To determine how fast you must travel to reach Sirius, which is about 9 light-years away, in 150 years of ship time, we can use the concept of time dilation from Einstein’s theory of relativity.
First, we need to find the speed required for the journey. The distance to Sirius is 9 light-years. If we want to get there in 150 years of ship time, we need to calculate the necessary speed.
The formula to calculate speed is:
Speed = Distance / Time
In this case:
Distance = 9 light-years
Time = 150 years (ship time)
However, due to the effects of relativity, the time experienced on the spaceship and the time in the rest frame (Earth) will differ. We’ll need to use the Lorentz factor (γ) to account for time dilation:
γ = 1 / √(1 – v²/c²)
Where v is the velocity of the spaceship and c is the speed of light.
We rearrange the time experienced on Earth:
Time (Earth) = γ × Ship Time
Let’s denote…
Earth Time = γ × 150 years
We also know:
Earth Time = Distance / Speed
Putting this together:
9 light-years / v = γ × 150
Solving this leads to:
v = 9 light-years / (γ × 150)
We also have the relationship for γ, so let’s substitute and solve that. The actual calculations can be quite complex, but generally, you would find that to reach near light-speed (around 0.995c) is necessary to make this journey feasible within the time frame given.
Indeed, the exact speed will depend on the calculations which factor in relativistic effects. However, reaching up to 99.5% of the speed of light would allow you to travel that distance quickly while experiencing only 150 years of time on the ship.
This means, essentially, that traveling at such high speeds would allow for significant time compression relative to an Earth observer.