TIME TO DESTINATION Are we there yet? Are you there yet? The time it takes to reach your favorite star and return depends on whether you’re on the starship or waiting back on Earth for good picture of your favorite solar system. This form allows to enter the distance (in light-years) to a star, nebula, galaxy etc. and returns the time it takes for the ship to get there as viewed from the Earth or perhaps more importantly, how long it takes to get there for passengers in the starship. Though our closest star (Proxima Centauri) is about 4.2 light-years and our galaxy is about a 100,000 light-years across, you don't have to travel faster than light to reach your favorite star (http://www.space.com/3380-constellations.html). Traveling to any star in our galaxy can be achieved within an average lifetime. The following forms allow you to determine how much time it would take, within a starship moving with constant acceleration, to travel a given distance. For example, travelling with a constant acceleration of 1g to the Andromeda Galaxy (2,538,000 light years) can take 2.5 million Earth years but will take less than 30 years for the passengers on our ideal starship. The outward bound part of your round-trip consists of two equal blocks of time. At the launch of the starship, the time t equals zero (t 0 = 0), the initial speed equals zero (v 0 = 0), the acceleration a is constant, and the ship will reach it maximum speed (v max ) just before its engines are turned around to decelerate halfway to its destination. Slowing down with constant acceleration (deceleration) this way will bring to ship to rest (v f = 0) at its chosen destination/distance. Representing the farthest distance reached by the ship with Χ, the Earth time to reach that distance T, and the Ship time to reach that distance T', these quantities can be expressed as functions of x:

The midway-engine-turnaround distance (x) as a function of time (farthest distance, x max = 2x): x = (c2/a)[√(1 + a2t2/c2) - 1] → farthest distance = x max = 2(c2/a)[√(1 + a2t2/c2) - 1] Rearranging and solving for the Earth time (t) and ship time (τ) as functions of x: t = √(x2/c2 + 2x/a) → Earth time to farthest distance = T = 2√((0.5Χ)2/c2 + Χ/a), τ = c/a sinh-1[√(a2x2/c4 + 2ax/c2)] → Ship time to farthest distance = T' = 2c/a sinh-1(a/c T) = 2c/a sinh-1[a/c √((0.5Χ)2/c2 + Χ/a)].