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Notes follow and parts taken from sources in Bibliography Light – Light can act as a ray, a wave, or a particle, depending on the circumstances and how you observe it. We’ll start by thinking of light as a wave, which naturally introduces other concepts. What do the peaks and troughs of the wave represent? Basically, they are peaks and troughs of the electric and magnetic fields which make up the light. These waves are transverse, like the ones you would make if you were trying to get a hose or electrical cord free from some obstacle and didn’t want to walk back to the point where it’s stuck. The vibrations back and forth are therefore perpendicular to the direction of travel of the wave. The distance from one peak to the next on the wave (or one trough to the next, etc.) is

called one wavelength, usually represented by the greek letter (lambda). If you stand in one place and count the number of peaks of the wave that go by you in one second, that’s

called the frequency of the wave and is usually represented by either (nu) or f . These two things (number of meters between peaks and number of peaks per second moving past one place), multiplied together, give the speed of the wave. We can write it as

c = f where c is the speed of the phenomenon. Usually, if you use c you’re talking about light. For other things (sound, etc.), it’s more common to use v, as in velocity. For sound, the wave’s speed is about 330 m/s. Middle C (musical note) is produced when the piano string (or vocal cords) vibrate about 261 times per second (the MKS unit for frequency is Hertz (Hz) which could also be written as 1/second). This means the wavelength of the sound is 330 (m/s) / 261 Hz = 1.26 meters. Light works the same way. The center of the visible spectrum (a green color) is at about 550 nm (5.5 x 10-7 m). The speed of light is 3 x 108 m/s. That means that the number of green-light wave peaks passing one place in a second is about 5.45 x 1014! The Spectrum – Not all electromagnetic waves are visible – in fact, only a small portion of the spectrum is visible light. Red light is the lowest-frequency (and therefore longest wavelength) of visible light – next are orange, yellow, green, blue, and violet. At frequencies below visible light (lower frequency = larger wavelength, since the speed is the same for all frequencies), we first come to the infrared. This is the kind of light used in remote controls, and it’s also felt as heat. Most night vision goggles are actually just sensing infrared light. Below the infrared is the microwave range. Wavelengths here are best measured in centimeters. At still lower frequencies, we get to the TV and radio bands. Also, since we use AC power which oscillates at 60 Hz in the US, everything you plug in (and the power lines overhead) send out electromagnetic waves with a wavelength of 300,000,000 (m/s) / 60 Hz = 5,000,000 m = 5,000 km. At the other end of the spectrum, starting with frequencies just above violet, is the ultraviolet (UV) region. UV light is what gives us tans, sunburns, and fades paint and fabrics. Above the UV lies the X-ray region, and above that is the gamma ray region. None of these parts of the spectrum is absolutely defined. There isn’t one precise value of wavelength which serves as the boundary between microwave and infrared, or UV and X-ray, or even violet and UV.

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Speed of Light – All electromagnetic radiation (radio waves & below to gamma rays & up) travels at the same speed through a vacuum. This speed, 299,792,458 m/s, is the ultimate speed limit for the universe. No information or object can exceed this speed. Actually, no object with mass can even reach the speed of light. The fact that this value is the same, no matter how you measure it (running away from it, towards it, or standing still) is the centerpiece of relativity. Einstein traded the idea of time as something unchangeable for this, and it works. While the speed is constant for all EM waves in a vacuum, they don’t all have the same speed through matter. That’s why the prism separates light into different colors – each different color (or wavelength or frequency) is slowed by a different amount and therefore bends by a different amount. Measurement of the speed of light has a long & difficult history, but we’re going to skip it. The thing that makes light unlike any other wave you’re familiar with is the fact that it doesn’t require a medium to “wave” in. Sound waves need air, and water waves need water, but light waves can travel through a vacuum. We don’t generally notice the wave character of light, because it’s usually only visible on the small scale. We said earlier that light doesn’t go around corners, but the real story is that is only goes around a little. This is what we would expect from a wave, not a ray. The reason this is so rarely observed is that it’s really only noticeable if light is doing something like going through a small (not huge compared to length of the waves) opening. An example of light going through a pinhole is shown below. You’d expect, using the ray theory, to see on solid light circle, the same size as the pinhole. You actually get a series of light and dark bands. On the left, you see what the pinhole would show. The image on the right is the same thing, but brightness is represented by height.

The idea here is that waves spread after going through small holes. What would happen at the beach if there were logs or sandbars which had only a small break in them? When waves hit the break, they will start more circular waves (left), not straight-line segments of waves (right).

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Interference – If light acts like a wave, we’ll notice strange things when the waves come together from different directions. If the waves are out of phase (the peak of one arrives at a point at the same time the trough of another arrives), they’ll combine to give no wave (destructive interference). On the other hand, if peaks arrive from two different directions at once, they’ll add to make a large peak (constructive interference). Look at the diagram below

to see what’s going on. Remember that if the wavelength is , we’ll see peaks every meters

away in every direction. We’ll see valleys (troughs) /2 meters away, and every meters after that in all directions.

Let’s say that the distance from the lower slit on the left to the movie screen on the right

(purple line) is equal to a whole number of wavelengths (say, 10 rather than 10.2 or 9.97). If the distance from the top slit on the left to the screen (red line) is also a whole number of

wavelengths (maybe 8 - we know it can’t be 10 because it’s closer than the lower slit), both waves will be at a peak (constructive interference) and that spot on the screen will be brighter than if just one of the slits was open and the other blocked. Adding a slit adds light, which makes sense. Here’s what doesn’t make sense. If the purple line is still 10 wavelengths long, but the red line is really 8.5 wavelengths long, a peak from the lower slit will hit at the same time as a trough from the upper slit. They’ll combine destructively, and the screen will be dark at that spot – darker than it would have been with just one slit open. We open another slit and we reduce the light that some parts of the screen get. It seems weird, but that’s how it works. (Incidentally, if you’re interested in this kind of weird behavior, it happens with electrons, too. It’s even stranger when you use particles (which we can think of light as made up of). If you’d like a very readable and math-free explanation of all this & more, take a look at QED: The Strange Theory of Light and Matter by Richard Feynman. He won the Nobel Prize in physics,

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and many people think he was about the best at taking difficult concepts and making them simple). Continuous Spectrum and Kirchoff’s 1st Law – If a solid, liquid, or dense gas is heated, it will give off a continuous spectrum. The filament in a light bulb does this – electricity gets it incredibly hot, and light pours out of it. It looks white because it’s really radiating all colors. In addition to radiating all the visible spectrum, it also radiates longer wavelengths (infrared). Both the energy emitted and the particular spectrum depend on the temperature of the object. For example, after turning on the element in a stove, the element will start to heat up. You’ll be able to feel the infrared radiation from the element before you can see it getting hot. After a few more seconds, it will start to get dull red in color, and then make its way up to a brighter red or orange. As heating continues, the peak in the spectrum (the color emitted most strongly) moves from infrared to red, orange, yellow, etc. You won’t see it turn green, because it’s emitting the lower-frequency colors as well. Our eyes perceive things that are really hot as being white. The other thing you’ll notice is that the energy output is wildly sensitive to temperature. When the temperature of an object goes up by a factor of two (measured in Kelvin), the energy output goes up by a factor of 16! Why do we have to use Kelvin? We need an absolute temperature scale. You’ll frequently hear people say something like “It was 25˚ yesterday, and today we doubled that and hit 50˚”. 25˚ F is NOT ½ the temperature of 50˚ F. If you think it is, imagine how much hotter 50˚ is than 1˚, 0˚, or –15˚. You’ll see that the idea falls apart at 0˚, and makes no sense at all at negative temperatures. We need to use a scale that is directly connected to the energy of the molecules that make up an object, and that’s the Kelvin scale. Two formulas we’ll need are shown below. On the left, we have Wien’s law, which describes where the peak in the spectrum is for a given temperature. On the right is the Stefan-Boltzmann law or Stefan’s law, which tells us how much power is emitted by a body at a given temperature.

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Wien’s law tells us that the wavelength of the peak in the spectrum (what color is emitted more strongly than any other) multiplied by the temperature of the object is a constant. That means that as temperature increases, the wavelength of the peak decreases (which means the frequency increases). That’s why things progress from emitting mostly heat to reddish light, then orange, yellow, etc. The Stefan-Boltzmann law describes the power radiated by the hot object. Power is the rate at which energy is produced. As an example, two cars which are the same size and are moving at the same speed have the same kinetic energy. If one car takes 15 seconds to get to that speed, but the other car can do it in 5 seconds, the quicker car has a more powerful engine. This also means that you don’t get a power bill every month, you get an energy bill. The electric company doesn’t care if you use all that energy in one day, or spread it out over the month – you’re billed for the energy, not the rate at which you use it.

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Unfortunately, the book misuses this and refers to Stefan’s law as giving energy per area. This is incorrect. It’s fine to talk about power/area, in which case the formula simplifies to

𝐹 = 𝜎 𝑇4 Since energy is a total and power is a rate, it doesn’t make sense to talk about the energy per area radiated by something. You can make the energy per area radiated by a night light as large as you want if you leave it on forever. It makes much more sense to talk about power or power/area, since that means we don’t have to specify a time frame. Power is measured in watts (or in horsepower in the English system). A 1200 watt hairdryer uses 1200 Joules of energy every second. Anyway, the power this hot body radiates depends on its temperature (very sensitively – that’s what the T 4 means in the formula) and its surface area. This is also reasonable – a small burning ember from a fire should give off less heat than the whole fire. A fact that makes life much simpler for us is that the spectrum is pretty much independent of everything else. This is the reason the “point-and-shoot” infrared thermometers can exist. It doesn’t really matter what you’re pointing at, since the spectrum reveals the temperature all on its own. A few examples of blackbody spectra are shown below.

A few things to notice about this: 1) As the temperature increases, the peak of wavelength (dotted line) moves to smaller values. This is what we’d expect from Wien’s law. 2) The power emitted is proportional to the area under the curves. Notice that 6000K is only one and a half times 4000K, but look at the difference in area under the curves – it’s huge. We can calculate the ratio of the areas – it should be (6000K / 4000K) 4 = 1.54 = about 5 times more area under the 6000K curve. Now we begin to see why a spectroscope is an important instrument: if we can get a look at a star’s spectrum, it will give us a good idea of its temperature (this is surface temperature, of course, because the surface is the final radiator before we see it).

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Doppler Effect – The Doppler effect is a sensation familiar to anyone who has ever heard a siren speeding by them. The pitch of the siren is higher as it approaches you and lower as it moves away from you. The explanation for this is relatively simple. We know sound (and light) are oscillations of something (air density for sound, electric & magnetic fields for light), and the oscillations go through peaks and valleys. The number of peaks and valleys that hit our ears or eyes in a given time determines the pitch we hear or the color we see. If we move towards the source, or it moves towards us, we will encounter more peaks and valleys per second than we would if we were standing still. That’s why we say the frequency (which measures the number of peaks/valleys per second) increases as we move towards the source, or it moves towards us. Similarly, if we move away from the source, fewer peaks and valleys strike us per second than if we were standing still. The frequency of the light or sound is lower. In the scene below, this is what the pattern of peaks (or it could be valleys – either one) would look like if the source was moving quickly. The source started at the center and was emitting peaks and valleys – let’s say the circles represent peaks. Each circle is centered on the point where the source was when it was emitted. The source is moving, so the centers of the circles move, too. The blue face will be hit by lots of peaks in a short time, meaning a higher frequency. The red face will be hit by fewer peaks in the same time, so it looks like a lower frequency. There is no motion towards or away from the yellow face, so it sees the “real” frequency of the source. This is the frequency that someone riding with the source (inside the ambulance) would hear (or see, or whatever). By the way, the source in this figure is moving half as fast as the waves themselves (that’s ½ the speed of light for light, or Mach 0.5 for sound).

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The shift to a lower frequency is called redshift and the shift to a higher frequency is called blueshift. These are universally used, but there’s something to keep in mind: we call it redshift because, if we are talking about light which is in the middle of the visible spectrum, shifting to a lower frequency means moving towards the red end of the spectrum. Similarly, moving to a higher frequency means moving towards the blue end. The thing to keep in mind is this: blueshift always means a shift to a higher frequency, and redshift always means a shift to a lower frequency. As an example, if a source of radio waves is moving away from you, you still say that it’s redshifted, even though it’s moving further away from the visible-light color red in the spectrum. Measuring speed – We can use the size of the Doppler shift to tell us the speed with which the object is moving towards or away from us. Keep in mind that only radial motion (towards or away from us) will show up here – we won’t see any shift if something is moving transversely to our line of sight. In other words, if you’re at the center of a circular race track, the cars around you are certainly moving, but they aren’t getting any closer to you or further from you – they’re moving transversely as far as you’re concerned, and the Doppler shift only happens for the part of an object’s motion that is towards or away from you (there actually is a small Doppler shift in this case, called the transverse Doppler shift – it’s explained by the theory of relativity, but it is unimportant as long as the object’s speed is small compared to the speed of light, so we’ll ignore it). Anyway, the thing that matters if we’re trying to find the speed of the source is the relative change in the wavelength (or frequency – it’s just usually done in wavelength terms) of the light. In other words, we need to know the difference between the wavelength of the light emitted by the source and its wavelength when the observer receives it, and we need it as a fraction of the original wavelength – i.e., the wave observed has a wavelength 10% larger than it was when it was emitted, or 25% shorter, or something like that. Here’s the formula for light (it’s a little more complicated for sound, but we get to ignore sound completely in astronomy!):

c

v

In this equation, is the original wavelength of the light, as emitted by the source. is the shift in the wavelength from emission to reception. That means the left side of the equation will be a length divided by a length, so it has no units – it’s just a number. On the right, v represents the radial velocity (velocity towards or away from the observer), and c represents the speed of the waves (speed of light, speed of sound, whatever we’re talking about). Light as a Particle – For many years, scientists debated the nature of light – was it a wave, or a particle? It turns out that light has some particle-like properties, and some wave-like properties. Depending on what you choose to measure, you can see one or the other. If we think of light as a particle, the particle is called a photon. Different colors of light (=different frequencies = different wavelengths) have different energies. Higher frequencies carry more energy per photon. That’s why visible light is harmless, and UV or X-rays or gamma rays can cause serious damage to people and objects. The photons can’t be divided – a UV photon

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can’t be broken into two lower-energy photons (like 2 red photons). The common unit of energy when discussing photons is called the electron-volt or eV. This is an incredibly small unit of energy. For example, you probably know that the purpose of a battery is to give electrons a boost of energy and move them along. If one electron goes from the bottom of a battery (most are 1.5 volt batteries) to the top (positive end), it will gain (1 electron) x (1.5 Volt) = 1.5 electron volts of energy. The fact that this amount of energy is so small, and that visible photons typically carry about only a few eV of energy, explains why light doesn’t seem to come in pieces when we look at it. It’s the same principle as matter being made of atoms – if the atoms were huge, it would be obvious that matter isn’t continuous. They’re so small (on our scale) that it seems like you can just keep dividing a piece of something into smaller & smaller pieces forever without having to stop. To get an idea of the energy scale, this is the formula for finding the energy of a photon, given its frequency (= color)

E = h f where E is the energy in eV, f is the frequency in Hz, and h is a quantity called Planck’s constant, which has a value of 4.14 x 10-15 eV sec. This idea of light as a particle got a modern boost with Albert Einstein’s explanation of the photoelectric effect. This effect occurs when light (usually UV light) falls on certain metals and ejects electrons from the surface. Since moving electrons produce a current, this effect is easy to measure. The strange thing was that if light was a wave, rather than a particle, we would expect to observe a few things – 1) there should be some delay between the time the light is turned on and the time the electrons have absorbed enough oscillations from the wave to have the energy to leave 2) with a wave, lower frequency light should be able to have the same effect as high frequency light if the intensity (brightness) is turned up. In other words, a dim UV light and a very bright infrared light should have the same effect. These two things didn’t happen, and Einstein explained it using photons. There is no delay because as soon as the photon hits the electron, it’s ejected (if the frequency of the light is high enough). Also, since frequency is connected to energy, that’s all that matters for ejection. Low frequency photons, even a great many of them (bright source) just don’t have the energy to eject the electrons. It was for this work that Einstein received the Nobel prize in physics, although he made at least 3 other huge contributions to physics. Waves or Particles? So which is light? Wave or particle? Neither one, and/or both. It turns out that light (and electrons and other small pieces of matter) can show either behavior, depending on what you’re looking for. This idea of things being waves and/or particles just doesn’t apply – we can’t take these concepts from the large scale and expect them to work at the small scale. Things are just different there, and we have to live with that. Spectra – The word spectrum (plural: spectra) refers to the EM radiation (visible light, UV, X-rays, etc.) absorbed or emitted by an object. This is the intensity of radiation at each wavelength. We will examine the sources of three different types of spectra, including their production and what observation of them can tell us. First, we need an instrument to observe these spectra, and the eye alone is not a good one. Our eyes are not good at breaking the

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light from an object into its various colors – we tend to see things as white if they radiate a few different colors. We could use something called a photometer, which measures the intensity of the light falling on it, and we could use various filters and get some of the data we want. A more efficient way to do it is to break the light into its components using either a prism or a diffraction grating. The diffraction grating sends different colors of light in slightly different directions – you can try this with a CD if you shine light on the grooves, which are so small that they function as a diffraction grating. These will be involved in our discussions of the three types of spectra, which are 1) Continuum (all colors – this is like a rainbow) 2) Emission (also called discrete emission – this spectra is made up of lines with large dark spaces in between them. The lines will have the same color as the continuum spectrum would have at that point. In other words, if your instrument shows red on the left and violet on the right for the continuum, the lines in an emission spectrum will be red if they’re on the right & violet on the left). 3) Absorption spectrum – this will look kind of like the continuum spectrum with an emission spectrum subtracted from it. You’ll have most of the rainbow, but little dark lines will appear where light was absorbed. It looks like the opposite of the emission spectrum (which is mostly dark but with colored lines in a few places). Inverse square law for light - Using geometrical arguments, we can quickly see how intensity will vary with distance. Imagine a 100 W light bulb inside a glass sphere with a radius of 1 m, which is itself inside a larger glass sphere of radius 2 m. The intensity of light

at the first sphere will just be 100 W / (4 (1 m)2), or about 7.95 W/m2. Exactly as much light passes through the second sphere (it has nowhere to go – we’re assuming perfect glass in a vacuum, so no light will be absorbed, scattered, or created) as the first, but its area is much

larger. This means the intensity will drop down proportionally to 100 W / (4 (2 m)2), or about 1.99 W/m2. If we double our distance, the intensity drops to ¼ of its previous value; at 10 times the distance, intensity will be 1/100th as much. This is known as the inverse square law for light. It’s no surprise that intensity drops off as 1/r2 since intensity is just power (which is a constant in our example) / area, and area increases as r2. In this sense, light is no different than paint. If you want to paint a 2 m sphere with the same amount of paint as a 1 m sphere, it will have to be ¼ as thick on the larger sphere. Discrete Spectra – Not all things that emit electromagnetic (EM) radiation have a blackbody spectrum – only solids, liquids, or dense gases that are radiating because they are hot. If a thin gas is excited somehow (electric fields can do it, or heat), the radiation pattern will be composed of isolated spectral lines separated by dark bands. This means that light is only being radiated at certain specific (discrete) wavelengths, rather than all across a large band of wavelengths. Where do the lines come from? Keep in mind that each line represents a very tiny range of wavelengths (it’s sometimes said that it represents a single wavelength, but that’s not possible – it’s just a tiny range). For our purposes, we can call it a single wavelength, which means it has a single frequency and its photons would have a single energy. We’ll come back to this. First, let’s look at the atoms that make up the gas. We’ve already said that they have been excited somehow (excited means that they are carrying more internal energy than usual – this can happen if they’re hot, and carrying a lot

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of kinetic energy, and they slam into each other frequently), and that means that one or more of the electrons in the atom has been moved further away from the nucleus than it would be otherwise. The electron is kind of like a ball on a staircase here – it’s been moved away from the bottom to one of the higher steps. It would “like” to come back down the stairs (move closer to the nucleus). One important difference is that the stairs aren’t evenly spaced – the first one is the highest, then each step above that is shorter & shorter. Anyway, the electron can only come back downstairs if it gets rid of some energy; specifically, it needs to get rid of exactly the same amount of energy that it absorbed when it was thrown away from the nucleus (up the staircase) in the first place. What happens almost all of the time is that the electron will radiate a photon & go back to the lower state it came from. Instead of taking the path in one long jump, it can also hit a couple of steps on the way down, and emit a photon at each step. The combined energy of all these photons will be the same as the energy of the single photon. Again like the staircase, there are only certain allowed energies the electron can have, and therefore give back. This is where the “quantum” in quantum physics comes from. The staircase analogy is not perfect, though, because if you try to throw the ball up 4 ½ steps, it will go up & come back down. If you try to give an electron an amount of energy between what it would need to go to the 4th / 5th steps, it will ignore the energy and let it pass by without interacting. This is why the spectrum is discrete: only certain energy levels are involved, so only certain energies can be emitted from the electron, which means only certain colors of light in the spectrum. Because all silicon atoms are the same, they all have the same energy level structure (same size of steps on staircase) and therefore have the same spectrum. Because all the different elements have different energy level structures, the spectra of different elements are different, making it an ideal way to determine what something is made of (notice the difference between this and the blackbody spectrum, where all the elements seem to look alike). Absorption Spectrum – Closely connected to the discrete emission spectrum is the absorption spectrum. If all colors of light ( = white light) are passed through a thin gas, some of those colors will have energies that match the energy levels (stairs) in the atoms in the gas. Those colors are then used to move electrons around, and so they aren’t there for you to see. If you pass the formerly white light through a spectrograph after it goes through the gas, you’ll notice that some of the colors are now missing. What happens to the colors? The electrons absorb them, and later they re-radiate the same colors as they make their way back down the staircase. The interesting thing to keep in mind is that, when the photons are re-radiated, they don’t have any memory of which way they were going when they were absorbed – they go off in random directions.

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Here’s what you will see from different positions:

Notice the three different views. If you’re looking directly the hot body without any intervening gas (not possible for the Sun, but we’ll skip that), you should see the full spectrum. If you’re looking at the hot body through the gas, some of that light will be absorbed and re-emitted in random directions (this is called scattering), so the spectrum is that of a black body with a few lines missing. If you’re looking at the now-excited gas and seeing the scattered radiation that comes from the electrons bouncing back down the staircase and emitting photons in all directions, you’ll just see a few lines of light against a dark background. Notice that these lines appear in the same place where the dark lines in the absorption spectrum are found. Add the discrete emission and absorption spectra together, and you get the blackbody (full rainbow) back. Just as we used the continuum spectrum (blackbody) to determine the temperature of an object, we can use the discrete emission and/or absorption to determine the composition of a body. We can make a list of the spectral lines (and their strengths, or brightness) for each element by exciting them in the lab. Once we have that list, we can look at light from a source in the sky and determine what elements are present, and how much of each one is present. Bright lines mean more photons, and therefore more of the atoms that produce the lines. This was how Helium was discovered – it was found in the Sun’s spectrum before it was discovered on Earth. Energy Level Diagrams – The different stairs on the energy level staircase are labeled from 1 on up to infinity. 1 represents the ground state, or the state when the atom has the lowest possible internal energy. This is also when the electron is closest to the nucleus (this concept is only meaningful as an average: the electron does not “orbit” the nucleus in a circle, or an ellipse, or any other kind of continuous curve. Because of the nature of quantum mechanics, rather than having a definite location or orbit, it just has a certain probability of being at a given place. You can think of the electron as disappearing & reappearing all over the place on incredibly short time scales. When an electron is more likely to be found near the nucleus (or even inside it!) than far away, that’s the ground state). These energy levels are negative,

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because it takes an input of energy from outside to remove the electron from the atom. The ground state has the most negative energy (large negative number). This is not really a foreign concept – if you have a heavy block of lead in your basement, you can think of it as having negative energy compared to a brick on the first floor; you’ll have to put some energy in it to get it up to the first floor. We’re all bound to the Earth the same way. You can watch a rocket liftoff & see that it takes quite a bit of energy to move us off of the Earth. As we’ve said before, the energy levels are not equally spaced; the lower levels are further apart from each other than the higher levels. The spacing is proportional to E0 / n2 , where E0 is the energy it would take to completely remove an electron in the ground state from the atom, and n is the level number. For n = 1, this means it will take E0 / (1)2 = E0 to remove the electron. If the electron is already in the second energy level (also called the first excited state), n = 2 and the energy it would take to remove it completely from that level would be E0 / (2)2 or E0 / 4. Taking the electron away if it’s in the 2nd excited state, where n = 3, would take only E0 / (3)2 or E0 / 9. You can see the first step is very high, but the others get closer and closer together. What you’ll also find is that the photons emitted have energies that match things like E0 / 4 - E0 or E0 / 9 - E0 / 4. The first one would represent a photon going from n = 2 to n = 1 and the second would be n = 3 to n = 2. That tells us an electron in the n = 3 state might jump to n = 2 (emitting a photon) and then from n = 2 to n = 1 (emitting a different colored photon) or it might make the jump from n = 3 to n = 1 directly (emitting only one photon, but with an energy different from either of the previous two we mentioned). You can see how an atom with a few electrons, each of which may have several different energy levels open to it, can create many different spectral lines. General rules about this would include: a photon emitted in the jump from n=5 (for example) to n = 3 will have the same energy as the combined energies of an n = 5 to n = 4 photon + an n = 4 to n = 3 photon. The jump from the n = 1 level to any other level will always be very large, which means high-energy photons (for hydrogen, this is somewhere in the ultraviolet and therefore not visible to the naked eye). Jumps from a higher level to the next level down (n = 8 to n = 7, for example) will produce low-energy and therefore long-wavelength photons. For a hydrogen atom, the n = 8 to n = 7 jump would give a photon in the far infrared (also not visible). Therefore, the visible-light portion of the spectrum will rarely tell the whole story about a situation. Before this was understood, people had found a way to group the spectral lines of different elements. For example, there is a simple formula that relates the wavelengths of all lines produced when an electron in a hydrogen atom drops from some level to the 1st excited state (n = 2) . This series is called the Balmer series of lines (shown below), and they occur in the visible region. Drops to the ground state (which would be higher in energy, and are therefore in the ultraviolet) make up the Lyman series, and they have the same kind of mathematical relationship. There are other series with names, but as soon as people figured out what was really happening, there was no need to group the lines by this method. These people were like Kepler, in that they could see what was happening, but they didn’t understand why it happened.

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The Balmer series of hydrogen (representing transitions from higher energy levels down to the first excited state). The red line is produced when the electron goes from n=3 to n=2. As you move to the left, the energies are larger so the steps represented are also larger. http://commons.wikimedia.org/wiki/File%3AEmission_spectrum-H.png Ions – what happens if an electron in the ground state receives a photon with an energy larger than E0 ? The electron will be knocked completely free of the atom and the atom is then called an ion. We could then say that High Energy Photon + Atom Ion + Free electron. We can also have the situation go in reverse. The ion would like to capture another electron (or the original one) & go back to being an atom, but for that to happen, the electron will have to get rid of some energy (= fall down the staircase). That means we could get a high-energy photon back out, or we could get several lower-energy photons back out. Breaking the atom into an ion and an electron is called ionization. The process of an electron and an ion getting back together is called recombination. There is typically some ionization and some recombination going on all the time in all gases. The Sun’s Spectrum – From what we know so far, we can look at the Sun’s spectrum and find its temperature (about 6000K at the surface – the surface of the Sun is called the photosphere). At that temperature, solids & liquids can’t exist, so the Sun must be a huge ball of gas. Because we get a blackbody spectrum, we know that it also must be a very dense gas, or we’d just get a few emission lines rather than the full blackbody spectrum. Looking closer, we see that lines are missing from the spectrum – these must represent absorption by a cooler, thinner gas in between the Sun’s surface and the Earth. This is the Sun’s lower atmosphere, and it’s probably about 4500K (cooler than the surface). We can also find out what this gas is made of (mostly hydrogen & helium) by examining the spectral lines. Above this, we have the chromosphere and then above that, the corona. The odd thing about these regions is that the temperature goes up as you move away from the Sun. We can see these regions during a total solar eclipse, when the much brighter continuum spectrum of the photosphere is blocked out, and all that’s left is the emission spectra of the hot, thin gases in these regions. In the corona, temperatures are over 1,000,000 K! We’ll see more about this when we discuss the Sun. Spectra of Nebulae – Nebula are fuzzy clouds of gas (the word actually means “cloud” in Latin) found in interstellar space. They have their own distinct spectra. Emission nebulae are clouds of gas which glow and have a discrete-line emission spectrum. We know that this spectrum requires something to excite the gas, so that the electrons are moved up the energy staircase. This has to happen, because seeing the emission lines means they’re coming back down it, and something had to put them up there. The usual source is a very hot star which is emitting a large number of ultraviolet (and therefore high energy) photons. When these high energy photons strike hydrogen atoms (and they’re the most likely targets), the atoms are ionized and the electrons are free. As we saw earlier, once an ion and electron are formed from an atom, they both are attracted to each other (or to another atom/electron) and some will be able to recombine. The recombination and the electron’s jump (or series of jumps) back down to the ground state produces the emission nebula spectrum.

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Nebulae also show an example of what is called forbidden-line emission. Certain spectral lines are seen in nebulae which are never seen in the lab. At first, astronomers believed this was evidence for a new element, as it was when helium was discovered. They later realized that the lines were a result of electrons making level transitions which don’t happen on the Earth. Although we’ve ignored this point so far, that’s one of the flaws with the staircase model: the ball can take pretty much any pattern back down the stairs, but the electron will only make certain transitions. Observation of this lead to the development of selection rules – in other words, we notice on the Earth that certain excited states of hydrogen (or any element) never make transitions to certain other states on the way back down to the ground state. The reason for this is that ordinarily, when an electron is excited, it will spontaneously jump back down the staircase in a few billionths of a second! For certain states, the electron can sit there for much longer periods of time –a few seconds or more. It’s like a step on the staircase that has a little dip in it that catches the electron. On Earth, one second is practically an eternity on the time scale of atom-atom collisions. For that reason, an electron that finds itself in one of these special states (called metastable states) will not be left alone for a second or two so that it can radiate & produce one of these forbidden lines. Instead, the atom will probably be involved in a collision with another atom (these happen very frequently) and that will take the energy away instead. Even in the best vacuums we can make, there are still LOTS of atoms slamming into each other. Only in the nebulae is the density low enough for the electron to radiate a photon from this metastable state & drop down to a normal state. For that reason, “forbidden” is probably a bad name, but it is much catchier than “never seen in the lab”. Similar techniques have also been used to discover interstellar clouds of gas – if a cloud is between Earth and a star, absorption lines will appear in the spectrum of the star. If the star is a one of the members of a binary system (these systems are very common), we will see a shift in the continuum spectrum as the star moves towards / away from us in its orbit around its companion. The absorption lines would also shift if the absorption was happening in the star’s atmosphere. If the absorption lines stay put while the continuum spectrum shifts back & forth, we know it’s not moving and therefore is gas between the star and the Earth. What about when light acts like a ray? If you’ve used a laser pointer, this is an example of light acting like a ray. It travels in a straight line and doesn’t bend around corners like sound does. When light hits something, it is either reflected, transmitted, or absorbed (or some combination of all three). Additionally, the different colors of light will sometimes behave differently under these conditions: when white light (really a mixture of all colors of light) hits a red object, the red light is strongly reflected and the other colors are mostly absorbed. If white light hits a piece of glass, some of the light will be reflected (that’s why the glass isn’t invisible), some will be transmitted, and some will be absorbed. The light that is transmitted will be refracted, or bent, by passing through the glass. Different colors are bent by different amounts – that’s why a prism breaks white light into the rainbow of colors. See the picture below (from www.howstuffworks.com) for an example of the prism.

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The violet light is bent most, and the red light is bent the least. Why does light bend? Because it travels at a different speed through different materials – fastest in a vacuum, slightly (very slightly) slower in air, significantly slower in glass or water. If the light strikes the surface of the new material at an angle instead of head-on, it will be bent. Look at the picture below & imagine you’re looking at one axle of a wheelchair (with the wheels) from above. How do you make it turn? To turn left, make the left side go slower than the right, and vice versa. In the same way, the light will go slower in water than air, so watch the way the bend

occurs. Notice that is smaller than if the light goes slower in the material where is measured. Also, notice that the light bends towards the normal (the line perpendicular to the surface) in the slower medium. We can reverse the direction of the light rays & have the same picture – light bends away from the normal in the faster medium.

In addition to the light ray that goes into the water (at a smaller angle from the normal because it moves slower in water), there is a ray (shown below) that is reflected from the surface. The angle between the normal and the reflected ray is the same as the angle between the normal and the incoming (or incident) ray.

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We know that it is the slowing down of light that makes it bend – the exact formula for the angle of the bend is this:

2211 SinnSinn

where each n represents the index of refraction of the material. The index of refraction is just the speed of light in a vacuum divided by the speed of light in the material. In other words, if light goes 1/3 as fast in some kind of glass as it does in the vacuum, the index of refraction of the glass is 3. The two angles are the angles (from the normal) of the light ray in each medium. For the picture above, we could write this as

SinnSinn waterair .

Because light moves almost as fast through air as it does through a vacuum, we might just use 1 for nair. Water has an index of refraction of 1.33. That’s enough for us to find the relationship between the two angles if we wanted to, but let’s move on.

Telescopes and Images – Because of the huge distances to the stars, there’s essentially no other way to get information about them other than the use of telescopes to collect the electromagnetic radiation from them. The basic idea of the telescope is to collect more light than your eye would catch without it. This will give you a brighter image. They will also give you a larger image than you would get without any aid. The simplest kind of image-magnifying device is called a pinhole camera. It’s literally just a very small hole (pinhole) in a piece of cardboard (or something else that is not transparent). The light rays from the object you’re interested in go through the pinhole and keep going until they hit a screen somewhere. The rays from above the pinhole will go through and end up near the bottom of the screen. The effect will be the same in every direction, so the image will appear backwards. See below.

This kind of device is commonly used to view solar eclipses, because it’s so easy to make one. All you need to do is poke a pinhole in aluminum foil and let the Sun shine through it onto a flat white surface (piece of paper). You’ll see a small image of the eclipse as it happens, without having to worry about being blinded! Lenses – In astronomy, using lenses as part of the observing equipment is far more common. The idea behind using a lens is shown below – you want to capture more light from the source, which involves catching light rays that would have missed your eye without the lens (and maybe would have landed on your head, shirt, etc.) and bending them down toward your eye.

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In the picture above, the red ray of light will hit the eye, but the blue rays (and the ones that aren’t drawn that would be between the red and blue) will go above or below the eye. We know that going into a medium with a higher index of refraction causes the light to bend, so let’s use that to bring the blue rays back down to your eye. Look at the picture of the prism above and you can see the shape and orientation of glass we need to use. Also, we’re going to have to do something about the travel time of the light – the blue rays have a much longer path than the red one, but we want all the light that leaves at one time to arrive at the same time. There’s no way to speed up the blue rays, so we’ll have to slow down the red ray. We can do that by putting a piece of glass in the way – we don’t want it bent, just slowed down, so we’ll use glass with parallel sides like in a window.

After going through the glass, (if we choose the size & shape right) we’ll get all the rays arriving at the eye at the same time. Notice we need thick glass in the middle and thin at the ends. The exact amount changes at each position, but when we do the math, we get the familiar shape below.

How does light behave when it goes through a lens? It depends on a few things – what kind of glass the lens is made of (index of refraction – how well does the glass bend light?), how far away the object is, and how much the lens is curved. There is an objective way to

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measure the curvature of the lens, but we aren’t going to worry about it. It’s connected to something else that we will spend some time on, and that’s called the focal length of the lens. The focal length for any lens is an unchanging property of it, just like its diameter or its mass – once you know it for a lens, it will not change regardless of how near or far the object is. To measure this focal length, we take parallel light and pass it through the lens. It will come to a focus (for converging lenses like the one above – they’re the ones whose sides poke outward from the center) some distance away, and that distance is the focal length (below, left). If you ever burned anything with a magnifying glass, the glass was one focal length away from what you were burning. Parallel light comes from objects that are very far away; as the object moves closer to the lens, its light becomes less parallel (the light that hits the top of the lens is moving in a different direction than the light that hits the bottom, like above). Because the lens only has so much ability to bend light, it will be able to bend light down to a focus more quickly (closer) if the light is parallel rather than spreading out. For non-parallel light, the image will focus further away from the lens than the focal point (below, right). Once you know the focal length of a lens (found by sending parallel light = light from a very distant source = sunlight through it & measuring distance from lens to focus), you can use it to predict where non-parallel rays will focus. The formula is below:

io ddf

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where f is the focal length, do is the distance from the object to the lens, and di is the distance from the lens to the focused image. For example, if the focal length of a lens is 10 cm, and a source is placed 40 cm away from it, the image will form 13.33 cm away from the lens (on the opposite side of the lens from the object) because 1/40 + 1/13.33 = 1/10. This formula also shows you why parallel rays come together at one focal length from the lens. If the rays are parallel, that’s the same thing (basically) as saying that the source they come from is very far away. To see this, imagine an ant on top of the lens and one on the bottom of the lens. If both ants point towards the source, and it is very far away, they’ll be pointing in the same direction so the “rays” from the ants would be parallel. For a very distant source, do will be a very large number, so 1/ do would be a very small number, or essentially zero. That

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makes the formula just 1/f = 1/di which we could then rewrite as f= di . That means the image (di) forms at one focal length (f ) from the lens. Telescopes – The main use for lenses in astronomy is obviously in telescopes. The basic design is to have a large lens (large diameter, large focal length) called the objective at one end of a light-proof tube and a small lens (small diameter & focal length) at the other end to examine the image formed by the objective. In reality, there are many other lenses in the telescope which try to counteract some effects we have skipped so far. One of them is called chromatic aberration, and it comes about because, as we have discussed, different colors of light are bent by different amounts when passing through the same lens. This means white light gets split up a little, and the focus is slightly spoiled (all colors don’t focus at the same point). Another flaw is a result of spherical aberration. This happens when you use lenses which have spherical face curves. A sphere won’t take every light ray & focus it at the same place – to do this, you need a different shape, called a parabola. A spherical shape is much easier to manufacture, though, and is probably good enough for small telescopes. The two basic kinds of telescopes are called refractors and reflectors. The kind we just described is called a refractor (because the objective is a lens, which refracts light). The other kind, the reflecting telescope, replaces the lens with a mirror which is also curved (unlike a bathroom mirror, which is flat and does not magnify things). The arrangement is also different, of course – the mirror is usually at the back of a telescope rather than the front. You also have to add another mirror somewhere, because we can’t use the mirror to send the light right back out the way it came in – your head would be in the way if you tried to look at the image with an eyepiece. The objective (mirror or lens) should be as large as possible for one simple reason: a larger objective will intercept more light than a smaller one, just as a larger bucket will catch more water in a rainstorm than a smaller one. Since this ability depends on the area of the lens, it is proportional to the radius squared. In other words, a 16 inch telescope, which has a mirror 2 times as big as an 8 inch telescope, has (16/8)2 or 22 = 4 times the light gathering power. The pupil in your eye is only about 5 mm in diameter in the dark – an 8 inch (203 mm) scope has (203/5)2 = 1650 times the light gathering power of your eye! Another important measure of a telescope’s quality is its resolving power. This is a measure of how close together two sources can be and still be seen as two sources rather than one big blob. This will have to be expressed as an angle, of course, since that’s the only thing we really see when we look at the sky. This is limited by diffraction through the objective, so it depends on the size of the objective. We saw earlier that diffraction is more important for small openings than for large ones. The formula for resolving power in degrees is:

)(

)(25.0sec)(

mDiameter

minarcR

where R is the resolving power in degrees (first formula) or in arc-seconds (second formula). A lower value of R is better – the sources can be closer together and still be seen as

independent objects. The wavelength of the light used is , and D is the diameter of the objective. If we apply the formula to your eye (assume 5 mm diameter pupils and 550 nm

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wavelength light = 0.55 m), we get 27.5 seconds of arc. A 10 meter telescope, using the same light, would give us an R 0.01375 seconds of arc. Again, a bigger objective makes a better telescope. Actually, the shimmering of the Earth’s atmosphere makes this theoretical limit unattainable. Although the index of refraction of air is small, it’s not the same as the vacuum, and light does get bent by air (that’s where mirages come from). The amount of bending fluctuates constantly on a very small scale, which is why the stars twinkle when you look at them without a telescope. This is why telescopes are frequently located on mountaintops – there’s less air between the telescope and the star. Typically, the air puts a limit on resolution of somewhere between 0.25 – 2 arc seconds. That’s why the Hubble is such an important telescope, even though its mirror is only 2.4 meters in diameter (that would be big on the Earth, but the largest ones now are 8-10 meters and up) – it’s above all of the air that spoils viewing. Magnification is the last important part of telescope quality. This tells you how much bigger the object appears with the telescope than without it. The formula for this is simple: M = F/f where M is the magnification, F is the focal length of the objective and f is the focal length of the eyepiece. We can’t just use a tiny eyepiece with a short focal length because of diffraction and atmospheric effects will give us a rotten image (although you’ll see huge magnifications advertised on cheap telescopes, they aren’t usable for this reason). Again, we really need a long focal length objective, and focal length goes up with the area of the lens (this is a rough rule-of-thumb – it also depends on other parameters). We’re back to needing a big objective – everything so far (except money) favors larger objectives. Mounting the telescope is an important point, but we’ll just quickly look at two ways to do it. The altitude-azimuth mount lets the telescope swivel up & down and left & right – fine for spotting things on the Earth, but not very helpful in astronomy, since things move in circles around the North Celestial Pole. For this reason, good quality amateur telescopes and essentially all professional ones use equatorial mounting. This way, a clock motor can just work on one axis, allowing you to keep objects centered for long periods of time in the simplest way possible. Reflectors – In a reflecting telescope, the objective is a curved mirror rather than a lens. This has several advantages: 1) No more chromatic aberration – reflection does not depend on wavelength like refraction does 2) There is only one surface to polish (that does not mean “clean”. Polish in this sense means the process of grinding away the tiniest little imperfections on the face of the mirror or lens so that the shape is exactly what you need. The imperfections need to be less than one wavelength of light – the smaller, the better) as opposed to two surfaces per lens in a multi-lens refractor. Less manufacturing effort = lower price 3) Because the telescope will be pointed in a wide range of directions, a large objective will tend to sag as the telescope is pointed to higher altitudes. This sagging changes the curve of the lens. As in the manufacturing process, even a slight change in the curve is important. Reflectors can be supported from all across the back of the mirror, as opposed to being held by the edges like lenses. This means that the largest refractor, which has a 1 meter lens, is much smaller than the largest reflectors (currently about 10 meters – the theoretical limit for this is uncertain, as it improves with improvements in manufacturing.). Also, reflectors are now being made of many smaller mirrors which are mounted on devices

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that can be moved by a computer to keep the image in focus. This should allow even larger telescopes to be built in the future. There are several ways to design a reflector. First, look at the shape of the mirror. While the reflector is not subject to chromatic aberration, a spherical mirror will still produce spherical aberration. This can be solved by grinding the mirror to a parabolic shape (expensive & more difficult) or by introducing a very thin correcting plate at the front of the telescope. This is kind of like a lens, in that it bends the light slightly. The light leaving the plate is bent so that when it hits the spherical mirror, all the rays come to a single point of focus. It therefore corrects the spherical aberration but still allows the use of a cheaper spherical mirror. This is called a Schmidt telescope. The other main types of reflectors are Newtonian, Cassegrain, and prime focus reflectors. These all represent different ways to get light rays from the focal point of the big mirror to an astronomer or a detector. The prime focus method involves placing the detector (or astronomer) inside the tube, actually blocking some of the light which would otherwise hit the mirror. This is only practical when the blockage (astronomer or detector) is small compared to the size of the mirror (so it doesn’t block much light). Future telescopes & detectors – Making a large mirror is a very difficult proposition. The glass has to be very thick for a large mirror, and it takes a long time to make one that big. The glass can spend a year or more cooling when it is that size. Additionally, every night when the dome is opened, the mirror has to cool to the same temperature as the air around it, and it takes a long time for a large mass like that to cool. There is a 240” telescope in Russia which is almost unusable because it takes most of the night to reach operating temperature (the image will shimmer until it does). A better answer, and the way modern telescopes are being made, involves multiple smaller mirrors which are cheaper to make and can be much thinner. They need to be controlled by computer, though, so that the overall shape does not get distorted. This design also lends itself more easily to use in adaptive or active optics, where the overall shape of the mirror is changed slightly in response to atmospheric conditions (which can be monitored using a laser). This requires very rapid adjustments in position, and this field is still very new. A way to get the resolution of a large telescope without the expense involves using two telescopes which feed synchronized images to a central station. The images are compared using interferometry mentioned before. The resolution becomes comparable to that which would be obtained by the use of a single mirror which is as large as the separation between the two mirrors(!). That means two telescopes 100 meters apart could have a combined resolution almost as good as a single telescope with a 100 meter mirror! The light gathering power, of course, will not change – that depends on how much light the mirrors can catch, not the distance between the two. Other changes to astronomy in the last few years/decades involve the movement away from film. Film is much better than the naked eye for viewing objects for a few reasons: 1) a permanent record can be obtained 2) the human eye “washes out” frequently – if you go outside and wait for your eyes to adjust to the darkness, you’ll either see something or you won’t. Staring at the same place all night will not enable you to see things you couldn’t see right away. Film doesn’t work like that – it will accumulate photons all night long, eventually resulting in a picture of something that could not have been seen with the naked eye. 3) the eye has two different kinds of structures involved in vision – rods and cones. The rods are very sensitive to light – possibly even individual photons! The cones are responsible for color vision, but they are far less sensitive to light (about 1000 times less). That’s why most of the

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stars appear to be white dots – you aren’t getting enough light to see the true color. Film will show that color, and as we’ll see later in the semester, the color of a star says a lot about it. We can find precise color information by using colored filters which only allow certain small ranges of wavelengths of light to pass. We let that light strike a photometer which can accurately measure the amount of light striking it. In this way, we can find intensity and wavelength information about a given star over a whole range of wavelengths (called the spectrum of the star). Knowing the spectrum of a star gives us a great deal of other information about it. The modern day replacement for film is called a CCD (charge coupled device). It works like many tiny little electronic buckets to catch light. When a photon hits the detector, it triggers the release of electrons which are amplified to produce an electronic signal. We end up with electronic film (the CCD is what makes modern digital cameras work). There are several advantages to the CCD’s - 1) they are several times more sensitive than film 2) there’s no time or money wasted in film developing 3) the images are already digital so they can be processed on the computer to get more information out of them. Astronomy in other wavelengths – As we’ve seen, the electromagnetic spectrum includes many kinds of radiation that aren’t visible. One important tool in astronomy is the radio telescope. This focuses radio waves much like an optical telescope focuses light waves. One difference is that all radio telescopes are reflectors. They also have to be much larger than optical telescopes, because the amount of radio energy striking the Earth is about 10-18 times smaller than the visible and infrared energy. This is OK, because radio telescopes are much easier to make than optical telescopes. All telescopes generally need to be smooth on a scale which is smaller than a wavelength – for optical light, this means tens of nanometers, which is very difficult to do. For radio telescopes, though, the surface only has to be smooth on a scale of centimeters or so – in fact, the surface can be made of what is essentially chicken wire and it will still “look” solid to the radio waves (that’s why you can look through your microwave oven’s door and see what’s cooking inside – the holes in it are large enough for light to come through, but the metal part of the door seems solid to the 3-cm microwaves inside). So, the advantages of radio astronomy include daytime/cloudy observations (radio waves penetrate clouds, and the Sun’s radio emissions (not very strong anyway) don’t get scattered all over the sky by the atmosphere like its light does) and the low tolerances required, which allow large dishes to be built. Disadvantages include the small amount of radio energy present, and the poor resolution of the dish (go back to our resolution formula – the largest radio telescope on Earth is about 300 meters across. Looking at 10-cm wavelength radio waves gives a resolution of about a minute of arc (that’s not too good). Low resolution pictures mean you get things which look kind of like the low/high pressure maps weather forecasters use – big blobs whose shape you can roughly determine. Interferometry is much easier to do with radio telescopes, though, and they have been combined so that the baseline (distance between telescopes) is about the size of the Earth! That gives resolution better than even optical telescopes! This is called VLBI for Very Long Baseline Interferometry. Radio telescopes and optical telescopes now provide complementary data. Observatory Locations – Optical observatories are generally located on mountaintops for a few reasons. At high altitudes, you’re above most of the atmosphere, which reduces the turbulence. The other requirement for optical astronomy is to be far from city lights. Many

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light fixtures are not designed for the most efficient use. Ideally, all of the light should be directed down. Otherwise, not only is the light wasted, but it tends to wash out the higher-magnitude stars. At the shorter-wavelength end of the spectrum, the telescopes need to be located even higher, which generally means high-altitude plane or balloon flights, or satellites. The lack of an atmosphere in the way improves resolution in the optical range, and allows observations that would be impossible at higher energies (shorter wavelengths).

The Solar System There are a number of regularities among the eight planets of the Solar System. If we closely examine the similarities and differences in the planets and their orbits, we should be able to learn enough to make an intelligent guess about the formation of the Solar System. Among the easier things to notice are 1) the planetary orbits are ellipses, but the eccentricities are very low – even if we were to include Pluto, its orbit is very hard to distinguish (visually) from a circle 2) the orbits are also in nearly the same plane as Earth’s orbit (the ecliptic). Again, the most extreme violation of this would be Pluto, and its orbit is still only at a 17˚ angle to Earth’s. 3) the planets all move around the Sun in the same direction (counterclockwise as seen from far above Earth’s North Pole). Additionally, most of the planets spin in the same direction they orbit. We can also say that the planetary axes are roughly perpendicular to the planetary orbits (again, this is an approximation – the angle is 23.5˚ for us, and about the same for Mars) for most of the planets 4) Most planetary satellites move around their primary (the planet they orbit) in the same direction (CCW) and generally in the plane of the ecliptic 5) the eight planets are roughly ordered by size and composition – close to the Sun, we have small, rocky planets while far from the Sun we have very large, gaseous planets.

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Angular Momentum Problem (A.M. Problem) – Any theory explaining the formation of the Solar System will have to explain the fact that, while the Sun has 99.8% of the Solar System’s mass, it has only about 2% of the angular momentum (in addition to explaining the facts mentioned in the previous section). The Sun has angular momentum because it is rotating on its axis. That’s just another kind of motion in a circle. The Earth also has some angular momentum due to its spin, but compared to its orbital angular momentum (which exists because it moves around the Sun), it’s very small. One of the earliest and most enduring ideas about the formation process includes a collapsing cloud of gas and dust which eventually formed the Solar System. If this cloud had any rotational motion at all (and it would take a strange arrangement of objects for it to not have at least some rotational motion), by the law of conservation of angular momentum, that should not change when the cloud collapses to form the Sun. As the radius of the cloud would have been much larger than that of the Sun, we would expect the Sun’s rotational speed to be very much larger than the cloud’s would have been. This assumption stems from the fact that we can look at the cloud as a collection of individual particles – each one has some angular momentum relative to the center of the cloud. Since most of the particles ended up in the Sun, it’s logical to think most of the angular momentum should end up there as well. We need to think about why it didn’t. Ideas about Solar System formation – A simple idea that could explain the angular momentum problem would be if we said the cloud was rotating only slowly, it collapsed to form the Sun, and the Sun later captured the speedy (therefore high angular-momentum) planets later as it traveled the galaxy. The biggest problem here is that while this would explain the A.M. problem, we then have to explain why all the planetary orbits are in approximately the same plane, why they all move CCW around the Sun, why they are split into two groups of very similar members, etc. There are too many coincidences to explain for this to be a believable theory. The Nebular hypothesis (a simple version of it) instead says that the cloud condensed with some rotational motion, which tended to flatten the cloud into a disk-like shape (this is what would happen physically, so this is a good sign). During the collapse, rings of material might be left behind at larger distances from the center. Particles in these rings would attract each other & gradually assemble into planets (which would all be moving in the same direction – another good sign). Problems with this idea include: 1) it’s not easy to get hot gases to collect anywhere – they have a tendency to spread out rather than contract (hot air balloons work because the air inside spreads out & takes up more space) 2) If the future Sun was spinning so fast that rings of material were being thrown off of it rather than continuing inward with the rest of the Sun, the Sun should still have a much greater percentage of the angular momentum of the Solar System than what we observe 3) This still doesn’t explain small rocky planets near the Sun and large, gaseous planets far away from it. Collision (or Catastrophic) hypothesis – This one says that a passing star managed to pull some material out of the Sun as the two got near one another. This might explain the common direction of the planets’ orbits around the Sun, but there is no explanation for why the material from the Sun would condense into planets, why there are differences in composition between the large & small planets, or even the A.M. problem. Modern Theories of Formation – Current ideas about the Solar System’s formation are based on a modified nebular hypothesis which still has the system forming from a condensing

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cloud, but adds features to help the model match reality. The cloud of gas and dust will begin to collapse under its own gravity, but it doesn’t happen instantly. The particles in the dust cloud (both gas and dust) have potential energy. Potential energy means that something could potentially release energy because of its position. For example, a brick on top of a building has potential energy – the brick isn’t moving, but if it falls from the building, it will hit the ground and release a significant amount of energy on what it hits. The brick on the ground (or what’s left of it) has no potential energy left because it has nowhere else to fall – there’s no lower point the brick can go to, so its potential energy is zero. Immediately before it hit the ground, the brick had a large amount of kinetic energy, or energy of motion. Everything which has mass and is moving has kinetic energy. The amount of kinetic energy the brick had immediately before striking the ground is the same as the amount of potential energy it had at the top of the building. The whole way down, the potential energy was decreasing and the kinetic was increasing in a smooth flow from one to the other. This process is called the conservation of energy. Energy is never created or destroyed – it just moves from one place or form to another. After it hits, that energy will go into destroying the brick and making sound and heat. The gas and dust in the cloud have potential energy because they are being pulled to the center of the cloud. If they get there, their potential energy will be zero and their kinetic energy will have increased. This increase in kinetic energy acts to slow the collapse, however. Rapidly moving particles (high kinetic energy) are more likely to have the speed necessary to escape the cloud (escape velocity) than slower moving ones. Also, since temperature is just a measure of the average speed of the molecules in something, faster gas molecules have a higher temperature. This means (from chemistry class) higher pressure. As the pressure increases, the gas wants to expand, not contract. So, as the cloud collapses and heats up, the collapse is slowed by the heating it caused. If the heat is radiated away (carried off into space by photons), the kinetic energy of the gas will be reduced (as will its temperature and pressure), and the gas will be able to collapse further. It was realized in the 1940’s that this rotating (and flattening) cloud would be turbulent – just like a stream flowing around rocks & trees, there should be little whirlpools & currents moving in all different directions. Where two of these whirlpools (moving in opposite directions) collide, particles may accumulate. Once they start to accumulate, their mass increases and they begin to attract other particles more strongly. The only problem with this is that condensation would probably be so slow that the Solar System would not have had time to form as early as we know it must have. Dust Grains - Dust particles seem to have played a major role in the ability of clumps to form in the early gas cloud. The dust particles (solids that are around 10-5 – 10-6 cm in size) would have been composed of things like Carbon and Silicon and their compounds. The reason is that, while hydrogen and helium made up 98% or so of the mass of the Solar System (and still do), they would not have contributed much to the dust grains. Helium will not combine with other elements (except under extreme conditions which didn’t exist in the early cloud) and it will not even condense to form a solid except near absolute zero while under high pressure (also not likely in the early Solar System). Hydrogen is also very volatile (fast moving & therefore not likely to condense) by itself, but it combines readily with other elements. It may have been present in solid ices of water, methane, and ammonia.

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The dust particles are more likely to stick to each other during collisions, and therefore form larger bodies. Also, the amount of dust increased as the cloud collapsed, radiated heat, and then cooled. As these elements condensed, they would do so around other dust particles like frost on outdoor objects. This would add to the size of the dust particles. Planetary Differentiation – The condensation process above will naturally occur at different temperatures for different elements. Things like iron, silicon, etc. will condense from the gas phase to the liquid or solid phase while still at very high temperatures (over 1000 K = more than about 1300 F), while the more volatile elements don’t condense even at our room temperature (300K), where oxygen, hydrogen, helium, etc. are all still gases. Only far away from the hot center would temperatures get so low that even these elements collect on planets. Even if they don’t condense to the liquid/solid form, colder gases have less kinetic energy than hotter ones (that’s what temperature means, remember) and are therefore easier for a planet to capture. As the planet gathers gases, its mass increases and so does the mass-gathering ability. The large, outer planets are not just hydrogen and helium – the same materials which formed the Earth & inner planets should be contained in the larger planets, but they make up a much smaller percentage of the total mass because so much hydrogen & helium was able to gather there. The dust and gas also tend to separate somewhat for the reasons discussed above – the dust collects near the Sun, the gas moves out to the cooler regions far from the Sun. This theory would also explain the reason that most satellites of planets move in the direction of their primary’s spin. Whatever doesn’t fall into the condensing planet would circle it in a band and possibly collect there to form moons. As the Sun formed, it may or may not have ejected a large amount of matter. If it did, this would tend to push gas away from the Sun and towards the outer parts of the Solar System (providing the material necessary to construct the gas giants). Even if it didn’t undergo this ejection, once it ignited & became a real star, the pressure of its emitted radiation (yes, light can exert a force on things – force over an area = pressure) would have pushed the lighter gases away from the center and towards the forming outer planets. These events would have probably removed whatever early atmospheres the inner planets may have formed by that time. Only the large outer planets, with their greater distance from the Sun and higher masses (= higher gravity) would have been able to hold on to their atmospheres. Tweaks to the theory – Planetary satellites which rotate in the same direction as their primary have a larger tendency to escape from the system than satellites which rotate oppositely. Also, a satellite far from its planet experiences a smaller attractive gravitational force. Therefore, we expect to see moons far from their planet rotating in a direction opposite to the planet’s spin. Finally, if a satellite forms close to its planet, its orbit will tend to be in the same plane as the planet’s equator, while if it is far from the planet, its orbit will be closer to the plane of the surrounding Solar System. Unexplained observations – The theory above doesn’t explain the fact that Venus spins in a direction opposite to its travel around the Sun, nor the odd orientation of the rotational axis of Uranus. Venus may have ended up in its weird rotational state due to tidal interactions with the Sun of the same kind that have locked the Moon’s rotation around the Earth so that it

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always keeps the same face towards us. A collision might explain Uranus’ odd spin, but there’s no real evidence for this. Angular Momentum – Do these theories explain the A.M. problem? Not alone. Somehow the Sun had to get rid of a great deal of angular momentum – it might have gone to the planets, or it might have gone out of the Solar System by some other means. One way it could be transferred to the planets is through the use of a magnetic field. The cloud of gas and dust probably had a significant amount of charged material – atoms tend to ionize in low-pressure situations like this cloud would have involved. Charged particles in motion create magnetic fields, and they respond to them as well. A small magnetic field in the large cloud would intensify as the cloud collapsed, and act to transfer angular momentum to more distant parts of the cloud (where the planets will form). This is a possible explanation, and the current favorite, although by no means the final answer. Best Current Theory? - To choose among the various theories, we can look at some evidence. To understand what will help us choose, we look at different isotopes of hydrogen. Ordinary hydrogen is the simplest atom – one proton, one electron, nothing else. The identity of an element is determined by its number of protons only, so if we form an atom of one proton, one neutron, and one electron, we still have hydrogen, but it’s obviously different from the earlier case. This kind of hydrogen is called deuterium. It’s heavier than the other kind, but because chemistry is almost completely determined by the electrons, it seems the same in almost every other way. That means that it finds its way into water just like ordinary hydrogen. About 1 molecule of water in 6,000 contains an atom of deuterium and one of hydrogen and one of oxygen. Deuterium is quickly destroyed or consumed in stars, so it seems unlikely that, given the amount of deuterium we see around us, it was pulled from the Sun – it wouldn’t last there. To further distinguish between various models, radioactive dating of rocks (here and on the Moon) demonstrate that the Sun formed less than about 100 million years before the planets. On the scale of the Sun’s lifetime, that means the Sun & the planets formed at essentially the same time. The only theories that survive both of these tests are nebular hypotheses of one form or another. Other Planetary Systems – If we assume that the nebular hypotheses (with current modifications) is correct, then planetary systems should be fairly common. In contrast, if we had decided that the collision model was more likely to be correct, we would not expect events like that to be very common throughout the galaxy. If there are other planetary systems, how could we detect them? One way would be to look for the “wobble” in the parent star’s path. This requires us to re-examine a subtle point we’ve ignored so far. The planets don’t actually move around the center of the Sun – instead, each planet AND the Sun move around a common center. It’s just that the location of that center is very close to the center of the Sun. In mathematical terms, if the distance from the Sun to the common center is RSun and the distance from the Earth to the common center is REarth, and the masses of each are MSun and MEarth, we have that MSunRSun = MEarthREarth . So, if Earth is about 150,000,000 km from the common center, and the Sun’s mass is about 333,000 times greater than Earth’s, the Sun’s center should be moving around a point just 450 km from its center towards Earth. This will also be true in any other planetary system (although the numbers will almost certainly be different, of course).

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There are a couple of possible ways to detect planets based on this idea. One way would be to measure the change in the star’s position through time as a result of this wobble. This method requires incredibly precise position data, though, because the stars are very far away and in the case of the Earth & Sun, the wobble is only a few hundred km per year. No planets have been detected this way yet. Another possibility is to measure the Doppler shift in the star’s light as it moves back and forth around the ellipse. The Doppler shift of sound is familiar to anyone who has noticed the change in pitch of a siren as it approaches you & then passes you & moves away. The sound is shifted to a higher frequency when it’s moving towards you and a lower frequency moving away. The amount of the shift depends on the velocity of the source relative to the speed of sound (interstate speed is about 10% of the speed of sound, or Mach 0.1). The same effect occurs with light. A green light (center of the visible spectrum) would seem more bluish (blue = higher frequency than green) when moving towards you, and more reddish (red = lower frequency than green) when moving away. We never notice this on the Earth because the amount of the shift for light depends on the speed of the object relative to the speed of light. Interstate speed is about one ten-millionth the speed of light, so the shift is not detectable for most things. From the math above, you can see that a large planet will have a greater effect on its star than a small one. For that reason, most of the planets that have been detected so far have been much more massive than Earth (typically at least 100 times Earth’s mass). One other important factor is the inclination of the planet’s orbit relative to the line of sight from Earth. If we are looking “down” on the planetary system, there won’t be a detectable change in the star’s color because it’s not moving back & forth towards/away from us. You can think of this by realizing that if you were in the center of a circular racetrack, an ambulance could go around it with the siren on & you’d never hear a change in pitch. It’s obviously moving, but it isn’t getting closer to you or further from you, so there’s no Doppler shift. One other way to look for planets is to carefully watch the light output of the star over a long period of time and look for changes in the light we receive. The idea is that planets passing between us and the distant star will block some of its light & we’ll notice the drop. This is called Transit Photometry since it’s measuring light output during transits (planet moving across the disk of a star). This method will be ineffective if the planet’s orbital plane isn’t parallel to our line of sight. If we go back to the case mentioned above where we look down on the planet’s orbital “North Pole”, we’ll never see it pass in front of the star. The advantage of this method is that is works best for planets close to the star, as opposed to the other methods. So far, hundreds of planets have been discovered orbiting stars outside of our solar system. Earth - The interior of the Earth can’t be explored by any direct means. The deepest hole ever drilled in the Earth (12 km or so) would only be two tenths of a percent of the way to the center. This is not even deep enough to get through the crust. One way the interior can be explored indirectly is to carefully study the way seismic waves (from earthquakes) spread outward from the center of an earthquake. There are two kinds of waves; one is called a P-wave, where the P can stand for Pressure, or Primary (since these are the fastest earthquake waves and therefore arrive first). This wave is very much like a sound wave in the air – rock is compressed and expanded in the direction of the wave’s travel (just like if you pushed a spring back and forth along its length – the wave travels in the direction of the compression).

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The other kind of wave is called an S-wave for shear or for secondary. The rock moves up and down as the wave travels along it (imagine a rope tied to a wall – take the other end and move it up and down rapidly – the wave travels from you towards the wall, but the rope moves up and down only. These are transverse waves, just like light). Because the relative speeds of P and S waves are known, measuring the different arrival times of the two waves will reveal the distance of the source from the measuring station. Another important point to notice is that the P-waves can travel through something without structure (like a liquid) – you prove this every time you make a sound or hear one. Air doesn’t have any structure, but it still allows sound waves to travel. S-waves are different – they won’t travel through liquid. The technical reason is that liquids don’t support shear. The more obvious way to state that is to notice that one region of water has no particular connection to any other region – water doesn’t mind sliding over itself. Solids resist that kind of thing. Observation of these waves (including the fact that some regions won’t receive the S-waves at all) have revealed a solid inner core (probably iron and nickel) covered by a molten liquid outer core. We can also use the P-waves to determine the sizes of these two cores - when a P-wave hits the boundary between the outer core and the mantle, or between the inner and outer cores, its speed must change. That speed changes bends the wave, just as the way light changes speed when it goes from water to air and causes optical illusions like “bent” spoons in water glasses. The outer core’s wave-bending properties keep P-waves away from some parts of the Earth after an earthquake. These regions are called the shadow zone. They aren’t completely calm, though, because the inner core actually bends other P-waves back towards the shadow zone! The outer core is liquid because of the extreme heat – the only thing keeping the inner core solid is that, while the heat is 3000-5000˚ C, the pressure is high enough (the rest of the Earth is sitting on it, after all) that the melting point is even higher. The core is under so much pressure that the iron and nickel in it are compressed to much greater densities than surface deposits of the same metals. The inner core has a density of about 13.3 gm/cm3 and the outer core’s density is about 11 gm/cm3. On top of the liquid core is the mantle, which is broken into different layers and has a density of about 4 gm/cm3. The layer next to the core is called the mesosphere, and it is made of hot rock under high pressure (so it’s stronger). Above it is a lower-pressure (and weaker) region called the asthenosphere. It’s sort of plastic & deformable, and other kinds of rock will be able to float on it. Above this layer is the lithosphere, which is strong and not as hot as the asthenosphere. The top of the lithosphere has the lightest rock, and it is light enough to float on the asthenosphere below. Also, the continental crust is marginally less dense than the oceanic crust (ocean floor). This series of different layers is called differentiation – it’s evidence that the Earth was molten at one time, and the most dense material settled to the center (the region of lowest potential energy). It pushed less dense material out of the way, and the less dense material therefore floats on top of it, like oil on top of water. Plate Tectonics – it has been noticed for some time that the west coast of Africa seems to match up with the east coast of South America like a jigsaw puzzle. Also, there are some interesting similarities between species on the different continents. That led to the idea that all of Earth’s continents were at one time part of a single huge continent called Pangaea, traced back to 200 million years ago. The age is based on the fossil record and the current

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measurable rate of continental drift. The asthenosphere was found to be something on which continents could float, which added support for the idea. Finally, the plates which carry the continents (and the ocean floor) have been observed to slide along, over, and under each other. What provides the force to move the continents from one place to another? The process of convection, which is the large-scale movement of matter as a way to transfer heat from one place to another. When a pot of water boils, it is because so much heat has built up at the bottom that conduction through the water can’t get rid of it fast enough. The water on the bottom gets too hot, expands (lowering its density) and rises (because it’s now less dense). When it reaches the surface and gets rid of the heat, its density increases again and it falls to the bottom of the pan. In this way, a cycle is started which moves heat by moving matter. The continents are riding circular patterns of molten rock across the surface of the Earth. As the continents move, they interact with one another in various ways. If they’re moving away from each other, new crust is formed which pushes them away – this is happening at the mid-ocean ridge. When they move towards each other, one piece can drop beneath another piece and create a subduction zone, returning the crust to deeper parts of the Earth. The collision can also create mountains as the plates buckle at the site of contact. If the plates are sliding past one another, they can sometimes get caught. They will remain still as the pressure builds and eventually, they will break free (earthquake). The famous San Andreas fault in California is an example of this kind of motion. What’s the source of the heat that drives this motion? The radioactive decay process used to date rocks and meteorites. In addition to releasing small particles (He atoms, etc.), a great deal of heat is also released during these decays. We’ll look at that next. Ages of Rocks - The age of a rock can be determined through radioactive dating. This is the same process anthropologists use to date early human settlements, except the anthropologists use different elements. The idea behind this is relatively simple – the atoms of a radioactive element like Uranium-238 (U238) gradually decay into Lead-206 (Pb206) over time. The half-life for this step of the process (time necessary for ½ of some quantity of U238 to decay into Th234, the next step in the chain) is about 4.5 billion years. A few things about this are worth mentioning – first, this idea is unlike the effect of aging on humans. We could talk to statisticians working for insurance companies, and they could tell us, very accurately, when half of the people born in any given year will have died – let’s say 70 years later. We know, without talking to the insurance company, that 70 years after that, all of the remaining people will have died. In the case of radioactive elements, it doesn’t work like that – after one half life, half of the original element will have decayed, and half will remain. After another half life, half of what was left will have decayed, and half will remain (now one quarter of the original amount) and it continues like that. What that means (this is one of the weirdest parts of quantum mechanics!) is that an atom of U238 that is 10 billion years old is just exactly as likely to decay today as is a brand new atom of U238! Definitely not like people! One thing the element does have in common with people – just as we can say what fraction of the element will decay in a given time, and the insurance company can say how many people will be alive after a certain time, when it comes to looking at one atom or one person, we have no idea when the atom will decay, and the insurance people have no idea when one person will die. This is the basic idea of statistics.

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One other interesting thing – notice how much the atomic weight changes during the decay – from 238 to 206! That means 32 neutrons/protons are now gone! Where did they go? They left, at different times, in the form of 8 Helium nuclei which pick up electrons and become ordinary He atoms. This is where all the helium on Earth originates. By looking at concentrations of different elements in a rock, it can be dated – for example, in an easy case, suppose you find a pocket of metal in the asteroid which is 50% Th234 and 50% U238 – it’s probably 4.5 billion years old. This assumes that this pocket of metal was all U238 when it formed – if there was already some thorium in it, that would affect your calculations. For that reasons, scientists generally try to use several radioactive elements to arrive at one common age. The oldest Earth rocks, when dated, tend to have ages which are about 4 billion years. Rock Types – Rocks are classified into 3 families. Most of Earth’s surface rocks are sedimentary, or built up layer by layer. Rocks formed by volcanoes form another family, the igneous rocks. The last group is called metamorphic rock, which is rock that has been modified by high pressure and/or high temperature. Most of the rock under the surface is igneous or metamorphic. The rocks participate in a cycle – they are eroded away and the bits form sedimentary rock. The rocks can be moved by the crust to areas where high temperature/pressure change them into metamorphic rock. If it’s hot enough to melt, it can form igneous rock. The melting point is sensitive to both temperature and pressure – just like if you’re cooking at a high altitude. If you want to boil something at a higher altitude, it will boil at a lower temperature because of the lower pressure, and therefore it will need to be cooked longer at this lower temperature. This melting & re-solidifying tends to reset the radioactive clocks on these rocks (Lead and Uranium can separate, for example). The oldest rock structures found on Earth are about 4 billion years old. Rocks are largely made of combinations of Silicon, Oxygen, and other elements – these are called silicates for that reason. Earth's crust is (numbers are by number of atoms, not mass) 46% Oxygen, 28% Silicon, 8% Aluminum, 6% Iron, 4% Calcium, and about 2% each of Sodium, Magnesium, and Potassium (keep in mind that the crust is only a small part of the volume of the Earth) There is plenty of evidence that the Earth is still changing geologically. The presence of volcanoes, earthquakes, and plate motion show that the surface of the Earth is being affected by the interior of the Earth. The fact that volcanoes occur in a “ring of fire” around the Pacific is not a coincidence – it’s the result of a subduction zone at the plate boundary. Production of volcanoes is also tied to the production of new land – the Hawaiian islands were formed by a similar process. As the Pacific plate they ride on moved over a hot spot, magma (molten rock) was able to burst through the thin crust. Because the plate was moving, a chain of islands formed rather than one large island. Earth’s Magnetic Field – As anyone who’s seen a compass knows, the Earth has its own magnetic field. Magnetic fields can be created by either of two processes: the orderly arrangement of certain elements in a solid allows the electrons which surround the atoms to cooperate and produce a magnetic field. This is how permanent magnets work. The other way you can get a magnetic field is by producing an electrical current. If you’ve ever wrapped a wire around a nail and then connected the ends to a battery to make a magnet, this is why it worked.

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The typical explanation given for Earth’s magnetic field is that it has a liquid iron core (good conductor of electricity) and it is rotating rapidly (stronger field). The only problem with this is that a rotating conductor doesn’t produce a current. You need to have some kind of relative motion between the electrons (negative charges) and the ions (positive charges) if you expect to produce a current. So far, there’s no explanation of how this happens. There are some other reasons to like the idea anyway (slowly rotating objects generally have weak/nonexistent magnetic fields, objects without conducting cores don’t usually have magnetic fields, etc.). Maybe someone will soon figure out how the charges get separated and a current appears. Anyway, Earth’s magnetic field is usually represented by lines which come out of the south pole and re-enter the Earth at the north pole (magnetic north and south - these aren’t in exactly the same place as the geographic north & south poles). These field lines stretch very far into space, and they tend to deflect the solar wind (charged particles streaming out from the sun). When the solar wind is very strong, these particles can be funneled to the parts of the atmosphere near the magnetic poles, and they create a light show called an aurora. Earth’s Atmosphere – The atmosphere of the Earth is made up of about 78% Nitrogen (in the form of N2), 21% Oxygen (O2), 1% Argon (a noble gas) and 0.03% Carbon Dioxide. Water Vapor is also present in the air in varying amounts (less than 2%). The atmosphere can be divided into four layers – close to the ground (and up to about the limit of plane flights) is the troposphere. This is the layer where our weather happens – the circulation of air in this layer is driven by the same convection process (warm air near ground rises, cools, falls) we saw when looking at continental drift. The air moving over large distances at reasonably high speeds (100-300 km/hr or so in the jet stream) is influenced by the Coriolis effect. Temperature and air pressure both drop as altitude increases in this layer – this is why airplanes have to be heated and pressurized. Above that layer (from about 10-50 km) is the stratosphere. This is where the ozone layer is located. Ozone, as you probably know, is a molecule composed of three oxygen atoms (O3) which absorbs ultraviolet (high-energy) light before it hits the ground. This absorbed energy goes into heating the stratosphere, so in this region temperature increases with height. Above the stratosphere is the mesosphere (50-80 km) – temperature starts dropping with height again. Finally, the ionosphere is above the mesosphere and is the top layer. This zone contains lots of ions (as you might expect) which are formed when ultraviolet (UV) radiation rips electrons from atoms. The fact that UV radiation can do that is what makes it so dangerous to life. The combined effect of all these layers is to largely filter out UV and to allow visible light to strike the Earth. When it hits the Earth, it causes the ground to heat up & re-radiate heat back into space. When clouds (water vapor) or carbon dioxide get in the way, the heat gets trapped and the temperature goes up. This is called the greenhouse effect because it works on the same principle as a greenhouse – let the light in to make heat, but don’t let the heat out. You might have noticed this effect if you watch the weather at all – clear nights in the winter are usually very cold because what little heat has been created by light striking the Earth is quickly re-radiated back into space. If clouds move in, it usually keeps the night significantly warmer. The greenhouse effect is not known to be stable – when we look at Venus & Mars later, we’ll see that Venus’ extremely high temperatures and Mars’ extremely low

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temperatures are examples of a runaway greenhouse effect. That’s the reason why there is so much concern about global warming and CO2 emissions from power plants, cars, etc. It’s not just the fact that summers might be a degree warmer on average or that sea level might rise a little due to ice melting in Greenland. The dangerous part is that pushing the system too far accelerates it – the warming gets to a certain point and higher temperatures keep more water vapor in the air. More water vapor leads to more heat trapping which leads to higher temperatures. The atmosphere we have now is probably not the original one. Earth probably had an atmosphere which was mostly hydrogen compounds (methane, ammonia, water vapor), some hydrogen, helium, and neon (a relatively abundant element in the Sun). Where did it all go? In any gas, there will be a distribution of molecular speeds, meaning that at any instant, you will find some molecules moving slowly, some moving more quickly, some moving extremely fast, etc. This distribution (a graph of it is shown on p. 262 in your book) is lopsided. There are more molecules at higher than average speeds than there are at lower than average speeds. The main reason is that there is no simple upper limit to the speed a molecule may have, but zero is a lower limit. The average velocity of the molecules in a gas depends only on their masses and the temperature (which is a measure of the average speed of molecules, if you remember). The formula is shown below:

𝑣𝑚𝑜𝑙𝑒𝑐𝑢𝑙𝑒 = √3 𝑘𝐵 𝑇

𝑚

In this formula, m is the mass of the molecule, T is the temperature of the gas in Kelvin, and kB is Boltzmann’s constant, which is 1.38 x 10-23 J/K. If we use 300 K (the temperature in a warm room) and the mass of a hydrogen molecule H2 (m = 3.34 x 10-27 kg), we get an average velocity of about 1900 m/s. The lighter gases would’ve had such high kinetic energies that the Earth would have been unable to hold them. Collisions between molecules will tend to leave some of the lighter molecules/atoms moving faster afterwards. They escape, and the molecules left behind keep colliding and the cycle continues. The methane and ammonia could have been broken up by the Sun’s energy and the hydrogen then escaped. Nitrogen and carbon would’ve been left, and carbon ends up in carbonate rocks or dissolved in the water. The atmosphere we have today was probably produced by outgassing from the rocks of the Earth. Volcanoes are a source of CO2, H2O, nitrogen, chlorine, etc. The oxygen was not present in large quantities until the plant life on Earth appeared about 2 billion years ago. Bibliography

Discovering Astronomy (Shawl, Robbins & Jefferys) UNIVERSE (Freedman & Kaufmann) 21st Century Astronomy, 4th ed., (Kay, Palen, Smith, and Blumenthal) Astronomy Today, 8th ed., (Chaisson, McMillan)