A local prescription for the softening length in self-gravitating gaseous discs

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Volume 507 / No 1 (November III 2009)

A&A, 507 1 (2009) 573-579

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Table 1:

Various prescriptions for the softening length adopted in some simulations of self-gravitating gaseous discs (see also Fig. 1).
Reference	Softening length $\lambda$
Papaloizou & Lin (1989)	$0.056 \times (a_{\rm out}-a_{\rm in})\times f(a,R,a_{\rm in},a_{\rm out})$
Adams et al. (1989)	$0.1\times R$
Saio & Yoshii (1990)	$\{0.01,0.02\} \times a_{\rm out}$
Shu et al. (1990)	$0.1\times R$
Morishima & Saio (1994)	$0.02 \times a_{\rm out}$
Sterzik et al. (1995)^a	$0.06 \times \lambda_{\rm c}$
Laughlin et al. (1997)	$0.1 \times a_{\rm in}$ , $0.1 \times a_{\rm out}$ , and $0.001 \times R$
Laughlin et al. (1998)	$0.0001+0.01 R \times f(a,a_{\rm in},a_{\rm out})$
Tremaine (2001)	$\beta \times R$ , with $\beta \approx 10^{-4} {-} 0.2$
Caunt & Tagger (2001)^b	H=2h
Baruteau & Masset (2008)	0.3 - 0.5 h (depending on scale height)
Li et al. (2009)^c	$\approx$ 0.17 to $0.33 \times \Delta a$
this work^d	$\displaystyle f\left(\frac{h}{a},\frac{R}{a}\right)$
	$\approx$ h/e at R=a, homogeneous case;
	see Eqs. (26), (32), (42) and (39)

^a $\lambda_{\rm c}$ is the critical wave length of disturbances.
^b concerns the magnetic potential.
^c $\Delta a$ is the grid spacing, 3D-disc.
^d Axisymmetric limit, finite size disc (inner edge $a_{\rm in}$ , outer edge $a_{\rm out}$ ), symmetry with respect to the mid-plane, finite size layer (thickness 2h=H), explicit function of vertical stractification, local validity.

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