PART 3 - THE PHYSICAL BASIS OF ASTRONOMY - GRAVITY AND LIGHT

Motion and gravity

• We have learned that astronomical objects move around in space. Now we wish to understand why objects move. This is important for two reasons:
  • We will find that all objects move according to the same fundamental laws. The motion of astronomical objects just provides one example of the universal laws of motion.
  • Learning out why objects move will enable us to determine a fundamental characteristic of astronomical objects, namely, their mass

• Before we can talk about the laws of motion, we need to define some terms which we will use to define motion.
  • The motion of an object is specified by stating how fast it is moving, and also in what direction it is moving. The combination of speed and direction is called velocity. In common usage, velocity often means the same thing as speed, but not for us - if an object keeps the same speed but changes the direction in which it is moving, its velocity changes because velocity is the combination of speed and direction.
  • If the velocity of an object is changing (either in speed or direction), the object is said to be accelerating.

• The understanding of motion was pioneered by Isaac Newton (1642-1727), one of the first great theorist in astronomy & physics. He invented calculus to provide the framework in which to understand motions. He was able to explain why Kepler's Laws described planetary motion using his laws of motion and gravity. In addition, these same laws also explained motions of objects on Earth.

• Newton's Laws of Motion:
  • 1st law: Law of Inertia - The velocity of an object will not change unless a force acts upon the object. An object at rest stays at rest unless something pushes it, and an object which is moving will continue to move in the same direction at the same speed unless something pushes it.

  • The 2nd law (force law) describes how the motion of an object will change if a force is applied.
    • If a force is applied, the object accelerates in the direction of the force. Acceleration just means that its velocity changes. The change in velocity can be a change in speed, a change in direction, or a combination of both.

    • The acceleration is larger if the force is stronger

    • For a given force, the acceleration is larger on a less massive object than on a more massive object. Mass just measures how much ``stuff'' there is in an object.

    • Mathematically, the acceleration is given by
      a = F/m
      where m is the mass of the object and a is the acceleration.

    • Whenever an object is observed to accelerate, you know that a force must be acting on it. For example, an apple accelerates when it drops to the ground, which implies that it is feeling a force - gravity.

  • 3rd law: Action/Reaction Law - For every applied force, a force of equal magnitude but opposite direction arises. Newton's 3rd Law has several additional consequences:
    • It is the basis behind the principle of Conservation of Momentum. Momentum (p) is defined as
      p = mv
      where v = velocity. When there are no other forces present, the total momentum is a constant (or conserved). That is, if an object, like a bullet, is shot from a gun (the action), the gun will recoil (reaction) in the opposite direction. The gun recoil is with slower speed than the bullet because its mass is greater.
    • The 3rd Law is the basis behind rocketry. The rocket releases gas molecules (which each have low mass) at high speeds; the rocket reacts by moving the opposite direction. This is true for a launch off the Earth or for maneuvering in space.

• Newton's laws of motion describe how motion is affected by the presence of forces. But where do forces come from? It turns out that we know of only four basic kinds of forces in nature: gravity, the electromagnetic force, the strong force, and the weak force. The strong and weak forces are forces that only operate inside the nucleii of atoms and are not important on large scales. For astronomical objects, gravity is the only force which affects their motion.

• The Law of Gravity
  • Gravity is an example of a force. It is a central force, that is, the force of gravity between two objects (for example, Earth & Moon) acts along a line between the centers of the two objects. When a mass is released in a gravitational field (like an apple near the Earth's surface), it drops in a direction toward the center of a planet.

  • Gravity causes every body which has mass to attract every other body which has mass with a gravitational force. The strength of the force is larger for objects which have more mass. However, the strength of the force also depends on the distance between the objects: objects which are further apart attract each other less strongly that objects which are close together.

    Mathematically, Newton described the force of gravity (F_g) acting between two masses (M₁ & M₂) as follows:

    F_g = GM₁M₂/R²
    where R is the distance separating the two masses and G is the gravitational constant (just a constant number). For example, the force of gravity on an apple would be the above where M₁ is the mass of the apple, M₂ is the mass of the Earth, and R is the radius of the Earth.
    • Note that F_g is directly dependent on the masses of the objects. Doubling the mass of one object will double the force of gravity.
    • Note also that F_g is inversely proportional to the square of the distance between the objects. Doubling the distance between the objects will decrease the force of gravity by a factor of 4.

• Example: gravity is the source of the force holding us down to the ground. It tries to make us accelerate downward, and does so until we hit the ground, when another sort of force holds us up. Gravity is the source of our weight, which is the force we exert on the ground. Consequently, our weight depends on where we are located with respect to the center of the earth, and we would weigh a different amount if we were on another planet. Our weight on other planets depends on two things: the mass of the planet and also the radius (size) of the planet. If a planet has more mass, that would make you weigh more, but if it is larger, then you are farther from the center of the planet, and that would make you weigh less. Note that weight and mass are very different quantities. Mass is a fundamental quantity that is the same everywhere in the Universe. However, weight is a measure of the force of gravity and so it depends on the mass and size of the planet that you are standing on.
  • Exercise: how much does your weight change if you go to another planet? Note this depends both on the mass of the planet (the more massive the planet, the stronger the force of gravity) and on the size of the planet (the larger the planet, the farther away you are from the center, which would make the force of gravity weaker).
  • Moon: mass is 0.01 mass of earth, radius is 0.25 radius of earth
  • Mars: mass is 0.1 mass of earth, radius is 0.5 radius of earth
  • Jupiter: mass is 300 times mass of earth, radius is approx 10 times radius of earth
  • Saturn: mass is 100 times mass of earth, radius is approx 10 times radius of earth

Orbits

• The same force that holds objects to the ground, gravity, keeps objects in orbit around others. This applies to planets orbiting the Sun, moons orbiting planets, and artificial satellites in Earth orbit.

• The difference between an object on Earth and one in orbit is that an orbiting object has an initial transverse velocity. Transverse velocity is the ``sideways'' velocity of an object. The amount of transverse velocity that an object has determines the type of orbit it will make.

• If an object has transverse velocity, the force of gravity causes it to orbit, even though gravity acts to pull objects towards one another. This is because the motion of an object is a combination between its initial velocity and the acceleration caused by gravity. If an object starts off with zero velocity, gravity will pull it directly towards another object. But if an object starts with some transverse velocity, gravity acts to pull it around another object.

• The laws of motion and the law of gravity can be manipulated to derive Kepler's laws of planetary motion. They explain why these laws apply. However, they are far more general: they apply not only to planetary motion, but to motions of other objects as well.

• Consequently, gravity can explain why planets orbit around the Sun if the planets have some transverse velocity. In fact, our ideas about how the solar system formed explain why we would expect that planets do have some initial transverse velocity.
  • We think that the solar system formed out of a large interstellar gas cloud. Such clouds are expected to be spinning very slowly.

  • The formation of the solar system started when one of these clouds started to collapse because of the force of gravity pulling it together.

  • As the cloud collapses, it starts to spin faster because of a principle called the conservation of angular momentum. This is the same reason an ice skater spins faster when he/she pulls her arms inward.

  • The combination of the increased spin with the attractive force of gravity makes the cloud flatten into a disk which is spinning relatively fast.

  • Inside this disk, the Sun forms at the center, and planets form around it. The spinning provides the necessary transverse velocity to keep planets from falling into the Sun. The planets will necessarily be rotating around the Sun all in the same direction and in the same plane - exactly as is observed.

  • Collisions between particles in the protoplanetary disk are likely responsible for making the orbits nearly circular.

• The law of gravity also provides the principle by which rockets go into orbit. To place an object into orbit, a spacecraft must have enough velocity along the orbit. Too little velocity and the spacecraft will fall back to Earth. Too much velocity and the object will escape into space. The exact orbital velocity needed depends on the mass of the planet & the orbital altitude (Law of Gravity equation). When a rocket is launched, it must be given transverse velocity: a rocket is never launched straight up!

• Why do astronauts appear to be ``weightless'' on the Space Shuttle if gravity is present? This is an illusion. The Space Shuttle is really free falling toward the Earth constantly but because of its orbital speed, it misses the edge of the Earth. Any free-falling objects appear to be weightless (person dropped from an airplane in an enclosed box) because both the object and the container are falling at the same rate.

Gravity as a Mass Probe

• We can use our understanding of gravity to measure masses of astronomical objects. The principle is simple: the presence of gravity makes objects fall towards each other or orbit each other, and the observed motion depends on the strength of the gravitational force. Since the gravitional force depends on the mass of the objects and the distance between them, we can measure masses of objects if we can measure how fast they move (or how long it takes for them to go around each other), and how big the orbits are.

• For example, measuring the accelaration an object dropped on the surface of the earth can tell us the mass if the Earth if we know the Earth's radius.

• For orbits, gravity tells us that the speed of an orbiting object depends on the masses of the objects and the distance between them. The speed of the orbiting object can be combined with the size of the orbit to determine the period of the orbit. We can then derive, using our understanding of gravity, a relation between the period of an orbit, its size, and the masses of the two objects. This leads to a generalization of Kepler's 3rd law which can be derived from Newton's laws:
  (M₁+M₂) = A³/P²
  where P is the orbital period (in years), A is the average distance between the bodies (in astronomical units), and M₁ and M₂ are the masses of the bodies (in solar masses). For our solar system, M₁ is the mass of the Sun and M₂ is the mass of a planet which is always much less than that of the Sun; this explains why Kepler's 3rd law works for planetary orbits.

• This version of Kepler's 3rd Law can now be used for any situation in which two objects are orbiting about each other and can be used to determine the total mass of the system (M₁ + M₂). In many cases, one object is much more massive than the other, and the sum of the two masses is essentially identical to the mass of the more massive object. For example, the Galilean satellites orbit about Jupiter. We can measure the orbital period by observing the motion of a Jovian moon over time and we can measure the orbital radius (A) from a photograph. Then, plugging into Newton's version of Kepler's third law above, we can determine the mass of Jupiter (which is much larger than that of any of the moons, M_Jupiter + M_moon $\approx$ M_Jupiter).

• This formulation of Kepler's 3rd law can be applied to many astronomical objects to determine their masses. The basic idea is that astronomical objects move because of gravity; the strength of gravity depends on the masses of objects, so by observing motions of objects and understanding gravity, we infer masses of objects

• Planetary masses are determined by looking at the time it takes satellites (either natural or man-made) to orbit the planets. We find that the masses of planets have a large range. In units of Earth masses, the planets have the following masses: Mercury 0.06, Venus 0.82, Earth 1.00, Mars, 0.11, Jupiter 317.9, Saturn 95.2, Uranus 14.5, Neptune 17.1, Pluto 0.003. One can immediately see that there are at least two generally different types of planets in the solar system, the inner planets and the outer planets (except for Pluto); the inner planets are substantially less massive than the outer planets.

• Masses of stars can also be determined by looking at motions. This is done using binary stars which orbit each other. By measuring the period and the sizes of the orbits, we can infer masses of the stars. We find that not all stars have the same mass. The most massive stars are about 100 times as massive as the Sun, while the least massive stars are about 1/10 the mass of the Sun. So in this respect, the Sun is roughly average. However, there are many more low mass stars than there are high mass stars.

• We can also use our understanding of gravity to infer masses of galaxies. The law of gravity applies to the rotation of stars around the center of spiral galaxies. If most of the mass is concentrated in the centers of these galaxies, similarly to the concentration of light, one would expect stars in the outer regions of galaxies to be revolving with slower velocities.
  • However, we observe that stars in the outer regions of galaxies are rotating with approximately the same velocity as stars in the inner regions.
  • This implies that there is a large amount of mass in the outer regions of galaxies, even though we don't see very much starlight in those regions. The presence of this mass is inferred entirely from gravity. This unseen matter is called dark matter. Something to ponder - the rotation velocities imply that there may be 10 times more dark matter than luminous, so we may be the abnormal matter!

• There is other evidence for dark matter in the Universe. For example, motions of galaxies in galaxy clusters imply that there is more mass in these clusters than can be accounted for in the luminous parts of the galaxies.

• Evidence of dark matter is also provided by another graviational effect, namely, the effect of gravity on light. Although Newton's laws provide a very accurate description of gravity in most cases, they do not provide a perfect physical theory. The best modern theory of gravitation is Einstein's theory of general relativity. For most locations in the Universe, the prediction of Einstein's theory is identical to those of Newton's theory, so it is fair to say that Newton's laws are ``correct'' almost everywhere. However, there are a few effects of gravity which occur in nature which Newton's laws don't describe. In particular, it turns out the the motion of light can be influenced by a strong gravitational force; Newton's laws don't predict this because light does not have any mass (it is a form of pure energy). However, Einstein's theory does predict that the path of light will be altered by a strong gravitational field.
  • We can observe the bending of light in what is known as gravitational lenses. These occur when we observe very distant galaxies which happen to have other nearer galaxies which lie along the same line of sight. Light from the distant galaxy goes out in all directions. Some of the light comes directly to us on a straight path, but we also observe other light which is directed towards us by the gravitational bending of light by the nearer galaxies. This causes us to see multiple images of the same galaxy.
  • Measuring the properties of these gravitational lenses allows us to infer something about the masses of the intervening galaxies. Again, we find significant evidence for a lot more mass in galaxies than we would infer from the observations of bright they are. This is supporting evidence for the existence of dark matter.

• What is the dark matter made of? Currently, we do not know. Some possibilities are:
  • ``Failed'' stars; stars which don't have enough mass to start nuclear reactions, and hence, shine. Planets could be considered to be failed stars. However, it doesn't seem likely that there are enough of these to make up the dark matter.
  • ``Burned-out'' stars, stars which have lived their lives and no longer shine. Also seems unlikely.
  • Black holes.
    • Einstein's theory of gravity, general relativity, allows for a very peculiar type of object called a black hole. A black hole is an object which has such a strong gravitational force at its surface that even light cannot escape from the object.
    • The gravity required to keep light from escaping is immense. To get a black hole, you need to have a lot of mass concentrated into a very small volume, so that the gravitational force at the surface is very large.
    • You cannot see a black hole directly, since by definition no light can escape a black hole. However, we can infer their presence, for example, in binary star systems, by their gravitational effect on surrounding objects.
    • At a distance from a black hole, the black hole behaves no differently that any other massive object; the force of gravity is exceptionally strong at its surface, but not necessarily strong at a distance. Consequently, black holes don't suck other objects into them; objects with transverse velocities can orbit a black hole just like they could orbit, for example, a star or the center of a galaxy.
    • Given our understanding of how black holes are likely to form, however, it does not seem likely that there are enough of them to make up the dark matter.
  • Some other sort of matter than ``normal'' matter (i.e. protons, neutrons, and electrons) which only weakly interact with normal matter and which do not produce light. A strange idea, but currently this is the leading candidate! There are some existing experiments that are trying to detect this stuff....

• Our current understanding of what makes up the bulk of our Universe is very limited! Not only do we have the existence of dark matter, we also believe there is a large component of ``dark energy'', which, as we discussed briefly some time ago, is suggested by observations of the expanding Universe and the microwave background.

Light

• We have considered two basic questions: why do objects move? and what can we learn about objects by studying their motion? We now proceed to ask some other basic questions: Why do things shine? What can we learn about them from studying the light they emit?

• Before we talk about this, we need to consider an underlying question: What is light? Light is a form of energy, in particular, a form of electromagnetic energy. Light is intimately related to the electromagnetic force. The presence of light makes charged particles accelerate, and the acceleration of charged particles makes light.

• How do we characterize light? Light comes in little packets called photons. Each of these can be thought of as a little wave which moves through space. Like any waves, you can characterize light waves with a wavelength, a frequency, a speed, and an energy. Light waves come in all different wavelengths. However, all wavelengths travel at the same speed, the so-called speed of light. In addition, the energy of each photon is also related to the wavelength. Photons with shorter wavelengths have more energy than those with longer wavelengths.

• Light of different wavelengths have different names. The whole range of different kinds of light is called the electromagnetic spectrum. Gamma rays ( 10^-12m) have the shortest wavelengths, followed by X rays( 10^-10m), ultraviolet light(10^-7m), visible light (violet, blue, green, yellow, red), infrared light (10^-4m), microwaves (10^-3m), and radio waves ( 0.1 - 10⁴m). All of these are forms of light.

• How do you describe what kind of light you see from a source? You can consider two things: how bright is an object, and what different wavelengths are being produced by an object. The combination of these two things, namely, measuring the brightess of each different wavelength, is called the spectrum of an object.
  • When your eye sees an object, it sees a combination of all the different wavelengths that the object produces. You can measure the different wavelengths individually by using an instrument called a spectrograph. A spectrograph takes incoming light and sends each different wavelength in a different direction where its intensity can be measured.
  • A prism is a simple type of spectrograph.
  • You get a crude idea of the spectrum of an object by looking at the color of an object: a blue object produces more blue light than other wavelengths, a red object produces more red light than other wavelengths, etc.

• What determines what wavelengths are produced by a light source? Does a light source produce all kinds of light simultaneously, or just particular wavelengths? It depends on the properties of the source. The answers to these questions were formulated in some simple principles in the 1800s by Gustav Kirchoff, who stated that:
  • dense warm substances produce continuous spectra, with some light at all wavelengths. There are many different types of continuous spectra, which we'll discuss in more detail later. Be aware that the light that is produced is not necessarily light that our eyes are senstive too, unless the object is hot!

  • low density, hot gases emit emission lines at distinct wavelengths. There are many different types of emission line spectra.

  • if a continuous source is placed behind a cooler gas, the cooler gas produces absorption lines. There are many different types of absorption line spectra.

  • A schematic representation

• How do these apply to astronomical objects?
  • Planets are dense objects, and produce continuous spectra. However, the bulk of the light from planets which we see is not light which is produced by the planet itself, but instead, it is light from the Sun which is reflected by the planet.

  • The inside of stars are hot, dense objects and thus produce continuous spectra. However, the outer layers of stars are composed of cooler gasses, so the combination of a cooler gas in front of a continuous source make stars produce absorption line spectra.

  • Some of the interstellar matter is composed of hot, low density gasses, so these regions produce emission line spectra. Cool regions in the interstellar matter can produce absorption lines in background objects.

• What can we learn from studying the light which comes from astronomical objects? First, we consider what we can learn by observing object with continuous spectra.
  • All warm dense objects produce light at all wavelengths, but they don't produce an equal amount at all wavelengths. Inside dense objects, there are large numbers of atoms, and photons of light don't travel far before they interact with other atoms. As a result, the energy distribution of the photons is closely connected with the energy distribution of the atoms, i.e. how fast the atoms are moving. The particular kind of continuous emission that warm objects produce is called thermal radiation (also called blackbody radiation), and the spectrum that such objects produce is called a thermal, or blackbody spectrum. In a thermal spectrum, although light comes out over a wide range of wavelengths, there is always a wavelength where the most light is emitted, and all objects emitting thermal radiation have spectra with the same characteristic shape.

  • The wavelength where most of the light comes out depends on the temperature of the source. This happens because the emission of light arises from the motions of charged particles; when particles move faster, they produce light with typically higher energies than when particles move slower. The energy of a photon of light is related to its wavelength; photons of shorter wavelengths carry more energy than photons of longer wavelengths. In addition, the motion of particles is related to the temperature of the object; in hotter objects, particles move faster than in cooler objects. Consequently, hotter objects produce more light at shorter wavelengths, while cooler objects produce more light at longer wavelengths. For objects where the peak wavelength falls around the optical part of the spectrum, like stars, hotter objects are bluer than cooler objects.
    • Astronomers and physicists use a temperature scale called the Kelvin scale which is directly related to the motions of atoms. Temperatures in degrees Kelvin are very similar to temperatures in degrees Celsius: K = C + 273. Degrees Kelvin are directly related to the motions of atoms; at 0 K, atoms don't move at all - this is called absolute zero. In degrees Kelvin, the temperatures of objects on Earth are roughly 300 degrees Kelvin.

  • We can measure colors of stars, hence we can infer temperatures. The sun is about 5700 degrees K, and puts out most of its light at yellow wavelengths. Hotter stars are bluer, cooler stars are redder. Objects at ``room temperature'' put out most of their light at infrared wavelengths; consequently, all objects around us (including us) are glowing, but in light which are eyes cannot see.

  • Light from objects around us on Earth and from planets is a bit complicated because it arises from two sources. All objects are glowing with continuous radiation because they are dense objects. In addition, however, many objects also reflect light from other sources. For example, the planets glow in infrared light, but also reflect visible light from the Sun.

  • Although in principle we can measure temperatures of stars by looking at their colors, there is a complication which makes this difficult. The problem arises because light from a star has to travel through space before it arrives at Earth. If there was nothing between the star and Earth, the light would be unaffected, but sometimes, the light passes through some interstellar matter which lies between the stars. This interstellar matter can affect the color of the star.
    • Light can be affected by the presence of intervening matter. In the simplest example, if you put something in front of a light source (e.g., a brick wall), you can block the light entirely. This occasionally happens in space when there are very dense interstellar clouds between us and a star.
    • More commonly, light has to pass through a region of space which has a relatively sparse number of particles. In this case, most of the light passes through fine, but some of the light is affected. Because of the nature of how light interacts with particles, blue light is usually more affected than red light; more blue light is lost when travelling through an interstellar cloud than red light. As a result, an object which is viewed through an interstellar cloud appears redder than it would have appeared if the cloud had not been there. This complicates the measurement of temperature from an observation of color; a red star may be red because it is cool, or it may be red because its light has passed through an interstellar cloud.
    • This same phenomenon is responsible for the changing color of the Sun from yellow at midday to red at sunrise and sunset. The Sun's light must pass through particles in the Earth's atmosphere before arriving at the Earth's surface. During the middle of the day, the Sun's light passes through a shorter path in the atmosphere than at sunset. Consequently, during the middle of the day, all wavelengths from the Sun except the shortest (bluest) make it through the Earth's atmosphere, and the Sun appears yellow. The blue light which is removed by the Earth's atmosphere is scattered around and can arrive at the Earth's surface from directions other than the direction of the Sun, which makes the sky appear blue. At sunset, the Sun's light must travel through more atmosphere, and only the reddest wavelengths make it to the surface directly, making the Sun appear red.

• Why do objects produced emission and absorption line spectra and what can we learn from these? We shall see that the production of emission and absorption lines depends on the characteristics of individual atoms, and will find that we can measure something about the compositions, temperatures, and motions of stars by studying their spectra.

• In low density objects, atoms are much farther apart, and the physics that explains light emission and absorption is closely related to the structure and energies of individual atoms.

• Atomic structure
  • At the most basic level, all matter is composed of elementary particles, the most common of which are protons, neutrons, and electrons.

  • These particles are found in groups known as atoms. Inside atoms, protons and neutrons are found together in a nucleus, and electrons are found orbiting around the nucleus.

  • The chemical properties of atoms are largely determined by the number of protons in the nucleus. Because of this, the number of protons determines what the element is. For example, hydrogen is the element with one proton in its nucleus, helium has two protons, lithium has 3 protons, carbon has 6 protons, uranium has 92 protons, etc.

  • Atoms normally have about the same number of neutrons as protons, but they can come in different forms with different numbers of neutrons. These different forms of an element are called isotopes.

  • Atoms normally have about the same number of electrons as protons, but they can come in different forms with different numbers of electrons. These different forms of an element are called ions.

  • Atoms can be found linked together into things called molecules.

• How do atoms produce or absorb light?
  • Inside atoms, electrons orbit around the nucleus, as we have discussed. Each electron orbit in an atom has a different energy associated with it. However, we find that electrons do not exist at all possible energy levels. Whenever we study a particular type of atoms, we find that electrons are always found in one of several possible energy levels, but never in between these levels. This is the basis for our current theory of atoms, quantum mechanics.

  • It is possible for electrons to switch from one energy level to another, and in the process, they either emit energy (if they switch to a lower energy orbit) or need to absorb energy (if they switch to a higher energy orbit). The energy that they emit or absorb comes in the form of light.

  • When electrons shift to lower energy orbits, they produce emission lines. If there is a background light source which in shining on atoms, absorption lines will be created at the energies which are used by the electrons to shift to higher energy orbits. Emission line sources are hot, low density gasses; in these gases, electrons are continually being knocked up to higher orbits by collisions with other atoms, after which the electrons fall down to lower orbits and produce emission lines.

  • The incredibly powerful thing about emission and absorption lines is that every different element has different electron orbit energies and consequently has different possible emission or absorption lines. This lets us determine the composition of any gas in which we see lines.

  • We can also use emission and absorption lines to determine the temperature of objects. The temperature of a gas is related to the speed at which atoms move around, and thus related to the strength of collisions between atoms. In hotter gases, collisions are stronger and electrons can be knocked to bigger orbits than in cooler gases. Although each element has its own unique set of spectral fingerprints which correspond to all possible electron transitions, in any object you will only see some subset of these depending which orbits are populated by electrons. Since this depends on the temperature, you can also determine the temperature of an emission or absorption line source by seeing which of the lines are present.

• How does this work for astronomical objects?
  • In the interstellar matter, atoms are heated by nearby stars. The temperature of an object is related to the speed of the atoms in the object, so when we say an object is hotter, that means its atoms are moving around faster. When atoms move around, they can collide with each other. These collisions can knock electrons up to higher energy orbits. Left in the higher energy states, electrons in an atom will always fall down to lower energy orbits; in the process they will emit light of precisely the wavelength that corresponds to the energy change between the two orbits. Consequently, these gas clouds will produce emission lines, with the pattern of lines which are emitted being determined largely by the chemical composition of the gas. However, the lines which are produced also depend on the temperature. If the gas is too cool, atoms won't move around fast enough so that collisions are able to excite the electrons to higher energy orbits. If the gas is too hot, collisions will knock the electrons entirely away from the atoms. At temperatures in between, only a subset of the possible electron orbits will be occupied. Thus, observations of emission lines can be used to determine both compositions and temperatures.

  • In stars, the central regions produce continuous spectra, but as this light passes through the outer layers of the star, certain wavelengths are absorbed by whatever type of atoms happens to be in these layers. The wavelengths which are absorbed are the photons whose energies precisely match the energy difference between electron orbits in the outer layers. Any photon which has an energy which does not match the exact energy differences passes through unaffected. However, a few photons have just the right energy to excite an electron, and these photons are removed from the light passing out of the star. This causes the appearance of absorption lines in stellar spectra; these are wavelengths where light appears to be missing. By studying which wavelengths are absorbed, we can measure what the outer layers are made of. The energy which is absorbed by the electrons is eventually re-emitted when the electrons fall back to the lower orbits, but this energy is re-radiated equally in all directions, and thus we still observe absorption lines. As with emission lines, we can also determine the temperature of absorption lines sources (at the layer where the absorption arises) by seeing what subset of possible lines are present.
    • In fact, in stars, one finds that all stars have very similar compositions. We find that most of the normal matter Universe appears to be made of hydrogen (approximately 90%. Of the remaining 10%, about 9% is helium, and all of the other elements only make of 1% of the total number of atoms. There is some important variations in the abundances, but these variations are small: they range from heavy element (everything except hydrogen and helium) abundances of nearly zero, to about 2-3 percent of the star's mass. Of course, these fractions do not necessarily apply to the dark matter which appears to dominate the mass in the Universe, since we do not at this time know what this dark matter is made of. It is possible that it is not made of protons, neutrons, and electrons at all, but is perhaps made of some other sort of basic particle.

    • However, different stars have very different spectra from each other because of a strong temperature dependence of absorption line strengths. The different types of spectra from stars lead to what astronomers call spectral classification, in which different stars are placed into different groups depending on the appearance of the absorption lines in their spectra. The different spectral types arise almost entirely from the temperature effect. Originally, astronomers didn't know this, and they just invented letter names for different groups of stars. Now we understand that the sequence is a sequence in temperature, and the groups can now be ordered by temperature into the sequence: OBAFGKMLT, where O and B stars are the hottest and M,L, and T stars are the coolest.

    • Observed spectrum of the Sun

    • Observed spectra of stars of different temperatures

• What can we learn from studying the total brightness of objects?
  • Up until now, we've been seeing what we can learn about objects by studying their spectra, or the relative amount of light coming out at different wavelengths. So far, we haven't talked about the total brightness of objects, partly because this is affected by several factors.

  • The total brightness of objects is affected primarily by three different things:
    • Hotter objects are brighter than cooler ones of the same size.

    • Bigger objects are brighter than smaller objects of the same temperature.

    • Objects of the same size and temperature appear brighter when they are closer to you than when they are further away.

  • Consequently, if you can measure the color, the brightness, and the distance to a star, you can infer a temperature and a size for the star. We have learned that you can measure temperatures by studying the spectra, and distances from parallax. Thus it is possible to measure the sizes of stars.

    We find that there is an interesting relation between the size of stars and their temperatures. A very useful tool for understanding stars is a diagram which plots the color against the luminosity, or intrinsic brightness (that is, after correcting for the effects of distance). This diagram is called a color-magnitude diagram, or sometimes, a Hertzsprung-Russell (HR) diagram (for the people who first used it). In such a plot, stars are only found at several locations:

    • Most stars are found along a line in this diagram, known as the main sequence. The hotter stars along this sequence are brighter, even more than expected from their temperature: they are also bigger.

    • Some stars are found which are cool but bright. The only way this can be is if such stars are very large. Consequently these stars are known as red giants. They are red because they are cool, and bright because they are very large.

    • Some stars are found which are hot but faint. These stars are known as white dwarfs.

    • Later we'll discuss why there appear to be these different types of stars.

  • Of course, the simplest way to measure the size of an object is to measure how large an area it appears to take up in the sky. This is called the angular size. If you know the distance to the object, you can easily calculate its true size, since the angular size is determined from the true size and the distance. However, stars are so far away that we cannot detect their angular size, hence inferring the size from the brightness is very important.

• We can also learn something about the motions of astronomical objects by studying the light which they emit.
  • For relatively nearby objects, we can measure the transverse velocity by observing how the star appears to move relative to background stars from night to night. Transverse velocity means the part of the velocity which is perpendicular to our line of sight. For example, if a star is moving directly towards or away from Earth, it will not appear to move with respect to more distant objects. However, even the nearest stars have very tiny observed transverse velocities because they are so far away.

  • We can measure the radial velocity of an object by a pheonomenon called the Doppler shift. The radial velocity is the part of the velocity directed towards or away from us.
    • The Doppler shift arises for objects which emit waves. Since light behaves like a wave, we can observe the Doppler shift in the light which is emitted from astronomical objects.

    • The motion of an object affects the wavelength of the light which it emits because if an object moves towards you, the crests of the emitted waves will be bunched closer together than if the object is standing still and emitting waves. In the same way, if an object is moving away from you, the crests of the emitted waves with be spread further apart. For light, a shorter wavelength (crests bunched closer together) means bluer light. So if an object moves towards you, its light appears slightly bluer, and if it moves away from you, its light appears slightly redder.

    • This wavelength shift of light is usually quite small. For example, it changes red light into slightly redder or less red light; it rarely significantly affects the color of an object. However, for objects with emission or absorption lines, it can clearly be detected because a characteristic pattern of lines from some element will be noticed to be shifted a small amount from the position where the lines are observed in the element when it is at rest. This small shift can be used to measure the radial velocity of astronomical objects.

    • The Doppler shift applies to any sort of wave-like phenomenon. Since sound is also a wave, the Doppler shift affects sound as well. This is the reason why a car or train horn appears to change pitch as it passes from coming towards you to going away from you.