The 3 Body Problem

In the realm of gravitational interactions, two bodies (like a binary star system) allow for analytic solutions to their motion equations. With known initial conditions, one can predict the future positions and velocities—this is referred to as the 2 Body Problem.

However, when three objects interact—say, star 1, star 2, and star 3—the complexity increases. Each star influences the others, leading to an intricate web of forces. Unfortunately, no analytical solution exists for this scenario; it remains unsolvable in equation form.

Instead, we can tackle the 3 Body Problem using numerical methods. By segmenting time into small intervals, we can compute gravitational forces, assuming they remain constant during these brief periods. With this approach, we can track how the momentum and positions of the three stars evolve, repeating the process iteratively.

While each calculation is straightforward, the sheer volume of computations required is substantial. To manage this, computers handle the heavy lifting—at least they don’t complain.

Here's a detailed numerical solution to the 3 Body Problem.

A Stable 3 Body Solution

Generally, numerical simulations are required to predict the motion of three objects. Yet, specific configurations can yield standard solutions. We need to introduce some fundamental physics concepts first.

The gravitational interaction between two masses can be expressed as follows:

Where M1 and m2 are masses separated by a vector r, G is the universal gravitational constant, and r-hat denotes a unit vector. If multiple masses are involved, the total force is the vector sum of each individual force.

Considering circular motion, the simplest arrangement for a tri-solar system involves circular orbits. An object moving in a circle experiences an acceleration directed towards the center, quantified by:

Assuming equal mass for all three stars, one can envision a scenario where one star remains stationary while the other two revolve around it.

The middle star experiences gravitational forces from both orbiting stars, which ideally balance each other out, keeping it at rest.

Conversely, star 2 feels two gravitational pulls toward the left, resulting in a net force that causes it to accelerate. We can determine the necessary speed (v) for it to maintain a circular trajectory, considering that star 1 is at a distance r from star 2, while star 3 is positioned at 2r.

With the orbit size (r) known, we can calculate the velocities of both star 2 and star 3.

Will this configuration hold? We can test it out. Below is a numerical simulation involving three stars of equal mass, where the central star starts at rest and the others at their calculated velocities. The full Web VPython code is available, and the animation depicts their motion:

It’s important to note that the tri-solar system is inherently unstable—this instability is a key challenge for the inhabitants of Trisolaris. Yet, within the narrative, the stars seem to achieve this arrangement.

Orbit of the Trisolaris Planet

Introducing a planet into this scenario does not transform it into a "four body problem," as we can assume its mass is negligible compared to the stars. Thus, the gravitational force exerted by the planet on the stars will be minimal, allowing for easier modeling.

From the planet's viewpoint, the three stars should remain aligned. This alignment occurs if Trisolaris matches the angular velocity of two stars, with the central star remaining stationary.

This schematic is not to scale. The planet experiences three gravitational forces from the stars, meaning its acceleration is influenced by its angular velocity (?) and orbital distance (r_p).

In this configuration, the gravitational forces align with the acceleration, allowing us to express Newton’s second law as follows (with M representing star mass and m for the planet):

The planet's angular velocity can be derived from one of the outer stars' angular velocities.

Despite the planet's mass (m) canceling out, solving for the orbital distance (r_p) proves complex. However, we can obtain a numerical solution by plotting both sides of the equation against varying values of r_p, identifying their intersection as the solution.

By experimenting with the scale, I propose an orbital distance of 0.21065 AU for the planet. Running the 3 Body Problem calculation with this fourth object yields the following results:

This illustrates the planet's trajectory, mostly aligning with the stars' rotation, although it drifts away toward the end—an expected outcome, as the tri-solar syzygy is not perpetual.

Modeling the Tri-Solar Syzygy

What causes beings (not actual humans) to be lifted from Trisolaris' surface? The hypothesis is that the combined gravitational forces from the three stars exceed the downward gravitational pull towards the planet.

Assuming the planet resembles Earth, the gravitational force would be approximately 9.8 N/kg for an entity standing on its surface. Meanwhile, the net force from the three stars would be about 1.5 N/kg—less than Earth's gravitational pull.

However, this difference doesn’t translate to a sensation of reduced weight on Trisolaris. Why? Because the planet itself is accelerating towards the stars as it moves in a circular trajectory.

We can still assess the cumulative gravitational influence of the three stars. The interaction with these stars results in an upward force on the planet, causing it to accelerate. However, this acceleration pertains to the planet's center rather than its surface. An entity on the surface experiences the planet's center acceleration while feeling a gravitational force closer to the stars.

The perceived gravitational force at Trisolaris' surface equates to the difference between the gravitational field at the center and that at the surface. Assuming a radius of 6.3 x 10^6 meters (similar to Earth), the effective upward force from the stars would be approximately 0.00074 N/kg.

Thus, a being weighing 70 kg would typically experience a weight of 686 N, but during the tri-solar syzygy, this would drop to 685.99 N—a negligible difference.

Interestingly, if a being were positioned on the opposite side of the planet, they too would experience a lighter sensation. Although the gravitational force from the stars would slightly decrease, the planet's center acceleration would create the illusion of it moving away from them.

This phenomenon parallels why Earth experiences two tidal bulges. The center of the Earth accelerates due to the moon's gravitational pull, resulting in two distinct water bulges rather than one. Below is a visualization comparing surface acceleration to the actual gravitational field on Earth.

For those interested in the details behind this visualization, further information is available.

Visualizing Tidal Forces With Python

As a final note, it's essential to remember that The 3 Body Problem is a work of fiction, adapted into a Netflix series. While the physics may not be flawless, the concepts explored are nonetheless thought-provoking, and the parameters chosen appear reasonable.