*My thanks to R.J. Salvador for this guest posting of his solar variation model based on planetary periods. It’s forecast is in good agreement with that made by Tim Channon back in Feb 2011 using a different technique and different data (Judith Lean’s TSI reconstruction). R.J.’s model is available to interested parties known to the talkshop, make a request in comments for a copy (7meg .xls). R.J. asked me to include Sparks plots of Uranus orientation to the Sun which is included into the model as the 1/4 period of its orbit. Click for full size.*

**A Mathematical Model of the Sunspot Cycle for the past 1000 Years**

By R.J. Salvador

**Summary**

Using many features of Ian Wilson’s Tidal Torque theory, a mathematical model of the sunspot cycle has been created that reproduces changing sunspot cycle lengths and has an 85% correlation to the sunspot numbers from 1749 to 2013. The model makes a reasonable representation of the sunspot cycle for the past 1000 years, placing all the solar minimums in their time periods. More importantly, I believe the model can be used to forecast future solar cycles out quantitatively for 30 years and directionally for 100 years. The forecast is for a solar minimum and quiet sun for the next 30 to 100 years. The model is a slowly changing chaotic system with patterns that are never exactly the same, much like a model of the weather. Inferences as to the causes of the sunspot cycle patterns can be made by looking at the models terms and relating them to aspects of the Tidal Torque theory and possibly Jovian magnetic field interactions.

**The Model **

The Tidal Torque theory proposed by Ian Wilson provides a system of interrelated consistent frequencies and now I believe a unique set within a narrow error range.

This model is simply four interacting waves but they are modulated to create an infinite possibility for sunspot formation.

The basic frequencies in years are:

A VEJ frequency of 22.14 (varying),

A VEJ frequency of 19.528 (varying and forming a beat frequency of 165.5 with 22.14),

Jupiter Saturn synodic frequency of 19.858,

One quarter Uranus orbital frequency equal to 21.005,

Two modulating frequencies of 178.8 and 1253.

Ian Wilson has shown the connection between the Hale cycle (22.1±1.9 yr) and the synodic period of Jupiter and Saturn (19.859) such that: ( 22*.*34 × 19*.*859)/ (22*.*34 − 19*.*859) =178.8 years which is the Jose cycle.

Individual sunspot cycles have varying cycle lengths and this is an impediment to obtaining a continuous mathematical model. The monthly sunspot data imply that frequencies and or phasing of the basic cycles are slowly changing over time.

The reader will recognize the 178.8 frequency as also being the time of rotation of the sun around the barycenter. This provided the idea that perhaps the Jovian 19.858, 21.005 and the 22.14 VEJ frequency and phases are changing over time to the barycenter rotation of 178.8. By correlation it is found that the 19.528 VEJ frequency is changing to a slower 1253 frequency presumably because the Earth and Venus rotate about the sun instead of the barycenter directly. These frequency and phase changing capabilities are built into the model and for the most part solve the cycle length problem for correlation.

The model does not reproduce the skewed Gaussian shape of the sunspot cycles as the model is attempting to simulate the forces activating the cycle and not the process of actual sunspot formation and disappearance. Since the time length of the formation of sunspot is unstated the phasing in the model is left open and determined by correlation.

Below is a description of the model.

The sunspot data was transformed into positive and negative oscillations by multiplying the monthly sunspot number by the sunspot cycle’s polarity of plus or minus one.

SNC=SN*POLARITY

The data was then correlated to the following equation:

Where the Fs, Ls, Ns, and Ps are all constants determined by a non linear least squares optimization,

SN=(F1*cos(w1*(t +ph1))+F2*cos(w2*(t+ph2))+ F3*cos(w3*(t+ph3))+F4*cos(w4*(t+ph4))

SN is the sunspot number and T is the time in Calendar years.

Ws are the modulated frequencies and are changed by either 178.8 or 1253.

w1=2*pi/(19.528*(1+n1*cos(2*pi/1253*(t+L1))))

w2=2*pi/(22.14*(1+ n3*sin(2*pi/178.8*(t+L2))))

w3=2*pi/(19.858*(1+n5*cos(2*pi/178.8*(t+L3))))

w4=2*pi/(21.005*(1+n7*sin(2*pi/178.8*(t+L4))))

Phs are the modulated phases of each component of the model and are changed by the frequency of 178.8 or 1253.

ph1=p1*(1+n2*cos(2*pi/1253*(t+L1)))

ph2= p2*(1+ n4*sin(2*pi/178.8*(t+L2)))

ph3=p3*(1+n6*cos(2*pi/178.8*(t+L3)))

ph4= p4*(1+ n8*sin(2*pi/178.8*(t+L4)))

When only monthly sunspot data from 1749 to 2013 are used for the correlation the model achieves an R^2 of 0.85 but with those correlated constants the model is unable to reconstruct the past and therefore has no credibility for forecasting. It’s not surprising the model fails since it contains a frequency of 1253 and the data covers only 20% of that cycle and only one and a half cycles of the 178.8 frequency.

To overcome this difficulty more sunspot data over a much longer time period is needed. Sami Solanki and co-workers have reconstructed ten year average sunspot numbers for the past 11000 years from radiocarbon 14 analysis. Since the model requires monthly data (not 10 year averages) and the polarity of the cycle, the Solanki data cannot be used in total. However the Solanki data does quantify three time periods in the past 1000 years when the sunspot number was zero. (Maunder, Sporer and Wolfe minimums) These monthly time periods as defined by Solanki can be used with the sunspot number set to zero and then the polarity becomes a non issue.

The graphic below shows the Solanki data from the year 1000 to 1895 where it stops due to interferences by activities of modern society.

Using this additional data still produces a strong correlation of R^2 =0.85 for the data between 1749 and 2013 and a very interesting and reasonable reproduction of sunspot cycles for the past 1000 years. Below are graphics of the 1000 year and 1749-2013 correlation shown in the more usual absolute value form instead of a cyclical wave.

Sunspot reconstruction from the year 1000.

Forty has been added to the Solanki sunspot average for illustrative purposes.

**Forecasting**

Having pointed out to the model where the solar minimums are, how do we know that any future forecast is valid? We can redo the correlation with data only up to 1950 and 1900 and determine what the forecast would be for the next 50 and 100 years and see if the model can forecast what we already know.

Below is a forecast made from a correlation of the model with data just up to 1950.

With data up to 1950 the model forecast a peaking sunspot cycle and a significant decline in sunspots around the turn of the century and an ongoing solar minimum. The model is a little early but directionally correct 50 years out.

Below is forecast made with data just up to 1990.

Although the model did not predict the magnitude of the increase in spot activity, 50 years past 1900 it did forecast an increasing and then decreasing sunspot activity with a minimum around the turn of the century.

I believe this shows the model has credibility in forecasting two to three sunspot cycles out and directionally for one hundred years.

Below is a forecast made with data up to 2013. The forecast is for a very quiet sun for the next 100 years. The model forecasts that the sunspot cycle will not produce sunspot values over 100 again until the cycle that starts around 2160.

The model forecasts that the existing cycle, 24, will end in 2018. The next cycle, 25, could prove to be very interesting as the model predicts it will be difficult to tell when it ends and the next one begins. Cycle 25 length will be either 10.5 years or 15 years long.

**Sunspot Activation **

What does the model telling us about how the solar system effects sunspot formation?

What are the destructive and constructive wave interactions that produce a Maunder minimum (see below) or a modern maximum in the model.

Below is a graphic of the sum of the two terms of the VEJ cycle (19.528 and 22.14) and the two terms of the Jovian cycles (19.585 and 21.005) from the year 1600 to 2200.

The model gives equal weight in magnitude to the VEJ and Jovian cycles. These cycles can hide, and interfere both constructively and destructively with each other.

The model has these two interference patterns that in turn interfere with each other to produce the minimums and maximums of the solar cycles. For example the Dalton minimum occurred at a minimum in both the VEJ and Jovian cycles. Yet the Maunder resulted from destructive wave interference when both cycles were near maximums. The Modern maximum is a result of constructive interference from a maximum in both cycles. The coming solar minimum is the result of wave pattern destructive interference between the VEJ and Jovian cycles and it is extended by minimized VEJ and Jovian internal destructive interference.

The VEJ and Jovian oscillations are changing through time so that the same precise pattern never repeats itself. At present the VEJ cycle has an oscillation of 165.5 years and the Jovian cycle of 363.6 years but this changes as the base frequencies are modulated.

Below are the constructive and destructive interference patterns for the VEJ and Jovian cycles for the past 1000 years.

**Beliefs without Proof**

I believe this model captures the fundamental relationships between a gravitational disturbance to the sun’s magnetic field through the Tidal Torque process and a magnetic disturbance to the sun’s magnetic field through the Jovian planets. The model will not work without the Uranus one quarter orbital frequency of 21.005. Given the unusual orbital rotation of Uranus around its equator, I believe this is a strong indication of a magnetic to magnetic field interaction.

I also believe this model describes a chaotic process much like the weather where the so called flap of a butterfly’s wings in Japan leads to a hurricane in the Atlantic Ocean.

I am sure many more planetary based disruptions to the sun’s magnetic field could be built into a model to improve accuracy but the system will still be a chaotic one over time.

Fortunately because the changes to the base frequencies occur slowly in terms of human life spans we can make forecasts that maybe useful.

Below is a forecast for the next 1000 years. It only has significance for the first 100 years. The future will be different and it will not be a mirror of the past 1000 years.

Thank you to,

Ian Wilson for the Tidal Torque theory and frequencies,

Paul Vaughan for laying out in detail the relationship between frequencies,

Tim Channon for a very timely comment,

Rog Tattersall for pointing out the Uranus one quarter frequency and helpful articles.

**Model Constants**

The model can be constructed in an excel spread sheet using the equations in this article and the parameters listed below. There is no significance to the number of digits other than I have not taken the time to reduce them.

L1=18.7423277309335;

L2=-28.3318257572537;

L3=588.787354023188

L4=-115.90353105987;

P1=9.43165318099019;

P2=11.0794326320698;

P3=-4.20093948386925;

P4=5.72557858242286;

N1=0.0393355303094293;

N2=4.15575999334559;

N3=0.0117113642671855;

N4=1.86070148778939;

N5=-0.00440097114681985;

N6=2.13680584387995;

N7=-0.0088086784704264;

N8=-3.2683621209398;

Scalars:

F1= -32.277327890156

F2= -65.6646990792377

F3= 51.9264185855531

F4= 40.9140438856888

**References and Sources**

Ian Wilson Tidal Torque Theory and many reference locations

Sami Solanki Data and Article

ftp://ftp.ncdc.noaa.gov/pub/data/paleo/climate_forcing/solar_variability/solanki2004-ssn.txt

http://cc.oulu.fi/~usoskin/personal/nature02995.pdf

Sunspot Monthly Data

In the world there are useful correlations as well as useful causations. Of course, the two are conflated, climate science and Al Gore being examples of what happens when correlations are seen to reflect causation. For predictive purposes, however, good correlations can be useful, even if piracy levels do “predict” temperature rises as well as CO2 (to be useful, correlations have to reflect the entire cycles, not just one half!).

The first part of all scientific learning is to suspect one thing is connected to another. The second part is establishing that this is true. The third part is determining why it happens. The correlation that speaks to a common cause instead of the causation itself is valuable. So is the correlation that tells us coincidence is involved: it tells us we need to be aware of how naive interpretations lead us into trouble (Al, are you listening?).

It is a sad reflection on our times that I am hoping that all the signs of a period of quiet sun, increased cloudiness and lowering global temperatures are correct. People experience pain when times get cold. But if sense doesn’t come back into political minds wrt climate-based social ideology, we face huge cultural problems as we fight eco-green fears with weapons of mass economic destruction.

This is a good study. It goes to the point that reasonable people can see and investigate what might be happening without having to have institutional layers determine what is worthy of discussion.

I see Svalgaard is busy waving his arms on WUWT

http://wattsupwiththat.com/2013/09/07/new-paper-says-no-evidence-of-planetary-influence-on-solar-activity

We gave this one a good kicking back in July

https://tallbloke.wordpress.com/2013/07/28/cameron-and-schussler-no-evidence-for-planetary-influence-on-solar-activity/

It’s the beginning of a busy week, so it may be several days before I can give RJ’s work the

very carefulattention it deserves.However, I have prepared an animation to mark this occasion:

.gif climatology map animation = annual cycle dynamic attractor:

visualizing & understanding terrestrial 200hPa= mean terrestrial jet streamssemiannual midlatitude westerly windsCredit: Climatology (average annual cycle) animations have been assembled using JRA-25 Atlas images. JRA-25 long-term reanalysis is a collaboration of Japan Meteorological Agency (JMA) & Central Research Institute of Electric Power Industry (CRIEPI).

This new animation is a strategic supplement to help everyone understand solar Schwabe modulation of annually cycling terrestrial insolation (heat engine) gradients.

That article emphasizes what’s

observed, but itcanbe extended to terrestrial forecasting using agoodsolar activity model — (maybe RJ’s….)Regards

RJ:

Thanks sincerely for your contribution.

Best Regards!

Rog, you mentioned that RJ’s model here is in good agreement with the lean model that Tim did.

But I note that RJ’s statement that

” The model forecasts that the sunspot cycle will not produce sunspot values over 100 again until the cycle that starts around 2160″is also in pretty good agreement with the extended version of Tim’s model that you guys kindly provided for me as a prize not long ago. In that extended version we see temperatures bottom out at 2140 somewhere below those of the Maunder.

This is the latest version of Uranus orientation to the Sun.

Sparks: Thanks already put both your updates on the headline post. Very interesting plots.

J Martin: Scary ain’t it? 🙂

Your welcome, I have a graph that I have just uploaded, its a more detailed version of the North South Polar orientation. http://thetempestspark.files.wordpress.com/2013/09/uranus-solar-4-rbg2.gif

I picked out the interesting point that we are heading into destructive interference much like the Maunder which seems to indicate we will be in similar conditions in the near future.

Thanks for your comments. And Paul that is some animation, I will have to use it at the end of future e-mails. 🙂

An update:

I have a work in progress for a future add on to this model, where I have taken the existing model’s output wave and used it as input to a model for sunspot kinetics.

The output from this enhanced model is the skewed Gaussian of the sunspot cycles. The enhanced model also tightens the permissible frequencies and raises the correlation factor R^2 to 0.915.

The output from this model can be seen here:

R.J. is keeping us on the edge of our seats I see; no forecast with the new improved 0.91 correlation model. Ah well.

I know he is very busy from our correspondence, so we’ll have to wait for an update. Meantime there’s plenty to look at in this new output. I like the ‘knee’ in cycle#4 around 1795 which pushes the solar minima around the Dalton into line, and the low cycle #20 in the 1970s.

Don’t keep us waiting too long R.J. !

A paper from a few years ago.

And yet the correlations are undeniably excellent.

Good puzzle. 🙂

Fortunately, all bets are off for mainstream solar theory:

http://wattsupwiththat.com/2012/07/09/weak-solar-convection-approximately-100-times-slower-than-scientists-had-previously-projected/

From TB’s link:

‘“If these motions are indeed that slow in the Sun, then the most widely accepted theory concerning the generation of solar magnetic field is broken, leaving us with no compelling theory to explain its generation of magnetic fields and the need to overhaul our understanding of the physics of the Sun’s interior.”’

‘Widely accepted’ aka ‘wrong’. Another one bites the dust 😉

Well one upshot is that if there are no big convection cells disrupting the solar surface, then the latitudinal waves resulting from planet-tidal interactions (which are many orders of magnitude bigger than the vertical displacements), will have more of an effect than previously assumed. The paper Tim C linked has a formula for calculating them. Though you’ll need to remember the Sun’s high surface gravity flattens the vertical component and widens the horizontal component. Hence Jupiter’s tidal effect on the Sun produces waves which are about 1mm high but many hundreds of kilometers wide.

Perhaps if we can see things are so but the explanation given by others who have never been there doesn’t seem to fit, something else is going on.

RJS is not presuming a solar mechanism, merely that an available pattern seems to fit an observed pattern.

Nothing wrong with that. There are several different ways of addressing a problem and arriving at an explanation.

Tim C: RJS is not presuming a solar mechanism, merely that an available pattern seems to fit an observed patternYes, but the ‘available pattern’ isn’t just constructed from ‘free parameters’ used to wiggle the elephants trunk. It is constructed from the periodicities of real celestial phenomena, modulated by the interaction periods of other real celestial phenomena. The exact mechanism of causality may still be unclear, but the quality of the correlation isn’t in much doubt in my opinion.

Tim C and TB:

When I said sunspot kinetics I was not referring to the sun’s core nuclear processes. The process of planetary disturbance as you have pointed out is a tide at the surface. The kinetics I am speaking about are also at the surface and not the core nuclear reactions of the sun. Consider the suns surface as a simmering plasma at (for want of a better term) its boiling point, which is disturbed by an energy wave. Like a pot of water at 100 C with a hot poker placed in it, Boltzmann kinetics can be assumed to apply. (a plasma being like a very hot gas). The suns reaction to this kind of disturbance could be rather quick. In any case I have a model that says it’s relatively quick and our eyes tells us the same thing.

I suspect that conventionally assumed laws governing tidal formation are at best highly confused when it comes to the sun.

I’m in the process of reading up and to cut things short, Rog, have you considered asking Lindzen if he has an opinion on possible solar tidal effects?

The reasons is this widely mentioned work http://www-eaps.mit.edu/faculty/lindzen/29_Atmos_Tides.pdf

ATMOSPHERIC TIDES

RICHARD S. LINDZEN

SYDNEY CHAPMAN

186 pages, dated 1969

This includes the history of trying to figure out the highly complex matter of atmospheric tides which turn out to be primarily thermal.

I expect he has considered what is going on other bodies or perhaps keeps out it.

The possibility exists of changing the optimiser software I use to directly or indirectly work on the proposed model and see what happens when it is shown input data. It looks complicated but not excessively, problem is more a functionality change.

The starting point would be the present solution, the aim being refining values to best fit known data. We know there are question marks over the data.

I don’t know if this is practical nor how long it would take.

Is it worth a try?

R.J. Tides act throughout the entire body they are acting on. The degree of deformation produced depends on the modulus of elasticity, gravity and very likely several other things. Clearly the plasma at the surface is going to be the focus of interest. But if you want to get into the theory of how the planets could affect the sun tidally or via angular momentum, you’ve got some serious reading and calculation ahead of you. I will be posting Ian Wilson’s latest insights soon, which are a bit more accessible and may give you some food for thought (and some relevant periods to think about).

Bottom line: steer clear of worrying about mechanism and concentrate on fitting periods and their modulation. We’ll reverse engineer what you find to be the best combination to see if we can find an underlying rationale.

Is it worth a try?Tim, only you can know the answer to that. Will it be of use in other problems?

Lindzen: Hmmmm. I suspect he’d look at me as if I’d just asked him for a lightly grilled stoat in a bun with fries. The problem with the Sun is we know so little about the fluid dynamics of the surface and subsurface conditions. Tidal theory is a heuristic mess anyway. Ian’s new article might help decide a couple of things about the torque/momentum issue.

So that would be served on Connolly leather bound wikipedia?

Lightly grilled is very restrained.

A reaction like that by L. is what I would expect, at least in public but I’ve never met him.

Never can tell what is useful with function generation. The general solar problem is not met by straightforward methods, problem has to be transformed into a tractable form.

Hi everyone, Can I just point out that I did a little forecasting graph of my own a few weeks back, it’s based on the orbits of Neptune-Uranus-Jupiter, I’m not 100% happy with it but it does appear to agree with forecasts that I’ve seen here on the talkshop, I manually added generic solar cycles from the sunspot area record that matched the trend of N+U+J orbital observations.

Here is the forecast;

This is what the orbital data I used as the guide looks like graphed against the Sunspot estimate.

Sparks: It’s worth putting a legend and title on your plots along with your website name and date. It’ll bring hits your way from Google images later on.

Rog, Thanks for the tip that’s a good idea.

While I’m here, do you know of any data (if it even exists) on the movement of the suns core?

The other thing I’ve been looking at lately is if there are small objects like planetoids and asteroids locked into the suns magnetic activity, I’ve found a few contenders, guess which way these objects orbit the sun?? they have a polar orbit South –> North –> South.

This shows the objects distance from the sun over the Sunspot area record. The graph doesn’t go far back because I’ve just looking for these objects at the moment.

This Asteroid is called Tanete it has a sidereal period of 5.2 years its distance is 3 AU from the sun and its inclination to the elliptic is 28°.

RJS, in SN=(F1*cos(w1*(t +ph1))+F2*cos(w2*(t+ph2))+ F3*cos(w3*(t+ph3))+F4*cos(w4*(t+ph4))

What are F1 through F4?

Tim C

They are scalars and I see why you asked since I overlooked the inclusion of their values at the end of the article.

F1= -32.277327890156

F2= -65.6646990792377

F3= 51.9264185855531

F4= 40.9140438856888

R J: I’ve added the scalars to the article.

Sparks: Motion of the solar core? What core? It’s red hot turtles all the way down. 😉

Actually, the high pressure at depth means the turtles are pretty tightly packed. A bit like the 8:15 into Euston.

Thanks. I’m trying to figure out what you are doing.

Annoy some folks, wassis?

That was fun getting WordPress to stop wriggling, took N tries, sigh.

Near enough an overlay plot. Might be mistakes. A starting point and we have replication.

So elegant. Nice work Tim.

RJ: The opportunity for fruitful collaboration is now on offer.

What if the Sun wasn’t ‘changing polarity’ every 11 years but one ‘pole’ was waxing and submerging the other ‘pole’? Then there would be two overlaying curves rather than one, and we would get realistic minima with varying lengths. Better than ‘upending’ the alternate opposite polarity signal creating a sharp ‘bounceback’?

What I have in mind is something like the way the north magnetic pole of Earth behaves. There are actually two north magnetic poles, and as the one in the Canadian archipelago waxes stronger, the other in Siberia gets weaker, pulling the resultant pole our compasses point to in the direction of Canada. Could it be that some analogous battle is going on in the sun/solar system, and that the Sunspot numbers are a ‘resultant’ of opposing forces?

Clearly, there are several different combinations of planetary signals we can use to get somewhat similar results. I think Oldbrew and I are in the middle of demonstrating why that should be with our ‘Why Phi?’ series of posts. Since various groups of planets ‘reflect’ each others periodicities within their own parameters, why would it not be the case that the Sun does the same? The Sun and the planets all exist in the same resonating bubble floating round the galaxy after all.

Tim, If your graph at 2:37 is a model with a good match then shouldn’t it be projected to 2040 or so in order to see how it suggests things might play out in the near future and to compare it to other projections ?

I always like to see models extended into the part of the future that is likely to be of immediate interest to society. That cycle 25 thing.

Rog., Waxing poles. Radical idea, email it to Leif and see what he says. I’m not sure that Leif entertains radical ideas though. Just imagining the look on his face would make it worth it.

JM: Tim has replicated RJ’s work, so the forecast is the same. I suggest you ask Leif, don’t mention me. You’ll get a saner response that way. He has mentioned a similar idea in the past himself, so it might be productive.

Given the eccentricity of Jupiter’s orbit, plotting the dates on which it crosses the solar equatorial plane against the occurrence of solar min and max might be of interest. It will go in and out of synch, but the regularity (or otherwise) of the rate at which it does so might tell us more about the relative effectiveness of the gravitational and eletromagnetic forces involved. I wonder what the rate of precession of Jupiter perihelion is. Time to consult JPL’s ephemeris again.

Nice work Salvador.!!

How do your results stack up against other modellers future projection models?

Like scafetta, Vukcevic , Doormann, Landscheidt etc

Do they use the same or similar technique?

You mention destructive interference . Is this a cancellation of force vectors causing maunder minimum?

This means net force = 0= no sunspots?

Weather Cycles:

I have not directly compared the results of this model with others.

The model implies that at times parts of the solar system work against each other with their effect on the sun keeping in mind that this is a very simple model trying to describe 300 to 1000 years of history of the sunspot cycle and the solar system.

TB:

I am coming to believe that the “Gnevyshev−Ohl (G−O) Rule” almost guarantees your musings about two cycles.

Hi RJ,

These equations superimposed are really interesting. Even though things go bad during the

Dalton minimum.

a:

-3 * Sin[ (2 pi / 11.0326)(year – 1913.625) ]

b:

3 * Abs{ Sin[ (2 pi /165.5)(year – 1913.625) ] } ^ 1/2

c:

add vertical lines for solar minimum dates

[…] TLMango on R.J. Salvador: Planetary model… […]

RJS, I’m head down on other things, why I didn’t say much, verify was a quick look to see I had it right. I’ll probably re-engage later after we have all had space to think.

I had been keenly looking forward to this article from RJ during weeks when I had extra free time. Like Tim, I hope to further this discussion when the time’s right …but that’s not this week and possibly it’s not even this month. No doubt we’ll all resynchronize at some point in time. Natural timing’s the best. We’ll see what that ends up being…

Cheers…