# Gravity Wells

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## WHAT IS IT?

This model is part of the Gravitational Model Suite. This model specifically displays "gravity wells" a visual representation of gravitational fields. While the other models in the suite simulate gravity in 3d space, this model simulates in 2d space, and uses the z axis to display the magnitude of the gravitational field at every point in space.

## HOW IT WORKS

Every particle has mass, position, and velocity. At every time step, every particle pulls on every other particle with a gravitational force proportional to its mass and inversely proportional to the distance between the two particles. Each particle sums up these forces to calculate its total acceleration at every tick, and updates its velocity accordingly.

Uniquely to this model, the magnitude of the gravitational field is also calculated at every point in space, and displayed as the distance the mesh plane is distorted along the z-axis. This provides an intuitive view of the gravity of an object - the deeper of a distortion it makes, the stronger its pull, and being captured by its gravity is analogous to falling down its gravity well.

## HOW TO USE IT

The NUMBER-OF-OBJECTS slider determines how many particles are spawned. The STARTING-BOX slider defines the coordinates they can be spawned inside - a smaller value spawns them closer to the center. MAX-STARTING-VELOCITY allows them to have a randomly determined starting velocity, and MAX-STARTING-MASS defines a maximum mass for the particles to be randomly assigned. SETUP sets up the model, and GO starts or pauses it. G and SoI are for display purposes only - G simply scales the amount that the mesh field is distorted, so higher values of G result in deeper gravity wells. SoI (or "Sphere of Influence") is the distance that a particle will affect visual updates for the gravity wells, so turning down SoI makes the model run faster, but loses visual accuracy farther away from the particles. Note that SoI DOES NOT affect physics calculations at all, only the visual gravity well display. ZOOM-FACTOR simply displays the amount that the model is zoomed in or out.

## THINGS TO NOTICE

Notice that along an orbit, kinetic energy and potential energy are in opposition. When a particle reaches the highest point in its orbit (i.e., has the most potential energy), it is moving the slowest. When the particle reaches its lowest point, it is moving fastest. Conservation of energy states that a particle's total energy (kinetic + potential) must be constant unless it gains or loses energy by interaction with something else. Therefore, as it is moving through its orbit, it converts kinetic energy to potential energy and vice versa.

This also means that if (for example) a particle gains kinetic energy at the lowest point of its orbit, its highest point will correspondingly have more potential energy, which requires it to be farther away. In simpler terms, speeding up widens the opposite end of the orbit, and slowing down brings the opposite end closer. These are all emergent properties that result naturally from Newtonian laws of gravitation.

## THINGS TO TRY

Try messing with G and SoI to make the gravity wells more apparent and visually interesting. Also try letting particles come near each other and seeing how their gravitational fields interact.

## EXTENDING THE MODEL

One of the biggest problems with this model is the simple fact that updating the mesh to display gravity wells is very computationally expensive and takes a long time. There are a bunch of ways that this could be made more efficient - the "sphere of influence" simplification is a pretty naive and simple one. It would also be possible to group together masses that are far away and treat them as one - this would simplify calculations a bit without completely disregarding things beyond an arbitrary distance. Or instead of doing calculations for every single point on the mesh, maybe a subset could be picked and the other point positions could just be extrapolated from there. However, while speeding this up would be great, I decided that accuracy is the #1 priority. This is a model, not a video game.

## NETLOGO FEATURES

One workaround that was used was "zooming" in and out to compensate for the fact that Netlogo doesn't support resizing the world. So when a particle tries to go outside the bounds of the world, rather than making the world bigger, I just scaled down all the positions. This is why every turtle has a separate x-pos, y-pos and z-pos defined in addition to the default xcor, ycor and zcor; the new -pos variables are not scaled down and are used in all physics calculations, while the default cor variables are modified by the global zoom-factor variable and are used for displaying turtles. As a result, the world can effectively become bigger and smaller as needed.

## RELATED MODELS

This model is part of a suite of models, along with the Basic Gravitation model and the Orbital Playground model. The code for the gravity well mesh was also heavily inspired by the code from the Hill Climbing Example 3D model in the Models Library.

## CREDITS AND REFERENCES

http://modelingcommons.org/browse/one_model/4670

Wilensky, U. (1999). NetLogo [computer software]. Evanston, IL: Center for Connected Learning and Computer-Based Modeling, Northwestern University. http://ccl.northwestern.edu/netlogo .

## Comments and Questions

globals [ zoom-factor ] ;; used to "zoom" by scaling the displayed positions of particles breed [ stayers stayer ] ;; used as points to form the mesh that displays the gravity wells breed [ particles particle ] ;; particles that interact and exert gravitational forces upon each other turtles-own [ x-pos y-pos z-pos ;; used as the basis for physics calculations - xcor, ycor, zcor only used for display ] stayers-own [ gravity ;; the magnitude of the total gravitational field at a given point ] particles-own [ x-velocity y-velocity ;; components of 2d velocity, used for physics calculations mass ;; also used for physics; the greater the mass, the greater the gravitational pull ] to setup clear-all ;; create stayers ask patches with [pzcor = 0] [ sprout-stayers 1 [ set gravity 0 create-links-with turtles-on neighbors set x-pos xcor set y-pos ycor hide-turtle ] ] set-default-shape turtles "circle" set zoom-factor 1 create-particles number-of-objects [ setxyz (random-float starting-box - random-float starting-box) (random-float starting-box - random-float starting-box) 0 set x-velocity ((random-float max-starting-velocity) - (random-float max-starting-velocity)) set y-velocity ((random-float max-starting-velocity) - (random-float max-starting-velocity)) ;set z-velocity ((random-float max-starting-velocity) - (random-float max-starting-velocity)) set mass random-float max-starting-mass set size mass ^ (1 / 3) set x-pos xcor set y-pos ycor set z-pos zcor ] compute-gravity-wells set-particle-display-positions reset-ticks end to go compute-acceleration-changes check-boundaries move-particles check-slide compute-gravity-wells set-particle-display-positions tick end to tick-once compute-acceleration-changes check-boundaries move-particles check-slide compute-gravity-wells set-particle-display-positions tick end ;; Update positions of stayers to correspond with gravitational field to compute-gravity-wells ask stayers [ set gravity 0 set x-pos xcor * zoom-factor set y-pos ycor * zoom-factor ] ask particles [ ask stayers with [distancexy [xcor] of myself [ycor] of myself < SoI] [ let xdist [x-pos] of self - [x-pos] of myself let ydist [y-pos] of self - [y-pos] of myself let scale-const max list ((xdist ^ 2 + ydist ^ 2) ^ .5) ([size] of myself / 2) set gravity gravity + [mass] of myself * G / scale-const ] ] ask stayers [ set z-pos gravity * -1 set zcor z-pos / zoom-factor ; set color scale-color red zcor 0 -50 ] end to set-particle-display-positions ask particles [ set z-pos [z-pos] of min-one-of stayers [distancexy [xcor] of myself [ycor] of myself] set zcor z-pos / zoom-factor if not hidden? [ set xcor x-pos / zoom-factor set ycor y-pos / zoom-factor set size mass ^ (1 / 3) / zoom-factor ] ] end ;; Check center of mass of the overall system, adjust view accordingly to check-slide let CoM center-of-mass particles slide-view (item 0 CoM) (item 1 CoM) (0) end ;; Calculate the center of mass of the entire system to-report center-of-mass [turtleset] ; R = 1/M sigma (mi * ri) let M 0 let totalx 0 let totaly 0 ;let totalz 0 ask turtleset [ let mi [mass] of self set M M + mi set totalx totalx + [x-pos] of self * mi set totaly totaly + [y-pos] of self * mi ;set totalz totalz + [z-pos] of self * mi ] report (list (totalx / M) (totaly / M)) end to check-boundaries let flag false ask particles [ if abs (x-pos + x-velocity) > max-pxcor * zoom-factor * .8 or abs (y-pos + y-velocity) > max-pycor * zoom-factor * .8; or ;abs (z-pos + z-velocity) > max-pzcor * zoom-factor * .8 [ set flag true ] ] ifelse flag [ ;print "zooming out" set-zoom 1.1 ] [ ;print "zooming in" set-zoom .9 ] ask particles [ ifelse abs (x-pos + x-velocity) > max-pxcor or abs (y-pos + y-velocity) > max-pycor or abs (z-pos) > max-pzcor [ set hidden? true ] [ set hidden? false ] ] end ;; Have particles compute their acceleration based on gravitational interactions with all other particles to compute-acceleration-changes ask particles [ let turtle1 self ask other particles [ gravitationally-affect turtle1 self ] ] end ;; Move particles according to their velocity to move-particles ask particles [ set x-pos x-pos + x-velocity set y-pos y-pos + y-velocity ;set z-pos z-pos + z-velocity ] end ;; Compute gravitational pull between a pair of particles to gravitationally-affect [ affected affecting ] ; g = m1 * m2 / r^2 ; r = sqrt(xdist^2 + ydist^2) -> r^2 = xdist^2 + ydist^2 let xdist [x-pos] of affecting - [x-pos] of affected ;print xdist let ydist [y-pos] of affecting - [y-pos] of affected ;print ydist ;let zdist [z-pos] of affecting - [z-pos] of affected let total-acc 0 let xyangle 90 ;let zangle 90 let xydist (xdist ^ 2 + ydist ^ 2) ^ .5 if xdist + ydist > 0 [ set total-acc [mass] of affecting / (xdist ^ 2 + ydist ^ 2) let angle-heading atan xdist ydist set xyangle (90 - angle-heading) mod 360 ;set zangle (90 - z-angle-heading) mod 360 ] if total-acc > 50 [ set total-acc 50 ] set total-acc total-acc / 20 ;let xyacc total-acc * (cos zangle) ask affected [ set x-velocity (x-velocity + total-acc * (cos xyangle)) set y-velocity (y-velocity + total-acc * (sin xyangle)) ;set z-velocity (z-velocity + total-acc * (sin zangle)) ] end ;; Move view laterally to slide-view [deltax deltay deltaz] ask particles [ set x-pos x-pos - deltax set y-pos y-pos - deltay set z-pos z-pos - deltaz ] end ;; Change zoom-factor to zoom in or out to set-zoom [new-zoom] set zoom-factor zoom-factor * new-zoom end

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**Parent:** Gravitation

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