LCP 9: Calculating the Age of the Earth and the Sun
This LCP is based on my article “Calculating the age of the Earth and the Sun”, which was published by the British journal Physics Education in 2002. It can be downloaded from my website. This LCP is essentially a text that provides information about the history of the efforts made to determine the age of the earth and the sun. It goes beyond conventional approaches by inviting the reader to follow the arguments and the calculations made in detail. I have added a section in which the modern approach to calculating the age of the earth is discussed in some detail
There are no explicit Questions and Problems sections here as in the previous LCPs. However, the text should provide the physics instructor with a rich background from which to set tasks for students that involve them in the history and the physics behind the estimates of the age of the earth and the sun. The instructor/student is reminded to look at LCP 6 “Solar Energy” before or while studying this LCP.
IL 1 A comprehensive discussion of the physics and astronomy of the sun
Fig. 1. The life cycle of the sun, according to contemporary physics and astronomy.
Fig. 2 A total solar eclipse, August 1999. (The author witnessed this event in
Munich. See my website, Pictures).
TH1E MAIN IDEA:
We will review the main attempts made to calculate the age of the earth and the sun, beginning with Newton’s thought experiment and ending with Hans Bethe’s thermonuclear model
of the sun’s energy. In Part One, special attention is paid to the protracted debate about the age of the earth in the second half of the nineteenth century that involved Kelvin and Helmholtz. We will also look into 20th century explanations and dating techniques, paying special attention to the thermonuclear model first proposed by Hans Bethe in the late 1930s.
The results of the calculations are given in the main text but details can be found in the boxes that will allow teachers and students to solve novel problems and generate interesting questions for discussion. SI units will be used throughout, but sometimes it may be expedient to mention the original units used (as in the case of Kelvin’s calculations in his celebrated
paper of 1862 “On the Secular Cooling of the Earth”).
THE DESCRIPTION OF THE CONTEXT:
Newton presented a thought experiment in the Principia to show that a large body like the earth made of molten iron would take about 50,000 years to cool. He first estimated the time it would take to cool for a “globe of iron of an inch in diameter, exposed red hot to open air ”. He then argued that, since the heat retained is in proportion to the volume and the heat radiated in proportion to the exposed area, the time for cooling would be proportional to the diameter. (taken from Dalrymple, 1991.p. 28).
However, such a high estimate was in direct contradiction to the theologians’ claim that the earth was created by God about 6000 years earlier. John Lightfoot, Vice-Chancellor of the University of Cambridge, first published his calculations of the age of the earth in 1644, thus anticipating Bishop Ussher’s famous statement made in 1650. Guided by a careful interpretation of Mosaic chronology, he succeeded in working out the date of the creation of the earth exactly. According to Dr. Lightfoot: The earth was created on October 26, 4004 B.C., at nine o=clock in the morning in Mesopotamia, according to the Julian calendar. (Dalrymple, 1991, p. 14).
Fig. 3 Bishop Ussher (1581-1656) Fig.4 Dr. John Lightfoot (1601-1675).
Fig. 5 Charles Lyell (1797-1875). Fig
. 6 James Hutton ( 1728-1797).
On the other hand
, James Hutton declared in 1798 :
“We find no vestige of a beginning, no prospect of an end”.
By about 1830, geologists, led by Charles Lyell, argued that the earth must be very old, if not infinite as Hutton thought, then certainly billions of years. Their reasoning was based on the discovery that very long times were required for geological processes to take place.
IL 2 *** Bishop Ussher’s chronology
Fig. 7 An eroded outcrop at Siccar Point showing Fig. 8 Dacite columns that formed tens of sloping red sandstone thousands of years ago when a flow cooled rapidly against a glacier
IL 3*** A short but comprehensive discussion of the ideas of James Hutton
IL 4 *** A biography of James Hutton
We will now digress a little and move briefly into the 19th century and check how good a guess Newton made by using Stephan’s radiation law of 1878, and not, as may be expected, Newton’s law of cooling. Textbooks generally state that Newton discovered experimentally that the rate of cooling of an object is proportional to the difference in the temperatures between that of the object and the surroundings. This proportionality statement leads to an exponential expression that most physics textbooks discuss and students use it to solve problems It turns out that Newton used the proportionality statement only to calibrate a linseed thermometer that could measure temperatures higher than 200C.(See French, 1993). Moreover, it is now known that this relationship does not hold for high temperatures (see Silverman, 2000).
The law states that H, the rate at which an object radiates heat per unit area (J/s m2) is proportional to the fourth power of the absolute temperature. The rate of energy loss per unit area then is given by
H = (T i4 - Tf 4 ).
providing that the ambient temperature is much lower than the final temperature of the object.
Another assumption we are making is that the temperature of the surface of the sphere is constant
at all time. As shown in Block 1, the cooling time from an initial temperature Ti to a final
temperature Tf is given approximately by
t = (1/Tf3 - 1/Ti3) m c / 3 A,
Applying this formula to the small sphere we find that the cooling time is about 47 minutes.
To find the time of cooling of a globe of the size of the earth that is made entirely of iron, we again assume that, as the globe cools, at any time the temperature of the large sphere is the same everywhere. We find that the time for cooling is about 45,000 years (See Block 1).
This is in very good agreement with Newton’s conclusion. Of course, the cooling time for such a large object would be very much longer, because it would take considerable time for the heat to be be conducted to the surface as the body cools. To solve the problem of temperature distribution for such a case would require the application of a Fourier analysis, later accomplished by Kelvin.
Fig. 9 The Stephan-Boltzmann law of radiation
Fig. 10 As the temperature decreases, the peak of the black-body radiation curve moves to lower intensities and longer wavelengths
Fig. 11 Georges-Louis Leclerc, Comte de Buffon, Fig. 12 Isaac Newton (1642-1727).
Count de Buffon tests Newton’s thought experiment
We will now go back to the 18th century, after a sneak preview of the second half of the 19th century.
The French scientist (natural philosopher) Buffon was one of the most productive of the eighteenth century. He made fundamental contributions to mathematics, biology and paleontology and wrote a monumental 12 volume encyclopedia of science . He was interested in determining the age of the earth and being a wealthy man asked his foundry to make him ten iron spheres, in increments of ½ inch up the 5 inches. He treated them to white heat and then observed the time required to cool, first to red heat, absence of glow, to the temperature when the surface could be touched, an finally to room temperature. He concluded that if the earth had been made of molten iron, the earth would require 42,964 years to cool below incandescence and 96,670 years to cool to the present temperature.
Today, we smile at such efforts. However, one must remember that in the mid eighteenth century alchemy was still in vogue and there was not even an elementary theory of heat established .Buffon believed and demonstrated that nature was rational and could be understood through physical processes. He was also the first to apply experimental techniques to the problem of the age of the earth. A century passed, however, until Helmholtz and Kelvin, equipped with more sophisticated physics and experimental procedures, tackled the problem again.
Helmholtz and the age of the sun
New insights into the physics of radiation was gained in the second half of the 19th century.
Spectroscopy was established by Kirchhof and Bunsen, Helmholtz estimated the age of the sun and Stephan discovered his radiation formula (see above).
Hermann von Helmholtz was the most famous German natural philosopher and cosmologist of his generation. He was one of the original contributors to the General Principle of the Conservation of Energy. Already in 1854 he argued that the sun’s energy must be supplied by gravitational contraction because no known chemical reaction could produce sufficient energy. His calculations showed that the sun could supply the energy we measure without us being aware of the contraction. According to his model the sun could be about 20 M years old.
Fig. 13 Hermann von Helmholtz (1821-1894) Fig. 14 Lord Kelvin (1824-1907)
*** About Lord Kelvin
IL 6 *** About von Helmholtz
The sun as a furnace burning coal
Helmholtz first calculated that if the sun’s energy were due to a chemical source the life expectancy would be about 5000 years. This is easy to show. It was known that coal contained about 25 BTU per ton of potential energy of combustion . This is about 3.0x107 J/kg. The energy output of the sun was thought to be about 7000 HP per square foot of surface, or about 3.6x1026 J/s for the whole surface. The mass of the sun is about 2.0x1030 kg. Therefore, the maximum life expectancy of the sun would be:
2.0x1030 x 3.0x107 / 3.6x1026 = about 5000 years.
Fig. 15 The “burning” sun.
The sun as a gravitationally collapsing body
Helmholtz immediately rejected this model and argued that gravitational collapse of the original cloud of material to the present size of the sun was the source of the sun’s size and its continued production of energy. He assumed that material fell into a proto mass from infinity and the sun grew by accretion and heated up to the present temperature.
In his famous “Popular Lectures”of 1857 he discussed this model and gave the value for the gravitational potential energy of the sun as shown in Appendix 1. He found that the approximate gravitational potential energy then is 2.3x1041 J . Since he knew the energy output of the sun , Helmholtz was able to estimate the age of the sun and found it to be about 21 million years.
Fig. 16 Helmholtz’s gravitational contraction model:
Fig. 17 Gravitational collapse
Fig. 18 Formation of the sun.
To find the age of the sun
Helmholtz also estimated the temperature of the sun by assuming that the specific heat of the sun was equal to that of water and that mechanical equivalent of heat (just recently published by Joule) was about 4.2 Joules per calorie, as shown in Appendix .
The temperature Helmholtz found was 28,611,000 °C. It is interesting to note that the estimated temperature of the center of the sun is about 108
K. Helmholtz also estimated the pressure at the center of the sun and obtained a value of about 1.3x1014
To show this only a very simple calculation is needed: One calculates the force produced by a 1 m2
column of the sun’s gas, having a height of 7x108
m, an average density of 1.4x103
, and the average gravity being that of the middle, or 270/2 m/s2
, or 135 m/s2
. The modern value of the density of the sun’s center is about 1.6x105
. Of course, he did not realize that at that pressure and temperature thermonuclear fusion would be initiated. Helmholtz then estimated the age of the sun by assuming that the energy output has remained constant. He arrived at a value of about 20 million years.
Finally, Helmholtz estimated the shrinkage of the sun necessary to produce sufficient energy to account for the observed 3.6x1026 joules per second. He came to the conclusion that about 250 feet (about 80 m) of shrinking per year is necessary. (These four results by Helmholtz are discussed in detail in Appendix 1.).
IL 7 **** A short article on the age of the sun by Kelvin (1862).
Fig. 19 Spectrographic analysis of the sun, already available to Helmholtz and Kelvin.
Fig. 20 In 868 an English astronomer, Joseph Norman Lockyer, discovered an unknown element in the Sun, i.e. a set of spectral lines which did not correspond to elements in the lab. He named this element helium (Latin for Sun element).
During the solar eclipse of October, 1868, Lockyer observed a prominent yellow line from a spectrum taken near the edge of the Sun from Vijaydurg. With a wavelength of about 588 nm, slightly less than the so-called "D" lines of sodium. the line could not be explained as due to any material known at the time, and so it was suggested by Lockyer that the yellow line was caused by an unknown solar element. He named this element helium after the Greek word 'Helios' meaning 'sun'. Helium was discovered on Earth in 1895. In March 1895, while examining the spectrum of gases given off by a uranium mineral called cleveite, the British chemist William Ramsay spotted a mysterious yellow line Lacking a good spectroscope, he sent gas samples to both Lockyer. Within a week it was confirmed that the gas was the same as the one Lockyer had observed in the sun more than 25 years earlier. Lockyer was beside himself with joy as he squinted through the spectroscope at the "glorious yellow effulgence" he had first seen on the Sun in 1868.
Lord Kelvin and the age of the sun
Like Helmholtz, Kelvin first established the age of the sun, before calculating the age of the earth. Like Helmholtz he quickly dismissed the chemical energy theory, because it allowed for less than 10,000 years. He then investigated the physics of his meteoric hypothesis. In this hypothesis he assumed that the energy of the sun is replenished by constant rate of meteoric bombardment.
He first calculated the kinetic energy of impact of a mass of 1 pound of matter falling into the sun at the escape velocity of the sun. Using SI units, a quick calculation shows that a 1 kg mass hitting the sun at the escape velocity of 624 km/s would have a kinetic energy of ½ x1x (6.24x105)2 J, or 1.94x1011 J.
A simple calculation then showed that about 1/5000 of the sun’s mass over a period of 6000 years would suffice to account for the energy given off by the sun. It must be remembered that this would represent about 70 times the mass of the earth! We can quickly check this:
The sun’s energy output was found to be about 3.6 x1026 J/s. Therefore in 6000 years time the sun will have liberated 6000x3.15x107x3.5x1026 or 6.61x1037 J. Dividing this by the kinetic energy of 1 kg mass falling in we get 6.61x1037/ 1.94x1011 or 3.40x1026 kg. This is 3.4x1026 / 2x1030, or 1/5000 of the mass of the sun. However, by about 1861, Kelvin rejected this theory also, because:
1. There was no spectroscopic evidence found that anything faster then about 1/20 of the escape velocity of the sun is found in the vicinity of the sun.
2. The effect of the additional mass of the sun on the period of rotation of the earth would have been detected.
Kelvin calculated this effect and found that it would be about 1/8 of a year in a 2000 year period. His calculations are quite sophisticated but we can check his value with a relatively simple approach, using Kepler’s third law and Newtonian mechanics. Kelvin then argued that such a discrepancy would have been found and therefore he finally accepted Helmholtz’s theory of gravitational contraction as the only viable one. (See Appendix 2)
However, his commitment to the gravitational model of Helmholtz emerged only gradually, starting with his early recognition of the soundness of Helmholtz’s ideas after reading his lecture of 1857 to the time Kelvin developed his model for the cooling of the earth in 1862. At the beginning he thought that his meteoric theory and Helmholtz’s gravitational collapse theory would complement each other, but by 1862 he grudgingly deferred to Helmholtz. Kelvin did calculate the age of the sun, using both a linear and exponentially decreasing density. He found that for the first case the age of the sun was 20 M years and for the second about 60 M years. So he was able to push the age of the sun to three times the value based on Helmholtz’s simple model. Kelvin’s models predicted that a gravitational shrinkage rate was very close to that calculated by Helholtz. It is interesting to note that he estimated that about 80% of the mass of the sun was contained inside a sphere of half the radius.
Kelvin calculates the age of the sun and of the earth
Having established that the sun is at least 20 million years old, Kelvin set out to determine the age of the earth. His seminal paper “On the Secular Cooling of the Earth” was published in 1862 and produced an instant sensation in both scientific and public domains. Kelvin made the following assumptions, as recorded in his paper:
1. Most of the earth’s heat was originally generated by gravitational energy.
2. The earth cooled from a temperature of about 3700°C to the present temperature.
of about 0° C very quickly, probably in about 40-50000 years.
3. The temperature of the earth’s surface has not changed significantly since then.
4. The interior of the earth is solid and therefore only conduction is significant.
5. In all parts of the earth a gradually increasing temperature has been found
in going deeper. This finding implies a continual loss of heat by conduction.
6. Since the upper crust does not become hotter from year to year, there must be a secular loss of heat from the whole earth.
The physical constants that he needed were :
1. The temperature gradient of the surface of the earth
2. The specific heat of the earth’s crust.
3. The thermal conduction coefficient for the crust.
Kelvin estimated an average for the temperature gradient to be 1/50 °F per foot of depth and decided that the thermal conduction coefficient of the earth’s crust was 400 BTU /y.ft. °F (see Table 1)
Further, he assumed that the temperature after solidification was 3700 ° C at the center and 0 °C at the surface. These temperatures, he argued, remained constant over millions of years and were located on the two sides of an arbitrary infinite plane in an infinite solid. Such a distribution provides the initial conditions for the discontinuity between the two planes. In his own words, he then
applied one of Fourier’s elementary solutions to the problem of finding at any time the rate of variation of temperature from point to point, and the actual temperature at any point, in a solid extending to infinity in all directions, on the supposition that at an initial epoch the temperature has had two different constant values on the two sides of a certain infinite plane
(Kelvin, p. 301)
(See Appendix 4 for details about Kelvin’s calculations).
In his paper he drew the following conclusions:
1. The limits of the earth’s age are between 20 million and 400 million years.
2. The temperature gradient will remain constant at about 1/50 ft per F for about
3. During the first 1000 million years the variation of temperature does not become
“sensible” at depth exceeding 568 miles (910 km).
4. The temperature gradient diminishes in inverse proportion to the square root of the time.
Thus we would have:
At 40,000 years 1 ° F per foot
“ 160,000 “ ½ ° “
” 4,000,000 “ 1/10 ° “
” 100,000,000 “ (now) 1/50 ° “
400,000,000 “ (future) 1/800 ° “
Finally, it is interesting to compare the physical parameters used by Kelvin to those geologists accept today. Kelvin would be pleased and would probably say that no major changes are necessary in his mode