FORTH Theory Calculation #002: Magnetic Field Structure Visualization in 4D Torus

Overview

We calculated the magnetic field configuration in a 4D torus structure through numerical simulation and visualized it via projection into 3D space. Using M87 black hole parameters, we analyzed the magnetic field distribution in a structure with major radius R=1000Rs and minor radius r=1Rs.

Simulation Setup

Physical Constants (CODATA 2018)

Gravitational constant G = 6.67430×10⁻¹¹ m³ kg⁻¹ s⁻²
Speed of light c = 299,792,458 m/s (defined value)
Solar mass M☉ = 1.98847×10³⁰ kg

M87 Black Hole

Mass: M = 6.5×10⁹ M☉ = 1.293×10⁴⁰ kg
Schwarzschild radius: Rs = 1.920×10¹³ m

Torus Parameters

Major radius: R = 1000 Rs = 1.920×10¹⁶ m
Minor radius: r = 1 Rs = 1.920×10¹³ m
R/r ratio: 1000

Main Calculation Results

1. Accurate Jet Velocity Calculation

Calculation Process:

v/c = √(1 - (r/R)²)
    = √(1 - (1/1000)²)
    = √(1 - 0.000001)
    = √0.999999
    = 0.999999500

Result: v/c = 0.999999500 (Lorentz factor γ = 1000)

2. Magnetic Field Structure Parameters

Helical Structure

Pitch angle: α = arctan(r/R) = arctan(0.001) = 0.057°
Number of helical turns: N = 2πR/r = 6283
Characteristic scale: λ = 2πr/Rs = 6.28 Rs

Magnetic Field Strength Distribution

• Reference field strength: B₀ = 100 Gauss (observational estimate for M87)
• Toroidal component: B_tor = B₀√(r/R) = 3.16 Gauss
• Poloidal component: B_pol = 0.1B₀sin(w) (W-axis phase dependent)
• W-axis component: B_w = 0.05B₀cos(φ+w) (generates helical structure)

3. W-Axis Period (Disappearance Time)

Calculation Process:

T_W = 2πr/c
    = 2π × 1.920×10¹³ / 299,792,458
    = 4.025×10⁵ seconds
    = 111.8 hours

Result: T_W = 111.8 hours = 4.66 days

4. Energy Analysis

Magnetic Energy Density

B = 0.01 Tesla (100 Gauss)
U_B = B²/(2μ₀) = (0.01)²/(2×4π×10⁻⁷) = 39.8 J/m³

Kinetic Energy Density

Electron density: n_e = 10⁸ /m³
Electron mass: m_e = 9.109×10⁻³¹ kg
Jet velocity: v = 0.999999500c

U_K = ½n_e m_e v² = 4.09×10⁻⁶ J/m³

Conversion Efficiency

η = U_K/(U_B + U_K) ≈ 0.01%

Note: This low efficiency is due to the simplified model. In actual black hole accretion systems, conversion from gravitational energy must be considered.

Observable Predictions

Characteristic Frequencies

1. Major radius frequency: f_R = c/(2πR) = 2.49×10⁻⁹ Hz
2. Minor radius frequency: f_r = c/(2πr) = 2.49×10⁻⁶ Hz
3. Frequency ratio: f_r/f_R = 1000 = R/r

Polarization Pattern

• Rotation measure variation due to helical structure
• Periodic change in polarization angle dependent on W-axis phase
• Helical pattern with pitch angle 0.057°

3D Visualization

Implementation Details

python
Torus coordinate system (Rs units)

X = (R/Rs + r/Rs × cos(V)) × cos(U)
Y = (R/Rs + r/Rs × cos(V)) × sin(U)
Z = r/Rs × sin(V)

Normalized magnetic field strength

B_normalized = √((r/R) / (1 + (r/R) × cos(V)))

Visualization Elements

• Torus surface: Magnetic field strength displayed as color map
• Field lines: 20 helical magnetic field lines shown in red
• Coordinate system: Schwarzschild radius units (Rs)

Verification Methods

Manual Calculation Confirmation

All calculations can be verified by the following steps:

1. Jet velocity:

   
   v/c = √(1 - 1/1000²) = √0.999999 = 0.999999500
   

2. W-axis period:

   
   T = 2π × 1.92×10¹³ ÷ 3×10⁸ ≈ 4×10⁵ seconds ≈ 112 hours
   

3. Helical pitch angle:

   
   α = arctan(1/1000) = 0.001 rad = 0.057°
   

Calculation Code

Complete calculation code is available on GitHub:

• GitHub: FORTH Calculations #002

Executable in Python as follows:

python
python calculation_002.py

Physical Interpretation

Role of Torus Structure

1. Magnetic field confinement: Stable confinement by toroidal magnetic field
2. Helical structure formation: Right-handed helix due to W-axis component
3. Jet acceleration: Particle acceleration by magnetic field gradient

Comparison with Observations

Consistency with M87 observational data:

Conclusion

Through accurate calculation of magnetic field configuration in 4D torus structure, we have confirmed:

1. Accurate jet velocity: v/c = 0.999999500 (R/r=1000)
2. Helical structure formation: Right-handed helix with pitch angle 0.057°
3. W-axis period: 111.8 hours (4.66 days)

These predictions can be verified by future high-resolution observations.

Calculation Script

The calculation code is available on GitHub:

• Calculation Script

Note: This calculation is a theoretical prediction requiring observational verification. The low energy conversion efficiency indicates limitations of the simplified model, requiring more detailed magnetohydrodynamic simulations in the future.