Stellar Nucleosynthesis

Main Sequence Page 2

The stardust hypothesis and stellar nucleosynthesis

Introduction of a Rambling Sort
My objective for this, page 2 of the Main Sequence: I'd like to investigate the stardust hypothesis, the ludicrous claim that

"Human beings are made out of the ashes of ancient dead stars."

Is there a credible argument? A viable time-sequence of events? Time seems to be a key issue... we've been occupying this corner of the galaxy for maybe five billion years... what happened in the preceding 9 billions years to set us in motion? To summarize the basic questions I have:

How does cosmology provide a credible argument for the stardust hypothesis?
Why do we suppose that heavy elements had to be "built" from lighter ones?
What are the basic requirements for alchemy?
Where does it take place?
What sort of simple physical results might we see in consequence?

And by that last question I don't mean Jerry Lewis films. I mean something directly observable that at least corroborates the sequence of ideas, particularly that nuclear fusion is at the heart of the matter. To peek ahead: I have a suspicion this will take the form of relative abundances of the elements throughout the cosmos.

Alas there are no good direct experiments to do (by me) in the matter of nuclear fusion. These pages are built on doing experiments but here at page two... I have no experiment to do. To begin with I do not have access to a Tokomak and low-energy fusion is but a dream. (Occasionally people make claims for low-energy fusion but to date they've all turned out to be incorrect.) Isaac Newton spent decades secretly trying to do alchemy, or transmutation of matter. Historians have determined that he was a bit susceptible to mysticism, so he tried for years but as the temperature of his furnace was too cold he had no success. (The biographer James Gleick points out that Newton lived in pre-Newtonian times.) To be fair, Sir Isaac did manage more productive inquiries into the nature of nature.

But back to the topic: Fusion requires high temperatures; we need millions of degrees (Kelvin) and the best way to do this is to search for dense clouds of hydrogen gas floating in space. The notion is quite clever: A sufficiently large cloud of hydrogen atoms will densify or congeal due to their mutual gravitational attraction, and the resulting very dense cloud (hypothesized to be spherical in shape) will become warm in its interior as the pressures therein become very high.

We have in fact discovered one such hydrogen sphere located not far from the earth. It is shown in this photograph.

It is inferred to be some 150 millions kilometers from earth and is periodically visible during the daytime. Since the sun is inferred to be much hotter inside than on its surface (only about 5800 Kelvins) and since the interesting physics starts happening at high temperatures, I should like to see what goes on in there. What would the sun look like could one peer inside?

(Here is the artist's conception.)

Naturally one way to proseed would be for someone to cut a hole in the sun and take some pictures.

Anyway things are just not so easy. Solar astronomy and stellar spectroscopy proceed by inferrence, by examining light emanating from the surface of the sun and the stars and passing through space and maybe or maybe not through the earth's atmosphere depending on where the camera is placed. Stars have the unfortunate habit of hiding the results of their work deep in their interiors. I should mention that there are some very nice tricks for looking inside our sun to a limited extent, particularly neutrino radiation and solar seismology. I'll have to take those up another time (but isn't it cool that Robert Leighton had the idea of looking for sun-quakes?)

Here I'll proceed with a little indirect experimentation--the Nullius in Verba stuff--and the rest will be reading and note-taking from reputable sources. I'll present these ideas as if they are infallible and incontrovertible, noting that as always they are subject to further investigation, scrutiny and verification. By way of warming up to the subject: A star ten times more massive than our sun burns 10,000 times hotter and uses up its fuel that much faster. This star (or any star) creates Helium in its core by fusing together Hydrogen nuclei. Under favorable circumstances a star may also build heavier elements that astrophysicists refer to as 'metals' for some mysterious reason. Building heavier elements from lighter ones is called nuclear fusion but sometimes I refer to it for fun as alchemy.

Superposition Digression
I'd like to indulge in a philosophical digression on an interesting idea: Superposition of information, a subject I know next to nothing about so therefore I feel qualified to be curious.

Above is a painting by the pointillist Georges Seurat, a very recognizeable scene--it is "The Lighthouse at Honfleur"--even though it is built up twice from a lot of individual dots, first in the original painting and then on a computer screen. Pointillism is cool because it emphasizes the contrast between the meaninglessness of individual dots in contrast with their very definite meaning as an ensemble. Having set many dots on the canvas Monsieur Serat has produced this very fine painting. This is synthesis of information (the scene) from source units (the dots) where the dots are spatially separated from one another, emphatically not superimposed, at least not much.

Now consider this photograph I took through a specially modified box of grape nuts yesterday...

This is what starlight looks like when passed through a prism. (To be more precise, this is what our sun's light looks like passed through a narrow slit and a transmission grating. But spectroscopy on stars follows the same procedure; I'm just using the sun because it's easier to capture a spectrum by a factor of about 100 in effort.)

Suppose this spectrum was obtained from a distant star. No telescope yet built is capable of resolving the physical width of a star; they are truly point sources of light, as dot-like as one can get. (Some stars appear larger than others on photographic plates because the light from brighter stars spills over in circular puddles on the film; the brighter the star the bigger the puddle.) But if a star is a dot and yet can produce a spectrum like the one above... Notice the various dark vertical bands interrupting the continuum of colors; is this an artifact of the imaging technique, or is this real information about the source? If it's really information about the star then I submit that spectrographs make a good case for the superposition of information. In dots.

Might we increase the information in a spectral light signature further? Might one compress all the information contained in The Lighthouse at Honfleur into a single spectrum, a single dot? If so we might then hang the dot in the Louvre so that people would walk past and and say "Ah! The Lighthouse at Honfleur! I love this painting!" And then they could walk across the river and buy a crepe.

Also if the lighthouse were still standing there at Honfleur it could transmit a powerful beacon of light carefully filtered to produce this signature spectrum, that is, the compressed version of the painting of the lighthouse itself. Sailors with prisms could decode the beam and determine that they were near the rocky shoals of Honfleur.

Information is an abstract idea, rather hard to define (like energy) but nevertheless something we traffic in quite naturally. That being the case how does it compare to other fundamental, hard-to-define things? We are accustomed to linear, planar, and three-dimensional space, and mathematically one can extend these to any number of dimensions, even to non-integer (fractal) dimensions. We have as well the dimension of time that isn't quite space and seems to be lacking in the reversibility department but is certainly part of our experience. Time multiplied by the speed of light gives a distance, and correspondingly there is some pretty physical and mathematical machinery that permits one to treat time as a fourth dimension. (I wonder what fractal time would be like...) In addition to these spatial and time fundaments there are apparently some further ground-floor characteristics of reality: Electric charge for example, and of course mass (which might be a good way of starting to define energy).

I wonder is information in some way its own kind of creature, neither time nor space nor electrical charge but something else that exists in physical reality? Or is it fated to live in the same abstract folder as the integers? Perhaps information is not fundamental because it only exists if there is an intelligence to perceive it. The tree falls in the forest and produces a noise; but someone must be present for it to make a sound. Finally it is important to acknowledge another debt to quantum mechanics, that information has a lower bound in the context of measurement. So whatever information turns out to be, it ought to be consistent with the physical reality that draws the line at 6.6x10^-34 Joule-seconds.

If information is its own thing then how does it work? What is representation or encoding of information, really? Is information compressed in superposition, or is superposition merely an illusion brought about by our more usual experiences of information distributed over the course of phone calls? Does information density behave like the density of matter? There exists a branch of math called 'information theory' that might concern itself with these issues... like Portugal I have a sense that information theory is out there while never having been any closer than the lighthouse at Honfleur.

In carelessly tossing about this terminology I should finally confess: One ulterior motive relates to the lives of stars. Some day I would like to understand Hawking's ideas on information loss in black holes, so I am starting to make little guesses about what 'information' means in that context. The suggestion does not originate with me that information in comparison with space and time and charge is another type of fundamental thing. Perhaps it can be folded or encoded into the other dimensions according to some interesting rules. They do say that time is money, after all. Perhaps information is equal to energy times the speed of light squared.

But enough of these remarks which are really just poetic imagery and not science or math in the strict sense. Sort of like the Drake equation. To get back on track, the fact that single dots of starlight contain enormous amounts of superimposed information is very counter-intuitive to me... and a big part of figuring out where all this carbon came from. Let's get down to business.

Spectra
The real smoking gun from a sun is the fact that its light output is not uniform. There is extra light in some colors and a lot of missing light in others; just take another look at that spectra picture up above, and notice the dark vertical bands. In fact if I really want to mind the gaps I should let the pros take the picture:

Credit: N.A. Sharpe / NSO / Kitt Peak FTS / NOAA / AURA / NSF

This is an amazing spectrogram. Each horizontal slice covers 60 angstroms and there are 50 of them, so that's 3000 angstroms or 0.3 microns. The visible spectrum runs from about 0.4 to 0.7 microns in wavelength so that's consistent. I kinda wish the picture had more violet in it and I also wish that the peaks in intensity were as obvious as the dropouts. But this is an absorption spectrum so we don't get much in the way of peaks.

These nice folks at Kitt Peak are trying to tel us that the light from the sun is not a perfect unblemished rainbow. It has lots of defects or gaps in its output, and they claim that these mean something important. The assertion is that this spectrum is like a fingerprint telling us what elements exist in the outer layers of the sun's atmosphere courtesy of the high-temperature chemistry on the surface of the sun. (Emphatically this is not a fingerprint from very high temperature alchemy in the interior of the sun.) Now nullis in verba let's see if we can see at least some of these gaps in the solar spectrum by building a simple spectroscope.

Building a spectroscope
Start with some kind of cardboard container maybe 20 cm across. The opposing sides are modified and the entire affair is covered with additional material to exclude stray light. That's it; all done! (Except some details.)

A first-attempt; the box could be covered better with the addition of some more aluminum foil. Is a cereal box necessary? Nope, just convenient. The constraints for this design-plan are:

Case can easily be cut/modified; cardboard is ideal.
Two opposing flat surfaces, one for a transmission grating window (shown above) and one for a thin slit.
The thin slit is formed from two razor blades taped down flat, edge to edge. This is a minimum size constraint for the flat surface.
Device is covered over by some heavy material to exclude as much stray light as possible.
Stray light will also enter through the transmission grating window; can minimize using a big black cape!

This device is easy to build and quite useful. I'll continue by describing the light-slit and grating-window modifications.

The two razorblades are taped down edge-to-edge above a hole cut in one end of the spectroscope. This makes a narrow slit-opening to admit light. First cut the hole, then place one razor edge across it and tape the razor blade in place. Carefully place the second razor edge right next to the first and secure it the second razor blade. Too close together the two blade edges will overlap and block light entry. Placing the blade edges farther apart permits more light in. This might be desirable for a dim source but I'd suggest starting with a very narrow slit.

On the side opposite the razor slit I install the transmission grating, courtesy of the nice folks at scitoys.com. This is a piece of transparency with parallel lines etched onto it, perhaps 2 cm square. It behaves like a prism. Scitoys sells sheets of grating transparency for next to nothing and one sheet of is easily sufficient for nine spectroscopes. They provide more complete building instructions and other designs using CDs and so on.

Last note on construction: It is necessary to align the transmission grating lines in parallel with the slit. You can't see these gratings so the easiest way to get the alignment right is trial and error. Hold the grating in place while looking through it; the orientation is correct when you look through the spectroscope at a bright light source shining through the opposite slit and then cast your gaze sideways and see a very distinct spectrum. I try both orientations to convince myself that I have it right before taping the grating in place. This photo shows both the slit and the spectrum in proper orientation, made using a simple digital camera.

This photo isn't good enough to convincingly show absorption features; I'll need to monkey around with it a bit; more on this later.

Building a Case for the Stardust Hypothesis
Here is why the experimentation--solar spectroscopy--is indirect: The exterior of the sun reflects consequences of interior processes and consequences of past history elsewhere. It is not hot enough to show heavy elements in production from lighter ones, an assertion that should be backed up by more details. At best a spectrogram of the sun might provide evidence that the sun could build carbon or oxygen or silicon or calcium. At the least such a spectrogram will show that the same heavy elements we're familiar with on earth are also to be found on the surface of the sun. How is this so?

[And at this stage I need to include a couple of plots comparing solar lines to Calcium.]

So far I'm avoiding the issue, the steps to the stardust hypothesis. So there are heavy elements and they are here on earth and can be seen in the outer layers of the sun and other stars. Where does the need for all this heavy-element building come from?

Why do we suppose that heavy elements had to be "built" from lighter ones?

The answer involves a circuitous chain of reasoning.
1. The universe is expanding. (Evidence: uniform red shifts of distant galaxies.)
2. Looking back in time we infer that the universe had an origination event, the big bang. (Evidence: Cosmic microwave background radiation.)
3. The big bang produced Hydrogen and Helium almost exclusively, not much in the way of lithium and heavier elements. (Evidence: This is the least-supported link in the chain. The result is primarily a theoretical result.)
4. Therefore in this universe there was originally a dearth of heavy elements. Whereas today we see an abundance of heavy elements, not to mention we notice this by using our minds which are built from heavy elements. Carbon and oxygen and nitrogen are quite common, lucky for us, but they were not present at the start.

Ergo something had to build up all those heavy elements over the timespan from the big bang until now, 13.7 billion years.

The non-uniformity of cosmic abundances
This brings spectra back into the inquiry process.

We look around the sky for evidence of element-building. The nearest candidate for an alchemistry heavy-element factory is our own sun. It must necessarily be quite hot in the interior, so that the repulsive forces between protons can be overcome now and again and they can get close enough to fuse into deuterons. The key idea is that the end product weighs less than the constitutive ingredients and the mass difference is cashed out as energy. Once we establish that it is possible to smush protons together it becomes a matter of figuring out recipes for building heavier elements.

Now my unsubtantiated report is that: Our sun is good at producing Helium from Hydrogen (really alpha particles from protons) but it is not quite hot enough in its interior to build heavier elements. So what do we do? We turn to the stars and look for other hotter furnaces. Fortunately there are many stars to see, and this leads to a number of situations where creative element building is going on.

Of course it is still quite a leap to invoke these various circumstances and recipes to arrive at... ourselves.

Fuel-Burning Alchemy
Just to start out I'll take a look at what is happening inside our sun. Helium is a great building block for heavier elements, so let's start off with hydrogen and build helium.

The premise is this: Stars begin as clouds of hydrogen collapsing under gravitational force, the mutual attraction of all those little atoms. As the hydrogen cloud collapses it heats up. When the cloud interior reaches--say--a few million degrees Kelvin it ignites in the center which means that the hydrogen starts burning. This burning is not in the sense of a candle burning because burning candles are really just swapping around electrons and breaking apart wax molecules and building soot molecules. This does not touch the atomic nuclei. Whereas the kind of hydrogen burning we're talking about inside of stars is a whole nother level of burning. This would be a good time to digress a little bit into a famous equation.

Equation Digression goes here, from paper clips and to 5e6 tons

Gradually this builds up helium and releases energy, and the energy pushes outward countering the gravitational pressure. Thus a star reaches an equilibrium between its gravitational collapse and counter-balancing nuclear burning forces. This can continue as long as there is fuel to burn.

Suppose that there were nothing more in the universe than hydrogen to begin with. How do we get heavier elements like the carbon our life is based upon? There is a two-step answer that proceeds from our premise.

NTS: Refine this in terms of strict hydrogen in Step 1 and Everything Else in Step 2.

Step 1. The star burns hydrogen into helium, and helium into heavier elements. Almost all of the reactions involved in this formation of heavier elements are exothermic; they release energy so the process continues as long as there is fuel.

Step 2. Once the star runs out of fuel the gravity is no longer counter-balanced. In some cases (larger stars) the star collapses or implodes, and this implosion generates a shockwave that radiates outward, blowing huge amounts of dead-star material out into the cosmos and superheating it in the process. During this superheating explosion, a supernova, there is enough energy available to fuse together the light elements into heavier elements, all the way up the periodic table to uranium.

The Hydrogen Chain
I've enjoyed reading several books in order to try and write this page. A couple thousand pages of dense astronomy and physics does not easily compress into a single web page, so I've necessarily had to leave out everything but a few interesting punchlines. The first of these is that

Fuel-burning alchemy is exothermic

That is, it produces an excess of energy. Equivalently we can weigh what we start with and find that the end-product weighs less. Here is the recipe for building ⁴He from ¹H atoms. Since it's very hot here the entire process proceeds in plasma; there are no atoms in the normal sense, just a soup of protons and electrons and other interesting objects. Here is the cast of characters.

H is a hydrogen atom, really a proton.
¹H is the same thing; a proton.
²H is deuterium, or a deuteron: A hydrogen proton with a neutron stuck to it somehow.
³H is tritium, a proton with two attached neutrons.
⁴H is getting ridiculous and does not appear here; it may not even exist.
He is Helium and that's 2 protons plus some number of neutrons, most commonly 2 neutrons.
³He is Helium-3, i.e. 2 protons and 1 neutron.
⁴He is regular Helium, 2 protons and 2 neutrons.
α is exactly the same as a ⁴He nucleus.
β is a beta particle, which is to say an electron or an anti-electron.
β^- is an electron.
β⁺ is an anti-electron or a positron.
ν is a neutrino. Neutrinos will be assumed to fly out of the sun immediately taking energy with them. This is a little vague as yet...
ν(0.42) is a neutrino carrying 0.42 MeV of energy.
γ is a gamma ray or a high-energy photon. Recall a blue photon carries about 3 eV of energy. Gamma rays carry typically an MeV. They do not carry charge. Gamma rays are the energy payoff of these processes so they often have an associated parenthetical energy value. For example:
γ(1.02) is a 1.02 MeV photon.

Step 0: Assumptions
Assume our starting materials are an unlimited supply, a sea of hot protons and electrons (H and β^-).
Assume any β⁺ particles created will annihilate with β^- particles producing energy as gamma rays.
Assume any product created is soon available in unlimited quantities as well.
Step 1: ¹H + ¹H -> ²H + β⁺ + ν(0.42)
Get to make anti-matter right off the bat! And one proton became a neutron...
Step 2: β⁺ + β^- ->   2 ν + γ(1.02)
So much for the antimatter... cashed in as energy (but why not .411 x 2?)
Step 3: ²H + ¹H ->   ³He + ν + γ(5.49)
Another proton transmuted into a neutron, nice energy payoff.
Step 4: ³He + ³He ->   ⁴He + ¹H + ¹ H + γ(12.86)
Two light Helium nuclei give us a regular Helium nucleus (alpha)
plus two hydrogens plus a lot of energy.

End product: A ⁴He nucleus, also known as an alpha particle. Energy yield: Take Energy { 2 x (Step 1 + Step 2 + Step 3) } and add Energy { Step 4 } -> 26.7 MeV. The equivalent mass is that of about 65 electrons.

The rationale for the energy yield goes like this: To do Step 4 we need to do Step 3 twice. To make two ²H deuterons we need Step 1 twice. This means that Step 2 is required to happen twice since we create two β⁺ particles to annihilate. In the process we convert the raw materials (6 x ¹H + 2 x β^-) into product material (⁴He + 2 x ¹H) plus the excess energy. Checking conservation of charge the beginning charge was +4 and the end charge is +4, so that's ok. However we have "lost" two electrons from the universe... but we've also converted two protons into neutrons. Which is pretty bizarre; I have no idea how or why that ought to happen. What's even stranger is that neutrons can survive for quite awhile in a nucleus but left to their own devices (i.e. as free particles) they decay in about a minute or so. (Into what???)

This page is incomplete. Sources include Wikipedia for quick reference and the following:
Structure and Evolution of the Stars by Martin Schwarzschild, 1957 by Princeton.
Timothy Ferris' books
Various additional including spectroscopy...

Implosion Alchemy
Here need to survey large star processes and supernovae with care to placing the different builders in proper context.

Staring Directly at the Sun

What Next?
I find it overwhelming to try and imagine events that got our solar system poised and ready to go with the right ingredients five billion years ago, even given a preceding 8.7 billions years of utter chaos in which to manage the task. Douglas Adams was fond of the term 'mind-boggling'... I don't know about you but I'm pretty mind-boggled at this point. The whole thing seems so complicated and so very very improbable.

This rather begs the question "How can I as a limited human being make heads or tails out of complexity?" To make progress I'd like to resort to one of the oldest tricks in the book, to coin a phrase. And by book I mean an imaginary slim volume How To Think. And by oldest I mean hardwired into our minds quite some years ago when trees served as homes. And by trick I mean an imperative written many times into this little book:

Ignore absolutely everything (almost).

Speaking of Douglas Adams, this imperative can be loosely paraphrased as Don't Panic. You may say that ignoring everything is simply a coping mechanism but I would counter that no, in fact ignoring everything is a cornerstone of sound science and mathematics. It permits us to shut out the complexity and reduce messy problems to simpler ones that we might be able to solve.

Let's use the Ignore Everything (Almost) principle to have a quick look at evolving complex structure from very simple rules. We'll go fishing (on the planet WaTor, among other places) for a sense or conviction that order can spontaneously arise out of seeming chaos. It's time to head to the computer and explore...

...little toy universes.

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