The Mathematics of Interstellar Space Travel: Voyager

One of the key equations needed to understand spacecraft navigation is Newton’s inverse square law of gravity:


We learned this in Physics 101 in the university as engineers. Essentially, the force measured (F) is gravity’s (G) work attracting two masses (m1, m2) that weakens/strengthens as the square of the distance between those two bodies changes. So gravity’s attraction effect is virtually infinite according to the inverse square law. The Sun’s mass is pretty huge.

Grasping this law we can further derive equations which describe the motion of the sun, the planets and the Voyager spacecraft flying between them. These equations are fairly trivial if we only consider two bodies—the Earth and the Sun. So we can find a simple solution which predicts exactly where the Earth and Sun will be at any point in time given information about their positions and velocities at some starting point. This two body problem, as it is called, is well known and fairly straightforward, but not very practical as the whole picture emerges.

Now, to simulate the reality of what Voyager will actually experience on its trip through and beyond the 11 billion miles thus far travelled, we add a third body to our equations of motion. The Voyager spacecraft, moving somewhere between and beyond the Earth and the Sun, will encounter other planets, moons and bodies as well and now we no longer have a simple analytical solution. The equations are now unsolvable! This problem is known as the three body problem —one of the most difficult problems in all of celestial mechanics.

For Voyager to travel past Neptune, for example, it must escape the gravitational pull of the Sun which has a mass so large that it still has an effect on the spacecraft so many billions of miles away. That trip would usually demand a very large rocket engine with lots of fuel (Remember that it took lots of this precious fuel, which is also added weight, just to escape Earth’s gravity pull). A flight past Neptune, more than four billion km (2.5 billion miles) away – could easily take 30 or 40 years because Voyager’s rocket engines are fighting against the tremendous forces of gravity from the Sun and the neighboring planets it will pass.

At the time Voyager was developed, NASA only had the technology to propel a spacecraft into space with, at most a few months of operational life, and so the outer planets (no less leaving our solar system) were considered out of reach.

That was an accepted fact until a 25-year-old mathematics graduate named Michael Minovitch accepted the challenge to understand and change this situation in 1961.

Minovitch was enthralled with UCLA’s new IBM 7090 computer, the fastest on Earth at the time, so he decided to take on the daunting “three body problem” of the Earth/planets, Sun and Voyager reacting in space with this new tool.


Jupiter is the largest planet in our solar system and exerts the strongest gravitational pull toward another body because of its mass

The problem is predicting exactly how the gravity of the Sun and a planet will influence Voyager’s trajectory on its journey out of our Solar system.

Renowned astronomers have struggled with this problem for at least 300 years, back when they tried plotting the path that comets took as they sped through our Solar System towards the Sun.

Some of the brightest scientists at the time, like Sir Isaac Newton, hadn’t been able to solve this complex problem. Minovitch had access to the IBM 7090 computer, unlike Newton and others, and endeavored to use the method of mathematical iteration to craft a solution that was not realizable without the processing power of such a computer.

In the 2-body problem, we are able to develop a set of equations that describes the orbit of two bodies in space. For example, the orbit of a planet around the sun is in the form of an ellipse, like a conical section similar to a skewed circle.

Keplers three laws of planetary motion2 are the basic theory behind the two body problem. First and foremost of these laws is that planets orbit the sun in an elliptical shape, not a circle, as was previously thought. The other two laws of planetary motion describe numerically how the orbits behave.

All bets are off for the 3-body problem. Minovitch understood that the only way to completely describe a system of 3 bodies is to observe how their behavior unfolds. This is where the number-crunching power of the IBM 7090 came into play.

Minovitch fed rough observed data on planetary orbits into his model at first and he had a data point. He later persuaded his boss at JPL to give him more accurate data on planetary positions to refine his model. After running the iterative1 simulations again, his solution still worked. This young novice had now made an extraordinary breakthrough in spacecraft propulsion.

The result showed that as a craft flew close to a planet orbiting the Sun; it would acquire some of that planet’s orbital speed, and be sling-shot away from the Sun. Such acceleration did not have to use a single drop of rocket propellant—an amazing discovery that changed the face of space travel and exploration!


 The domain of the Sun’s influence is called the heliosphere: Both Voyagers are approaching the edge of this enormous balloon of charged particles thrown out into space by our Sun. (Image Courtesy of NASA)

After much doubt by Minovitch’s peers, in 1965, another summer intern named Gary Flandro, a spacecraft engineer, familiar with the intricacies of spaceflight appeared on the scene. Flandro knew that any mission to the outer planets would have to be flown as fast as possible; otherwise the craft might not last long enough to reach its destination. He drew graphs of where the planets were going to be in future years and found that Jupiter, Saturn, Uranus and Neptune would all be on the same side of the Solar System in the late 1970s.

Using a solution to the three-body problem, a single mission, launching from Earth in 1977, could sling-shot a spacecraft past all four planets within 12 years. Such an opportunity would not present itself again for another 176 years. Time was of the essence so as to not miss this window.

With further lobbying from Minovitch and high level intervention from Maxwell Hunter, who advised the President on US space policy, NASA eventually embraced Minovitch’s slingshot propulsion and Flandro’s idea for a “grand tour” of the planets.


The three body solution was only made possible using the most powerful computer of the day, the IBM7090 (Image courtesy of NASA)

Here is Voyager going through the “Magnetic Highway” towards the end of July 2012, a newly discovered region of our solar system. Here, it was discovered that the Sun’s magnetic field lines are connected to interstellar magnetic field lines. This allowed particles from inside the heliosphere to move away and particles from interstellar space to quickly fill into that void.

An old Star Trek episode has the Voyager (known as “Vger” since part of the name on the body of the craft had worn away) as a type of demi-god, stranded on a remote planet somewhere in the cosmos—still with its light flashing and some power remaining to keep it functional hundreds of years in the future.

As our imaginations conjure up exciting encounters that these amazing spacecraft might stumble upon, we come to realize the power of “digitally-assisted” mathematics with a computer and some innovative thinking coupled with tenacity.

  1. An “iterative process” is one in which the same rule is applied repeatedly, using the output as the input for the following iteration. For example, if we take the rule “add 1″, and apply it to 0, we get 1. If we iterate again, we get 2, then 3, and so forth.

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