Basically, these notes are frozen as of the afternoon of Sept 22. There are several missing sections for which there are placeholders. These missing sections will be on a new link so that you can print this one out once and for all. These notes are not intended to be a stand-alone set of words. They are not intended to take the place of the Tuesday/Thursday classes. We highly recommend that you read a real textbook if the combination of the classwork, homework, notes are not enough. One such real textbook is the one we have on reserve (Impey). There are many many "Astro 100" texts in the main library. Classes of Sept 9,11,16,18 In this set of notes we'll summarize the rest of physics of light emitting objects, physics of how atoms emit light, the H-R diagram, inferences from the H-R diagram, age of the Earth... I. More on Laws of Light a) Wien's Law tells us that the color of a solid or dense gas (I always use the phrase "hot iron rod" but it can be anything) depends on its temperature. As we've said before, this is really important because it gives us a way to derive temperatures of astrophysical objects. We all know what "color" means in everyday life. It means the same thing in astronomy, but we need to ask how we actually measure this color. A red object gives off more red light than blue light, for instance. So, if you put a red filter in front of a digital camera and take a picture, and then put a blue filter in front of your camera and take a picture, you'll quantify the fact that more red light is received. Similarly, for a blue star, the blue picture would contain more light than the red picture. So astronomers can measure colors very straightforwardly. We'll remind you about the invese-square law of light below. For now, it should be easy to imagine that if you move a star farther away, BOTH the red light and the blue light will decrease, but the COLOR won't change. b) Stefan-Boltzman Law Solids of different temperature have another important difference besides their color. Take two iron balls of exactly the same size, heat them to two different temperatures. The hotter ball will emit more light at all colors! If you add up all of the light from the hotter ball, and compare that light output to the light from the cooler ball, you'll discover that Light output (luminosity) is proportional to T**4 (remember, this is a shorthand for "T to the fourth power"). So L propotional to T**4 Furthermore, it's easy to imagine that if you have two hot steel balls of the same temperature, that the bigger (in radius) one will emit more light. It has more surface area from which to emit! So L propotional to R**2 Combining these two ideas we get L proportional to (R**2)(T**4) So you can make an object brighter either by making it bigger or heating it up. This idea will give us great power once we've introduced the H-R diagram! c) Inverse-square law First, let's introduce the square law of pizzas. How many more pepperoni pieces does a 16" pizza have than an 8" pizza? That's right, 4 times as many! That's because a 16" pizza has 4 times the area as an 8" pizza. Now, on to inverse-square laws. We see these a lot in astronomy. Let's simplify the problem, for the sake of argument. It works just the same in the general case... Put a flash bulb in the center of a hollow ball. POOF! Light's emitted in a burst. The light is really bright an inch from the burst, but is really faint far away. Why? Well, imagine that this amount of light is paint, and it needs to coat the inside of the ball. Make the ball bigger, you have the same amount of paint, but a bigger surface to paint. So the paint would have to be thinner. In fact, it goes as the surface area of the ball, which goes as R**2. So the intensity of light goes down as 1/(R**2), by the inverse-square. The farther you are away from a car headlight, the fainter is appears to your eye. So L proportional to 1/R**2 We note in passing that gravity is a 1/R**2 force, too, for much the same reason. Whatever is doing the pulling has to cover more area when an object is farther away. d) Perfect-Gas Law In order to discuss the structure of the Sun, we need to know a little bit about how gases behave. Most of what you need to know can be learned from balloons and car tires. You know that when your car tire is flat, you add air to make it have the right pressure. So the pressure has something to do with the number of particles in the volume you're talking about. You know that if you try to squeeze a balloon to a tiny size, that it resists mightily. So apprently the pressure is related to the volume; if you squeeze a bunch of air into a small volume, the pressure will be higher. If you heat up a car tire by driving down the road (friction), the pressure goes up. If you cool a balloon with liquid nitrogen, the pressure goes way down and the balloon collapses. Take away the liquid nitrogren, the balloon heats up and reinflates. We can write this mathematicall as PV=nkT P=pressure V=volume n= number of particles (density of particles) k="physics" T=temperature e) Balls of gas under the influence of gravity (stars and atmospheres of planets) Let's do a thought experiment about the atmosphere of Earth. the atmosphere is neither falling on the floor nor escaping into space. It's in equilibrium. What does that mean??? Well, whatever forces, like gravity, that are trying to pull the atmosphere down onto the floor are being balanced by the random motions of the gas particles trying to make the gas bigger and bigger. So, let's imagine a column of Earth's atmosphere extending from the floor of the lecture hall to outer space. Gravity pulls down, gas pressure holds it up. Well, this column of gas weighs something! So a basketball-sized hunk of this column at the bottom has to hold up a lot of weight, the entire column of air above it. A basketball-sized hunk of air near outer space hardly has to hold anything up. How can a gas hold up a weight? How does it do it in your car tire? Well, if you put enough molecules of air into your tire, it'll unflatten and hold up the weight of the car. So, a gas can hold itself up against gravity by pressure. Using PV=nkT we see P is proprtional to n and T (all basketballs are the same volume). So the bottom of the atmosphere is hotter and denser than the top! The center of a star is hotter and denser than its outside! So, at least in principle, we have a way to derive the temperature structure of a star. A one-solar-mass star has a central temperature of around 10 million K. Note, this central temperature just comes from equilbrium and the properties of gases, NOT FROM ANYTHING ELSE. Since a more massive star has more material to hold up, its central temperature is going to be hotter than the central temperature of a less massive star. Again, this fact comes from equilibrium and the gas laws. f) Radioactivity Some atomic nuclei are unstable against decay. When they decay, they break into two or more smaller nuclei, thus changing from element xx to element yy. This decay is a statistical process. In class, we described it as similar to buying life insurance: actuaries set the rate knowing the statistical properties of an ensemble of people, they have no useful predictive power over what's going to happen to an individual. WE HIGHLY RECOMMEND STUDYING THE DISCUSSIONS OF RADIOACTIVITY THAT WE'VE PLACED ON THE WEBSITE. By knowing that half of a sample of radioactive particles decays in one half life, and by knowing a little bit of geology, we can use this decay to calculate the age of a rock. SEE THE FIGURES IN THE BENNETT DISCUSSION ON THE WEBSITE (Sept 9 figures). We discover that the oldest objects in the solar system are moon rocks and meteorites, which are pretty uniformly 4.5 billion years old. The oldest rocks on Earth are younger than that, but we realize that EROSION has destroyed all of the oldest rocks, so the Earth is not a good place to look for oldest things in the solar system. Jumping a little ahead of our story, since the planets and Sun formed together, we now know the current age of the Sun. Using everything we know about solar system formation, the age of the Sun is 4.5 billion years plus a little bit = 4.6 billion years. So, at some point, we'll have to explain how the Sun has shined for 4.6 billion years ("billion years" is often written "Gyr", which stands for Giga Years") g) Properties of atomic nuclei, the forces of nature SOON h) Quantum mechanics and the emission of light SOON i) Laws of gravity SOON II. The H-R diagram and inferences from it Don't forget, there are lots of figures on the website in the figures area. a) An example of an "H-R-diagram-like" thing Imagine that you're a scientist from the planet Zorgon. You land on Earth and decide, in a non-invasive way (you don't get to cut people apart), to figure out what makes people the way they are. You take a few prisoners, say this class of students, and try to deduce which properties of people control their structure. So you plot hair color against weight. You discover that there are no trends in such a plot. There are heavy people with blonde hair, and light people with blonde hair. There are heavy people with blue hair and light people with blue hair. If the entire plot is filled, you're not on the right track. Finally, you stumble across height versus weight. Almost everyone in the room falls in a very narrow band of height versus weight. Even without knowing about bones and DNA, you've now learned something important about people. And now you can measure the height of a person and deduce their weight without weighing them, because you've calibrated this diagram using the few people in this room. b) The H-R diagram At about the time of World War I, Hertzsprung and Russell made a similar diagram for the properties of stars. If you plot luminosity against temperature, 90% of all stars fall on a well-defined line. We call this line the MAIN SEQUENCE. Just as above, knowing the temperature of a main-sequence star, you can deduce its luminosity. So without measuring the distance to a random star (I had to measure a few distances to set up the H-R diagram), I can now tell you its luminosity. That's a pretty powerful trick. But wait, there's more! Let's think about how the first H-R diagram was set up. You need temperatures. Well, one way that astro 203 students could measure temperatures is, as above, by using Wien's Law. You need luminosities. Astro 203 students can convert apparent brightnesses into luminosities by using parallax to measure the distances to the stars in question. So we've used a little simple physics to turn apparent quantities into intrinsic quantities. Well, we know some other tricks that will make the H-R diagram even more useful. We can use the laws of gravity to measure the masses of stars in binary star systems. Then we can label the H-R diagram with masses. (and we can learn, as we argued above, that mass and central temperature are related). So indeed, the main sequence is a mass sequence. More massive main sequence stars are hotter and more luminous. See the figures. If we know the temperature and the luminosity, we can use L proportional to R**2 times T**4 to derive R, the radius of the star. So now we know sizes! Again, see the figures. As long as we can somehow know that a star is a main-sequence star, and if we know its temperature, we can derive its luminosity, mass, and size. That's quite a height versus weight diagram! c) Lifetimes of stars Knowing the lifetime of the Sun of 10**10 years, which comes from its current age of 0.46x10**10 years coupled with computer models, we can learn how long stars of other masses will live, EVEN WITHOUT UNDERSTANDING WHAT KEEPS STARS SHINING FOR SUCH LONG TIMES. The way we make this calculation is actually well known to you. It is identical to how one calculates the miles between fillups of cars. All you need to know is the size of the fuel tank and the mileage (miles per gallan). So if a volkswagon beetle goes 300 miles between fillups (10 gallon gas tank and 30 miles per gallon)... and if you know that a Corvette has twice as big a fuel tank and gets 1/3 the miles per gallon... Distance between fillups = volkswagon distance TIMES relative size of fuel tank TIMES relative fuel efficiency distance_corvette= 300 TIMES 2 (bigger tank) TIMES 1/3 (crummier mileage)= 200 miles. Even though the corvette has a bigger fuel tank, it cannot go as far on one tank of fuel as can a volkswagon beetle. For stars, lifetime = (lifetime of Sun) TIMES mass in solar masses DIVIDED BY luminosity in solar luminosities As an example, a 100 solar mass star shines at 1 million solar luminosities (see the H-R diagram figure) lifetime of this star= 10**10 years (lifetime of Sun) TIMES 100 (mass of fuel compared to Sun) DIVIDED BY 10**6 (luminosity in solar units) = 10**6 years. So a 100 solar mass star only has enough fuel to last 1 million years. Putting it another way, any such star you see was BORN YESTERDAY. Putting it another way, if as one such star died you formed another, you'd have to have 4600 such stars back to back to make it the current age of the Sun, and 10,000 back to back to make it the lifetime of the Sun. Massive stars were all born yesterday, they don't live long enough to have been born a long time ago.So, to a decent approximation, all massive stars "live but a day or so" and are "but a day or so old." So, still, without knowing how stars work, just by knowing that the Sun has a finite lifetime, we come to the inescapable conclusion that star formation is going on as we speak, since we see massive main sequence stars. d) "Red Star, Blue Star" Now, again without knowing how stars really work, we can explore the 10% of stars that aren't on the main sequence. There is a small set of cool, luminous stars. Our job is to understand why they're called RED GIANTS. There is a set of relatively hot, very low luminosity stars. Our job is to see why they're called WHITE DWARFS (yes, they're "DWARFS" and not "DWARVES"). By looking at an H-R diagram, we see that there are main-sequence stars that are the same temperature as red giants. But the red giant might be 10,000 times more luminous than the main sequence star. Go back up to the top of this set of notes. Two hot candmium balls of the same temperature can be different luminosities if they're different sizes. So the red giant is a different size than the same-temperature main-sequence star. Since L proprtional to R**2, this hypothetical giant is the square root of 10,000, or ONE HUNDRED, times as big. So this star would be the size of the Earth's orbit, not the size of the sun! A white dwarf of the same temperature as the Sun is around 1/100,000 th as bright. That makes it 300 times smaller than the sun (300 times 300 is 100,000). that's about the size of the Earth, not the size of the Sun. e) General properties of stars So these objects we lump under the name "stars" seem to have a wide range of properties. They range in luminosity from 1 million solar luminosities to about 10**-6 solar luminosites, a range of a factor of a trillion! They range from 100 solar masses to 1/15th of a solar mass, a range of over a factor of 1000. They range in radius from about 1/1000th of a solar radius to about 1000 solar radii, a factor of a million. Why do we call all of these objects by the same name? We'll get there... f) Nuclear Fusion in the cores of stars Around 150 years ago, it was calculated that the luminosity of the Sun could be made by burning 1500 pounds of coal per hour per square foot of the Sun's surface. Unfortunately, spectroscopy has showed us that the Sun is not made of coal! Even if the sun WERE coal, we could only keep the Sun shining for about 100,000 years this way, assuming I did my math correctly. So we're only off by a factor of around 100,000 in the needed energy. So the huge age of the Sun means that ordinary means of making energy don't work. Luckily for us, Nature didn't leave us hanging. In around 1930 all of the physics was available to understand the structure of stars and to understand how to make vastly more energy than is available from simple chemical reactions. So in 2003, say, the way the Sun makes its energy seems very natural and right. And, to top it off, IT'S THE RIGHT ANSWER. We'll be talking more about this process this week. In the meantime, let's remind ourselves about the structure of the Sun, and about the properties of atomic nuclei. The sun is massive, one solar mass, 2x10**33 grams or 2x10**27 tonnes (a metric tonne is almost identical to an english system ton). From spectroscopy, it is ROUGHLY a 5000K gas on its surface. Again from spectroscopy, it is roughly 75% Hydrogen, 25% Helium, and 2% everything else (yup, that adds up to 100% to within roundoff error). It's the same composition in its middle, which has a temperature of about 10 million K. One of the other laws of Nature is that heat flows from a hot place to a cool place. Since the center of the Sun is really really hot, and the outside is only 5700K, heat is flowing from the inside to the outside. This means that the structure of the Sun has to be constantly changing, because gases change as they gain or lose energy. THE FACT THAT THE SUN GIVES OFF LIGHT MEANS THAT IT MUST CHANGE, EVOLVE. more on new page...