Question

The current Sun is said to be on the main-sequence of its life. It shines by fusing hydrogen into helium. This requires an ignition temperature of T ∼ 107 K – to estimate this, insist that two hydrogen nuclei (protons, singly charged), if they were to fuse via the magic of quantum tunnelling, still have to have sufficient kinetic energy such that they can resist Coulumb repulsion and reach each other within a distance of order their de Broglie wavelength (λ ∼ h/(mpv)). A fusion of this kind releases 0.7% of the rest mass and fuels the solar luminosity.

1) Consider the center of the Sun where hydrogen is being steadily turned into helium. At the same temperature and density, how does the central pressure change when an initially 100% hydrogen (assumed to be fully ionized) is converted into 100% helium (also fully ionized)? How does the Sun react to this change?

2) In a later stage in its life, the Sun will instead fuse helium into carbon. Burning helium will require a higher temperature than that of hydrogen. Using the same logic that yields the 107K ignition temperature for hydrogen fusion, obtain the threshold temperature for helium fusion.

3) Recall that in the last problem set, using Virial theorem, you have obtained a central temperature for the Sun of Tc ∼ 107 K. This, intriguingly, “coincides” with the above estimate for the ignition temperature of hydrogen burning. Convince your grader that this is NOT a coincidence. In particular, consider the following two scenarios and argue that the Sun will return to Tc ∼ 107K even if it started elsewhere: A) if the Sun was initially larger and therefore has a lower Tc. B) if the Sun was initially smaller with a higher Tc. Lastly, when the Sun exhausts its central hydrogen and has to start fusing helium, what will happen to its radius? (for this, assume that the mass of the Sun is unchanged from its main-sequence stage, and that the Sun is more or less a single-temperature plasma ball with a constant density)

Answer 1

Answer 1. Nuclear fusion, the source of all the energy so generously radiated by the Sun, does two things: it converts hydrogen into helium (or rather, makes helium nuclei from protons) and it converts mass to energy.

The mass-to-energy conversion is described by Einstein's famous equation: E = mc², or, in words, energy equals mass times the square of the velocity of light. Because the velocity of light is a very large number, this equation says that lots of energy can be gained from using up a modest amount of mass.

The energy created by the fusion processes within the core of the Sun (or any other star) exerts an outward pressure. Unless contained, such pressure would produce an explosion (as happens in the hydrogen bomb, on a much smaller scale). The inward pressure that keeps a star from exploding is the gravitational attraction of the gas mantle surrounding the core (which is most of the volume of the Sun, and is very hot but does not burn itself).

The outward pressure from the fusion reactions keeps the stars from collapsing. The inward pressure from gravitation keeps the stars from exploding. If the fusion reactions in the core become too weak, a star can and does collapse. Such collapse can provide new conditions in a core that result in new types of fusion reactions, so that expansion follows. If fusion reactions in the core become too strong, a star can and does explode.Such events can be observed. When a star explodes it shines with extreme brightness for a while; it turns from an unnoticed to a "new" star, a "nova". Stars, like our Sun, where inward pressure and outward pressure is nicely balanced, fluctuate but little in brightness and give off a steady stream of energy. The balance is achieved by self-regulation: a slight decrease in fusion energy would result in contraction that would heat up the core and increase fusion rate, and vice versa.

Thus, the reason that the Sun neither expands (from the ongoing explosion within) nor collapses (from its own weight) is that the two forces keep the balance. In the distant future, when this balance is disturbed because most of the hydrogen is used up, the Sun will expand.