Orbit, the board game
Jun. 28th, 2006 10:24 pm![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
bnewmanI've been musing for a while about designing a tabletop orbital simulator — a board game system for Newtonian gravitational motion around a single, massive object (i.e. the Sun). I can think of a couple of different ways to do it.
The first way is to extend the simplest system for Newtonian motion without gravity: Each object is represented by two tokens, representing its position at successive moments. The displacement between them is the object's velocity. An object moves by "leapfrogging" these two tokens — delta-V can then be applied to the token just moved. This works with both gridded and ungridded boards.
However, with gravity, you must also apply acceleration due to gravity every turn. In this case, the solution is to rule concentric circles on the board whose separation is equal to the acceleration due to gravity at a given distance (relative to the time step). Time- and space- resolution differences between regions (things happen much faster towards the center of the board) can be resolved by assigning different time steps to different regions.
I like this system, because it uses space (on the board) as space — thus, you can tell when two objects are close enough to interact by looking at the board. This is, of course, very important. However, I have an idea for another system which is more elegant in a certain esoteric sense, and I wonder if there is any reasonable way to implement it.
An object moving in two dimensions occupies a phase space of four dimensions — two of position, and two of velocity. In the above system, all four must be updated every turn. However, three values determine an orbit — total energy, total angular momentum, and azimuth at perihelion. The fourth value required to fix an object simply locates it along its orbit. If we could use this coordinate system, we'd only need to update one value for an object on any turn in which it didn't interact.
Unfortunately, this coordinate system makes it a bit tricky to determine when two objects are close enough to interact with each other. Every fix I can think of involves representing space as space, as in the simple system, and representing the phase space in question as some kind of chart. Most reasonable ways to do this are equivalent to the first system. Oh, well.
The first way is to extend the simplest system for Newtonian motion without gravity: Each object is represented by two tokens, representing its position at successive moments. The displacement between them is the object's velocity. An object moves by "leapfrogging" these two tokens — delta-V can then be applied to the token just moved. This works with both gridded and ungridded boards.
However, with gravity, you must also apply acceleration due to gravity every turn. In this case, the solution is to rule concentric circles on the board whose separation is equal to the acceleration due to gravity at a given distance (relative to the time step). Time- and space- resolution differences between regions (things happen much faster towards the center of the board) can be resolved by assigning different time steps to different regions.
I like this system, because it uses space (on the board) as space — thus, you can tell when two objects are close enough to interact by looking at the board. This is, of course, very important. However, I have an idea for another system which is more elegant in a certain esoteric sense, and I wonder if there is any reasonable way to implement it.
An object moving in two dimensions occupies a phase space of four dimensions — two of position, and two of velocity. In the above system, all four must be updated every turn. However, three values determine an orbit — total energy, total angular momentum, and azimuth at perihelion. The fourth value required to fix an object simply locates it along its orbit. If we could use this coordinate system, we'd only need to update one value for an object on any turn in which it didn't interact.
Unfortunately, this coordinate system makes it a bit tricky to determine when two objects are close enough to interact with each other. Every fix I can think of involves representing space as space, as in the simple system, and representing the phase space in question as some kind of chart. Most reasonable ways to do this are equivalent to the first system. Oh, well.
