Which Point do the Jovian Planets Orbit?

This is an intriguing question, I have read many different opinions but nothing to clearly substantiate their claims. Others in the Scientific arena when pressed are afraid to give an answer. Checking the JPL data which is an ephemeris produced by the Jet Propulsion Laboratory shows that Jupiter certainly doesn’t orbit around the SSB (although at first because of an error I made, it looked as if it did) and probably orbits around the Jupiter/Sun barycenter. After checking the Jupiter/Sun distances (which I will refer to as the radius vector which is quite different from the semi-major axis) through JPL it became obvious there was a substantial variance each orbit that was measured each time in the same place. Originally I saw this as an opportunity to look for a solid link between Angular Momentum and the Solar modulation re the planets, but soon discovered Jupiter and all the other planets have a modulating Perihelion/Aphelion distance. The quest was on the find out why and involved many weeks searching for any data I could find. Others on this site got involved and I even emailed an Astronomer but to no avail, very sparse detail available. Dr Svalgaard suggested it was a result of planetary perturbations and while correct his understanding was also far from complete. Below is an account of the progress along with what I think is the complete answer to what perturbs Jupiter and how.

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My research shows the Jupiter/Sun distance varies on average 300,000 km each orbit of Jupiter. Check graphic above.
The Earth is seen to orbit the Sun directly. If we use NASA’s JPL data it shows the Earth/Sun distance is varying by 0.0001AU per year (approx 15000 km). The AU measure is the average distance between the Sun and Earth. If we measure The Earth to SSB (solar system barycenter) distance it shows a much larger variance.
What I have done is measure the Jupiter/Sun distance at exactly the same point in the elliptical orbit of Jupiter each 4331.572 days, this should isolate any aphelion/perihelion changes. I also measured the Jupiter/SSB distances.

As can be seen in the first graphic there is a large variance each orbit between BOTH measurements (Sun & SSB). This makes it hard to pinpoint any point of orbit, although the variances seem smaller on average with the Jupiter/Sun data. I also compared the Heliocentric longitude (angle away from Sun) and found each Jupiter/Sun reading had an angle of 359 deg, but the solar position at perihelion does return very closely to the same position each orbit.

The above graph was made from reliable data by blogger JimP (thanks for the spreadsheet). Once blown up it shows some remarkable detail. It shows the Perihelion distance (closest point) for each Jupiter orbit over 400 years. There is a definite pattern, what causes this modulation? The movement between the closest and furthest perihelion is 1.2 million km’s , the Sun is capable of moving 1.5 million km’s from the barycenter, Neptune’s radius vector moves by 1.4 million km. Importantly the corresponding movement at the other end (aphelion) is the same value on each orbit but in the opposite direction, if one end shortens the other end lengthens.


Jupiter/Sun distances 1600-2020 with Jup/Sat conjuncture, Jup/Sat opposition and Jup/Sat quadrature (square) positions plotted for every third occurrence. Note the phase change around 1880. If the Variance in distance is caused by angular momentum J/S conjunction & J/S opposition both produce high angular momentum. Regular reader lgl and myself suspect the other Jovians need to come into the equation, but what is needed is a new method of displaying angular momentum other than Carl’s graph. A total strength with both J/S conjuncture and J/S opposition shown as high points instead of a sine wave is required, manipulation of the current JPL data via a spreadsheet can do it. Update: There is a new article covering this.


Saturn/Sun distance showing the same 60 yr pattern as Jupiter. After 1880 they are in unison with Jupiter but not before. There is a reason for these fluctuations but I have not found any literature on this phenomena.

UPDATE 13/05/09:


This diagram has been adapted from “Linkages between solar activity, climate predictability and water resource development” Alexander et al. Bailey has been criticized for claiming the Sun-Earth distance varies by the Sun-SSB distance, although not correct there is a fluctuation caused in the same manner as per Jupiter but to a smaller degree….0.0001 AU.

UPDATE 27/05/09: Here is another suggestion Dr. Svalgaard has provided to explain the modulating Jupiter Perihelion.

“The general principle is that to change the orbit, you have to apply a force along the orbit, so Saturn’s effect is largest when it is ‘to the side’ (approximately quadrature I think the astrological term is), rather than in conjunction. All of this has been understood for 250 years, and there are no other forces involved. Saturn doesn’t ‘push’ (gravity is attractive, not repulsive). And the Sun is not ‘dragging’: the force is always along the line connecting two bodies.
To recapitulate how Kepler’s second law works: To move an orbit out a bit (increase perihelion distance if you are near perihelion), you apply a force along the orbit in the same direction as the movement of the planet, e.g. by Saturn being ahead of Jupiter. That force produces a ‘delta-v’ (google it), which ‘lifts’ the orbit a bit out of the gravitational well, to another (higher) orbit, where it actually moves slower than before. Because of the periodic movements, the lifting is counteracted over time by a similar but oppositely directed movement, so that the semi-axis stays constant. This takes a couple of orbits to accumulate, so you have to consider the integrated effects over many years.”

There is substantial merit in Dr. Svalgaard’s rejigged explanation, but there still remains some pertinent questions. The explanation works well for perturbations occurring in the 1/2 to 3/4 region before Perihelion, but perhaps not so well in others. Dr. Svalgaard explains its a matter of the smaller background oscillations that make it hard to track the source of the perturbation. This is plausible but maybe a little weak and may be the reason it is near impossible to find any reliable data on this topic via the web. In particular I have questions why the “force along the orbit” fails to work when Jupiter approaches Perihelion and actually looks to have an opposite effect. Remember, when Jupiter & Saturn are in conjunction the Perihelion distance is at its shortest, but as you can see in the following diagrams the substantial pull along the orbit fails to elevate Jupiter’s orbit to a higher state.

The red dots signifies Perihelion in this 1762 example:

I wanted to see the perturbation affects plotted with successive orbits to see how the planets contribute to Jupiter’s orbit changes. Although fairly easy when you know a couple of tricks (thanks to Dr. Svalgaard) this task took me many days. I plotted the JPL data into excel and once the orbit was expanded to a size too big to display here it was easy to see each individual orbit. I was lucky with my data selection because it included a very strong perturbation of the 1987 Perihelion. The orbit has Jupiter doing a fly past Saturn, Uranus and Neptune before Perihelion which manages to take it away from its more normal path by 6 million kilometers. Once Jupiter passes all 3 planets its orbit is dramatically “braked” which brings it back into line around Perihelion, but still manages a very long perihelion distance. This is a good example of the perturbation theory in practice.

The corresponding planetary view shows the relative positions of planets which line up with the above average perturbation.

So do we have a situation where the perturbation theory works in some parts of the orbit only? Could other forces be overriding this perturbation? The orbit changing mechanics have been described as speeding up the planets velocity (I have serious reservations that this occurs) which raises it up the gravity well to a higher orbit before settling into a slower overall velocity.
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UPDATE:After much research I think I finally have the cause for Jupiter’s Perihelion distance modulation….and it surprising how little is known or published in this area. The above example is not worded very well and leads you to think an acceleration lifts the orbit higher then slows the velocity (that’s at least how I read it), from what I now understand it actually happens in reverse…the acceleration moves the orbit closer (following Kelper’s 2nd Law) but then unlike a rocket using boost, Jupiter is then immediately subjected to deceleration from Saturn’s gravity which slows velocity in turn lengthening the radius vector or raising Jupiter to a higher orbit. As Jupiter moves further away from Saturn the Sun takes over and reins in its sibling and waits for the next perturbation. So now it becomes obvious, and completely explains why a Jupiter/Saturn conjunction causes a short Perihelion, Jupiter is always in acceleration mode during this lineup which causes it to move closer and the plotted orbit agrees. Also of note, Neptune and Uranus also help out and can be strong when together as we see elsewhere on this site.

The orbit spreadsheet is able here: http://users.beagle.com.au/geoffsharp/jup_orbit1940-2009.xls

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25 comments on “Which Point do the Jovian Planets Orbit?

  1. THE QUESTION NOW IS: DO WE HAVE ENOUGH EVIDENCE TO PROVE THE SUN/JUPITER DISTANCE IS VARYING?

    I would say yes. Jupiter is orbiting the Sun-InnerPlantes-Ju BC, not the SSBC, in the meaning the center of the orbit is one of the focal points of the orbit.
    Since the Sun is revolving around the Su-IP BC much faster than Ju around the Su-IP-Ju BC, I guess it’s safe to say that Su-Ju distance is varying.

    REPLY: I am tending to agree…the JPL data does not lie? I think we may finally have a scientific link for planetary theory. But still need to quantify how much rotation change can occur with a 300,000 km semi-major axis change….I bet Ian Wilson could work this one out.

  2. … but, adding to my previous, this is probably minor to the effect of the Ju,Sa,Ne/Ur conjunctions.
    http://virakkraft.com/60-yr-conj.ppt
    Also ask Ian to calculate the rotational energy of the solar system in june 1984 and june 1990 (around the SSBC).

    REPLY: This 60 year pattern is elusive but I came across the same conclusions as the link you provided. There does seem to be a correlation with overall angular momentum at the point of perihelion. Stronger angular momentum providing closer perihelion. I will have to find a method of graphing this as Carl’s graph doesn’t quite do it.

  3. Geoff,
    You have to show that the average radius changes, not just two points.

    REPLY: That would be nice and I ‘m sure it shows the same trend, but might be a job for a programmer as doing manually would be horrific.

  4. Geoff said “That would be nice and I ‘m sure it shows the same trend, but might be a job for a programmer as doing manually would be horrific.”

    What? Manually? Any real scientists working in this field would know how to use the software Mathematica or Matlab to do exactly what you are asking a ‘programmer’ to do. This is disturbing.

    REPLY: I am not a scientist…hate to burst your bubble, but I have been spending a lot of time trying to get a result on the Sun/Jovian distances. I might have some news soon.

  5. This topic is proving harder than expected. There is something here but I cant quite tease it out. With all perihelia, Jupiter has the SSB between it and the Sun, this is a consequence of angular momentum. When angular momentum is highest (Jupiter/Saturn together)the Sun is moved furthest away from the SSB AND Jupiter, BUT this also coincides with Jupiter’s closest perihelia which is opposite to expectations. To make it even harder the other option that provides very short perihelia is times of very low angular momentum (which the Sun uses in times of high activity)when Jupiter and Saturn are opposite with the Sun close to the SSB. One can only conclude that Jupiter is being dragged aggressively in the high angular momentum case and perhaps dragging it past its “normal” position?

    I have been searching for any relevant data on the changing perihelion/aphelion distances but so far coming up short….has anyone else found anything?

  6. Regarding the varying perihelion and aphelion distances of Jupiter, I don’t think we need to look beyond Newtonian mechanics. If S, U, and N are all on the same side as J, then they will be perturbing J to be further away from the Sun. But if S, U, N are all on the opposite side of the Sun to J, then they will pull J a little closer to the Sun. J’s distance to the barycentre will vary by a lesser amount, because the perturbing gravitational effect is in the same direction as the influence on the barycentre.

    Rich.

    REPLY: Do you have any documentation to support your claim? I have been searching for perturbing influences of Jupiter’s orbit but so far have found none.

  7. REPLY: Do you have any documentation to support your claim? I have been searching for perturbing influences of Jupiter’s orbit but so far have found none.

    Do I have any documentation on Newton’s laws of mechanics? Or is the question whether those laws fit the observed data on Jovian perihelia? It seems as if Carsten Arnholm’s program might be valuable here, from the sound of it. Take a date of perihelion, calculate all the planetary positions and then the Sun’s from the invariance of the barycentre, and calculate Jupiter’s distance from the Sun, and check against the observation. I am convinced it will agree, apart from tiny relativistic effects.

    Regarding perturbations to Jupiter, my understanding is that Neptune was inferred and located from perturbations, though I suppose they were a greater effect on Uranus. But Jupiter’s perihelic distance is as much due to “perturbations” to the Sun by planets S,U,N.

    Rich.

    REPLY: I have done your test using JPL and it shows a variance in perihelia distance as you can see, but it is not conclusive how Jupiter’s orbit is changing. I am looking for something concrete here, a paper showing/stating how Jupiter’s orbit is affected by the solar system, the actual path not the velocity. As far as I can see Neptune was discovered because the velocity of Uranus changes when nearing and leaving Neptune. This is very different to path change.

  8. See – owe to Rich said,
    May 2, 2009 @ 12:45 am
    “calculate all the planetary positions and then the Sun’s from the invariance of the barycentre”.

    Surely, in absolute (galactic) terms, the position of the barycentre is not invariant, it changes all the time? The only times I can imagine the barycentre not moving would be when a stellar system had one sun and no planets or one sun and all the planets were of the same mass and were sitting in one another’s Trojan Points.

    As the Barycentre is moving all the time, anything orbiting it will be affected by this movement. Orbiting bodies towards which the barycentre moves will accelerate in their orbits and those becoming more distant from the Barycentre will slow down.

    The total energy of the system will no doubt be preserved but perhaps someone with more maths than myself can say whether this movement will transfer energy from one body to another.

    REPLY: Your last statement is where the proof will come. But I must correct you on the barycenter statement. There is a difference in the barycenter of the solar system and where the current solar system determines the point of greatest angular momentum (or where the Sun is at present). The SSB or barycenter is the central mathematical point of the solar system mass as averaged over time or central point of the Sun’s orbit. This is exactly how JPL treat the SSB.
    Dr. Svalgaard also made the same mistake as you.

  9. Roger,
    “the position of the barycentre is not invariant”
    The barycenter is travelling through space but that doesn’t mean it’s jumping around like some persons seem to think. The gravity of the galaxy is acting on the center of mass of the solar system. The BC is the hub of the solar system wheel, carrying the enormous momentum of the system through space so it can’t jump around, and the Sun is forced to counterweigh the planets. Like it’s the Earth-Moon BC that is following a ‘smooth’ orbit around the Sun, the SSB does the same around the galactic center, so in the context we are discussing here the SSB is the fixed point, not the Sun.
    Agree Geoff?

    REPLY: Looks good to me. Some say the path of our solar system snakes above and below the central galactic plane which could add another nice “wobble” to the equation.

  10. Being a notional point, the sum of a large number of different masses with different orbital velocities, it still feels like the barycentre should be moving all the time!

    Anyway I’ll stop bugging you all now. Perhaps I should try and plot something – that may help with my understanding.

  11. [~snip]behave
    “Not even wrong”. Jupiter is not pulled or dragged or responding to anything. The SSN is not an entity that exerts any forces and pushes and pulls.

    [WORDPRESS HASHCASH] The poster sent us ‘0 which is not a hashcash value.

  12. As I said, you wouldn’t understand anything anyway. It are the same forces that work on the long and short term. There are no other.
    ‘nice’ seems to be anything that avoids the truth. I was stating the simple fact that your ‘theory’ is nonsense, if you have anything constructive to say, say it, otherwise I can see why you snip.

    The planets move in the combined gravitational field of the Sun and all of the planets, there are no other [excepting that of the Galaxy and the rest of he Universe, which we have not been able to measure so far].

    And, of course, you do have a barrel to push; it is called ego. And it is more than just the ‘facts’. I have told you the facts many times. As has Carsten and others, including Shirley.

    Explanation [probably wasted too, but let me try]: When Saturn is in the position indicated on your second graph, its gravity is pulling Jupiter forwards in Jupiters orbit. This works like firing a rocket on the ISS space station to lift it out to a higher orbit, hence increases the distance between the Sun and Jupiter. If S were ‘below’ Jupiter [as it was ‘above’ in the lower graph], S would brake Jupiter and J’s distance to the Sun will decrease, until such point when S no longer breaks [i.e. when aligned with the Sun as in the first graph]. At that point the distance would be smallest after all that breaking. Now, what is so hard to understand?

    Anyway, such handwaving arguments are no use. Only simple, straightforward calculation [as in the tutorial I referred you to – did you read it? did you study it? did you understand it? Must I quizz you on it to verify?] shows how things really are.

    [WORDPRESS HASHCASH] The poster sent us ‘0 which is not a hashcash value.

  13. Leif,
    “Now, what is so hard to understand?”

    When Jupiter is at one side of the Sun and all the rest lined up at the opposite side the resulting gravity is 272E21 N in the direction of Jupiter and the Sun is close to the barycenter.
    When Jupiter has moved to the same side as the rest the gravity is 566E21 N in the direction of Jupiter and the Sun-Ju distance has increased, right? Both gravitational force and distance increase. Then it is a bit hard to understand that gravity is moving the Sun.
    Numbers from here: http://arxiv.org/ftp/arxiv/papers/0903/0903.5009.pdf page 52

    REPLY: you make a good point. What this says is that when it comes to orbits you cannot exclude angular momentum, they co exist.

  14. Re Leif #12, I am glad to see that he is around here, as I am near to posting some mathematical calculations in an attempt to understand the modulation of Jupiter’s perihelic distance. Unfortunately, although by a combination of Saturn perturbing Jupiter and the Sun moving around a barycentre, I am able to see why in theory J’s perihelion might be closest at both JS conjunction and JS opposition, the numbers don’t come out quite right. Perhaps when I post these Leif will spot what I am missing (or say what is significant about small things I am intentionally missing).

    Rich.

    [WORDPRESS HASHCASH] The poster sent us ‘0 which is not a hashcash value.

  15. As a postscript, Geoff, though your graphs are pleasing to the eye, a numerical table of J’s perihelic distances would be valuable to me to assess my (probably inadequate) theory. Would it be easy for you to provide that?

    Thanks,
    Rich.

    REPLY: It might be easier if you get the tables direct from JPL. Use this link http://ssd.jpl.nasa.gov/horizons.cgi and select the “Observer option”. If you have any drama’s let me know. That link will only give Jupiter info to about 1927, if you want data further back I have monthly figures to 1600.

  16. Roger Edmunds [post #10] said, “Being a notional point, the sum of a large number of different masses with different orbital velocities, it still feels like the barycentre should be moving all the time!”

    It depends on the reference frame one is using.

    Note:
    This can result in misunderstandings in interdisciplinary communications since specialists are used to being able to ‘assume’ everyone knows very-well to what frame they are referring, even if they have not been explicit.

    [WORDPRESS HASHCASH] The poster sent us ‘0 which is not a hashcash value.

  17. Leif [post #12] addressing Geoff: “[…] the tutorial I referred you to […]”

    Where is the link to this tutorial?

    REPLY: http://stjarnhimlen.se/comp/tutorial.html
    Its all a bit academic now. After long discussions and the plotting of Jupiter’s orbit it was found that Dr. Svalgaard’s initial explanation was incorrect.

  18. I have now done some mathematics on the question of Jovian perihelia, as I mentioned I would in a comment of the previous thread. The results show an effect of the sort I predicted from combining perturbation and barycentre effects, but not of the right relative magnitude between Saturnine conjunctions and oppositions. In fact, even the sign is wrong for the case of opposition. Still, I shall record the workings here, and hope that Leif Svalgaard, whose qualitative suggestions above are similar to mine, may be able to help. Along the way, I shall record various assumptions I am making.

    Assumption 1: The planet with the dominating effect on Jovian perihelia is Saturn.

    This is justified by the fact that the data show a strong 60-year periodicity.

    Assumption 2: The perturbation which Saturn (S) has on Jupiter’s (J’s) elliptical orbit is the same to first order as it would be if the orbit were circular.

    Assumption 3: Apart from this perturbation, the only relevant factor is the way S modulates the Sun’s (=Helios’s=H’s) distance from the HJS barycentre B.

    Assumption 4: The component of HB which is orthogonal to HJ has a negligible effect on the distance HJ.

    This is justified by the fact that HB is very small compared to HJ.

    To begin the analysis, choose the unit of length to be Saturn’s orbital radius, and let that of Jupiter be r (=0.5454). Let a be the angle subtended at H by J and S, increasing with time, and let j (with |j|=r) be the vector from H to J, and s (with |s|=1) be the vector from H to S. Let H, J, and S also denote the masses of these bodies.

    The inverse square gravitational force which S applies to J is GJS(s-j)/|s-j|^3, where G is the gravitational constant, so the acceleration applied is

    GS(s-j)/|s-j|^3

    Now, we may resolve this acceleration into a radial and a tangential component (to J’s orbit). For a while I was tempted into

    Assumption 5: The tangential component is a second order effect which we ignore,

    but later on I included it and discovered this assumption was false.

    Let v_R be J’s radial velocity out from the Sun, let x_R be the radial perturbation from a circular orbit, and let v_T be the tangential velocity. Starting with the radial component away from H, let

    A = GS/|s-j|^3

    where by the Cosine Rule |s-j|^2 = 1 + r^2 – 2r cos a.

    Then the radial component of acceleration from S’s perturbation is

    GSj.(s-j)/(|j||s-j|^3) = A(|s||j|cos a – |j|^2)/|j| = A(cos a – r)

    since |s|=1 and |j|/|s|=r. This has to be added to the radial component from the Sun, H, which is

    B = -GH/r^2

    The tangential component of acceleration is likewise

    -A sin a

    Thus, if a is small and positive, J has just passed S and S is pulling J back from the direction it is travelling. Now consider what happens when the angle through which J moves around H increases by a small amount e during a small interval t; the angle a then changes by
    e*P_J/P_JS where P_J is J’s period, and P_JS = P_J P_S/(P_S-P_J) is the JS synodic period.

    Let v_R’ and v_T’ be the new velocities after the movement, in the original axes, and let v_R” and v_T” be these velocities in the new axes rotated by e. Since J moves a distance r*e (to first order), this must be v_T*t, so

    t = re/v_T

    Then, to first order,

    v_R’ = v_R + (A(cos a – r)+B) re/v_T
    v_T’ = v_T – A sin a (re/v_T)
    v_R” = v_R’ cos e + v_T’ sin e = v_R’ + e v_T’
    = v_R + e(A(cos a – r)+B)r/v_T + v_T)
    v_T” = v_T’ cos e – v_R’ sin e = v_T’
    = v_T – e A sin a (r/v_T)

    since v_R is of small order compared with v_T. The change in radial distance x_R is then given by

    x_R” = x_R + v_R t = x_R + e v_R r/v_T

    These then are the basic equations which we integrate using the Trapezium Rule to maintain symmetry about a = 0, using say 400 points. The value of GH can be estimated by seeing what would happen if Saturn were not there (so A=0) and v_R=0 and v_T = 2 pi r/P_J. Then

    Br/v_T + v_T = 0

    so B = -GH/r^2 = -4 pi^2 r/P_J^2, hence

    GH = 4 pi^2 r^3/p_J^2

    and A can be deduced from this. The initial conditions of the integration have to be set very carefully so that after a changes by 2*pi radians times P_JS/P_J, i.e. one synodic revolution, v_R, v_T, and x_R are all unchanged. For 400 points these conditions turn out to be

    v_R = 0
    v_T = 1.0001557224 * 2 * pi * r / PJ
    x_R = -0.0008064236

    Saturn’s movement of the Sun

    In addition to Saturn perturbing Jupiter, in a complicated way, Saturn perturbs the Sun but in a simple way expressed by a barycentre. So, with only H, J, and S in play, and if, without loss of generality, J is at (r,0) then S is at 1(cos a,sin a), so relative to the HJS barycentre H has to be at

    -(1,0)rJ/H – (cos a,sin a)S/H

    Assumption 6: The variation in r is an insignificant contributor to this

    Along with Assumption 4 this means that the extra distance from H to J from this effect is

    (S/H)cos a

    (If a=0 then J and S are aligned and H is on the opposite side of the barycentre.) The following R code implements these equations and numerical integrations.


    r = 0.5454 # r_J / r_S
    skm = 1427e6 # r_S in km
    S = 95.16/333000 # S/H
    J = 317.83/333000 # J/H
    PJ = 11.862
    PS = 29.457
    PJS = PJ*PS/(PS-PJ) # 19.859

    n = 400
    del = 2*pi/n # angle to S
    delc = PJS/PJ # angle to H increases faster than angle to S
    th = 0:n * del
    ath = (cos(th)-r) / (1+r^2-2*r*cos(th))^1.5

    vt = 0 * 0:n
    vr = vt
    xr = vt
    if( n == 400 ){
    vt[1] = 1.0002722850 * 2 * pi * r / PJ;
    vr[1] = 0;
    xr[1] = -0.0008064236;
    }

    GH = 4 * pi^2 * r^3 / PJ^2
    B = GH / r^2
    A = GH * S / (1+r^2-2*r*cos(th))^1.5
    Ar = A*(cos(th)-r)
    At = -A*sin(th)
    t = 0

    # if J is at (r,0) and S is at 1(cos a,sin a)
    # then H is at -(1,0)rJ/H - (cos a,sin a)S/H
    hx = cos(th)*S

    vt1 = .5 * At[1] * r / vt[1]
    vr1 = .5 * ((Ar[1] - B) * r / vt[1] + vt[1])
    xr1 = .5 * vr[1] * r / vt[1]

    # integrate over angle from 0 to 2*pi using trapezium rule
    for( i in 1:n ){
    vt2 = .5 * At[i+1] * r / vt[i];
    vt[i+1] = vt[i] + del*delc * (vt1 + vt2);
    vr2 = .5 * ((Ar[i+1] - B) * r / vt[i+1] + vt[i+1]);
    vr[i+1] = vr[i] + del*delc * (vr1 + vr2);
    xr2 = .5 * vr[i+1] * r / vt[i+1];
    xr[i+1] = xr[i] + del*delc * (xr1 + xr2);
    t = t + del*delc * r / vt[i];
    vt1 = vt2; vr1 = vr2; xr1 = xr2;
    }

    print(t)
    print(skm*xr[1]) # -8.064e-4
    # J at S opp - S con
    print(skm*(xr[1+n/2]-xr[1])) # 2.031e6
    # H at S opp - S con
    print(skm*(hx[1+n/2]-hx[1])) # -8.156e5
    # JH at S con
    print(skm*(xr[1 ]+hx[1 ])) # -7.430e5
    # JH at S qua
    print(skm*(xr[1+n/4]+hx[1+n/4])) # 1.163e5
    # JH at S opp
    print(skm*(xr[1+n/2]+hx[1+n/2])) # 4.723e5

    Results

    The results are frankly disappointing, as they don’t give close to equal values for Saturn at conjunction and opposition. On the other hand, the general magnitudes are not far from the correct size. Here are the predicted anomalies of distance from Sun to Jupiter:

    -743000km at Saturn conjunction (correct sign, i.e. closer)
    +116300km at Saturn quadrature (correct sign)
    +472300km at Saturn opposition (incorrect sign)

    The only thing I can think of to do to double-check on this, is to integrate the equations of motion for all 3 bodies, and see if the results differ much from above. I am unlikely to do this for several months. In the mean time, suggestions as to what has gone wrong here would be most welcome.

    I appreciate that there is a fair bit of maths here, but some people, for example Leif Svalgaard, should be able to help if they feel inclined.

    I hope that this article will at least be of some interest,
    Rich.

    REPLY: Thanks for the effort Rich, but the answer is clear now….there is an update to the article at the end.

  19. Geoff,

    Regarding your UPDATE beginning “After much research”, it sounds as if you are saying qualitatively what I am trying to calculate quantitatively. And I can give another qualitative explanation which is as follows. If you imagine a ball moving vertically up and down on an elastic string, or a simple harmonic pendulum, there is a time lag between when the maximum force is applied and when the object reaches the extreme position from that force. So, when the ball feels maximum upthrust it is actually at its most “down” position. Likewise, when Jupiter is in conjunction with Saturn, it is feeling the maximum force away from the Sun, but is at its nearest position to the Sun.

    But there is a problem with the case of JS opposition. Here, J experiences its maximum force towards the Sun and so should, by the analogy and my mathematics, then be at its maximum distance from the Sun. This contradicts the close perihelia of the years 1950 and 2010, and I’d be interested to see how you explain this.

    My own explanation, which was in the right direction, involved the Sun-Saturn barycentre B, via the following diagram:


    S-------------J------------B-H
    S--------------------------B-H----------J

    So at opposition H (=Helios=Sun) is on the same side of B as J, and this reduces the distance HJ – but not by enough to match the observed data. As I said, please explain this…

    Rich.

    REPLY: Maybe your going down the path I went down at first. What seems to be occurring is Jupiter moves in real time as the Sun orbits the SSB ie. the gravity effect is near instantaneous, Jupiter must follow the Sun. But the position of the other Jovian’s causes Jupiter to speed up or slow down which in turn forces a radius vector change according to Kepler’s law.

    If you havent done so already, have a look at the Jupiter orbit plotted via excel, track it against a solar system viewer and it becomes clear.
    http://users.beagle.com.au/geoffsharp/jup_orbit1940-2009.xls

  20. Frederick Bailey’s book “Textbook of Gravity, Sunspots and Climate”, gives full details, arguments, diagrams etc. of the method used to determine the true Earth – Sun distance at any point in the ecliptic plane and hence allowing one to calculate the variation in energy received by the earth.

    In this mathematical construct, using just Newtonian laws of motion, as mentioned above and in the diagram reproduced above, the Solar System Centre of Mass is a fixed point travelling at a constant speed in an (almost) straight line, as this is a balanced, closed sytem with all the forces resolving to zero at the SSCM. This was used to prove how sunspots are produced but the unitentional discovery was, that Fred realised that the diagrams showed that the earth – sun distance is constantly varying but none the less predictable; further more, it is a simple matter to then represent the varying distances in terms of variation in energy received by the earth.

    In reading the above articles it is obvious that the methods used above always result in unanswered questions or some degree of ‘fudging’ to make things fit the argument. Fred’s method suffers none of these problems beacause everything fits perfectly and he continues to do further research that dovetails perfectly with accepted events, whether they be sun spot records/predictions or historical weather events.

    The following blog gives a lot of inside information regarding the methododology and pragmatic arguments used;

    http://omniclimate.wordpress.com/2010/02/21/about-frederick-baileys-textbook-of-gravity-sunspots-and-climate/

    REPLY: Thanks Howard, I have read your fathers et al paper many times, and the orbit spiral diagram is enlightening. The solar acceleration theory has merit but the varying Earth/Sun distance due to an Earth orbit axis point of the SSB does not hold weight scientifically. Sometimes one part of your theory can be off base with the rest remaining intact.

  21. Please excuse my interuption of what are obvoiusly much better equipt minds than my own, but I wonder stumbled in while searching for info on this topic under the heading of “radius vector inertia” and was thinking that the better approach might be to literally look at things from a different perspective, that being from the side as the elliptical plane must appear most narrow on an average over a long period of time from one of the bodies, this seems to be the most practical starting point for a calculation. Then factoring in the distance of that body from the sun, take mercury as the opposite extreme, and seek a fundamental principle to lash both extremes to. Then apply these factors to the other dimension to calculate THAT BC. Just an uneducated suggestion!:)

  22. Pingback: Scafetta’s New Paper Linking Mid-Latitude Aurora to the 60 Year Temperature Cycle. « Landscheidt Cycles Research

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