## How does the Sun shine? (Part 1: Chemistry)

In this article series, I am going to retrace the history of astronomy/astrophysics in trying to answer the question of how is the energy radiated from the Sun being produced? Even if this question can be traced back to ancient Greece, I will only start from the 19th century, in which, the most common fuel was coal(anthracite) and by its combustion, the most common source of energy known. We will see that there are 4 periods of time which correspond to new theories on Sun’s energy production being published and gaining more and more experimental evidence for or against these theories: chemistry, thermodynamics, nuclear and quantum physics.

This first article is about the first period of time, in which, scientists tried to answer the question through chemistry, and the next articles will look at the other periods of time respectively. Of course, gravity discovered well before the the 19th century, affects theories in all of these 4 periods of time.

(1) How long has the sun been burning? It must have been burning at least as long as the age of the earth. If we consider the earth as a hot ball of iron, what is the age of the earth? How long would it take for the earth to cool to its present temperature? This idea originated first from Newton but the equation to make the calculation is only available in the late 19th century (Stefan Boltzmann’s radiation law of 1879). Newton made a thought experiment in order to find the age of the Earth. He hypothesized that the time for cooling a hot sphere of iron would be proportional to the diameter of the sphere without having real knowledge of temperature and heat. Knowing the approximate diameter of the Earth, Newton found that it would take 50000 years for the Earth to cool. Now, using the Boltzmann’s equation, which states that the rate of heat energy radiated by an object per unit of time is proportional to the fourth power of its absolute temperature:

$\frac{dQ}{dt}=\epsilon\sigma{T}^{4}{A}$

$\epsilon=1$
$\sigma={5.67}\times{10}^{-8}\frac{{W}}{{m}^{2}{K}^{4}}$
${A}_{earth}={5.14}\times{10}^{14}{ m}^{2}$ Surface of the earth

Now, combine this with the relationship between how much heat it takes to raise the temperature of x grams of something by a given amount of temperature, with the specific heat capacity (hcap) and integrate the 2 sides of the equation:

${dQ}={m}{h}_{cap}{dT}$

${m}_{earth}=8.7\times{10}^{24}{ kg}$ Mass of the earth
${h}_{cap}=450\frac{J}{{kg K}}$ Specific heat capacity of Iron

$\int_{{T1}}^{{T2}} {T}^{-4}\ {dT}={a}\int_{0}^{t}{dt}$

$\frac{1}{3}(\frac{1}{{T2}^3}-\frac{1}{{T1}^3})={a}{t}$

${a}=0.0074\times{10}^{-18}$ (which represents all the constants)
${T1}=1811{K}$ The temperature of melting iron
${T2}=288{K}$ The average temperature of the earth today
$1{year}=3.153\times{10}^{7}{seconds}$

We found ${t}=60000{years}$ for the age of the earth as a ball of cooling iron, which is in the range of what you can find in many textbooks. Geologists have known that Earth is much older than that and in fact, very, very old (and by inference the Sun) since the early 19th century.

(2) Is the Sun a gigantic globe of burning coal?

Hermann von Helmholtz was the first to show that if the Sunâ€™s energy
were due to a chemical source, the life expectancy of the Sun would be about 5000 years. It is easy to show that for coal it is about 4000 years.

Knowing the heat of combustion of coal (anthracite), the total energy output of the sun per second, and the mass of the sun, one can calculate that the age of the Sun as a ball of burning coal is 40000 years.

${Lifetime}=\frac{{M}\times{E}_{comb}}{L}$ Lifetime in seconds

$M = 2\times{10}^{30}\ {kg}$ The mass of the sun
$L = 4\times{10}^{26}\ {J/s}$ The luminosity of the sun, which is also the total energy output per second
${E}_{comb} = 3\times{10}^{7}\ {J/kg}$ The energy released by the combustion of coal per kg
$1{year}=3.153\times{10}^{7}{seconds}$

In conclusion, the answers to these questions were too simple, but given the knowledge of the 18th-19th century, they were not easy questions to answer and not even easy to ask! Simple calculations show that without an internal source of energy, the Sun would cool down much too quickly and would not even reach 60000 years as it was believed to be the (wrong) age of the earth.

The origin of the warmth of the earth and the growing evidence that the earth was very old became central problems until the 20th century.