1. How close is the position of the barycentre to an arithmetic sum of each individual planetary influence?
2. How close an approximation of the barycentre plot will be obtained by calculating the pull of each individual planet on the position of the sun and graphing the combined result?
3. Is combined planetary gravity significant in addition to individual planetary effects?

Wiki page here shows tabular calculations and diagrams for two-body barycentres, but in the actual solar system the barycentre moves in response to gravity of all the planets.

Very grateful for any answers
Thanks

A good simulation and evolution of the Solar systems barycenter can be founh here :
http://www.orbitsimulator.com/gravit...arycenter.html

Fairly close, but since gravity isn't a linear function, there will be differences between simultaneous computation of all the planets and the summation of individual computations.

Sounds like the first question, except it's obtained graphically, rather than arithmetically - same answer, but with the induced graphic error (by hand? or by computer?)

Sounds similar, but this isn't really a question, as there's no qualification of "significant," as in significant as compared to what? Again, combined calculations will differ slightly from the summation of individual effects.

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If I set the budget, we'd have Ares and more. Unfortunately, I don't set the budget, and Ares is just too expensive and too far out for us to accomplish our goals within the budget we were given.

If we halt the ISS, all versions of Ares, and transport Orion and Altair aboard DIRECTv3's Jupiter family of Shuttle-Derived Launch Vehicles, we just might make it back to the Moon by 2020.

But only to the extent that you need the accuracy of a non-Newtonian situation or the point-mass approximation isn't adequate, and those are really minute effects compared to the magnitude of the solar barycentric motion. For Newtonian point masses, the combined force is the (vector) sum, and since in this case the displacement is always extremely small compared to even Mercury's distance, adding the individual planetary displacements is a really good approximation. (This breaks down when the perturbing masses are larger, for example, so that the variation in force due to one perturber changes significantly over the distance of the perturbation caused by another). Again, it depends on what "significant" means in a particular context.

Thank you very much, this is precisely what I was looking for. I had never seen a picture of the SSB epitrochoid before. It removes the planets one by one starting from Jupiter and ending with Pluto, ranked by their affect on the SSB, to provide simple graphical pictures of the gravitational influence of each planet.

If I may, I have two further questions from looking at this material. Many thanks again for any possible advice.

It states "The Sun's period around the barycenter and Uranus' period around the barycenter match." I don't understand what this means. Can anyone explain it?

The SSB plot when Jupiter is removed returns exactly to its origin after three loops, rather than producing a set of overlapping 'spirograph' cycles. Is this correct, and if so why?

The simulation was done by Tony , also a forum-member , but reading through the link he states that after deleting the bigger planets as Jupiter , Saturn and Neptune , the barycenter of the rest of the system is mainly dictated by the sun and Uranus due to the Mass*Distance relationship . The other smaller planets (Mercury,Vneus, Earth, Mars , Pluto) contribution to the barycenter position is small . So the path of the sun around the barycenter is mainly given by Uranus .

The loops after removing Jupiter are continuiing I think . The graph was stopped after a certain amount of time , giving the nice picture . While there is not a close resonance between the Saturn/Neptune and Uranus orbit , continuing the simulation should give the spirograph cycle you mention .

Thanks. Looking again at the SSB minus Jupiter, you can see the line starts and ends in the middle where the loop is not exact. I would be interested to know the period of this graph. Assuming it primarily shows the Saturn-Neptune opposition as marking the points where the SSB is closest to the sun and the Saturn-Neptune conjunctions as the points where SSB is furthers from the Sun, at 35.8 years per cycle the graph seems to show 107.4 years of the SSB. Is this right?

This means that the amount of time it takes the Sun to trace that circle is the same amount of time as Uranus' orbital period.

It's far from exact. Because none of the heavy planets are in resonance with each other, they will never align the same way twice. So the pattern has some chaos in it. If I let the simulation run longer before taking a screen shot, rather than overlapping spirographs, The image would just get busier and busier until all you saw was one big yellow blob in the middle of the screen.

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Are you sure about this? It depends what you mean by resonance. There is a fairly strong 179 year resonance between Jupiter, Saturn and Neptune with drift of just over one part in a thousand (0.1% per cycle). Of course nothing is exact in such matters, but this ‘resonance’ produces very strong similarity between SSB patterns separated by 179 years. Surely you could run a lot more iterations on the simulation without Jupiter before the pattern got too busy?

[QUOTE=ngc3314;1301220]But only to the extent that you need the accuracy of a non-Newtonian situation...[quote]

I was only considering the Newtonian effects. No relativistic effects were considered.

Yes and no. Although the Sun's pertubations in position are minute compared to the Earth's orbit around the sun, the Sun's mass is as much greater as it's pertubations are small. However, since the force between two bodies is proportional to the masses (which doesn't change) and inversely proportional to the square of the distance between them (which changes very little for such a relatively large mass as the Sun), the variance in the Force is small.

But it is nevertheless there, and it most certainly does affect planetary orbits on down the road, which is what I think the OP was getting at.

You're right - it's a really good approximation. But that's limited to what's happening now, for example, determining the forces between the objects, or their departure vector. When it comes to the effects of gravity over time (again, back to the OP), however, errors are compounded, and a "really good approximation" doesn't hold a candle to the precision required to compute where the planets would be a hundred years from now.

In fact, nothing, including relativistic effects, is approximated for long-term gravitational models. And when such models are put through their computational paces, we find two curious things. The first is a much tighter approximation of a planet's gravity. The second are unknown masses and locations required to make the mathmatical model match what's been observed. This "missing mass" is, in part, one of the bases for the existance of the Oort cloud. The remainder were observations of cometary patterns by Ernst Öpik, and later, by Jan Hendrik Oort.

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If I set the budget, we'd have Ares and more. Unfortunately, I don't set the budget, and Ares is just too expensive and too far out for us to accomplish our goals within the budget we were given.

If we halt the ISS, all versions of Ares, and transport Orion and Altair aboard DIRECTv3's Jupiter family of Shuttle-Derived Launch Vehicles, we just might make it back to the Moon by 2020.

Thank you for these informative answers. I am wondering what the 'non-linearity' of gravity amounts to in this context. To explain, we can plot the Jupiter effect on the barycentre as a sine wave with period of the Jupiter orbit and amplitude the distance the sun is perturbed by Jupiter. Overlaying Saturn Neptune and Uranus sine waves on this chart, the barycentre function seems to be mainly derived from the sum of these four wave functions by linear addition.
I have just made the attached picture to illustrate. It is derived from a spreadsheet with three sine functions with frequencies corresponding to the orbits of Jupiter, Saturn and Neptune, plotting the sum of the three sine values. For this chart, sine amplitudes are equal so it treats the three planets as having the same gravity, exaggerating the relative effect of Saturn and Neptune compared to Jupiter. Nonetheless, the resulting line graph has the same shape as the SSB data provided by JPL, including the apparent 179 year resonance marked by the sets of arrows.

Attached Images

JSN SSB Sine Function.GIF (28.6 KB, 111 views)

I don't see 0.1% when I run the numbers. Using orbital periods of 11.85920, 29.657296, 164.79 years for Jupiter, Saturn & Neptune respectively, I see an approximate 15:6:1 resonance. But the real numbers are
15.093766864544 : 6.03561430549838 : 1.08623096061654
which miss a perfect resonance by about 5%. This will certainly cause drift in the plot.

Here's 2000 years of the planets sans Jupiter curving the path of the center of the sun. It's pretty busy, and if I let it run 10s of thousands of years, it would be a solid yellow circle (or perhaps a doughnut as it looks like the middle won't get filled in) because of the width of the plotted line.


I was wondering about this statement. I guess we'd have to quantify "hold a candle", but I thought I'd try it anyway.

I know that JPL's Horizons integrator includes lots of stuff that a Newtonian-only point mass simulator would not include, such as non-spherical bodies and relativity. So a comparison can be made by creating a simulation of the solar system, propogating it forward 100 years, and then comparing the results with what JPL says they should be based on their integrator. In every case except Venus and Pluto, the simulated positions of my planets were within one planet diameter of where JPL Horizons said they should be. And Venus and Pluto weren't very far off either:

Mercury: 3094 km
Venus: 13061 km
Earth: 8707 km
Mars: 4696 km
Jupiter: 1974 km
Saturn: 1325 km
Uranus: 394 km
Neptune:146 km
Pluto: 19255 km
Makemake: 330 km

These numbers are very small compared to the size of the solar system. For example, the ratio between Earth's miss distance of 8707 km, and its distance to the sun is about the same as a golf cup placed over 2000 yards from the tee box. Imagine making a hole-in-one from 2000 yards away. Or in the case of Neptune, it would be like making a hole-in-one from 2000 miles away.

Additionally, propogating the solar system backwards correctly predicts (or postdicts is probably a better term) the time and date of transits of Venus more than a century ago. These can be verified by historical record.

I've never heard this before. I can't seem to find it in the wiki link you provide. I've been under the impression that since the Oort cloud is exterior to us, evenly distributed, and not very massive (~5 earth masses according to wiki), that it can't create significant perturbations. It's also not mentioned as a source of error in this paper by JPL's Jon Giorgini: http://neo.jpl.nasa.gov/apophis/Apop...D_PREPRINT.pdf . In it he talks about the difficulty of predicting how close asteroid Apophis will pass Earth in the year 2036. The uncertainties in the planets' positions ranks high, but uncertainties from the exclusion of the Oort cloud are not mentioned.

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My comment wasn't directed towards a difference between newtonian and non-newtonian calculators. It was directed towards the difference between a newtonian calculator and an x-y (arithmetic mean) calculator.

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If I set the budget, we'd have Ares and more. Unfortunately, I don't set the budget, and Ares is just too expensive and too far out for us to accomplish our goals within the budget we were given.

If we halt the ISS, all versions of Ares, and transport Orion and Altair aboard DIRECTv3's Jupiter family of Shuttle-Derived Launch Vehicles, we just might make it back to the Moon by 2020.

Thanks tony873004, my statement was too condensed. What I meant was that Jupiter-Saturn meet nine times every 178.7 years, while Saturn-Neptune meet five times every 178.9 years, and Jupiter-Neptune meet 14 times every 178.9 years. Considering this period as a cycle, JS drifts against SN at a rate of 0.1% per cycle (0.2 years per 180). btw your numbers, ~ 12:30:165 look more like 1:2.5:14 than 1:6:15

Thanks! This looks to have over 100 loops, so is already busy. Your picture showing only three loops did not show what would happen when the loops repeat the cycle, which a diagram with about ten or twenty loops would do.

Based on material presented here, I have a further question. Is my understanding of gravity as presented below correct?

The attached picture illustrates the relative gravitational effect of the four gas giant planets at the sun. I explain the mathematics here. These four waves are the main decomposed sine functions that produce the pattern of the solar system barycentre. The wavelengths correspond to the orbital period of each gas giant, while the amplitude shows the Jupiter function multiplied by the gravity of each planet at the sun, as per the following table, calculated by the inverse square law. This calculation indicates that as a proportion of Jupiter’s gravity, Saturn’s gravitational effect on the sun is 30% as strong, Uranus is 4.6% and Neptune is 5.4%. Eight hundred years are presented.
The planetary gravity at sun is calculated by mass of planet x mass of sun ÷ Distance squared ÷ Jupiter result

Attached Images

Jupiter Saturn Uranus Neptune Gravity at Sun.GIF (55.2 KB, 94 views)

You can't have a decimal in resonance. They have to be integers. Only at integer multiples of their orbits completed will they line up again. For example, when Jupiter and Saturn are at conjunction, 1:2.5 means that after Saturn completes 1 orbit that Jupiter completes 2.5 and is on the opposite side of the Sun from Saturn.

If the pattern repeats every 179 years, then you get the resonance by dividing 179 by the orbital periods:

179/11.85920 = 15.093766864544
179 / 29.657296 = 6.03561430549838
179/164.79 =1.08623096061654

None of these are integers. After 179 years, Neptune has overshot the starting position by 9%, Saturn by 4% and Jupiter by 9%. So there's a 5% difference between Jupiter and Saturn, and between Neptune and Saturn, which is where I got the value of 5% I reported.

Saturn is 1.84 times farther from the Sun than Jupiter (9.58201720 / 5.204267). This makes its gravity (1/1.84^2) = 0.295 as strong as Jupiter's at the distance of the Sun. But Saturn is only 0.3 times as massive as Jupiter, weakening its gravity again.
So at the sun, Saturn's gravity is only 0.295 * 0.3 = 0.088 times as strong as Jupiter's.

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Thank you. What the numbers you provide show is that all three planets have overshot by an amount which brings them together again at a point one twelfth of the way further around the ecliptic. The resonances are shown by comparing the periods of each combined planetary cycle: ie Jupiter-Saturn = 19.85 years, Jupiter-Neptune = 12.78 years, and Saturn-Neptune = 35.8 years. These periods recur at 179 year intervals to within 0.2 years. At this general science thread I provide diagrams which illustrate how Jupiter, Saturn and Neptune return to conjunction every 179 years, which seems to me to indicate a resonance. These three planets are always about 31 degrees of arc further around the ecliptic than their positions 179 years ago. These orbital JSN periods produce a clear gravitational pattern in the wave function of the solar system barycentre as shown here.
Thank you. My understanding was that to calculate the gravitational effect of a body on another body, the mass of both was multiplied and the product divided by the square of the distance. Hence, because the sun has disproportionate mass, Saturn's effect on it is actually very close to 29.5% of Jupiter's. (Inverse square law)

You persist is seeing resonances in cases in which the orbital period ratios are only rough approximations of what are needed for a true resonance. Perhaps you should study some references on the topic of orbital resonance, such as this one:
http://en.wikipedia.org/wiki/Orbital_resonance
I know Wiki articles should not be taken as gospel, but this one looks good. Perhaps Tony and others would like to comment.

But why cannot you have a decimal in resonance? The fact is that every 179 years, Jupiter Saturn and Neptune are in the same relative positions, with error around 0.1%, at a point one twelfth of the way further around the sun. Because this is a function of the four biggest objects of our solar system (sun + 3 gas giants), it produces a readily visible pattern in the plot of the solar system barycentre. I don't understand why this is not classed as resonance. Integer resonances apply when you have two planets, but this is looking at three. On the model of this wiki table of orbital resonances (thank you Hornblower), putting in the numbers Tony gave of J:S:N = 15.094: 6.036 : 1.086 produces, as I calculate it, a mismatch after one cycle of only about 0.3° of arc for Jupiter's position relative to both Saturn and Neptune. This is much less than any of the listed binary mismatches, and has a very long randomisation time of around 100,000 years. Adding in Saturn-Neptune, the mismatch is about 1° of arc per cycle with randomisation period for the whole JSN group of about 30,000 years.

Jupiter-Saturn: 9 cycles = 178.65 years
Jupiter-Neptune: 14 cycles = 178.90 years
Saturn-Neptune: 5 cycles = 179.35 years

JS : JN : SN = 9 : 14 : 5. This relationship between the three largest planets is among the closest to an exact integer of all the resonances in the solar system.

Because you have not shown any convincing evidence that it meets the gravitational dynamic criterion for a resonance. Please go back and read the opening paragraph.

I don't know how you got this link to read "wiki table of orbital resonances" in the final display. They call it "Coincidental 'near' ratios of mean motion", and that is how the URL reads in the raw editing window. They say that even the closest ones listed are dynamically insignificant and you have given no gravitationally based reason to think otherwise.

Close, but still no demonstration of the gravitational dynamics of a true resonance. You appear to be focused on numbers and pretty pictures, rather than on gravitational consequences, if any, of the positions that yield those numbers.

If you really want to see a resonance, go to the Jovian system, where three of the galilean satellites: Ganymede, Europa and Io, have a resonance of 1:2:4.

Secondly, I have not the foggiest idea what the purpose of this thread is, except to show that Jupiter has the greatest influence on the solar system barycenter and all other planets have relatively "minor" contributions.

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Tusenfem, one answer to my questions indicated that Saturn had only 8% of Jupiter's gravitational effect on the sun, but my calculation is that it is 30%, with Uranus and Neptune around 5%. Which is right?

Well, let's see, gravity is proportional to the mass of the planet and inversely proportional to the distance squared of the planet. Taking Jupiter as 1/1 then Saturn is 0.30/1.83, which leads to the ratio 0.30/1.83² = 9×10^-2.

Methinks, you are wrong, you only considered the mass ratio between Jupiter and Saturn, and not the difference in radial direction.

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善數, 不用籌策 (shàn shù, bù yòng chóu cè)
He who is good at counting, uses no counting tools
“A good scientist has freed himself of concepts and keeps his mind open to what is”
道德經, 二十七 (dào dé jīng, 27)

The only thing that one has to consider to find the answer to this question is that the Sun and the planets are particles. Taking a certain point as origin, taking already into account that the Sun is not in the origin, one can easily calculate the location of the "center of mass" of the system r_cm for any given time using the equation:

r_cm(t) = Σ_i m_i r_i(t) / Σ_i m_i

This is the calculation for the center of mass, or barycenter, nothing more, nothing less, it does not have anything to do with gravity, and that is where the confusion comes in between the 8% influence and the 30% influence difference between Jupiter and Saturn. The gravitational pull of Saturn on the Sun is only 8% of that of Jupiter, however, in determining the location of the center of mass of the system Saturn will have 30% of the influence of Jupiter.

Simple comme bonjour.

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善數, 不用籌策 (shàn shù, bù yòng chóu cè)
He who is good at counting, uses no counting tools
“A good scientist has freed himself of concepts and keeps his mind open to what is”
道德經, 二十七 (dào dé jīng, 27)

And yet, the barycentre wave function primarily follows the Jupiter-Saturn difference in radial direction as its main gravitational factor, given that the distance from the sun to the centre of mass is greatest every 19.8 years on the JS conjunction, in a simple underlying pattern that is perturbed by the effects of Uranus and Neptune. Understanding and explaining this observation is the reason for my questions here. If planetary gravitational effects on the sun's position are J=1, S=0.3, U=0.04, N=0.05 and the rest of the planets total .01 as per my table above, Saturn contributes 21% of the overall movement of the sun's position against the barycentre.
Yes, the 'pull' of Saturn is 8%, but the gravitational attraction between two objects is a function of their combined mass, by Newton's inverse square law, and Saturn influences the position of the sun with 30% of the effect of Jupiter. Is not the vector from the centre of mass to the sun a purely gravitational function? I would be interested in references to explain the barycentre formula.

Well, the fact that the three biggest planets combine to produce a very regular pattern in the barycentre resulting from their 5:9:14 combined orbital ratios is easily demonstrated, but I take your point that this is quite a different thing from orbit-clearing resonances.
Thank you for the correction. What I am looking at is a regular relationship between the three biggest planets which have a clear dynamic effect on the barycentre, and in which the integer ratios are more exact than any of the listed 'coincidental near ratios'. I am not trying to claim anything about these other ratios, just trying to understand the data about the gravitational function of the barycentre.
That’s not right! This pretty picture summarises the biggest planetary gravitational relationships of the solar system by graphing their relative amplitude and frequency. And this pretty picture derives the actual 179 year SSB phasing apparent in JPL data purely from the sine functions of the three biggest planets. In the search for a true resonance, the very regular SSB JSN pattern is a strong candidate.

Sorry, you are wrong here, the calculation of the "barycentre" (which is centre of mass) has absolutely nothing to do with gravitational pull, it is just the vector averaged over mass, nothing more, nothing less.

The "barycentre" formula, as you want to call it, is just the calculation of the centre of mass of an object (and then the "object" can exist of separate masses, like the solar system or e.g. the centre of mass of the cluster spacecraft) and is just very very basic mechanics, the formula is given above in my previous post and can be found in any introductory physics book and there is a not too bad wiki page on the topic.

What, however, seems to bother you is the motion of the barycentre. This motion is governed by the motion of the planets which is given by Keppler's laws. The main interaction is the gravitational interaction between planets and the sun, and this gives the periods of the planets.

You keep on talking about "pull" in the 30% relative contribution to the location of the barycentre of Saturn compared to Jupiter. But it is NOT a pull, the barycentre is the location where you can put a stick and rest the system on it, so that the system is in balance, just like the simple (not completely correct) example of a two-armed balance, if you put equal amounts of mass on both sides, then you have balance.

So, at one point in time you find the centre of mass, then you jump in time, and calculate the CoM again, and naturally you find that the Sun has the main influence, and then there will be Jupiter and the Saturn, which contribute their Mj*rj and Ms*rs to the sum etc. and note that although Ms is smaller than Mj, rs is almost double rj, so that again increases the influence of Saturn.

That is the way you calculated the CoM, gravity only gives you the periods with which the planets move.

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善數, 不用籌策 (shàn shù, bù yòng chóu cè)
He who is good at counting, uses no counting tools
“A good scientist has freed himself of concepts and keeps his mind open to what is”
道德經, 二十七 (dào dé jīng, 27)

Please go back and read the Wiki article again. You are showing no evidence that you paid any attention to the type of gravitational action that is indicative of a true resonance.

If you think their treatment of this topic is wrong, please show us a mathematical treatment of gravitational perturbations to back up your case.

Okay, the suggestion that the observed 5:9:14 ratio between the orbits of Jupiter, Saturn and Neptune can be described as a planetary resonance looks to be against the mainstream, so if I wish to defend it I will do need to so in the ATM part of the BAUT forum rather than here.

When Jupiter and Saturn are conjunct, the barycentre is about one solar radius from the centre of the Sun (ie at the surface) due to nothing else than the combined pull of these planets. When Jupiter and Saturn are opposite each other, the Sun is at the barycentre due to their combined pull. My discussion of plotting the barycentre has really been about the position of the sun with respect to the barycentre. The plot of the barycentre is nothing but a function of the combined gravitational effects of the planets and the sun. What is the real difference between "vector averaged over mass" and "gravitational pull"? Are they not just two ways of describing exactly the same thing?
But Jupiter and Saturn do pull the barycentre away from the sun when they are conjunct. This is a simple empirical observation. The barycentre is dynamic, moving in relation to the sun primarily according to the Jupiter-Saturn cycle. I see what you are saying about the weighted balance, but that is a static example. For the solar system, the inverse square law of gravity provides the balance of forces, so the vector is gravitational.

Oh Puhlease! get yourself a introductory book on physics and look in mechanics for Centre of Mass (e.g. in Alonso & Finn, Physics, section Systems of Particles) or look at the Wiki page where the CoM is explained through the equation that I gave in my post above. Do you see any mention there of a force, in that equation?

The plot of the "barycentre" that you showed us shows us where the CoM of the whole solar system is, you can calculate it through the equation that I wrote down above. If it would be gravitationally determined then you would not find such a big influence of Saturn with respect to Jupiter (i.e. the 30 vs 8 % discussion)

And in the solar system CoM, gravity only acts to make the planets move around in their orbits around the sun (Kepler, you might have heard of him) and the motion of the CoM is created by the motion of the planets but the location of the CoM is given by the location of the planets and their mass only and no gravitational interaction.

And I am lost when you say "so the vector is gravitational" I have not the foggiest what vector you are talking about here.

I really cannot make it any simpler that this. You seem hooked on putting gravity in there, for some reason. I also said that the weighted balance was a bad example.

So to summarize:
- CoM of the solar system is given by: r_cm(t) = Σ_i m_i r[/sub]i[/sub](t) / Σ_i m_i
- the location of the planets r_i(t) is given by Kepler's law

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善數, 不用籌策 (shàn shù, bù yòng chóu cè)
He who is good at counting, uses no counting tools
“A good scientist has freed himself of concepts and keeps his mind open to what is”
道德經, 二十七 (dào dé jīng, 27)

As you can see I have some glaring holes in my understanding of astrophysics! At least I am eager to learn, so thank you very much for this explanation. It still leaves me wondering though, gravitysimulator.com says Is this right?

No, technically it is not, because if the pull of the planets would influence where the sun is, why is it not displaced in the direction of maximum pull and have the barycentre at the opposite side?

S------J-------Sun--BC The planets pulling on the Sun
S------J-------BC--Sun The planets orbiting the CoM

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善數, 不用籌策 (shàn shù, bù yòng chóu cè)
He who is good at counting, uses no counting tools
“A good scientist has freed himself of concepts and keeps his mind open to what is”
道德經, 二十七 (dào dé jīng, 27)