How big will the Sun be once it becomes a red giant? How much of the solar system will it engulf?
[Physics] What size will the Sun become once it is a red giant
astrophysicssolar systemsun
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There's a nice paper by Drs. Klaus-Peter Schroder and Robert Smith on the distant future of the Sun and Earth; it's available at the arXiv:
Table 1 in that preprint summarizes a number of parameters, but in simplified form the radii (in terms of the current value) at different times (given in billions of years) are:
Age Radius
ZAMS 0.00 0.89
present 4.58 1.00
MS:final 10.00 1.37
RGB:tip 12.17 256.
ZA-He 12.17 11.2
AGB:tip 12.30 149.
(hopefully that will render correctly.) For comparison, the current orbit of the Earth is 215 times the current solar radius. ZAMS is the zero-age Main Sequence, present is today, MS:final is the end of the Main Sequence, RGB:tip is the maximum size during the Red Giant branch, ZA-He is the start of core Helium burning and AGB:tip is maximum size during the asymptotic giant branch phase. After that the Sun will fade away as a white dwarf.
While there is 2.17 billion years between the end of the Main Sequence and the start of core Helium burning (which also marks the end of the Red Giant phase), for more than two billion years the Sun is less than ten times its current radius - it's only during the last 200 million years when the expansion towards the Earth's current orbit happens. This is plotted in Figure 1 of the preprint, which the radius of the Sun during the final three hundred million years.
So in the context of the Sun's overall lifetime, the expansion in the giant phase is extremely rapid. Of course, on our timescales it's a very long time...!
This is a really rough calculation that doesn't take into account the realistic direction of the bow shock, or calculation of the drag force. I just take the net momentum flow in the solar wind and direct it so as to produce the maximum decceleration and see what happens.
Apparently the solar wind pressure is of the order of a nanoPascal. As I write this it's about $0.5\ \mathrm{nPa}$. You can get real time data from NASA's ACE satellite or spaceweather.com (click through "More data" under "Solar wind"). During periods of intense solar activity it can get up to an order or magnitude or so more than this. Let's take this worst case and assume, unrealistically, that all of the pressure is directed retrograde along the Earth's orbit. This will give the maximum deccelerating effect. I get a net force of $\sim 10^6\ \mathrm{N}$. Dividing by the Earth's mass gives a net acceleration $2\times 10^{-19}\ \mathrm{m/s^2}$. Let's fudge up again and call it $10^{-18}\ \mathrm{m/s^2}$. The time it would take for this to make a significant dint the the Earth's orbital velocity ($30\ \mathrm{km/s}$) is of the order of $10^{15}\ \mathrm{yr}$. I think we're safe.
For the other planets there is a $1/r^2$ scaling of the solar wind with the distance from the sun (assuming the solar wind is uniformly distributed) and an $R^2$ scaling with the size of the planet. So for Mercury the former effect gives an order of magnitude increase in drag and the latter effect takes most of that increase away again. There is an additional $R^{-3}$ increase in effect due to the decreased mass of a smaller body (assuming density is similar to the Earth). Then there is the $r^{-1/2}$ increase in orbit velocity due to being closer to the sun. So the total scaling factor for the time is $ R r^{3/2} $, which for Mercury is about 0.1. So the end result is not much different for Mercury.
This site always causes me to learn new Mathematica features. It made really quick work of this since it has all sorts of astronomical data built in:
Note that the number of digits displayed in the final column is ludicrous. :)
Best Answer
This is answered in How fast will the sun become a red giant?. I'm just adding a note here because it's not answered directly in a form a non-expert might spot.
The maximum size of the sun is estimated to be 256 times it's current radius, the Earth's orbit is 215 times the sun's radius - so it will consume Mercury, Venus, Earth and a bit of the way toward mars.
It's a little complicated because as the Sun expands it losses mass - large stars blow off their outer atmosphere. With the Sun having less mass it's gravity is weaker and Earth's orbit moves further out. The linked paper says (if I read it correctly) that the Sun will expand first, passing the Earth, before it has lost enough mass for the Earth to move far enough away.