{
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  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<font size=\"5\">作业11、假设在红移为1处去观测一个现在红移为2的星系，其红移为多少？"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "解：z=1时，相对尺度因子$a1 = \\frac{1}{1+z} = 0.5$\n",
    "\n",
    "   z=2时，相对尺度因子$a2 = \\frac{1}{1+z} = \\frac{1}{3}$\n",
    "\n",
    "   从红移1的地方观测红移为2的星系，红移2的尺度因子是红移1的$\\frac{1}{1+z^*}$倍，即\n",
    "\n",
    "   $a1 \\times \\frac{1}{1+z^*} = a2$，所以$z^* = 0.5$\n",
    "\n",
    "   从红移1的地方观测红移2的星系，其红移为0.5\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<font size=\"5\">假设宇宙的膨胀是一个线性过程，请根据今天的哈勃常数估算宇宙的年龄"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "解：$H_0 = 70 km/s/Mpc = 2.27 \\times 10^{-18} s$\n",
    " \n",
    "$t(z) = \\int_z^{\\infty} \\frac{1}{H(z)} \\frac{1}{1+z} dz$\n",
    "\n",
    "$t(0) = \\int_0^{\\infty} \\frac{1}{H_0} dz$\n",
    "\n",
    "$t = \\frac{1}{H_0} = 4.4 \\times 10^{17} s = 139 亿年$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<font size=\"5\">作业12、一团1000个太阳质量的中性气体云（全部为H原子）, 温度为30K，当云的密度大于多少时，气体云将发生塌缩？塌缩（自由下落）时标为多少？\n",
    "\n",
    "• 云塌缩：势能 > 热能"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "解：Jeans mass是在引力作用下，气体云或气体团块保持稳定的最小质量。如果气体云的质量小于Jeans mass，它会因为热运动而膨胀；如果大于Jeans mass，则会发生引力塌缩\n",
    "\n",
    "Jeans mass $M_J = \\frac{(5kT)^{3/2}}{(G \\mu m_H)^{3/2}} (\\frac{3}{4 \\pi \\rho})^{1/2} $\n",
    "\n",
    "k 是玻尔兹曼常数\n",
    "\n",
    "T 是气体的温度\n",
    "\n",
    "G 是引力常数\n",
    "\n",
    "$\\mu$ 是气体的平均分子量（对于中性氢，$\\mu≈1$）\n",
    "\n",
    "$m_H$ 是氢原子的质量\n",
    "\n",
    "$\\rho$ 是气体云的密度\n",
    "\n",
    "当$M_{cloud} > M_J$时，气体云将发生塌缩，即$1000M_{sun} > \\frac{(5kT)^{3/2}}{(G \\mu m_H)^{3/2}} (\\frac{3}{4 \\pi \\rho})^{1/2}$\n",
    "\n",
    "$\\rho > (\\frac{1}{1000M_{sun}} \\times (\\frac{5kT}{Gm_H})^{3/2} \\times (\\frac{3}{4 \\pi})^{1/2})^2 =  3.73 \\times 10^{-19} kg/m^{3}$\n",
    "\n",
    "坍缩的时标$t_{collapse} = \\sqrt{\\frac{3 \\pi}{32G \\rho}} = 1.08 \\times 10^{14} s = 3.4 \\times 10^6 yr$\n",
    "(https://physics.stackexchange.com/questions/94746/gravitational-collapse-and-free-fall-time-spherical-pressure-free)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<font size=\"5\">这团气体中能形成的最大质量的恒星的质量是多少？\n",
    "\n",
    "• 假设分子云中的恒星形成遵循Salpter初始质量函数（最小质量的恒星为0.08太阳质量）\n",
    "\n",
    "• 这道题不是很好解析求解（需要估算一个合理的ΔM），或者采用MC方法\n",
    "\n",
    "• http://cluster.shao.ac.cn/~shen/Lecture/IMF.pdf"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$N = \\int_{M_{min}}^{M_{max}} kM^{-2.35}dM$\n",
    "\n",
    "$M_{tot} = \\int_{M_{min}}^{M_{max}} MkM^{-2.35}dM$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<font size=\"5\">太阳中的氢大概有10%在主序阶段被燃烧，请估算太阳处于主序阶段的时间？太阳表面的温度大概是~5500K，请由此估算地球表面的温度。\n",
    "\n",
    "• 地球的反射率~0.3"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "解：lifetime = energy/(rate [energy/time])\n",
    "\n",
    "太阳释放能量的速率是$3.8 \\times 10^{26} W$。\n",
    "\n",
    "太阳通过核心反应发光，这些反应将4个氢原子转化为1个氦原子，大约0.7%的原始质量消失了，这些消失\n",
    "的质量转化为能量，这就是太阳发光的能量。\n",
    "\n",
    "$E = 0.7 \\% \\times Mc^2$ （M是太阳中能进行上述核反应的质量）\n",
    "\n",
    "在主序阶段只有10%的氢燃烧，$M = 0.1M_{sun}$\n",
    "\n",
    "$E = 0.7 \\% \\times 0.1M_{sun}c^2 (M_{sun} = 2 \\times 10^{30}kg)$\n",
    "\n",
    "$t = \\frac{E}{3.8 \\times 10^{26} W} = 10^{10} yr$\n",
    "\n",
    "太阳每秒释放的能量$L = \\sigma T_{sun}^4 \\times 4 \\pi \\times R_{sun}^2 = 3.8 \\times 10^{33} erg/s$\n",
    "\n",
    "太阳释放的能量到达地球的单位面积流量$S_d = \\frac{L}{4 \\pi d^2}$ (d为日地距离)\n",
    "\n",
    "地球吸收的总能量$S_d \\times \\pi \\times R_{earth}^2(1-0.3)$\n",
    "\n",
    "地球辐射出的能量$\\sigma T_{earth}^4 \\times 4 \\pi \\times R_{earth}^2$\n",
    "\n",
    "将地球看成一个黑体，吸收的能量等于辐射出的能量：\n",
    "\n",
    "$S_d \\times \\pi \\times R_{earth}^2 (1-0.3) = \\sigma T_{earth}^4 \\times 4 \\pi \\times R_{earth}^2$\n",
    "\n",
    "$T_{earth} = 255 K$\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<font size=\"5\">一个星系的恒星形成历史可以用exp(-t/τ)来描述（其中τ=3Gyr），该星系的年龄为10Gy，请计算该星系的V波段的恒星质光比（随着时间的演化）。（选做）\n",
    "\n",
    "• SSP的质量随着时间的变化\n",
    "\n",
    "• http://202.127.29.3/~shen/Lecture/bc2003_hr_m162_salp_ssp.4color"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "星系年龄为10Gyr时\n",
    "$Vmag = 7.1919   $\n",
    "$M_* = 6.8765E-01$\n",
    "\n",
    "$SFR(t) = \\Psi(t) = SFR_0 \\cdot e^{(-t/ \\tau)}$\n",
    "\n",
    "星系在时间t时形成的总质量M(t)为\n",
    "\n",
    "$M(t) = \\int_0^t SFR(t^{'})dt^{'} = SFR_0 \\cdot \\tau \\cdot (1 - e^{(-t/\\tau)})$\n",
    "\n",
    "$L = L_{sun} \\cdot 10^{-0.4 \\cdot (V-V_{sun})}$\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "在10 Gyr时的质光比为: 6.0552\n"
     ]
    }
   ],
   "source": [
    "import numpy as np\n",
    "\n",
    "\n",
    "tau = 3  # Gyr\n",
    "t_age = 10  # Gyr\n",
    "V_mag = 7.1919  # V波段mag\n",
    "M_star = 0.68765  # 恒星质量\n",
    "\n",
    "\n",
    "t = np.linspace(0, t_age, 100)\n",
    "\n",
    "# 假设恒星形成速率 SFR_0\n",
    "SFR_0 = M_star / (tau * (1 - np.exp(-t_age / tau)))\n",
    "\n",
    "# 计算恒星形成历史\n",
    "SFR = SFR_0 * np.exp(-t / tau)\n",
    "\n",
    "# 计算总质量\n",
    "M = SFR_0 * tau * (1 - np.exp(-t / tau))\n",
    "\n",
    "# 计算光度 L\n",
    "L = 10 ** (-0.4 * (V_mag - 4.83))  \n",
    "\n",
    "# 计算质量光比\n",
    "M_L = M / L\n",
    "\n",
    "# 输出最终结果\n",
    "print(f\"在10 Gyr时的质光比为: {M_L[-1]:.4f}\")\n"
   ]
  }
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