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Fix layout in docs
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FredDeCeuster committed Aug 8, 2024
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15 changes: 8 additions & 7 deletions docs/src/background/forward_problem.ipynb
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},
"source": [
"## Spectral line formation\n",
"\n",
"---"
]
},
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"metadata": {},
"source": [
"### Physical model\n",
"\n",
"---\n",
"In pomme, we use Cartesian coordinates, $\\boldsymbol{x}=(x,y,z)$, and assume the line of sight to be along the positive $z$-axis, such that the plane of the sky corresponds to the $(x,y)$-plane.\n",
"Furthermore, we will assume that the observer is located at $z=0$, and that the model box has a depth, $L$, along the $z$-axis.\n",
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"\\tau_{\\text{obs}}(\\nu; x, y, z)\n",
"\\ \\equiv \\\n",
"\\int_{0}^{z} \\text{d}z' \\ \\chi\\big( \\nu_{\\text{com}}(\\nu; \\, x, y, z'); \\, x, y, z' \\big) .\n",
"\\end{equation*}"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"\\end{equation*}\n",
"\n",
"\n",
"Since we look along the $z$-axis, the non-relativistic Doppler-shifted frequency in the co-moving frame is given by,\n",
"\\begin{equation*}\n",
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"metadata": {},
"source": [
"### Numerical implementation\n",
"\n",
"---"
]
},
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"metadata": {},
"source": [
"#### Line optical depth\n",
"\n",
"---\n",
"Velocity gradients play a key role in line radiative transfer. Since spectral lines are narrowly peaked in frequency space, they are very sensitive to Doppler shifts, and thus motion (gradients), along the line-of-sight.\n",
"Therefore, when numerically solving a line transfer problem, it is key to properly trace the velocity (gradient) along the line-of-sight. \n",
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"metadata": {},
"source": [
"#### Line radiative transfer\n",
"\n",
"---"
]
},
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"metadata": {},
"source": [
"## Observation\n",
"\n",
"---\n",
"In the previous section, we presented a theoretical model for spectral line formation.\n",
"In practice, however, spectral line observations are affected by instrumentation effects, such as binning and noise, the particular form of which highly depends on the way the object is observed.\n",
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