{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "99879Z5Y1K"
    },
    {
      "field": "uses",
      "target": "YJ1M0FYXJD"
    }
  ],
  "body": "\n## Example\n\nLet $D \\subseteq \\R^3$ be a region filled with a medium, and let $S^2$ be the unit sphere of directions.\nThe unknown is the specific intensity $I(x, \\Omega)$ of radiation at a point $x \\in D$ travelling in the direction $\\Omega \\in S^2$, at one fixed frequency.\nLet $\\kappa(x) \\ge 0$ and $\\sigma(x) \\ge 0$ be the absorption and scattering coefficients, $B(x)$ the emission source, and $p(\\Omega' \\cdot \\Omega) \\ge 0$ the phase function, normalized by $\\frac{1}{4\\pi} \\int_{S^2} p(\\Omega' \\cdot \\Omega) \\, d\\Omega' = 1$.\nThe stationary radiative transfer equation is\n$$\\Omega \\cdot \\nabla_x I(x, \\Omega) + \\big(\\kappa(x) + \\sigma(x)\\big) I(x, \\Omega) = \\kappa(x) B(x) + \\frac{\\sigma(x)}{4\\pi} \\int_{S^2} p(\\Omega' \\cdot \\Omega) \\, I(x, \\Omega') \\, d\\Omega'.$$\nIt contains a derivative of $I$ along the direction $\\Omega$ and an integral of $I$ over all directions.\nSo it is an [[integro-partial differential equation|TH4NGC4016]].\n\nCheck: if $\\kappa$, $\\sigma$ and $B$ are constant, the constant intensity $I = B$ is a solution.\nThe left side is $(\\kappa + \\sigma) B$, and by the normalization of $p$ the right side is $\\kappa B + \\sigma B$.\n\n## Remarks\n\nThe left side describes the loss of intensity along a ray by absorption and by scattering out of the direction $\\Omega$.\nThe right side describes thermal emission and scattering into $\\Omega$ from all other directions.\nAlong each ray $x + s\\Omega$ the equation is a linear ordinary differential equation, and integrating it turns the problem into an integral equation for $I$.\nFor a plane-parallel atmosphere with isotropic scattering this leads to equations such as the [[Milne problem|KKT4Y0TBRZ]].\n\n## Explanations\n\n### L1 human\n\nLight travelling through fog, a star or a furnace is partly absorbed, partly scattered into new directions, and new light is emitted.\nThe equation keeps track of how much light goes in each direction at each point.\nThe scattering term collects light from all directions, which is why an integral over directions appears next to the derivative along the ray.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "instance_of": [
      "TH4NGC4016"
    ],
    "kind": "example",
    "msc": [
      "85A25",
      "45K05",
      "35Q49"
    ],
    "name": "Radiative transfer equation",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Radiative_transfer"
      }
    ],
    "tag": "EX9YYRMAJA"
  },
  "kind": "example",
  "link": "https://tamnd.github.io/math2/node/EX9YYRMAJA/",
  "links": [
    {
      "field": "instance_of",
      "target": "TH4NGC4016"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/EX9YYRMAJA.md",
  "msc": [
    "85A25",
    "45K05",
    "35Q49"
  ],
  "name": "Radiative transfer equation",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/EX9/EX9YYRMAJA.md",
  "status": null,
  "tag": "EX9YYRMAJA",
  "uri": "m2://EX9YYRMAJA",
  "url": "https://tamnd.github.io/math2/wiki/radiative-transfer-equation/",
  "writing": {
    "tag": "EX9YYRMAJA",
    "kind": "example",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}