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In principle, all orbits in the universe should progressively decay due to the emission of gravitational waves. However, does this always happen? Are there any kind of orbits that do not decay as they do not emit gravitational waves? Perhaps if the orbit is perfectly axially symmetric with an even distribution of mass?...

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    $\begingroup$ What do you mean by a perfectly axially symmetric orbit? $\endgroup$
    – ProfRob
    Commented Jul 16, 2023 at 13:46
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    $\begingroup$ Electrons orbiting a nucleus do not emit gravitational waves, but that is because they don't actually orbit. They are in a quantum superposition of distributions (wave functions). Also, some orbits may not decay even as they emit gravitational waves because there could be tidal friction that is increasing orbital energy. $\endgroup$
    – eshaya
    Commented Jul 16, 2023 at 17:03
  • $\begingroup$ Might be helpful physics.stackexchange.com/questions/33132/… $\endgroup$ Commented Jul 17, 2023 at 13:12
  • $\begingroup$ Oh this is such a beautiful and answerable question! It will no doubt bring forth very interesting answers, so +1 $\endgroup$
    – uhoh
    Commented Jul 17, 2023 at 22:54

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All orbits as you would normally think of them emit gravitational waves. You mention an axially symmetric orbit. A hypothetical rotating rigid hoop would not, since the distribution of mass is not changing over time.

The difference between the gravitational waves produced by neutron stars or black holes vs planets or stars orbiting the galactic center is just a huge difference in degree. From what I have read about the long term future of the universe, on trillion-year timescales, Newtonian interactions between planets lead to them getting ejected before orbits can decay from gravitational waves. (This is a statistical analysis; orbits cannot be forecast nearly that far in detail.)

On the galactic scale, it remains true that energy loss due to gravitational waves is such a weak phenomenon that other effects will determine the ultimate fate of the (by then mostly extinguished) stars. See this link (from the comments - thanks to Sten).

In a sense it is almost true that ordinary orbiting planets, or stars orbiting galactic centers, don't emit gravity waves because the waves they emit are so weak. The energy in the waves has to do with the masses of the bodies and the period of revolution, and longer periods mean less energy. The stately progression of stars around the galactic center are not going to produce very energetic gravity waves.

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    $\begingroup$ So then, would even galaxies collapse in the very far future due to gravitational waves being emitted by all its components in rotation? (assuming their planets and stars do not get ejected)? @MarkFoskey $\endgroup$
    – vengaq
    Commented Jul 16, 2023 at 21:19
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    $\begingroup$ @vengaq Two-body relaxation effects are way more important than gravitational wave emission in galactic contexts; stars will (probably) sink due to dynamical friction with the dark matter long before gravitational waves are relevant. See astronomy.stackexchange.com/a/53780/47607 $\endgroup$
    – Sten
    Commented Jul 16, 2023 at 22:18
  • $\begingroup$ @Sten and if a galaxy did not have dark matter?. Also in your answer you mention that a symmetric "hoop" around a planet would not emit gravitational waves. Would rings around planets (as it happens in Saturn) be examples of this? $\endgroup$
    – vengaq
    Commented Jul 17, 2023 at 11:35
  • $\begingroup$ Wikipedia has an equation for gravitational radiation power for a simple orbiting system. FWIW, the Earth+Moon orbiting the Sun emits ~201.5 watts, the Jupiter system orbiting the Sun emits ~5284.6 watts, and the Moon orbiting the Earth emits ~7.4 microwatts. $\endgroup$
    – PM 2Ring
    Commented Jul 17, 2023 at 13:20
  • $\begingroup$ sagecell.sagemath.org/… $\endgroup$
    – PM 2Ring
    Commented Jul 17, 2023 at 13:31

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