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In principle every object orbiting another (e.g. a planet revolving around a star) would emit gravitational waves, relaxing the orbit over time.

However, this would not happen if the orbits had a time-invariant and symmetric quadrupole moment. As it is indicated in this question (https://www.reddit.com/r/askscience/comments/5lzgwq/is_orbital_energy_lost_through_gravitational/), it appears that if the masses were perfectly symmetrically ordered around a star (e.g. 4 planets separated by 90º from each other orbiting the same star), then the system would not emit gravitational waves. Are there any other examples of orbits that would not emit gravitational waves?

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  • $\begingroup$ Keep in mind that quadrupole radiation isn't the only contribution to gravitational radiation, it's just the leading term. You can engineer orbits that have suppressed gravitational radiation but I'd be surprised if you could get rid of it entirely. $\endgroup$
    – knzhou
    Commented Jul 28, 2023 at 21:50
  • $\begingroup$ relaxing the orbit over time What do you mean by “relaxing”? $\endgroup$
    – Ghoster
    Commented Jul 28, 2023 at 21:53
  • $\begingroup$ @Ghoster that there would be a loss of orbital energy with time so the orbits would be reduced until the bodies in orbit collide $\endgroup$
    – vengaq
    Commented Jul 29, 2023 at 16:55

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Having a "time-invariant and symmetric quadrupole moment" is not sufficient to not have gravitational wave (GW) production. While this would ensure that there is no GW production in the quadrupole ($l=2$) modes of the system, there may still be production in the higher harmonics.

The only sure fire way of getting no gravitational wave production at all is if all mass and current multipole moments are constant. (Technically, you would only the higher order time derivatives (with the order depending on the mode number $l$. to vanish. However, since in anything that could be called an "orbital configuration" you would expect the multipole moments to bounded in time, the only way to achieve this is if they are all constant.)

The way to achieve this is to have a configuration that is axially symmetric around its axis of rotation. An accretion disk comes close to satisfying this (but will dissipate energy and angular momentum through other channels). A Dyson ring would also come close.

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