breed [ planets ] turtles-own [mass xvelocity yvelocity accx accy newx newy radius avg-radius radius-ratio oldy orbitcount total_orbitcount orbitcount_old revolutions mylabel sundirection_init sundirection clock] globals [sun earth mars mercury venus jupiter saturn time scale G display_on NearPlanet dist2jupiter dist2saturn jupiterplot saturnplot] to setup display ca clear-output set display_on true set NearPlanet false set Zoom 15 set scale 1 set scale Zoom * max-pxcor / 150 set time 10 set G 1.205 * 10 ^ (0 - 10) ;Gravitation constant in units of newton*m^2/kg^2 converted to units of Astronomical Units, ;AU and earth masses, Me, giving units of AU^3/(Me sec^2) crt 1 create-planets 6 ask planets [ pen-down set color green ] set sun turtle 0 ask sun [ set color red set mass 329390 set mylabel "Sun " ] ask turtles [set shape "circle" set size 3] set mercury turtle 1 ask mercury [ set mass .0549 set avg-radius 0.388 set mylabel "Mer " ] set venus turtle 2 ask venus [ set mass .8073 set avg-radius .722 set mylabel "V " ] set earth turtle 3 ask earth [ set mass 1 set avg-radius 1.00 set mylabel "E " set color yellow ] set mars turtle 4 ask mars [ set mass .1065 set avg-radius 1.53 set mylabel "Mar " ] set jupiter turtle 5 ask jupiter [ set mass 314.5 set avg-radius 5.2 set mylabel "J " ] set saturn turtle 6 ask saturn [ set mass 94.07 set avg-radius 9.54 set mylabel "Sat " ] ;planets are randomly positioned, except for the comet, but at the correct distance from the sun, in Astromonical Units. Masses are in earth-masses. ;calculations are done with newx and newy as actual coordinates, then translated into NetLogo screen units scaled to show more or fewer planets orbiting, as desired ask planets with [who < 7][ ifelse random-int-or-float 2 = 0 [set newx random-int-or-float avg-radius] [set newx 0 - random-int-or-float avg-radius] ifelse random-int-or-float 2 = 0 [set newy sqrt (avg-radius ^ 2 - newx ^ 2)] [set newy 0 - sqrt (avg-radius ^ 2 - newx ^ 2)] ] ask planets [ ifelse abs (newx * scale) > max-pxcor or abs (newy * scale) > max-pxcor [set hidden? true setxy newx * scale newy * scale] [set hidden? false setxy newx * scale newy * scale] ] ask planets [ set radius sqrt(newx ^ 2 + newy ^ 2) set sundirection_init 360 - ((180 + towardsxy 0 0) mod 360) set orbitcount 0 set total_orbitcount 0 set revolutions 0 set clock 0 set label mylabel ] ;Compute the initial velocity in x and y directions to produce an approximate circular orbit of radius calculated above regardless of where start point is. ;This comes from a = v^2/r for tangential velocity and a = -G*M/r^2. Solve for v at x = r. Then v = -sqrt(G*M/r). Then compute v in x and y directions. ;Using sin ((towards-nowrap sun) - 180) and cos ((towards-nowrap sun) - 180) to compute y and x components of velocity, respectively, gives the correct ;sign for the resulting value regardless of which quadrant the planets are located. ask planets with [who < 7][ set yvelocity sqrt ((mass-of sun * G) / radius) * sin ((towards-nowrap sun) - 180) set xvelocity 0 - sqrt ((mass-of sun * G) / radius) * cos ((towards-nowrap sun) - 180) ] ;compute acceleration in x and y directions due to sun's gravity only ask planets [ set accx 0 - ((newx * mass-of sun * G) / radius ^ 3) set accy 0 - ((newy * mass-of sun * G) / radius ^ 3) ] ;compute acceleration in x and y directions due to gravities of other planets - each planet computes influence of other planets excluding itself ;and combine accelerations due to sun and other planets ;;switch allows turning off effects of other planets and use gravity of sun only ask planets [ ask planets with [who != who-of myself] [ set accx-of myself accx-of myself - (((mass * G ) * (newx-of myself - newx))/(sqrt ( (newx - newx-of myself ) ^ 2 + (newy - newy-of myself) ^ 2)) ^ 3) set accy-of myself accy-of myself - (((mass * G ) * (newy-of myself - newy))/(sqrt ( (newx - newx-of myself ) ^ 2 + (newy - newy-of myself) ^ 2)) ^ 3) ] ] ;compute the average velocity that will apply for the time period used to compute the next position, estimated by computing velocity at one half the time period ;and applying it for the entire time period. The new velocity is the initial velocity plus the acceleration applied over t/2 seconds. ask planets [ set xvelocity xvelocity + accx * time / 2 set yvelocity yvelocity + accy * time / 2 ] setup-plots end to go orbit-sun end to orbit-sun set scale Zoom * max-pxcor / 150 ifelse Vieworbits = true [ask turtles [pen-down]] [ask turtles [pen-up]] ask planets [ ;;compute new positions for all orbiting bodies set newx newx + xvelocity * time set newy newy + yvelocity * time ;;hide any turtle whose new position is off-screen ifelse abs (newx * scale) > max-pxcor or abs (newy * scale) > max-pxcor [set hidden? true] [set hidden? false setxy newx * scale newy * scale] set radius sqrt(newx ^ 2 + newy ^ 2) ;;compute new accelerations for all orbiting bodies based on sun's gravity. Equations are the same as in setup. set accx 0 - ((newx * mass-of sun * G) / radius ^ 3) set accy 0 - ((newy * mass-of sun * G) / radius ^ 3) ;;unless only the sun's gravity is to be considered, each orbiting body asks all other orbiting bodies for their contribution to its acceleration ask planets with [who != who-of myself] [ set accx-of myself accx-of myself - (((mass * G) * (newx-of myself - newx))/(sqrt ( (newx - newx-of myself ) ^ 2 + (newy - newy-of myself) ^ 2)) ^ 3) set accy-of myself accy-of myself - (((mass * G) * (newy-of myself - newy))/(sqrt ( (newx - newx-of myself ) ^ 2 + (newy - newy-of myself) ^ 2)) ^ 3) ] ] ask planets [ ;;compute new velocities set xvelocity xvelocity + accx * time set yvelocity yvelocity + accy * time set label mylabel ] ;; ask planets [ ;;keep track of number of orbits each planet makes ;; if oldy < 0 and newy > 0 ;; [set orbitcount orbitcount + 1] ;; set oldy newy ] ask planets [ ;;keep track of number of orbits each planet makes ;; if ( (360 - ((180 + towardsxy 0 0) mod 360)) < sundirection ) ;; [set clock clock + 1] ;;;360 - ((180 + towardsxy 0 0) mod 360) set sundirection 360 - ((180 + towardsxy 0 0) mod 360) set orbitcount_old orbitcount if sundirection > sundirection_init [set orbitcount (sundirection - sundirection_init) / 360] if sundirection < sundirection_init [set orbitcount (360 + sundirection - sundirection_init) / 360] if orbitcount - orbitcount_old < 0 [set revolutions revolutions + 1] set total_orbitcount revolutions + orbitcount ] ;ask planets [set oldy newy] ;;keep track of ratio of current radius to initial average radius for both earth and comet so they can be plotted ask earth [set radius-ratio ((radius / (avg-radius)))] do-plot end to do-plot ;; set-current-plot "Earth - Sun Distance" ;; ask earth [set-current-plot-pen "radius-ratio earth"] ;; ask earth [plot radius-ratio] end to setup-plots ;; set-current-plot "Earth - Sun Distance" ;;scale the earth orbit plot based on anticipated ranges. Determined by previous observations ;; set-plot-y-range precision (1 - (2 * sqrt(time))/ 1000) 4 precision (1 + (2 * sqrt(time))/ 1000) 4 ;; set-plot-x-range 0 1000 end to ToggleDisplay ifelse display_on = true [no-display set display_on false] [display set display_on true] end @#$#@#$#@ GRAPHICS-WINDOW 239 12 733 527 60 60 4.0 1 10 1 1 1 0 1 1 1 -60 60 -60 60 CC-WINDOW 5 541 742 636 Command Center 0 BUTTON 39 10 124 43 Setup Setup NIL 1 T OBSERVER T NIL BUTTON 11 50 69 83 Go Go T 1 T OBSERVER T NIL SLIDER 13 135 124 168 Zoom Zoom 1 100 15 1 1 NIL SWITCH 13 95 125 128 vieworbits vieworbits 0 1 -1000 BUTTON 75 51 137 84 Go once go NIL 1 T OBSERVER T NIL TEXTBOX 141 93 218 196 Slow down the planets using the slider at the top of the Graphics window. MONITOR 17 290 126 339 Earth years total_orbitcount-of earth 3 1 MONITOR 19 182 126 231 Mercury years total_orbitcount-of mercury 3 1 MONITOR 18 236 127 285 Venus years total_orbitcount-of venus 3 1 MONITOR 16 343 123 392 Mars years total_orbitcount-of mars 3 1 MONITOR 18 396 125 445 Jupiter years total_orbitcount-of jupiter\n 3 1 MONITOR 18 450 127 499 Saturn years total_orbitcount-of saturn 3 1 TEXTBOX 146 190 232 304 RADII:\nMercury 0.388 \nVenus 0.722 \nEarth 1.000\nMars 1.53\nJupiter 5.2\nSaturn 9.54 @#$#@#$#@ This model is a modification of "Solar System" by Jerry James (2/13/03). It allows one to compare the orbiting times of the six historical planets. The original model focuses on the effect of a comet. This version focuses on the relative orbit times of the planets. This model mimics the solar system. Setup creates the sun and five planets. Unlike in the real solar system, these orbiting bodies are in the same plane, and setup distributes the planets randomly. However, the planets are at the correct relative distances from the sun and have correct relative masses. As the model is run, the number of orbits of each planet is given to three decimal places. To Run the Model Just press Setup. Setup resets Zoom to 15, which shows all six planets. To view the motions of the inner planets, increase Zoom. The initial angle of each planet is preserved with a radial line. Press Go. It may be useful to slow down the model using the speed slider above the graphic window. Or to search for a specific value, use the "go once" button. If you want to confirm Kepler's Law that relates the radius of the orbit to the period, compare each planet to the earth. The Procedures show the radius of each planet's orbit, and the year monitors show the number of orbits (years) compared to the earth years. Notes: The speed adjustment actually changes the size of time intervals at which the new planet positions are updated. This affects the accuracy of the orbits. At a speed setting of 10 there are about 1000 updates of the position of the earth per revolution of the sun, but only about 250 for Mercury. At a speed setting of 1 there are ten times as many updates per revolution. The effects of speed settings on the orbit calculations can be seen by setting the Sun_Only switch to "on" and observing the orbital plots for both earth and comet at different speed settings. The ellipticity of the earth's orbit decreases as the number of updates per year increases. While the Model is Running To increase the speed of execution without affecting computation accuracy, turn the display off by pressing Display On/Off. This is a toggle which alternately stops the display from showing the changing planet positions, or resumes showing their positions. Calculations Computing the acceleration of object #1 towards object #2 due to gravity: From F = ma = -GMm/r^2 and solving for the x and y components of acceleration a we get accx = -GMx/r^3 and accy = -GMy/r^3 where accx and accy are the accelerations in the x and y directions, x and y are the distances in x and y directions between the objects #1 and #2, M is the mass of object #2 (the mass m of object #1 cancels out), G is the gravitational constant, and r = sqrt(x^2 + y^2). G is expressed in terms of Astronomical Units, AU and earth masses, Me, giving units of AU^3/(Me sec^2). Computing the inital conditions After the planets are randomly placed at their proper orbiting distances from the sun, the initial velocities are computed from the equation v = -sqrt(G*M/r) where v = tangential velocity. The velocities in the x and y directions are then x-velocity = v * sin (angle) and y-velocity = v * cos (angle) where angle is reported from (towards-nowrap sun - 180). The initial accelerations are computed as described above, with each planet having the effects of the sun's and all the other planets' gravity on itself computed. At each time interval a new x and y position for each planet and the comet is computed based on the x and y velocities for that time period. Since the velocities at the beginning of the time interval increase by (acceleration * time) the average velocity for each time period is used. 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