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This is an old question, but I thought it might be worth chewing on a bit. The loss of mass due to fusion in the sun is piffle. The Earth’s orbital radius will change more likely due to interactions with the other planets. The first order perturbations in the orbital elements of the Earth are its eccentricity and right ascension. The change in the orbital radius or the sem-major axis distance is higher order. However, that can occur and there is an over all orbital drift in planetary orbits which is chaotic in nature. The Earth is in a near 1/12 orbital resonance with Jupiter. The Earth may over the next billion years shift away from this and enter into a near 1/11 orbital resonance with Jupiter, where our orbital radius is about 1.06AU.
This early Earth may have been at .83AU relative to today’s orbit very early on. This is an orbital resonance of about 16 with Jupiter. The sun had a power output of 70% of current power. If you factor these together you get a solar irradiance on the Earth comparable to today. If the Earth had the same orbital radius as today, even factoring in a $\mathrm{CO_2}$ atmosphere temperatures would be $30~^\circ\mathrm C$ cooler than today. Curiously if Earth does drift outwards this delays the solar death of the Earth. If Earth remains at the current radius temperatures will become intolerant in 500 million years for complex life.
Some numerical analyses of this I have run. The interaction with Jupiter results in a periodic oscillation, and a computation over a longer period of time result in a drift which pushes the Earth outwards on average by about $4.2~\mathrm{km/sec}$.
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This is in part due to alpha Centuri’s commets. One big uncertainty is with understanding the early Earth. I did some homework on this and at 1AU about the warmest the Earth could have been is about -25C with various estimates. Of course this is my interpretation of geo-modelling.
The orbital dynamics is based on computer modeling. This is a general plot of 45,000 years. I should have posted this image. This illustrates the “signal” in these long runs, where the low frequency stuff has the largest amplitude. This is the main signal for an outwards drift.
This does extend the future for life on Earth. If this planet stays at 1AU the prognosis becomes grim about 500 million years from now. The planet will start to reach temperatures 30C higher than today and complex life will begin to die out, and further in a billion years oceans will start to boil. That will really foul things up. However, with the outwards drift these time frames are almost doubled. The luminosity increase in the sun will accelerate faster in time and over take this. The outwards range on this is 2.5 billion years before the oceans start boiling. Once the oceans start boiling this planet will transform into a 400C version of Venus. So I figure complex life on this planet, life which emerged with the Cambrian revolution 550 million years ago, might have a good 750 to maybe 1000 million years ahead of it.
When I first read about the future time frame of life on Earth my mind instantly questioned what happened going back in time. It implies a very cold early Earth; one where it seems the development of life would have been far more difficult.
If we want to change the distance to the sun, in which direction do we have to change the gravity?
The diagram shows the Earth's orbit in comparison with a circle centered on the Sun. The foci of the... [+] orbital ellipse and the positions of aphelion and perihelion are indicated. The count of ecliptical longitude starts at the First Point of Aries (the direction of the vernal equinox). Image credit: wikimedia user Sch, CC BY 3.0 A-SA
In any direction! The distance between the Earth and the Sun is a careful balance between the mass of the Earth, the mass of the Sun, the speed with which the Earth orbits the Sun, and the strength of gravity. If any of these things change in any direction, the orbit of the Earth would change. If the orbit of the Earth changes, then the distance between the Earth and the Sun changes.
If we kept the strength of gravity the same, and increased the mass of the Sun, the Sun would exert a stronger gravitational force on everything that orbits it, and in the absence of any other changes in the solar system, would disturb the orbits of all of the planets, pulling them closer for at least part of their orbits. Our Earth has an almost perfectly circular orbit around the sun, but that’s not required to be the case - just look at the long, looping orbits of comets around our sun. If you tug on an object once (as you would, if you suddenly increased the mass of the sun), you’ll likely pull a circular orbit into a more oval, looping orbit.
If the Earth’s mass suddenly increased, we’d have a few problems here on earth, as our bodies are calibrated to work best when we’re dealing with exactly the amount of gravitational force that we have. But on an orbital sense, the gravitational force between two objects depends on the mass of both the larger and the smaller object. So if we increase the mass of the smaller object (the Earth), that will increase the gravitational force between the Sun and the Earth, probably still pulling the Earth off of its circular orbit, but also pulling more strongly on the Sun, making the Sun wobble very slightly more as the Earth orbits around it.
Orbits of 3 periodic comets: Halley, Borrelly and Ikeya-Zhang. Image credit: wikimedia user Morgan... [+] Phoenix , CC BY 3.0 A-SA
It’s not hard to imagine ways of changing the distance between two objects before you get around to playing with the strength of gravity. However, changing the strength of gravity does the exact same kind of things to an orbit as changing the masses of the objects you’re interested in. Let’s say we increase the strength of gravity; that’s equivalent to making everything in the solar system more massive at once. That change would, in turn, mean that the strength of the gravitational pull between all objects gets stronger. The exact response of the orbits will depend on how different the masses are, but we can safely say that the Sun would be pulled into a more wobbly rotation around its own axis, and that the Earth would wind up closer to the Sun for at least part of the year.
If, on the other hand, we let gravity get fainter, the opposite happens. The pull between Earth and Sun grows weaker, and the planets would drift farther from the Sun, spending much more time at much greater distances from the Sun. If you’d like to play around with this more directly you can try out this solar system simulator (there are a lot of other setups you can tinker with).
So really, any change to the strength of gravity, or any change to the masses of the objects involved, would change the distance between the Earth and the Sun. However, changing the strength of gravity would have relatively catastrophic consequences for a whole series of physical processes that we currently rely on, so I wouldn’t recommend it as a scenario.