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When @anna v explained why the planetary model of the atom does not make sense in this post, she said the electron in an orbit is accelerating continuously and would thus radiate away its energy and fall into the nucleus.

However, when the Earth orbits the Sun, isn't it accelerating continuously toward the Sun? If this is the case, should the Earth radiate away its energy via gravitons and fall into the Sun eventually?

I also checked questions about Why doesn't the Moon fall onto the Earth?, but the answers didn't talk about radiating away gravitons.

Qmechanic
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戴淯琮
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    Wikipedia; The Earth-Sun system radiates gravitational waves. The average power radiated away in such waves is only about 200 watts, so the tiny energy loss has negligible effect on the dynamics. “At this rate, it would take the Earth approximately $3×10^{13}$ times more than the current age of the universe to spiral onto the Sun.” This is a common homework problem in a GR course. – Ghoster Oct 11 '23 at 00:31
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    There is no need to think about gravitons. The numbers of them radiated per second by the Earth-Sun system are so vast that the classical approximation of a gravitational wave is excellent. – Ghoster Oct 11 '23 at 00:41
  • Note that whether or not "gravitons" even exist is disputed. Many models for gravity do fine without them. – Philipp Oct 11 '23 at 10:55

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The earth stores a very large amount of energy, and very little is lost in gravitational waves. According to the Wikipedia article on gravitational waves, the orbital energy amounts to $1.14*10^{36}J$ and only $200J$ is lost every second. Hence, for the earth to lose 1% of its orbital energy would take about $5*10^{31}s$, or more than $10^{24}$ years. As the universe is of the order of $10^{10}$ years old, that 1% loss would take about 100 trillion times longer than the age of the universe - and about 300 trillion times longer than the age of the earth.

In other words, the loss is negligible.

hdhondt
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  • Just a slight nitpick - the "total orbital energy" of the earth should be negative, not positive. And for the earth to collide with the sun, that energy would have to be more negative, and by much more than a factor of two (infinitely more negative in the limit where the sun's radius is small, and it is pretty small). The 1% calculation is probably still right though, for the earth moving 1% closer to the sun and its energy becoming 1% more negative. – AXensen Oct 11 '23 at 08:49
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    @AXensen Thanks for the correction. The reason why I chose 1% is because a small change let me assume a linear relation. That's much simpler than going through all the calculations. – hdhondt Oct 11 '23 at 09:00