , R_{0} , n ] }$. Following the same argument as in Sect. 3.4 for the razor-thin case, we may assume that the response matrix is diagonal w.r.t. the indices ${ [k_{\phi } , k_{r} , R_{0} ] }$. As a consequence, it then only remains to check whether or not for a given set ${ [ k_{\phi } , k_{r} , R_{0} ] }$, the response matrix is diagonal w.r.t. the index $k_{z}^{n}$. The expression (5.24) of the symmetric diagonal basis elements is straightforward to generalise to the non-diagonal ones and gives

$$\begin{aligned} \widehat{\mathbf {M}}_{pq} = \frac{2 \pi G \Sigma \alpha _{p} \alpha _{q}}{h \kappa ^{2} \sqrt{(1 \!+\! (k_{z}^{p}/k_{r})^{2}) (1 \!+\! (k_{z}^{q} / k_{r})^{2})}}&\, \sum _{\ell _{z} \mathrm {even}} \!\! \exp \!\bigg [\! - \frac{(k_{z}^{p})^{2} \!+\! (k_{z}^{q})^{2}}{2 \nu ^{2} / \sigma _{z}^{2}} \!\bigg ] \, \mathcal {I}_{\ell _{z}} \!\bigg [\! \frac{k_{z}^{p} k_{z}^{q}}{\nu ^{2} / \sigma _{z}^{2}} \!\bigg ] \nonumber \\&\, \times \frac{1}{(1 \!-\! s_{\ell _{z}}^{2})} \bigg \{\! \mathcal {F} (s_{\ell _{z}} , \chi _{r}) \!-\! \ell _{z} \frac{\nu }{\sigma _{z}^{2}} \frac{\sigma _{r}^{2}}{\kappa } \mathcal {G} (s_{\ell _{z}} , \chi _{r} ) \!\bigg \} \, . \end{aligned}$$

(5.106)

As already underlined in Eq. (5.28), starting from Eq. (5.106), it is straightforward to obtain the expression of the associated antisymmetric non-diagonal coefficients thanks to the substitution ${ \alpha \!\rightarrow \! \beta }$ and the restriction of the sum on $\ell _{z}$ to odd values. Because it is a symmetric matrix, showing that the response matrix is diagonal amounts to proving that for ${ p \!\ne \! q }$, one has ${ \widehat{\mathbf {M}}_{pq} \!\ll \! \widehat{\mathbf {M}}_{pp} }$. In order to perform such a comparison, let us focus in Eq. (5.106) on the quantities which depend on $k_{z}^{p}$ and $k_{z}^{q}$. We introduce the quantity $K_{pq}^{(\ell _{z})}$ as

$$\begin{aligned} K_{pq}^{(\ell _{z})} = \frac{1}{\sqrt{(1 \!+\! (k_{z}^{p} / k_{r})^{2}) (1 \!+\! (k_{z}^{q} / k_{r})^{2})}} \, \exp \!\bigg [\! - \frac{(k_{z}^{p})^{2} \!+\! (k_{z}^{q})^{2}}{2 \nu ^{2} / \sigma _{z}^{2}} \!\bigg ] \, \mathcal {I}_{\ell _{z}} \!\bigg [\! \frac{k_{z}^{p} k_{z}^{q}}{\nu ^{2}/\sigma _{z}^{2}} \!\bigg ] \, . \end{aligned}$$

(5.107)

One can note that the definition from Eq. (5.107) does not involve the prefactors $\alpha _{p}$ and $\alpha _{q}$, as they are always of order unity. In addition, Eq. (5.107) does not involve the terms ${ \mathcal {F} (s_{\ell _{z}} , \chi _{r}) }$, ${ \mathcal {G} (s_{\ell _{z}} , \chi _{r}) }$, and ${ 1 / (1 \!-\! s_{\ell _{z}}) }$ from Eq. (5.106), as they do not depend on the choices of $k_{z}^{p}$ and $k_{z}^{q}$. Figure 5.23 illustrates the behaviours of the reduction functions ${ s_{\ell _{z}} \!\mapsto \! \mathcal {F} (s_{\ell _{z}} , \chi _{r}) }$, ${ \mathcal {G} (s_{\ell _{z}} , \chi _{r}) }$ defined in Eq. (5.27). We note in Fig. 5.23 that these functions are ill-defined when computed for integer values of $s_{\ell _{z}}$. In order to regularise these diverging behaviours a small imaginary part is added to $s_{\ell _{z}}$. While this procedure works for exactly integer values, this does not however prevent the divergences of $\mathcal {F}$ and $\mathcal {G}$ in the neighbourhood of integers. As illustrated in Fig. 5.23, in order to avoid these divergences, let us assume that the functions $\mathcal {F}$ and $\mathcal {G}$ can be approximated by the smooth functions

Fig. 5.23

Illustration for ${ \chi \!=\! 1 }$ of the behaviour of the reduction functions ${ s \!\mapsto \! \mathcal {F} (s , \chi ) }$ (left panel) and ${ s \!\mapsto \! \mathcal {G} (s , \chi ) }$ (right panel) given by the black curves, along with their approximations from Eq. (5.108) given by the grey lines. One should note the divergences of these functions in the neighbourhood of integers. However, these functions are well defined when evaluated for integer values of s, provided one considers ${ \lim _{\eta \rightarrow 0} \text {Re} [\mathcal {F} (n \!+\! \mathrm {i}\eta , \chi )] }$ (similarly for $\mathcal {G}$), as illustrated with the black dots

$$\begin{aligned} \mathcal {F} (s_{\ell _{z}} , \chi _{r}) \simeq f_{r} \;\;\; ; \;\;\; \mathcal {G} (s_{\ell _{z}} , \chi _{r}) \simeq - g_{r} s_{\ell _{z}} \, , \end{aligned}$$

(5.108)

where $f_{r}$ and $g_{r}$ do not depend on $s_{\ell _{z}}$. As already underlined in Eq. (5.29), when computing the collisionless diffusion coefficients from Eq. (2.32) or the dressed susceptibility coefficients from Eq. (2.50), the frequency $\omega$ should be considered at resonance so that ${ \omega \!=\! \varvec{m} \!\cdot \! \varvec{\Omega } }$. Following Eq. (5.29), the value of $s_{\ell _{z}}$ is either an integer (for ${ \ell _{z} \!=\! m_{z} }$) or far from one provided that ${ \nu / \kappa }$ is of high rational order. This distance from the exact resonance justifies the approximations from Eq. (5.108). Thanks to these approximations, the sum on $\ell _{z}$ in Eq. (5.106) may then be cut out according to the resulting powers of $\ell _{z}$. In order to prove that for ${ p \!\ne \! q }$, one has ${ \widehat{\mathbf {M}}_{pq} \!\ll \! \widehat{\mathbf {M}}_{pp} }$, one should therefore prove that

$$\begin{aligned} S_{\gamma } (p , q) = \sum _{\ell _{z}} \frac{\ell _{z}^{\gamma } K_{pq}^{(\ell _{z})}}{1 \!-\! s_{\ell _{z}}^{2}} \ll S_{\gamma } (p , p) \, , \end{aligned}$$

(5.109)

where the power index $\gamma$ is such that ${ \gamma \!\in \! \{ 0, 1, 2 \} }$.

In order to further non-dimensionalise the problem, let us introduce the typical dynamical height of the disc, ${ d \!=\! \sigma _{z} / \nu }$, as well as the dimensionless quantities

$$\begin{aligned} \ell _{p} = k_{z}^{p} d \;\;\; ; \;\;\; \ell _{q} = k_{z}^{q} d \;\;\; ; \;\;\; \ell _{r} = k_{r} d \, , \end{aligned}$$

(5.110)

which allow us to rewrite Eq. (5.107) as

$$\begin{aligned} K_{pq}^{(\ell _{z})} = \frac{\mathcal {I}_{\ell _{z}} \big [ \ell _{p} \ell _{q} \big ] \, \mathrm {e}^{- (\ell _{p}^{2} + \ell _{q}^{2})/2}}{\sqrt{(1 \!+\! (\ell _{p} / \ell _{r})^{2}) (1 \!+\! (\ell _{q} / \ell _{r})^{2})}} \, . \end{aligned}$$

(5.111)

As was already illustrated in Fig. 5.4, let us recall that the fundamental symmetric frequency is significantly different from the other quantised frequencies (both symmetric and antisymmetric), as it is the only frequency inferior to ${ \pi / (2 h) }$. In order to emphasise this very specific property, in this Appendix only, let us renumber the vertical indices p, such that ${ p \!=\! 0 }$ corresponds to the quantised fundamental symmetric mode, while ${ p \!\ge \! 1 }$ corresponds to the rest of the quantised frequencies, all superior to ${ \pi / (2 h) }$. With such a choice, the numbering of the antisymmetric elements only starts at ${ p \!=\! 1 }$. Following Fig. 5.4, one has the inequalities

$$\begin{aligned} 0< \ell _{0}< \frac{\pi }{2 \sqrt{2}} \;\; ; \;\; \frac{(p \!-\! \tfrac{1}{2}) \, \pi }{\sqrt{2}}< \ell _{p} < \frac{(p + \tfrac{1}{2}) \, \pi }{\sqrt{2}} \;\; \text {(for}\, { p\!\ge \! 1}\text {)} \, , \end{aligned}$$

(5.112)

where, following Eq. (5.74) for the Spitzer profile, we relied on the relation ${ h \!=\! 2d }$, with h the height of the WKB sharp cavity (see Fig. 5.3). Similarly, one has the relation ${ \ell _{r} \!=\! (k_{r} h) / \sqrt{2} }$.

Let us note that the expression (5.111) of $K_{pq}^{(n)}$ involves a modified Bessel function ${ \mathcal {I}_{n} [ \ell _{p} \ell _{q} ] }$ that needs as well to be approximated carefully. Equivalents in 0 and ${ + \infty }$ of these Bessel functions are immediately given by

$$\begin{aligned} \mathcal {I}_{n} (x) \underset{0}{\sim } \frac{1}{n!} \bigg (\! \frac{x}{2} \!\bigg )^{n} \;\;\; ; \;\;\; \mathcal {I}_{n} (x) \underset{+ \infty }{\sim } \frac{\mathrm {e}^{x}}{\sqrt{2 \pi x}} \, . \end{aligned}$$

(5.113)

As illustrated in Fig. 5.24, for a given value of n and x, one has to determine which approximation (polynomial or exponential) is relevant for ${ \mathcal {I}_{n} (x) }$. Let us therefore define for each ${ n \!\ge \! 0 }$, the quantity $x_{n}$ such that for ${ x \!\le \! x_{n} }$ (resp. ${ x \!\ge \! x_{n} }$), one uses the asymptotic development from Eq. (5.113) in 0 (resp. ${ + \infty }$). Because in the expression (5.111) the Bessel functions are only evaluated in ${ \ell _{p} \ell _{q} }$, for p and q given, there exists an integer $n_{pq}$ such that

$$\begin{aligned} \forall \, \ell _{z} < n_{pq} \, , \;\; \mathcal {I}_{\ell _{z}} \big [ \ell _{p} \ell _{q} \big ] \simeq \frac{\mathrm {e}^{\ell _{p} \ell _{q}}}{\sqrt{2 \pi \ell _{p} \ell _{q}}} \;\;\; ; \;\;\; \forall \, \ell _{z} \ge n_{pq} \, , \;\; \mathcal {I}_{\ell _{z}} \big [ \ell _{p} \ell _{q} \big ] \simeq \frac{1}{\ell _{z}!} \bigg (\! \frac{\ell _{p} \ell _{q}}{2} \!\bigg )^{\ell _{z}} \, . \end{aligned}$$

(5.114)

Fig. 5.24

Illustration of the asymptotic behaviours of the modified Bessel function $\mathcal {I}_{n}$ as given by Eq. (5.113). The full lines are the four first Bessel functions, along with their polynomial approximations in zero (dashed curves). The black dashed curve is their common exponential approximation. The transition between the two approximations is given by the quantity $x_{n}$

In Fig. 5.24, let us finally note that, except for ${ \ell _{z} \!=\! 0 }$, the exponential approximation of the Bessel function is significantly bigger than the actual value of $\mathcal {I}_{\ell _{z}}$. This does not impact the upcoming discussion, as, when proving ${ \widehat{\mathbf {M}}_{pq} \!\ll \! \widehat{\mathbf {M}}_{pp} }$, the exponential approximation is applied for $\widehat{\mathbf {M}}_{pq}$ alone, or for $\widehat{\mathbf {M}}_{pq}$ and $\widehat{\mathbf {M}}_{pp}$ simultaneously with similar errors, so that the comparisons between the approximations also hold for the exact values. Following Eq. (5.109), a naive approach to compare the terms ${ S_{\gamma } (p , q) }$ and ${ S_{\gamma } (p,p) }$ would be to compare the sum on $\ell _{z}$ term by term, i.e. to prove that ${ K_{pq}^{(\ell _{z})} \!\ll \! K_{pp}^{(\ell _{z})} }$ for all $\ell _{z}$. However, this is not sufficient and one should therefore be more cautious. In Eq. (5.109), one cuts out the sum on $\ell _{z}$ appearing in ${ S_{\gamma } (p,q) }$ between three different contributions, for which one can straightforwardly show:

• For the first terms, with ${ |\ell _{z}| \!<\! n_{pp} }$ and ${ |\ell _{z}| \!<\! n_{pq} }$:

  $$\begin{aligned} K_{pq}^{(\ell _{z})} \ll K_{pp}^{(1)} \, . \nonumber \end{aligned}$$

• For the intermediate terms, with ${ n_{pp} \!\le \! |\ell _{z}| \!<\! n_{pq} }$:

  $$\begin{aligned} \sum _{n_{pp} \le |\ell _{z}| < n_{pq}} \!\! \frac{\ell _{z}^{\gamma } K_{pq}^{(\ell _{z})}}{1 \!-\! s_{\ell _{z}}^{2}} \ll K_{pp}^{(1)} \, . \nonumber \end{aligned}$$

• For the last terms, with ${ |\ell _{z}| \!\ge \! n_{pq} }$:

  $$\begin{aligned} \sum _{|\ell _{z}| \ge n_{pq}} \frac{\ell _{z}^{\gamma } K_{pq}^{(\ell _{z})}}{1 \!-\!s_{\ell _{z}}^{2}} \ll K_{pp}^{(1)} \, . \nonumber \end{aligned}$$

This last relation holds whenever ${ k_{r} h \gtrsim 0.03 }$, but gets violated for ${ q \!=\! 0 }$ in the limit of a razor-thin disc. The previous comparisons are straightforward to obtain thanks to the step distances between consecutive basis elements from Eq. (5.112) and the use of the approximations of the Bessel functions from Eq. (5.113). The combination of these relations shows that for ${ k_{r} h \!\gtrsim \! 0.03 }$, for all p and q, one has ${ \widehat{\mathbf {M}}_{pq} \!\ll \! \widehat{\mathbf {M}}_{pp} }$. The same conclusion also holds for ${ k_{r} h \!\lesssim \! 0.03 }$, but only for ${ q \!\ne \! 0 }$. We therefore reached the following conclusions:

• The antisymmetric response matrix can always be assumed to be diagonal.

• For ${ k_{r} h \!\gtrsim \! 0.03 }$, the symmetric matrix response can be assumed to be diagonal

• For ${ k_{r} h \!\lesssim \! 0.03 }$, i.e. in the limit of a razor-thin disc, the symmetric response matrix takes the form of an arrowhead matrix.

As a last step of this Appendix, let us finally justify why for a sufficiently thin disc, for which the symmetric response matrix takes the form of an arrowhead matrix, the diagonal response matrix can still be assumed to be diagonal. In this limit, the symmetric response matrix takes the form

$$\begin{aligned} \widehat{\mathbf {M}} = \left( \begin{array}{cccc} \alpha &{} z_{1} &{} \cdots &{} z_{n} \\ z_{1} &{} d_{1} &{} &{} \\ \vdots &{} &{} \ddots &{} \\ z_{n} &{} &{} &{} d_{n} \end{array} \right) \, , \end{aligned}$$

(5.115)

where thanks to the previous calculations, one has the comparison relations ${ \alpha \!\gg \! z_{i} }$ and ${ z_{i} \!\gg \! d_{i} }$. Let us assume that ${ \forall i \, , z_{i} \!\ne \! 0 }$ and that ${ \forall i \!\ne \! j \, , d_{i} \!\ne \! d_{j} }$. Following O’Leary and Stewart (1990), it can be shown that the eigenvalues ${ (\lambda _{i})_{0 \le i \le n} }$ of the arrowhead matrix from Eq. (5.115) are the ${ (n \!+\! 1) }$ solutions of the equation

$$\begin{aligned} f_{\widehat{\mathbf {M}}} (\lambda ) = \alpha - \lambda - \sum _{i = 1}^{n} \! \frac{z_{i}^{2}}{d_{i} \!-\! \lambda } = 0 \, . \end{aligned}$$

(5.116)

In addition, provided that the $d_{i}$ are in descending order, these eigenvalues are interlaced so that

$$\begin{aligned} \lambda _{0}> d_{1}> \lambda _{1}> ...> d_{n} > \lambda _{n} \, . \end{aligned}$$

(5.117)

Finally, the eigenvectors $\varvec{x}_{i}$ associated with the eigenvalue $\lambda _{i}$ are proportional to

$$\begin{aligned} \varvec{x}_{i} = \bigg ( 1 \, ; \, \frac{z_{1}}{\lambda _{i} \!-\! d_{1}} \, ; \, ... \, ; \, \frac{z_{j}}{\lambda _{i} \!-\! d_{j}} \, ; \, ... \, ; \, \frac{z_{n}}{\lambda _{i} \!-\! d_{n}} \bigg ) \, . \end{aligned}$$

(5.118)

Accounting for the comparison relations ${ \alpha \!\gg \! z_{i} }$ and ${ z_{i} \!\gg \! d_{i} }$, Fig. 5.25 illustrates the behaviour of the function ${ \lambda \!\mapsto \! f_{\mathbf {\widehat{M}}} (\lambda ) }$ from Eq. (5.116).

Fig. 5.25

Illustration of the behaviour of the function ${ \lambda \!\mapsto \! f_{\widehat{\mathbf {M}}} (\lambda ) }$, whose roots are the eigenvalues of the arrowhead response matrix from Eq. (5.115)

In order to justify why the arrowhead response matrix from Eq. (5.115) may be considered as diagonal, one has to justify that, despite its first line and column, the matrix eigenvalues remain close to the matrix coefficients, so that

$$\begin{aligned} \lambda _{0} \simeq \alpha \;\;\; \text {and} \;\;\; \lambda _{i} \simeq d_{i} \; \text {(for}\, {i \!\ge \! 1}\text {)} \, . \end{aligned}$$

(5.119)

In addition, one must also ensure that the associated eigenvectors $\varvec{x}_{i}$ remain close to the natural basis elements so that

$$\begin{aligned} \varvec{x}_{i} \simeq (0 \, ; \, ... \, ; \, 1 \, ; \, 0 \, ; \, ...) \, , \end{aligned}$$

(5.120)

where the non-zero index is at the ith position. As illustrated in Fig. 5.25, the determination of the eigenvalues $\lambda _{i}$ requires to solve Eq. (5.116), which may be rewritten as

$$\begin{aligned} 1 - \frac{\lambda _{i}}{\alpha } - \sum _{i = 1}^{n} \frac{(z_{i} / \alpha )^{2}}{(d_{i} / \alpha ) \!-\! (\lambda _{i} / \alpha )} = 0 \, . \end{aligned}$$

(5.121)

Because we have ${ (z_{i} / \alpha ) \!\ll \! 1 }$, in order for Eq. (5.121) to be satisfied, one must necessarily either have ${ \lambda _{i} / \alpha \!\simeq \! 1 }$ or ${ ((d_{i} / \alpha ) \!-\! (\lambda _{i} / \alpha )) \!\ll \! 1 }$. It then follows immediately that ${ \lambda _{0} \!\simeq \! \alpha }$ and ${ \lambda _{i} \!\simeq \! d_{i} }$. Equation (5.119) therefore holds and the matrix eigenvalues $\lambda _{i}$ remain close to the matrix diagonal coefficients ${ (\alpha , d_{1} , ... , d_{n}) }$. The eigenvectors $\varvec{x}_{i}$ from Eq. (5.118) may then be rewritten as

$$\begin{aligned} \varvec{x}_{i} = \bigg ( 1 \, ; \, \frac{(z_{1} / \alpha )^{2}}{(\lambda _{i}/\alpha ) \!-\! (d_{1} / \alpha )} \frac{1}{(z_{1} / \alpha )} \, ; \, ... \, ; \, \frac{(z_{j} / \alpha )^{2}}{(\lambda _{i} / \alpha ) \!-\! (d_{j} / \alpha )} \frac{1}{(z_{j} / \alpha )} \, ; \, ... \bigg ) \, . \end{aligned}$$

(5.122)

Let us consider the first eigenvector associated with ${ i \!=\! 0 }$. Following Eq. (5.119), one has ${ \lambda _{0} \!\simeq \! \alpha }$, so that, because ${ d_{j} \!\ll \! \alpha }$, the generic term from Eq. (5.122) becomes

$$\begin{aligned} \frac{(z_{j} / \alpha )^{2}}{(\lambda _{0} / \alpha ) \!-\! (d_{j} / \alpha )} \frac{1}{(z_{j} / \alpha )} \simeq \frac{(z_{j} / \alpha )}{1} \ll 1 \, , \end{aligned}$$

(5.123)

where we relied on the fact that ${ z_{j} \!\ll \! \alpha }$. As consequence, for ${ i \!=\! 0 }$ in Eq. (5.122), all the terms except the first one are negligible in front of 1, and one gets ${ \varvec{x}_{0} \!\simeq \! (1;0;...;0) }$. Similarly, in Eq. (5.122), one can consider the case ${ i \!\ne \! 0 }$, for which the $i^\mathrm{th}$ term of Eq. (5.122) takes the form

$$\begin{aligned} \frac{(z_{i} / \alpha )^{2}}{(\lambda _{i} / \alpha ) \!-\! (d_{i} / \alpha )} \frac{1}{(z_{i} / \alpha )} \simeq \frac{1}{(z_{i} /\alpha )} \gg 1 \, , \end{aligned}$$

(5.124)

where we relied on the same argument as in Eq. (5.121). It states that for ${ i \!\ne \! 0 }$, there is only one dominant term in the sum from Eq. (5.121), given by ${ \tfrac{(z_{i}/\alpha )^{2}}{(d_{i} / \alpha ) - (\lambda _{i} / \alpha )} \!\simeq \! 1 }$. As a consequence, for ${ i \!\ne \! 0 }$, the eigenvector $\varvec{x}_{i}$ from Eq. (5.122) is dominated by its $i^\mathrm{th}$ coefficient and the eigenvector may therefore be assumed to be proportional to (0; ... ; 1; 0; ...) , where the non-zero index is at the $i^\mathrm{th}$ position. We may therefore assume that the response matrix eigenvectors remain close to the natural basis elements. As a conclusion, even in the limit of a razor-thin disc, the symmetric arrowhead response matrix from Eq. (5.115) may still be assumed to be diagonal. We therefore justified why one may limit oneself to the diagonal coefficients of the response matrix, as in Eq. (5.23). The thickened WKB basis elements therefore allowed us to diagonalise the disc’s response matrix. This is a crucial step in the explicit calculations of the collisionless and collisional diffusion fluxes as shown in Sects. 5.5 and 5.6.

C From Thick to Thin

In this Appendix, let us detail how one can recover all the razor-thin expressions obtained in Chap. 3, starting from the thickened WKB basis.

5.1.1 C.1 The Collisionless Case

Let us first consider the case of the collisionless diffusion presented in Sect. 5.5 and show one now may compute the collisionless diffusion coefficients when the disc is too thin to rely on the continuous expression from Eq. (5.50). We will show that this second approach is fully consistent with the one used in Eq. (5.50). We will also show how one can recover the razor-thin expressions previously obtained in Sect. 3.5.

We noted in Eq. (5.49) that in order to use Riemann sum formula w.r.t. the index $k_{z}^{p}$, one should ensure that the typical step distance ${ \Delta k_{z} \!\simeq \! \pi / h }$ from Eq. (5.17) remains sufficiently small compared to the scale on which the function ${ k_{z} \!\mapsto \! g_{\mathrm {s}} (k_{z}) }$ varies. In the limit of a thinner disc, one has ${ h \!\rightarrow \! 0 }$, so that ${ \Delta k_{z} \!\rightarrow \! + \infty }$. As a consequence, the continuous approximation cannot be used anymore, and one should keep the discrete sum on the quantised $k_{z}^{p}$ in Eq. (5.49). Of course, it is also within this limit of a thinner disc, that one can recover the razor-thin results from Sect. 3.5.

Starting from Eq. (5.49), the expression (5.52) of the symmetric collisionless diffusion coefficients becomes

$$\begin{aligned}&\, D_{\varvec{m}}^\mathrm{sym} (\varvec{J}) = \delta _{m_{z}}^\mathrm{{even}} \, \frac{1}{(2 \pi )^{2}} \sum _{n_{p} , n_{q}} \!\! \int \!\! \mathrm {d}k_{r}^{p} \, \mathcal {J}_{m_{r}} \!\bigg [\!\! \sqrt{\!\tfrac{2 J_{r}}{\kappa }} k_{r}^{p} \!\bigg ] \, \mathcal {J}_{m_{z}} \!\bigg [\!\! \sqrt{\!\tfrac{2 J_{z}}{\nu }} k_{z}^{n_{p}} \!(k_{r}^{p}) \!\bigg ] \, \frac{\alpha _{p}^{2}}{1 \!-\! \lambda _{p}} \nonumber \\&\times \!\! \int \!\! \mathrm {d}k_{r}^{q} \, \mathcal {J}_{m_{r}} \!\bigg [\!\! \sqrt{\!\tfrac{2 J_{r}}{\kappa }} k_{r}^{q} \!\bigg ] \, \mathcal {J}_{m_{z}} \!\bigg [\!\! \sqrt{\!\tfrac{2 J_{z}}{\nu }} k_{z}^{n_{q}} \!(k_{r}^{q}) \!\bigg ] \, \frac{\alpha _{q}^{2}}{1 \!-\! \lambda _{q}} \, \widehat{C}_{\delta \psi ^{\mathrm {e}}} [m_{\phi } , \varvec{m} \!\cdot \! \varvec{\Omega } , R_{\mathrm {g}} , k_{r}^{p} , k_{r}^{q} , k_{z}^{n_{p}} \!(k_{r}^{p}) , k_{z}^{n_{q}} \!(k_{r}^{q})] \, , \end{aligned}$$

(5.125)

where the perturbation autocorrelation, $\widehat{C}_{\delta \psi ^{\mathrm {e}}}$, was introduced in Eq. (5.51). Let us recall that the antisymmetric analog of Eq. (5.125) is straightforward to obtain thanks to the substitutions ${ \alpha _{p} \!\rightarrow \! \beta _{p} }$ and ${ \delta _{m_{z}}^\mathrm{{even}} \!\rightarrow \! \delta _{m_{z}}^\mathrm{{odd}} }$. For the antisymmetric case, as already noted in Eq. (3.73), one should pay attention to the fact that the perturbation autocorrelation involves the odd-restricted vertical Fourier transform of the potential perturbations. As in Eq. (5.55), the next step of the calculation is to diagonalise the perturbation autocorrelation, where one should pay attention to the fact that ${ k_{z} \!=\! k_{z} (k_{r} , n) }$ is no longer a free variable but should be seen as a function of the considered $k_{r}$ and n. Following Appendix D in Fouvry et al. (2017), Eq. (5.55) becomes here

$$\begin{aligned} \big < \delta \widehat{\psi ^{\mathrm {e}}}_{m_{\phi } , k_{r}^{1} , k_{z}^{n_{1}}} [ R_{\mathrm {g}} , \omega _{1} ] \, \delta \widehat{\psi ^{\mathrm {e}}}^{*}_{m_{\phi } , k_{r}^{2} , k_{z}^{n_{2}}} [R_{\mathrm {g}} , \omega _{2}] \big > = 2 \pi h \delta _{\mathrm {D}} (\omega _{1} \!-\! \omega _{2}) \, \delta _{\mathrm {D}} (k_{r}^{1} \!-\! k_{r}^{2}) \, \delta _{n_{1}}^{n_{2}} \, \widehat{\mathcal {C}} [m_{\phi } , \omega _{1} , R_{\mathrm {g}} , k_{r}^{1} , k_{z}^{n_{1}}] \, , \end{aligned}$$

(5.126)

where the diagonalisation w.r.t. the vertical dependence is captured by the Kronecker symbol ${ \delta _{n_{1}}^{n_{2}} }$. This diagonalised autocorrelation allows us to rewrite the diffusion coefficients from Eq. (5.125) as

$$\begin{aligned} D_{\varvec{m}}^\mathrm{sym} (\varvec{J}) = \delta _{m_{z}}^\mathrm{{even}} \frac{1}{4 h} \sum _{n_{p}} \!\! \int \!\! \mathrm {d}k_{r}^{p} \, \mathcal {J}_{m_{r}}^{2} \!\!\bigg [\! \sqrt{\!\tfrac{2 J_{r}}{\kappa }} k_{r}^{p} \!\bigg ] \, \mathcal {J}_{m_{z}}^{2} \!\!\bigg [\! \sqrt{\!\tfrac{2 J_{z}}{\nu }} k_{z}^{n_{p}} \!(k_{r}^{p}) \!\bigg ] \, \bigg [\! \frac{\alpha _{p}^{2}}{1 \!-\! \lambda _{p}} \!\bigg ]^{2} \, \widehat{\mathcal {C}} [m_{\phi } , \varvec{m} \!\cdot \! \varvec{\Omega } , R_{\mathrm {g}} , k_{r}^{p} , k_{z}^{n_{p}} \!(k_{r}^{p})] \, . \end{aligned}$$

(5.127)

Equation (5.127) is the direct discrete equivalent of Eq. (5.56), and both expressions are in full agreement. Indeed, starting from Eq. (5.127), the continuous expression w.r.t. $k_{z}^{p}$ can immediately be recovered by using Riemann sum formula with the step distance ${ \Delta k_{z} \!\simeq \! \pi / h }$ from Eq. (5.17). Similarly to Eq. (5.57), one can also simplify Eq. (5.127) thanks to the approximation of the small denominators, which gives here

$$\begin{aligned} D_{\varvec{m}}^\mathrm{sym} (\varvec{J}) = \delta _{m_{z}}^\mathrm{{even}} \frac{1}{4 h} \!\sum _{n_{p}}\! \Delta k_{r}^{n_{p}} \mathcal {J}_{m_{r}}^{2} \!\!\bigg [\! \sqrt{\!\tfrac{2 J_{r}}{\kappa }} k_{r , n_{p}}^\mathrm{max} \!\bigg ] \, \mathcal {J}_{m_{z}}^{2} \!\!\bigg [\! \sqrt{\!\tfrac{2 J_{z}}{\nu }} k_{z , n_{p}}^\mathrm{ma x} \!\bigg ] \, \bigg [\! \frac{(\alpha _{n_{p}}^\mathrm{max})^{2}}{1 \!-\! \lambda _{n_{p}}^\mathrm{max}} \!\bigg ]^{2} \, \widehat{\mathcal {C}} [m_{\phi } , \varvec{m} \!\cdot \! \varvec{\Omega } , R_{\mathrm {g}} , k_{r , n_{p}}^\mathrm{max} , k_{z , n_{p}}^\mathrm{max}] \, . \end{aligned}$$

(5.128)

In Eq. (5.128), for a given value of the index $n_{p}$, we consider the behaviour of the function ${ k_{r}^{p} \!\mapsto \! \lambda (k_{r}^{p} , k_{z}^{n_{p}} (k_{r}^{p})) }$, and assume that it reaches a maximum value $\lambda _{n_{p}}^\mathrm{max}$ for ${ k_{r} \!=\! k_{r , n_{p}}^\mathrm{max} }$ on a domain of typical extension ${ \Delta k_{r}^{n_{p}} }$. In Eq. (5.128), we also used the shortening notation ${ k_{z , n_{p}}^\mathrm{max} \!=\! k_{z}^{n_{p}} (k_{r , n_{p}}^\mathrm{max}) }$. The antisymmetric analogs of Eqs. (5.127) and (5.128) are straightforward to obtain by considering the antisymmetric quantised frequencies $k_{z}$ from Eq. (5.100) and performing the substitution ${ \alpha _{p} \!\rightarrow \! \beta _{p} }$. As already emphasised in Eq. (5.56), one should pay attention to the fact that in these antisymmetric analogs, $\widehat{\mathcal {C}}$ involves an even-restricted vertical Fourier transform of the autocorrelation, despite the fact that one is interested in antisymmetric diffusion coefficients.

Starting from the discrete expression of the diffusion coefficients obtained in Eq. (5.127), let us now illustrate how one can recover the razor-thin WKB diffusion coefficients from Sect. 3.5 by considering the limit of a thinner disc. As already noted in Fig. 5.4, let us recall that except for the fundamental symmetric frequency $k_{z , \mathrm {s}}^{1}$, one always has ${ k_{z}^{n} \!>\! \pi / (2 h)}$. As a consequence, in the infinitely thin limit, for which ${ h \!\rightarrow \! 0 }$, one has ${ k_{z}^{n} \!\rightarrow \! + \infty }$, except for ${ k_{z , \mathrm {s}}^{1} }$. Let us also recall that in Eq. (5.127), the dependence of ${ \widehat{\mathcal {C}} [k_{z}^{p}] }$ takes the form

$$\begin{aligned} \widehat{\mathcal {C}} [k_{z}^{p}] = \!\! \int _{- 2 h}^{2 h} \!\!\! \mathrm {d}v \, \widehat{\mathcal {C}} [v] \, \cos [k_{z}^{p} v] \, . \end{aligned}$$

(5.129)

One therefore gets the majoration ${ |\widehat{\mathcal {C}} [k_{z}^{p}] | \!\le \! 4 h \, \widehat{\mathcal {C}}_\mathrm{max} }$, which, in the razor-thin limit, cancels out the prefactor 1 / (4h) present in Eq. (5.127). Because $\forall n \!\ge \! 0 \, , \lim _{x \rightarrow + \infty } \mathcal {J}_{n} (x) \!=\! 0$, it immediately follows from Eq. (5.127) that

$$\begin{aligned} \lim \limits _\mathrm{thin} D_{\varvec{m}}^\mathrm{anti} (\varvec{J}) = 0 \, . \end{aligned}$$

(5.130)

In addition, Eq. (5.127) also implies that for symmetric diffusion coefficients, the sum on $n_{p}$ may be limited to the only fundamental term ${ n_{p} \!=\! 1 }$. Equation (5.16) gives us that in the razor-thin limit, one has ${ k_{z , \mathrm {s}}^{1} \!\simeq \! \sqrt{k_{r} / h} }$. Equation (5.127) therefore also implies that for ${ m_{z} \!\ne \! 0 }$, one has ${ \lim _\mathrm{thin} D_{\varvec{m}}^\mathrm{sym} \!=\! 0 }$. Therefore, in the infinitely thin limit, only the symmetric diffusion coefficients for ${ m_{z} \!=\! 0 }$ does not vanish. In addition, from Eq. (5.127), it is also straightforward to obtain that in order to have a non-vanishing symmetric diffusion coefficient, one should also restrict oneself to ${ J_{z} \!=\! 0 }$. In the razor-thin limit for ${ m_{z} \!=\! 0 }$ and ${ J_{z} \!=\! 0 }$, one can therefore write

$$\begin{aligned} \lim \limits _\mathrm{thin} D_{\varvec{m}}^\mathrm{sym} (\varvec{J}) = \lim \limits _\mathrm{thin} \frac{1}{4 h} \!\! \int \!\! \mathrm {d}k_{r}^{p} \, \mathcal {J}_{m_{r}}^{2} \!\bigg [\!\! \sqrt{\!\tfrac{2 J_{r}}{\kappa }} k_{r}^{p} \!\bigg ] \, \bigg [ \frac{\alpha _{1}^{2}}{1 \!-\! \lambda _{p}} \bigg ]^{2} \, \widehat{\mathcal {C}} [m_{\phi } , \varvec{m} \!\cdot \! \varvec{\Omega } , R_{\mathrm {g}} , k_{r}^{p} , k_{z , \mathrm {s}}^{1}] \, . \end{aligned}$$

(5.131)

The definition of the prefactor $\alpha _{p}$ in Eq. (5.21) immediately gives us ${ \lim _\mathrm{thin} \alpha _{1} \!=\! 1 }$. In addition, we also obtained in Eq. (5.34) that ${ \lim _\mathrm{thin} \lambda _{p} \!=\! \lambda _{p} ^\mathrm{thin}}$. The last step of the present calculation is to study, in the razor-thin limit, the behaviour of the term ${ \widehat{\mathcal {C}} [k_{z , \mathrm {s}}^{1}] }$ from Eq. (5.129). Equation (5.129) takes the form of an integral of length 4h of a function oscillating at the frequency ${ k_{z , \mathrm {s}}^{1} \!\simeq \! \sqrt{k_{r} / h} }$. In this interval, the number of oscillations of the fluctuating term is of order ${ k_{z , \mathrm {s}}^{1} h \!\sim \! \sqrt{k_{r} h} }$, so that in the razor-thin limit the number of oscillations of the function ${ v \!\mapsto \! \cos [k_{z , \mathrm {s}}^{1} v] }$ tends to 0. In the razor-thin limit, Eq. (5.129) then becomes ${ \lim _\mathrm{thin} \widehat{\mathcal {C}} [k_{z , \mathrm {s}}^{1}] \!=\! 4 h \widehat{\mathcal {C}} [v \!=\! 0] }$. Using this relation in Eq. (5.131), one finally gets

$$\begin{aligned} \lim \limits _\mathrm{thin} D_{\varvec{m}}^\mathrm{sym} (\varvec{J}) = \!\! \int \!\! \mathrm {d}k_{r}^{p} \, \mathcal {J}_{m_{r}}^{2} \!\!\bigg [\! \sqrt{\!\tfrac{2 J_{r}}{\kappa }} k_{r}^{p} \!\bigg ] \, \bigg [\! \frac{1}{1 \!-\! \lambda _\mathrm{thin}} \!\bigg ]^{2} \, \widehat{\mathcal {C}}_\mathrm{thin} [m_{\phi } , \varvec{m} \!\cdot \! \varvec{\Omega } , R_{\mathrm {g}} , k_{r}^{p}] \, , \end{aligned}$$

(5.132)

where ${ \widehat{\mathcal {C}}_\mathrm{thin} [m_{\phi } , \varvec{m} \!\cdot \! \varvec{\Omega } , R_{\mathrm {g}} , k_{r}^{p}] }$ stands for the local razor-thin power spectrum of the external perturbations in the equatorial plane as defined in Eq. (3.67) in the razor-thin case. In Eq. (5.132), we fully recovered the razor-thin result previously obtained in Eq. (3.68).

5.1.2 C.2 The Collisional Case

Let us now follow the same approach for the collisional diffusion. We will especially show how one should estimate the system’s susceptibility coefficients in the case where the disc is too thin to rely on the continuous expression from Eq. (5.67), and how this approach allows for the recovery of the razor-thin results previously obtained in Sect. 3.6.

As already noted in the previous section, let us recall that in the razor-thin limit, for which ${ h \!\rightarrow \! 0 }$, the quantised vertical frequencies $k_{z}^{n}$ are such that ${ k_{z}^{n} \!\rightarrow \! + \infty }$, except for the fundamental symmetric frequency $k_{z , \mathrm {s}}^{1}$. In the expression (5.66) of the dressed susceptibility coefficients, let us also note the presence of a prefactor 1 / h, so that in the razor-thin limit, one has to study the behaviour of a term of the form

$$\begin{aligned} \frac{1}{h} \frac{1}{k_{r}^{2} \!+\! (k_{z}^{n_{p}})^{2}} \underset{\mathrm{thin}}{\longrightarrow } {\left\{ \begin{array}{ll} \begin{aligned} \displaystyle &{} \frac{1}{k_{r}} &{} \text {if} &{} \;\;\;k_{z}^{n_{p}} \!=\! k_{z,\mathrm {s}}^{1} \, , \\ \displaystyle &{} 0 &{} \text {if} &{} \;\;\; k_{z}^{n_{p}} \!\ne \! k_{z,\mathrm {s}}^{1} \, . \end{aligned} \end{array}\right. } \end{aligned}$$

(5.133)

In the razor-thin limit, because all the other terms appearing in Eq. (5.66) are bounded, one therefore gets

$$\begin{aligned} \lim \limits _\mathrm{thin} \frac{1}{\mathcal {D}_{\varvec{m}_{1} , \varvec{m}_{1}}^\mathrm{anti}} = 0 \, . \end{aligned}$$

(5.134)

In addition, in the razor-thin limit, the sum on $n_{p}$ appearing in Eq. (5.66) may also be limited to the only fundamental term ${ n_{p} \!=\! 1 }$. Moreover, in order to have non-vanishing susceptibility coefficients, as already justified in the collisionless case, only symmetric diffusion coefficients associated with ${ m_{1}^{z} \!=\! 0 }$ and ${ J_{z}^{1} \!=\! 0 }$ will not vanish in the razor-thin limit. Finally, let us recall that in the razor-thin limit, one has ${ \lim _\mathrm{thin} \lambda _{p} \!=\! \lambda _\mathrm{thin} }$ and ${ \lim _\mathrm{thin} \alpha _{1} \!=\! 1 }$. Thanks to these restrictions, in the razor-thin limit, the symmetric susceptibility coefficients from Eq. (5.66) become

$$\begin{aligned} \lim \limits _\mathrm{thin} \frac{1}{\mathcal {D}_{\varvec{m}_{1} , \varvec{m}_{1}}^\mathrm{sym}} \sim \frac{1}{\mathcal {D}^\mathrm{thin}_{\varvec{m}_{1} , \varvec{m}_{1}}} \, \mathcal {J}_{0} \!\bigg [\! \sqrt{\!\tfrac{2 J_{z}^{2}}{\nu _{1}}} k_{z , \mathrm {s}}^{1} \!\bigg ] \, , \end{aligned}$$

(5.135)

where ${ 1/\mathcal {D}_{\varvec{m}_{1} , \varvec{m}_{1}}^\mathrm{thin} }$ stands for the razor-thin WKB susceptibility coefficients obtained in Eq. (3.80).

In order to recover the razor-thin WKB Balescu-Lenard diffusion flux, let us now consider the expression (5.69) of the thickened WKB drift coefficients and study their behaviour in the razor-thin limit. Let us first rewrite the thick quasi-isothermal DF from Eq. (5.5) as

$$\begin{aligned} F_\mathrm{thick} (J_{\phi }^{1} , J_{r}^{1} , J_{z}^{1}) = F_\mathrm{thin} (J_{\phi }^{1} , J_{r}^{1}) \, \frac{\nu _{1}}{2 \pi \sigma _{z}^{2}} \, \exp \!\bigg [\! - \frac{\nu _{1} J_{z}^{1}}{\sigma _{z}^{2}} \!\bigg ] \, , \end{aligned}$$

(5.136)

where we wrote $F_\mathrm{thin}$ for the razor-thin quasi-isothermal DF from Eq. (5.69). In order to illustrate the gist of this calculation, let us now focus only the remaining dependences w.r.t. $J_{z}^{2}$ in Eq. (5.69). This corresponds to an expression of the form

$$\begin{aligned} \frac{\nu _{1}}{2 \pi \sigma _{z}^{2}} \!\! \int \!\! \mathrm {d}J_{z}^{2} \, \exp \!\bigg [\! - \frac{\nu _{1} J_{z}^{2}}{\sigma _{z}^{2}} \!\bigg ] \, \mathcal {J}_{0}^{2} \!\bigg [\! \sqrt{\!\tfrac{2 J_{z}^{2}}{\nu _{1}}} k_{z , \mathrm {s}}^{1} \!\bigg ] = \frac{1}{2 \pi } \, \mathcal {I}_{0} \!\bigg [\! \frac{(k_{z , \mathrm {s}}^{1})^{2}}{\nu _{1}^{2} / \sigma _{z}^{2}} \!\bigg ] \, \exp \!\bigg [\! - \frac{(k_{z , \mathrm {s}}^{1})^{2}}{\nu _{1}^{2} / \sigma _{z}^{2}} \!\bigg ] \; \underset{\mathrm{thin}}{\longrightarrow } \; \frac{1}{2 \pi } \, , \end{aligned}$$

(5.137)

where we relied on the formula from Eq. (3.42), as well as on the fact that in the razor-thin limit ${ (k_{z , \mathrm {s}}^{1})^{2} / (\nu _{1}^{2} / \sigma _{z}^{2}) \!\sim \! h \!\rightarrow \! 0 }$. Using Eq. (5.137) into the general expression (5.69) of the drift coefficients, one gets

$$\begin{aligned} \lim \limits _\mathrm{thin} A_{\varvec{m}_{1}}^\mathrm{sym} (\varvec{J}_{1}) = - \frac{4 \pi ^{3} \mu }{(\varvec{m}_{1} \!\cdot \! \varvec{\Omega }_{1})'} \!\! \int \!\! \mathrm {d}J_{r}^{2} \, \frac{\varvec{m}_{1} \!\cdot \! \partial F_\mathrm{thin} / \partial \varvec{J} (J_{\phi }^{1} , J_{r}^{2})}{| \mathcal {D}_{\varvec{m}_{1} , \varvec{m}_{1}}^\mathrm{thin} (J_{\phi }^{1} , J_{r}^{1} , J_{\phi }^{1} , J_{r}^{2} , \varvec{m}_{1} \!\cdot \! \varvec{\Omega }_{1}) |^{2}} \, , \end{aligned}$$

(5.138)

where one has to restrict oneself to ${ m_{1}^{z} \!=\! 0 }$ and ${ J_{z}^{1} \!=\! 0 }$. Following the same approach, the razor-thin limit of the collisional diffusion coefficients from Eq. (5.70) is straightforward to compute and reads

$$\begin{aligned} \lim \limits _\mathrm{thin} D_{\varvec{m}_{1}}^\mathrm{sym} (\varvec{J}_{1}) = \frac{4 \pi ^{3} \mu }{(\varvec{m}_{1} \!\cdot \! \varvec{\Omega }_{1})'} \!\! \int \!\! \mathrm {d}J_{r}^{2} \, \frac{F_\mathrm{thin} (J_{\phi }^{1} , J_{r}^{2} )}{| \mathcal {D}_{\varvec{m}_{1} , \varvec{m}_{1}}^\mathrm{thin} (J_{\phi }^{1} , J_{r}^{1} , J_{\phi }^{1} , J_{r}^{2} , \varvec{m}_{1} \!\cdot \! \varvec{\Omega }_{1}) |^{2}} \, . \end{aligned}$$

(5.139)

This concludes our calculations, as we note that the two razor-thin limits obtained in Eqs. (5.138) and (5.139) are in full agreement with the razor-thin results previously obtained in Eqs. (3.83) and (3.84).