The solar spectrum

The electromagnetic spectrum
Light (and all EM radiation) has two complementary properties. We can describe it as waves of electricity and magnetism, or as a stream of particles of essentially zero size containing fixed amounts of EM energy. Waves have physical extent; particles do not. Particles carry fixed mass-energies; waves do not. If we build an experiment to test the wavy properties of light, we will find it is, indeed, composed of waves. If we build an experiment to test the particle properties of light, we will find it is, indeed, composed of particles. We call the EM particles photons. This fundamental wave-particle duality is the hallmark of quantum mechanics. All elementary 'particles' (electrons, quarks, photons, etc) are both wavy and particly.

We will begin with the wavy nature. Electromagnetic radiation is a "pure" wave of electric and magnetic forces which travels freely through the vacuum. There is no underlying material which is waving. If such a wave encounters a charged particle like an electron, the particle will oscillate back and forth in response to the forces. The wave is a solution of Maxwell's equations of electricity and magnetism, perhaps the greatest physics accomplishment of the 19^th century. The waves have five properties:

Direction of travel: EM radiation travels on a straight path in the vacuum.
Speed: EM radiation travels at 299,790 km/s in the vacuum for all observers.
This speed of EM radiation is always represented with the letter symbol c.
Wavelength: the distance between repeating orientations of the electric force.
Polarization: the orientation of the electric oscillation, always perpendicular to the travel direction.
Amplitude: the strength of the electric oscillation; the number of photons.

The spectrum of EM radiation refers to the variety of wavelengths possible. All wavelengths are possible in a continuous progression from zero to infinity meters. Since some of the wavelengths are very short, here are the metric system designations for fractions of a meter:
cm: centimeter, 1/100 meter
mm: millimeter, 1/1000 meter
μm: micrometer or micron, 1/1000 mm = 10⁻⁶ meter
nm: nanometer, 1/1000 μm = 10⁻⁹ meter.
Another unit slowly going out of favor but often used by astronomers is the Ångstrom (Å), equal to 1/10 nanometer.

The EM spectrum is divided into seven wavelength bands:

gamma-ray	X-ray	ultraviolet	light	infrared	microwave	radio
<0.01nm	0.01nm-10nm	10nm-400nm	400nm-700nm	0.7μm-1mm	0.3mm-1cm	>1cm
nuclear bomb	medicine	sunburn	vision	"heat" radiation	ovens, comm.	communication

The boundaries are kind of fuzzy.

Light is specifically defined as visible EM radiation, but sometimes people refer to 'ultraviolet light' or 'infrared light'. The visible spectrum is the familiar rainbow of colors produced by a prism or a rainstorm:

400 nm =violet; 550 nm =green; 700 nm =red.

(4000 - 7000 Å)

Summary
Light is a form of electromagnetic radiation. EM radiation can be described as oscillating waves of electric and magnetic forces. Different kinds of radiation have different wavelengths.

Blackbody radiation
All EM radiation is produced by vibrating charged particles; almost all of the radiation astronomers analyze is produced by vibrating electrons. Electrons are very common (all atoms have them), lightweight, and easy to set into vibration.

What we perceive as heat or cold is just the vibrations of the particles in a substance. If those particles are vibrating faster than the ones in our finger nerves, something feels hot. If they are vibrating slower it feels cold.

The average energy of motion of each particle in a warm object is about kT, where k = 1.381×10⁻²³ Joules/°K is Boltzmann's constant (the Joule is the metric unit of energy) and the temperature is measured in Kelvins. The Kelvin temperature scale begins at 0° where particles have no vibrational motion (so-called absolute zero), and uses Celsius divisions from there. Water freezes at 273.15 °K and boils at 373.15 °K.

Electrons vibrating with energy E tend to emit photons with similar energies. The relation between the energy E of a photon and its wavelength λ (Greek letter lambda) is E = hc/λ, where h = 6.626×10⁻³⁴ Joule-sec is Planck's constant and c is the speed of light (see above).

So, objects with temperature T tend to radiate photons with wavelength λ ≈ hc/kT = 0.015/T meters. Not all particles in an object vibrate with the same energy; some vibrate faster, some slower, as they collide with each other and constantly exchange bits of energy. The result is that a warm body emits photons with a wide distribution of wavelengths. Below is a plot of the energy flux leaving a hot, opaque object of uniform temperature as a function of wavelength. The quantity plotted is the number of Joules of radiation energy leaving each square meter of the object each second in each nanometer band of the spectrum.

Notice that:
a) Increasing the temperature increases the brightness rapidly. The total radiation coming from a hot body is proportional to T ⁴.
b) The most luminous wavelength band, like the average wavelength, is proportional to 1/T: λ_max (in microns) = 3000/T.
c) The apparent color of hot bodies changes from reddish at low temperatures (more 700 nm than 400 nm photons), to white (more-or-less flat across the visible spectrum), to bluish (more 400 nm than 700 nm photons) at high temperatures.

These distributions are called Planck or blackbody distributions. Max Planck figured out why hot bodies radiate this way at the end of the 1800s and presented his paper on 14 December 1900. The reason is related to the particle nature of light, and Planck's work ushered in an entirely new way of looking at the world, quantum mechanics. The term blackbody refers to a perfect absorber, an object that absorbs all radiation falling on it and converts the energy into heat before re-radiating it.

No exposed object is a perfect blackbody (the interior of a uniformly heated body does contain this distribution of radiation). However, many objects closely approximate blackbodies, including the Sun, most other stars, rocks like the Moon and asteroids, incandescent lightbulbs, and the human body. We humans have a temperature about 300 °K and we radiate profusely at about 10 μm.

Below is a graph showing the spectral distribution of the Solar irradiance at the top of the Earth's atmosphere in Watts (=Joules/sec) per square meter per nanometer.

Data from The 1985 Wehrli Standard Extraterrestrial Solar Irradiance Spectrum at http://rredc.nrel.gov/solar/standards/am0/

The overall distribution of radiation is more-or-less like that of a blackbody, but there are significant departures especially in the violet and ultraviolet parths of the spectrum. These departures introduce us (and the physicists of the 19^th century) to the next section.

Summary
Hot, opaque, objects like stars and planets radiate a continuous range of wavelengths with most of the radiation coming out near λ_max (in microns) = 3000/T. This radiation is produced by electrons vibrating with similar energies.

Atomic physics and spectroscopy
As early as 1814, the German optician Fraunhofer cataloged 754 dark features in the solar spectrum. The title square at the top of the page is a recent picture. These "missing wavelengths" are signature absorptions by atoms in the Sun's atmosphere.

Electrons in atoms
Students should be familiar with the general structure of atoms. At the center of an atom is the nucleus, composed of roughly equal numbers of protons (+1 charge unit) and neutrons (0 charge unit). Almost all of the mass of an atom is in the nucleus. Protons and neutrons have roughly equal mass, both being about 1830× the electron mass.

A charge-neutral atom has as many electrons oscillating about the nucleus as there are protons in the nucleus. The cloud of electrons determines the chemical properties of the atom. Each chemical element is then distinguished by the proton number, called the atomic number Z, of the nucleus. There are 92 naturally occuring elements, hydrogen through uranium. The elements have periodic chemical properties. For example, beryllium (Z=4), magnesium (Z=12), and calcium (Z=20) all behave similarly. The periodic table of the elements reflects this behavior. The great achievement of quantum mechanics was the explanation of both the periodic behavior and the dark features in the spectrum.

Most of us are taught in grade school that the electrons in an atom orbit the nucleus somewhat like the way planets orbit the sun. This picture is not very accurate. Electrons, like photons, are waves when they are not being particles. How are we to understand a wavy electron? Every particle has a wave function which describes a probablility amplitude over all possible locations and at all times. The likelyhood (or expectation value) for finding the 'particle' at a particular location is the amplitude squared.

Because the positively charged nucleus pulls on the negative electron, the wave of the electron is confined to the neighborhood of the nucleus. Imagine this wave circling around and bouncing back and forth against this confinement. It will overlap on itself over and over again. Unless the wave exactly overlaps crest on crest each time it comes back, negative portions of the wave will eventually cancel positive portions. Only those waves that meet the condition of perfect overlap will exist. A better analogy than orbits are the allowed vibrations of a tightened string.

The patterns of allowed waves depend on a number of factors, including the presence of other electrons. However, we can make certain general statements about which wave patterns are allowed. Each pattern can be described by four quantum numbers:

n:	the principle number, which describes radial properties of the wave. It may have any integer value from 1 to ∞. An electron's average distance away from the nucleus increases with increasing n.
l :	the azimuthal number, which describes angular properties. It may have any integer value from 0 to n −1. An electron's average distance away from the nucleus also increases with increasing l, although not as much as with n.
ml:	the magnetic number, which describes orientation properties. It may have any integer value from −l to +l.
ms:	the spin orientation number of the electron spin. It may have a value −½ or +½.

The Pauli exclusion principle states that no two electrons in an atom can have the same set of four quantum numbers. From that principle we can build the periodic table of the elements. In the table below, electrons fill the boxes starting close to the nucleus at n = 1. At each n, there can be two electrons with l = 0: the two spin states in the single m_l = 0 state. There can be six electrons with l = 1: the two spins for the three different m_l values −1, 0, and +1.

Element			n:	1	2		3
		Z	l:	0	0	1x1x	0	1x1x	xxx2xxx
Hydrogen	H	1
Helium	He	2
Lithium	Li	3
Beryllium	Be	4
Boron	B	5
Carbon	C	6
Nitrogen	N	7
Oxygen	O	8
Fluorine	F	9
Neon	Ne	10
Sodium	Na	11
Magnesium	Mg	12
Aluminum	Al	13
Silicon	Si	14
Phosphorous	P	15
Sulfur	S	16
Chlorine	Cl	17
Argon	A	18
...

The chemical properties do not depend very strongly on the value of n. They do depend on the l value of the outermost electron and the number of other electrons with the same l. Thus beryllium is similar to magnesium because both have two l = 0 electrons on the outside.

Spectroscopy
The boxes above that are empty (with all values of n up to ∞) represent quantum wave states that are possible but empty. It is possible to take an electron from one of the occupied states and cause it to vibrate in one of these other states. There are two ways to do that. First, if another electron collides with the atom, it might kick the confined electron into the empty state. Second, if a photon comes through the atom it can kick the electron into the new state. Remember that photons are also a wave of electricity and magnetism; the waving forces stimulate the new vibration of the electron.

Not just any photon can interact with an atomic electron. Each photon is an indivisible particle of energy; the atom has to accept all or nothing. We mentioned above that the energy of a photon is inversely related to its wavelength λ: E = hc / λ.

Each available vibration state for an electron in an atom has a particular energy. Generally, states with higher quantum number n have higher energy. An electron in a high n state is on average farther away from the nucleus; it takes work to pull it away from the positive charge. Only those photons with just the right energy (equivalently just the right wavelength), equal to the energy change necessary to lift the electron exactly into a new state, will be absorbed.

One way to picture the vibration states and their energies is like a staircase. On the left we illustrate hydrogen, the simplest element. The [n, l = 1, 0] state has the lowest energy. To lift the electron up to the [n, l = 2, 1] state requires an ultraviolet photon of just the right energy, or the electron cannot find itself on a "stair". Once it is on an upper stair the electron can fall back again, either by emitting a new photon, or by giving the energy to another electron in a collision. If the electron is in the [n, l = 2, 1] state, it is possible for a red photon of just the right energy to kick it up one more step to the [n, l = 3, 2] state. It is these red photons which are "missing" in the spectrum of the Sun. A blue-green photon can kick the electron from [n, l = 2, 1] directly up to [n, l = 4, 2]. In hydrogen, all the l values of a given n value have the same energy. To go from [n, l = 3, 2] to [n, l = 4, 3] requires a lower energy (longer wavelength) infrared photon.

Each different element has a different number of protons in the nucleus and a different number of electrons. Thus, for each element the "active" electron (usually the outermost with the highest n and l) sees different forces and has different available vibration state energies. The result is that each element absorbs a different set of photons. For example, the faintish dark line in the yellow part of the Solar spectrum results from sodium [n, l = 3, 0] to [n, l = 3, 1] absorptions. At this website is a nice display of the emission spectra of various elements. Emission occurs when the electron falls from an excited state to a lower energy state closer to the nucleus.

Imagine the following simplified picture of a star. Deep inside, there are enough vibration and motion states for electrons to produce a blackbody distribution of electromagnetic radiation, independent of the composition. Near the surface, the gas is cooler and the atoms are less excited than they are in the interior. When the blackbody radiation shines out through the cooler layers, the atoms there selectively absorb only those wavelengths which can excite their electrons.

By measuring how much radiation is "missing" in each dark feature in the spectrum, astronomers can measure the relative abundances of the various elements in distant objects.

In other modules, we will see how we can measure velocities of objects through the Doppler shift of the wavelengths of an element's dark features in the spectrum.

Summary
The electrons in atoms follow very specific waves of probability, that only exist at particular energies. Only one electron can occupy each wave state. The periodic table of elements is arranged according to which wave states are filled with electrons. If a photon with just the right energy (proportional to 1/λ) comes along, it can kick an electron up into an empty state farther from the nucleus. Each element has a different pattern of possible absorbing photons.