Dark Matter, Dark Energy; Now There's "Dark Gulping"
Written by Nancy Atkinson
For all you dark matter and dark energy fans out there, now there's another new "dark" to add to the list. It's called "dark gulping," and it involves a process which may explain how supermassive black holes were able to form in the early universe. Astronomers from the University College of London (UCL) propose that dark gulping occurred when there were gravitational interactions between the invisible halo of dark matter in a cluster of galaxies and the gas embedded in the dark matter halo. This occurred when the Universe was less than a billion years old. They found that the interactions cause the dark matter to form a compact central mass, which can be gravitationally unstable, and collapse. The fast dynamical collapse is the dark gulping.
Dr. Curtis Saxton and Professor Kinwah Wu, both of UCL's Mullard Space Science Laboratory, developed a model to study the process. They say that the dark gulping would have happened very rapidly, without a trace of electro-magnetic radiation being emitted.
There are several theories for how supermassive black holes form. One possibility is that a single large gas cloud collapses. Another is that a black hole formed by the collapse of a giant star swallows up enormous amounts of matter. Still another possibility is that a cluster of small black holes merge together. However, all these options take many millions of years and are at odds with recent observations that suggest that black holes were present when the Universe was less than a billion years old. Dark gulping may provide a solution to how the slowness of gas accretion was circumvented, enabling the rapid emergence of giant black holes. The affected dark mass in the compact core is compatible with the scale of supermassive black holes in galaxies today.
Dark matter appears to gravitationally dominate the dynamics of galaxies and galaxy clusters. However, there is still a great deal of conjecture about origin, properties and distribution of dark particles. While it appears that dark matter doesn't interact with light, it does interacts with ordinary matter via gravity. "Previous studies have ignored the interaction between gas and the dark matter," said Saxton, "but, by factoring it into our model, we've achieved a much more realistic picture that fits better with observations and may also have gained some insight into the presence of early supermassive black holes."?
According to the model, the development of a compact mass at the core is inevitable. Cooling by the gas causes it to flow gently in towards the center. The gas can be up to 10 million degrees at the outskirts of the halos, which are few million light years in diameter, with a cooler zone towards the core, which surrounds a warmer interior a few thousand light years across. The gas doesn't cool indefinitely, but reaches a minimum temperature, which fits well with X-ray observations of galaxy clusters.
The model also investigates how many dimensions the dark particles move in, as these determine the rate at which the dark halo expands and absorbs and emits heat, and ultimately affect the distribution of dark mass the system.
"In the context of our model, the observed core sizes of galaxy cluster halos and the observed range of giant black hole masses imply that dark matter particles have between seven and ten degrees of freedom,"?said Saxton. ?"With more than six, the inner region of the dark matter approaches the threshold of gravitational instability, opening up the possibility of dark gulping taking place.?
The findings have been published in the Monthly Notices of the Royal Astronomical Society.
Source: RAS
Filed under: Black Holes, Dark Energy, Dark Matter
Related stories on Universe Today
Comment policy: Be nice and brief. Don't advertise your stuff, or promote your personal theories. We'll delete any comments that break these policies. Click here for more details.
April 22nd, 2009 at 8:28 pm
The most creative name for another "epicycle" in the Big Bang theory.
April 22nd, 2009 at 8:28 pm
Yes lol. Black holes are merely entrances and exits of the nutrino universe. How do I know this. Because the universe is just a giant version of an atom. Material surrounded by energy bound to a center point of something we haven’t figured out yet. One should assume that the universe is a giant sphere where the surface is near equidistance from the center point. Surrounding the universe is a nutrino enrgy field that is probably very similar to the same thing that is in the center point holding it all together. Energy flows from and back to the center point. Within space there are Megagalaxies, combined galaxies, combining galaxies, singular galaxies, star clusters, star groupings, star pairings, and single stars. The same can be said for material in space. Out there somewhere there could be a huge system of planets, then planet groupings, planetarty pairs, singular planets, moons, astroids and meteors, meteorites, space rocks, and space dust.
April 22nd, 2009 at 8:30 pm
It is now known that stars and matter go into and out of black holes. So why is that?
April 22nd, 2009 at 8:40 pm
Riiiight.
Because considering something that has been previously ignored, but was (presumably) thought to happen anyway is /so/ the same thing as adding another epicycle.
April 22nd, 2009 at 8:45 pm
I really wish they picked a better name. At least they didn't call it "dark slurping". (The eating-related metaphors relating to black holes sometimes annoy me).
Oh, and congratulations Ms. Atkinson. Your article has finally broken solrey. =)
April 22nd, 2009 at 8:51 pm
JustinnitsuL, once again I can only stare in amazement at your post with my mouth agape. Wow. Just… wow.
April 22nd, 2009 at 9:04 pm
holy crackpot, batman…i think this article just broke a whole lot of people.
Might as well just consider this article's comment section as lost once our resident oily crackpot starts up his BS in this one. Like a moth to a candle flame…
April 22nd, 2009 at 9:11 pm
"We are unfamiliar with plasma because we are blind to it. Modern astrophysicists are in this sense correct to claim that 90% of the universe is undetectable 'dark' substances. Their error is to fill in the blank with mathematical extrapolations from familiar theories and to leave their thinking blind to plasma." — Mel Acheson, physicist, December 2008
April 22nd, 2009 at 9:27 pm
But hey, the article is fun.
Then again, because there are so few constraints in the over all so-called "black hole" hypothesis,anything goes…
It's hard to say one "black hole" hypothesis has any better claim to reality than any other.
After all, infinity can't be quantified.
Sooo…
You get these kinds of theories…
Kind of like looking in a fun-house mirror…
April 22nd, 2009 at 9:53 pm
Amd yet today we also have an article elsewhere that essentially states that electromagnetic only theories can't cut it, so what gives you two.
:3
April 22nd, 2009 at 10:02 pm
LOL trippy, do you honestly think those guys would actually take a thoughtful consideration of anything that refutes their beliefs?
April 22nd, 2009 at 10:11 pm
Dark Slurpies…LOL
Strange things are afoot at the Circle K!
So…what next? Dark Monty's?
Man: "Strange dark maidens lying in tenth dimension HII clouds distributing mass is no basis for a system of science."
King: "BE QUIET"
Man: "I mean, you can't expect to…"
King: "SHUT UP"
Man: "Just because some farcical simitar…"
King: Grabs Man, shakes him vigorously
Man: "Ah, now we see the dogma inherent in the system… 'elp, 'elp…I'm being repressed."
Scene cuts to lightning in the distance as the "king" and his minions ride off on imaginary horses, while the jesters bang coconuts together, and claim to be galloping.
Patsie, the kings jester, is evaporated by a stroke of lightning.
Trumpets blare and angels sing…
April 23rd, 2009 at 12:31 am
Joe: Not in the slightest :3
April 23rd, 2009 at 1:02 am
"Because the universe is just a giant version of an atom."
I knew something odd was going on when I made a transition to another quantum state and emitted a photon. Oh – actually, I think that was just me farting.
solrey Says:
April 22nd, 2009 at 10:11 pm
You should write fiction solrey. Aw wait – you already do!
April 23rd, 2009 at 1:06 am
Seriously though – they have a serious study, so why do they have to cheapen it by calling it 'dark' something? It's like the Watergate scandal started every f__ing journo in the world calling every slip up by a politician '(insert stupid name here)-gate'. Man it pisses me off when journos do that.
In fact, being pissed off about that reminds me of how I'm pissed off about still being at work despite the fact it's 6:06pm AEST. I'm getting the hell out of here.
April 23rd, 2009 at 2:04 am
It’s called “dark gulping,” and it involves a process which may explain how supermassive black holes were able to form in the early universe.
There are several theories for how supermassive black holes form.
My theory is that we live in a cyclic universe, which is at the end concentrating all the matter into black holes, which only at the end are able to merge into one Big CRUNCH. If this Big Crunch explodes again and become an evaporating big bang, black holes will be the first result.
http://bigbang-entanglement.blogspot.com/2007/03/backreaction-lee-smolins-trouble-with.html
Conclusion the Big Bang singularity does not exist in the form of nothingness.
The most obvious nuclear component is compressed Higgs particles, able to change form by mutual collision into couples of entangled electrons and positrons.
April 23rd, 2009 at 3:47 am
If Dark Gulping is a proven hypothesis, then it follows that Dark Flatulence is a phenomenon of equal proportions.
April 23rd, 2009 at 3:49 am
Leo Vuyk Says:
April 23rd, 2009 at 2:04 am
Why do people insist on posting non-peer-reviewed blogsites and websites with the theories that they have devised?
All I can say is:
Leo Vuyk, meet Oilsmastery, Solrey and Anaconda. O,S & A, meet Leo.
Round 1 – FIGHT!
April 23rd, 2009 at 4:58 am
EARTH HAS 4 CORNER
SIMULTANEOUS 4-DAY
TIME CUBE
IN ONLY 24 HOUR ROTATION.
4 Corner TIME, CUBES EARTH.
Americans are dumb, educated ONEness
stupid and they worship ONEism Evil.
Americans don't know that they are dumb, stupid and evil, for they think they are in the image of a GOD. Okay, but this also means that their God is in
their own image – and just as stupid.
Supermassive blackholes are ONEess and stupid and do not follow Dr. Gene Ray's most HARMONIOUS and genius quad spiraling helix and you realize that 1/4 rotation of cube equals FULL day.
April 23rd, 2009 at 5:49 am
HarmoniousFourDayTimeCube, I couldn't have said it better myself – more sense was never made than in your post.
April 23rd, 2009 at 5:54 am
Any (free) links to the entire paper? Couldn't find anything with Google.
April 23rd, 2009 at 7:33 am
Dark Gulping? I thought, at first, that it was a title for a really bad porn movie *giggle*
April 23rd, 2009 at 8:39 am
"Why do people insist on posting non-peer-reviewed blogsites and websites with the theories that they have devised?"
The dark gulping model was presented in a very long paper (33 pages!) published in a prestigious peer-reviewed journal "Monthly Notices of the Royal Astronomical Society". The full reference is
"Radial structure, inflow and central mass of stationary radiative galaxy clusters" (C. J. Saxton and K. Wu, 2008, MNRAS, vol. 391(3), p.1403-1436).
It is not easy to understand, too many equations, but does not look like a wacky article.
April 23rd, 2009 at 8:45 am
"Any (free) links to the entire paper? Couldn’t find anything with Google."
To read the article we need pay-subscription to the journal "Monthly Notices of the Royal Astronomical Society". However, the pre-print version of it can be found in this web-page:
http://arxiv.org/abs/0809.3795
Just click PDF in the download box, and you will get a copy. It took quite a while to download it.
April 23rd, 2009 at 8:49 am
Beware of geeks bearing formulas.
Any math that points towards unobserved Dark Gulping is pseudoscience and occult and has no correspondence to actual physical material reality.
Dark Gulping is a mental artifact of Meinong's Jungle whereas plasma is 99.999% of the visible universe.
April 23rd, 2009 at 9:45 am
Thanks for that link, science_fan.
After looking at the paper and noticing all of the variables that can be tweaked, I'm so reminded of a quote by Tesla:
“Today's scientists have substituted mathematics for experiments, and they wander off through equation after equation, and eventually build a structure which has no relation to reality"
Compare with this approach:
http://plasmascience.net/tpu/downloads/AdvancesII.annotated.pdf
April 23rd, 2009 at 10:39 am
@science_fan, Many thanks for the link to the preprint. It is a pretty dense read (38 pages) and most of the math is beyond my 3rd year college skills. I think I can still follow their argument (this is a paper on computer modeling & theoretical astrophysics, so go fig, newt) I was curious as to why my Google search for 'dark gulping' returned no hits on the paper at all. I went through the entire 38 pages and (Astrofiend will love this) found no mention of "dark gulping" ! There were references to "dark growth" and "dark physics" (call it dark-gate). If it's in there, please correct me, I might have missed it, but since this was presented at a meeting, I suspect the authors wanted to jazz up their presentation or the RAS PR machine came up with the term, but I agree with Astrofiend, enough of dark-gate, back to more sensible names like those used in particle physics (like the charmed quark).
April 23rd, 2009 at 10:46 am
So, if I understand correctly, we now have
1) Dark Matter
a) …that may or may not have a "Dark Electric Charge"
i) …which may imply a "Dark Electromagnetic Spectrum" that we can't see or detect.
ii) …which may imply some form of "Dark Electromagnetism" and "Dark Electric Currents" are possible.
2) Dark Energy
3) Dark Flows
4) Dark Gulping
Is anyone holding the astronomers responsible for anything anymore?
The queen of the sciences is running about stark naked and half-mad. Can someone please give her some new clothes, a few antipsychotics, and a hug?
Astronomers need to stop putting so much faith in abstract maths not based in either direct observation or experiment and go back to the lab, study plasma physics and electrodynamics and stop making up "new physics" when the existing physics of plasma and electromagnetism aren't yet even being applied properly!
Every news release seems to seek to trump the last in absurdity. How much longer will responsible scientists stand for the continued maddening nonsense proceeding from the standard model's falsified theories and continual patchwork to cover them up?
Best of luck,
~Michael
April 23rd, 2009 at 10:48 am
Why not call it "dark gulping?" You people need to realize that those people put in the work – they can call it whatever the hell they want. Leave your misconceptions about the "high and mightyness" of naming physics ideas with Hollywood.
April 23rd, 2009 at 11:04 am
@solrey, above you quote Tesla as saying “Today’s scientists have substituted mathematics for experiments, and they wander off through equation after equation, and eventually build a structure which has no relation to reality”. Could this not be said of the Peratt paper you linked to, also? I'm sure thousands, if not tens of thousands of lines of computer code were written to produce these computer simulations. And hundreds of equations must surely be involved to try to make the simulations produce some semblance of reality. The difference in these 2 papers is mainly style of presentation, not mathematical rigor.
April 23rd, 2009 at 11:09 am
Solrey said:
"Today’s scientists have substituted mathematics for experiments, and they wander off through equation after equation, and eventually build a structure which has no relation to reality."
Sadly, at this point, I have to agree. The astronomers' equations bear no relation to actual experimentally observed reality (I'm talking "in the lab" experimentation, not abstract guesswork and flicking spaghetti at the wall to see what sticks)…
The first thing they need to realize is that plasma is not a superconductor. Its efficiency at conducting a current is not 100%. Its resistance is non-zero. It does not "instantly charge-neutralize," and non-zero electric fields *do* exist between differing regions of charge within a plasma. Anyone who says otherwise is selling you a bill of goods.
Granted many sites out there still erroneously state that plasmas are superconductors. Still, ACTUAL lab experiments have shown this NOT to be the case. The less random (thermal) motion there is, the better the conductivity. The only superconductors created to date have been ultra-cold (nearing absolute zero). The ions and electrons of a plasma are ultra-hot.
Since plasmas are not ideal conductors, magnetic field lines are not "frozen in" to it or "carried along with it." They are dynamically generated. They do in fact change over time. Why? Because the underlying electric currents change. Electric charges are the source of the electric field. Electric currents (the net differential motion of those charges) are the source of the magnetic field. Look it up!
All magnetic fields on earth and in space are derived from electric currents. Period. End of story.
Google: Richard Fitzpatrick Classical Electromagnetism
Then look up his articles on "magnetic monopoles" and "Helmholtz's theorem." (Node35 and Node37, I believe, of the professor's lecture series.)
Once that one simple misconception is done away with, many other vistas open up for understanding how things ACTUALLY work, seeing as how 99.999% of the OBSERVABLE matter in the universe (not any of this wonky "dark matter" or other kludges based upon falsified theories) is in the plasma state.
I don't expect anyone to take my word for it, I expect them to look it up on their own to independently verify.
Regards,
~Michael
April 23rd, 2009 at 11:23 am
John Hanford said:
"Could this not be said of the Peratt paper you linked to, also?"
It COULD, but would it be accurate? Peratt works at LLNL and is trained in actual LAB physics. IE, they have to actually WORK with the stuff. So,they generally know what does and does not work and which equations are actually applicable and which are NOT.
While I see what you're saying (computers are a GIGO system; garbage in, garbage out), I'd generally trust an experimentalist over an abstract mathematician most days of the week.
"Dark Matter" and "Dark Energy" don't come from lab experimentation. We can't get out hands on them. (They probably DON'T actually exist, but you didn't hear it from me. Mum's the word! ;o] ) Ockham famously said "entities should not be multiplied beyond necessity," or something very close to that. IE, the fewer assumptions, entities and hypotheticals necessary to explain a system the better. It's also best to "go with what you know," rather than invent new hypotheticals at every turn. Should we not exhaust what can be learned from "known" physics before venturing into "new physics?"
So, yes, one could say that we should be very careful with computer models and understanding that they're only as good as what's fed into them. That said, plasma physics has been a fairly useful discipline for many years and much *is* known about how plasmas act and what behaviors can be expected from them. Plasma physicists generally know what equations *do* work and what the results might be as well as what equations *don't* work (those implying plasmas are superconductors or can "freeze in" magnetic field lines, for one!)…
Can astronomers claim the same level of knowledge of what "dark matter" is or how it works? Methinks not.
Just my 2c.
Regards,
~Michael
April 23rd, 2009 at 11:30 am
Just to be clear, the reference to Richard Fitzpatrick doesn't deal with the superconductivity issue, just with magnetic fields and electric currents. As I was reading it, I noticed that the arrangement of paragraphs made that a bit ambiguous. Just wanted to clear that up 'fore anyone else got confused.
April 23rd, 2009 at 1:46 pm
@ Michael Gmirkin, While I agree with your statement "Peratt works at LLNL and is trained in actual LAB physics. IE, they have to actually WORK with the stuff.", some of the stuff in the paper being discussed are galaxies and quasars. We know of course no one at LLNL has brought IC 342 or Cygnus A into their labs. The paper mentions the monochromatic power output of QSOs and double radio galaxies as being between 10^33 – 10^39 Watts.They're surely not generating anything at LLNL at such powers. So he's not actually handling these sources in the lab, he's extrapolating from terrestrial experiments. And that extrapolation has a number of variables that can greatly affect the outcome. One other aspect of this research appears puzzling. The paper that we're discussing was published in 1998. With the tremendous increase of computing power over the intervening 11 years, why have no other computer simulations been performed and published, either by Dr Peratt or other researchers at LLNL. They have access to some of the best supercomputers in the world. I would think this would be a boon to the study /simulation of plasmas on a galactic scale? Are plasma scientists being denied access to LLNLs supercomputers or would such simulations be redundant and a waste of time, as solrey claims? It just seems strange that more research on plasma cosmology or "plasma universe" is not being intensively pursued by scientists with access to some of the worlds most sophisticated supercomputers.
April 23rd, 2009 at 5:13 pm
Oils is like an oil slick, it creeps in everywhere and spreads its poison.
In spite of oil slick's nonsense, this is an important find. The very early universe had extremely low entropy, near zero in fact, but massive black holes were produced pretty quickly within 500 million years. How these thunderous BHs formed is a matter of consierable interest. Dark matter "particles" are maybe supersymmetric pairs of the neutrino, Higgs, and photon in a mixed or condensate state or neutralino. Dark matter in halos provides the mechanism for large BH formation in the post radiation dominated phase.
Lawrence B. Crowell
April 24th, 2009 at 3:22 am
Jon Hanford,
as I understand from my own reading the method that the two authors used is not computer simulation at all. They used the computers to obtain numerical solutions to the equations, which were derived analytically, following … hmmm ….. a very standard procedure: conservation of mass, momentum and energy. The complexities arise because it is a system of multiple components. The only parameters that one can play with in this formulation is the degrees of freedom of the dark matter. The others are standard variables obtained from observations, as in other research papers of similar kind. (We cannot call the others as parameters as they serve just to give the normalisation of the variables in the equations.)
From what I see, it is the dynamical interactions between the two components that give rise to the interesting behaviours. In their case for some situations the dark matter component can collapse to make a big dense mass which does not emit electro-magnetic radiation. The paper itself had no big claim.
I doubt that the authors had made big claims either — there have not be press conferences on this subject from any of the authors. The paper was published last year. If they want to they can make press release from their own institution. What we have seen so far in many news sources was that the authors asked to answer some questions by science writers and they explained what the finding would imply, in the context of black-hole formation in the early universe.
Sometimes, I appreciated the time and efforts that scientists put into their research. If those two guys are willing take such a length to make a long papers like this it is inconsistent that they are wacky nuts who just want to have a helicopter rise in the fame game. It is not very cost effective.
There is a slightly different article on the same subject in the website space.com .
April 24th, 2009 at 6:53 am
@science_fan, Sorry if you took my comments on their published paper the wrong way. I actually think they're on to something that may help explain the early appearance of SMBHs, and well worthy of a vigorous presentation of their work. I was mainly responding to comments on this article about the actual origin of the term 'dark gulping', that's all. I read all 38 pages of the paper and could not find the term. At no time did I state or imply "they are wacky nuts who just want to have a helicopter rise in the fame game". They may just be on to something, so now let's see if other researchers can duplicate or refute their arguments. Again, no slight against these scientists or their paper. Also, I generally don't take the length of a paper into consideration when considering its' validity. Content counts, not word counts.
April 25th, 2009 at 5:42 am
Jon Hanford
Sorry that I caused the misunderstanding. I should split my previous e-mail into two. Some of the comments were not intended to direct to your previous posts.
"Also, I generally don’t take the length of a paper into consideration when considering its’ validity. Content counts, not word counts."
I agree with you that writing a scientific paper is not a word-count contest. The value of a piece of scientific work should be judged by its content.
What made I bring up the length was that I noticed that if we are not serious about something we are unable to have the discipline to sit down to make something that demand hours and hours of labour. I studied science in college and in graduate school. The longest calculations that I did, for a course in my graduate student years, was about 20 pages. It was a take-home examination. My classmates and I were allowed for 2-week to finish it. That calculation was like a torture for me already, and I still remember the fourteen days and nights that I ploughed pages and pages of xyzabc.
The paper by those two scientists has 33 pages in the final printed journal version. The equations presented there were already cleaned up, i.e. most of the calculations for the intermediate steps were not included. I can imagine how much time that it would take to construct the conceptual model, translate it into mathematics, manipulate the mathematics to get a form suitable for computer programming, write the computer programme and run the computer programme. This does not count the time that it takes to check the mathematical calculations and to debugg the computer programmes.
In an analogy, good musicians do not need to play a master piece in order to demonstrate their artistic level. However, if someone can compose a grand opera that takes a half day to execute, the composer should have some solid musical capability.
Going back to the issue related to the "dark gulping" discussion. There are several predictions mentioned in that paper. The predictions can be tested by observations and experiments. It is just a matter of time whether the model of SMBH produced by "dark gulping" would or would not work. A good model does not only explain. It should make testable predictions.
April 27th, 2009 at 3:54 pm
I dunno about anyone else, but * I'm * still waiting on them to prove the prediction that black holes exist.
I'm not a scientist, I'm a layman. I've looked at several different views on the matter. I see two strong viewpoints:
1. A gravity-dominated Universe that says electromagnetism doesn't exist away from planet Earth, and contains Black Holes as a fundamental facet of it's body.
2. A plasma cosmology that attempts to apply known physics to cosmological phenomena without relying on mathematical conjecture as the basis of statements regarding various observations.
I'm leaning towards the Plasma Cosmology. I have a basic understanding of electrical theory & physics from my time in the Navy. I knew then (as I know now) that electrical physics is scalar- you can arc 250 millivolts off your cat's nose wearing wool socks, and you can arc 17,500 volts off a powerline if you let wet treelimbs get too close to the power pole.
I saw some pics they took of the closest yet view to the center of the galaxy. They told me I was looking at a black hole, surrounded by "swirling bands of superheated plasma"… but didn't explain why all the plasma arcs were twisted into helical structures of two strands. Also, they say that the "Helix Nebula" is a body of gas that is located directly above the 'black hole' at the center.
Well, the plasma guys say that the "twisting helical bands of plasma" are the wires that carry electricity throughout the Universe and we see them at the busy core of the galaxy., And they say that the galaxy is powered by one of them, straight through down the galactic pole, and when they were able to see the core up close here they see this 'Helix Nebula'… but I sure didn't see any 'black hole.'
I'm sorry guys, but if I'm forced to choose between explanations based upon relative, observable phenomena from lab experiments, or believing explanations based upon mathematical conjecture…
… I'll go with the plasma. I can *see* plasmas.
April 27th, 2009 at 8:06 pm
@ mharratsc, If they told you that "the “Helix Nebula” is a body of gas that is located directly above the ‘black hole’ at the center." while in the Navy, you got some bad advice or misunderstood what they were telling you. The 'Helix Nebula', NGC 7293, is a planetary nebula with a hot white dwarf star at its center. You may want to study this page on the Helix at Wiki (and reading some of the links. Great article!) here: http://en.wikipedia.org/wiki/Helix_Nebula . As for Plasma Cosmology………
April 29th, 2009 at 1:08 pm
@ science_fan: I've done a few multipage calculations myself, so I hear what you're saying. And the point about developing the discipline to carry out this work is spot on. Rigor is certainly necessary to flesh out and back out one's position.