The aim of the investigation was to estimate the energy required to travel to Kepler 62E in a reasonable period of time, and to see if it would be realistic to use antimatter as an energy source for the journey.
The results obtained from our teammates showed that, according to the theory of Relativity, by travelling at speeds close to c (that is, the speed of light), there is a change in relative time, meaning more distance can be covered in less time (please refer to the post on relativity for details). If a speed of 0.999965 c is reached then a ‘gamma factor’ of 120 is reached, meaning that the journey to Kepler 62E (1200 ly) takes only 10 years.
Our calculations then would be based on the assumption that a spaceship of mass of the order of 10*10^3 kg would travel at 0.999965 c. By using the formula Ek = 1/2mv^2 we can find out the necessary energy input for this specific travel, as follows:
Ek = 1/2 * (10*10^3) * (0.999965*3*10^8)^2 = 4.5*10^20 J
That is a great amount of energy. The most dense energy source we know of is antimatter. When a gram of antimatter fuses with a gram of matter it gives away 180 TJ. Hence to reach this energy we would need 2.5 tonnes of antimatter to be used by an engine with 100% efficiency, as shown: (4.5*10^20)/180*10^16 = 2500 kg
To reach such a high-speed would take around 0.23 years if the acceleration is not grater than 41ms^-2 which is equal to 4g, the maximum acceleration humans should withstand. This was calculated using the formula v = u + at, as follows:
[(0.999965*3*10^8) – 7800]/41*60*60*24*365 = 0.23 years.
[Note that 7800ms^-1 is taken as the original speed of a spaceship taking off from Earth]
If the same time is used to decelerate, then the total journey would take about 1 and 1/2 years.
Given that between 1 and 10 ng of antimatter is produced yearly (according to Pennsylvania State University), the chances of travelling to Kepler 62E are slim, at least in the near future. Yet the use of antimatter is our only hope as other sources such as unstable atoms would require much higher quantities. The energy given by the decay of a gram of Uranium 238 can be calculated as follows:
(6.02*10^23)/238 = 2.5*10^21 nucleons in a gram of Uranium 238.
The average binding energy of Uranium 238 is 7.5 MeV, which equals 1.2*10^-12 J.
(1.2*10^-12)*(2.5*10^21) = 3100 MJ
Hence 1 gram of Uranium 238 can give away 0.0031 TJ, compared to the much greater 180TJ given by 1 gram of antimatter.
This investigation has shown the amount of energy required to reach Kepler 62E in ten and a half years (including the greatest rate of acceleration humans can withstand), and the possibility of using antimatter as an energy source or fuel is discussed. Three concepts of physics have been applied to this investigation, and the work has been shared among the team members.