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A Stellar Astronomy Toolbox In order to understand stars there are important aspects about stars that we would like to know. These aspects we will call stellar parameters. Here are some questions: 1) 2) 3) 4) 5) 6) 7) How big are stars? What are stars made of? How hot are stars? How old are they? How much energy do they produce? How far away are they? How do they move? Here are the stellar parameter names: Radius or Mass Chemical Composition Surface temperature Age Luminosity Distance Velocity Stellar parameters are fundamental aspects of stars. They are determined by analyzing data by either making calculations or through comparison to theoretical models. None of the stellar parameters are directly measureable. They have to be determined using data which we will refer to as observables. Here are some stellar observables: 1) The color of a star 2) The brightness of the star as seen from Earth 3) Characteristics of the stellar spectral lines (This includes the identification, strength, width, and location of spectral lines) Defining stellar parameters and why they are important. Size of the star: There are two ways we will define the size of a star in this class. a) Radius. This refers to the size of the star in kilometers. This is similar to how much space the star takes up (which we call volume). For a spherical star (the kind we will be concerned with) the volume is related to the radius as follow: 4 π = ππ 3 3 b) Mass is another way to characterize the size of a star. Mass is the amount of matter in an object. In our everyday world when an object has a large volume it is normally very massive also. For example an elephant is much bigger than a mouse in mass and in volume. This is not always the case for stars. For stars such as main sequence stars there is a direct relation between the mass and the volume. But there are also evolved stars called giants, supergiants and white dwarfs for which there is not a clear relationship between radius and mass. c) How can objects with a large radius have a small mass, or an object with a small radius have a large mass? The amount of matter contained in an objectβs volume is called its density. Density is defined as mass divided by volume. π·πππ ππ‘π¦ = πππ π ππππ’ππ Knowing how dense a star is tells us a great deal about the age and energy processes occurring in the star. Determining Mass: To determine the mass of a star we need to use Newtonβs laws of motion and law of gravity or else Keplerβs third law. The basic concept is this: How long it takes for an object to complete one orbit around another object depends only on the mass of the two objects and the distance between them. π + π π2 = 4π 2 πΊ π3 Where d is the size of the orbit and P is the orbital period. The orbital period is just the time it takes an object to orbit once. For instance, the Earth orbits the Sun in one year. So the orbital period of the Earth is 1 year or 3.14 x 107 seconds. Finding the distance between the objects is somewhat harder, especially for stars that are very far away. We will find that in some cases we can measure the orbital velocity of an object. If we know the velocity and we know the period we can find d because π£ππππππ‘π¦ = πππ π‘ππππ π‘πππ If time is the orbital period, then distance is just the circumference (C) of the orbit. π£ππππππ‘π¦ = πΆππππ’ππππππππ ππππππ = 2ππ π To compute the mass of stars we use binary stars. These are stars that orbit around each other. The period can be determined as the time it takes to complete one orbit and the velocity can be found using the Doppler shift (something we will discuss). Stellar Brightness and Law of Gravity (Inverse square laws) How bright an object appears to use depends on two variables. One variable is how much energy the object is emitting. The other variable is the distance to the object. Here are two thought experiments. 1. Suppose you have a 100 watt light bulb and a 40 watt light bulb. They are both the same distance away from you. Which will appear brighter? (Note: a watt is a unit of power. It is the amount of energy being emitted each second. Where energy is measured in Joules) Answer: the 100 watt light bulb 2. Suppose you have two 100 watt light bulbs and one is a few feet away and the other is 50 feet away. Which one will appear brighter to you? Answer: the one that is a few feet away. So using these two observables, we can come up with an equation to describe brightness. How bright an object appears is directly proportional to its luminosity. (In astronomy we refer to power as luminosity). In other words, the more luminous an object the brighter it will appear. How bright an object appears is inversely proportional to its distance. In other words the greater the distance the less bright the object will appear. From this we can begin to guess an equation. Let B stand for brightness, L stand for luminosity and D stand for distance. Then our equation must look something like this: π΅ ππππ ππππ πΏ π· There is one problem with our equation. That is how light spreads out when it leaves a source. Imagine a flash bulb that gives one bright flash and then is gone. The light from the flash moves out in all directions and is spread over the surface of a sphere. As the distance from the original flash moves outward the sphere grows in size. The light is spread out over a larger and larger surface area. See image below: So we need to relate the distance from the flash to the surface area of this growing sphere. The surface area of a sphere is ππ’πππππ π΄πππ = 4ππ·2 So the true brightness equation is: π΅ = πΏ/4ππ·2 This is a very important equation in this class. Notice that brightness is something we can readily measure here on Earth using a telescope and a digital camera. So, 1. If we know how far away a star is (D) and we measure its brightness (B) then we can calculate how luminous (L) it is. 2. Or if we know how luminous a star is (L) and we measure its brightness (B) then we can calculate how far away (D) the star is. Newtonβs Law of Gravity. When Newton formulated his law of gravity he assumed that the gravitational force must spread out the same way the light spreads out, over the surface of a sphere. He also realized that the force of gravity must go like the product of the two masses. The law is: πΉ = βπΊππ/π·2 Where D is the distance between two massive objects, M is mass of one object and m is mass of the other. G is the universal gravitational constant and is the same everywhere in the universe. Its value is G = 6.67 x 10-11 N m2/kg2 Both the Brightness law and the Law of Gravity are called inverse square laws, because the brightness (or the force) is inversely proportional to the square of the distance. What is temperature? We all use the word temperature often in our lives but you may not know how it is defined. Temperature is defined in terms of kinetic energy. So before I continue lets talk about what kinetic energy is. Kinetic energy is the energy of motion. Any object which is moving, or in other words, has velocity, also has kinetic energy. The amount of kinetic energy that an object has depends on how fast it is moving. Kinetic energy is directly proportional to the velocity, which means that the faster you move the more kinetic energy you have. To be more precise the amount of kinetic energy actually goes like the square of the velocity. Kinetic energy also depends on how much mass an object has. If you were running alongside a small child you would both have the same velocity, but you are likely to be more massive than the child. That means that you actually have more kinetic energy than the child does. So kinetic energy is directly proportional to the mass of an object. Here is the equation: πΎππππ‘ππ πΈπππππ¦ = 1 ππ£ 2 2 Where m is the mass and v is the velocity. You can see in this equation, that if v = 0 then you have no kinetic energy. Now letβs change the subject just a bit. You are right now surrounded by air molecules. They are mostly Nitrogen and Oxygen molecules with a little carbon dioxide and methane thrown in. If we think back to Galileoβs law, all objects fall at the same rate regardless of their mass, then there should be no reason that air molecules shouldnβt fall to the ground at the same rate that a book would fall. But we know that this is not the case. The atmosphere of the Earth is not lying on the ground. This must mean that the molecules in the atmosphere are moving. In fact, the air molecules around moving very rapidly and constantly colliding with each other. IF they are moving, then they also have kinetic energy. As you might expect, not all of the molecules are moving at the same speed. In fact some are moving very fast and some are moving very slowly. When a fast moving one collides with a more slowly moving one, the slower molecule can pick up speed and the fast molecule can slow down. There are billions of molecules around you that are moving and colliding all the time. This means it is virtually impossible to keep track of the velocity, or kinetic energy, of all the individual particles. Fortunately, we can keep track of the average kinetic energy of a very large group of particles. The plot above shows the distribution of velocity for a bunch of gas molecules. On the vertical axis is the number of particles, n, and on the horizontal axis is their velocities, v. Letβs first just concentrate on the blue line. What blue line shows is that there are virtually no particles with zero velocity. Then the number increases as the velocity increases. The number of particles reaches a peak at around v = 650 m/s. Then it decreases once again. When you get to very high velocities you can see that n becomes small once again. So, there is a very large range of velocities. But it is possible to assign an average velocity to the blue curve. The average velocity is very close to the peak in the curve. So v_ave ~ 650 m/s. knowing this and knowing the mass of the molecules we could determine an average kinetic energy. π΄π£πππππ πΎππππ‘ππ πΈπππππ¦ = 1 π(ππ£πππππ π£ππππππ‘π¦)2 2 Now consider the other two curves. What do you think the average kinetic energy of the red curve and the green curve are compared to the blue curve? Since the average velocity is near the peak, it must be the case that the average kinetic energy of the molecules represented by the red curve is less than the molecules represented by the green curve. And the green curve is less than the blue curve. Now on to temperature. Temperature is directly proportional to the average kinetic energy of a group of particles. The equation is: 3 π΄π£πππππ πππππ‘ππ ππππππ¦ = 2 ππ Where T is the temperature and k is the boltzman constant (k = 1.38 x 10-23 J/K). So what is important here? The higher the average kinetic energy of a group of particles, the higher the temperature. They are directly proportional. Why does a stove burner feel hot? When a stove burner has a high temperature it means that the molecules in the burner have a large average kinetic energy. Or in other words, the molecules are moving around very rapidly. Your hand (on the other hand) is at a much lower temperature. This means that your hand molecules are not moving nearly as fast. When your hand comes in contact with the stove, the molecules in the burned smash violently into the slowly moving molecules in your hand. The results is that your hand molecules begin to move very rapidly and you sense this as something hot. If you leave your hand on the burner for even a couple of seconds, the molecules in your hand will move so violently that they will be torn apart into smaller molecules or even individual atoms. The result is a nasty wound on your hand.