I thought I might do a smaller/quicker post about Bob’s trip from Earth to Saturn before he heads out to Epsilon Eridani. Since Bob isn’t moving very fast, no relativistic physics is required, you can get away with go ol’ Newtonian mechanics (kinematics really, we aren’t considering forces here).

The relevant excerpts are:
Chapter 13: Bob – August 17, 2133 – Enroute

“The side trip would take a bit over six days at a constant two-g acceleration”

…

“I was travelling at over 5000 km/s by the time I reached the second-largest planet in the solar system.”

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

The trickiest part of this calculation is to determine the distance from the Earth to Saturn in August of 2133. The Jet Propulsion Lab (JPL) puts out large data sets, called ephemerides, that provide high precision location and motion information for the planets and their moons (among other data). I used DE440, which is the most recent dataset, published in February of 2021. This article in The Astronomical Journal discusses the dataset and is pretty dense. Here’s a download link to DE440 and some other datasets.

Once you have the dataset, you need to figure out how to use it to calculate the data point you are interested in. Fortunately, there is an excellent Python module called Skyfield that makes it very easy to load these datasets and then use them to calculate things like the distance between planets at any given time covered by the dataset.

Here’s the Python code I used to calculate the distance to Saturn for April of 2133:

You need to setup your Python environment properly (with Skyfield installed), after that it should be pretty straight forward. Here’s a link to the code on Github.

The output of the script looks like this:

planet,date,distance
Saturn,2133-08-17,9.810694167148197
Saturn,2133-08-18,9.827132134235871
Saturn,2133-08-19,9.843605975500804
Saturn,2133-08-20,9.86011150432379
Saturn,2133-08-21,9.876644525518076
Saturn,2133-08-22,9.893200822811535

The distances are measured in astronomical units (AUs) and I ended up just using the last one, 9.893200822811535. The definition of an AU can be found here:

1 AU = 149597870700 meters (m)
9.893200822811535 AU = 1480001777500 m

Here's the setup I use to solve standard kinematics type problems:
d = 1480001777500 m
a = 2g = 2 * 9.80665 = 19.6133 m/s/s
t = ?
vi = 0 m/s
vf = ?

We determined the distance to Saturn for the correct date/time, we are told Bob was accelerating at a constant 2g, and I’m assuming he started from zero initial velocity when leaving Earth. This is not exactly true, as we know Bob was trying to avoid the missiles but we don’t know anything about the actual vectors and speeds involved in that, so this was easiest and won’t have a big impact on the final answer. The kinematics equation that relates d, a, and solves for t is:

 (1)    
Subbing in the values above gives us:
 (2)    
Solving this for t:
 (3)    
 (4)    

Here’s the same treatment for the variables d, a, and :

 (5)    
Subbing in the values above gives us:
 (6)    
Solving this for :
 (7)    
 (8)    

Hmm. While Dennis’ values are certainly in the same ballpark as the ones calculated above, there is some discrepancy. There are any number of explanations as to why that might be, so I’m not going to pick this apart any further.

Updated:

Here is a plot showing the positions of the planets for the same timeframe (Apr 17, 2133). Not sure if Dennis planned it this way but Saturn is in a pretty favorable position.

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 away with an error of only . Hmm, this wouldn’t get you to Dennis’ 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)    
Ship's time to the star system:
 (2)    
Speed at the midpoint to the star system:
 (3)    
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 (). 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 
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 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 🙂

Welcome to The Bobiverse – Explained. I, like you, am a big fan of Dennis E Taylor’s Bobiverse books, well, the first three at least. Since Dennis writes in the genre known as hard science fiction, I thought it would be fun to write a series of blog posts that explore the science behind the novels and to provide some detail by showing calculations for some of the values Dennis describes (e.g. how long it takes to get to a certain star system). Most of these posts confirm of Dennis’ accurate work although I will occasionally raise some questions regarding some of the science. Future posts may veer off into more conjecture but I’ll try to keep those to a minimum.

Please note this is not meant to be a critique or criticism of Dennis’ work in any way, I LOVE these books, have read them MANY times, and continue to reread them. It’s simply an exercise for me to brush up on some Physics and has been something I’ve been thinking of doing ever since a made a post in the Fans of the Bobiverse Facebook group (please note, this is a private group) about Dennis’ use of tau ( ).

I don’t plan to cross post these on FB except for this introduction with some slight modifications, so if you’d like to comment, discuss or have a suggestion, please feel free to do so on any of the posts. Also, if have a question about some aspect of the science in any of the books, add a comment or contact me via the Contact page and I’ll do my best to answer.

I’d like to write one new post per week but we’ll see how that goes. And now, on to the first post, examining Bob’s flight to Epsilon Eridani.

NB: I am uploading most code found in the subsequent posts to GitHub, which can be found here.