ner_tags: O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O B-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O
tokens: while gradually decreases with increasing , as can be seen from Equation ( 57 ). Figure 14. Normalized local effective sound speed () and normalized effective local vertical half-thickness () as functions of the normalized external pressure . The value of is maximum (=1.061) at (as shown by the dashed horizontal and vertical lines). Appendix E Notes on the Generalized Toomre Criterion In the limit of flux-freezing, the dispersion relation isTo minimize ω 2 , the criteria are d ( ω 2 )/ dk = 0 and d 2 ( ω 2 )/ dk 2 > 0 at , yieldingNow, to obtain the instability criterion, we set ω 2 0 at , which givesyieldingIn the dimensionless form, it becomeswhereSee Section 2.6 for further discussion. Appendix F Notes on the Stationary Field Limit In the limit of stationary magnetic fields, , the resulting dispersion relation can be obtained using Equations ( 42 )–( 45 ). It follows that where A 1 , A 2 , θ , and γ are explicitly written in Section 2.4 . Recall that θ and γ represent the case of AD and OD, respectively. Simplifying each individual term of Equation ( F2 ) yields For the case of only OD (set θ = 0), combining Equations ( F3 ), ( F5 ), and ( F6 ), we obtain the resulting dispersion relation and corresponding growth timescale as shown in Equations ( 91 ) and ( 92 ). For the case of only AD (set γ = 0), combining Equations ( F3 ), ( F4 ), and ( F6 ) we obtain the resulting dispersion relation and corresponding growth timescale as shown in Equations ( 95 ) and ( 96 ). Appendix G Additional Figures of the Normalized Growth Timescale versus Length Scale Figure 15 shows the curves of normalized growth timescale as a function of normalized length scale for different normalized rotation Q = 0, 0.1, 0.2, 0.3, 0.4. This figure represents our model of the protostellar disk for distinct normalized mass-to-flux ratios μ 0 = 0.5, 1, 1.1, 2, and 6, with and , corresponding to n n ,0 = 10 11 cm −3 . This is one of our fundamental results, which can be obtained by plotting the normalized form of the full dispersion relation as seen in Equation ( 63 ). Because of the combination of both nonideal MHD effects, the timescale versus length-scale curve attains a minimum at a smaller value of even for the subcritical case μ 0 = 0.5 as compared to the case when only one nonideal MHD effect is present. With the transition of μ 0 from subcritical to supercritical, these curves gradually approach that of the hydrodynamic case. See Sections 3.2 and 3.3 for a detailed discussion. Figure 15. Normalized growth time of the gravitationally unstable mode as a function of the normalized wavelength for models with a fixed normalized ohmic diffusivity and neutral–ion collision time for different normalized mass-to-flux ratios μ 0 = 0.5, 1, 1.1, 2, and 6. Each figure shows the timescale curves for models with different normalized rotation Q = 0 (black), 0.1 (red), 0.2 (blue), 0.3 (green), and 0.4 (magenta). Appendix H Monochromatic Perturbation Here we focus on the form of