Orbit simulations, tidal forces and planetary oblateness

[madsmh]

How does one correct for tidal forces when (classically) simulating the orbits of solar system bodies?

 

[mfb]

Tidal forces don't affect the motion of the center of mass unless the object is notably different from a sphere. If that is relevant, just calculate how large the effect is and take it into account.

 

[madsmh]

Thanks, can you point me to some resources on how to do that?

I am making a solar system simulator, and the errors I get when simulating the larger moons of Jupiter are about the diameter of their orbits when simulating three years. I have made some animations and can see that they fall behind their reference positions quite early in the simulations. And my current hypothesis is that it's due to tidal effects.

 

[sophiecentaur]

Tidal forces don't affect the motion of the center of mass unless the object is notably different from a sphere. If that is relevant, just calculate how large the effect is and take it into account.

Calculating and predicting the change in lunar rotation and orbit seems to be fairly do-able or so they claim.

 

[mfb]

Thanks, can you point me to some resources on how to do that?

I am making a solar system simulator, and the errors I get when simulating the larger moons of Jupiter are about the diameter of their orbits when simulating three years. I have made some animations and can see that they fall behind their reference positions quite early in the simulations. And my current hypothesis is that it's due to tidal effects.

I would expect an issue with the integration scheme. How does the difference change if you change the step size? How many moons with mutual interactions do you consider? Do you consider the other planets?

 

[madsmh]

I don't experience any significant changes in accuracy when changing using 15 steps/hour compared to 10 steps/hour. And as most bodies are simulated quite accurately, I don't expect it to be a issue with the integrator. I am using Velocity Verlet,

Below is a exhaustive list of the bodies that interact in my simulation along with the absolute error of the predictions over about 3 years.

Also I have found out that the oblateness of the satellites might be a fator since I have assumed the bodies to be spherical. I have yet to find a way to correct the accerlerations.

Absolute error for the bodies are
0 Sun: 5.08786 km
1 Mercury: 848.865 km
2 Venus: 259.957 km
3 Earth: 153.673 km
4 Mars: 82.7782 km
5 Jupiter: 83.7391 km
6 Saturn: 179597 km
7 Uranus: 792.117 km
8 Neptune: 384.748 km
9 Pluto: 87264.3 km
10 Luna: 208.759 km
11 Ceres: 36.4271 km
12 Pallas: 22.929 km
13 Vesta: 49.2801 km
14 Hygiea: 46.0951 km
15 Eros: 180.347 km
16 Psyche: 16.794 km
17 Kalliope: 36.4285 km
18 Ida: 31.9049 km
19 Mathilde: 12.6696 km
20 Interamnia: 46.6359 km
21 Itokawa: 61.7569 km
22 Bennu: 143.937 km

Jovian moons:
23 Io: 845850 km
24 Europa: 1.0181e+06 km <----
25 Ganymede: 169375 km
26 Callisto: 49408 km
27 Amalthea: 365301 km
28 Himalia: 304.113 km
29 Elara: 365.324 km
30 Pasiphae: 115.442 km
31 Sinope: 97.3601 km
32 Lysithea: 341.638 km
33 Carme: 30.7982 km
34 Ananke: 105.603 km
35 Leda: 556.354 km
36 Thebe: 451371 km
37 Adrastea: 259997 km
38 Metis: 258080 km
39 Callirrhoe: 60.3905 km
40 Themisto: 1204.33 km
41 Megaclite: 63.9931 km
42 Taygete: 219.793 km
43 Chaldene: 65.5604 km
44 Harpalyke: 141.156 km
45 Kalyke: 178.079 km
46 Iocaste: 145.36 km
47 Erinome: 100.23 km
48 Isonoe: 40.5988 km
49 Praxidike: 46.3703 km
50 Autonoe: 102.283 km
51 Thyone: 38.8438 km
52 Hermippe: 59.777 km
53 Aitne: 213.869 km
54 Eurydome: 80.6264 km
55 Euanthe: 57.0632 km
56 Euporie: 78.7794 km
57 Orthosie: 136.569 km
58 Sponde: 192.518 km
59 Kale: 136.733 km
60 Pasithee: 133.47 km
61 Hegemone: 104.588 km
62 Mneme: 112.059 km
63 Aoede: 95.9567 km
64 Thelxinoe: 124.509 km
65 Arche: 100.992 km
66 Kallichore: 169.745 km
67 Helike: 64.0277 km
68 Carpo: 229.495 km
69 Eukelade: 35.7214 km
70 Cyllene: 152.035 km
71 Kore: 125.362 km
72 Herse: 86.1234 km

Saturninan moons:
73 Mimas: 534415 km
74 Enceladus: 652793 km
75 Tethys: 661220 km
76 Dione: 932684 km
77 Rhea: 703536 km
78 Iapetus: 536944 km
79 Titan: 486999 km
80 Hyperion: 349000 km
81 Phoebe: 492243 km
82 Janus: 478768 km
83 Epimetheus: 478336 km

Plutonian moons:
84 Charon: 113572 km
85 Nix: 138768 km
86 Hydra: 210743 km
87 Kerberos: 173800 km

Martian moons:
88 Phobos: 19077.7 km
89 Deimos: 19806.4 km

Neptunian moons:
90 Triton: 17231.6 km
91 Nereid: 818.001 km
92 Naiad: 96933.8 km
93 Thalassa: 100705 km
94 Despina: 105511 km
95 Galatea: 124375 km
96 Larissa: 147478 km
97 Proteus: 235669 km
98 Halimede: 534.517 km
99 Psamathe: 63.2979 km
100 Sao: 423.781 km
101 Laomedeia: 299.331 km
102 Neso: 10.007 km
103 2004N1: 210822 km

Uanian moons:
104 Ariel: 104619 km
105 Umbriel: 45315.1 km
106 Titania: 13825 km
107 Oberon: 8247.51 km
108 Miranda: 233671 km
109 Cordelia: 104734 km
110 Ophelia: 110910 km
111 Bianca: 119732 km
112 Cressida: 128302 km
113 Desdemona: 129935 km
114 Juliet: 128796 km
115 Portia: 136793 km
116 Rosalind: 140571 km
117 Belinda: 152412 km
118 Puck: 172812 km
119 Caliban: 1364.51 km
120 Sycorax: 1256.41 km
121 Prospero: 2049.09 km
122 Setebos: 648.569 km
123 Stephano: 2552.68 km
124 Trinculo: 1443.74 km
125 Francisco: 854.111 km
126 Margaret: 1388.9 km
127 Ferdinand: 276.821 km
128 Perdita: 155092 km
129 Mab: 196198 km
130 Cupid: 149637 km

 

[mfb]

6 Saturn: 179597 km

Huh?

Jupiter is notably oblate, that could contribute.

Are these numbers the largest deviations that occur over a three year period? Too many of them are too close to the diameter of the orbit to be pure coincidence for the final state.

Earth/Moon have a complex system of strong tides, but they fit nicely - If they don't show anything I wouldn't expect an issue at Jupiter from that aspect.

 

[madsmh]

Yes I've noticed Saturns error is very high which I can't explain.
The numbers I posted are the largest deviations that occur in a 3 year period. Europa deviates by about 100 degrees while with Io, it's deviation is >180 degrees, it it remains it the same orbit as the reference.

I get your point about tidal forces, should I make a new thread where the issue about oblatenes can be discussed?

 

[mfb]

I can also add the oblateness to the title here.

 

[madsmh]

Please do :)
Do you have any idea on how to compensate the oblateness of the bodies?

 

[madsmh]

I've managed to find a way to correct for the obleateness, and the results are pretty remarkable with regards to to Gallian moons, The absolute error on Europa's position has been reduced to about 32000 km, a vast improvement.

However I am at a loss what is causing the huge errors on Saturn, maybe I should add more of its moons to the simulation?

The full error tabel is below.

0 Sun: 5.14231 km
1 Mercury: 848.873 km
2 Venus: 259.934 km
3 Earth: 155.667 km
4 Mars: 82.7718 km
5 Jupiter: 149.619 km
6 Saturn: 179610 km
7 Uranus: 792.884 km
8 Neptune: 384.544 km
9 Pluto: 87264.3 km
10 Luna: 175.672 km
11 Ceres: 36.0613 km
12 Pallas: 23.002 km
13 Vesta: 49.4094 km
14 Hygiea: 46.0786 km
15 Eros: 180.262 km
16 Psyche: 16.7899 km
17 Kalliope: 35.475 km
18 Ida: 32.1831 km
19 Mathilde: 12.699 km
20 Interamnia: 46.6507 km
21 Itokawa: 61.7391 km
22 Bennu: 143.931 km

Jovian moons:
23 Io: 63278.1 km
24 Europa: 32203.2 km
25 Ganymede: 4257.91 km
26 Callisto: 1106.63 km
27 Amalthea: 363740 km
28 Himalia: 234.613 km
29 Elara: 247.837 km
30 Pasiphae: 548.243 km
31 Sinope: 482.96 km
32 Lysithea: 156.687 km
33 Carme: 465.288 km
34 Ananke: 512.96 km
35 Leda: 171.167 km
36 Thebe: 444726 km
37 Adrastea: 258226 km
38 Metis: 256367 km
39 Callirrhoe: 584.401 km
40 Themisto: 239.298 km
41 Megaclite: 555.278 km
42 Taygete: 350.124 km
43 Chaldene: 678.057 km
44 Harpalyke: 339.637 km
45 Kalyke: 256.896 km
46 Iocaste: 319.982 km
47 Erinome: 440.694 km
48 Isonoe: 237.873 km
49 Praxidike: 536.72 km
50 Autonoe: 142.663 km
51 Thyone: 592.346 km
52 Hermippe: 601.08 km
53 Aitne: 223.225 km
54 Eurydome: 721.749 km
55 Euanthe: 743.731 km
56 Euporie: 985.634 km
57 Orthosie: 476.155 km
58 Sponde: 477.298 km
59 Kale: 148.728 km
60 Pasithee: 317.449 km
61 Hegemone: 450.883 km
62 Mneme: 461.759 km
63 Aoede: 128.746 km
64 Thelxinoe: 371.268 km
65 Arche: 612.66 km
66 Kallichore: 583.245 km
67 Helike: 727.543 km
68 Carpo: 194.279 km
69 Eukelade: 462.406 km
70 Cyllene: 202.673 km
71 Kore: 171.663 km
72 Herse: 206.78 km

Saturninan moons:
73 Mimas: 502255 km
74 Enceladus: 524371 km
75 Tethys: 708282 km
76 Dione: 501068 km
77 Rhea: 397256 km
78 Iapetus: 512190 km
79 Titan: 463536 km
80 Hyperion: 352955 km
81 Phoebe: 492018 km
82 Janus: 467471 km
83 Epimetheus: 460140 km

Plutonian moons:
84 Charon: 113572 km
85 Nix: 138768 km
86 Hydra: 210743 km
87 Kerberos: 173800 km

Martian moons:
88 Phobos: 18973.9 km
89 Deimos: 11524.9 km

Neptunian moons:
90 Triton: 17774.6 km
91 Nereid: 850.916 km
92 Naiad: 96807.7 km
93 Thalassa: 100451 km
94 Despina: 105364 km
95 Galatea: 124233 km
96 Larissa: 147253 km
97 Proteus: 200616 km
98 Halimede: 534.407 km
99 Psamathe: 63.4044 km
100 Sao: 423.617 km
101 Laomedeia: 298.837 km
102 Neso: 9.99487 km
103 2004N1: 210894 km

Uanian moons:
104 Ariel: 170622 km
105 Umbriel: 56578.6 km
106 Titania: 22768.4 km
107 Oberon: 14810.6 km
108 Miranda: 259896 km
109 Cordelia: 104889 km
110 Ophelia: 110390 km
111 Bianca: 119722 km
112 Cressida: 128471 km
113 Desdemona: 130054 km
114 Juliet: 128623 km
115 Portia: 136771 km
116 Rosalind: 140476 km
117 Belinda: 153542 km
118 Puck: 172150 km
119 Caliban: 1328.52 km
120 Sycorax: 1253.25 km
121 Prospero: 2049.8 km
122 Setebos: 648.268 km
123 Stephano: 2550.94 km
124 Trinculo: 1435.3 km
125 Francisco: 877.728 km
126 Margaret: 1389.24 km
127 Ferdinand: 277.038 km
128 Perdita: 154671 km
129 Mab: 196092 km
130 Cupid: 149027 km