Part III: Solar Magnetic Activity | Chapter 9

Magnetic Fields of the Sun

Zeeman effect, magnetograph measurements, and magnetic field topology

9.1 The Zeeman Effect

Derivation 1: Zeeman Splitting in a Magnetic Field

The Zeeman effect is the primary tool for measuring solar magnetic fields. When an atom is placed in a magnetic field, its energy levels split due to the interaction between the magnetic moment and the external field.

Step 1. The magnetic moment of an electron in an atom with total angular momentum $\mathbf{J} = \mathbf{L} + \mathbf{S}$ is:

$$\boldsymbol{\mu} = -g_J \mu_B \frac{\mathbf{J}}{\hbar}$$

where $\mu_B = e\hbar/(2m_e)$ is the Bohr magneton and $g_J$ is the Lande g-factor:

$$g_J = 1 + \frac{J(J+1) + S(S+1) - L(L+1)}{2J(J+1)}$$

Step 2. The interaction energy with a uniform magnetic field $\mathbf{B}$ is:

$$\Delta E = -\boldsymbol{\mu} \cdot \mathbf{B} = g_J \mu_B m_J B$$

where $m_J = -J, -J+1, \ldots, J$. Each level with angular momentum $J$splits into $2J+1$ sub-levels.

Step 3. The wavelength splitting between adjacent $\sigma$ components:

$$\boxed{\Delta\lambda = \frac{e\lambda^2 g_{\text{eff}} B}{4\pi m_e c} = 4.67 \times 10^{-13} \lambda^2 g_{\text{eff}} B \text{ [nm]}}$$

where $\lambda$ is in nm and $B$ in Tesla. For Fe I 617.3 nm with $g_{\text{eff}} = 2.5$ in a 0.3 T sunspot field:$\Delta\lambda \approx 0.013$ nm, measurable with modern spectropolarimeters. George Ellery Hale first detected sunspot magnetic fields using the Zeeman effect in 1908.

9.2 Magnetograph Measurements

Derivation 2: Stokes Parameters and Magnetic Field Inference

Step 1. The polarization state of light is described by the four Stokes parameters$(I, Q, U, V)$:

$$I = \text{total intensity}$$

$$Q = I_0 - I_{90} \quad \text{(linear polarization, 0/90 deg)}$$

$$U = I_{45} - I_{135} \quad \text{(linear polarization, 45/135 deg)}$$

$$V = I_R - I_L \quad \text{(circular polarization)}$$

Step 2. For the Zeeman effect, the line-of-sight field component produces circular polarization (Stokes V), while the transverse component produces linear polarization (Stokes Q, U). In the weak-field limit ($\Delta\lambda_B \ll \Delta\lambda_D$):

$$\boxed{V(\lambda) \propto g_{\text{eff}} B_{\parallel} \frac{\partial I}{\partial \lambda}}$$$$\boxed{Q(\lambda) \propto g_{\text{eff}}^2 B_\perp^2 \frac{\partial^2 I}{\partial \lambda^2}}$$

The Stokes V signal (circular polarization) is linear in $B_\parallel$, making longitudinal magnetograms straightforward. The transverse field measurement from Stokes Q, U is quadratic in $B_\perp$ and suffers from a 180-degree ambiguity. Instruments like SDO/HMI and Hinode/SOT/SP routinely measure the full Stokes vector.

9.3 Sunspot Magnetic Structure

Derivation 3: Magnetic Pressure Balance in a Sunspot

Step 1. A sunspot is a concentration of vertical magnetic flux. The sunspot is cooler than its surroundings because the magnetic field inhibits convection. Pressure balance across the sunspot boundary requires:

$$P_{\text{ext}} = P_{\text{int}} + \frac{B^2}{2\mu_0}$$

Step 2. Since the internal gas pressure is lower (cooler gas), the magnetic pressure makes up the deficit. For $B \approx 0.3$ T in a sunspot umbra:

$$\boxed{P_B = \frac{B^2}{2\mu_0} = \frac{(0.3)^2}{2 \times 4\pi \times 10^{-7}} \approx 3.6 \times 10^4 \text{ Pa}}$$

This is comparable to the photospheric gas pressure ($\sim 10^4$ Pa), confirming that sunspots are regions where magnetic energy dominates. The Wilson depression (the sunspot being geometrically depressed by ~400 km) allows us to see deeper, hotter layers, explaining why sunspot penumbrae are brighter than umbrae.

9.4 Potential and Force-Free Fields

Derivation 4: Potential Field Source Surface (PFSS) Model

Step 1. A current-free (potential) magnetic field satisfies:

$$\nabla \times \mathbf{B} = 0 \quad \Longrightarrow \quad \mathbf{B} = -\nabla\Phi$$

Combined with $\nabla \cdot \mathbf{B} = 0$: $\nabla^2 \Phi = 0$.

Step 2. In spherical coordinates, the solution using spherical harmonics:

$$\boxed{\Phi(r,\theta,\phi) = R_\odot \sum_{\ell=1}^{\infty}\sum_{m=0}^{\ell} \left(a_\ell^m r^{-(\ell+1)} + b_\ell^m r^\ell\right) P_\ell^m(\cos\theta)(c_m\cos m\phi + d_m\sin m\phi)}$$

Step 3. The PFSS model uses boundary conditions: observed $B_r$ at$r = R_\odot$ and radial field at the source surface $r = R_{ss} \approx 2.5 R_\odot$.

For non-potential fields, the force-free approximation ($\nabla \times \mathbf{B} = \alpha\mathbf{B}$) is used, where $\alpha$ parameterizes the current density. Linear force-free ($\alpha = \text{const}$) and nonlinear force-free (NLFFF) extrapolations are the workhorses of coronal magnetic field modeling.

9.5 Magnetic Flux Distribution

Derivation 5: Magnetic Flux Tubes and Equipartition

Step 1. Magnetic flux is conserved along a flux tube:$\Phi = B \cdot A = \text{const}$. As a flux tube rises through the convection zone:

$$B(r) = B_0 \frac{A_0}{A(r)}$$

Step 2. The equipartition field strength, where magnetic energy density equals kinetic energy density of convective motions:

$$\boxed{B_{\text{eq}} = \sqrt{\mu_0 \rho v_{\text{conv}}^2} \approx 0.04 \text{ T (photosphere)} \sim 10 \text{ T (base of CZ)}}$$

At the photosphere, the observed field in network elements (~0.1-0.15 T) exceeds $B_{\text{eq}}$, indicating that magnetic flux is concentrated by convective collapse (Parker 1978) into kilogauss flux tubes with diameters of ~100 km.

Numerical Simulation

Solar Magnetic Fields: Zeeman Effect, Stokes V, Field Topology, Flux Distribution

Python

script.py154 lines

import numpy as np
import matplotlib
matplotlib.use('Agg')
import matplotlib.pyplot as plt

fig, axes = plt.subplots(2, 2, figsize=(14, 10))
fig.suptitle('Solar Magnetic Fields', fontsize=16, fontweight='bold', color='white')
fig.patch.set_facecolor('#0a0a0a')

for ax in axes.flat:
    ax.set_facecolor('#0a0a0a')
    ax.tick_params(colors='white')
    ax.xaxis.label.set_color('white')
    ax.yaxis.label.set_color('white')
    ax.title.set_color('white')
    for spine in ax.spines.values():
        spine.set_color('#333')

# --- Panel 1: Zeeman splitting ---
ax1 = axes[0, 0]
lam0 = 617.3  # nm
dlam = np.linspace(-0.05, 0.05, 500)

# No field
I_no_B = np.exp(-dlam**2 / (2 * 0.005**2))

# With B = 0.3 T, g_eff = 2.5
B = 0.3
g_eff = 2.5
delta_lam = 4.67e-13 * lam0**2 * g_eff * B  # nm, B in Tesla
sigma_line = 0.005

I_sigma_plus = 0.5 * np.exp(-(dlam - delta_lam)**2 / (2 * sigma_line**2))
I_sigma_minus = 0.5 * np.exp(-(dlam + delta_lam)**2 / (2 * sigma_line**2))
I_pi = 0.5 * np.exp(-dlam**2 / (2 * sigma_line**2))

ax1.plot(dlam, I_no_B, color='white', linewidth=2, linestyle='--', alpha=0.5, label='No field')
ax1.plot(dlam, I_sigma_plus, color='#f43f5e', linewidth=2, label='sigma+')
ax1.plot(dlam, I_pi, color='#fbbf24', linewidth=2, label='pi')
ax1.plot(dlam, I_sigma_minus, color='#60a5fa', linewidth=2, label='sigma-')

ax1.axvline(x=delta_lam, color='#f43f5e', linestyle=':', alpha=0.5)
ax1.axvline(x=-delta_lam, color='#60a5fa', linestyle=':', alpha=0.5)
ax1.annotate('delta_lam = ' + str(round(delta_lam*1000, 2)) + ' pm', xy=(0.02, 0.7),
            fontsize=8, color='white')
ax1.set_xlabel('delta lambda [nm]')
ax1.set_ylabel('Intensity')
ax1.set_title('Zeeman Splitting (Fe I 617.3 nm, B=0.3 T)')
ax1.legend(fontsize=7, facecolor='#0a0a0a', edgecolor='#333', labelcolor='white')

# --- Panel 2: Stokes V profile ---
ax2 = axes[0, 1]
# Stokes V ~ g_eff * B_par * dI/dlambda
dI_dlam = np.gradient(I_no_B, dlam)
B_values = [0.05, 0.1, 0.2, 0.3]
colors_B = ['#4ade80', '#fbbf24', '#fb923c', '#f43f5e']

for B_val, col in zip(B_values, colors_B):
    V = g_eff * B_val * dI_dlam * 0.5
    ax2.plot(dlam * 1000, V, color=col, linewidth=2,
            label='B = ' + str(int(B_val * 1000)) + ' mT')

ax2.axhline(y=0, color='white', linestyle=':', alpha=0.3)
ax2.set_xlabel('delta lambda [pm]')
ax2.set_ylabel('Stokes V (relative)')
ax2.set_title('Stokes V Profiles')
ax2.legend(fontsize=8, facecolor='#0a0a0a', edgecolor='#333', labelcolor='white')

# --- Panel 3: PFSS-like dipole field lines ---
ax3 = axes[1, 0]

# Dipole magnetic field lines
theta = np.linspace(0, 2*np.pi, 100)
r_circle = np.cos(theta)
z_circle = np.sin(theta)
ax3.plot(r_circle, z_circle, 'w-', linewidth=1.5)

# Dipole field lines: r = r_0 * sin^2(theta)
for r0 in [1.5, 2.0, 2.5, 3.0, 4.0, 5.0]:
    th = np.linspace(0.1, np.pi - 0.1, 200)
    r_fl = r0 * np.sin(th)**2
    x_fl = r_fl * np.sin(th)
    y_fl = r_fl * np.cos(th)
    # Only plot if within source surface
    mask = r_fl < 2.5
    if np.any(mask):
        ax3.plot(x_fl[mask], y_fl[mask], color='#f43f5e', linewidth=1, alpha=0.7)
        ax3.plot(-x_fl[mask], y_fl[mask], color='#60a5fa', linewidth=1, alpha=0.7)

# Source surface
th_ss = np.linspace(0, 2*np.pi, 100)
ax3.plot(2.5*np.cos(th_ss), 2.5*np.sin(th_ss), 'w--', linewidth=1, alpha=0.3, label='Source surface (2.5 R_s)')

# Open field lines
for th0 in [0.2, 0.4]:
    th_open = np.linspace(th0, np.pi/2, 50)
    r_open = np.linspace(1, 5, 50)
    x_o = r_open * np.sin(th0)
    y_o = r_open * np.cos(th0)
    ax3.plot(x_o, y_o, color='#fbbf24', linewidth=1.5, alpha=0.7)
    ax3.plot(-x_o, y_o, color='#fbbf24', linewidth=1.5, alpha=0.7)
    ax3.plot(x_o, -y_o, color='#fbbf24', linewidth=1.5, alpha=0.7)
    ax3.plot(-x_o, -y_o, color='#fbbf24', linewidth=1.5, alpha=0.7)

ax3.set_xlim(-5, 5)
ax3.set_ylim(-5, 5)
ax3.set_aspect('equal')
ax3.set_title('PFSS Dipole Field Topology')
ax3.set_xlabel('x / R_sun')
ax3.set_ylabel('z / R_sun')

# --- Panel 4: Magnetic flux distribution ---
ax4 = axes[1, 1]
np.random.seed(42)
B_values_dist = np.concatenate([
    np.random.exponential(5, 5000),    # weak field (network)
    np.random.normal(150, 30, 500),     # kG flux tubes
    np.random.normal(300, 50, 200),     # sunspot umbra
])
B_values_dist = B_values_dist[B_values_dist > 0]

ax4.hist(B_values_dist, bins=np.logspace(0, 3, 60), color='#f43f5e', alpha=0.7,
        edgecolor='#333')
ax4.set_xscale('log')
ax4.axvline(x=40, color='#fbbf24', linestyle='--', label='Equipartition (40 mT)')
ax4.axvline(x=150, color='#4ade80', linestyle='--', label='kG flux tubes')
ax4.axvline(x=300, color='#60a5fa', linestyle='--', label='Sunspot umbra')
ax4.set_xlabel('|B| [mT]')
ax4.set_ylabel('Count')
ax4.set_title('Magnetic Flux Distribution')
ax4.legend(fontsize=7, facecolor='#0a0a0a', edgecolor='#333', labelcolor='white')

plt.tight_layout()
plt.savefig('output.png', dpi=150, bbox_inches='tight', facecolor='#0a0a0a')
plt.close()
print("Plot saved to output.png")

print("\n" + "=" * 60)
print("SOLAR MAGNETIC FIELDS")
print("=" * 60)
print("\n--- Zeeman Splitting ---")
print("  Fe I 617.3 nm, g_eff = 2.5, B = 0.3 T:")
print("  delta_lambda = " + str(round(delta_lam * 1000, 2)) + " pm")
print("\n--- Typical Field Strengths ---")
print("  Quiet Sun network: 100-150 mT (kG flux tubes)")
print("  Internetwork: 1-10 mT")
print("  Sunspot umbra: 200-400 mT")
print("  Sunspot penumbra: 100-200 mT")
print("  Active region corona: 1-10 mT")
print("  Quiet Sun corona: 0.1-1 mT")
print("\n--- Magnetic Pressure (B = 0.3 T) ---")
print("  P_B = B^2/(2*mu_0) = " + str(round(0.3**2 / (2 * 4 * np.pi * 1e-7), 0)) + " Pa")
print("=" * 60)

Click Run to execute the Python code

Code will be executed with Python 3 on the server

9.6 Detailed Zeeman Splitting Derivation

Full Energy Shift: $\Delta E = \frac{e\hbar B}{2m_e}\,m_J\,g_J$

We derive the energy shift rigorously from the Hamiltonian of the magnetic interaction.

Step 1. The perturbation Hamiltonian for an atom in a uniform field $\mathbf{B} = B\hat{z}$:

$$H' = -\boldsymbol{\mu}\cdot\mathbf{B} = \frac{e}{2m_e}(\mathbf{L} + 2\mathbf{S})\cdot\mathbf{B} = \frac{eB}{2m_e}(L_z + 2S_z)$$

Step 2. We use the projection theorem: since $\mathbf{L}+2\mathbf{S}$ is not parallel to $\mathbf{J}$ in general, we project onto $\mathbf{J}$:

$$\langle J,m_J| L_z + 2S_z |J,m_J\rangle = g_J\,m_J\,\hbar$$

Step 3. To derive the Lande g-factor, note that $\mathbf{L}+2\mathbf{S} = \mathbf{J}+\mathbf{S}$. The expectation value of the projection of $\mathbf{S}$ onto $\mathbf{J}$:

$$\frac{\langle\mathbf{S}\cdot\mathbf{J}\rangle}{J(J+1)\hbar^2} = \frac{J(J+1)+S(S+1)-L(L+1)}{2J(J+1)}$$

Step 4. The complete energy shift is therefore:

$$\boxed{\Delta E = \frac{e\hbar B}{2m_e}\,m_J\,g_J, \qquad g_J = 1 + \frac{J(J+1)+S(S+1)-L(L+1)}{2J(J+1)}}$$

Step 5: Stokes Parameters for Polarimetry. The Zeeman components have definite polarization states relative to the magnetic field direction. Consider the transition $\Delta m_J = 0, \pm 1$:

$$\Delta m_J = 0 \;\;(\pi): \quad \text{linearly polarized } \| \mathbf{B}, \quad \text{contributes to } I, Q$$

$$\Delta m_J = +1 \;\;(\sigma^+): \quad \text{right circular polarization}, \quad \text{contributes to } I, V$$

$$\Delta m_J = -1 \;\;(\sigma^-): \quad \text{left circular polarization}, \quad \text{contributes to } I, -V$$

In the weak-field regime where the Zeeman splitting is much smaller than the Doppler width, one can show from the Taylor expansion of the absorption profile:

$$\boxed{V(\lambda) = -C\,g_{\text{eff}}\,B_\parallel\,\frac{\partial I_0}{\partial\lambda}}$$$$\boxed{Q(\lambda) = -\frac{C^2}{4}\,G\,B_\perp^2\,\frac{\partial^2 I_0}{\partial\lambda^2}\cos 2\chi, \quad U(\lambda) = -\frac{C^2}{4}\,G\,B_\perp^2\,\frac{\partial^2 I_0}{\partial\lambda^2}\sin 2\chi}$$

where $C = 4.67\times10^{-13}\lambda^2$ (cgs), $G$ is an effective Lande factor combination, and $\chi$ is the azimuth of the transverse field in the plane of the sky. The key insight: Stokes V gives $B_\parallel$ linearly (easy to measure), while Q,U give$B_\perp^2$ quadratically (harder, with a 180-degree azimuth ambiguity).

9.7 Potential and Force-Free Field Extrapolation

Linear Force-Free Solution: $\nabla\times\mathbf{B}=\alpha\mathbf{B}$

Step 1. The force-free condition states that the Lorentz force vanishes:

$$\mathbf{J}\times\mathbf{B} = 0 \quad\Rightarrow\quad \nabla\times\mathbf{B} = \alpha\mathbf{B}$$

Step 2. Taking the divergence: $\nabla\cdot(\nabla\times\mathbf{B}) = \alpha\nabla\cdot\mathbf{B} + \mathbf{B}\cdot\nabla\alpha = 0$. Since $\nabla\cdot\mathbf{B}=0$, we need $\mathbf{B}\cdot\nabla\alpha=0$: the parameter $\alpha$is constant along each field line.

Step 3. For the linear (constant-$\alpha$) case, take the curl of both sides:

$$\nabla\times(\nabla\times\mathbf{B}) = \alpha\nabla\times\mathbf{B} = \alpha^2\mathbf{B}$$

$$-\nabla^2\mathbf{B} + \nabla(\nabla\cdot\mathbf{B}) = \alpha^2\mathbf{B} \quad\Rightarrow\quad \nabla^2\mathbf{B} + \alpha^2\mathbf{B} = 0$$

Step 4. This is a vector Helmholtz equation. In Cartesian coordinates with a half-space geometry ($z>0$), the solution for $B_z$ is:

$$\boxed{B_z(x,y,z) = \sum_{m,n} a_{mn}\cos(k_x x)\cos(k_y y)\,e^{-\kappa z}, \quad \kappa = \sqrt{k_x^2+k_y^2-\alpha^2}}$$

When $\alpha=0$ this reduces to the potential field. Non-zero $\alpha$ introduces twist and shear in the field. The field is physically meaningful (decaying with height) only when$\alpha^2 < k_x^2+k_y^2$. The maximum allowed $|\alpha|$ sets the maximum free energy in the force-free extrapolation.

9.8 Sunspot Magnetic Field Structure

The following diagram shows the cross-section of a sunspot with its characteristic magnetic field geometry: vertical field in the umbra transitioning to nearly horizontal field in the penumbra.

Cross-section of a sunspot showing vertical magnetic field in the umbra and inclined field in the penumbra. The Wilson depression (~400 km) results from reduced gas pressure inside the magnetic flux concentration.

Extended Simulation: Zeeman Patterns & Potential Field Extrapolation

Extended: Anomalous Zeeman Patterns, Potential Field Extrapolation, Force-Free Decay

Python

script.py165 lines

import numpy as np
import matplotlib
matplotlib.use('Agg')
import matplotlib.pyplot as plt

fig, axes = plt.subplots(2, 2, figsize=(14, 10))
fig.suptitle('Zeeman Splitting Patterns and Field Extrapolation', fontsize=16, fontweight='bold', color='white')
fig.patch.set_facecolor('#0a0a0a')

# --- Panel 1: Anomalous Zeeman pattern (J=2 -> J=1) ---
ax1 = axes[0, 0]
# Upper level J=2, L=1, S=1: g_J = 1 + (6+2-2)/12 = 1.5
# Lower level J=1, L=1, S=1: g_J = 1 + (2+2-2)/4 = 1.5
# For Fe I 617.3nm: J_u=2, L_u=2, S_u=2, g_u=1.5 approximately
g_u = 1.5
g_l = 2.0
J_u = 2
J_l = 1
B_field = 0.3  # Tesla

transitions = []
colors_trans = []
for m_u in range(-J_u, J_u+1):
    for m_l in range(-J_l, J_l+1):
        dm = m_u - m_l
        if abs(dm) <= 1:
            shift = g_u * m_u - g_l * m_l
            transitions.append((shift, dm))

dlam_unit = 4.67e-13 * 617.3**2 * B_field  # nm per unit g*m

sigma = 0.003
lam = np.linspace(-0.06, 0.06, 1000)
I_total = np.zeros_like(lam)
for shift, dm in transitions:
    center = shift * dlam_unit / 2.5  # normalize
    amp = 0.3
    comp = amp * np.exp(-(lam - center)**2 / (2*sigma**2))
    if dm == 0:
        ax1.fill_between(lam*1000, 0, comp, color='#fbbf24', alpha=0.3)
    elif dm == 1:
        ax1.fill_between(lam*1000, 0, comp, color='#f43f5e', alpha=0.3)
    else:
        ax1.fill_between(lam*1000, 0, comp, color='#60a5fa', alpha=0.3)
    I_total += comp

ax1.plot(lam*1000, I_total, color='white', linewidth=1.5)
ax1.set_xlabel('delta lambda [pm]')
ax1.set_ylabel('Intensity')
ax1.set_title('Anomalous Zeeman Pattern (B=0.3 T)')
from matplotlib.patches import Patch
legend_elements = [Patch(facecolor='#f43f5e', alpha=0.5, label='sigma+'),
                   Patch(facecolor='#fbbf24', alpha=0.5, label='pi'),
                   Patch(facecolor='#60a5fa', alpha=0.5, label='sigma-')]
ax1.legend(handles=legend_elements, fontsize=8, facecolor='#0a0a0a', edgecolor='#333', labelcolor='white')

# --- Panel 2: Potential field extrapolation (2D bipolar) ---
ax2 = axes[0, 1]
x = np.linspace(-5, 5, 200)
z = np.linspace(0.1, 5, 200)
X, Z = np.meshgrid(x, z)

# Two point charges (positive and negative poles)
d = 1.5  # half-separation
depth = 0.5
Bx = (X - d)/((X - d)**2 + (Z + depth)**2) - (X + d)/((X + d)**2 + (Z + depth)**2)
Bz_field = (Z + depth)/((X - d)**2 + (Z + depth)**2) - (Z + depth)/((X + d)**2 + (Z + depth)**2)

ax2.streamplot(X, Z, Bx, Bz_field, color=np.sqrt(Bx**2 + Bz_field**2),
              cmap='RdYlBu_r', density=2, linewidth=1.2, arrowsize=1)
ax2.plot(d, 0, 'o', color='#f43f5e', markersize=10, label='+')
ax2.plot(-d, 0, 'o', color='#60a5fa', markersize=10, label='-')
ax2.set_xlabel('x [arcsec]')
ax2.set_ylabel('z [arcsec]')
ax2.set_title('Potential Field Extrapolation (Bipolar)')
ax2.set_ylim(0, 5)
ax2.legend(fontsize=10, facecolor='#0a0a0a', edgecolor='#333', labelcolor='white')

# --- Panel 3: Force-free field with shear ---
ax3 = axes[1, 0]
alpha_vals = [0, 0.5, 1.0, 1.4]
colors_a = ['#4ade80', '#fbbf24', '#fb923c', '#f43f5e']
kx = np.pi / 5
ky = 0
z_arr = np.linspace(0, 5, 200)

for alpha_val, col in zip(alpha_vals, colors_a):
    kappa_sq = kx**2 - alpha_val**2
    if kappa_sq > 0:
        kappa = np.sqrt(kappa_sq)
        Bz_profile = np.exp(-kappa * z_arr)
        lbl_str = 'alpha=' + str(alpha_val) + ', kappa=' + str(round(kappa, 2))
    else:
        lbl_str = 'alpha=' + str(alpha_val) + ' (oscillatory)'
        Bz_profile = np.cos(np.sqrt(-kappa_sq) * z_arr)
    ax3.plot(z_arr, Bz_profile, color=col, linewidth=2.5, label=lbl_str)

ax3.set_xlabel('Height z [arcsec]')
ax3.set_ylabel('B_z / B_z(0)')
ax3.set_title('Force-Free Field Decay with Height')
ax3.axhline(y=0, color='white', linestyle=':', alpha=0.3)
ax3.legend(fontsize=7, facecolor='#0a0a0a', edgecolor='#333', labelcolor='white')

# --- Panel 4: g_eff vs quantum numbers ---
ax4 = axes[1, 1]
lines = []
labels = []
for L in range(0, 4):
    for S_val in [0, 0.5, 1]:
        for J in [abs(L-S_val)]:
            if J == 0:
                continue
            g = 1 + (J*(J+1) + S_val*(S_val+1) - L*(L+1)) / (2*J*(J+1))
            lines.append((L, S_val, J, g))

L_arr = np.arange(0, 4)
S_labels = {0: 'S=0', 0.5: 'S=1/2', 1: 'S=1'}
for S_val, col in zip([0, 0.5, 1], ['#60a5fa', '#fbbf24', '#f43f5e']):
    g_vals = []
    L_valid = []
    for L in range(0, 4):
        J = abs(L - S_val)
        if J == 0:
            continue
        g = 1 + (J*(J+1) + S_val*(S_val+1) - L*(L+1)) / (2*J*(J+1))
        g_vals.append(g)
        L_valid.append(L)
    if len(g_vals) > 0:
        ax4.plot(L_valid, g_vals, 'o-', color=col, markersize=8, linewidth=2,
                label=S_labels[S_val] + ', J=|L-S|')

ax4.set_xlabel('Orbital quantum number L')
ax4.set_ylabel('Lande g-factor')
ax4.set_title('Lande g-factor for Different L, S')
ax4.set_xticks([0, 1, 2, 3])
ax4.set_xticklabels(['S (L=0)', 'P (L=1)', 'D (L=2)', 'F (L=3)'], color='white')
ax4.legend(fontsize=8, facecolor='#0a0a0a', edgecolor='#333', labelcolor='white')

plt.tight_layout()
plt.savefig('output.png', dpi=150, bbox_inches='tight', facecolor='#0a0a0a')
plt.close()
print("Plot saved to output.png")

print("\n" + "=" * 60)
print("EXTENDED: ZEEMAN PATTERNS AND FIELD EXTRAPOLATION")
print("=" * 60)
print("\n--- Anomalous Zeeman Effect ---")
print("  Multiple pi and sigma components for J>1/2 transitions")
print("  g_eff determines the effective splitting")
print("\n--- Force-Free Fields ---")
print("  alpha=0: potential field (current-free)")
print("  alpha>0: linear force-free (constant alpha)")
print("  NLFFF: nonlinear (alpha varies along field lines)")
print("  Constraint: alpha^2 < k_x^2 + k_y^2 for decay with height")
print("=" * 60)

Click Run to execute the Python code

Code will be executed with Python 3 on the server

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Magnetic Fields of the Sun

9.1 The Zeeman Effect

Derivation 1: Zeeman Splitting in a Magnetic Field

9.2 Magnetograph Measurements

Derivation 2: Stokes Parameters and Magnetic Field Inference

9.3 Sunspot Magnetic Structure

Derivation 3: Magnetic Pressure Balance in a Sunspot

9.4 Potential and Force-Free Fields

Derivation 4: Potential Field Source Surface (PFSS) Model

9.5 Magnetic Flux Distribution

Derivation 5: Magnetic Flux Tubes and Equipartition

Numerical Simulation

Solar Magnetic Fields: Zeeman Effect, Stokes V, Field Topology, Flux Distribution

9.6 Detailed Zeeman Splitting Derivation

Full Energy Shift: \(\Delta E = \frac{e\hbar B}{2m_e}\,m_J\,g_J\)

9.7 Potential and Force-Free Field Extrapolation

Linear Force-Free Solution: \(\nabla\times\mathbf{B}=\alpha\mathbf{B}\)

9.8 Sunspot Magnetic Field Structure

Extended Simulation: Zeeman Patterns & Potential Field Extrapolation

Extended: Anomalous Zeeman Patterns, Potential Field Extrapolation, Force-Free Decay