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The idea I have is to have a planet that, at its core, has a being that gradually eats away at the planet's substance and converts it into massive amounts of energy (think matter-antimatter annihilation) that just get funnelled away. The result is that the overall mass of the planet gradually decreases over time. Would this mean that, based on conservation of momentum, that the planet would speed up or would the annihilated mass also take kinetic energy out of the system, causing the planet to stay at the same speed? What about the planet's own angular velocity?

Furthermore, the gravitational pull of the planet would slowly weaken as the mass decreases. Would this change the orbit of the planet around its star or would this cancel out in some way and not change the orbit? How about moons orbiting the planet?

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Assuming that the mass of the planet is substantially lower than the mass of the star, then there won't be any change in the system's orbital dynamics (depending on the way the energy from the annihilated matter is distributed - see below). Unless the two masses are comparable, we can treat the planet as a test particle when studying its orbit and ignore its mass. The lowest mass stars are a bit under $0.07M_{\odot}$, or about 75 times the mass of Jupiter. The most massive planets (yes, the mass limit of a planet is a bit fuzzy) are generally around 13 Jupiter masses, so you would expect to see $M_p/M_*\lesssim1/6$. In this regime, Kepler's third law$^{\dagger}$ predicts that the planet will have perhaps a non-negligible influence on the orbits of both bodies. On the other hand, this is quite the edge case, and I'm not confident that you'd expect to see many ultra-high-mass planets orbiting ultra-low-mass stars. There are surely some, but not many.

Now, all of this assumes that the energy from the annihilated matter is released isotropically, the same in all directions. If not - for instance, if it's emitted in a beam of light in some direction - then conservation of momentum would mean that the planet will move in the opposite direction. But if we can assume that the emission is isotropic, like a star, there shouldn't be any effects on the orbit.

John Dallman makes a point that there would be issues with the internal structure of the planet - certainly true. Removing matter would constantly perturb it from from hydrostatic equilibrium. Assuming the process is gradual, the planet will then contract, reaching a new equilibrium when a chunk of mass is lost, then contracting into a new equilibrium when the next chunk is lost, and so on. I'd argue that in this sense, it's analogous to - though not the same as - a quasistatic process

The orbits of any moons around the planet would expand. Assuming that the moons retain their angular momentum (and I don't have any reason to say that they won't), the quantity $M_pa_m$ is conserved, with $a_m$ the semimajor axis of a moon. In other words, cut the mass of the planet in half and the semimajor axis of each moon will double. This is analogous to how Earth's orbit will expand as the Sun loses mass when our star enters the red giant, and, subsequently, asymptotic giant branch phases of its life.


$^{\dagger}$By the relation $a^3\propto(M_p+M_*)P^2$, which $a$ the semi-major axis and $p$ the orbit period. If $M_P\ll M_*$, we can approximate $M_P+M_*\approx M_*$, which is done if you want a quick-and-dirty model of a planet's orbit. If $M_P\approx M_*/6$, then I'd argue the planet's mass can no longer be ignored.

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    $\begingroup$ If the matter that’s being annihilated is converted to photons then it will melt or vaporise the planet, if any significant mass is being lost. It really has to be converted to neutrinos, not photons. $\endgroup$
    – Mike Scott
    Jul 6, 2021 at 21:05
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    $\begingroup$ @MikeScott I'd say it strongly depends on where the photons are being emitted - the surface vs. the core makes a difference. It's also extremely unlikely that - in the case of electron-positron annihilation, for instance - the annihilation will lead to neutrino-antineutrino production; photon-photon pairs are by far the favored pathway. I guess it also depends on how quickly annihilation is taking place, too. $\endgroup$
    – HDE 226868
    Jul 6, 2021 at 21:09
  • $\begingroup$ I actually have a bit more magic in my world than the question probably suggests, I just needed a hint on what happens if a planet "magically" loses mass, so I turned to annihilation as a way it could happen with "real" physics. To know that the momentum etc. is, broadly speaking, conserved across the sum of leftover matter and emitted energy is enough for me. In-universe, that being uses up all of the energy to fuel vast magical processes, so assuming an isotropical release is probably what makes the most sense. $\endgroup$
    – Sacchan
    Jul 6, 2021 at 21:42
  • $\begingroup$ Let us continue this discussion in chat. $\endgroup$
    – HDE 226868
    Jul 6, 2021 at 22:18
  • $\begingroup$ There's a significant side-effect of the planet loosing mass. Holding empty spaces open at the pressure of a planetary core is impractical without serious magic. if that is not provided, the planet's radius will shrink as mass is lost. The effects of this are basically earthquakes, and their severity depends on the rate at which mass disappears. If that process is rapid, the planet will melt. $\endgroup$ Jul 6, 2021 at 22:22

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