Theory behind traveling to Kepler 62E

In order to understand the theory behind the travel, two conclusions of special relativity need to be introduced.

They are:

  1. Time Dilation
  2. Length Contraction

 

  1. Basically – time slows down the faster you go. A clock of a moving frame will be seen to be running slow, or “dilated” according to the Lorentz transformation. The time will always be shortest as measured in its rest frame. The time measured in the frame in which the clock is at rest is called the “proper time”; which means “the shortest time between events as measured by all inertial observers”.
  2.  Basically – distance shortens the faster you go. A ruler of moving frame will be seen to be “contracted” in the direction of movement. The idea is similar, and the concept of “proper length” refers to “the longest distance/length as measured by all inertial observers”.

The aim of this investigation was to estimate the speed we would have to travel with in order to get to Kepler 62E in reasonable time. We investigated relativistic effects while traveling at speeds close to the speed of light.

From Lorentz transformation we derived an equation for speed:

γ=1/√(1-(v^2/c^2 ))

γ√(1-(v^2/c^2) )=1

γ^2 (1-v^2/c^2 )=1

γ^2-( (γ^2 v^2)/c^2) =1

v^2 (γ^2/c^2 )=γ^2-1

v^2=(c^2 (γ^2-1))/γ^2

v^2=c^2 (1-1/γ^2 )

v=√(1-(1/γ^2 ))c

 

We calculated needed velocities for different gamma factors. These are the results we obtained:

γ  factor

Needed velocity

[ms-1]

Total time [Years]

15

0.997775c 80.50

30

0.999444c 40.50

60

0.999861c

20.50

120 0.999965c

10.50

240

0.999991c

5.50

480 0.999998c

3.00

960 0.999999c

1.75

How to calculate the needed velocity and total time?

Example: γ factor = 120

v=(√(1-(1/120^2) )c=0.999965c

Time of travel = (distance to Keplar 62 e)/(γ factor)=(1200 light years)/120=10 light years

Maximum acceleration (due to human comfort) = 4g = 40 ms-2 

u=0

v=0.999965c

t=?

v=u+at

t=v/a=0.999965c/40=7.49974*10^6=0.24 years

Total time = 10 + 2(0.24) ≈ 10.5 years

If we were to travel at 0.999965 of the speed of light, it would be possible to get to Kepler 62E in around 10.5 years. However the energy needed for such a journey is huge and will be investigated in a different post.

 

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