Therefore, the inertia tensor elements scale as:
IijV(x)xixjd3xR2+DHI_{ij} \sim \int_V \rho(\mathbf{x}) x_i x_j \, d^3x \sim R^{2 + D_H}
C.5. Spin Alignment Scaling
From LiijkTjlIlkL_i \propto \epsilon_{ijk} T_{jl} I_{lk}, we now estimate the angular momentum magnitude variance:
L2T2I2R(1+DH)R4+2DH=R3+DH\langle L^2 \rangle \sim \langle T^2 \rangle \langle I^2 \rangle \sim R^{-(1 + D_H)} \cdot R^{4 + 2D_H} = R^{3 + D_H}
Thus, fractality enhances angular momentum buildup at smaller scales.
The alignment between L\mathbf{L} and the local filament direction is typically quantified by:
cos()=Le^3L\cos(\theta) = \frac{\mathbf{L} \cdot \hat{e}_3}{|\mathbf{L}|}
where e^3\hat{e}_3 is the eigenvector of the tidal/shear tensor corresponding to the smallest eigenvalue (filament axis).
For a fractal field, the increased anisotropy and long-range correlations induce stronger preferential alignment:
Alignment strength increases with decreasing DHD_H (more fractal),
Variance in cos()\cos(\theta) narrows, meaning tighter distribution around the filament axis.
These analytical insights were confirmed in Section 4B through N-body simulations.
