How long until we fall into the Sun? As a planet moves through the solar system, a bow shock is formed as the solar wind is decelerated by the magnetic field of the planet. Presumably the creation of this shock wave would cause drag on the planet, certainly in the direction of orbit but possibly rotation as well.
Is there an estimate for the amount of drag on the Earth as it orbits the Sun? Based on the drag, how long would it take before the orbital velocity slows to the point that we spiral slowly into the Sun? Would any planets fall into the Sun prior to the Sun expanding into a Red Giant, gobbling them up?
 A: This question is different from, but related to another question
How is it that the Earth's atmosphere is not “blown away”?.
In answering that question with respect to solar wind, I remarked that
the orbital speed of Earth is 30 km/s while the speed of the solar
wind varies between 300 km/s and 800 km/s in a nearly orthogonal
directions (fully orthogonal if the orbit is considered circular).
Hence the apparent wind is mostly a side wind, slightly in front (a
slightly close reach in sailing terms). As a first small angle
approximation, the dragging effect of the solar wind on the planet
orbital speed does not come from the solar wind speed, but only from
the planet own speed, which is at best a tenth of the solar wind
speed.
Hence the actual effect of the solar wind on braking down Earth orbital
speed is at best one tenth of the effect computed by Michael Brown,
which makes it even less significant.
Another point is that the pressure due to the speed of the solar wind
itself is pushing the planet outwards, away from the Sun. I am not
sure how this should be analyzed, I mean to give the best insight. One
way to do it is to consider that it reduces the centripetal force
towards the Sun due to gravity. Furthermore, its effect must also decrase like the solar wind density in proportion to
the square of the distance from the Sun, as does gravity. However the effect
is proportional to the cross-section surface of the Earth, rather than
its mass.
The energy output of the Sun is thought to have increased by about 30%
since it formation (some 4.6 billion years ago). So the pressure from
the solar wind should have increased in proportion, being equivalent
to a minute reduction of the centripetal force that keeps the Earth in
orbit. But it also increases the orbital drag in the same proportion.
This energy output should continue to increase slowly.
Note that I assumed in these last  remarks that the increased output is due to a greater amount of particles being output at the same speed. Some of the energy could be due to a greater speed of the solar wind which would increase the outward push, but not the orbital drag. I do not know which actually occurs.
More detailed calculations, which I have not done, should tell which
of the two effects dominates, though they are probably both very negligible.
A: 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. :)
