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        "description": "The stability analysis uses an idealized thin stellar disk, local tightly wound perturbations and the stated collisionless velocity distribution. The local axisymmetric criterion is not a universal global/nonaxisymmetric stability theorem. Finite thickness, gas, nonlinear evolution and different distributions require additional analysis. Earlier and later disk models remain separate.",
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        {
          "citation": "Alar Toomre1964: On the gravitational stability of a disk of stars, original1217-1238",
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        {
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      "statement": "# Toomre and Kalnajs: when random stellar motion resists disk clumping\n\nToomre analyzes the competition between rotation, stellar random motion and self-gravity, deriving with Kalnajs a local neutral-stability relation and the Gaussian stellar-disk threshold sigma_R approximately3.36G Sigma/kappa.\n\nA galaxy can rotate and still be vulnerable to gravitational clumping. Rotation helps resist long disturbances, while random stellar motions suppress short ones. Toomre explains why both effects are needed and calculates the dividing line for small, axisymmetric disturbances in an idealized thin stellar disk. The calculation uses nearly circular stellar orbits and an approximately Gaussian velocity distribution. Its familiar threshold compares radial velocity dispersion with the disk surface density and epicyclic frequency; the coefficient is3.36, not the pi appearing in the corresponding gas calculation. The paper expressly credits A. Kalnajs for the final neutral-wave formula after he identified an error in an earlier version. Toomre also gives numerical ring examples and a solar-neighborhood comparison, but he does not establish that every disk above the local threshold is globally stable or immune to bars. The work supplies a useful diagnostic with declared assumptions, rather than a universal certificate for a galaxy.\n\n## Scientific scope\n\nThe stability analysis uses an idealized thin stellar disk, local tightly wound perturbations and the stated collisionless velocity distribution. The local axisymmetric criterion is not a universal global/nonaxisymmetric stability theorem. Finite thickness, gas, nonlinear evolution and different distributions require additional analysis. Earlier and later disk models remain separate.\n\n## People and systems\n\nAlar Toomre\n\n\n\n## Original sources\n\n- [Alar Toomre1964: On the gravitational stability of a disk of stars, original1217-1238](https://adsabs.harvard.edu/pdf/1964ApJ...139.1217T)\n- [James Binney2019: The shearing sheet and swing amplification revisited](https://arxiv.org/abs/1906.11696)\n",
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      "title": "Toomre and Kalnajs: when random stellar motion resists disk clumping"
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