Potentially habitable planet TRAPPIST-1b may have a carbon dioxide-rich atmosphere, The innermost Earth-like planet in the famous TRAPPIST-1 system might be capable of supporting a thick atmosphere

    
import numpy as np
import matplotlib.pyplot as plt

# Constants
c = 3e8  # Speed of light (m/s)
target_speed = 0.5 * c  # Target speed (0.5c)
ship_mass = 1e5  # Mass of the spacecraft without fuel (kg)
fuel_efficiency = 1e-3  # Fuel conversion efficiency (e.g., antimatter ~0.1, fusion ~0.001)
exhaust_velocity = 1e7  # Exhaust velocity of the propulsion system (m/s)
specific_impulse = exhaust_velocity / 9.81  # Specific impulse (seconds)

# Functions
def relativistic_mass(speed, rest_mass):
    """Calculate relativistic mass."""
    gamma = 1 / np.sqrt(1 - (speed / c) ** 2)
    return gamma * rest_mass

def fuel_needed(final_speed, ship_mass, exhaust_velocity):
    """Estimate fuel required using the Tsiolkovsky rocket equation."""
    delta_v = final_speed
    mass_ratio = np.exp(delta_v / exhaust_velocity)
    fuel_mass = ship_mass * (mass_ratio - 1)
    return fuel_mass

# Optimization
def optimize_propulsion(target_speed, ship_mass, exhaust_velocity, fuel_efficiency):
    fuel_mass = fuel_needed(target_speed, ship_mass, exhaust_velocity)
    energy_required = fuel_mass * fuel_efficiency * c**2  # Total energy for propulsion
    return fuel_mass, energy_required

# Perform calculations
fuel_mass, energy_required = optimize_propulsion(target_speed, ship_mass, exhaust_velocity, fuel_efficiency)

# Display results
print(f"Target speed: {target_speed / c:.2f}c")
print(f"Required fuel mass: {fuel_mass:.2e} kg")
print(f"Energy required: {energy_required:.2e} joules")

# Visualization
speeds = np.linspace(0, 0.5 * c, 100)
fuel_masses = [fuel_needed(speed, ship_mass, exhaust_velocity) for speed in speeds]

plt.figure(figsize=(10, 6))
plt.plot(speeds / c, fuel_masses, label='Fuel Mass Required')
plt.xlabel('Speed as a Fraction of Light Speed (v/c)')
plt.ylabel('Fuel Mass (kg)')
plt.title('Fuel Requirement vs. Speed')
plt.legend()
plt.grid()
plt.show()

  

        Relativistic Mass: Accounts for relativistic effects as the spacecraft approaches significant fractions of the speed of light.
    Tsiolkovsky Rocket Equation: Estimates fuel requirements based on the propulsion system's exhaust velocity and initial mass.
    Energy Requirement: Evaluates energy needs assuming the specified fuel efficiency.


  

        Idealized Assumptions: Assumes perfect efficiency and ignores engineering constraints.
    Simplified Physics: Doesn't account for external factors like gravity, drag, or realistic propulsion systems.
    Energy Storage: No consideration for how energy would be stored or delivered.