This is the fourth in my series of killing time at work using math (here are the others 1, 2, and 3).

As before, there are a few prerequisites for this post:

• You like recreational math.
• You work at a computer and have access to Microsoft Excel.
• You are bored out of your tree.
• You don't want to get caught slacking off.

Travel to the stars has long been a dream of humanity but we don't really yet have the technology to do that just yet.

This post is a calculation of how long it would take to get there with just a few rules:

1. We will be using real physics as we currently understand it. This means that the speed of light cannot be violated so there will be none of that warp speed nonsense.
2. Humans have developed the tech to accelerate at 1 gravity ('1 g' or 9.8 m/s²) for as long as they want.

So I would say that this post falls under the category of hard science fiction.

Setting Up The Spreadsheet

The spreadsheet we need to do this calculation will look like the one shown above. I will go through each cell one by one and explain what each calculates.

Column B - Time As Seen By Earth (seconds): Start at time = 0 and then simply increment it to simulate the time steps. Increment slowly at first and then makes those increments larger and larger as the time value gets larger. The amount you increase the time steps is at your preference.

Column C - Time As Seen By Earth (days): This is the same thing except it is Column B divided by 86,400 to get the time in days (24 hours x 60 minutes x 60 seconds). To save space and because it is so simple I don't need to show you how to do that.

Column D - Time As Seen On The Ship (seconds): Time on the ship will become dilated according to the rules of Special Relativity. I did a post on that about a month ago and it is here: Intuitive Special Relativity - Time Dilation.

In essence the clocks on the ship will slow down and the time increment will decrease by a factor called the Lorentz term. That term is calculated in Column M which I will be getting to.

Column E - Time As Seen On The Ship (days): This is just Column D divided by 86,400 to get the elapsed time in days.

Column F - Acceleration As Seen By Earth (m/s^2): As the title says, this is just the acceleration as seen by observers on the Earth. The value of the acceleration will be 9.8 times the reciprocal Lorentz factor to account for the effects of special relativity (ship mass increases as you approach light speed). As the ship velocity increases the acceleration will appear to be less as seen by Earth.

The occupants on the ship however will see a steady and comfortable Earth Standard of 1 g where 'g' stands for gravity. On Earth this is 9.8 m/s².

Column G - Velocity Increment (m/s): The velocity as seen on the ship will increase with each time step. This spreadsheet is set up so that you can choose whatever time steps you want so I formulated it to adapt for that.

Column H - Composition Of Velocities (m/s): This is the heart of the Special Relativity aspect of the spreadsheet. If you go to the Wikipedia page on Special Relativity they have conveniently provided this equation for us:

Column I - Velocity As A Fraction Of c (m/s): This is simply Column H divided by 299,792,458 m/s. The term 'c' is what STEM types use to describe light speed.

Column J - Distance Traveled (m): This computation uses discrete mathematical techniques in a simple way. To calculate the distance traveled so far, just take the distance from the previous time step and add to it the average velocity over the current time step times the amount of time elapsed in the current time step. Not perfect but good enough for this simple calc.

Column J - Distance Traveled (ly): This is the distance traveled in light years ('ly'). This is just Column J divided by the number of metres in a light year (1 ly = 365 x 24 x 60 x 60 x 299,792,458 = 9,454,254,955,488,000 m).

Column L - Speed of Light (m/s): Just set this column to 299,792,458 which is the speed of light in m/s.

Column M - Reciprocal Lorentz Factor: This is a unitless term and it calculates the time dilation on the ship. Here is the equation (taken from Wikipedia)

The Results

The whole idea is that you accelerate your ship at 1 g for one half of the trip and then at the half way point you turn it around and decelerate it at 1 g for the remaining half of the trip. The screenshot below gives the results of my calc.

Alpha Centauri is about 4.3 light years away so we only have to calculate one half of the trip (2.15 light years) and then double the resulting time.

As you can see from the figure above, Column C tells me that the time to get halfway (2.15 ly) is 1082.521 days as seen by Earth but Column E tells me that it was only 650.240 days as seen by people on the ship.

Doubling each number gives a total trip time of:

• 2165 days or 5.9 years as seen by Earth.
• 1300 days or 3.6 years as experienced by everyone on the ship.

Not bad, only 3.6 years to get to the nearest star system.

But Wait, There's More!

If you extend this calculation out to 5 light years (which means a total trip distance of 10 light years) we find that the total trip time is:

• 4300 days or 11.8 years as seen by Earth.
• 1773 days or 4.8 years as experienced by everyone on the ship.

So, once you get up to near-light speed then the phenomenon of time dilation makes the extra time to further distances really negligible.

In this case it takes only 1.2 more years (ship time) to get another 5 or 6 light years further.

The Time to 100 Light Years

If you extend this calculation out to 50 light years (which means a total trip distance of 100 light years) we find that the total trip time is:

• 37201 days or 101.9 years as seen by Earth.
• 3296 days or 9.0 years as experienced by everyone on the ship.

This is amazing, it takes only another 4.2 years (ship time) to get another 90 light years.

Of course everyone you knew on Earth is dead but hey you are a colonist on a New Earth so who cares (right?).

Spreadsheet Validation

Double checking my results against the relativistic rocket calculations (here and here) I see that they get 3.6 years to get to Alpha Centauri at an acceleration of 1 g for the first half the trip and a deceleration of 1 g for the second half of the trip.

This agreement lends confidence that the calculations in the spreadsheet are valid and can be used in whatever science fiction calculation you wish to make.

Closing Words

So this calculation will not only waste your time productively at work but it will also teach you some basic facts about Special Relativity and time dilation.

Maybe one day humans will have developed compact and powerful fusion reactors or they can bottle antimatter for their power source and also have some really kick-ass ion drive engines that can get large ships up to 1 g accelerations.

When that happens there is no reason that we won't be able to set out and explore our galaxy and make it to the stars before the occupants die.

Postscript: My original version of this article neglected to adjust the acceleration (Column F) by the reciprocal Lorentz factor. I fixed that and this brings my calc in alignment with all of the other ones posted on the internet and cited below. Special Relativity is subtle indeed.

Thank you for reading my post.

Post Sources

Special Relativity (SR)
Composition of velocities in SR
Lorentz factor
Speed of Light
Wikipedia article on the topic
Relativistic Rocket
YouTube Video on same topic
Stack Exchange thread on same topic
Distance to Alpha Centauri
Online Calculator