Core relaxation

Figure 4.1: Peak density evolution of the relaxing cores. The peak density has been nor-malized to the initial value for each model.

Starting from the initial conditions described in chapter 3.1, in all the three models, the core does not spontaneously collapse. Instead, after the first few vigorous oscillations, the contraction and the expansion of the core gradually becomes milder and milder. Define

the largest density value in the domain as the peak density ρ_peak, we record the evolution of peak density as figure 4.1, illustrating the aforementioned core behavior over time.

We further define the core to be fully-relaxed if its peak density at the most contracted phase is≤ 10% to the average value of the previous period. Once the full relaxation is achieved, we halt the simulation run and use this fully-relaxed core to study its response to wave-like perturbations. To enter this stage, the core undergoes 43 oscillations in model M, 38 in model M2, and 58 in model H. The imposed magnetic field apparently speeds up the relaxation process. The peak density profile over the last few periods is shown in figure 4.2. With the same initial core density profile, the relaxed core seems to be more concentrated in model M2 than the core in model M1.

Figure 4.2: Peak density evolution of the relaxing cores during the last few oscillatory periods. The peak density has been normalized to the initial value for each model.

Next, we display the resultant profile of the fully-relaxed core. The result is shown in 4.3. The initial z-direction rotation and magnetic field destroy the system isotropy. Hence, we display the profile both along the x-axis and the z-axis. The peak hydrogen number density of the relaxed core ranges from 5-10×10⁴cm⁻³ and the central core temperature is about 10 K, both are consistent with the observational results introduced in chapter 1.2 (Lippok et al., 2013). As for the outer regions of the core, from observation the hydrogen

number density drops to∼ 10⁴cm⁻³at r∼ 0.1 pc. The corresponding regions in our runs are r = 0.6–0.8 r_c. Here the temperature is about 40–50 K, higher than the 15 K suggested by the observational results.

Figure 4.3: The core profile after the relaxation process. Top panel: the hydrogen number density profile from r = 0–3 r_c. Bottom panel: the H₂ core temperature within r = r_c.

In the meantime, we capture the evolutionary characteristics of the relaxation process for each model by visualizing the z = 0 and the y = 0 planes near the isolated core. Four time-steps are chosen – (1) in the beginning of the relaxation, (2) at the end of the first few vigorous oscillations, (3) after 17–19 core oscillations, and (4) when the fully-relaxed state is achieved. The 2D slice plots of model M, model M2, and model H are shown as figure 4.4–4.7, figure 4.8–4.11, and figure 4.12 to 4.14, respectively.

First of all, from the density maps one can easily observe the deformation effect of the magnetic field at the early stage of the relaxation. The edge-on snapshots in figure 4.4 and 4.8 shows that a oblate spheroid is the favorable structure for the magnetized cores. The symmetric axis is parallel to the direction of the initial magnetic field. At the beginning of the relaxation process, the core strongly contracts due to the gravitational attraction. In response to the dramatical mass infall, the system reconfigures itself to establish the

pres-sure support against the gravity. The timescale of the responding mechanism is determine by the speed of the fast magneto-sonic wave (hereafter, the fast wave) in the ideal MHD framework. In the direction perpendicular to the background magnetic field, the phase speed of the fast wave reaches its maximum. Therefore, provided the system is Jeans sta-ble, the established pressure support initiates the outward expansion near the equatorial plane while the gravity still dominate the fluid motion elsewhere. Therefore, the matter contracting near the z-axis is likely to be dragged outward along the equatorial plane since the fluid has the continuity property, resulting in the oblate spheroid core shape.

Besides, in the early stage of the relaxation process, the core also develops a pressure gradient towards the core. As for the total pressure, the ratio of the core value to the ambient value reaches≥ 2 in model M and model M2, while in model H the resultant pressure gradient between the core and the ambient is 50% weaker (see figure 4.6, 4.10, and 4.14). The additional pressure support possibly originates from the distortion of the magnetic field lines in the core center. The field lines are bending toward the core by the gravitationally induced fluid contraction, enhancing the magnetic pressure. The increase of the core magnetic field can also be observed in the edge-on poloidal magnetic field map, as shown in figure 4.7 and 4.11.

As the core relaxed for about 20 periods, the oscillation becomes more gentle, featuring the substantial decrease of Mach number in the velocity field map (figure 4.5, 4.9, and 4.13). At this stage, the velocity value drops to 20% of the local√

c²_S+ c²_Avalue. As for the rotational motion, from figure 4.15 and 4.16 one can see that in model M and model M2, the initial rigid body rotation along the +z-axis has been changed in the direction at least twice during the relaxation. The corresponding timescale is about 8 oscillatory periods. With the torsional magnetic force induced by the initial rotation gradually damps out the rotational motion in the system, the rotation dies out at the final stage. In model H, without the magnetic field interplay, the initial rotation persists until the end of the relaxation process.

Figure 4.4: The log10|ρ| contour maps (normalized to the ambient gas value) during the relaxation process in Model M. From top to bottom, the core has experienced 0, 8, 17, and 43 oscillatory periods. Figure (a) – (d) show the y = 0 plane near the core, while figure (e)–(h) are the z = 0 plane near the core.

Figure 4.5: The poloidal velocity field during the relaxation process in Model M. The magnitude is in the unit of local√

c²_S+ c²_Aand arrows represent the direction. From top to bottom, the core has experienced 0, 8, 17, and 43 oscillatory periods. Figure (a) – (d) show the y = 0 plane near the core, while figure (e)–(h) are the z = 0 plane near core.

Figure 4.6: The total pressure (normalized to the ambient gas value) during the relaxation process in Model M. From top to bottom, the core has experienced 0, 8, 17, and 43 oscil-latory periods. Figure (a) – (d) show the y = 0 plane near the core, while figure (e)–(h) are the z = 0 plane near the core.

Figure 4.7: The poloidal magnetic field (normalized to the initial magnetic strength) dur-ing the relaxation process in Model M, with arrows representdur-ing the direction. From top to bottom, the core has experienced 0, 8, 17, and 43 oscillatory periods. Figure (a) – (d) show the y = 0 plane near the core, while figure (e)–(h) are the z = 0 plane near the core.

Figure 4.8: The log10|ρ| contour maps (normalized to the ambient gas value) during the relaxation process in Model M2. From top to bottom, the core has experienced 0, 8, 17, and 38 oscillatory periods. Figure (a) – (d) show the y = 0 plane near the core, while figure (e)–(h) are the z = 0 plane near the core.

Figure 4.9: The poloidal velocity field during the relaxation process in Model M2. The magnitude is in the unit of local√

c²_S+ c²_Aand arrows represent the direction. From top to bottom, the core has experienced 0, 8, 17, and 38 oscillatory periods. Figure (a) – (d) show the y = 0 plane near the core, while figure (e)–(h) are the z = 0 plane near the core.

Figure 4.10: The total pressure (normalized to the ambient gas value) during the relaxation process in Model M2. From top to bottom, the core has experienced 0, 8, 17, and 38 oscillatory periods. Figure (a) – (d) show the y = 0 plane near the core, while figure (e)–(h) are the z = 0 plane near the core.

Figure 4.11: The poloidal magnetic field (normalized to the initial magnetic strength) during the relaxation process in Model M2, with arrows representing the direction. From top to bottom, the core has experienced 0, 8, 17, and 38 oscillatory periods. Figure (a) – (d) show the y = 0 plane near the core, while figure (e)–(h) are the z = 0 plane near the core.

Figure 4.12: The log10|ρ| contour maps (normalized to the ambient gas value) during the relaxation process in Model H. From top to bottom, the core has experienced 0, 8, 19, and 58 oscillatory periods. Figure (a) – (d) show the y = 0 plane near the core, while figure (e)–(h) are the z = 0 plane near the core.

Figure 4.13: The poloidal velocity field during the relaxation process in Model H. The magnitude is in the unit of local cSand arrows represent the direction. From top to bottom, the core has experienced 0, 8, 19, and 58 oscillatory periods. Figure (a) – (d) show the y = 0 plane near the core, while figure (e)–(h) are the z = 0 plane near the core.

Figure 4.14: The total pressure (normalized to the ambient gas value) during the relax-ation process in Model H. From top to bottom, the core has experienced 0, 8, 19, and 58 oscillatory periods. Figure (a) – (d) show the y = 0 plane near the core, while figure (e)–(h) are the z = 0 plane near the core.

Figure 4.15: The evolution of the ρv_ϕ(normalized to the initial r = r_cvalue) during the relaxation process. All the figures are the y = 0 plane near the core. In figure (a)–(d) the core has experienced 0, 8, 17, and 43 oscillatory periods in model M, while in figure (e)–(h) the core has experienced 0, 8, 19, and 58 oscillatory periods in model H.

Figure 4.16: The evolution of the ρv_ϕ(normalized to the initial r = r_cvalue) during the relaxation process. All the figures are the y = 0 plane near the core. In figure (a)–(d) the core has experienced 0, 8, 17, and 38 oscillatory periods in model M2, while in figure (e)–(h) the core has experienced 0, 8, 19, and 58 oscillatory periods in model H.