Post 1b – Bob’s Trip From Earth to Saturn

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.

NB: The scale for Mercury and Venus is a little janky in order to make the Sun noticeable. Distances are accurate from the Sun’s centre.