How do the particles in the solar wind avoid getting slowed down by the sun's gravity?

They ARE slowed down, and can take days to get as far as the Earth whereas light only takes 8 minutes. But they're still small enough and fast enough to keep going - certainly fast enough to achieve escape velocity.

You have misunderstood one or two things.

First, you are thinking, apparently, of gravity in Newtonian rather than Einsteinian terms. Gravity as a force is simply how we interpret the curvature of space-time caused by the presence of mass. Light is affected by that curvature, too. For example, that is how 'gravitational lensing' works. (If you don't know about gravitational lensin, look it up.)

Second, and most significantly, although alpha and beta particles are affected by gravity, gravity is not the only force, and not the most important force, acting upon them. They are affected far more by electro-magnetic forces - which, for example, is how the solar wind gets deflected towards the earth

s magnetic poles to cause the aurorae.

Third, you are making an assumption about the velocities of the particles which make up the solar wind. If they were acted upon solely by gravity, their trajectories would generally be one of the conic sections - but besides parabolae, hyperbolae (particles escaping from the solar system) and ellipses would also be possible.

<< SInce photons have no mass, I appreciate that they'll shoot out at the speed of light and carry on going, unaffected by the sun's gravity. >>

-- Actually they're affected by gravity, too. They follow the path that takes the least time, so their paths are bent by gravity.See general relativity:

http://en.wikipedia.org/wiki/General_relativity

Particles don't avoid getting slowed down by gravity. They get accelerated by electromagnetic field enough to escape.

Photons can take thousands even hundreds of thousands of years to make it from the sun's interior to the corona but once there, they hit light speed, literally. The sun's gravitational pull has a negligible effect on photons and it takes them eight minutes to reach us.

Particles such as hydrogen, helium and those of other elements are more severely affected and travel considerably more slowly. The matter spewed out in a CME, coronal mass ejection take days, even weeks to arrive.

The alpha particles are affected by the sun's magnetic field, and are pushed by photons hitting them. These forces are much much more that the sun's gravity.

They are slowed down by the Sun's gravity. They're just going fast enough that it doesn't matter much.

No you are right they are slowed down by gravity. The photons suffer from gravitational redshift