r^1 Orbits

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19683
Posts: 151
Joined: Wed Jun 05, 2013 12:15 pm

r^1 Orbits

Post by 19683 »

Load this code:
Gravity Fun at TestTubeGames.com: [ForceG: 1,Qual: 1,Zoom: 1,xSet: 35,ySet: 23], [x0: 0,y0: 0,vx: 0,vy: 0,t0: 0,who: 1,m: 1000], [x0: 300,y0: 0,vx: 0,vy: 1,t0: 0,who: 3,m: 0]

r^1 seems to give stable, non-preccessing, symmetrical orbits with the star at the center.
Last edited by 19683 on Tue Jul 09, 2013 1:01 pm, edited 1 time in total.
Binomial Theorem: ((a+b)^n)= sum k=0->k=n((n!(a^(n-k))(b^k))/(k!(n-k)!))
A Random Player
Posts: 523
Joined: Mon Jun 03, 2013 4:54 pm

Re: r^-1 Orbits

Post by A Random Player »

You mean r^1? I agree, r^1 is probably one of my favorite rules/exponents (besides r^-2, of course).
  • It acts like springs are joining each planet/star
  • This means all orbits have equal periods, if the star's mass is held constant.
  • Also, all orbits are ellipses, with the star at the center of the ellipse (where in r^-2, it's at the focus). The reason is left as an exercise for the reader. (Hint: Consider the properties of springs, sine waves, circles, and ellipses)
  • Any multistar orbit acts like the object orbits a single star at the stars' barycenter (though I am unsure on the relationships of the masses):

    Code: Select all

    Gravity Fun at TestTubeGames.com: [ForceG: 1,Qual: 1,Zoom: 1,xSet: -2277,ySet: -242], [x0: 2180,y0: 238,vx: 0,vy: 0,t0: 0,who: 1,m: 1000], [x0: 2408,y0: 178,vx: 0,vy: 0,t0: 0,who: 1,m: 1000], [x0: 2324,y0: 190,vx: -1.2,vy: -1.6,t0: 0,who: 3,m: 0]

    Code: Select all

    Gravity Fun at TestTubeGames.com: [ForceG: 1,Qual: 1,Zoom: 1,xSet: -2276,ySet: -242], [x0: 2142,y0: 144,vx: 0,vy: 0,t0: 0,who: 1,m: 1000], [x0: 2458,y0: 60,vx: 0,vy: 0,t0: 0,who: 1,m: 1000], [x0: 2368,y0: 360,vx: 0,vy: 0,t0: 0,who: 1,m: 1000], [x0: 2349,y0: 182,vx: -1.35,vy: -1.35,t0: 0,who: 3,m: 0]
  • Resonance! Asteroids' orbits can grow infinitely large if they are orbiting a star prograde to a planet, and their orbits stretch if they're going retrograde! (They sometimes collide with the star, and other times may reverse direction and start going prograde)

    Code: Select all

    Gravity Fun at TestTubeGames.com: [ForceG: 1,Qual: 1,Zoom: 1,xSet: -2276,ySet: -242], [x0: 2271,y0: 239,vx: 0,vy: 0,t0: 0,who: 1,m: 1000], [x0: 2286,y0: 54,vx: -5.85,vy: -0.47,t0: 0,who: 2,m: 100], [x0: 2308,y0: 242,vx: 0.09,vy: -1.17,t0: 0,who: 3,m: 0], [x0: 2187,y0: 236,vx: 0.1,vy: -2.6,t0: 0,who: 3,m: 0]
    (Delete one asteroid if you want.) Nothing special happens when the asteroid crosses the planet's orbit - It doesn't matter where the planet it, the orbits will still resonate. This also comes from the properties of springs.
r^1 is actually a lot simpler than r^-2.

Edit: After more investigation, prograde motion can also cause decay:

Code: Select all

Gravity Fun at TestTubeGames.com: [ForceG: 1,Qual: 1,Zoom: 0.86,xSet: -2176.65,ySet: -253.4], [x0: 2112.93,y0: 295.07,vx: 0,vy: 0,t0: 0,who: 1,m: 1000], [x0: 2374.6,y0: 112.29,vx: -5.78,vy: -8.27,t0: 33.3,who: 2,m: 100], [x0: 1964.34,y0: 291.72,vx: -0.11,vy: 4.7,t0: 56.7,who: 3,m: 0]
$1 = 100¢ = (10¢)^2 = ($0.10)^2 = $0.01 = 1¢ [1]
Always check your units or you will have no money!
19683
Posts: 151
Joined: Wed Jun 05, 2013 12:15 pm

Re: r^-1 Orbits

Post by 19683 »

A Random Player wrote:You mean r^1? I agree, r^1 is probably one of my favorite rules/exponents (besides r^-2, of course).
  • It acts like springs are joining each planet/star
  • This means all orbits have equal periods, if the star's mass is held constant.
  • Also, all orbits are ellipses, with the star at the center of the ellipse (where in r^-2, it's at the focus). The reason is left as an exercise for the reader. (Hint: Consider the properties of springs, sine waves, circles, and ellipses)
  • Any multistar orbit acts like the object orbits a single star at the stars' barycenter (though I am unsure on the relationships of the masses):

    Code: Select all

    Gravity Fun at TestTubeGames.com: [ForceG: 1,Qual: 1,Zoom: 1,xSet: -2277,ySet: -242], [x0: 2180,y0: 238,vx: 0,vy: 0,t0: 0,who: 1,m: 1000], [x0: 2408,y0: 178,vx: 0,vy: 0,t0: 0,who: 1,m: 1000], [x0: 2324,y0: 190,vx: -1.2,vy: -1.6,t0: 0,who: 3,m: 0]

    Code: Select all

    Gravity Fun at TestTubeGames.com: [ForceG: 1,Qual: 1,Zoom: 1,xSet: -2276,ySet: -242], [x0: 2142,y0: 144,vx: 0,vy: 0,t0: 0,who: 1,m: 1000], [x0: 2458,y0: 60,vx: 0,vy: 0,t0: 0,who: 1,m: 1000], [x0: 2368,y0: 360,vx: 0,vy: 0,t0: 0,who: 1,m: 1000], [x0: 2349,y0: 182,vx: -1.35,vy: -1.35,t0: 0,who: 3,m: 0]
  • Resonance! Asteroids' orbits can grow infinitely large if they are orbiting a star prograde to a planet, and their orbits stretch if they're going retrograde! (They sometimes collide with the star, and other times may reverse direction and start going prograde)

    Code: Select all

    Gravity Fun at TestTubeGames.com: [ForceG: 1,Qual: 1,Zoom: 1,xSet: -2276,ySet: -242], [x0: 2271,y0: 239,vx: 0,vy: 0,t0: 0,who: 1,m: 1000], [x0: 2286,y0: 54,vx: -5.85,vy: -0.47,t0: 0,who: 2,m: 100], [x0: 2308,y0: 242,vx: 0.09,vy: -1.17,t0: 0,who: 3,m: 0], [x0: 2187,y0: 236,vx: 0.1,vy: -2.6,t0: 0,who: 3,m: 0]
    (Delete one asteroid if you want.) Nothing special happens when the asteroid crosses the planet's orbit - It doesn't matter where the planet it, the orbits will still resonate. This also comes from the properties of springs.
r^1 is actually a lot simpler than r^-2.

Edit: After more investigation, prograde motion can also cause decay:

Code: Select all

Gravity Fun at TestTubeGames.com: [ForceG: 1,Qual: 1,Zoom: 0.86,xSet: -2176.65,ySet: -253.4], [x0: 2112.93,y0: 295.07,vx: 0,vy: 0,t0: 0,who: 1,m: 1000], [x0: 2374.6,y0: 112.29,vx: -5.78,vy: -8.27,t0: 33.3,who: 2,m: 100], [x0: 1964.34,y0: 291.72,vx: -0.11,vy: 4.7,t0: 56.7,who: 3,m: 0]
Yes, I mean r^1. :oops:

Wow! Can't wait to try some of those cool things you found!
Binomial Theorem: ((a+b)^n)= sum k=0->k=n((n!(a^(n-k))(b^k))/(k!(n-k)!))
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robly18
Posts: 413
Joined: Tue Jun 04, 2013 2:03 pm

Re: r^1 Orbits

Post by robly18 »

Woah, I already knew some of these, but others were unknown to me.
Also, I wonder. If r^-2 gives you orbits with one focus outside the star, r^1 with two foci, then does that mean there's a value where there are three?

Could you imagine a triangular orbit?
Convincing people that 0.9999... = 1 since 2012
A Random Player
Posts: 523
Joined: Mon Jun 03, 2013 4:54 pm

Re: r^1 Orbits

Post by A Random Player »

robly18 wrote:Woah, I already knew some of these, but others were unknown to me.
Also, I wonder. If r^-2 gives you orbits with one focus outside the star, r^1 with two foci, then does that mean there's a value where there are three?

Could you imagine a triangular orbit?
An ellipse only has two foci. There isn't an ellipse with more than 2. Otherwise, it's not an ellipse...
$1 = 100¢ = (10¢)^2 = ($0.10)^2 = $0.01 = 1¢ [1]
Always check your units or you will have no money!
19683
Posts: 151
Joined: Wed Jun 05, 2013 12:15 pm

Re: r^1 Orbits

Post by 19683 »

Try this:

Gravity Fun at TestTubeGames.com: [ForceG: 1,Qual: 1,Zoom: 1,xSet: 0,ySet: 0], [x0: -11,y0: 9,vx: 0,vy: 0,t0: 0,who: 1,m: 1000], [x0: 142,y0: 8,vx: -0.03,vy: -4.84,t0: 0,who: 2,m: 100], [x0: 267,y0: 9,vx: 0,vy: -8.79,t0: 0,who: 2,m: 100], [x0: -129,y0: 7,vx: -0.06,vy: 3.73,t0: 0,who: 2,m: 100], [x0: -272,y0: 6,vx: -0.09,vy: 8.25,t0: 0,who: 2,m: 100], [x0: 63,y0: 8,vx: -0.03,vy: -2.34,t0: 0,who: 3,m: 0], [x0: 203,y0: 11,vx: 0.06,vy: -6.77,t0: 0,who: 3,m: 0], [x0: 341,y0: 11,vx: 0.06,vy: -11.13,t0: 0,who: 3,m: 0], [x0: -73,y0: 10,vx: 0.03,vy: 1.96,t0: 0,who: 3,m: 0], [x0: -202,y0: 7,vx: -0.06,vy: 6.04,t0: 0,who: 3,m: 0], [x0: -345,y0: 9,vx: 0,vy: 10.56,t0: 0,who: 3,m: 0]
Binomial Theorem: ((a+b)^n)= sum k=0->k=n((n!(a^(n-k))(b^k))/(k!(n-k)!))
A Random Player
Posts: 523
Joined: Mon Jun 03, 2013 4:54 pm

Re: r^1 Orbits

Post by A Random Player »

19683 wrote:Try this:

Gravity Fun at TestTubeGames.com: [ForceG: 1,Qual: 1,Zoom: 1,xSet: 0,ySet: 0], [x0: -11,y0: 9,vx: 0,vy: 0,t0: 0,who: 1,m: 1000], [x0: 142,y0: 8,vx: -0.03,vy: -4.84,t0: 0,who: 2,m: 100], [x0: 267,y0: 9,vx: 0,vy: -8.79,t0: 0,who: 2,m: 100], [x0: -129,y0: 7,vx: -0.06,vy: 3.73,t0: 0,who: 2,m: 100], [x0: -272,y0: 6,vx: -0.09,vy: 8.25,t0: 0,who: 2,m: 100], [x0: 63,y0: 8,vx: -0.03,vy: -2.34,t0: 0,who: 3,m: 0], [x0: 203,y0: 11,vx: 0.06,vy: -6.77,t0: 0,who: 3,m: 0], [x0: 341,y0: 11,vx: 0.06,vy: -11.13,t0: 0,who: 3,m: 0], [x0: -73,y0: 10,vx: 0.03,vy: 1.96,t0: 0,who: 3,m: 0], [x0: -202,y0: 7,vx: -0.06,vy: 6.04,t0: 0,who: 3,m: 0], [x0: -345,y0: 9,vx: 0,vy: 10.56,t0: 0,who: 3,m: 0]
Woah. The planets and asteroids do get out of phase though.

Edit: And the inevitable:

Code: Select all

Gravity Fun at TestTubeGames.com: [ForceG: 1,Qual: 1,Zoom: 1,xSet: 0,ySet: 0], [x0: 0,y0: 0,vx: 0,vy: 0,t0: 0,who: 1,m: 1000], [x0: 10,y0: 0,vx: 0,vy: 0.316227766017,t0: 0,who: 3,m: 0], [x0: 20,y0: 0,vx: 0,vy: 0.632455532034,t0: 0,who: 3,m: 0], [x0: 30,y0: 0,vx: 0,vy: 0.948683298051,t0: 0,who: 3,m: 0], [x0: 40,y0: 0,vx: 0,vy: 1.26491106407,t0: 0,who: 3,m: 0], [x0: 50,y0: 0,vx: 0,vy: 1.58113883008,t0: 0,who: 3,m: 0], [x0: 60,y0: 0,vx: 0,vy: 1.8973665961,t0: 0,who: 3,m: 0], [x0: 70,y0: 0,vx: 0,vy: 2.21359436212,t0: 0,who: 3,m: 0], [x0: 80,y0: 0,vx: 0,vy: 2.52982212813,t0: 0,who: 3,m: 0], [x0: 90,y0: 0,vx: 0,vy: 2.84604989415,t0: 0,who: 3,m: 0], [x0: 100,y0: 0,vx: 0,vy: 3.16227766017,t0: 0,who: 3,m: 0], [x0: 110,y0: 0,vx: 0,vy: 3.47850542619,t0: 0,who: 3,m: 0], [x0: 120,y0: 0,vx: 0,vy: 3.7947331922,t0: 0,who: 3,m: 0], [x0: 130,y0: 0,vx: 0,vy: 4.11096095822,t0: 0,who: 3,m: 0], [x0: 140,y0: 0,vx: 0,vy: 4.42718872424,t0: 0,who: 3,m: 0], [x0: 150,y0: 0,vx: 0,vy: 4.74341649025,t0: 0,who: 3,m: 0], [x0: 160,y0: 0,vx: 0,vy: 5.05964425627,t0: 0,who: 3,m: 0], [x0: 170,y0: 0,vx: 0,vy: 5.37587202229,t0: 0,who: 3,m: 0], [x0: 180,y0: 0,vx: 0,vy: 5.6920997883,t0: 0,who: 3,m: 0], [x0: 190,y0: 0,vx: 0,vy: 6.00832755432,t0: 0,who: 3,m: 0]
And the denser version:

Code: Select all

Gravity Fun at TestTubeGames.com: [ForceG: 1,Qual: 1,Zoom: 1,xSet: 0,ySet: 0], [x0: 0,y0: 0,vx: 0,vy: 0,t0: 0,who: 1,m: 1000], [x0: 4,y0: 0,vx: 0,vy: 0.126491106407,t0: 0,who: 3,m: 0], [x0: 8,y0: 0,vx: 0,vy: 0.252982212813,t0: 0,who: 3,m: 0], [x0: 12,y0: 0,vx: 0,vy: 0.37947331922,t0: 0,who: 3,m: 0], [x0: 16,y0: 0,vx: 0,vy: 0.505964425627,t0: 0,who: 3,m: 0], [x0: 20,y0: 0,vx: 0,vy: 0.632455532034,t0: 0,who: 3,m: 0], [x0: 24,y0: 0,vx: 0,vy: 0.75894663844,t0: 0,who: 3,m: 0], [x0: 28,y0: 0,vx: 0,vy: 0.885437744847,t0: 0,who: 3,m: 0], [x0: 32,y0: 0,vx: 0,vy: 1.01192885125,t0: 0,who: 3,m: 0], [x0: 36,y0: 0,vx: 0,vy: 1.13841995766,t0: 0,who: 3,m: 0], [x0: 40,y0: 0,vx: 0,vy: 1.26491106407,t0: 0,who: 3,m: 0], [x0: 44,y0: 0,vx: 0,vy: 1.39140217047,t0: 0,who: 3,m: 0], [x0: 48,y0: 0,vx: 0,vy: 1.51789327688,t0: 0,who: 3,m: 0], [x0: 52,y0: 0,vx: 0,vy: 1.64438438329,t0: 0,who: 3,m: 0], [x0: 56,y0: 0,vx: 0,vy: 1.77087548969,t0: 0,who: 3,m: 0], [x0: 60,y0: 0,vx: 0,vy: 1.8973665961,t0: 0,who: 3,m: 0], [x0: 64,y0: 0,vx: 0,vy: 2.02385770251,t0: 0,who: 3,m: 0], [x0: 68,y0: 0,vx: 0,vy: 2.15034880891,t0: 0,who: 3,m: 0], [x0: 72,y0: 0,vx: 0,vy: 2.27683991532,t0: 0,who: 3,m: 0], [x0: 76,y0: 0,vx: 0,vy: 2.40333102173,t0: 0,who: 3,m: 0], [x0: 80,y0: 0,vx: 0,vy: 2.52982212813,t0: 0,who: 3,m: 0], [x0: 84,y0: 0,vx: 0,vy: 2.65631323454,t0: 0,who: 3,m: 0], [x0: 88,y0: 0,vx: 0,vy: 2.78280434095,t0: 0,who: 3,m: 0], [x0: 92,y0: 0,vx: 0,vy: 2.90929544735,t0: 0,who: 3,m: 0], [x0: 96,y0: 0,vx: 0,vy: 3.03578655376,t0: 0,who: 3,m: 0], [x0: 100,y0: 0,vx: 0,vy: 3.16227766017,t0: 0,who: 3,m: 0], [x0: 104,y0: 0,vx: 0,vy: 3.28876876658,t0: 0,who: 3,m: 0], [x0: 108,y0: 0,vx: 0,vy: 3.41525987298,t0: 0,who: 3,m: 0], [x0: 112,y0: 0,vx: 0,vy: 3.54175097939,t0: 0,who: 3,m: 0], [x0: 116,y0: 0,vx: 0,vy: 3.6682420858,t0: 0,who: 3,m: 0], [x0: 120,y0: 0,vx: 0,vy: 3.7947331922,t0: 0,who: 3,m: 0], [x0: 124,y0: 0,vx: 0,vy: 3.92122429861,t0: 0,who: 3,m: 0], [x0: 128,y0: 0,vx: 0,vy: 4.04771540502,t0: 0,who: 3,m: 0], [x0: 132,y0: 0,vx: 0,vy: 4.17420651142,t0: 0,who: 3,m: 0], [x0: 136,y0: 0,vx: 0,vy: 4.30069761783,t0: 0,who: 3,m: 0], [x0: 140,y0: 0,vx: 0,vy: 4.42718872424,t0: 0,who: 3,m: 0], [x0: 144,y0: 0,vx: 0,vy: 4.55367983064,t0: 0,who: 3,m: 0], [x0: 148,y0: 0,vx: 0,vy: 4.68017093705,t0: 0,who: 3,m: 0], [x0: 152,y0: 0,vx: 0,vy: 4.80666204346,t0: 0,who: 3,m: 0], [x0: 156,y0: 0,vx: 0,vy: 4.93315314986,t0: 0,who: 3,m: 0], [x0: 160,y0: 0,vx: 0,vy: 5.05964425627,t0: 0,who: 3,m: 0], [x0: 164,y0: 0,vx: 0,vy: 5.18613536268,t0: 0,who: 3,m: 0], [x0: 168,y0: 0,vx: 0,vy: 5.31262646908,t0: 0,who: 3,m: 0], [x0: 172,y0: 0,vx: 0,vy: 5.43911757549,t0: 0,who: 3,m: 0], [x0: 176,y0: 0,vx: 0,vy: 5.5656086819,t0: 0,who: 3,m: 0], [x0: 180,y0: 0,vx: 0,vy: 5.6920997883,t0: 0,who: 3,m: 0], [x0: 184,y0: 0,vx: 0,vy: 5.81859089471,t0: 0,who: 3,m: 0], [x0: 188,y0: 0,vx: 0,vy: 5.94508200112,t0: 0,who: 3,m: 0], [x0: 192,y0: 0,vx: 0,vy: 6.07157310752,t0: 0,who: 3,m: 0], [x0: 196,y0: 0,vx: 0,vy: 6.19806421393,t0: 0,who: 3,m: 0]
And the code:

Code: Select all

import math
output = "Gravity Fun at TestTubeGames.com: [ForceG: 1,Qual: 1,Zoom: 1,xSet: 0,ySet: 0], [x0: 0,y0: 0,vx: 0,vy: 0,t0: 0,who: 1,m: 1000]"
step = 10
for a in range(step,200,step):
 output = output + ", [x0: " + str(a) + ",y0: 0,vx: 0,vy: " + str(.001**.5*a) + ",t0: 0,who: 3,m: 0]"
print output
$1 = 100¢ = (10¢)^2 = ($0.10)^2 = $0.01 = 1¢ [1]
Always check your units or you will have no money!
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