A New Method of Measuring Magnetic Field Strength in Highly Structured Protostellar Envelopes

Abstract

Magnetic fields play a fundamental role in protostellar collapse and disk formation, yet direct measurements of magnetic field strength in deeply embedded protostellar envelopes remain difficult. We present a new method to estimate both the vertical and total magnetic field strength in collapsing, pseudodisk- or sheetlet-dominated protostellar envelopes, derived directly from the magnetohydrodynamic momentum equation. The method relates the magnetic field strength to two observationally accessible quantities: the projected gravitational acceleration toward the center of collapse and the face-on column density of the pseudodisk, and two dimensionless parameters, $a_{b, R}$ and $γ_{zR}$, which characterize magnetic contribution to the force balance and the field geometry, respectively, through $|B_z|=(2πa_{b,R}γ_{zR}g_RΣ)^{1/2}$. Using non-ideal magnetohydrodynamic simulations, we verify the assumptions underlying the method, justify the adopted approximations, and calibrate the two key dimensionless parameters. We provide canonical estimates of these two parameters, and show that they exhibit only weak spatial and temporal variations, allowing robust field strength estimates even when detailed gas kinematics or high-resolution polarization information is unavailable. We show that the method is applicable in both turbulent and non-turbulent envelopes and is insensitive to the ambipolar diffusion coefficient, making it robust against uncertainties in the local turbulence strength and ionization rate. We apply the method to the Class 0 source L1157, using column-density and gravitational-acceleration estimates from the literature to estimate the magnetic field strength for L1157. Our result is broadly consistent with previous estimates from independent methods, demonstrating the utility of this approach for constraining magnetic fields in embedded protostellar systems.

Figures (12)

• Figure 1: Cartoon of the envelope structure in a collapsing magnetized protostellar system. The center of the system is the star and its circumstellar disk; In the models where the core is initially laminar (i.e., the M0.0 models), the protostellar envelope is dominated by the "pseudodisk"; in the initially turbulent core models (i.e., the M1.0 models), the protostellar envelope is dominated by the "gravo-magneto-sheetlets" Tu2024a. Both the pseudodisk and sheetlets dominate mass and magnetic field transport within the protostellar envelope, and our goal here is to estimate the magnetic field strength on these pseudodisk/sheetlets.
• Figure 2: Force balance in the M0.0AD1.0 model when $M_\star=0.2M_\odot$ (40% of total core mass). Panel (a): Force balance on the pseudodisk excluding the time-dependence term (see equation \ref{['equ:alpha_t']}- \ref{['equ:mhd alpha equ']}). The notations here only show the physical meaning of this figure, and the averaging is defined by equation \ref{['equ:weight_definition']}. Panel (b): Force balance on the pseudodisk excluding both the time-dependence and the pressure gradient terms. Panel (c): azimuthally-averaged force balances in each annulus, excluding the time-dependence term only (blue line), and excluding both the time-dependence and pressure gradient terms (red line).
• Figure 3: Ratio between the magnetic tension force and the total magnetic force in the cylindrical-$\hat{R}$ direction. Panel (a) shows the ratio on the pseudodisk in the M0.0AD1.0 model; Panel (b) shows the ratio in the M0.0AD2.0 model, and panel (c) shows the azimuthal-averaged ratio in both models, indicating the total magnetic force can be approximated well with the magnetic tension force alone.
• Figure 4: Measurement of $\alpha_{b, R}$ in the M0.0AD1.0 model in the upper panels and the M0.0AD2.0 model in the lower panels. Panel (a) and (d) show $\alpha_b$ through its definition (equation \ref{['equ:alpha_b']}, method 1 in section \ref{['sec:est_alpha_b_direct']}); Panel (b) and (e) show $\alpha_b$ through an approximation using the tension assumption and column density (equation \ref{['equ:alpha_b_uplow']}, method 2 in section \ref{['sec:est_alpha_b_direct']}). Panels (c) and (f) show their azimuthal average and a comparison between these two methods, revealing remarkable agreements. There is a characteristic "V" shape in Panels (c) and (f) due to a change in the magnetic field geometry responding to gas dynamics, as discussed in section \ref{['sec:est_alpha_b_direct']}. An animated version of this figure is available at https://doi.org/10.6084/m9.figshare.31445326.
• Figure 5: Illustration of magnetic field geometry using the estimated curvature radius of magnetic field ($R_c$, equation \ref{['equ:Rc']}) in the M0.0AD2.0 model. A large $R_c$ indicates the field is relatively straight, whereas a small $R_c$ indicates the field is highly pinched. At large radii, outside the pseudodisk ($\gtrsim 600~\mathrm{au}$), $R_c$ is very large as the field is straight. In the intermediate radii on the pseudodisk, $R_c$ stays around $70~\mathrm{au}$. At small radii $(\approx200~\mathrm{au}$), $R_c$ decreases sharply to $\lesssim15~\mathrm{au}$.