By David J French
Although long, I believe that the following analysis is worth pursuing to the end.
While browsing through Wikipedia on the Internet I recently came across this interesting observation about the Sun:
“The power production by fusion in the core varies with distance from the solar center. At the center of the Sun, theoretical models estimate it to be approximately 276.5 watts/m3, a power production density that more nearly approximates reptile metabolism than a thermonuclear bomb.[b] Peak power production in the Sun has been compared to the volumetric heats generated in an active compost heap. The tremendous power output of the Sun is not due to its high power per volume, but instead due to its large size.”
http://en.wikipedia.org/wiki/Sun – (under “Core”)
What is this? I always thought the Sun was a continuously self-fueled hydrogen bomb. Not only are these levels far below that of a hydrogen bomb, but the amount of heat being produced on a unit of volume basis is indeed a trickle.
A cubic meter contains 1,000,000 or 100 X 100 X 100 cubic centimeters. Therefore, according to this reference, the rate at which heat is flowing out of a cubic centimeter of the Core at the center of the Sun is 0.2765 milliwatts! This would hardly light an LED. But we must check the footnote reference; after all this is Wikipedia.
Footnote 55 links to a website operated by the Fusion and Plasmas Group of the Contemporary Physics Education Project (CPEP). CPEP is a non-profit organization of teachers, educators and physicists, with substantial student involvement. CPEP creates educational materials on contemporary physics topics for use in introductory physics classes. This website addresses introductory educational materials on fusion energy and the physics of plasmas. http://fusedweb.llnl.gov/CPEP/Chart_Pages/5.Plasmas/Sunlayers.html
This link starts by explaining that the Core, the innermost layer of the Sun where energy originates, has a density of 160 g/cm3, 10 times that of lead. At this density it might be expected that the Core would be solid. However the Core’s temperature of 15 million degrees Kelvin, virtually identical to degrees Centigrade at this temperature, or 27 million degrees Fahrenheit. This high temperature keeps the Core in a fluid plasma state.
This reference also includes a chart based on a Computer Model of the Sun at 4.5 Billion Years into its lifetime, i.e., today. This chart can be viewed at the end of the last link referenced above.
The key figure that we’re looking for is the rate at which heat is being produced in the center of the Sun, and there it is under the title: Fusion Power Density (joules/sec-m^3). At the very center of the Sun, the value is 276.5 joules/sec-m^3. This means 276.5 Watts per cubic meter just as cited in the Wikipedia article.
According to that chart, the production of energy peters out by about one quarter of the radius of the Sun (24% shown on the chart shows heat production at the rate 0.67 Watts per cubic meter.) This turns out to be a very important factor.
But wait a minute, this data is the result of a “Computer Model of the Sun”, attributed to B. Stromgrew (1965) reprinted in D. Clayton Principles of Stellar Evolution and Nucleosynthesis. New York: McGraw-Hill, 1968, and others. Maybe these mathematicians have gotten it wrong. There must be another way to verify if this set of data is correct.
United States National Aeronautics and Space Association – NASA
The Marshall Space Flight Center’s Solar Physics web site, operated as part of NASA, is an authoritative source for research about the Sun. At this site background facts about the Sun are given here: http://solarscience.msfc.nasa.gov/ . On the very opening page the following key data is provided:
Solar radius = 695,990 km = 432,470 mi = 109 Earth radii
Solar mass = 1.989 1030 kg = 4.376 1030 lb = 333,000 Earth masses
Solar luminosity (energy output of the Sun) = 3.846 1033 erg/s
Surface temperature = 5770 K = 9,930° F
Surface density = 2.07 10-7 g/cm3 = 1.6 10-4 Air density
Surface composition = 70% H, 28% He, 2% (C, N, O, …) by mass
Central temperature = 15,600,000 K = 28,000,000° F
Central density = 150 g/cm3 = 8 x Gold density
Central composition = 35% H, 63% He, 2% (C, N, O, …) by mass
Solar age = 4.57 109 yr
Now we can do some calculations.
Objective: to calculate the energy flux/power density at the Core of the Sun per unit volume arising from nuclear synthesis
Volume of a sphere = 4/3 X 3.14 X radius3
Radius of Sun (from above) = 695990 km = 700000 km = 7 X 1010 cm
Radius of Core = 1/4 Radius of Sun = 1.75 X 1010 cm
Volume of Core = 4/3 X 3.14 X (1.75 X 1010 )3 cm = 22.437 X 1030 cm3
Solar luminosity (from above) = 3.846 X 1033 ergs/sec = 3.846 1026 joules/sec
Solar Heat Flux per unit volume = total heat flow/ volume = 3.846 X 1026 joules/sec / 22.437 X 1030 cm
= 0.01714 milliwatts/cm3 (or 17 Watts/m3)
Note: this is the heat flux averaged-out over the entire Core. Nuclear syntheses does not occur evenly throughout the Core. It is at a maximum at the center and tapers-off towards its outer limit at about one quarter of the solar radius, cf:
“The temperature at the very center of the Sun is about 15,000,000° C (27,000,000° F) and the density is about 150 g/cm³ (about 10 times the density of gold or lead). Both the temperature and the density decrease as one moves outward from the center of the Sun. The nuclear burning is almost completely shut off beyond the outer edge of the core (about 25% of the distance to the surface or 175,000 km from the center). At that point the temperature is only half its central value and the density drops to about 20 g/cm³.”
This decline in the heat flux is not necessarily linear. The chart above shows an output power of 19.5 watts per cubic meter at a distance of 14% of the solar radius and 6.9 W per cubic meter at a distance of 19% of the solar radius, with heat generation tapering off to nothing at 25% of the solar radius. Accordingly, this calculated value from NASA as a source is consistent with the article and footnote in Wikipedia
Analysis – How can this be true?
Remarkable as this appears, it seems to be absolutely true: the matter at the Core of the Sun is generating heat at a rate that is less than a milliwatt per cubic centimeter. Indeed, the average rate at which heat is being generated within the Core, from the center of the Sun out to 25% of the Sun’s radius, is on the order of 0.01714 milliwatts/cm3 (or 17 Watts/m3). Astounding!
Someone else has noticed this fact and provided an annotation in the paragraph in the Wikipedia referenced above. That annotation reads as follows:
“A 50 kg adult human has a volume of about 0.05 m3, which corresponds to 13.8 watts, at the volumetric power of the solar center. This is 285 kcal/day, about 10% of the actual average caloric intake and output for humans in non-stressful conditions.”
Essentially, this says that human beings generate heat, or consume calories, at a rate that is 10 times greater than that at the center of the Sun.
How can this be true?
There are several factors that contribute. The first explanation is that the Core of the Sun is surrounded by a very large amount of matter that does not generate heat: three quarters of the solar radius. The solar radius is 700,000 km and therefore the heat generated at the center of the Sun has to pass through 525,000 km of matter in order to escape.
The NASA website states:
“Although the photons travel at the speed of light, they bounce so many times through this dense material that an individual photon takes about a million years to finally reach the interface layer. The density drops from 20 g/cm³ (about the density of gold) down to only 0.2 g/cm³ (less than the density of water) from the bottom to the top of the radiative zone. The temperature falls from 7,000,000° C to about 2,000,000° C over the same distance.”
This reference is with respect to photons traveling from the bottom to the top of the “radiative zone” between the Core of the Sun and the next layer up. This does not represent the distance to the surface of the Sun. Again, from the NASA website:
“The radiative zone extends outward from the outer edge of the core to the interface layer or tachocline at the base of the convection zone (from 25% of the distance to the surface to 70% of that distance). The radiative zone is characterized by the method of energy transport – radiation. The energy generated in the core is carried by light (photons) that bounces from particle to particle through the radiative zone.
“Although the photons travel at the speed of light, they bounce so many times through this dense material that an individual photon takes about a million years to finally reach the interface layer.”
Accordingly, this 1 million years travel time applies to a mere 45% of the solar radius. However, this is a part of the Sun where the matter is very dense.
Now the Sun is 4.5 billion years old and if we will assume that it has been radiating at the same rate (not necessarily so) over that period of time, we can imagine that a lot of heat, in the form of photons, has spent a lot of time making the trip from the Core to the outer surface where it can escape. One million years is a long time for heat to accumulate even if it is only being generated at the rate of 100 or so watts per cubic meter in the Core. And 4 1/2 billion years is a very long time. Seen from this perspective, the phenomena is a little more believable.
And there is still another way to look at it.
The number of cubic meters inside a sphere can be much greater than the number of square meters on the surface. Imagine a square meter of the Sun’s surface sitting on a pyramidal wedge that extends 700,000 km all way back into the center of the Sun. Only the last quarter of this distance is generating heat. But one quarter of the radius of the Sun is still 175,000 km. Therefore, even though the pyramid is tapering to a point, there are 175,000,000 meters of heat-generating Core material backing up that single meter on the surface.
The same analysis can be carried out for all of the square meters on the surface of the Sun. On this basis, the value for the rate of heat generation within the Core of the Sun as contrasted with the rate of heat radiation on the surface of the sun at the surface of the sun becomes more understandable.
So the proposition that we started with, that the Core of the Sun generates heat at a rate that is less than 1 milliwatt per cubic centimeter, is probably true.
Why have we done all this calculating? The answer is that we are concerned to compare solar fusion with cold fusion. But first a further observation on the issue of the “quality” of heat. Then we can compare hot and cold fusion.
My first reaction was that my concept that the Sun was a continuously self-fueled hydrogen bomb was totally wrong. Instead it represents the embers from a fire that has been smoldering for 4.5 billion years.
These are not ordinary embers however.
While the rate of heat generation in the center of the Sun is modest, the temperature is not. The NASA data provided above indicates that photons proceeding outwardly from the Core start on their journey with the temperature equivalent of 7,000,000°C. By the time they reach the surface, the temperature equivalent has dropped to 5600°C. The heat from the Core is then released into space in the form of high temperature photons. In this sense, the heat being generated in the center of the Sun is different in quality from the same amount of heat being generated in a heap of rotting manure. But this quality is lost when we use the heat of sunshine to warm our swimming pools.
One difference between hot and cold fusion is the quality of the heat being produced, at least so far. But at what cost?
This lead me to explore the efforts being made to create energy for mankind using fusion. A little bit about this topic can be found here: http://en.wikipedia.org/wiki/Fusion_power
For more than 30 years scientists have aspired to create usable energy using fusion. The latest version of effort is that of the International Thermonuclear Experimental Reactor – ITER: http://www.iter.org/
The costs have been remarkable:
“ITER was originally expected to cost approximately €5billion, but the rising price of raw materials and changes to the initial design have seen that amount more than triple to €16billion. The reactor is expected to take 10 years to build with completion scheduled for 2019.” http://en.wikipedia.org/wiki/ITER
The figures quoted are simply for this single project. Many billions more have been spent over the years by countries around the world to advance the goal of achieving useful energy output from hot fusion. There has been a lot of talk in support of this process of bringing the energy source of the Sun down to the surface of the earth. But these kinds of aspirations do not seem compatible with the calculated values for the rate of output of energy being generated within the Sun as examined within this article.
Essentially, the hot fusion scientists are not trying to emulate the Sun. They are trying to emulate a supernova! With that thought in mind, it is understandable why the United States withdrew as a primary participant from the international ITER project in 1998 although it did rejoin as a junior 9% partner in 2008.
Next is the issue of Cold Fusion
Cold Fusion has been in disrepute over the last 24 years. This is largely due to a rush-to-judgment that occurred in 1989 at a time when many laboratories around the world could not duplicate the effect.
However, particularly in the last 20 years, numerous scientists have been able to demonstrate the presence of “unexplained excess energy” arising from the Cold Fusion effect. Generally this comes from super-loading Palladium with deuterium, and more recently, Nickel with hydrogen and then stimulating the generation of unexplained heat energy by applying electrical current, ultrasound, magnetic fields or simply even higher gas pressures within the metal hydride. There is no doubt that unexplained excess energy is being produced. Now that sufficient experiments have ruled-out experimental errors and chemical effects, it is hard to imagine where this energy could come from if it were not for some form of fusion effect.
Experimental results have been producing energies at rates ranging from milliwatts to watts and even some assertions of kilowatts of output thermal power from this unexplained source of energy. The apparatus producing these outputs has always been of a table-top character. Focusing on the actual source of the reaction, the Nickel or Palladium, energy has been produced in these experiments at rates or power levels that are far higher than mere milliwatts per cubic centimeter.
The quality of this heat has been generally low, e.g. under 100°C. But recently, indications have appeared (without naming them) that much higher temperatures can be achieved, e.g. 600, 700°C. Heat of this quality is indeed valuable. Such temperatures can be used to make electricity!
Consequently, Cold Fusion has been achieving “stellar” performances over the past 24 years, at least in terms of specific power being generated! And there are now signs that the temperature potential of this process to deliver commercially valuable results is real. By these standards, it is incomprehensible why governments have not invested further support to bring this phenomenon to commercial availability.
This is probably the most important conclusion to be drawn from the very interesting facts explored in this essay. The disparity between the support for hot versus cold fusion is extreme, indeed scandalous. But this is already known, at least in one of these two communities.
David J. French Ottawa, Canada