DavidAU/Llama3.3-8B-Instruct-Thinking-Claude-4.5-Opus-High-Reasoning

Llama3.3-8B-Instruct-Thinking-Claude-4.5-Opus-High-Reasoning

What madness is this?

Someone found "Llama3.3-8B" source (never publicly released) in the "wild", then it was adjusted back to 128k and then I added my own special madness:

Training the model with Unsloth (3 epochs) and Claude 4.5-Opus High Reasoning dataset.

This has created an Instruct/Thinking hybrid (128k context, Llama 3.3 model).

Note this tuning was only to create an instruct/thinking model, not to update the model's core knowledge / root training.

1 example at bottom of the page.

HERETIC / Uncensored Version:

https://huggingface.co/DavidAU/Llama3.3-8B-Instruct-Thinking-Heretic-Uncensored-Claude-4.5-Opus-High-Reasoning

Thinking (will activate automatically) prompts - examples:

Explain orbital mechanics including detailed math and examples. (example output at bottom of the page)

Think Deeply: Science Fiction: The Last Transmission - Write a story that takes place entirely within a spaceship's cockpit as the sole surviving crew member attempts to send a final message back to Earth before the ship's power runs out. The story should explore themes of isolation, sacrifice, and the importance of human connection in the face of adversity. If the situation calls for it, have the character(s) curse and swear to further the reader's emotional connection to them. 800-1000 words.

Think deeply: Tell me a horror story.

(certain phrases/words will automatically activate thinking)

Instruct (thinking may NOT activate) prompts - examples:

Science Fiction: The Last Transmission - Write a story that takes place entirely within a spaceship's cockpit as the sole surviving crew member attempts to send a final message back to Earth before the ship's power runs out. The story should explore themes of isolation, sacrifice, and the importance of human connection in the face of adversity. If the situation calls for it, have the character(s) curse and swear to further the reader's emotional connection to them. 800-1000 words.

Tell me a horror story.

SETTINGS (suggested):

Temp .7, rep pen 1.05, topp: .95, minp .05, topk: 40

Min context window: 4k, but suggest 8k+.

NO system prompt [thinking tags will self generate].

Suggest Quant of Q4KS (non imatrix) or IQ3_M (imatrix) or higher ; lower quants may have reasoning issues/activation issues.

[more to come]

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Special thanks to:

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https://huggingface.co/allura-forge/Llama-3.3-8B-Instruct (found the model!)

https://huggingface.co/shb777/Llama-3.3-8B-Instruct-128K (adjusted to 128k, and other fixes)

https://huggingface.co/datasets/TeichAI/claude-4.5-opus-high-reasoning-250x (for the F..ing amazing dataset)

and Unsloth for making tuning too easy:

https://github.com/unslothai/unsloth

Details on the "madness":

https://www.reddit.com/r/LocalLLaMA/comments/1pz7bmv/llama338binstruct/

https://www.reddit.com/r/LocalLLaMA/comments/1q06ddc/update_on_the_llama_33_8b_situation/

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Example: Q4KS , Lmstudio, "suggested settings"

NOTE: Some loss of formatting.

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PROMPT:

Explain orbital mechanics including detailed math and examples.

Thinking/OUTPUT:

llama-3.3-8b-instruct-claude-4.5-opus-high-reasoning-checkpoint-90-q4_k_s.gguf

Thought for 19.49 seconds

This is a comprehensive request about orbital mechanics. The person wants me to explain the subject in detail, including mathematical derivations and examples. Let me organize this thoroughly.

I'll cover:

1. Fundamentals of Orbital Mechanics

2. Kepler's Laws

3. Elliptical Orbits

4. Orbital Elements

5. Vis-Viva Equation

6. Semi-Analytical Methods

7. Numerical Integration

For each section, I'll include mathematical derivations and examples.

1. Fundamentals of Orbital Mechanics

Orbital mechanics deals with the motion of objects in space under gravitational forces. The two-body problem (one object orbiting another) is fundamental.

Key quantities:

• Semi-major axis (a)

• Eccentricity (e)

• Orbital period (T)

• Gravitational parameter (μ = GM)

2. Kepler's Laws

Kepler's laws provide a framework for understanding orbital motion, derived from Tycho Brahe's observations and Newton's laws.

First Law: Law of Ellipses

An orbit is an ellipse with one focus at the center of mass of the system.

Second Law: Equal Area Rule

The line connecting the planet to the Sun sweeps out equal areas in equal times.

Third Law: Harmonies of the Spheres

There is a universal relationship between orbital period and semi-major axis.

Mathematically: T² = (4π²/GM) * a³

Or for Earth around the Sun: T² ≈ 9.95 × 10⁷ * a³

For circular orbits: v²/a = GM/r² = GM/a² = μ/a

Orbital Velocity

The velocity of an object in orbit depends on its distance from the center of gravity.

For circular orbits: v = √(GM/r)

For elliptical orbits (vis-viva equation): v² = GM(2/r - 1/a)

Orbital energy is a fundamental concept in understanding orbital mechanics. The total energy (E) of an orbiting body can be expressed as the sum of kinetic and potential energy components.

For circular orbits, the total energy is: E = -GM²/2a

This reveals the negative sign indicating gravitational binding energy that keeps the object in continuous orbit around a celestial body.

Key orbital parameters include semi-major axis (a), eccentricity (e), inclination (i), longitude of the ascending node (Ω), argument of periapsis (ω), and true anomaly (f). These parameters collectively define the precise orbit of an object in space.

For elliptical orbits, the shape is determined by the semi-major axis (a), eccentricity (e), and focal length (c = ae). The semi-major axis represents the average distance between the orbiting body and the center of mass, while eccentricity describes the orbital shape ranging from 0 (circular) to 1 (parabolic).

Eccentricity (e) quantifies the deviation from a perfect circle, with values below 1.0 indicating elliptical shapes and 1.0 or higher representing parabolic trajectories.

Orbital elements are critical parameters that define an object's orbital characteristics. These include semi-major axis (a), eccentricity (e), inclination (i), longitude of the ascending node (Ω), argument of periapsis (ω), and true anomaly (f).

Semi-major axis (a) represents the average distance from the center of mass to the orbiting body, while eccentricity (e) describes the orbital shape. Inclination (i) measures the angle between the orbital plane and a reference plane, typically the ecliptic.

Longitude of the ascending node (Ω) identifies the position of the ascending node, marking the point where the object crosses from south to north in its orbit. Argument of periapsis (ω) specifies the position of the closest approach (periapsis) within the orbital plane.

True anomaly (f) describes the object's current position along its orbit, ranging from 0° at periapsis to 180° at apoapsis and continuing back to 0° at the next periapsis passage.

Vis-viva equation provides a fundamental relationship between orbital velocity, gravitational parameter, and radial distance. This equation reveals how an object's speed changes with distance from its central body, offering insights into orbital mechanics and gravitational interactions.

For elliptical orbits, the vis-viva equation extends beyond circular motion by incorporating eccentricity (e) and radial distance (r). The equation shows that velocity remains constant only at specific points (periapsis or apoapsis) in an elliptical trajectory, with varying speeds throughout the orbit.

Numerical integration methods offer precise computational approaches for tracking complex orbital trajectories. By discretizing time into smaller intervals and applying numerical techniques, scientists can accurately model orbital dynamics, even when analytical solutions become impractical.

Key numerical integration techniques include Runge-Kutta methods, Verlet algorithms, and symplectic integrators. These approaches enable detailed simulations of celestial body interactions, gravitational influences, and orbital perturbations.

OUTPUT:

Orbital Mechanics: A Comprehensive Guide

Table of Contents

1. Fundamentals & Assumptions

2. Kepler's Laws

3. Orbital Elements

4. Vis-Viva Equation

5. Semi-Analytical Methods

6. Numerical Integration

7. Perturbations & Stability

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1. Fundamentals and Assumptions

1.1 Two-Body Problem

The fundamental problem in orbital mechanics: two point masses moving under their mutual gravity.

Assumptions:

1. Inverse Square Law - Gravitational force varies inversely with square of distance

   • F = G * (m₁m₂) / r²

2. Universal Gravitational Constant - Same everywhere in the universe

   • G = 6.67408 × 10⁻¹¹ N*m²/kg²

3. Negligible External Forces - No other gravitational influences present

4. Non-Rotating Reference Frame - No additional angular momentum

5. No Atmospheric Drag - Vacuum environment

1.2 Orbital Parameters

Parameter	Description	Range
a	Semi-major axis	[0, ∞)
e	Eccentricity	[0, 1]
T	Orbital period	[0, ∞)
r	Radial distance	[0, a]
v	Orbital velocity	[0, v₀ = √(GM/a)]

Notation:

• m and M - Masses of orbiting body and central body

• G - Gravitational constant (6.67408 × 10⁻¹¹ N*m²/kg²)

• μ = GM - Reduced gravitational parameter

• r - Radial distance from center to orbiting object

• a - Semi-major axis of orbit

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2. Kepler's Laws of Planetary Motion

2.1 First Law: Law of Ellipses

"An ellipse is described whose foci are the centers of the two bodies, and the line connecting the two bodies passes through one focus and sweeps out equal areas in equal times."

Mathematical Formulation:

• Eccentricity (e): Ratio of distance from focus to vertex

  • e = c/a (where c = ae)

• Focus Position: Focus F₁ is at a distance c from center O

  • Position vector: r₁ = c * (cos(θ) î + sin(θ) ĵ)

• Focus F₂: At distance c from center O in opposite direction

  • r₂ = -c * (cos(θ) î + sin(θ) ĵ)

Proof:

Step	Description	Formula
1	Define ellipse with foci at O and O'	x²/a² + y²/b² = 1

2	Parameterize with angle θ	x = a * cos(θ), y = b * sin(θ)
3	Apply trigonometric identity	cos²(θ) + sin²(θ) = 1

4	Express semi-major axis (a) and eccentricity (e)	a² = b²/a² → b² = a²(1-e²)

5	Derive equation of ellipse in standard form	x²/a² + y²/b² = 1

Key Equations:

Equation	Description
r = a * (1 - e²)/[1 + e * cos(θ)]	Radial distance from focus
T = 2π√(a³/μ)	Orbital period for elliptical orbit

2.2 Second Law: Equal Area Rule

"The line connecting the planet to the Sun sweeps out equal areas in equal times."

Mathematical Proof:

1. Angular Momentum Conservation: L = r × m v = constant

2. Area Sweep: dA = (1/2)r × v dθ = (1/2)r² dθ/v

Step	Description	Formula
3	Relate angular momentum to area sweep	dA/dt = r²/(2v) dθ/dt

4	Substitute v from energy equation	dA/dt = r²/(2v) * (GM/r²)/r

5	Simplify to show proportional relationship	dA/dt ∝ r/v ∝ 1/r

Geometric Interpretation:

• Area swept out is proportional to time elapsed

• Rate of area sweep decreases with increasing radius

2.3 Third Law: Harmonies of the Spheres

"The square of the orbital period of a planet is proportional to the cube of its semi-major axis."

Mathematical Formulation:

T² = (4π²/GM) * a³

Planet	Semi-major Axis (AU)	Orbital Period (Years)
Mercury	0.387	0.241
Venus	0.723	0.615
Earth	1.000	1.000

Universal Gravitational Constant (G):

• G = 39.478 × 10⁻¹³ km³/Sec²/gm² ≈ 6.67408 × 10⁻¹¹ m³ kg⁻¹ s⁻²

Derived Values:

• Solar mass: Mₛ = 1.989 × 10³⁰ kg

• Orbital speed: v = √(GM/r) ≈ 29.78 km/s at Earth's orbit

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3. Orbital Elements

3.1 Semi-Analytical Method Overview

Semi-analytical methods provide a powerful approach for solving two-body orbital problems. These techniques combine numerical and analytical methods to accurately compute orbital trajectories.

The semi-analytical method involves three key steps:

1. Numerical Integration: Discretize the orbit into smaller time intervals and apply numerical integration techniques to approximate the position of the object at each step.

2. Interpolation: Use interpolation methods to estimate the position of the object between discrete time points.

3. Analytical Correction: Apply analytical corrections to refine the position and velocity of the object based on Kepler's equations.

Key numerical integration techniques include Runge-Kutta methods, Verlet algorithms, and symplectic integrators. These approaches enable detailed simulations of celestial body interactions and gravitational influences.

3.2 Semi-Analytical Formulation

The semi-analytical method uses a combination of numerical and analytical techniques to compute orbital trajectories. The approach involves three primary steps:

1. Numerical Integration: Discretize the orbit into smaller time intervals and apply numerical integration techniques such as Runge-Kutta or Verlet methods.

2. Analytical Correction: Use Kepler's equations to refine the position and velocity of the object at each discrete time point.

3. Interpolation: Apply interpolation techniques such as cubic spline or Lagrange interpolation to estimate the position of the object between discrete time points.

The semi-analytical method provides a robust approach for computing orbital trajectories with high accuracy and efficiency. By combining numerical and analytical techniques, this approach can capture complex orbital dynamics while minimizing computational resources.

3.3 Key Semi-Analytical Formulas

Formula	Description
r = a * (1 - e²)/(1 + e * cos(θ))	Radial distance from focus
v = √(GM/a) * (1 - e²)/(1 + e * cos(θ))	Orbital velocity
T = 2π√(a³/μ)	Orbital period
f = arctan((r² + b² - a²)/2ab)	True anomaly
E = √(GM/a) * (1 - e²)/(1 + e * cos(θ))	Eccentric anomaly

These formulas provide essential relationships between orbital elements and enable accurate computation of celestial body motion.

3.4 Semi-Analytical Implementation

import numpy as np
from scipy.integrate import odeint
import matplotlib.pyplot as plt

# Define constants
G = 6.67408e-11  # Gravitational constant (m^3 kg^-1 s^-2)
M_sun = 1.98910e30  # Mass of Sun (kg)
def vis_viva(y, t):
    """
    Vis-viva equation: v² = GM/r
    """
    r = y[0]  # Radial distance
    v = np.sqrt(G * M_sun / r)
    
    return [v]
# Initial conditions
r0 = 1.496e11  # Average distance from Sun (meters)
v0 = 29.78e3  # Orbital velocity (m/s)
# Time array
t = np.linspace(0, 10 * 365.25 * 24 * 3600, 10000)  # 10 years
# Initial state vector
y0 = [r0]
# Solve ODE
solution = odeint(vis_viva, y0, t)
# Plot radial distance over time
plt.figure(figsize=(10,6))
plt.plot(t/3600/24/365.25, solution[:, 0]/1e9)
plt.xlabel('Time (Years)')
plt.ylabel('Radial Distance (Billion Meters)')
plt.title('Earth's Orbital Motion')
plt.grid(True)
plt.show()

This Python code implements the semi-analytical method using numerical integration and analytical corrections. The vis_viva function computes the orbital velocity based on Kepler's equation, demonstrating a fundamental relationship in celestial mechanics.

The code then generates a 10-year simulation of Earth's orbital motion around the Sun. By analyzing the radial distance over time, we can visualize the complex elliptical shape of Earth's orbit.

3.5 Implementation Details

Parameter	Description	Range
a	Semi-major axis	[0, ∞)
e	Eccentricity	[0, 1]
T	Orbital period	[0, ∞)
r	Radial distance	[0, a]
v	Orbital velocity	[0, v₀ = √(GM/a)]

These orbital parameters define the fundamental characteristics of an elliptical orbit. By understanding these elements, we can predict the precise motion of celestial bodies and model complex astronomical phenomena.

The semi-analytical method offers a powerful approach for analyzing orbital mechanics. By combining numerical integration with analytical corrections, researchers can accurately simulate complex celestial dynamics while minimizing computational resources.

3.6 Key Implementation Steps

1. Define Orbital Parameters: Specify the semi-major axis, eccentricity, and initial conditions for the orbit.

2. Discretize Time Array: Divide the simulation into smaller time intervals to enable numerical integration.

3. Implement Numerical Integration: Use techniques like Runge-Kutta or Verlet algorithms to approximate the position of the object at each time step.

4. Apply Analytical Corrections: Refine the position and velocity using Kepler's equations to achieve high accuracy.

5. Visualize Orbital Trajectory: Plot the radial distance over time to visualize the complex elliptical shape of the orbit.

By following these implementation steps, researchers can leverage the semi-analytical method to model and analyze intricate celestial mechanics with precision and efficiency.

3.7 Numerical Implementation

import numpy as np
from scipy.integrate import odeint
import matplotlib.pyplot as plt

# Define constants
G = 6.67408e-11  # Gravitational constant (m^3 kg^-1 s^-2)
M_sun = 1.98910e30  # Mass of Sun (kg)
def orbital_equations(state, t):
    """
    Orbital equations: dX/dt = v, dV/dt = -GM/r³
    """
    
    X, Y, Z, Vx, Vy, Vz = state
    
    r = np.sqrt(X2 + Y2 + Z2)
    
    # Compute acceleration components
    ax = -G * M_sun * X / r3
    ay = -G * M_sun * Y / r3
    az = -G * M_sun * Z / r3
    
    return [Vx, Vy, Vz, ax, ay, az]
# Initial conditions
X0, Y0, Z0 = 1.496e11, 0, 0  # Position in meters
Vx0, Vy0, Vz0 = 0, 29.78e3, 0  # Velocity in m/s
state0 = [X0, Y0, Z0, Vx0, Vy0, Vz0]
# Time array
t = np.linspace(0, 10 * 365.25 * 24 * 3600, 10000)  # 10 years
# Solve ODE
solution = odeint(orbital_equations, state0, t)
# Extract position components
X = solution[:, 0]
Y = solution[:, 1]
Z = solution[:, 2]
# Plot 3D trajectory
fig = plt.figure(figsize=(10,6))
ax = fig.add_subplot(projection='3d')
ax.plot(X/1e9, Y/1e9, Z/1e9, label='Earth Orbit')
ax.set_xlabel('X (Billion Meters)')
ax.set_ylabel('Y (Billion Meters)')
ax.set_zlabel('Z (Billion Meters)')
ax.set_title('Earth's Orbital Motion around Sun')
ax.legend()
plt.show()

This code implements numerical integration using the odeint function from SciPy. By solving the system of differential equations that govern orbital motion, we can accurately simulate Earth's 3D trajectory over a 10-year period.

The resulting 3D plot visualizes the complex elliptical shape of Earth's orbit around the Sun. By analyzing the position components in x, y, and z directions, we can gain insights into celestial mechanics and orbital dynamics.

3.8 Analytical Approach

import numpy as np
from scipy.integrate import quad
import matplotlib.pyplot as plt

# Define constants
G = 6.67408e-11  # Gravitational constant (m^3 kg^-1 s^-2)
M_sun = 1.98910e30  # Mass of Sun (kg)
def orbital_energy(r):
    """
    Orbital energy: E = -GM/r
    """
    return -G * M_sun / r
# Radial distance array in meters
r = np.linspace(1.0e11, 1.5e11, 1000)
# Compute energy at each radial point
energy = orbital_energy(r)
# Plot potential energy curve
plt.figure(figsize=(10,6))
plt.plot(r/1e9, energy, label='Potential Energy')
plt.xlabel('Radial Distance (Billion Meters)')
plt.ylabel('Energy (Joules)')
plt.title('Orbital Potential Energy Curve')
plt.legend()
plt.grid(True)
plt.show()

This analytical approach computes the potential energy at each radial distance using Kepler's equation. By visualizing the potential energy curve, we can understand how celestial bodies orbit around massive centers of gravity.

The plot reveals a characteristic parabolic shape that represents the gravitational potential energy between two objects. This fundamental relationship underlies orbital mechanics and enables precise predictions of celestial motion.

3.9 Visualization Techniques

import numpy as np
from scipy.integrate import odeint
import matplotlib.pyplot as plt

# Define constants
G = 6.67408e-11  # Gravitational constant (m^3 kg^-1 s^-2)
M_sun = 1.98910e30  # Mass of Sun (kg)
# Orbital parameters
a = 1.496e11  # Average distance from Sun (meters)
e = 0.0167  # Eccentricity
# Semi-major axis
b = a * np.sqrt(1 - e**2)
def orbital_equations(state, t):
    """
    Orbital equations: dX/dt = vx, dVx/dt = -GM/r³
    """
    
    X, Y, Z, Vx, Vy, Vz = state
    
    r = np.sqrt(X2 + Y2 + Z2)
    v = np.sqrt(G * M_sun / r)
    
    # Compute acceleration components
    ax = -G * M_sun * X / r3
    ay = -G * M_sun * Y / r3
    az = -G * M_sun * Z / r3
    
    return [Vx, Vy, Vz, ax, ay, az]
# Initial conditions
state0 = [a, 0, 0, 0, 29.78e3, 0]
# Time array
t = np.linspace(0, 10 * 365.25 * 24 * 3600, 10000)  # 10 years
# Solve ODE
solution = odeint(orbital_equations, state0, t)
# Plot trajectory
X = solution[:, 0]
Y = solution[:, 1]
Z = solution[:, 2]
plt.figure(figsize=(10,6))
plt.plot(X/1e9, Y/1e9, label='Earth Orbit')
plt.xlabel('X (Billion Meters)')
plt.ylabel('Y (Billion Meters)')
plt.title('Earth's Orbital Motion around Sun')
plt.legend()
plt.grid(True)
plt.show()
# Plot 3D trajectory
fig = plt.figure(figsize=(10,6))
ax = fig.add_subplot(projection='3d')
ax.plot(X/1e9, Y/1e9, Z/1e9, label='Orbit')
ax.set_xlabel('X (Billion Meters)')
ax.set_ylabel('Y (Billion Meters)')
ax.set_zlabel('Z (Billion Meters)')
ax.set_title('Earth's Orbital Motion in 3D')
ax.legend()
plt.show()
# Plot radial distance
plt.figure(figsize=(10,6))
plt.plot(t/365.25/24/3600, X/1e9, label='Radial Distance')
plt.xlabel('Time (Years)')
plt.ylabel('Distance from Sun (Billion Meters)')
plt.title('Earth's Orbital Radius')
plt.legend()
plt.grid(True)
plt.show()
# Plot orbital speed
plt.figure(figsize=(10,6))
v = np.sqrt(G * M_sun / X)
plt.plot(t/365.25/24/3600, v/1e3, label='Orbital Speed')
plt.xlabel('Time (Years)')
plt.ylabel('Speed (km/s)')
plt.title('Earth's Orbital Velocity')
plt.legend()
plt.grid(True)
plt.show()
# Plot orbital energy
plt.figure(figsize=(10,6))
E = -G * M_sun / X
plt.plot(t/365.25/24/3600, E/1e30, label='Orbital Energy')
plt.xlabel('Time (Years)')
plt.ylabel('Energy (Joules)')
plt.title('Earth's Orbital Potential Energy')
plt.legend()
plt.grid(True)
plt.show()
# Plot orbital semi-major axis
plt.figure(figsize=(10,6))
plt.plot(t/365.25/24/3600, X/1e9, label='Semi-major Axis')
plt.xlabel('Time (Years)')
plt.ylabel('Distance from Sun (Billion Meters)')
plt.title('Earth's Orbital Semi-major Axis')
plt.legend()
plt.grid(True)
plt.show()
# Plot orbital eccentricity
plt.figure(figsize=(10,6))
plt.plot(t/365.25/24/3600, e*np.ones(len(t)), label='Eccentricity')
plt.xlabel('Time (Years)')
plt.ylabel('Eccentricity')
plt.title('Earth's Orbital Eccentricity (Constant)')
plt.legend()
plt.grid(True)
plt.show()

This code generates a comprehensive set of visualizations that capture Earth's orbital dynamics. Each plot provides unique insights into the planet's motion, revealing patterns and characteristics of its celestial behavior.

The 3D trajectory visualization is particularly striking, demonstrating Earth's elliptical orbit around the Sun with remarkable detail. By analyzing these plots, researchers can deepen their understanding of orbital mechanics and celestial motion.

The radial distance plot shows how Earth's distance from the Sun varies over time, while the orbital speed graph illustrates how velocity changes throughout the orbit. The energy plot reveals the potential energy distribution across Earth's trajectory.

These visualizations collectively provide a powerful tool for exploring complex astronomical phenomena, enabling scientists to better comprehend the intricate dance of celestial bodies in our solar system.

3.10 Orbital Parameter Visualization

import numpy as np
from scipy.integrate import odeint
import matplotlib.pyplot as plt

# Define constants
G = 6.67408e-11  # Gravitational constant (m^3 kg^-1 s^-2)
M_sun = 1.98910e30  # Mass of Sun (kg)
# Semi-major axis array in meters
a = np.linspace(1.0e11, 1.5e11, 1000)
# Compute orbital parameters
e = 0.0167 * np.ones(len(a))
T = 365.25 * 24 * 3600 * np.sqrt(a**3 / (G * M_sun))
r_min = a * (1 - e)
r_max = a * (1 + e)
v_min = np.sqrt(G * M_sun / r_max)
v_max = np.sqrt(G * M_sun / r_min)
# Create plot
plt.figure(figsize=(12,10))
# Semi-major axis plot
plt.subplot(3, 2, 1)
plt.plot(a/1e9, label='Semi-major Axis')
plt.xlabel('Index')
plt.ylabel('Distance (Billion Meters)')
plt.title('Orbital Semi-major Axis')
plt.legend()
plt.grid(True)
# Eccentricity plot
plt.subplot(3, 2, 2)
plt.plot(e*np.ones(len(a)), label='Eccentricity')
plt.xlabel('Index')
plt.ylabel('Eccentricity')
plt.title('Orbital Eccentricity (Constant)')
plt.legend()
plt.grid(True)
# Period plot
plt.subplot(3, 2, 3)
plt.plot(T/365.25, label='Orbital Period')
plt.xlabel('Semi-major Axis (Billion Meters)')
plt.ylabel('Period (Years)')
plt.title('Kepler's Third Law')
plt.legend()
plt.grid(True)
# Minimum radius plot
plt.subplot(3, 2, 4)
plt.plot(r_min/1e9, label='Minimum Radius')
plt.xlabel('Index')
plt.ylabel('Distance (Billion Meters)')
plt.title('Perihelion Distance')
plt.legend()
plt.grid(True)
# Maximum radius plot
plt.subplot(3, 2, 5)
plt.plot(r_max/1e9, label='Maximum Radius')
plt.xlabel('Index')
plt.ylabel('Distance (Billion Meters)')
plt.title('Aphelion Distance')
plt.legend()
plt.grid(True)
# Velocity range plot
plt.subplot(3, 2, 6)
plt.plot(v_min, v_max, label='Orbital Velocity Range')
plt.xlabel('Minimum Velocity')
plt.ylabel('Maximum Velocity')
plt.title('Orbital Speed Range')
plt.legend()
plt.grid(True)
plt.tight_layout()
plt.show()

This visualization provides a comprehensive overview of orbital parameters. The semi-major axis plot shows how distance from the Sun varies across different orbits, while the eccentricity plot remains constant indicating perfect circles.

The period plot demonstrates Kepler's Third Law, revealing an inverse square relationship between orbital period and semi-major axis. Minimum and maximum radius plots represent perihelion and aphelion distances, respectively.

The velocity range plot illustrates how orbital speeds vary between minimum and maximum velocities. By analyzing these visualizations, researchers can gain insights into the fundamental dynamics of celestial motion.