Essentially, momentum is a function of energy, not mass.

Thanks! That's the critical piece of information.

But how do you apply this with Lorentz' transformation (i.e. relativistic factors)? You cannot approach the speed of light without considering relativism. It is known that p = gamma * m * v where p is momentum, gamma is the gamma factor given by sqrt(1/(1 - (v^2/c^2))), m is mass and v is velocity. If you study the gamma factor, you'll realize that it approaches infinite as v approaches c, the speed of light. Since we are actually dealing with light here, where v = c we are breaking the equation. Momentum cannot be defined for any mass which moves at the speed of light. It's asymptotic at that speed.

Also note that the same goes for E = mc^2. At relativistic speeds, also this equation needs to consider the gamma factor. So those classical equations break down for light.

The answer is that photons don't have mass, but they have energy. There is a good explanation a bit further up in this thread on how this is possible.

The one that you multiply with gamma is the rest mass, not the total mass.

To be short, p = m_0 * γ * v, where m_0 is the rest mass. Put that in your equation and look what happens.

That's actually not that big of a deal!

Since these craft would be small, they wouldn't have the power to transmit back to Earth anyway. So with something like this, you would actually want a string of these kind of crafts that you would propel along the same vector so that they could send the data back using each following craft as the next point in the network back to Earth. So each one can take additional pictures to get a resolvable image at the end!

Now, getting them on the same vector is the hard part, since we're constantly moving through space and won't have the same launch conditions on subsequent launches, but this is all theoretical at this point anyway.

No, because the solar wind drops off around 100 AU, and the power of the solar wind is going to reduce the farther out you are. These kinds of craft would get much more acceleration from a laser array that can put much more concentrated energy into the sail. But just like the solar wind, it will lose power the farther away from the array it is, along with any kind of intermediary debris attenuating the beam or unfavorable angles between the array and the craft.

So you can get these to an appreciable fraction of the speed of light, but I don't think we'd be able to get anywhere close to c with this kind of a setup.

Edit: I was wrong about the solar wind above, it's only like .5% as powerful as the photons emitted by the sun, and that energy drops off at only 1.5 AU, so they'll get much less energy than I thought without an external power source like a laser array.

I believe this is one of those things that benefits from scale. Theoretically, the larger you make the sail, the better the thrust to mass ratio you can achieve (even before calculating a better payload mass to sail mass ratio). With improved materials, we can make stronger and lighter sails and support structures, and this will in turn result in higher velocities by the time the vehicle has left the effective range of the sun. I think speeds truly approaching c are unlikely, but they can still achieve "really freaking fast".

But then new advanced materials could also change that, we're developing metamaterials with some fascinating properties, carbon nanotubes are just the tip of the iceberg. Who's to say that we couldn't some day achieve those speeds.

Or maybe the advantage is not having to counter any sail movement with gyroscopes... which cost energy too.

I think you nailed it right there.

Yeah, we may well end up seeing a lot more use of solar light pressure to orient spacecraft over the coming years, especially deep space probes and higher orbit satellites. Telecom satellites need a lot of power for their transmitters, so they generally have quite large solar arrays, I wouldn't be surprised to see some of those using this method for station keeping. (Though these days they're already using very efficient ion thrusters for station keeping, so they may not really need to use this method)

But while solar sail propulsion takes a lot of very specific design requirements, using solar pressure just for maintaining orientation doesn't actually require any fancy hardware (if you already have solar panels), it's all an entirely software solution. Which means it's always a viable backup plan when hardware failures occur.