Part II: Solar Atmosphere | Chapter 7

The Solar Corona

The million-degree corona: heating problem, Alfven waves, scaling laws, and X-ray emission

7.1 The Coronal Heating Problem

Derivation 1: Why the Corona Cannot Be Heated by Thermal Conduction Alone

The corona has a temperature of 1-3 MK, yet it sits above the 5800 K photosphere. This violates naive thermodynamic expectations and requires a non-thermal energy source.

Step 1. The radiative loss function for an optically thin plasma is:

$$E_{\text{rad}} = n_e^2 \Lambda(T) \quad \text{[W/m}^3\text{]}$$

where $\Lambda(T)$ is the radiative loss function, peaking near $10^5$ K (transition region) and relatively flat around $10^{-23}$ W m$^3$ at coronal temperatures.

Step 2. For a typical coronal loop with $n_e \approx 10^{15}$ m$^{-3}$, length $L \approx 10^8$ m, the total radiative loss rate is:

$$P_{\text{rad}} = n_e^2 \Lambda(T) \cdot V \approx (10^{15})^2 \times 10^{-23} \times 10^{22} \approx 10^{19} \text{ W}$$

Step 3. The total coronal energy requirement:

$$\boxed{F_{\text{corona}} \approx 300 \text{ W/m}^2 \text{ (quiet Sun)} \sim 10^4 \text{ W/m}^2 \text{ (active regions)}}$$

This is only $\sim 10^{-6}$ of the solar luminosity, but no single mechanism has been universally accepted as the primary heating source despite 80+ years of research since the discovery of the hot corona by Grotrian (1939) and Edlen (1942).

7.2 Nanoflare Heating (Parker 1988)

Derivation 2: Energy Release from Magnetic Braiding

Parker (1988) proposed that the corona is heated by a multitude of small magnetic reconnection events (nanoflares) driven by photospheric footpoint motions braiding the coronal magnetic field.

Step 1. Photospheric motions with velocity $v_\perp$ displace magnetic footpoints, building up tangential field components $B_\perp$ over time $t$:

$$B_\perp \approx B_0 \frac{v_\perp t}{L}$$

Step 2. The energy density stored in the tangled field is:

$$u_B = \frac{B_\perp^2}{2\mu_0} = \frac{B_0^2 v_\perp^2 t^2}{2\mu_0 L^2}$$

Step 3. The Poynting flux injected at the photospheric boundary:

$$\boxed{S = \frac{B_0 B_\perp v_\perp}{\mu_0} \approx \frac{B_0^2 v_\perp^2 t}{\mu_0 L} \sim 300 \text{ W/m}^2}$$

For $B_0 = 100$ G, $v_\perp = 1$ km/s, this gives sufficient energy flux. When tangential discontinuities form, magnetic reconnection releases the stored energy as heat. Each nanoflare has energy $\sim 10^{17}$ J (compared to $\sim 10^{25}$ J for a large flare).

7.3 Alfven Wave Heating

Derivation 3: Alfven Wave Energy Flux and Dissipation

Step 1. The Alfven speed in the corona:

$$v_A = \frac{B}{\sqrt{\mu_0 \rho}} \approx \frac{10^{-3}}{\sqrt{4\pi \times 10^{-7} \times 10^{-12}}} \approx 1000 \text{ km/s}$$

Step 2. The energy flux carried by Alfven waves with velocity perturbation $\delta v$:

$$F_A = \rho (\delta v)^2 v_A$$

Step 3. Alfven waves can be dissipated by phase mixing (in inhomogeneous plasma) or by turbulent cascade. The phase mixing damping length is:

$$\boxed{L_{\text{damp}} \sim \left(\frac{6 v_A}{\omega^2 \eta}\right)^{1/3} l_\perp^{2/3}}$$

where $l_\perp$ is the transverse scale of the Alfven speed gradient and$\eta$ is the magnetic diffusivity. Observations from CoMP and SDO/AIA have confirmed the presence of ubiquitous transverse (Alfvenic) oscillations in the corona with amplitudes of ~20 km/s, carrying sufficient energy flux.

7.4 Coronal X-ray Emission

Derivation 4: Bremsstrahlung Emission from Coronal Plasma

Step 1. The thermal bremsstrahlung emissivity (free-free emission) for a hydrogen plasma:

$$\varepsilon_{ff} = 6.8 \times 10^{-38} Z^2 n_e n_i T^{-1/2} \bar{g}_{ff} \exp\left(-\frac{h\nu}{k_BT}\right) \text{ [W m}^{-3}\text{ Hz}^{-1}\text{]}$$

Step 2. Integrating over frequency gives the total bremsstrahlung power:

$$\boxed{P_{ff} = 1.43 \times 10^{-27} Z^2 n_e n_i T^{1/2} \bar{g}_{ff} \text{ [W m}^{-3}\text{]}}$$

Step 3. At coronal temperatures, line emission (from Fe, O, Si, etc.) dominates over bremsstrahlung. The emission measure determines the observable flux:

$$EM = \int n_e^2 \, dV \quad \text{[m}^{-3}\text{]}$$

The differential emission measure DEM(T) = $n_e^2 dV/dT$ is the fundamental observable that encodes the temperature distribution of coronal plasma. Modern instruments (SDO/AIA, Hinode/EIS, Solar Orbiter/SPICE) provide multi-wavelength imaging and spectroscopy to constrain the DEM.

7.5 RTV Scaling Laws

Derivation 5: Rosner-Tucker-Vaiana Loop Scaling

The RTV scaling laws (1978) relate the temperature, density, and length of static coronal loops in energy balance between uniform heating, thermal conduction, and radiation.

Step 1. The energy equation for a symmetric coronal loop of half-length $L$:

$$\frac{d}{ds}\left(\kappa_0 T^{5/2} \frac{dT}{ds}\right) = n_e^2 \Lambda(T) - H_0$$

where $\kappa_0 T^{5/2}$ is the Spitzer conductivity and $H_0$is the volumetric heating rate.

Step 2. The boundary conditions are: $dT/ds = 0$ at the loop apex ($s = 0$) and $T = T_0$ at the footpoints ($s = L$). Approximating $\Lambda(T) \approx \chi T^\alpha$ with $\alpha \approx -1/2$:

Step 3. Dimensional analysis and integration yield the RTV scaling laws:

$$\boxed{T_{\max} \approx 1.4 \times 10^3 (p_0 L)^{1/3} \text{ K}}$$$$\boxed{H_0 \approx 10^5 p_0^{7/6} L^{-5/6} \text{ W/m}^3}$$

These scaling laws predict that longer loops are hotter and that the heating rate scales steeply with pressure. Observations generally confirm $T \propto L^{1/3}$for active region loops, though many loops appear to be in a state of impulsive heating (nanoflare storms) rather than steady equilibrium.

Numerical Simulation

Solar Corona: Radiative Losses, RTV Scaling, Loop Profiles, DEM Distributions

Python

script.py117 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 Corona Physics', 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: Radiative loss function ---
ax1 = axes[0, 0]
log_T = np.linspace(4, 8, 500)
T = 10**log_T

# Approximate radiative loss function (Klimchuk 2008)
Lambda = np.where(T < 1e4, 1e-26 * (T/1e4)**(-1),
         np.where(T < 5e4, 1e-26 * (T/1e4)**2,
         np.where(T < 1e5, 5e-22 * (T/5e4)**(-0.5),
         np.where(T < 3e5, 3e-22 * (T/1e5)**3,
         np.where(T < 1e6, 3e-22 * (T/3e5)**(-1),
         np.where(T < 1e7, 1e-22 * (T/1e6)**(-0.5),
         1e-22 * (T/1e7)**0.5))))))

ax1.loglog(T, Lambda, color='#f43f5e', linewidth=2.5)
ax1.axvline(x=1e6, color='#fbbf24', linestyle='--', alpha=0.5, label='1 MK')
ax1.axvline(x=1e5, color='#4ade80', linestyle='--', alpha=0.5, label='0.1 MK (TR)')
ax1.set_xlabel('Temperature [K]')
ax1.set_ylabel('Lambda(T) [W m^3]')
ax1.set_title('Radiative Loss Function')
ax1.set_xlim(1e4, 1e8)
ax1.legend(fontsize=8, facecolor='#0a0a0a', edgecolor='#333', labelcolor='white')

# --- Panel 2: RTV scaling laws ---
ax2 = axes[0, 1]
L_arr = np.logspace(7, 10, 200)  # loop half-length in m
p0_vals = [0.01, 0.1, 1.0]  # pressure in Pa
colors_p = ['#f43f5e', '#fb923c', '#fbbf24']

for p0, col in zip(p0_vals, colors_p):
    T_max = 1.4e3 * (p0 * L_arr)**(1.0/3.0)
    ax2.loglog(L_arr / 1e6, T_max / 1e6, color=col, linewidth=2.5,
              label='p = ' + str(p0) + ' Pa')

ax2.set_xlabel('Loop half-length [Mm]')
ax2.set_ylabel('T_max [MK]')
ax2.set_title('RTV Scaling: T vs L')
ax2.legend(fontsize=8, facecolor='#0a0a0a', edgecolor='#333', labelcolor='white')

# --- Panel 3: Coronal loop temperature profile ---
ax3 = axes[1, 0]
s = np.linspace(0, 1, 200)  # normalized along loop (0 = footpoint, 1 = apex)

# Approximate temperature profile for different heating models
T_uniform = 1.0 * np.sin(np.pi * s / 2)**0.4  # uniform heating
T_foot = 1.0 * np.sin(np.pi * s / 2)**0.8     # footpoint heating
T_apex = 1.0 * (0.3 + 0.7 * np.sin(np.pi * s / 2)**0.2)  # apex heating

ax3.plot(s, T_uniform, color='#f43f5e', linewidth=2.5, label='Uniform heating')
ax3.plot(s, T_foot, color='#fb923c', linewidth=2.5, label='Footpoint heating')
ax3.plot(s, T_apex, color='#4ade80', linewidth=2.5, label='Apex heating')
ax3.set_xlabel('Position along loop (0=foot, 1=apex)')
ax3.set_ylabel('T / T_max')
ax3.set_title('Loop Temperature Profiles')
ax3.legend(fontsize=8, facecolor='#0a0a0a', edgecolor='#333', labelcolor='white')

# --- Panel 4: DEM distribution ---
ax4 = axes[1, 1]
log_T_dem = np.linspace(5, 7, 200)
T_dem = 10**log_T_dem

# Quiet Sun DEM
DEM_qs = 1e22 * np.exp(-((log_T_dem - 6.0)/0.3)**2) + 5e21 * np.exp(-((log_T_dem - 5.5)/0.2)**2)

# Active region DEM
DEM_ar = 5e22 * np.exp(-((log_T_dem - 6.3)/0.4)**2) + 1e22 * np.exp(-((log_T_dem - 5.8)/0.2)**2)

# Coronal hole DEM
DEM_ch = 5e21 * np.exp(-((log_T_dem - 5.9)/0.2)**2)

ax4.semilogy(log_T_dem, DEM_qs, color='#f43f5e', linewidth=2.5, label='Quiet Sun')
ax4.semilogy(log_T_dem, DEM_ar, color='#fbbf24', linewidth=2.5, label='Active region')
ax4.semilogy(log_T_dem, DEM_ch, color='#60a5fa', linewidth=2.5, label='Coronal hole')
ax4.set_xlabel('log T [K]')
ax4.set_ylabel('DEM [cm^-5 K^-1]')
ax4.set_title('Differential Emission Measure')
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("SOLAR CORONA")
print("=" * 60)
print("\n--- Coronal Parameters ---")
print("  Temperature: 1-3 MK (quiet Sun), up to 10 MK (flares)")
print("  Density: 10^14 - 10^16 m^-3")
print("  Alfven speed: ~1000 km/s")
print("  Plasma beta: ~0.01 (magnetically dominated)")
print("\n--- Heating Requirements ---")
print("  Quiet Sun corona: ~300 W/m^2")
print("  Active region corona: ~10^4 W/m^2")
print("  Coronal hole: ~100 W/m^2")
print("\n--- RTV Scaling (typical) ---")
print("  L = 100 Mm, p = 0.1 Pa: T_max = " + str(round(1.4e3 * (0.1 * 1e8)**(1.0/3.0) / 1e6, 1)) + " MK")
print("=" * 60)

Click Run to execute the Python code

Code will be executed with Python 3 on the server

7.6 Coronal Energy Budget — Detailed Derivation

We derive the energy flux required to maintain the corona against radiative and conductive losses, and compare it with the available photospheric Poynting flux.

Energy Flux Requirements

Step 1. The total coronal radiative loss per unit area for a loop of half-length$L$ and density $n_e$ is:

$$F_{\text{rad}} = n_e^2 \Lambda(T) \cdot 2L$$

where $\Lambda(T) \approx 10^{-23}$ W m$^3$ at $T \sim 10^6$ K.

Step 2. For an active region: $n_e \sim 10^{16}$ m$^{-3}$,$L \sim 5 \times 10^7$ m:

$$F_{\text{rad}} \approx (10^{16})^2 \times 10^{-23} \times 10^8 \approx 10^{17}\text{ W/m}^2 \cdot\text{m} = 10^7 \text{ erg/cm}^2\text{/s}$$

Step 3. For the quiet Sun: $n_e \sim 10^{14}\text{--}10^{15}$ m$^{-3}$, giving $F_{\text{rad}} \sim 3 \times 10^5$ erg/cm$^2$/s.

Comparison with Photospheric Poynting Flux

Step 4. The Poynting flux injected by photospheric motions shuffling magnetic footpoints is:

$$F_P = \frac{B_\perp B_0 v_\perp}{\mu_0} \sim \frac{B_0^2 v_\perp^2 t}{\mu_0 L}$$

For $B_0 = 100$ G = 0.01 T, $v_\perp = 1$ km/s = 10$^3$ m/s:

$$\boxed{F_P \sim \frac{B_0^2 v_\perp}{\mu_0} = \frac{(0.01)^2 \times 10^3}{4\pi \times 10^{-7}} \approx 8 \times 10^4 \text{ W/m}^2 \sim 10^7 \text{ erg/cm}^2\text{/s}}$$

Photospheric Poynting flux — sufficient for active region heating

This confirms that the available magnetic energy flux from photospheric motions is sufficient to power both quiet-Sun and active-region coronae, provided an efficient dissipation mechanism (reconnection, wave damping, turbulence) operates.

7.7 RTV Scaling Laws — Full Derivation

The Rosner-Tucker-Vaiana (1978) scaling laws are derived from the static energy balance equation for a coronal loop.

Energy Balance Equation

Step 1. For a symmetric loop with coordinate $s$ along the field (footpoint at $s = 0$, apex at $s = L$), the steady-state energy equation is:

$$\frac{d}{ds}\left(\kappa_0 T^{5/2}\frac{dT}{ds}\right) = n_e^2 \Lambda(T) - H_0$$

where $\kappa_0 \approx 10^{-11}$ W m$^{-1}$ K$^{-7/2}$(Spitzer conductivity coefficient) and $H_0$ is the uniform volumetric heating rate.

Step 2. Boundary conditions: at the apex ($s = L$),$dT/ds = 0$ (symmetry). At the footpoint ($s = 0$),$T = T_0 \ll T_{\max}$.

Step 3. Approximating $\Lambda(T) \approx \chi T^\alpha$ with$\alpha \approx -1/2$ and $\chi \approx 10^{-19}$ W m$^3$ K$^{1/2}$for coronal temperatures, the heating exactly balances radiation:$H_0 = n_e^2 \Lambda(T_{\max})$.

Step 4. The conductive flux at the footpoint must carry away the net heating. Dimensional analysis of the energy equation gives:

$$\frac{\kappa_0 T_{\max}^{7/2}}{L^2} \sim n_e^2 \chi T_{\max}^\alpha$$

Step 5. Using the ideal gas law $P = 2 n_e k_B T$ to eliminate$n_e$: $n_e = P / (2 k_B T)$. Substituting:

$$\frac{\kappa_0 T_{\max}^{7/2}}{L^2} \sim \frac{P^2 \chi T_{\max}^\alpha}{4 k_B^2 T_{\max}^2}$$

Step 6. With $\alpha = -1/2$, collecting powers of $T_{\max}$:

$$T_{\max}^{7/2 + 2 + 1/2} = T_{\max}^6 \sim \frac{P^2 \chi L^2}{4 k_B^2 \kappa_0}$$

Therefore $T_{\max}^3 \sim P L$, giving:

$$\boxed{T_{\max} = 1400 \,(P \cdot L)^{1/3} \text{ K}}$$

where $P$ is in dyn/cm$^2$ and $L$ in cm

Step 7. Inverting for the pressure:

$$\boxed{P = \left(\frac{T_{\max}}{1400}\right)^3 \frac{1}{L}}$$

Numerical Example

A coronal loop with $L = 10^{10}$ cm (100 Mm) and$P = 1$ dyn/cm$^2$ (0.1 Pa):$T_{\max} = 1400 \times (1 \times 10^{10})^{1/3} = 1400 \times 2154 \approx 3.0$ MK. This is typical of active region loops observed with SDO/AIA.

7.8 Alfven Wave Heating — Detailed Analysis

Alfven Speed Derivation

Step 1. The Alfven wave is a transverse MHD wave propagating along the magnetic field. From the linearized MHD equations, the dispersion relation for incompressible perturbations is $\omega = k_\parallel v_A$, with the Alfven speed:

$$\boxed{v_A = \frac{B}{\sqrt{4\pi\rho}} = \frac{B}{\sqrt{\mu_0 \rho}}}$$

CGS (left) and SI (right) forms of the Alfven speed

Step 2. In the corona with $B \sim 10$ G = $10^{-3}$ T,$n_e \sim 10^{15}$ m$^{-3}$($\rho \sim n_e m_p \sim 1.7 \times 10^{-12}$ kg/m$^3$):

$$v_A = \frac{10^{-3}}{\sqrt{4\pi \times 10^{-7} \times 1.7 \times 10^{-12}}} \approx 700 \text{ km/s}$$

Wave Energy Flux

Step 3. An Alfven wave with velocity perturbation amplitude$\langle\delta v^2\rangle^{1/2}$ carries an energy flux:

$$\boxed{F_A = \rho\,v_A\,\langle\delta v^2\rangle}$$

For $\langle\delta v^2\rangle^{1/2} \sim 20$ km/s (observed by CoMP):

$$F_A \sim 1.7 \times 10^{-12} \times 7 \times 10^5 \times (2 \times 10^4)^2 \approx 500 \text{ W/m}^2$$

Damping Length

Step 4. Alfven waves dissipate via phase mixing in a transversely inhomogeneous corona. The damping length for a wave of frequency $\omega$ in a medium with transverse gradient scale $l_\perp$ and resistive diffusivity$\eta$ is:

$$\boxed{L_{\text{damp}} \sim \left(\frac{6\,v_A}{\omega^2\,\eta}\right)^{1/3} l_\perp^{2/3}}$$

For coronal conditions ($\eta \sim 1$ m$^2$/s,$l_\perp \sim 10^6$ m, $\omega \sim 0.1$ rad/s):$L_{\text{damp}} \sim 10^{10}$ m, comparable to loop lengths. Turbulent cascade to small scales can dramatically reduce this length, making Alfven wave heating viable.

Diagram: Coronal Loop Temperature and Density Profiles

Advanced Simulation: RTV Scaling, Alfven Waves, and Loop Profiles

RTV Scaling Law Explorer, Alfven Wave Propagation, and Coronal Loop Temperature Profiles

Python

script.py144 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('Coronal Loop Physics: RTV Scaling, Alfven Waves, Temperature Profiles',
             fontsize=14, fontweight='bold', color='white')
fig.patch.set_facecolor('#0a0a0a')

# === Panel 1: RTV scaling law explorer ===
ax1 = axes[0, 0]
L_Mm = np.logspace(0.5, 2.5, 200)  # Mm
L_cm = L_Mm * 1e8  # cm

pressures = [0.03, 0.1, 0.3, 1.0, 3.0]
colors_p = ['#60a5fa', '#4ade80', '#fbbf24', '#f43f5e', '#a78bfa']

for P_cgs, col in zip(pressures, colors_p):
    T_max = 1400.0 * (P_cgs * L_cm)**(1.0 / 3.0)
    ax1.loglog(L_Mm, T_max / 1e6, color=col, linewidth=2,
              label='P = ' + str(P_cgs) + ' dyn/cm2')

ax1.axhline(y=1.0, color='white', linestyle=':', alpha=0.2)
ax1.axhline(y=3.0, color='white', linestyle=':', alpha=0.2)
ax1.set_xlabel('Loop half-length L [Mm]')
ax1.set_ylabel('T_max [MK]')
ax1.set_title('RTV Scaling: T_max vs L')
ax1.legend(fontsize=7, facecolor='#0a0a0a', edgecolor='#333', labelcolor='white')
ax1.set_ylim(0.3, 30)

# === Panel 2: Alfven wave propagation ===
ax2 = axes[0, 1]
# Height above photosphere
h_km = np.linspace(0, 50000, 500)  # km
h_m = h_km * 1e3

# Model magnetic field and density profiles
B_phot = 100.0  # G at photosphere
B_h = B_phot * np.exp(-h_km / 50000)  # slow decrease in G

# Density profile (exponential with scale height increasing)
H_low = 150.0   # km in photosphere
H_high = 50000.0  # km in corona
H_eff = H_low + (H_high - H_low) * (1.0 - np.exp(-h_km / 2000.0))
rho_0 = 3e-4  # kg/m^3 at photosphere
rho_h = rho_0 * np.exp(-np.cumsum(1.0 / H_eff) * (h_km[1] - h_km[0]))

# Alfven speed
B_SI = B_h * 1e-4  # T
mu_0 = 4.0 * np.pi * 1e-7
v_A = B_SI / np.sqrt(mu_0 * rho_h) / 1e3  # km/s
v_A = np.clip(v_A, 0, 5000)

ax2.semilogy(h_km / 1000, v_A, color='#f43f5e', linewidth=2.5, label='Alfven speed')
ax2.axhline(y=1000, color='#fbbf24', linestyle='--', alpha=0.4, linewidth=1)
ax2.text(30, 1200, '1000 km/s', color='#fbbf24', fontsize=9)
ax2.set_xlabel('Height [Mm]')
ax2.set_ylabel('Alfven speed [km/s]')
ax2.set_title('Alfven Speed Profile v_A(h)')
ax2.set_ylim(0.1, 5000)
ax2.legend(fontsize=9, facecolor='#0a0a0a', edgecolor='#333', labelcolor='white')

# === Panel 3: Coronal loop temperature profiles ===
ax3 = axes[1, 0]
s_norm = np.linspace(0, 1, 300)  # 0 = footpoint, 1 = apex

# Solve approximate RTV temperature profile
# T(s) ~ T_max * [sin(pi*s/2)]^(2/7) for uniform heating
T_uniform = np.sin(np.pi * s_norm / 2.0 + 1e-6)**0.286
T_foot = np.sin(np.pi * s_norm / 2.0 + 1e-6)**0.667
T_apex_heat = 0.3 + 0.7 * np.sin(np.pi * s_norm / 2.0 + 1e-6)**0.15

# Normalise
T_uniform = T_uniform / T_uniform[-1]
T_foot = T_foot / T_foot[-1]
T_apex_heat = T_apex_heat / T_apex_heat[-1]

ax3.plot(s_norm, T_uniform * 3.0, color='#f43f5e', linewidth=2.5, label='Uniform heating (3 MK)')
ax3.plot(s_norm, T_foot * 2.0, color='#fb923c', linewidth=2.5, label='Footpoint heating (2 MK)')
ax3.plot(s_norm, T_apex_heat * 1.5, color='#4ade80', linewidth=2.5, label='Apex heating (1.5 MK)')

ax3.set_xlabel('Position along loop (0 = footpoint, 1 = apex)')
ax3.set_ylabel('Temperature [MK]')
ax3.set_title('Loop T(s) for Different Heating Locations')
ax3.legend(fontsize=8, facecolor='#0a0a0a', edgecolor='#333', labelcolor='white')

# === Panel 4: Alfven wave energy flux vs perturbation amplitude ===
ax4 = axes[1, 1]
dv_arr = np.linspace(1, 50, 200)  # km/s

# Different coronal densities
n_vals = [1e14, 5e14, 1e15, 5e15]
colors_n = ['#60a5fa', '#4ade80', '#fbbf24', '#f43f5e']
m_p = 1.673e-27

for n_e, col in zip(n_vals, colors_n):
    rho_val = n_e * m_p  # kg/m^3
    B_val = 10e-4  # 10 G in T
    vA_val = B_val / np.sqrt(mu_0 * rho_val)
    F_wave = rho_val * vA_val * (dv_arr * 1e3)**2  # W/m^2
    ax4.semilogy(dv_arr, F_wave, color=col, linewidth=2,
                label='n_e = ' + str("%.0e" % n_e) + ' m^-3')

ax4.axhline(y=300, color='white', linestyle='--', alpha=0.3, linewidth=1)
ax4.text(35, 400, 'Quiet Sun requirement', color='white', fontsize=8)
ax4.axhline(y=1e4, color='white', linestyle=':', alpha=0.3, linewidth=1)
ax4.text(35, 1.3e4, 'Active region requirement', color='white', fontsize=8)
ax4.set_xlabel('Wave amplitude dv [km/s]')
ax4.set_ylabel('Wave energy flux F_A [W/m^2]')
ax4.set_title('Alfven Wave Energy Flux')
ax4.legend(fontsize=7, facecolor='#0a0a0a', edgecolor='#333', labelcolor='white')
ax4.set_ylim(1, 1e7)

plt.tight_layout()
plt.savefig('output.png', dpi=150, bbox_inches='tight', facecolor='#0a0a0a')
plt.close()
print("Plot saved to output.png")
print("")
print("=== RTV Scaling Law Examples ===")
for L_val in [10, 50, 100, 200]:
    L_c = L_val * 1e8
    for P_val in [0.1, 1.0]:
        Tm = 1400.0 * (P_val * L_c)**(1.0/3.0)
        print("  L = " + str(L_val) + " Mm, P = " + str(P_val) + " dyn/cm2 => T_max = " + str(round(Tm/1e6, 2)) + " MK")
print("")
print("=== Alfven Wave Heating ===")
rho_cor = 1e15 * 1.673e-27
B_cor = 10e-4
vA_cor = B_cor / np.sqrt(4 * np.pi * 1e-7 * rho_cor)
print("  v_A = " + str(round(vA_cor / 1e3)) + " km/s (B=10 G, n_e=10^15 m^-3)")
dv_obs = 20e3  # 20 km/s
F_obs = rho_cor * vA_cor * dv_obs**2
print("  F_A (dv=20 km/s) = " + str(round(F_obs)) + " W/m^2")

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Code will be executed with Python 3 on the server

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