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State variables used in the differential equations The first thing we need to do is to declare an array that will store that values of those twelve u variables: // Current state of the system We are finally done with algebra and will turn our equations into code. We use this origin because, as it turns out, the center of mass of any number of bodies remains motionless or moves with constant velocity.Īgain, this looks horrific, but in the program this turns into a nice loop that takes care of repetitions. We will be using a coordinate system with its origin located at the center of mass of the three bodies, as shown in Fig. The simulation is so sensitive to small changes that it may even look different when you run it with the same settings but in different browsers. When the simulation is running on the computer, small calculation and rounding errors accumulate over time and make the system unpredictable. This is an example of how an orderly system can quickly become erratic. The simulation tells us that the system appears to be stable, at least in the short term. This is precisely how this planet was discovered. Consequently, when the planet moves in front of the stars, it blocks some of their light. The orbital plane of the planet is located edge-on to us. Both binary stars are smaller than the Sun. This is a simulation of a binary star system that also has a planet with a mass of 1/3 of Jupiter’s. However, the combined gravity from Jupiter and the Sun traps the Earth, and it is destined to remain at L5 point behind Jupiter. We can check this by decreasing Jupiter’s mass and clicking the Reload button on the bottom right of the simulation screen.
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Therefore, normally, the Earth would overtake Jupiter. Planets that are closer to the Sun have shorter orbital periods. Notice that the radius of the Earth’s orbit is smaller than that of Jupiter initially. Here the Earth is located near a special point in space called the Sun-Jupiter L5 Lagrange point. This means that the simulation is working correctly, because it is run at one year per second. We can measure one period of the Earth’s orbit in the simulation to be around one second (may depend on computer speed and refresh rate of the monitor). This simulation uses true masses, velocities and distances of the Sun, Earth and Jupiter. At certain speeds you will see weird stroboscopic effects. Just for fun, try increasing the speed of this animation by clicking the clock icon and moving the slider. The system remains stable even if we change the masses of all bodies a little bit, to 0.99 for example. This is a stable three-body system discovered by Cris Moore. The little buttons under the slider run the following simulations.
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As always, feel free to check the full source code, and use it for any purpose. This work is based on Rosmary’s ideas and code. I want to say huge thanks to Dr Rosemary Mardling, who taught me astrophysics in Monash University. This work is built upon the two-body simulation code. In this tutorial we will program motion of three bodies in HTML and JavaScript. Figure eight Sun, Earth and Jupiter Lagrange point L5 Kepler-16 Chaotic