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earth_moon_transfer.py
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272
earth_moon_transfer.py
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"""
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地月转移轨道：有效载荷与燃料消耗（能量）关系分析
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基于齐奥尔科夫斯基火箭方程和霍曼转移轨道模型
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"""
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import numpy as np
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import matplotlib
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matplotlib.use('Agg') # 使用非交互式后端
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import matplotlib.pyplot as plt
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from matplotlib import rcParams
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# 设置中文字体支持
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rcParams['font.sans-serif'] = ['Arial Unicode MS', 'SimHei', 'DejaVu Sans']
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rcParams['axes.unicode_minus'] = False
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# ============== 物理常数 ==============
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G0 = 9.81 # 地表重力加速度 (m/s²)
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# 地月转移轨道典型ΔV预算 (m/s)
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DELTA_V_TLI = 3100 # 地月转移注入 (Trans-Lunar Injection)
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DELTA_V_LOI = 800 # 月球轨道捕获 (Lunar Orbit Insertion)
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DELTA_V_TOTAL = DELTA_V_TLI + DELTA_V_LOI # 总ΔV
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# 不同发动机类型的比冲 (秒)
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ENGINES = {
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'固体火箭': 280,
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'液氧煤油': 350,
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'液氧液氢': 450,
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'离子推进 (仅供参考)': 3000,
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}
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# 结构系数 (结构质量/燃料质量)
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ALPHA = 0
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def exhaust_velocity(isp):
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"""
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计算排气速度
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参数:
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isp: 比冲 (秒)
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返回:
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排气速度 (m/s)
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"""
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return isp * G0
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def mass_ratio(delta_v, isp):
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"""
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计算质量比 R = m0/mf
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基于齐奥尔科夫斯基方程: ΔV = ve * ln(m0/mf)
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参数:
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delta_v: 速度增量 (m/s)
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isp: 比冲 (秒)
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返回:
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质量比 R
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"""
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ve = exhaust_velocity(isp)
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return np.exp(delta_v / ve)
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def fuel_mass(payload_mass, delta_v, isp, alpha=ALPHA):
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"""
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计算所需燃料质量
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公式推导:
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m0 = mp + ms + mf (初始质量 = 有效载荷 + 结构 + 燃料)
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mf_final = mp + ms (最终质量)
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ms = alpha * m_fuel (结构质量与燃料成比例)
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由火箭方程: R = m0 / mf_final
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解得: m_fuel = (mp + ms) * (R - 1)
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m_fuel = mp * (R - 1) / (1 - alpha * (R - 1))
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参数:
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payload_mass: 有效载荷质量 (kg)
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delta_v: 速度增量 (m/s)
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isp: 比冲 (秒)
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alpha: 结构系数
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返回:
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燃料质量 (kg)
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"""
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R = mass_ratio(delta_v, isp)
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denominator = 1 - alpha * (R - 1)
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if denominator <= 0:
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# 物理上不可能: 结构质量太大
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return np.inf
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return payload_mass * (R - 1) / denominator
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def total_energy(fuel_mass, isp):
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"""
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计算燃料的动能（近似能量消耗）
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能量 E = 0.5 * m_fuel * ve²
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参数:
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fuel_mass: 燃料质量 (kg)
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isp: 比冲 (秒)
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返回:
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能量 (焦耳)
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"""
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ve = exhaust_velocity(isp)
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return 0.5 * fuel_mass * ve ** 2
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def characteristic_energy(delta_v):
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"""
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计算特征能量 C3 (用于轨道能量分析)
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C3 = V_infinity² (相对于地球的双曲线超速平方)
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对于地月转移，C3 ≈ -2 km²/s² (束缚轨道)
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简化计算: C3 ≈ (ΔV)² - V_escape²
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"""
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# 近地轨道逃逸速度约 10.9 km/s，轨道速度约 7.8 km/s
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# 从LEO出发需要的额外速度约 3.1 km/s
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return (delta_v / 1000) ** 2 # km²/s²
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def plot_payload_fuel_relationship():
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"""
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绘制有效载荷与燃料消耗的关系图
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"""
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# 有效载荷范围: 100 kg 到 50000 kg
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payloads = np.linspace(100, 50000, 500)
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fig, axes = plt.subplots(2, 2, figsize=(14, 12))
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# ========== 图1: 不同发动机的燃料消耗 ==========
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ax1 = axes[0, 0]
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for engine_name, isp in ENGINES.items():
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fuels = [fuel_mass(p, DELTA_V_TOTAL, isp) for p in payloads]
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fuels = np.array(fuels)
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# 过滤掉无穷大值
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valid = fuels < 1e9
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ax1.plot(payloads[valid]/1000, fuels[valid]/1000,
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label=f'{engine_name} (Isp={isp}s)', linewidth=2)
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ax1.set_xlabel('有效载荷质量 (吨)', fontsize=12)
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ax1.set_ylabel('燃料质量 (吨)', fontsize=12)
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ax1.set_title('地月转移：有效载荷 vs 燃料消耗\n(ΔV = 3.9 km/s)', fontsize=14)
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ax1.legend(loc='upper left')
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ax1.grid(True, alpha=0.3)
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ax1.set_xlim(0, 50)
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# ========== 图2: 燃料/载荷比 ==========
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ax2 = axes[0, 1]
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for engine_name, isp in list(ENGINES.items())[:3]: # 排除离子推进
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ratios = [fuel_mass(p, DELTA_V_TOTAL, isp) / p for p in payloads]
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ax2.axhline(y=ratios[0], label=f'{engine_name} (Isp={isp}s)', linewidth=2)
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ax2.set_xlabel('有效载荷质量 (吨)', fontsize=12)
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ax2.set_ylabel('燃料/载荷 质量比', fontsize=12)
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ax2.set_title('燃料效率比较\n(质量比与载荷无关，仅取决于Isp和ΔV)', fontsize=14)
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ax2.legend()
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ax2.grid(True, alpha=0.3)
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ax2.set_ylim(0, 10)
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# ========== 图3: 能量消耗 ==========
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ax3 = axes[1, 0]
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for engine_name, isp in list(ENGINES.items())[:3]:
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fuels = np.array([fuel_mass(p, DELTA_V_TOTAL, isp) for p in payloads])
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energies = total_energy(fuels, isp) / 1e12 # 转换为TJ
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valid = energies < 1e6
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ax3.plot(payloads[valid]/1000, energies[valid],
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label=f'{engine_name}', linewidth=2)
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ax3.set_xlabel('有效载荷质量 (吨)', fontsize=12)
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ax3.set_ylabel('能量消耗 (TJ)', fontsize=12)
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ax3.set_title('能量消耗 vs 有效载荷\nE = ½ × m_fuel × ve²', fontsize=14)
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ax3.legend()
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ax3.grid(True, alpha=0.3)
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# ========== 图4: ΔV 敏感性分析 ==========
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ax4 = axes[1, 1]
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delta_v_range = np.linspace(3000, 5000, 100) # 3-5 km/s
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payload_fixed = 10000 # 10吨有效载荷
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isp_reference = 450 # 液氧液氢
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fuel_vs_dv = [fuel_mass(payload_fixed, dv, isp_reference) for dv in delta_v_range]
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ax4.plot(delta_v_range/1000, np.array(fuel_vs_dv)/1000,
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'b-', linewidth=2, label='燃料质量')
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# 标记关键点
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ax4.axvline(x=DELTA_V_TOTAL/1000, color='r', linestyle='--',
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label=f'标准任务 ΔV={DELTA_V_TOTAL/1000} km/s')
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ax4.set_xlabel('ΔV (km/s)', fontsize=12)
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ax4.set_ylabel('燃料质量 (吨)', fontsize=12)
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ax4.set_title(f'ΔV敏感性分析\n(有效载荷=10吨, Isp=450s)', fontsize=14)
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ax4.legend()
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ax4.grid(True, alpha=0.3)
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plt.tight_layout()
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plt.savefig('/Volumes/Files/code/mm/20260130_b/earth_moon_transfer_analysis.png',
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dpi=150, bbox_inches='tight')
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print("图表已生成!")
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# plt.show() # 非交互式环境下注释掉
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return fig
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def print_summary():
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"""
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打印关键数据摘要
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"""
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print("=" * 60)
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print("地月转移轨道分析摘要")
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print("=" * 60)
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print(f"\n任务参数:")
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print(f" - 总ΔV需求: {DELTA_V_TOTAL/1000:.1f} km/s")
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print(f" - TLI (地月转移注入): {DELTA_V_TLI/1000:.1f} km/s")
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print(f" - LOI (月球轨道捕获): {DELTA_V_LOI/1000:.1f} km/s")
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print(f" - 结构系数 α: {ALPHA}")
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print(f"\n不同发动机的燃料效率 (每吨有效载荷所需燃料):")
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print("-" * 50)
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payload_ref = 1000 # 1吨参考载荷
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for engine_name, isp in ENGINES.items():
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R = mass_ratio(DELTA_V_TOTAL, isp)
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fuel = fuel_mass(payload_ref, DELTA_V_TOTAL, isp)
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ve = exhaust_velocity(isp)
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if fuel < 1e9:
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print(f"\n {engine_name}:")
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print(f" 比冲 Isp = {isp} s")
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print(f" 排气速度 ve = {ve:.0f} m/s")
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print(f" 质量比 R = {R:.2f}")
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print(f" 燃料/载荷比 = {fuel/payload_ref:.2f}")
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print(f" 1吨载荷需燃料: {fuel/1000:.2f} 吨")
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else:
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print(f"\n {engine_name}:")
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print(f" 比冲 Isp = {isp} s (化学火箭无法实现)")
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print("\n" + "=" * 60)
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print("关键数学关系:")
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print("=" * 60)
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print("""
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齐奥尔科夫斯基方程:
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ΔV = ve × ln(m₀/mf)
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质量比:
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R = exp(ΔV / ve)
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燃料-载荷关系:
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m_fuel / m_payload = (R - 1) / (1 - α(R - 1))
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能量消耗:
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E = ½ × m_fuel × ve²
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其中:
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ve = Isp × g₀ (排气速度)
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α = 结构系数 (约0.1)
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""")
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if __name__ == "__main__":
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# 打印分析摘要
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print_summary()
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# 绘制关系图
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print("\n正在生成图表...")
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fig = plot_payload_fuel_relationship()
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print("图表已保存至: earth_moon_transfer_analysis.png")
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