Lecture 20

The Stars

The sight of a starry sky is one of the more fascinating displays of nature, that has fueled the fantasy of human beings since the earliest days of their existence. Even more fascinating is the knowledge that we get when we try to understand the nature of stars and of the energy they emit in a seemingly inexhaustible flow.

Among all the different branches of Science, Astrophysics presents particular challenges, since it is not amenable to direct experimentation. In the standard scientific procedures, the validity of a theory or a model can be tested by verifying their predictions in a controlled experiment. When dealing with the stars, obviously we cannot reproduce them in a laboratory, and we must base our deductions on whatever little information we can collect from the distant objects. And such a feat is even more remarkable, if we think that the objects under study are separated from us by (litterally) astronomical distances !!

On the other side, it is intriguing to realize that the main contributions to the understanding of the biggest and furthest objects in the universe have come from the study of the smallest entities and the shortest distances. As we will learn from these last two chapters, discoveries in the fields of Nuclear and Elementary Particle Physics have been instrumental in advacing our understanding of the nature of stars and of the whole universe.

The Sun's Energy

To start understanding the nature of stars, it is better to begin with the nearest one, the Sun.

Since the advent of modern science, several physicists have been puzzled by the origin of the energy continuously produced by the Sun, but the knowledge of the time could not provide a satisfactory explanation. How can we estimate the total energy output of the Sun? In a sense, it is easier than you might think. The first step is to determine how much energy from the sun reaches the earth. This can be measured to a good accuracy: the result is that the energy received every second on a one square meter surface is 750 Joules, i.e. the energy flux measured on earth is about 750 W/m². Note that this value is the energy measured at the earth surface, and is affected by the absorption of some of the sun's energy by the atmosphere (we have seen how, the atmosphere absorbs both UV and IR radiation). If we were to perform the measurement above the atmosphere, we would get a value twice as big, 1.5 kW/m².

Next, we have to make some hypothesis about the distribution of energy output: the most logical model is that energy is emitted from the sun uniformly in all directions. Can we verify whether such a hypothesis is correct? Well, not really; we can certainly verify that the measured energy does not vary as the earth circles around the sun, but, in principle, we cannot be sure that the sun does not emit more (or less) energy in the "vertical" direction (i.e. perpendicularly to the plane of the earth's orbit).

On the other side, in view of the symmetry of the problem, it is hard to come up with a model that would predict a non-uniform energy output, therefore it is quite reasonable to adopt the most natural hypothesis of uniform emission. This rather straightforward hypothesis is all that we need to estimate the sun's total energy output, in the following way : take a sphere of arbitrary radius r centered on the sun; by conservation of energy, all of the energy emitted by the sun must cross the surface of such a sphere. If we now consider an arbitrary surface element of unit area on this sphere, the amount of energy crossing it will be the same regardless of its location on the sphere. Then the total energy crossing the surface (i.e. the total energy emitted by the sun) is given by the energy crossing the unit surface times the total area of the sphere's surface, i.e.

$E_{tot} = E_{us}\times 4\pi r^{2}$,

where E_us is the energy per unit surface. Inserting the value measured on earth for E_us and the sun-earth distance for r, we get that the sun's total energy output is 4.24 x 10²⁶ Joules every second, an enormous quantity!! Be sure you understand that this is the total energy emitted by the sun in all directions, not the energy received by the earth. To evaluate this second quantity, we would need to multiply E_us by the area of the earth exposed to the sun, i.e. $\pi r_{earth}^{2}$.

Using the same procedure and the earth's data, we can also estimate the amount of energy received by any other celestial object at an arbitrary distance from the sun. Let E_us be the energy measured on earth, r the earth-sun distance and E'_us the nergy per unit area recorded by an observer at a distance r' from the sun. Because of the constancy of E_tot, one has:

$E_{us}\times 4\pi r^{2}=E'_{us}\times 4\pi r'^{2}$, i.e. E_us/E'_us=(r'/r)²:

doubling the distance the energy per unit surface is 4 times smaller, etc.

The Process of Fusion

How can we account for the sun's enormous energy production? The answer can not be found unless we take into account the transformation of mass into energy occurring in nuclear reactions. Earlier attempts, based on traditional energy sources (coal, etc.), to estimate how long the sun could last reached the conclusion that the sun should run out of fuel in times of the order of 10,000 years (see problems 7 and 8). This was in clear disagreement with all geological and paleontological data, even though creationists might not dislike such a conclusion. Nowadays we know that the process that powers the sun, as well as all the stars, is the one of nuclear fusion.

Even though the energy production in the stars is controlled by the nuclear (strong and weak) forces, their birth and death are governed mainly by the force of gravity.

"A star is born" when some local higher concentration of matter (mostly Hydrogen) in the universe attracts, by gravitational pull, more and more material from its surroundings, forming larger and larger clumps. As the accretion process continues, the central cluster is squeezed together more and more by gravity. The tremendous crushing force accelerates the Hydrogen atoms to higher and higher energies; interatomic collisions separate the electrons from the protons in the atoms, (so that the central core of the newborn star is in a plasma state). The free protons acquire enough energy and are sufficiently squeezed together to overcome the electrostatic repulsion, and occasionally get close enough to allow the nuclear force to intervene.

The first nuclear reaction that occurs is

$p + p \rightarrow D + e^{+} + \nu$,

where e⁺ is the familiar positron and D is Deuterium, a Hydrogen isotope consisting of one proton and one neutron (if you have ever heard the term heavy water, it is water made with Deuterium rather than Hydrogen, D₂O rather than H₂O).

How do we know that it is this, and not some other reaction that takes place? Obviously we cannot peer into the interior of a star, but what we can do, under controlled conditions, is to accelerate protons up to the energies we expect to find in a star's interior, and send them against a proton (i.e. Hydrogen) target. We can then observe which reactions occur and with what probability. Based on this knowledge, we can deduce what is likely to happen inside a star.

This continuous interchange between nuclear/particle research and astrophysics is the typical process by which we advance our knowledge in both fields.

The production of Deuterium is not the end of the energy production chain. A free proton getting close enough to a D nucleus has a good probability of being captured, to produce a Helium-3 nucleus. Next, two Helium-3 nuclei will fuse to form a Helium-4 nucleus plus two free protons. In summary, starting with 6 protons one ends up with a Helium nucleus, two protons and some amount of energy, since in each of the successive steps some of the mass of the reacting particles is transformed into energy.

Even though, as we will discuss later, further fusion processes can take place within the star, most of the star energy production comes from the "burning" of Hydrogen into Helium. This is confirmed by the copious presence of Helium in the sun: not by chance, the name Helium was derived from Helios, the greek word for Sun, since it is there that the element was first observed.

Armed with the correct explanation for the sun's energy source, we can then re-evaluate its predicted lifetime : the result is about 10¹⁰ years, to be compared with the 10⁴ obtained under the assumption of conventional fuels. Nuclear energy allows the life of the sun to be a million times longer than what it would be possible with pure chemical processes.

The Structure of the Sun

After we realize that the sun's energy originates from nuclear reactions, one question may come up to mind. We have learnt that the typical energies carried by products of nuclear reactions (e.g. $\gamma$ rays) are very high, of the order of million of electron-Volts. On the other side, most of the energy that reaches us is in the visible range, with energies of a few electron-Volts. How can we explain this discrepancy? The answer is that the nuclear processes that fuel the sun only occur in the innermost core of the star, where gravitational pressure is at its highest. High energy particles generated in the fusion processes have to travel through most of the star's volume and in doing so will transfer most of their energy to other particles by collisions. As the density of matter decreases going towards the outer layers, collisions among atoms become less frequent, and the main mechanism of energy transfer is convection (like in a pot of boiling liquid, but with very hot gases playing the role of the boiling water).

The volume of the sun does not have a well defined boundary, but it extends over considerable distances with ever decreasing density and temperature. Similar to flickering flames, the outer layers can become visible during solar total eclipses.

In addition to electromagnetic radiation, the sun also emits a constant stream of ions (the so called "solar wind"). These charged particles are the ones responsible for the phenomenon of northern (or southern) lights, aurora borealis (or australis). Aurorae are due to the interaction of the solar wind with the upper atmosphere; they are concentrated around the polar regions because of the earth's magnetic field.

Solar Neutrinos

Electromagnetic radiation and solar wind are not the only particles emitted by the sun. Among the products of the fusion reactions occurring in the central core we also find neutrinos. Due to their very low probability of interacting with matter, solar neutrinos can reach us directly from the sun's core, without undergoing the large amount of collisions to which other particles are subjected.

Studying the neutrino flux from the sun can yield real-time information on the working of the inner core. In the recent years, large underground detectors capable of recording neutrinos went into operation and, while they confirmed the existence of a solar neutrino flux, they also recorded a total flux that was much lower than what expected from the sun's energy output. This so called "solar neutrino problem" has been a puzzle for several years, and still is. A possible explanation is provided by the theory of elementary particles, that allows transitions between different types of leptons.

According to the theory, an electron type neutrino (which is the one generated inside the sun) has a probability of transforming itself into a $\mu$ or $\tau$ neutrino. Given that the exisiting detectors are only sensitive to electron neutrinos, this transformation can explain the observed deficit. According to the theory, the transformation can only take place if the neutrino has a non-zero mass. Studying the sun's output can then be used to infer the neutrino's mass, a quantity otherwise too small to be measured. This is a case of astrophysics coming to the aid of particle physics.

Measuring the stars properties.

The spectrum of electromagnetic radiation emitted by the stars covers a very broad range of frequencies, even though it peaks in the region of visible wavelengths. For the major part of the history of astronomy, information about the stars could only be gathered with optical devices (bare eye to start with, and telescopes from the time of Galileo onward). During this century, a new window of observation opened up, thanks to the development of radio technology: the field of RadioAstonomy was born in the '30s. Even so, observation over the whole spectrum was limited by the effect of the atmosphere. When discussing the greenhouse effect, we have learnt the atmosphere is mostly opaque to UltraViolet and InfraRed radiation. It turns out that it is also opaque for most other wavelengths, except for a "window" in the radio wave range : this is the one that was explored by radioastronomers.

In the most recent years, the advent of space age has allowed to install observing stations in outer space, so that the whole range of electromagnetic spectrum can be explored. I don't expect you to remember the names and the features of all the installations mentioned in the book, but you should be aware that nowadays celestial phenomena can be studied over a range extending from the longer radio wavelengths to the highest energy $\gamma$ rays. This has spurred a tremendous progress in the field of astrophysics, and not much time goes by before one hears of some important new discovery.

But let's go back to the early days, and see what was learnt just by studying the optical properties of the stars.

The very first observation one can make just by looking at the night sky is that stars have a wide range of brightnesses. The obvious question is then: is the difference in brightness due to the intrinsic light output from the star or to their distance? Both factors are actually at play, but in order to make any progress and try to classify the stars it is necessary to find some way of assessing their distance. If the location of a star is known, then its observed brigthness (usually called apparent magnitude) allows to determine its absolute magnitude and its total energy output, or luminosity.