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Modern chemical rockets have so far traveled relatively short distances (i.e. to the moon) and there are many newer, more efficient ideas being theorised and tested for longer journeys.

However, without considering factors such as cost or fuel efficiency, is there going to be a limit to where such rockets can go, or will newer fuels and larger craft be capable of travelling to wherever the next furthest destination is?

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    $\begingroup$ Older chemical rockets, with gravity assists, have managed to fling objects clear out of the solar system. "is there going to be a limit to where such rockets can go" The biggest limit at the moment is 'time'. Voyager 1 should be in the region of another star ..in around 40 millennium. $\endgroup$ – Andrew Thompson Nov 13 '16 at 16:44
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In practice, yes, chemical rockets are very limited in what they can achieve, because they consume fuel to accelerate, and by bringing more fuel they grow in weight, yielding diminishing returns from each additional kilogram of fuel. The Tsiolkovsky rocket equation describes exactly how that works.

With chemical propulsion, we can do one way trips to the edge of our solar system in about 30-40 years (the Voyagers are there now; New Horizons will be, before very long). The nearest stars are about 1000 times further away than that, and we don't have any practical way of building a machine that can last for 30,000 years.

For the next phases of exploration of our own solar system, various electrical rocket propulsion technologies are attractive; their exhaust velocities are much higher than those of chemical rockets, so for the same mass of propellant they can achieve much higher speeds. That still requires thousands of years for interstellar travel.

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  • $\begingroup$ Electric propulsion is of course also limited by the ideal (or Tsiolkovsky) rocket equation. $\endgroup$ – David Hammen Nov 14 '16 at 10:57
  • $\begingroup$ Electrical rocket propulsion requires a powerful source of electric energy. Solar cells are useful only if the distance to the sun is not too large. Plutonium thermoelectric generators are limited by the half life of the used isotope. The electrical propulsion may be used for some years, but not for thousands of years. But if the data transmission does not work for real interstellar distances, sending a space probe does not make sense. $\endgroup$ – Uwe Nov 14 '16 at 16:29
  • $\begingroup$ I haven't run the math, but I was assuming running the electrical thruster for ~30 years off RTG, coasting interstellar, and switching to solar power when reaching the destination star. Then you "only" need to design a spacecraft that can wake from hibernation after ~3000 years of radiation and micrometeoroid exposure. ;) $\endgroup$ – Russell Borogove Nov 14 '16 at 20:35
  • $\begingroup$ Given that we can barely manage on solar power around Jupiter, we'd probably need some orders of magnitude improvement in solar panels before we can generate sufficient power to decelerate even from a fraction of a percent of the speed of light before overshooting the star... $\endgroup$ – a CVn Nov 15 '16 at 14:34
  • $\begingroup$ Who said anything about decelerating? (Also, there aren't orders of magnitude improvement in solar panels to be had; we're already getting 20-30%.) $\endgroup$ – Russell Borogove Nov 15 '16 at 16:22

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