Can we send a space probe to Proxima Centauri?
Sure could. Unfortunately, if it were anything like the probes we've been sending around our own solar system, it would take about 75,000 years to get there.
For some technology that has been proposed to get the job done a little more efficiently, see:
These projects have, respectively ~50 and 100 year travel times to a destination like proxima centauri. So it's not really feasible at the moment, but it certainly could be done.
Very interesting thread about this exact subject over at the aerospace engineering forums not long ago. https://www.physicsforums.com/showthread.php?t=244869
Not presently, but there are plenty of paper starships to choose from. What sort of mission? Fly-by? Orbital?
Someone mentioned "Longshot" and "Daedalus" which are really the same probes designed with different assumptions - what I mean is they have the same propulsion system and "Longshot" is really just a minature version of "Daedalus". The problem with both is that we don't have an inertial confinement ignition system that we know will work using deuterium/helium-3 propellant. We can fire off D-T reactions, but they make too many neutrons. Somewhat better is D-D fusion which is harder to start, but we have achieved it in bombs.
The really problem is striking the metaphorical match hard enough. Pure deuterium reactions require a large and very quick energy input to remain confined while fusion is happening. The best option is using a small D-T "spark-plug" to start a much larger D-D fusion reaction. To get the highest exhaust velocity - what we need for interstellar probes - the fusion pulse units have to be very large, thus the vehicle itself needs to be large too. Kind of makes sending just a probe kind of pointless.
The alternative is beamed energy propulsion, but that requires a large space-based power supply. Maybe once we build a few terawatts worth of solar power satellites we'll be ready for beamed power probes. Perhaps the easiest - if the right material can be found - is Jordin Kare's Sail-Beam, which uses an ultra-powered laser to rapidly fire off small laser-sails. These transfer their momentum to the star-probe via being blasted into plasma as they approach and running into a magnetic field wrapped around the probe. Viola! Interstellar propulsion. Just needs gigawatts of laser-power.
So can we launch a probe to Proxima Centauri? Not yet. Konstantin Tsiolkovsky theorised about chemical rockets reaching space in the 1890s and in just 60 years or so they did. We haven't done everything he dreamed of yet, but our dreams may only be as far away from their fulfillment. We may just have to get over our fixation on being on just one planet to do it though.
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