The most urgent thing is to invent better launches. Chemical rocket launches (not to mention space shuttle launches) are far too inefficient expensive to seriously reach space! One good idea is to test whether or not graphene can generate a Casimir effect.
The path to space seems obvious to many: colonize the Moon, then Mars, then the solar system, and then move outwards to the stars. Simple. The problem with this view is that it takes a linear approach: first the Moon, then Mars, then the rest of the solar system, then out to the stars in order. While it is certainly true that we will have to colonize the solar system before moving on to other star systems, due to the limited time and energy we have available to us here on Earth, we need to be colonizing the Moon, and Mars, and the other relatively stable parts of the solar system at the same time.
There are several reasons for this. Firstly, as noted elsewhere in this Wiki, we are consuming our energy resources here on Earth so quickly that we will run out of cheap energy soon, and if by that time we have only just started to colonize the Moon, we will not have enough resources to develop any of the other colonization projects. Secondly, if we do begin to colonize the Moon and Mars at the same time, then the resources we glean from the one may be used to accelerate the colonization of the other (see: Resources of the Inner Solar System). Thirdly, unlike energy consumption here on Earth where less and less energy is returned to us through our own energy usage, there is a law of increasing returns from energy usage in space (see: Law of Increasing Returns).
So how do we go about colonizing the Moon and Mars at the same time (see: the plan for greater detail)? The strategies for colonizing each, while having some similarities, also have their own unique requirements. For example, the Moon is very close to us (just a few days travel), whereas Mars is quite far away in human terms (six months travel one way, with a round trip time of maybe three years by the time the orbits of Earth and Mars bring them close together again). It is possible to use BTC travel (BTC=Better Than Chemical) to get there faster. Ion engines, for example. Also, the Moon has no atmosphere to speak of but Mars has a thin, mostly carbon dioxide atmosphere, meaning that manufacturing processes on each will be different. In the same vein, each has some resources that the other does not, and so each will be limited in what it can produce by itself with our current levels of technology.
The issue of distance is what causes the biggest headaches when it comes to the colonization of Mars. At its closest point to the Earth, radio signals (which travel at the speed of light) take around six minutes to get from the Earth to Mars and back again, assuming they are re-transmitted immediately. For a robotic explorer in need of an immediate decision from Earth on what to do next, this would be an unreasonable amount of time that could lead to its destruction, and could cost a huge amount of money for no product. Far better would be to have human explorers on Mars who can make split-second decisions by themselves that could ultimately save any colonization mission. Of course, when Mars is on the other side of the solar system (assuming the sun isn't in the way), radio signals would take much longer, meaning that if we were going to use robots to begin the colonization process, they would have to be seriously intelligent robots that can spontaneously react by themselves, and we simply do not have that technology yet.
The issue of distance is what makes rapid colonization of the Moon a viable possibility. The problem thus far has been the political climate: we only put men on the Moon to prove that men could actually go there and come back alive. The technologies of the day would not have been able to put manned colonies on the Moon, but that has changed now, and we all realize that we are running out of time before we lose that ability. So, do we actually need manned missions to the Moon at all yet? Well, one or two manned missions would certainly look nice, and if actual, permanent Moon bases were set up for scientific experiments then that would look nice too. But that would be too slow. If the Encyclopedia Britannica is correct and we only have 100 years of oil left (see we're running out of fossil fuels for the full detail), and bear in mind that NASA's Vision for Space Exploration will only be really beginning by 2020, and will certainly (without help) take longer than the so-called 2050 tipping point of planet Earth where our population has grown to nine billion and our planet can no longer sustain us (see this Guardian article for just one of many that highlights the year 2050), we really need to speed things up.
So if manned missions to colonize the Moon would be too slow, the only other option is a technological colonization that paves the way for us. Now, this does not mean that we need fully artificially intelligent robots to be built at huge cost and technological development and sent to the Moon to build habitats for us. That would take too long. But what we can do is build hundreds (or thousands) of simpler robots that we can remote control from here. Teenagers with computer game-pads could remotely control robots on the Moon, and could easily take care of the three second delay in reaction times due to the Moon's distance from the Earth. And further, the cost of launching robot missions rather than manned is diminished because the amount of money and work that needs to be spent on a human life support system on the launcher itself is zero. Indeed, we wouldn't even have to bring the robots back from the Moon, so we could re-use the materials from the launchers to make other products, such as walls for buildings.
The point of having a simpler technological colonization of the Moon is that, if we can organize ourselves into a substantial workforce then the manufacturing time required for everything we will need to produce will be dramatically reduced, and with mass production, so will the cost. Further, with so many machines being built cheaply, if some of them go wrong it won't mean the end of the project for another few years; the rest will be able to take up the slack. All the while, projects such as Mars Direct will have their chance to begin colonization of Mars at the same time. And if this substantial workforce can, through economies of scale, grow to help with other space colonization projects, for example making use of the resources of the inner solar system, then through the law of increasing returns the human race can begin to solve many of the problems we will be facing in the next 100 years.
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