Where does the energy of the stars come from ?

Where does the energy of the stars come from ?
In order to shine, a star must be hot, and it needs energy to maintain its heat.
Historically, it was natural at first to consider chemical reactions inside the Sun. Unfortunately, as these reactions have a very low efficiency, the Sun would have been extinguished after a while, having run out of fuel.

In the middle of the 19 th century, two physicists, Kelvin and Helmholtz, suggested that the Sun could collapse by its own weight, and that this gravitational collapse, by heating the internal gas, could make this gas radiate.
Although this process actually works in proto-stellar clouds, it can't be useful to sustain a star : calculations show that, less than 25 million years ago, the Sun would have needed to be bigger than the Earth orbit.

The solution came after Einstein's works, who established an equivalence relation between mass and energy, the famous formula E=mc².

The atom and the nuclear reactions

In order for the nucleus of an atom to be stable, there must exist a force greater than the electrostatic repulsion between the protons which make it up.
The heavier is the nucleus, the weaker is its potential energy, because of this force. This loss of potential energy is called the binding energy. And, by equivalence, it corresponds to a loss of mass. This means that a nucleus is lighter than the sum of its particles, if they were isolated.

On this drawing, the binding energy by nucleon is drawn, depending upon the number of nucleons in the nucleus.
The stronger energy is for Iron (Z=56), so, it's the most stable element.

If we begin with the left part of the curve, i.e. light elements, these elements are able to merge together, up to the iron, by releasing a part of their binding energy.
Conversely, heavy elements can only break into lighter elements, a mechanism called fission.

Hydrogen is the main component of a star. So, it's the fusion of this element into helium which is the source of energy, at least when the star is on the main sequence.
What is described symbolically :

4 H--> He + energy.

The Sun radiates a luminous energy of 4x1026 Watt. This is equivalent to 4 million tons of lost mass. In order to do so, it must use 600 million tons of hydrogen each second.

The proton-proton chain

To achieve this conversion, the easiest way is the proton-proton chain.
Two hydrogen nuclei (protons) merge together, and the result is a deuterium nucleus, a positron and a neutrino.
This deuterium merges with another proton to give a helium3 nucleus, by releasing a gamma photon, associated with a high energy.
Finally, two helium3 nuclei merge together to obtain a helium4 nucleus and two protons.

Proton-proton cycle inside a star.

On this drawing, the typical reaction times are indicated. The slower one is the initial fusion of two protons, it's the one which rules the global rate of the reaction.
Although this typical time is tremendous (a billion years), you must keep in mind that it's a statistical figure, and the number of protons that you can find greatly compensate the weak probability of the reaction.

The CNO cycle

The second way to obtain the transformation of hydrogen into helium is the CNO cycle - which stands for Carbon, Nitrogen, Oxygen.

This reaction is much more complex that the former one, and it needs six stages :

A carbon nucleus merges with a proton. The result is a nitrogen13 nucleus, and a highly energetic gamma photon,
This nitrogen nucleus is unstable, and spontaneously breaks apart into a carbon13 nucleus, a positron and a neutrino, with a half-life of 10 minutes,
The carbon13 nucleus merges with a proton to give a nitrogen14 nucleus and another gamma photon,
By merging with another proton, there appears an oxygen18 nucleus and a photon,
In the same way as nitrogen13, oxygen18 breaks apart and releases a positron and a neutrino,
Finally, the resulting nitrogen15 merges with a proton, and the result is a carbon nucleus and a helium nucleus.

CNO cycle inside a heavy star.
The overall result is the same as the proton-proton cycle : 4 protons produce a helium nucleus, along with some other particles.

In this reaction, carbon is regenerated. Its only -but important- use is to catalyse the reaction.
The necessary condition, of course, is to have carbon before the reaction : such a chain will mainly appear in population I stars.

Comparison of the two cycles

What's the ratio of energy release between these two cycles ?
The answer appears in the following drawing :

Compared efficiency of the two fusion cycles against température.

The hotter is the temperature, the more important is the production of energy from the CNO cycle. As the internal temperature of a star is a direct function of its mass, we can say that the p-p cycle dominates for a star of less than one solar mass. Above 1.3 solar masses, the CNO cycle will supply the main part of its energy.

In order for these reactions to be maintained, we must note that a minimal temperature of about 8 to 10 million degrees Kelvin is necessary.

Beyond helium

We've learned that when hydrogen is exhausted in the core of the star, it begins to shrink, and its temperature increases.
The star is becoming a 'red giant'.

At this moment, helium nucleii can merge together and change into beryllium. This beryllium nucleus fuses with another helium nucleus to give carbon. This reaction is called the "triple alpha" process.

If the star is big enough, reactions are able to go on as far as temperature gets higher. These reactions end with iron.

At the end, when the temperature is around 6 billion degrees Kelvin , iron nucleii are broken apart by the gamma rays. This reaction takes energy, it will trigger the core implosion, and we get a supernova.

Réaction	Ignition temperature (million of K)
Hydrogen burning 4 (¹H) -> ⁴He	10
Helium burning 2(⁴He)-> ⁸Be ⁸Be +⁴He-> ¹²C ¹²C+⁴He-> ¹⁶O	100
Carbon burning 2(¹²C)-> ⁴He+²⁰Ne ²⁰Ne+>⁴He-> n+²³Mg	600
Oxygen burning 2(¹⁶O)-> ⁴He+²⁸Si 2(¹⁶O)-> 2(⁴He)+²⁴Mg	1500
Silicon burning 2(²⁸Si)-> ⁵⁶Fe	4000
Iron photodistintegration ⁵⁶Fe-> 13(⁴He)+4n	6000