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# Erratum: Channels for streaming instability in dusty discs

Etienne Jaupart, Guillaume Laibe

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Etienne Jaupart, Guillaume Laibe. Erratum: Channels for streaming instability in dusty discs. 2022, pp.752-753. 10.1093/mnras/stac1010. insu-03711516

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# Erratum: Channels for streaming instability in dusty discs

by Etienne Jaupart* and Guillaume Laibe

Univ Lyon, Univ Lyon1, Ens de Lyon, CNRS, Centre de Recherche Astrophysique de Lyon UMR5574, F-69230, Saint-Genis-Laval, France

**Key words:** errata, addenda – instabilities – planets and satellites: formation – protoplanetary discs.

In our original paper Jaupart & Laibe (2020), we did an error of sign of order St³ in the term G which propagates in the calculation, especially in the Euclidian division [old equation (38)] – parameters α₃ and the parameters β₀, β₁, β₂. This error has been identified and communicated to us by L. Pan. This results in a mistake of 5 terms in equation (A1), namely, the (5, 1) element, the second term of the (6, 1) element, and the first terms in the (5, 5), (6, 6) and (7, 7) elements. Hence the approximations done for both the secular and the epicycle modes are correct only to the order St². The consequences of this mistake are the following

(i) The imaginary part of the (unstable) epicycle mode [old equation (46)] should be corrected according to

$$ \Im(\omega_e) = \frac{\varepsilon}{2} S_t \left\{ \frac{|\kappa_x|\varepsilon}{\cos\theta} S_t - \kappa_x^2 \left(3 + \varepsilon + \frac{(\varepsilon - 1)^2}{\cos^2\theta}\right) S_t^2 - \sin^2\theta \times \left(1 + \frac{|\kappa_x|(\varepsilon + 2)}{2\cos\theta} S_t + (\varepsilon(2\varepsilon + 3)\sin^2\theta - f_g^{-2}) S_t^2\right) \right\}. \quad (1) $$

(ii) The approximated instability criterion for the epicyclic mode [old equation (47)] now reads

$$ |\kappa_x|\varepsilon S_t \ge \sin^2\theta + \kappa_x^2(3 + \varepsilon + (\varepsilon - 1)^2)S_t^2. \quad (2) $$

The previous fig. 2 should be replaced by Fig. 1 ($\kappa_x = 0.6$ instead of $\kappa_x = 0.9$ to get unstable mode at $f_p = 0.1$). The qualitative analysis of the epicycle mode remains the same since the growth rate is dominated by contributions of order St² unaffected by the mistake.

(iii) The imaginary part of the secular mode [old equation (42)] should be corrected according to

$$ \Im(\omega_s^{(3)}) = \kappa_x^2 ([\varepsilon - 1] \left[ \frac{\varepsilon(\varepsilon - 1)}{\cos^2\theta} + \varepsilon \right] + 2\varepsilon). \quad (3) $$

Hence, the secular mode is always unstable, contrary to the original claim. In particular, the secular mode is unstable at $\varepsilon \equiv f_p/f_g = 1$ (Pan 2021), as evidenced in Fig. 2 that should therefore replace the old fig. 1. The correct expressions for the secular mode are

$$ \omega_s = \Re(\omega_s) + i \Im(\omega_s), \quad (4) $$

$$ \Re(\omega_s) = \kappa_x(\varepsilon - 1)S_t + r^{(3)}S_t^3, \quad (5) $$

$$ r^{(3)} = -\kappa_x \left( 2\varepsilon + (\varepsilon - 1)\tilde{r}_{\varepsilon-1}^{(3)} \right), \quad (6) $$

<figure><img src="image_5.png" /><figcaption>**Figure 1.** Imaginary part of the unstable epicycle of the streaming instability calculated numerically from the full set of hydrodynamical equations $P_8$ for dust fractions of $f_p = 0.1, 0.3, 0.4$ (from light to dark blue lines respectively) for varying Stokes numbers. The growth rates varies as $S_t^2$, as expected. Dots indicate the corresponding values predicted by the third-order expansion $P_8^{(3)}$ [and equation (1)] and shows almost perfect agreement. The vertical solid grey lines indicate the analytic stability criterion given by equation (2). We choose $\kappa_x = 0.6$ and $\kappa_z = 30$ for the epicyclic mode to be unstable at $f_p = 0.1$.</figcaption></figure>

$$ \tilde{r}_{\varepsilon-1}^{(3)} = 1 - \frac{\varepsilon}{f_g} \sin^2\theta + \varepsilon \left\{ \frac{(1 - \varepsilon)\kappa_x^2}{\cos^2\theta} + \frac{\sin^2\theta}{f_g} \right\}, \quad (7) $$

$$ \Im(\omega_s) = \kappa_x^2 \left( [\varepsilon - 1] \left[ \frac{\varepsilon(\varepsilon - 1)}{\cos^2\theta} + \varepsilon \right] + 2\varepsilon \right) S_t^3. \quad (8) $$

By the light of this new corrections, the conclusion of this study are

(a) The Euclidian factorisation method allows to split the secular and the epicycle modes and obtain their Taylor expansion with respect to St when $St \ll 1$. It provides a quantitative condition of validity for this expansion, namely $\cos^2\theta \gg St^2$, $\kappa_x^2 St \lesssim 1$, $|\kappa_z| St \lesssim 1$. Outside of this domain of validity, this mode splitting does not hold anymore [see Fig. 2 and e.g. Squire & Hopkins (2018) for $\varepsilon \ll 1$].

(b) Epicycle mode can be unstable with significant growth rates.

<footnote>* E-mail: ejaupart@gmail.com</footnote>
<figure><img src="image_6.png" /><figcaption>**Figure 2.** Imaginary part of the secular mode of the streaming instability calculated numerically from the full set of hydrodynamical equations $P_8$ for $\epsilon = 1$, Stokes numbers of $S_t = 5 \cdot 10^{-3}$, $1 \cdot 10^{-2}$, $5 \cdot 10^{-2}$ (from light to dark blue lines respectively) and varying $\kappa_x$. We choose $\kappa_z = 30$. The growth rates varies as $\kappa_x^2 S_t^3$, as expected from equation (3). Dots indicate the corresponding values predicted by the third-order expansion equation (3) and shows almost perfect agreement in the domain of validity of this expansion, here given by the condition $\kappa_x^2 St \lesssim 1$. Discrepancies can be observed at $\kappa_x \ge 10$ for $S_t = 5 \cdot 10^{-2}$, since these points are outside the domain of validity of the expansion.</figcaption></figure>

(c) The secular mode is always unstable, even at $\epsilon = 1$ in the $(\kappa_x, \kappa_z)$ domain where the mode splitting holds (Pan 2021). Outside of this domain, identifying the unstable mode as the epicycle or the secular is a matter of semantic (see e.g. Squire & Hopkins 2018).

(d) The analysis in terms of Green functions still holds: for comparable growth rates, epicycle mode is excited more efficiently than to the secular mode.

## ACKNOWLEDGEMENTS

We acknowledge Liubin Pan for having figured and pointed out this mistake.

## REFERENCES

Jaupart E., Laibe G., 2020, MNRAS, 492, 4591

Pan L., 2021, ApJ, 920, 80

Squire J., Hopkins P. F., 2018, ApJ, 856, L15

## APPENDIX A: DETERMINANT OF THE LINEAR SYSTEM $P_8$

The correct equation (A1) should read

$$ \left| \begin{array}{lcccccccc} -i\omega & i\kappa_x & 0 & i\kappa_z & 0 & 0 & 0 & 0 & 0 \\ \frac{i\kappa_x S_t^2}{(S_t^2+1)^2} + 1 & -i\omega & -2 & 0 & -\frac{2i f_p \kappa_x S_t}{S_t^2+1} & 0 & -\frac{i f_p \kappa_z S_t}{S_t^2+1} & i\kappa_x & 0 \\ -\frac{i\kappa_x S_t^3}{2(S_t^2+1)^2} & \frac{1}{2} & -i\omega & 0 & \frac{i f_p \kappa_x S_t^2}{2(S_t^2+1)} & -\frac{i f_p \kappa_z S_t}{S_t^2+1} & \frac{i f_p \kappa_z S_t^2}{2(S_t^2+1)} & 0 & 0 \\ 0 & 0 & 0 & -i\omega & 0 & 0 & -\frac{i f_p \kappa_x S_t}{S_t^2+1} & i\kappa_z & 0 \\ \frac{i\kappa_x S_t^3}{f_g(S_t^2+1)^2} - \frac{S_t}{f_g(S_t^2+1)} & -\frac{i\kappa_x S_t^2}{f_g(S_t^2+1)} & 0 & 0 & -\frac{i(2f_p-1)\kappa_x S_t^2}{f_g(S_t^2+1)} & -i\omega S_t + 1 & -2S_t & 0 & -\frac{i\kappa_x S_t}{f_g} \\ \frac{S_t^2}{2f_g(S_t^2+1)^2} - \frac{i\kappa_x S_t^4}{2f_g(S_t^2+1)^2} & 0 & -\frac{i\kappa_x S_t^2}{f_g(S_t^2+1)} & 0 & \frac{S_t}{2} & -\frac{i(2f_p-1)\kappa_x S_t^2}{f_g(S_t^2+1)} & -i\omega S_t + 1 & 0 & 0 \\ 0 & 0 & 0 & -\frac{i\kappa_x S_t^2}{f_g(S_t^2+1)} & 0 & 0 & -\frac{i(2f_p-1)\kappa_x S_t^2}{f_g(S_t^2+1)} & -i\omega S_t + 1 & -\frac{i\kappa_z S_t}{f_g} \\ \frac{\kappa_x S_t}{S_t^2+1} & \kappa_x & 0 & \kappa_z & -f_p \kappa_x & 0 & -f_p \kappa_z & 0 & 0 \end{array} \right| \quad \text{(A1)} $$

This paper has been typeset from a TeX/LATEX file prepared by the author.