Week 4 -- What the Sun does in our solar system

I. Starlight
The most important thing the Sun does is warm the planets with starlight. Starlight controls the temperature of the surface of planets, and also the type of light is important in maintaining life via photosynthesis and by allowing vision. But why does the Sun emit light, and why does it emit more yellow light than blue light or red light? These are questions for statistical mechanics and quantum mechanics.

To the first question, let us not say the Sun is luminous because of fusion, because actually the Sun emits light because it is hot, and it had to already be hot in order to undergo fusion in the first place. Also, we must distinguish the core temperature, the "guts" of the Sun that experience its "prevailing" temperature of some 10^7 K, from the much cooler surface layers, at around 6000 K. But which of those T is responsible for the luminosity of the Sun? (The luminosity L is the total rate of energy release, and by "responsible" I refer to the cause and effect relationship.)

It is a reasonable question, because when the Sun was very young (long before undergoing fusing in its core, when it was considered a "protostar"), it was actually its surface T that controlled its luminosity L. But after awhile, and still before the onset of fusion, the Sun made a shift from being fully convective to being dominated by radiation transport, and at that point the answer became "neither the core T nor the surface T ruled L." Instead, the surface T became ruled by L, while the core T, as always, continued to be ruled by the virial theorem, and still had no direct effect on L. Then when fusion initiated, all that happened was the Sun had a balancing energy source, preventing further contraction, but with no new consequences for L-- the L continued to be ruled by radiative diffusion and little else. We will return to this topic in more detail when we talk about solar evolution.

II. The Statistical Mechanics of Light-- the Stefan-Boltzmann Law

Next we must understand why hot things emit light in the first place. The answer is best found in statistical mechanics, where we discover that hot objects emit light for the simple reason that it is more likely they would do that than that they would not do that. This is pretty much the answer to all things in statistical mechanics-- if you look at all the things that are allowed to happen, the ones that actually do are simply the ones that can happen in the most ways, and are therefore more likely to happen. The situation is much like a poor dart player throwing darts at a board-- where the darts hit is in proportion to the area, and hence the chance.

As soon as it was understood by Huygens that light is a kind of wave, and therefore had a frequency associated with it, it was expected that it would obey the statistical mechanics of oscillators. An oscillator will steal average energy kT from a thermal reservoir it is in contact with, in essence because 1D motion takes 1/2 kT of energy (just like a free particle in 1D would; remember that 3/2 kT is what a particle takes from the reservoir in 3D), and the virial theorem of an oscillator says average KE = PE so average E = KE+PE = 2 KE = kT. Each of these oscillating modes propagates in 1D, but there are 3D directions they can take, so we have a 3D phase space in frequency nu, with a number of states per dnu proportional to nu^2 (just like a volume of a spherical shell of radius r and thickness dr is proportional to r^2*dr). So it was expected the energy per dnu (i.e., the thermal spectrum) of the thermal radiation field from a blackbody at given T would be proportional to nu^2*kT. This was in fact the case, but only for small nu (at given T). In the ultraviolet, we would have an energy "catastrophe", as the brightness just goes up and up at high nu.

If we ignore this catastrophic problem, we can just truncate the integral over frequency at some max nu proportional to T, and then you can easily see the integral gives a total radiant energy (per volume) that is proportional to T^4. Measuring the constant of proportionality then results in the "Stefan-Boltzmann law" (though conventionally the energy per volume is first converted to an energy flux density at the surface by multiplying by c/4pi and integrating over angle, but this doesn't matter because without quantum mechanics all we can do is say the surface flux density is also proportional to T^4 and measure the constant of proportionality).

III. The Quantum Mechanics of Light-- Planck's Constant and the Wien Law

The solution of this conundrum is also what provides the answer to the question of why the Sun emits more yellow and green light than blue or red. We know that hotter objects are bluer, and if they are not hot enough at all to emit in the visible then they emit in the infrared. But why is this true? Planck gave a heuristic derivation of the correct spectrum, but it was not understood why this held until the discovery of the photon.

The discovery of photons, in part motivated by Einstein's Nobel prize from the photoelectric effect, provides the considerably irony that in Newton and Huygens' debate on the nature of light, they were in some sense both right-- light is a particle (a la Newton), but it follows the rules of waves (a la Huygens). What is even more surprising is that this is not a special property of light-- it is true of all particles and all waves. All waves are quantized into "particles" of energy, and all particles are ruled by the mechanics of waves. Even what we think of as a "particle trajectory" is actually something that a wave with a very tiny wavelength is perfectly capable of doing! This is what is known as "wave/particle duality", it does not mean everything is sometimes a particle and sometimes a wave, it means everything is always made of particles and always moves according to the rules of waves, in every case, every situation. This is the fundamental lesson of quantum mechanics.

The consequences on the radiation field are decisive. To calculate the thermal radiative energy (per unit volume) at T, we weight each photon state in the 3D momentum phase space (where each state has volume h^3 as the usual consequence of the Heisenberg Uncertainty Principle, delta x * delta p = h in each dimension, times 2 for the photon polarizations) by the expectation value of the energy of the photons in that state. Summing over states amounts to an integral over 4pi p^2 dp (as always), times 2, divided by h^3, and weighted by the photon energy pc times the expectation value of the number of photons with momentum p at temperature T (the latter is called the "occupation number"). The integrand is the energy (per unit volume) within momentum interval dp, which is normally converted to a frequency interval dnu using p = h*nu/c. That gives a "spectrum" in the energy density, and all we require is the occupation number in each p state at T.

So far the only quantum mechanics we have is that the volume of a state in the 3D momentum space (per unit volume of real space) is h^3. Had we known the number of particles, this is all we'd need, because we get the probability the particle is in each state from the Boltzmann factor. But photons are being created and destroyed at T, so we don't know the particle number, and quantum mechanics appears again when we determine the occupation number in each state because we have to do a sum over all possible integer values of the number of photons that could be in the p state in question. So we use the Boltzmann factor to give the relative probability of each integer number of photons (provided by the thermal reservoir that is losing heat to create these photons). As usual, we get the expectation value for the number of photons by weighting each possible integer number of photons by that relative probability, and normalize to a true probability by dividing by the sum of the relative probabilities (the normalization in the denominator is called the "partition function"). I showed in class the result for the occupation number is
expected n_p = 1/(e^(pc/kT) - 1)
and when we combine that with counting the photon states, we find the integrand of KE/V is 8pi p^2 / h^3 times 1/(e^(pc/kT) -1). When converted to nu using pc = h*nu, and converted to a brightness per solid angle by multiplying by c/4 pi, we obtain the familiar form of the "Planck function." You can always look up that function, but now you can also manipulate it because you know where it comes from-- nothing but counting the states of the system using the phase space, and the states of the reservoir using the Boltzmann factor, all against the backdrop of a range of possible integer numbers of photons. The Planck function exhibits a turnover frequency where the brightness begins to decline (because the thermodynamics resists creating photons with energy much larger than kT per photon), and the fact that this turnover frequency is proportional to kT is called the "Wien displacement law". This turnover frequency solves the "UV catastrophe" in a way that could only be guessed at prior to quantum mechanics (as it requires the Planck constant). Aren't we lucky we live at a time where this is understood? Many great minds in the history of physics were never privy to this remarkable knowledge!

IV. Effect on Earth temperature
For us, the main importance of the Sun is that its light warms the Earth and makes life possible. We can easily determine the equilibrium surface T of the earth, relative to the surface T of the Sun, by using the Stefan-Boltzmann law at both surfaces and balancing the inward and outward energy fluxes at the Earth's surface. If we assume perfect blackbodies, we find immediately that the ratio of T^4 at the Earth surface to T^4 at the Sun's surface is the fractional solid angle subtended by the Sun in the sky, times a geometric correction factor that accounts for the foreshortening of the area perpendicular to a solar ray as the Sun gets lower in the sky. This geometric correction is normally handled by balancing the total flux over the entire surface of the Earth, creating a kind of day/night mean temperature that is enforced by the way the atmosphere shares heat from the day side to the night side. The result tends to produce a T that is a little low, like the freezing point of water, because it neglects the greenhouse effect that breaks the blackbody assumption (greenhouse gases are transparent in the optical regime where the energy comes in, but absorb some of the infrared light trying to leave). The net result is an additional warming that is essential to life (but too much of a good thing can lead to climate change and strife, as we are seeing right now from hurricanes in Cuba, Florida, and Newfoundland).