*Ed Fix has been back in touch about his solar activity simulation model. Ed couldn’t reveal too much last time around as the paper was pending publication in an Elsevier book. My thanks to Ed for being true to his word and returning here to the talkshop armed with a full explanation of his model and data. The spreadsheet and supporting info are here*.

A couple of months back, David Archibald generated a bit of commotion with a post on WUWT about an as-yet-unpublished paper of mine. At the time, I said I’d talk more about it after the paper had actually been published.

The book _Evidence Based Climate Science_, Dr. Donald Easterbrook, ed. (Elsevier, 2011) has been published as an e-book (hardcover to follow in Sept), and a preview is available on Elsevier’s website, (http://www.elsevierdirect.com/ISBN/9780123859563/EvidenceBased-Climate-Science). So now I am prepared to talk about my paper, included as chapter 14 (beginning on page 335 in the preview). The paper’s title is “The Relationship of Sunspot Cycles to Gravitational Stresses on the Sun: Results of a Proof-of-Concept Simulation”.

This paper presents what I believe is a new approach to linking the motion of the sun around the barycenter of the solar system to the sunspot cycle. I consider this paper to be a progress report after the preliminary phase of a work in progress. This effort differs from earlier work in three main respects.

First, I used a signed sunspot cycle (in my paper, I called it “polarized”). I later learned that this approximately 22 year cycle is called the Hale cycle. A well-known solar physicist told me that it is not correct to plot it this way, because there is some overlap at the minimum between sunspots belonging to the preceding and following cycles. Of course, that same physicist’s well-known Cycle 24 predictions completely missed the long Cycle 23-24 minimum, and only predicted Cycle 24 to start increasing in early 2010 AFTER that had already happened; my model called it in 2008.

Second, this model uses only the <em>radial</em> component of the sun’s motion about the barycenter–that is, the distance of the barycenter from the center of the sun. Specifically, it uses the second derivative wrt time of the radial distance–that is the radial acceleration of the sun wrt the barycenter. This first iteration of the model ignores rotation completely, giving this version of the model some fundamental limitations. That’s why the sub-title says “proof-of-concept”, rather than “42″.

I plotted the radial velocity of the sun along with the polarized sunspots and noticed intriguing similarities. In a blinding flash of inspiration (or maybe indigestion) I realized that the second derivative of distance is acceleration, and that acceleration is simply force scaled by mass (F=ma). If that’s so, why not treat it like a force and see what happens.

The third difference between this approach and others is the use of a dynamic model. I used the simplest possible physical analog; a mass oscillating on a spring with damping. The force equation for that is: F=-kx-bv, where k is the spring constant (which defines the resonant period), x is the displacement from center, b is the damping coefficient, and v is the velocity of the mass. In this sunspot model, x represents the sunspot number, and v is the rate of change of the sunspot number.

There is a third term in the force equation for this application: “+z (d^2 r/dt)” (add a coupling constant, z, multiplied by radial acceleration — second derivative of r wrt time). This adds a forcing function. Now, a=F/m, where a is the rate of change of the rate of change of the sunspot number, and the sunspot number is:

I like the analogy of a small child bouncing a ball, where the child hasn’t yet mastered the finer points of ball-bouncing. His bouncing is likely to be erratic, he may bounce the ball at varying heights, miss the ball sometimes, or hit it on the way down instead of on the way up, etc. If we record the height of the ball at, say, 0.1 sec intervals, can we then build a model that duplicates the trajectory of the ball? We could probably build a reasonably good model of a bouncing ball, and derive the characteristics empirically–air friction, gravitational acceleration, elasticity and rebound damping, etc. However, if we don’t duplicate the forcing function of the child’s hand, the resulting model output will look nothing like the actual trajectory.

The equation was implemented on a spreadsheet (BIG spreadsheet). The time interval between calculations is 1 month, so t = t^2 = 1 in the equation, and v0 = x_t-1 – x_t-2. The equation I used for each iteration in the spreadsheet was:

To set the various coefficients, I started the model in April, 1749 with a value of -90 and rate of change of 0 9 (the approximate peak of Cycle 0) and let it oscillate from that point forward. I adjusted the coefficients until the model output matched the historical sunspot data. I was amazed to discover that it did, at least for a few cycles.

Between 1800 and about 1812, there was an unusually weak and long Cycle 5, followed by a long pause before the start of Cycle 6. Again, in 1890-1912, Cycle 13 Was unusually long and Cycles 13 and 14 were relatively weak, followed by a long pause before Cycle 15 started up. This model didn’t do that; there was obviously something going on that this simple model couldn’t model. So to simulate, in a sense, an aspect of the sunspot cycle generator that this model couldn’t follow, I clamped the output to 0 from 1807 to mid-1811, and again from Nov., 1904 to Sept., 1911, and let the model do its thing after the forced quiescence. In both cases, it started up and followed the subsequent sunspot cycles, with the exception that between 1811 and 1904, the model output is 180 degrees out of phase with the polarized sunspot cycle. To better show the correlation, in this figure I’ve flipped the polarity of the sunspot numbers during the 19th century

This model is one-dimensional; it only operates in the radial direction. However, the latest astronomical observations show that the solar system occupies at least two dimensions, and it rotates. My suspicion was that the rotation had flipped the polarity of the forcing function, rather than the polarity of the sunspots flipping. I’ve since modified that suspicion a bit. However, we do know that the Cycle 15 was in fact reversed in polarity from Cycle 14–that’s when Hale first observed the polarity reversal.

Remember the bouncing ball analogy. Up until this point, the model had always been started at -90 in 1749. What happens if I start the model at 0 in, say, 1500 and let it run? What happens to it in the 18th century and later? As the next figure shows, not much changes. Seeing this was my EUREKA moment.

As I see it, there are a couple of take-aways from this. First, there are two parts to the problem: the forcing function that activates the sunspot cycle, and the resonant characteristics of the sunspot generator itself. Concentrating on only the driving force, as many barycentrists do, or only on the internal mechanisms of the sun, as most solar physicists do, would cause you to miss crucial aspects of the overall cycle.

Second, I suspect there may be a couple of coincidences tied up in all of this. First coincidence; I believe the fact that Jupiter’s orbital period is nearly the same as the traditional Schwabe sunspot cycle is pure coincidence and probably inconsequential. The base period of the force produced by the sun’s acceleration toward and away from the barycenter is about 20 years–the synodic period of Jupiter and Saturn’s orbits. This is strongly modulated by the motions of Uranus and Neptune.

Second coincidence: the sun’s intrinsic resonant frequency is near the frequency of the driving force. I think of the barycenter’s movement as exciting the sunspot generator, rather than driving it. If you were to pull on the sun only once and never again, it might produce 2 or 3 weak eleven-year cycles, and then never spit out another sunspot. However, if the sun has it’s own intrinsic resonant frequency, and it’s excited by an outside quasi-periodic force, that force’s frequency needs to be near the sun’s resonant frequency, or nothing much will happen. Just speculating here, but if Jupiter or Saturn were in different orbits, we might not have much of a sunspot cycle at all.

So that’s it. Any comments? Unlike some in the climate science world, I can take criticism.

Hi Ed,

many thanks for coming back here to discuss your work.

Preliminary comment regarding your ‘restarts’ in 1807 and 1904, you probably need to read this paper:

http://arxiv.org/pdf/astro-ph/0507269 which was kindly linked by Bill yesterday.

That paper also has some points of interest concerning the ‘phase reversal’ issue. they find one at 1945. Lanscheidt has interesting things to say about phase reversals too, and his ideas linking them to zero point crossings at times when higher order cycles have changed sign is worth considering.

The paper also find some solar rotation parameters which coincide with ~19-21 year periods.

I find your idea that we are dealing with both the planetary cycles and the Sun’s natural oscillatory frequency gels with my own thoughts on the probable coupling of the solar system dynamics and the ‘solar dynamo’.

Excellent…

Hi Ed, I can see that you have done a lot of work getting to this point. I have also traveled down many of these same roads.There are a lot of interesting things in it, but also I think a major defect.

(1) The 2nd differential you are using is indeed solar acceleration, or the total force of the planets on the Sun as a vector. If you study the cycles in this you will find that it includes 11.86 years (J), 11.07 years (J-V-E) and perhaps 9.93 years (1/2 19.8g J-S), which are close to real solar cycles. But the real solar cycles have 11.07 years dominating while the forces are dominated by 11.86 years and 19.86 years. You have “fixed” this by doing the restarts twice during the period. However the real problem is that the forces have the wrong period. Jose also “fixed” this by arbitrarily changing signs whenever it got out of line. This is not actually sound.

(2) You have used a memory system for the Sun and a forced oscillation. I did a similar thing in my model and achieved r=0.66 over the entire period without any restarts. However I used a calculation that is equivalent to using the z component of the solar motion. It really works. Not only that, it is based on previously uncalculated consequences of GR effects on the solar interior. Everyone just assumes that the effects are far too small. But they never looked at the results of 1/2 *a *t^2 when t gets large. Tallbloke has info about this in his first article on his blog (I think). The memory of the Sun is important I think in achieving the times when the Solar cycle almost stops because the forces act head on to the cycle.

Regards, Ray

Ray,

Timo Niroma found that solar cycle lengths tend to cluster around 10.4 and 12 years, rather than 11.07, the long term average.

But you know why this is anyway.

https://tallbloke.wordpress.com/2010/07/20/timo-niroma-on-solar-cycle-lengths/

Roy Martin’s 55 year cycle deserves a look too:

https://tallbloke.wordpress.com/2010/08/13/roy-martin-new-planetary-solar-cyclicity-hypothesis/

And my development of his work:

https://tallbloke.wordpress.com/2010/08/21/breakthrough-major-discovery-on-planetary-solar-connection/

We should do a post together on the z axis work, recapitulating your original discoveries and my additional discoveries. Let’s make the time.

Posted on “Breakthrough: major discovery on planetary – solar connection”:

Tenuc says:

August 23, 2010 at 9:28 pm

“Interesting take on what SC24 will look like, with a prediction of cycle max April 2011…”

BTW: the two links provided both seem to be bad links – GP

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Yes, it certainly does look like the SC24 SSN and F10.7 peaks were back in April 2011:

http://leif.org/research/TSI-SORCE-2008-now.png

Even if there is a second peak later, as there was in SC23, it will most likely be lower, as it was then, because of the impending Grand Minimum of cycles 24 and 25. Cycle 23 peaked in April. 2000:

http://www.solen.info/solar/solcycle.html

So, it looks like it will be 11 years, SC23 peak to SC24 peak – Just about the only normal characteristic of these two cycles.

-Gerry Pease

Gerry, which links are bad?

These links, in Tenuc’s August 23, 2010 comment:

http://psc.suijs.info/images/SC24_predict_200911.jpg

Full info here:-

http://psc.suijs.info/sc24predict1.htm

Brilliant match between model and record. Now, I only have one quibble. The past is all very well, but you stopped your graphs at 2010.

Can we have the future please ? To 2100 would be nice.

tallbloke says:

July 25, 2011 at 11:33 am:

The paper you linked to is essentially identical to one I referenced in the paper (Javaraiah, J., 2005). One thing I mentioned in passing in the paper, and left out of this summary, is that the forcing function I used is assymetrical (the positive-going peaks of the radial acceleration are much higher and sharper than the negative-going excursions) (reference figures 9 and 10 in the paper–I’ll send you a copy of Fig 9 that you can insert here if you want). This assymetry may well explain the G-O rule Javaraiah mentions. Also, the periods of highest radial acceleration coincide exactly with the periods of greatest orbital angular velocity/orbital speed/sharpest orbital curvature mentioned in other studies.

My working hypothesis is that the periods around 1807 and 1904 were disrupted by the driving function slipping out of phase with the sun’s oscillations, and those oscillations were temporarily suppressed by the driving force, rather than reinforced. I’m not actually convinced that with only two examples, we can state a general periodicity as he does. I guess we’ll find out in another century or two.

Ray Tomes says:

July 25, 2011 at 12:15 pm

Ray, thanks for taking the time to post a thoughtful critique. I agree that the arbitrary phase reversal I introduced into the sunspot cycle is open to criticism, and I understand the impulse to disregard the whole concept on that basis. And I could be completely wrong–it has happened before. Just ask my wife.

This effort actually began from a clean slate. I had absolutely no knowledge of the existing body of work. I just read a comment in a book about some sort of supposed link between the barycenter’s motion and the sunspot cycle, so I decided to try this out and see what I could see. The simple spring analog model is only a place-holder for a more realistic model (TBD), to test whether this was worth pursuing. The paper presents my results at the point where I couldn’t push further ahead using only a spreadsheet.

I have done some further work where the solar cycle phase reversals are not necessary, and achieved tantalizing-but-inconclusive results. The main thing I have done is to introduce the concept that when the angular position of the forcing function speeds up, that vector slips ahead of the oscillatory vector of the sun, to the point that the phase of the force reverses temporarily. That reversal doesn’t hold (as my 19th century phase reversal implies), because the forcing function is assymetrical, and so the resulting oscillation will settle back to the previous pattern after a short time. This has the effect of obviating the necessity of arbitrarily reversing the phase of the sunspots.

About the many periods near what you term the 11.07 year “real solar cycle”, I disagree with your interpretation. The ~11 year Schwabe cycle is certainly the traditional cycle, but I believe it’s an incomplete picture. Before I’d even heard of the Hale cycle, I noticed that the minima are typically sharper than the maxima; a characteristic typical of the absolute value of a sinusoid. Then I learned that the magnetic polarities of the sunspots of succeding cycles were reversed, and the idea clicked that the “real solar cycle” is actually the ~22 year Hale cycle, and the traditional Schwabe cycle is just an artifact of the astronomers’ inability to measure the magnetic fields for the first 300 years of observing and counting sunspots. If that’s not true, then my whole paradigm evaporates. That might still happen.

About your point 2, you and I are thinking along similar lines, I believe. Where you think in terms of the “memor of the sun”, I think in terms of an internal oscillation. The sun is always recovering from whatever perturbed it last, as well as reacting to the present perturbation. To clarify, by “z axis” do you mean in the direction of the sun’s poles (perpendicular to the ecliptic)? This is a component of the motion I haven’t even contemplated modeling.

J Martin says:

July 25, 2011 at 7:19 pm

“Brilliant…”

That might be overstating things a bit.

“The past is all very well…Can we have the future please”

But predicting the past is so much safer!

If you look at figure 10 in the paper on the Elsevier website, you’ll see I did run the model out a few more years. HOWEVER, I have no confidence the next couple of cycles will play out the way they’re shown on that graph. I’ve run a simulation where the phase of the forcing function reverses in 2013 during the barycenter’s next close approach through the sun, and the resulting sunspot cycle gets a kick in the negative direction, prolonging and strengthening the current cycle.

That’s just a guess at this point, since the model still doesn’t replicate the Maunder Minimum. When I can predict all the known past activity, I’ll feel a lot more confident predicting the future.

This makes sense given the recent posit that the size of a solar cycle is a result of the previous cycle during a time of sensitivity. About the conveyor.

Keep looking “past” the 11 year cycle – I think you will find that the 22 year cycle (one positive plus one negative cycle) is the beginning of the your solution. But keep in mind that the positive (first half of the 22 year cycle) does not actually “end”, but rather “coasts” to a low value (not creating the visible sunspots that come from the negative half becoming dominant) but not “zero” either.

Then, at the beginning of the next positive half of the 22 year cycle, the rate of increase of the “old” positive ramps up and becomes dominant for 11 (approximately) year. But try not to separate the “positives” into unique little events. You may find it illuminating to try your sequence with two 22 year cycles.

Look also for a 66 year forcing: I have seen sunspot counts seem to come in groups of two high, one low-to-medium, then two low count cycles, then another medium count, then two more high counts, one or two medium cycles … Note this is by observation of quantity of of the maximum number of spots in a cycle, not by time or duration of the cycles. But again, it seems to indicate a pattern in the cyclic nature of the currents in the upper atmosphere of the sun.

And, it takes little to move a flow of repulsive charges of very light ions moving at high speed near the surface of the sun’s plasma. One need not “move” the entire mass of the sun to affect circulating currents in a plasma – only a much smaller amount of light particles near the surface of the sun need be affected by small amounts.

Thank you for your time.

I think “cycle” studies are the way to go. Roy Martins work looks very good.

The sun contains 99.8% of the SS mass. Even if the mass of all the planets were in a lump touching the surface of the sun, the ssb would be <.2% of the sun's radius from the center of the sun. In reality the distance that the ssb could deviate from the suns center is a small fraction of that hypothesized <.2%.

Then the position of the ssb is only a vague underpinning for the surface actions of the sun.

Bill, the sun moves up to nearly a full solar diameter from the SS-COM when the alignments are at max.

Jupiter’s mass is 2.5 times that of all the other planets in our Solar System combined—this is so massive that its barycenter with the Sun lies above the Sun’s surface at 1.068 solar radii from the Sun’s center.

http://en.wikipedia.org/wiki/Jupiter

@tallbloke says:

July 25, 2011 at 12:28 pm

“Timo Niroma found that solar cycle lengths tend to cluster around 10.4 and 12 years, rather than 11.07, the long term average.”

Which is what one would expect. Odd numbered cycles are at Ea/Ve oppositions in line with Jupiter, and even numbered cycles are at Ea/Ve conjuncts in line with Jupiter, so the periods would have to group at 6.5 or 7.5 Ea/Ve synodic periods ~ 10.39yrs and 11.99yrs.

Ed, could you comment on your impossibly short SC25 ? (2017-2025)

http://wattsupwiththat.files.wordpress.com/2011/05/sosoon3.png

” tend to cluster around 10.4 and 12 years, ”

Which also fits a chaotic oscillator with two modes.

Ulric Lyons says:

July 26, 2011 at 1:12 pm

“Ed, could you comment on your impossibly short SC25 ?”

Certainly, I can comment. But…”impossibly short”? How about “very improbably” or “completely unprecedentedly”? But surely it’s not impossible!

Actually, I don’t believe it. I am very cautious of making predictions at this stage of the model’s development. I have, however, received one confirmation. In mid 2008, I noted that if the sunspot activity remained near 0 until the end of 2008, then it wouldn’t start up again until very late 2009 to early 2010. It was gratifying to see that happen.

However, the underlying oscillator model is a simple damped, forced spring analog, and that’s certainly not anything like what happens in the sun. For another thing this model doesn’t rotate. It’s strictly one-dimensional, while recent astronomical observations suggest the solar system occupies at least two dimensions. In other words, this result represents the projection of a two-dimensional process onto one-dimensional space.

Here’s one conceptual consequence. When the SS Barycenter swoops in near the sun’s center, the angular speed is very much faster than normal. If the acceleration force vector slips far ahead of the rotation vector, the resultant force, instead of pulling from one direction is suddenly pulling from the opposite direction which is equivalent to “pushing” from the original direction. In other words, the polarity of the forcing function has flipped. In 2013, just such an event will occur, and I am speculating that a weak Cycle 24 will strengthen after 2013, and overall, Cycle 24 will be somewhat longer than normal. It ought to be interesting in any event.

bill says:

July 26, 2011 at 11:24 am

Bill, as tallbloke says, the sun moves on a surprisingly large, chaotic orbit around the barycenter of the solar system. I’m attaching a graph showing that orbit over the 20th century. The gray circle in the center of the graph shows the size of the sun.

I didn’t mention it before, but the barycenter positions I used in this work are calculated only from the sun and the outer giant planets: Jupiter, Saturn, Uranus, and Neptune. The rocky inner planets were not used in the calculation, since their contribution to the barycenter’s position is miniscule at best.

Looks like I can’t stick a graph in the middle of a reply. Here’s a URL for the graph I meant to attach to the last comment.

http://www.mediafire.com/imgbnc.php/1f20fe417ed77873274be6c65c63e5f6e278e34ef177fe9bd812b51465ea12b66g.jpg

Rog, yes, individual solar cycles cluster around 10.4 and 12.0 years as expected from the J-V-E cycles as shown in my “Towards a Unified Theory of Cycles” paper and also by Jean-Pierre Desmoulins on his great web site. However in a spectrum, the peak near 11.07 years (excatly where is dependent on exactly what data you use) is always the strongest peak in the SSN spectrum. The peaks near 11.8 and 10.0 years are weaker. It is tempting to relates these two other peaks to J perihelion period (11.86 years) and J-S conjunctions (9.93 years).

Yes, let us do an update on the z axis work. I got some x,y,z axis data from Horizons, but I am not sure that it is in the solar equator plane frame. Do you know? If we have this data for 11,000 years, then I can easily process in CATS against the various other series.

Regards, Ray

It is true that the barycentre position is largely determined by the 4 gas giants. However, if you use the 2nd time differential of the distance, then the inner planets do become significant as you are then actually calculating acceleration. Please use actual positions data from say Horizons, rather than calculate. See my paper http://www.cyclesresearchinstitute.org/cycles-general/tomes_unified_cycles.pdf for a comparison table (Table 2) of planetary effects under the various proposed mechanisms. When you take 2nd difference, then even Mercury and Mars have bigger effects than Uranus and Neptune.

Ray,

An aside, beware aliasing in long period orbital ephemeris, been caught by this, so today I am reluctant to go longer than 10 day step (not sure even that is safe given Mercury and our moon). This might mean properly decimating afterwards, or alias again. Can look sane, just don’t delve deep, artefacts appear.

Which ephemeris do you use for long period? Seems a bit of a mess and I have seen too many conflicting texts.

Feel free to disagree.

The dominant period of solar barycentric radial acceleration is J-N.

Ed: Only site admins can insert graphs in comments, feel free to request, but I tend to use links to graphics to put in the actual graphics when I spot them anyway. An interesting coincidence between your model having the next zero crossing point at 2017 is that the JEV cycles do the same. See the thread I linked earlier:

https://tallbloke.wordpress.com/2010/08/21/breakthrough-major-discovery-on-planetary-solar-connection/

Ignore the cycle 24 ‘prediction’, I was using Roy Martin’s spreadsheet which is tidally based for my own purpose of investigating the E/M aspects along the Parker Spiral.

Ray: Horizons gives x,y,z data relative to the solar rotational plane, but only from -3000 to +3000. I’ll contact you about the write up for the z-axis. Interesting obs regarding the solar spectrum, it tends to confirm my thoughts about the inner planets refining the timing of the solar cycle, while the outer planets have more effect on amplitude as well as being the main timing driver. I think this is why things get a bit out of kilter when the Barycentre is close to the solar core, and agree with you that the inner planets should not be neglected, especially at these times.

Paul: Could you expand a little? Is this because Neptune is slowest moving, so Jupiter departs from it quicker than from Saturn an Uranus?

Ed Fix says:

July 27, 2011 at 1:40 am

“, and I am speculating that a weak Cycle 24 will strengthen after 2013,”

From what I see as the causes of past solar minimum`s, it should weaken from 2014.

Tallbloke, acceleration emphasizes CHANGES – i.e. all the little vertical wiggles become peaks & troughs (and all the longer, smoother variations get flattened out, reducing their power). The highest-frequency large-amplitude changes are J on N. With the whole jovian family running around the track, the pair meeting most frequently are the fast grandson (J) and the slow grandfather (N) since their speeds are most strongly contrasted. With increasing derivative order, the power spectrum shifts towards higher frequencies, but at the 2nd derivative (i.e. acceleration) the inner planets remain subordinate to J-N.

EF: “The base period of the force produced by the sun’s acceleration toward and away from the barycenter is about 20 years–the synodic period of Jupiter and Saturn’s orbits.”2 problems here:

1. Accelerating my car towards an object does not exert a force on the object.

2. J-S is the most dominant term in the rate of change, but

NOTin the rate of changeof the rate of change.EF: “I believe the fact that Jupiter’s orbital period is nearly the same as the traditional Schwabe sunspot cycle is pure coincidence and probably inconsequential.”Your model is a

linear combination. It’seasy enoughto design a linear combination that spits out J+N on average. Then a few tweaks help match amplitude. Stat 101 in action.Best Regards.

Paul Vaughan says:

July 27, 2011 at 1:19 pm

Paul, thanks, that’s very clear. See, you can when you want to. ;)

Ulric Lyons says:

July 27, 2011 at 12:31 pm

Ed Fix says:

July 27, 2011 at 1:40 am

“, and I am speculating that a weak Cycle 24 will strengthen after 2013,”

From what I see as the causes of past solar minimum`s, it should weaken from 2014.

Yep, I agree withe Ulric on this. The ‘normal rules’ don’t apply when the Sun goes into a big minimum. Working out what the abnormal rules are is a high priority at the moment, because otherwise we can’t makev good predictions during the next 20 years, and that’ll make us look like we have nothing to say. :(

TB , Ed and all

I got stuck on visualizing a system BC inside a body. When the system BC went though the “wall” like a ghost, I blinked and my head bounced back toward center – perhaps inadvertently groping along the trail of gravity without stopping to think that gravity “walks” backward to the movement of the system BB.

I’ll wear that one for a while to remind myself with ! !

It presently appears that SC24 SSN and F10.7 peaks were back in April 2011:

http://leif.org/research/TSI-SORCE-2008-now.png

Though there will likely be a second peak later, as there was in SC23, I believe it will probably be lower, as it was for SC23, because of the impending Grand Minima of cycles 24 and 25. Cycle 23 peaked in April. 2000:

http://www.solen.info/solar/solcycle.html

So, it looks like it may be exactly 11 years, SC23 peak to SC24 peak.

-Gerry Pease

P.S. Tallbloke, please remove or correct my previous comment, dated July 25, 2011 at 5:34 pm. It had a typo in the last sentence (10 years instead of 11).

[Fixed - TB]Bill, no worries, we’ve all managed to tie our brains in knots trying to visualize barycentric motion at some point. :)

Gerry, I think cycle 24 solar max will be hard to call for some time yet. We may get a brief peak which goes higher, but with lower activity either side. Could be a question of criteria.

Wow, I’m getting really behind on replies.

Ray Tomes says:

July 27, 2011 at 2:21 am

“…if you use the 2nd time differential of the distance, then the inner planets do become significant as you are then actually calculating acceleration.”

Ray, I pretty much completely disagree with that sentence. The inner planets contribute a negligible amount to the position of the barycenter–too small and too close in–and the first, second, third, or any derivative cannot possibly change that.

I did, in fact, download the Horizons barycenter ephemeris, and found it completely un-usable in its raw form for my purposes. As tchannon mentioned, when you take the first and especially the second difference, you get spikey, quasi-periodic noise that I finally figured out was aliasing due to mismatches between calculation and sampling rates.

My guess is the Horizons ephemerides are not calculated on the fly. The positions are precalculated at standard intervals, and stored in tables. When you request a specific calculation interval at specific dates (I used the 15th of each month), the program gets the nearest date positions of the required bodies and interpolates a position for your specific date. This introduces errors which are negligibly small in absolute position. But when you, for instance, subtract previous position from the current to get velocity, and subtract previous velocity from current to get acceleration, the errors become as large as the signal.

To get an ephemeris without the aliasing, you really should start with a good position and calculate the ephemeris directly from orbital parameters. However, I, being me, decided to take door number 3. I fitted a smooth curve to each of the four outer planets’ orbits, varying the orbital radius sinusoidally between perihelion and aphelion, likewise the angular position, developed the coefficients empirically to get a good fit between the calculated approximation and ephemeris data (without the aliasing). the resulting planetary positions were within 0.5% in radius and 0.5 degrees angular position wrt the downloaded ephemeris throughout the calculation period (1500-2200). I then calculated a 5-body barycenter to use as my input, and as a final check I compared that to the downloaded barycenter position.

The inner planets do, however have a tidal effect on the sun. To produce tidal effects, the body needs to be close and lots of mass helps. As I understane it, Earth and Venus produce measurable tides on the sun. Apparently even Jupiter produces tides due to its mass, even though it is so far out. These effects are small, but are measurable because the body of the sun is a fluid, not a solid. Tidal effects move some parts of the sun more than others, in effect stirring it up a bit. Hung (Hung, C., 2007. Apparent Relations Between Solar Activity and Solar Tides Caused by the Planets. NASA/TM-2007-214817, Cleveland, OH:NASA John H. Glenn Research Center) has shown correlations between the positions of the inner planets and flares from existing sunspots. Tides are caused almost exclusively by the inner planets. I don’t think it’s credible that this effect drives the sunspot cycle.

The motion of the barycenter, on the other hand, is caused almost totally by the outer giant planets, and moves the whole body of the sun, shaking it around an astounding amount. Fairbridge and Shirley (Fairbridge, R. W., J. H. Shirley, 1987, Prolonged Minima and the 179-Yr Cycle of the Solar Inertial Motion:Solar Physics, vol. 110, p. 191-220.) made this distinction long before I thought of it.

You can say that for my purposes I needed the sun shaken, not stirred.

tallbloke says:

July 27, 2011 at 2:36 pm

“Ulric Lyons says:

July 27, 2011 at 12:31 pm

Ed Fix says:

July 27, 2011 at 1:40 am

“, and I am speculating that a weak Cycle 24 will strengthen after 2013,”

From what I see as the causes of past solar minimum`s, it should weaken from 2014.

Yep, I agree withe Ulric on this. ”

Well, we have two competing predictions, and we only have to wait a few years to see the sun do something nobody expects :)

ED and all

would a model be more likely to give “DSN” values rather than actual sunspot count ?

Addendum:

J+N = how long it takes a combined grandson/grandfather effort to make 1 full sweep of the circular track (with the grandson doing most of the work, but the grandfather saving the grandson a bit of the work).

J + N ~= Schwabe

2*(J+N) = harmonic mean of J & N ~= Hale

Paul Vaughan says:

July 27, 2011 at 1:19 pm

Paul, I’m not following you here. First, I should have been more precise in my terminology. When I said “base period” I was referring to the dominant frequency component. A power spectral density of the barycenter position would show by far the strongest spike at about 20 years; the J-S component. I don’t see how differentiation could change the PSD. The first derivative of a sine wave is a cosine, and the derivative of that (second derivative of the sine) is a sine, frequency unchanged. The SS BC’s motion (being the combination of elliptical orbits) is necessarily a linear combination of sinusoids. Here’s a figure from my paper where the radial acceleration is shown as the dashed line. I haven’t done a frequency analysis, but visually I see a pretty strong ~20 year periodicity.

http://www.mediafire.com/imgbnc.php/446b750f9ae8903699252e5882974c8e5182257a698ca351ac2b3868a22f5f2c6g.jpg

Anyway, where I was talking about the possible coincidence of the barycenter’s motion matching the resonant frequency of the sun, that was just speculation, and peripheral to

“PV: Accelerating my car towards an object does not exert a force on the object.”

If your car has the mass and gravitation of the sun, it sure does. The sun “sees” a mass equal to its own equidistant opposite the barycenter, and that virtual mass is moving toward and away from the sun, moving the sun toward and away from the barycenter an equal amount. The mass is virtual, but the force is real.

We see the sun moving toward and away from the barycenter and we can calculate its acceleration. If an acceleration is observed, then a force is acting. F=ma has no exceptions.

bill says:

July 28, 2011 at 4:15 am

“ED and all

would a model be more likely to give “DSN” values rather than actual sunspot count ?”

bill, I don’t know what you mean by “DSN”. In general, though, you’ve got to be generally skeptical of the output of any model, especially if it hasn’t been validated in some way. Many times, I’ve seen modelers fall in love with their models, even to the point of believing if reality doesn’t follow the model, there must be something wrong with reality.

I try not to be like that.

tallbloke says:

July 27, 2011 at 8:10 pm

“Gerry, I think cycle 24 solar max will be hard to call for some time yet. We may get a brief peak which goes higher, but with lower activity either side. Could be a question of criteria.”

Yes, it really is to early to call but I’m sticking my neck out anyway, and if I’m wrong it’s not a big deal to me because I know that nobody else really knows at this point exactly what is going tol happen. Indeed, we will possibly get many brief peaks that are higher, but I actually expect the monthly SSN to be progressively lower even on those months. This has been the trend since April, and I think the trend may well continue all the way through brief peaks in 2012, 2013, and 2014 because of the persistent (and apparently increasing) suppression of spots and radio activity through both the early and later stages of SC24.

Even more interesting is the severe suppression of SC25 activity currently being predicted by mainstream solar observers. That will not be terribly surprising if there is a very long history of increasing activity suppression in SC24 before SC25 even starts. This would, in turn, constitute a replay of sorts of the prolonged minimum of SC23 that led up to the unexpected lack of activity in SC24.

Paul Vaughan says:

July 28, 2011 at 4:51 am

“Addendum:…”

Paul, thanks for the clarification. I kind of figured that was what you had in mind.

tallbloke, you really should get more sleep!!

I don’t know if you’ve gotten my e-mails, but I tried to send you the data a couple of days ago, and it bounced for being too big. Here’s a link to it; you might want to move it up to the original post to make it easy for folks to find. I’ll leave it up to you whether you want to host it on your own server.

http://www.mediafire.com/file/pma8d5cft38wkbm/Sunspot%20Simulation.zip

Now, I’m going to bed.

Ed, too busy to check personal mail in the last 48 hours! I’ve uploaded the file to

http://tallbloke.net/edfix/Sunspot_Simulation.zip

Thanks, Ed, for a thought provoking paper and a simple model which seems to fit reality regarding historical observational data, which although of poor quality is all we’ve got! I agree with your model of the solar system behaving a complex damped forced oscillator and that changes in the dynamics of the total sysem could either trigger the solar cycle through resonant effects, or the solar cycle has ‘entrained’ the solar systems observed orbital motions over very long periods of time. It is amazing how the precise harmony of the estimate 25,690y of the precession of the terrestrial equinox divided by 144 gives the length of the 178.5y barycentre cycle and is an indication of the intricate linkages of the solar system ‘machine’.

Current orbital theory states that all objects in the solar system go round an ever changing center of mass which is close to, or within the sun – the barycentre. However, as Fred Hoyle once stated, the solar system is not isolated from the effects of gravity from other massive objects in the rest of the galaxy. This is one of many potential external influences which need to be considered when looking for answers.

The big question, I think, is how the sun responds to gravity and could changes to the position of the barycentre result in a force being felt by the sun? My view is that the movement in position of the Sun, as a result of changes to the gravitational field from planets/universe, is translational, with all parts of it moving in unison. This means the Sun will “feel” no internal force, but there will be a very small amount of tidal effect, which may or may not have any effect on its electromagnetic activity.

Perhaps a better understanding of gravity and EM effects would help understanding?

Ed Fix

Compliments on your insight and preliminary development. I posted a brief notice at Climate Etc.

I was thinking to suggest exploring a full 3D vector motion analysis (following tallbloke’s fascinating graph on the LOD, temperature and Solar SSB), and then find Ray Tomes was thinking such thoughts ahead of me. The length of the 3D vector displacement may be helpful compared to your current radial distance. With such a 3D analysis, I expect you may not need to change reference planes for the orbital data. A 4th order Runga Kutta integration method or better may help.

Later incorporating the Livingston & Penn analysis (with disappearance of sunspots below a given magnetic field) may help with the solar cycle magnitude around the Maunder Minimum. Happy hunting.

Rog, you write “Horizons gives x,y,z data relative to the solar rotational plane, but only from -3000 to +3000.”

No problem, if they have x, y, z for some other system for longer periods (back 11,000 years) it is easy enough to convert the rest based on the relationship in the last 3000 years. I don’t think that the Sun axis precession is more than a tiny fraction of the earth’s (but it would be nice to know this for sure).

Hi Tim, yes, I am aware of aliasing effects. Always use intervals that give at least 3 measyrements per shortest cycle under consideration. So 10 days copes even with the Moon.

Ed, lots of good points in what you say. Many of these things are known:

(1) Edward Dewey used alternate cycles reversed and log (sunspots + 20) to correct for asymmetry of peaks and troughs. See http://cyclesresearchinstitute.wordpress.com/2010/07/16/the-lambda-function/

(2) Data from 2500 years of aurora observations gives sunspot cycle as 11.1 years average.

(3) I agree with your spring concept – I also used this. It is the means to get the times when spots die out.

(4) The spiky stuff from 2nd differences is the inner planets. You probably didn’t use short enough time intervals.

(5) I reiterate, if you are taking 2nd difference of COM motion, then you are actually calculating acceleration of gravitational force (not tidal, not COM). That is a fact.

(6) Now you just need to change to z axis and you will get the right results. :-)

Ed, forgot to mention that I get a spring natural period of 10.5 years. Do you have a measured period for that?

@Ed Fix (July 28, 2011 at 4:59 am)

Ed, I encourage you to invest the effort needed to see that the dominant term in solar barycentric radial acceleration is J-N. As an exercise, I can suggest generating a simple succession of time series plots for incrementing derivative orders of solar barycentric radius and noting the progression of relative power. (Don’t stop at the 2nd derivative.)

Ray, I think the JPL DE405/406 ephemeris only gives +/- 3000 years whichever frame of reference you select. I’ll do some hunting around.

[EDIT]The swiss ephemeris is for 12k years if we can find a public domain copy.Ed

DSN Total pixel area X average spot darkness.

http://www.landscheidt.info/?q=node/185

bill, the output of my model is an analog to theinternational sunspot number, available from NOAA/NGDC or from the SIDC in Belgian. If I can ever get it to replicate things like the Maunder Minimum, etc, then I’ll worry about which specific sunspot count is the best one to use.

Paul Vaughan says:

July 28, 2011 at 2:15 pm

“I encourage you to invest the effort needed to see…”

Paul, it actually didn’t require much effort. I got out my CRC math tables to review derivatives of sinusoids, and discovered I forgot something during my late-night off-the-top-of-my-head blogging. The derivative of a sine is indeed a cosine of the same frequency (and vice versa for cosines), but I’d forgot you also must multiply the resulting cosine by the derivative of the sine’s argument:

d/dt (sin(ft)) = d(ft)/dt cos(ft) = f cos(ft)

For a fixed frequency (f), the effect is to multiply the resulting cosine by the frequency. That means that when dealing with a linear combination of sinusoids, the higher frequency components are enhanced more than the lower frequency components, with each derivative, just as you said.

Now, that’s the general case. In this specific case, I’m still not sure if the second derivative of the barycenter position boosts the J-N component enough to overtake the huge J-S component in the position function.

I’ve downloaded a FFT library for my compiler, but it may be a while before I can get to it. As I said, this question is actually preipheral to what I’m trying to do.

Anyway, thanks for setting me straight.

Ed

Ray Tomes says:

July 28, 2011 at 12:16 pm

“I get a spring natural period of 10.5 years”

Ray, my paradigm only works on the Hale cycle, not the Schwabe, so your 10.5 year resonant period would correspond to a 21 year Hale cycle. My latest efforts settled in on periods from 21-22 years, so we’re in the same ballpark there.

Ray Tomes says:

July 28, 2011 at 12:15 pm

“(4) The spiky stuff from 2nd differences is the inner planets. You probably didn’t use short enough time intervals.”

I used one month intervals to match the monthly sunspot cycle. That is easily enough samples to satisfy Dr. Nyquist when dealing with frequencies measured in years or decades per cycle.

I tried an experiment with the ephemerides I have already downloaded. I had downloaded a SS Barycenter from the Horizons server, and I had downloaded individual ephemerides for each of the four outer planets and used them to construct my smoothed BC ephemeris. Today, I also took the four planets and calculated a radial acceleration ephemeris using the raw data (unsmoothed).

This graph shows the three radial acceleration curves for the 20th century:

http://www.mediafire.com/imgbnc.php/2e34f65e7c9b3e2085390b48046182c6c867a53c2f4284b2eef31cd5339b04786g.jpg

and this one shows the 50′s:

http://www.mediafire.com/imgbnc.php/c6d43686b765d9135d086144623afb470eb813f5283c7fcc3f583d483b0266826g.jpg

Since the outer planets ephemerides has no information on the inner planets’ orbits, the spiky stuff should have vanished if it were really from the inner planets. There definitely seems to be something else going on here. I suspect the difference between the red and green curves is the real inner-planet influence.

The other half of the experiment would be to download the inner planets’ ephemerides, smooth them,and build a 9-body barycenter.

I re-read the Horizons documentation, but didn’t find an explanation there.

Any thoughts?

Ed

Hi Ed

The fact that there are 5 peaks per annum with one really big one is a bit odd. Mercury is the fastest Planet and would make only 4 (+ a smidgeon) circuits relative to barycentre. Venus should lap Jupiter every 237 days, but no sign of that. It should be the largest factor in 2nd differential of COM. The horizon data should be accurate enough and not need smoothing. It might be a little while before I have all this in a form to see what I get. But it will happen. What happened in 1951 and 1990 (and almost in 1913)?

Ray

Ray, the barycenter and the solar core converge at those dates.

racookpe1978 says:

July 26, 2011 at 2:25 am

re your 66 year cycle

I fould a very decided approximate 66 year cycle in the NCDC land and ocean global temperature index, as it existed several years ago.

(They keep mucking up with the past figures so I eventually gave up in disgust).

The cycle was more a series of approximate 33 year linear zig zag functions than a sine wave, cycling around a linear increasing line of about 0.7 degrees celcius per 100 years.

I fitted a stock market chartist style channel to the data and found every single item of annual data fitted inside my channel, which was plus and minus 0.15 degrees on either side of the 33 year linear trend lines.

All this suggests to me that this 66 year cycle in temperature is driven by the trends in the sunspot cycles. What causes the long term (since 1880) underylying linear trend is not clear.

Is it CO2 (unlikely), longer term sunspot cycles (possible) or just growing UHI (also possible).

Ray Tomes says:

July 30, 2011 at 8:19 am

“The fact that there are 5 peaks per annum with one really big one is a bit odd…”

When I saw the values bouncing up and down on pretty much every single sample, I figured I was dealing with either rounding error, or some fundamental limitation of the underlying data, so decided to sidestep the problem by making a smoothed approximation of each of the 4 outer planets and calculating my own barycenter.

This morning, I found this in the Horizons documentation:

“Cartesian state vectors are output in all their 16 decimal-place glory. This does not mean all digits are physically meaningful…”

No kidding. The cartesian coordinates for Jupiter’s position, for instance, imply precision +/- 0.5 mm.

“For osculating element output, GM is rarely known to better than 5 significant figures.”

And this may be the clue. When I take the second difference, I’m bumping up against (or exceeding) that limitation.

Ed

What are you trying to do?

I calculate barycentre data locally.

Out of curiosity yesterday I computed 1700-2050 with 1 day step for DE406 and DE421. Whilst they are different and in surprising ways this is orders of magnitude below the useful number.

I also computed at 10 days, compare with filtered 1 day version. Once again is practically the same.

However, anyone delving into high pass characteristics is asking for trouble.

Keep in mind ephemeris is an chebchev approximation being interpolated by integration and is not a functional model.

The approximation is valid for a limited time range but even there that depends which version, for example there are commonly used subsets of ephemeris which work over a more limited timespan, hence are smaller.

**the information on valid ephemeris timespan is ambiguous/confused even from official sources.**

If you want orbital based a whole different data is involved and is not generally accessible.

The accuracy of all this is limited.

Now, if there is a real thing going on why does it take stretching maths past breaking point to find anything?

If you want barycentre affecting solar activity I already have that but putting together for in effect publication is a lot of work some of which is beyond me. I suspect there are surprises waiting, already have signs of that from literature search.

Ed, I can suggest that you carefully consider the factors affecting osculating elements. Regards, Paul.

Ed, the planetary orbits are definitely know to far higher precision than 5 digits. More like 10 digits. I can believe that the method used is limiting it to that though. However the jerky stuff still seems odd to me. But you can only work with what you have. However if you are calculating 2nd difference of COM position, then it remains true that Venus has more effect than Saturn, and even Mars has more effect than Uranus and Neptune.

Ray, all,

Done some experiments seeing what is true.

Assuming I correctly understand the maths processing being applied, which is 2nd order differentiation, eg. (a1-2.a2+a3) / d^2

The rippled effect is real.

Could make an article of this with plots but whether this is interesting enough I know not, at least for Tallbloke’s place.

Tried this on 10 day sampled (36.5 per annum), produces red noise like spectra and if the spectra software is violating Nyquist shows aliasing of itself.

Did the same with 1 day sampled. Different but it would be, however there are the same valid peaks. (**)

I then low pass filtered the 10 day at 0.2y, >90dB rejection at Nyquist limit which I take to be 3 x 2 * 1/36.5 = 0.136y

Plotting 10 day 2nd diff overlay is exactly that. On taking the spectra there *is* a difference, you can see the signature of the filter in the noise level but the difference is immaterial as far as what is seen as ripple.

An unexpected surprise came on ploting the 2nd diff with and without filtered data, but I ought to have known. The filter shows up. In a way I suppose it has done a reconstruction filter. This spins of into an awful lot of detail about sampled data, PCM.

Ray Tomes says:

July 30, 2011 at 11:01 pm

“the planetary orbits are definitely know to far higher precision than 5 digits.”

Ray, I was just quoting from the Horizons documentation there. Actually what tchannon was saying at “July 30, 2011 at 5:56 pm” was closer to my first impression, that the “ephemeris is an chebchev approximation being interpolated by integration and is not a functional model”, and the ripple I was seeing was just an artifact of the sampling/calculation method used. I didn’t really try to analyze it further, I just smoothed the four planets’ orbits and calculated my own barycenter ephemeris.

Now, however, tchannon seems on the verge of convincing himself there’s something real going on there. tchannon, I’m not quite following your thinking; some graphs would be helpful :)

You’re right about the process being applied. I calculated the radial distance, then r1-2*r2+r3, where r1, r2, and r3 are the present distance, distance at the previous time, and distance two time periods ago, respectively. It’s in the

I’m still leaning toward “artifact”.

Ray, you said, “…it remains true that Venus has more effect than Saturn, and even Mars has more effect than Uranus and Neptune”.

I’m not seeing that conceptually or in the data. Regardless of the source of the ripple we see, the difference between my red and green curves above is the effect of the inner planets, and this is in the second derivative. There’s just not much there.

Ed

tchannon–

It looks like my editing leaves something to be desired. I started to say, my calculations are all in the spreadsheets linked at the top of the article.

Ed

I saved the work in case it was needed. On looking freshly and starting to clean it up for showing, trying to figure out what gives clarity I can see this needs a lot more explanation.

After trying various things I think it would take too long to really explain.

Differentiation is equivalent to high pass filtering (integration to low pass).

Assuming significant aliasing is avoided by good practice.

Applying differentiation raises the faster end of the signal, drags up noise, artefacts and anything actually there so you see it. The “noise” floor is high because these are very tiny items.

Figure 1

Data in this case was best integration DE421 by Solex v11 at 10 day sample.

Analyser on a dataset fragment says the following, days based on Julian, 365d year.

236.88

398.36

89.78

44.43

1.09y was there, 236 days has been raised even more, those two are what you are seeing.

Maths and times have to be very accurate but that gets into arguments over what is meant and what reference. Therefore assume +-1 day or so.

These are to do with Venus, perhaps the Earth, not sure it is clear what.

I could say more.

Tim, I’m off to bed. If Leif responds, approve it if it’s polite. leave it for me to deal with if not.

Cheers

Synodic periods in days:

J-Me=89.79, (J-Me)/2=44.90

J-V=236.99

J-E=398.87

S-Me=88.69, (S-Me)/2=44.35

Each step of differentiation increases high frequencies relative to low. That is why inner planets become much more important with 2nd differential.

It seems that horizons takes some short cuts to get 5 digit accuracy. That is perfectly normal as accurate positions require integration of all bodies very accurately with small time steps. I don’t think there is any mystery. But there is the need to use more accurate data if you think that 2nd differential of COM motion is important.

Ed and Ray

Regarding accuracy/noise, while I’m not an astronomer, here are a few references I found that may help, including software.

US Naval Observatory Astronomical Information Center posts The International Celestial Reference System (ICRS) is the fundamental celestial reference system adopted by the International Astronomical Union (IAU) for high-precision positional astronomy.

Standards of Fundamental Astronomy

SOFA Issue: 2010-12-01

Standards of Fundamental Astronomy Catherine Hohenkerk (2011), Scholarpedia, 6(1):11404.

D. J. Champion et al. MEASURING THE MASS OF SOLAR SYSTEM PLANETS USING PULSAR TIMING Draft version August 24, 2010

See Table 2 for planetary masses. Saturn 5 2.858872(8)×10−4 M⊙

The Planetary and Lunar Ephemeris DE 421

William M. Folkner,* James G. Williams,† and Dale H. Boggs† IPN Progress Report 42-178 • August 15, 2009 Jet Propulsion Lab

“The orbits of Jupiter and Saturn are determined to accuracies of tens of kilometers as a result of spacecraft tracking and modern ground-based astrometry.”

Happy hunting

Ed and Ray

On current accuracy of planetary orbits, see: W. M. Folkner

Relativistic aspects of the JPL planetary ephemeris, Relativity in Fundamental Astronomy, Proc. IAU Symp. No. 261, 2009, ISU 2010 doi:10.1017/S1743921309990329

Just checked 10day 1700-2050 DE406 vs DE421, maximum difference is 10e-8au so it’s <<0.1%

DE406 is +-3000 years and I know there longer if you can find them, can't remember the details.

This may help a little.

http://www.projectpluto.com/jpl_eph.htm

I've not currently installed Fortran here but was toying with trying to compile the French code to see what happens over the insolation timescales.

Tim and David

I understood the accuracy to be at the level of 1 part in 10^10 or about 200 m in inner solar system. The two quote figures are 50 times greater and 10,000 times smaller. Well, I suppose that knowing where a spacecraft is to a few centimeters does not tell you where a planet is to that accuracy, as planets often have fuzzy edges. Anyway, even at 10^-8 AU, that is 3 digits beyond the Horizon data and more than enough for Tim’s purposes.

On a related note, satellites using radar are used to map the Earth. By placing 3 sided reflectors (like they put on the Moon) they can get reference points. In a test where the reference points were moved vertically, the movement could be measured to 1mm accuracy relative to the landscape. On another occasion I read that after an Australian earthquake the difference between before and after showed variations of a couple of centimeters over much of the country.

Actually that would be 0.02 km or 20 meters (not 200 m)