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As the Sun nears the end of its life and expands, would the Earths orbit get bigger as it moves out or would it stay in the same orbital plane and not move out? Therefore in the far off future, we'd have 512 days instead of 365 days?

As the Moon is moving out at a distance of 3.8 cms a year (ref: http://curious.astro.cornell.edu/about-us/37-our-solar-system/the-moon/the-moon-and-the-earth/111-is-the-moon-moving-away-from-the-earth-when-was-this-discovered-intermediate), are we moving out or in towards the Sun or is this not possible to work out because there is nothing to measure against?

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  • $\begingroup$ When you say "as the sun dies", do you mean, when it goes red giant or as it ages prior to red giant? The red giant part of this question is significantly different than the slowly grows larger and loses a little bit of mass part. $\endgroup$ – userLTK Jan 15 '17 at 0:47
  • $\begingroup$ @userLTK I feel that "As the Sun dies" is pretty self explanatory. Regardless of whether or not the Sun is a RG or MS star, the total amount of mass remaining within the Sun is going to determine the orbital relevance to this question. Besides, if the Earth can't relax its orbit enough, it'll be inside the RG envelope of the Sun and then this question really doesn't matter. $\endgroup$ – LaserYeti Jan 15 '17 at 3:43
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The answer is yes. As the Sun ages, it will become a red giant and the mass loss rate from its surface will increase. This effect will increase (dramatically) further when the Sun enters the asymptotic giant branch phase, where thermal pulsations drives a cool wind that may carry away a millionth of a solar mass per year, eventually leaving a burned-out core in the form of a white dwarf with about half a solar mass.

At any point in this evolution we can model the evolution of the Earth's orbit using some simple approximations - that the wind from the Sun escapes to infinity, that a negligible proportion is actually accreted by the Earth and nor does it exert a torque, that the mass loss takes place on a timescale much longer than the Earth's orbit and that the mass of the Earth $m$ is always much less than the time-dependent mass of the Sun $M(t)$.

In which case we consider the orbital angular momentum of the Earth: $$ m a \omega^2 \simeq G\frac{M m}{a^2},$$ where $a$ is the semi major axis. So the angular momentum $J = m a^2 \omega$ is given by $$ J^2 = m^2 a^4 \frac{G M m}{m a^3} \propto M a$$

As the angular momentum of the Earth's orbit is conserved, the $M(t) a(t)$ is constant and as the Sun loses mass, the semi major axis increases by the same factor.

Coming to the specifics - when the Sun is a half solar mass white dwarf, the semi-major axis will be 2 au (assuming the giant Sun did not quite engulf it - it will be a close-run thing) and Kepler's third law $(P^2 \propto a^3/M)$ can be used to estimate an orbital period of 4 years.

The tidal effects of the Sun on the Earth's orbit are quite negligible compared with these mass loss effects.

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  • $\begingroup$ I read somewhere that the red giant Sun, expanded presumably past Venus, the tidal effects on Earth would be significant cause the Earth would create a big bulge in the expanded sun and, even though the density of the outer bulge would be extremely low the proximity would cause the tides to be significant enough to pull the Earth inwards. I have no idea if that's accurate, but I remember reading it. $\endgroup$ – userLTK Jan 18 '17 at 23:27
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Rob Jeffries has already answered the question, but has left out a detail that should be included.

The mass that the sun loses to solar wind will cause the Earth to migrate outward. Increased drag from moving through a denser solar wind will slow the Earth and partly counteract the effect. Initially not by much, but as the sun expands the Earth will find itself moving through an increasingly dense medium. According to this paper, drag in the lower chromosphere will eventually be high enough for the planet to fall into the sun.

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  • $\begingroup$ This is a good paper. I need to understand the tidal interaction formalism they use, as it doesn't agree with other work I've seen. $\endgroup$ – Rob Jeffries Jan 25 '17 at 19:44
  • $\begingroup$ The link now goes to a login page instead of to a paper. Is there a citation for the paper, including a DOI number? $\endgroup$ – David Hammen Aug 8 '18 at 19:27
  • $\begingroup$ The login wall was put up more recently. It can still be viewed in the Wayback Machine at present, though I question the legality if the university has now blocked access. I will edit in a better citation. The paper is easy to find in a web search but I don't know how many of these sites are legal. $\endgroup$ – user25972 Aug 10 '18 at 5:48

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