Post 1 – Bob’s Trip to Epsilon Eridani

I’m going to keep this first post pretty simple. It’s based on these two excerpts:

Chapter 13: Bob – August 17, 2133 – Enroute

“Epsilon Eridani is 10.52 light-years away from Sol. The specs indicated that the ship could run at 2g indefinitely with no ill effects, which would get me to my target star in a little over eleven years.”

“With a mental sigh, I adjusted my heading for Epsilon Eridani and cranked the drive back up to 2 g. The trip would take just under eleven and a half years to the universe at large, but only three years ship’s time. At midpoint, I would be travelling within a hair of light speed.”

Excerpts From: We Are Legion (We Are Bob) Copyright © Dennis E. Taylor 2016

I don’t know what database or star catalogue Dennis used for his data, simply looking up Epsilon Eridani in Wikipedia provides a huge amount of astronomical data, nicely summarized down the right sidebar. That page lists Epsilon Eridani as being \text{10.475 lightyears (ly)} away with an error of only \pm \text{0.004 ly}. Hmm, this wouldn’t get you to Dennis’ \text{10.52 ly} but that doesn’t really matter given the accuracies we are dealing with in the book. There is also the question of what values Dennis used for constants, like the speed of light, c. I’ll create a post of constants and astronomical data so you know where mine came from.

So how does Dennis come up with “just under eleven and a half years” for a stationary observer, like someone on Earth and also “three years ship’s time”?

First, let’s talk about how one gets to a star system. There are basically two options:

  1. Accelerate your ship to its maximum velocity, cruise along at that maximum velocity for most of your voyage, then decelerate to arrive at your destination, at some relatively small velocity.
  2. If your ship does not have a maximum speed and is able to accelerate at some constant value indefinitely, then you just keep accelerating until you reach the midpoint of your journey, then start decelerating until you reach your destination.

Option 1 is how most of us take trips and it’s also the way NASA uses for the probes it sends to other planets. However, based on the excerpt above, it appears the Bob’s are able to use method 2. Dennis never talks about a limit to the speed a HEAVEN vessel can attain (or maintain). The challenges associated with this may be discussed in a future post but for now, we’re going to assume that we can accelerate for the first half of the trip, then decelerate for the second half.

Okay, so how do we do this calculation. You may have heard of Special Relativity, proposed by Albert Einstein in September of 1905. One aspect of Special Relativity is called time dilation, which states that time slows down when you move at very high speeds (a significant portion of the speed of light). Special Relativity typically deals with “inertial reference frames”, which are situations where objects are traveling at CONSTANT speed, not accelerating. When you have a spaceship like HEAVEN 1, it accelerates at a constant rate of 2g (2 times the rate of gravity), at least that is what the excerpt tells us. Evolving Special Relativity’s equations to work with constant acceleration involves calculus and hyperbolic functions and is beyond the scope of what I’m looking to do here. For those interested, here’s a paper that shows the derivation of uniform relativistic acceleration.

These equations are summarized on a great webpage, the Relativistic Rocket. I will use three of the equations found on this page to calculate the relevant values Dennis talks about. These equations are:

Earth time to the star system:

(1)   \begin{equation*}   t = \sqrt{\left(\frac{d}{c}\right)^2 + \frac{2d}{a} \end{equation*}

Ship's time to the star system:

(2)   \begin{equation*}   T = \frac{2c}{a} cosh^{-1} \left( \frac{ad}{2c^2} + 1 \right) \end{equation*}

Speed at the midpoint to the star system:

(3)   \begin{equation*}   \upsilon = \frac{at}{\sqrt{1 + \left(\frac{at}{c}\right)^2}} \end{equation*}

Where the various values are: d = distance to the star system c = the speed of light a = the ship's acceleration t = stationary/earth time to the star system (from equation 1)

There are two tricks to actually doing these calculations, one is to make sure you get the units for the various values correct. If we use lightyears (ly) for distance and years (y) for time, then speed will be lightyear / year (ly / y) and acceleration will be lightyear / year^2 (ly / y^2). The other is to realize that for the earth time equation, that’s the time if we accelerate all the way there. In order to stop at the star system, we have to calculate the earth time to the midpoint and then double that time to get the total time to the system. This is not required in the ship’s time calculation, as it was already taken into account in the formula.

Please note: a lightyear is a measure of distance not time. It’s the distance light travels in one year.

Using the units outlined above, these values become:
d = 10.475 ly
c = 1 ly / y
a = 2g = 2.0645816 ly / y^2
t = 11.4026586 y

I was going to write out the above equations, substituting the values above but writing out equations in WordPress is clumsy at best, so I don’t really feel like doing that. You could plug those values into your calculator, that works. To calculate the results here, I decided I’d write a small python script and embed it below. The best part about this solution is the code is interactive, so you can play around with the values, like the ship’s acceleration, to see what effect that has on the results. Any changes you make aren’t saved, so if you change some things and want to go back to the original values, click on the menu button on the top left and then select “Reset”, then click “Yes, I Am sure”. Clicking the play button below runs the code. Let’s do that now.

NB: The above code is now available on GitHub here.

Here's a summary of the results, rounding to 3 decimal places:
Earth time: 11.403 years
Ship's time: 3.062 years
Speed at midpoint: 0.996 fraction of c

Comparing the results to the Dennis’ excerpt above, you can see that his values are right on the money.

These formulae and the python script can be used to calculate times and speeds for any of the trips Dennis outlines in the book, just by updating the distance and ship’s acceleration. For example, the colony ships travel from Earth to \text{Omicron}^2 Eridani (also known as 40 Eridani), which is a distance of 16.34 ly and their acceleration is only 1g. If you plug these into the script above, you get:

Earth time: 18.174 years
Ship's time: 5.686 years
Speed at midpoint: 0.994 fraction of c

These numbers line up reasonably well with Dennis’, given here:

Chapter 58: Riker – April 2171 – Sol

“The colony ships had a maximum sustained acceleration of 1 g, so the trip would take slightly longer than it would have for a version 1 Bob. They would be on the road for a little over eighteen years. About six years would pass on-board, but no time at all for the colonists in their stasis pods.”

Excerpts From: We Are Legion (We Are Bob) Copyright © Dennis E. Taylor 2016

That’s all for this post. Let me know what you think. See you in the funny papers 🙂

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