ASTR 1210 (O'CONNELL) SUPPLEMENTS


There are three special supplements which are required reading for this class. Supplement I (a PDF file) was linked to Lecture 1 and contains reference material on units of measurement, powers of ten, nomenclature, and special symbols. It should be consulted as necessary throughout the remainder of the course.

Supplements II and III are intended to replace the material in the textbook covering the basic properties of matter, the nature of forces, the electromagnetic spectrum, and how astronomers exploit the properties of the EM spectrum to deduce the physical nature of distant objects. They also serve as the Study Guides for the lectures where this material is covered. Most of the concepts presented in II and III should already be familiar to you. These supplements are available in this HTML file.


ASTRONOMY 1210: SUPPLEMENT II


A. Elementary Particles

You should already be familiar with three elementary particles: the electron, the proton, and the neutron. These are the primary constituents of atoms. Atoms were first proposed as the basic components of matter by Greek philosophers (e.g. Democritus, ca. 400 BC), and their reality was debated for thousands of years. Their existence was proven beyond doubt in the 19th century.

Protons and neutrons are bound together in the nucleus of the atom, an object about 10-13 centimeters in diameter. The electrons orbit the nucleus in a cloud which is 100,000 times larger, 10-8 cm in diameter. There is a multitude of other elementary particles, most of which exist only for very short periods of time; most of these will not be discussed further in the course.

Two properties of elementary particles are relevant for us: their mass and charge.

Roughly speaking, mass is the "amount of matter" in an object. Quantitatively, it is the inertial mass and measures the "resistance" of an object to an applied force (as expressed in Newton's Second Law). The larger the mass, the harder it is to move the object. Mass is measured in grams:

Charge is a property of particles which determines their behavior when acted on by electromagnetic forces . "EM" forces are described in more detail in section B.

All elementary particles which have been isolated are found to have only certain definite amounts of charge: none, exactly one positive unit of charge, or exactly one negative unit. (Protons and neutrons are constructed of even more basic subunits called "quarks", which have fractional charges. But these are strongly bound together inside protons and neutrons. Free quarks should rarely exist and have never been observed in the laboratory.)

Since the nucleus of an atom contains only protons and neutrons, it will have a positive charge. The basic property that distinguishes the various elements from one another is the number of protons in their nuclei or, equivalently, the amount of positive charge on the nucleus.

There are 92 naturally occurring elements on the earth, ranging from hydrogen (with only one proton in its nucleus and no neutrons) to uranium (with 92 protons and 146 neutrons). [Other elements, e.g. plutonium, have been artificially produced, but all of these are unstable and have finite lifetimes.] All materials, despite their myriad appearances, are made of these 92 types of atoms. In fact, most everyday materials, including living things, are made of no more than about 20 types of atoms.

In their undisturbed or "neutral" state, atoms have as many electrons as protons and therefore have a net charge of zero. Under many circumstances, however, electrons can be stripped out of the cloud which surrounds the nucleus. The atom is then said to be ionized and is referred to as a "positive ion" since it has a net positive charge.

The last elementary particle we will discuss is the photon, which is not a "regular" particle at all. The photon has zero mass and zero charge. Instead, the photon is a packet of electromagnetic energy. The concept of the photon is a product of the quantum theory, which describes the behavior of elementary particles. The photon is the "particle" which transmits the electromagnetic force between charged particles. Light waves consist of photons, as do radio waves, and the discrete, particle-like impact of photons on special detectors can actually be readily demonstrated in the laboratory. Photons are discussed further in Supplement III.

B. Forces

If particles did not exert forces on each other, there would be no organization in the universe: no planets, no stars, no people. Particles would move throughout space in random but unchanging paths, never altering their motion even when they passed through one another.

However, particles do exert forces on each other, and the nature of these forces depends on the mass, charge (and other "internal" properties we will not discuss), and state of motion of the particle.

Isaac Newton was the first to formulate a quantitative description of how matter responds to forces which are applied to it. Historically, Newton's laws were a major stimulus for both the scientific revolution and the industrial revolution since they demonstrated that the universe is understandable through rational analysis. His description is encapsulated in his second law of motion :

where F is the force applied to an object, m is the object's inertial mass, and a is the object's acceleration.

Newton's formulation is now known not to apply on microscopic scales, such as to sub-atomic particles. On macroscopic scales, however, it is an excellent approximation and is the basis, for example, of structural, mechanical, and aeronautical engineering.

Acceleration is the rate of change of velocity. An object is said to accelerate if either its speed or direction of motion changes. It is the only one of the three parameters in the second law (force, mass and acceleration) which can be readily observed. So, it is useful to solve the equation for acceleration:

This expression implies that for a given mass, the larger the force the larger the acceleration. Also, for a given force, the larger the mass the smaller the acceleration. A boxcar is more massive than your little brother and is therefore harder to push around.

As the rate of change of a quantity (velocity), acceleration is the mathematical derivative of velocity (which is itself the derivative of position). Also, the force may depend on position, velocity, and other properties of the system. Thus, this expression is actually not a simple algebraic expression but instead is a differential equation. In order to solve it, Newton was forced to develop calculus, one of the most powerful and valuable intellectual tools ever invented.

Newton recognized only two kinds of force: (i) gravity, and (ii) the force produced by physical contact (e.g. you pull on something, a spring pushes something, the wind shifts something, etc.). We now realize that all varieties of this "physical contact" force are manifestations of electromagnetic force. But this was not understood until over 150 years after Newton published his Second Law. Fortunately, it was not necessary to know how objects produced the forces they exerted on one another in order to predict how they would move in response to those forces.

There are three categories of forces exerted by particles on each other that are now recognized: gravity, nuclear forces, and electromagnetic forces. Forces can be characterized by their strength, their range, and their coupling (that is, which properties of particles determine the response to a particular force).

Gravity

The gravitational force is an attractive force between any two objects containing mass. All particles in the universe attract each other; any pair of particles which start from rest will eventually move together (in the absence of other nearby particles). Gravity seems strong and inexorable in everyday life, but in fact it is by far the weakest of the known forces. Gravity falls off with distance, but is a long-range force compared to the nuclear forces (e.g. it can act over intergalactic distances).

Newton's theory was shown to be inadequate by Albert Einstein (1916) in the presence of large masses or over large distances, and has been replaced by the General Theory of Relativity in such situations. Relativity theory profoundly changed our understanding of space and time. But as a practical matter, Newton's theory is an entirely satisfactory description of "everyday" gravity. Only very minor corrections to the Newtonian predictions are necessary, for example, to send spacecraft with high accuracy throughout the solar system.

Nuclear Force

This is a short range force, acting over distances of only 10-13 cm (i.e. about the size of an atomic nucleus). It has no direct influence over significantly larger distances. (The longer range effects of nuclear materials, e.g. in radiation therapy, are produced by particles or photons ejected from atomic nuclei by the nuclear forces, not the forces themselves.) Because this is the case, the effects of nuclear force are described by the quantum theory and not Newton's Second Law. Technically, nuclear force is subdivided into the strong and weak forces (and hence you will often find references to the "four forces" of nature, rather than the three described here).

Electromagnetic Force

The ubiquity of the effects of electromagnetic (EM) force make it the most important in everyday life. It is, of course, responsible for electricity and magnetism. But it is also the source of combustion, solidity, friction, capillary action, etc. In short, it holds our everyday world together. EM force is responsible for all chemical reactions, including those occurring in biological organisms.

Atoms contain a nucleus with positive charges and a surrounding electron cloud of negative charges. It is the EM forces between these charged particles that hold atoms together. Michael Faraday's (ca. 1830) investigations of the electrical properties of materials provided the first clues to this fundamental discovery, and incidentally made it possible to utilize electrical power as the basic tool of modern civilization (see Study Guide 9). Nuclear forces do not play a major role on the scales corresponding to the sizes of atoms; they are important only inside the nucleus.



ASTRONOMY 1210: SUPPLEMENT III

A. Electromagnetic Waves

Based on the experimental studies of Faraday and others, especially Faraday's discovery of electromagnetic induction and his concept of a magnetic force field, James Clerk Maxwell managed to summarize (ca. 1865) all electromagnetic phenomena in a set of elegant and interconnected equations now known as Maxwell's equations. These represent the third great generalized physical law, the first two being Newton's equations of motion and his theory of gravity.

Maxwell was able to show that his equations implied that electric and magnetic force fields can propagate through space. A moving electron, for example, sends out a disturbance which can cause other electrons to move; "target" electrons at successively greater distances will feel the disturbance at successively later times. The propagating disturbance is similar to a water wave produced by a rock dropped into a pond and is therefore called an electromagnetic wave.

One of Maxwell's great discoveries was that he could use his equations to predict the velocity of an EM wave as it propagated through a vacuum. He found the velocity to be the same as the speed of light, which had already been determined with fair precision. The immediate inference was that light is one form of electromagnetic wave.

Maxwell's theoretical prediction of EM waves was later confirmed experimentally by Heinrich Hertz (1887), who detected, for the first time, artificially generated radio waves. Only a few years later in 1895, the physicist Wilhelm Röntgen discovered high energy EM waves in the form of X-rays and simultaneously recognized their tremendous penetrating power and usefulness as medical diagnostics. Maxwell's prediction of EM waves has had profound practical consequences.

Wave disturbances

EM waves are classified by their wavelength or frequency. By analogy to water waves, the wavelength of an EM wave is the distance between crests (or places where the EM effects are strongest). See the figure above. The frequency is the number of crests passing the observer per second. If the waves move with a definite velocity, then it is easy to see that wavelength and frequency must be inversely related to each other. The longer the wavelength, the lower the frequency; and the shorter the wavelength, the higher the frequency. Wavelength is measured in whatever unit of length is most convenient: from centimeters or millimeters at long wavelengths to Ångstroms (where 1 Å = 10-8 cm) for optical light. Frequency is measured in cycles ("crests") per second or Hertz (1 Hz = 1 cps).

All frequencies from zero to infinity are possible for EM waves, and this total range is called the EM spectrum. See the figure below; the textbook presents other illustrations and more information. Radio, television, microwaves, infrared, visible light, ultraviolet light, X-rays, and gamma rays are all forms of EM waves. All of these waves travel at the same speed (at least in a vacuum): the speed of light, which is 300,000 kilometers per second (186,000 miles per second).

Full EM spectrum with visible spectrum enlarged
(Units marked are microns. 1 micron = 10,000 Å)

The human eye is directly sensitive only to a very small range of wavelengths in the EM spectrum. This is called the "visible" or "optical" region. It extends roughly from wavelengths of 4000 Å in the deep violet to 7000 Å in the deep red. Green light has a wavelength around 5000 Å, or about 0.0005 mm.

The wavelengths or visible light are far smaller than sizes encountered in everyday life. Therefore, we are not conscious of light's wavelike character. However, radio waves are long enough that wave effects such as diffraction are obvious: the fact that radio waves can reach you even if the transmitter is hidden behind a building is a result of the waves diffracting around the building, just as water waves can move around a rock near the beach. (For more on diffraction, see
Study Guide 14.)

Although our eyes are not sensitive to most EM wavelengths, sensors in our skin can detect infrared EM waves (10,000-100,000 Å), which we perceive as radiant heat. Shorter wavelengths in the ultraviolet and X-ray cannot be sensed directly by our bodies; but these can cause damage to cells. For instance, damage resulting from exposure to ultraviolet light appears as "sunburn". Fortunately, most of this harmful cosmic X-ray and UV is screened out by the Earth's atmosphere and does not reach sea level.

In fact, the Earth's atmosphere is opaque to most wavelengths in the EM spectrum. The chart below shows the ability of different wavelengths to penetrate the atmosphere. The screening is protective for lifeforms on Earth's surface. But obviously, it is not convenient for astronomers who want to monitor the universe across the full EM spectrum. (This is the main motivation for space astronomy.) The atmosphere also absorbs large parts of the Earth's infrared spectrum, which leads to the atmospheric heating phenomenon known as the "greenhouse effect" (see Study Guides 15 and 19.)

Penetration depth in Earth's atmosphere
for different kinds of EM waves
Notes:


B. Photons

Although EM radiation does have many wavelike properties, it is not simply a wavelike disturbance. It is absorbed or emitted from particles in tiny packets, called photons , which themselves behave more like particles than waves. The energy of a photon depends on its frequency (in the wave analogy): the higher the frequency, the larger the photon's energy.

Photons are created or destroyed by atoms through interactions with the charged particles within the atom. According to the quantum theory, the internal energy states of an atom are discrete or quantized. This means that the electron cloud around the nucleus of the atom can exist only in certain definite energy states. Whenever the cloud changes its energy state, a certain definite amount of energy must be lost (if the energy drops) or gained (if the energy increases). One of the most important ways in which atoms lose or gain energy is through photons. The electron cloud can spontaneously lose energy by emitting a photon, which then escapes from the atom. Alternatively, it can absorb a photon which is near the atom, thereby gaining energy (and destroying the photon).


C. Spectroscopy

As a consequence of sponteneous photon creation by atoms, all objects which are above a temperature of absolute zero emit electromagnetic waves---even textbooks and ice cubes. There is a great deal of information about the physical properties of the object contained in the distribution of wave energy as a function of wavelength or frequency in its electromagnetic spectrum. For short, this distribution is called the object's spectrum. "Spectroscopy" is the study of EM spectra.

The spectrum of a dense object like the Earth or Lake Superior or the inner regions of the Sun is smooth or "continuous" and changes only slowly with wavelength. You can think of the atoms and electrons in such an object as interacting so strongly that the individual signatures of each type of atom are blended away. Such an object is said to be in thermal equilibrium, and the resulting spectrum is sensitive only to its temperature. This kind of spectrum is called a blackbody spectrum, named after ideal, laboratory sources which are purely self-luminous and reflect no light. Such sources have a spectrum with a smooth, well-defined, characteristic shape, as shown in the figure below:


This diagram shows how temperature affects the distribution of the emitted blackbody spectrum of a dense object.
(The wavelength scale is in nanometers. 500 nm on this scale corresponds to 5000 Å or 0.5 microns.)

As shown here, hotter objects emit more energy at shorter wavelengths while cooler objects emit more energy at longer wavelengths. Hence, hot stars will look blue-white to the eye while cool stars look orange-red. The spectrum of the Sun, with a surface temperature of almost 6000o Kelvin peaks at about 5000 Å --- i.e. in yellow-green light.

On the other hand, the Earth, with an overall mean surface temperature of only 287o Kelvin (14o C), peaks at a wavelength of 100,000 Å (10 microns) in the infrared part of the spectrum. Since our eyes are not sensitive at this wavelength, the Earth looks "dark". Objects at room temperature (about 300o Kelvin) also emit their peak radiation in the infrared. They are self-luminous at those wavelengths, but the light we see from them with our eyes is not their own self-radiation but rather reflected light from higher temperature sources (such as the Sun or incandescent light bulbs). For instance, see this image comparing the visible and "thermal-infrared" appearance of a cat.

The total radiative output from a blackbody source (summed over all wavelengths) is a very strong function of its temperature and scales as the temperature (on the Kelvin scale) to the fourth power (T4).

So far, we have discussed dense systems. By contrast, in a dilute or thin gas---the outer atmosphere of the Sun or an interstellar gas cloud, for example---the atoms do not interact strongly, and if there is a large enough number of atoms of a given type (e.g. hydrogen or oxygen), they will impress their individual signatures on the emergent spectrum.

EmSp Types

The important practical implication stems from the fact that the spectrum of each type of atom is different because its electronic structure is different. A hydrogen atom has a different spectrum than a helium atom, and carbon atoms have a yet different spectrum. Obviously, then, the emission or absorption line spectrum of a star or planet is related to its chemical composition.

Spectroscopy is therefore an exceedingly powerful tool. It is the source of most of the physical information we have about astronomical objects which are too distant to sample directly with spacecraft (i.e. most of the universe).


D. Luminosity and Flux

The luminosity of an object is defined to be the total amount of electromagnetic energy radiated by the object in a given time. It may refer to the whole EM spectrum, in which case it is denoted a "bolometric" luminosity, or it may refer only to a limited part of the spectrum. For example, the "visual" luminosity of a star refers to the total energy output in the visible part of the spectrum. It is denoted LV.

Luminosity is an intrinsic property of the object; that is, it does not depend on how far away the observer is. "Intrinsic brightness" or "power" are synonyms for luminosity. In the cgs system, luminosity is measured in ergs per second.

The response produced by an object of given luminosity in a detector (for example the human eye) depends both on the luminosity and the distance of the object. More distant objects produce smaller responses. The response of a detector to a given source of EM radiation is a function of the flux of radiation from the object. The flux is defined to be the energy crossing a given area in a given time. It is measured in ergs per second per square cm (ergs/sec cm2) in the cgs system. Flux is not an intrinsic property of the source because it depends on the observer's distance. The greater the flux, the greater the response of a given instrument. Flux can be defined with respect to part or all of the EM spectrum.

A little thought will lead you to realize that flux = (total energy output of source per unit time)/(area of sphere passing through observer with center on source). Or,

where F is the flux from the source, L is its luminosity, and R is its distance from the observer. Like the expression for Newtonian gravitational force, this is an "inverse square law".


E. The Doppler Effect

The Doppler Effect is a change in the apparent wavelength or frequency of an EM wave caused by the motion of the source of the wave toward or away from the observer. It also occurs with sound waves, where it is most familiar as the change in pitch of a police or ambulance siren as it passes you on the street. When the siren is approaching you, the pitch is higher (i.e. the frequency is higher and the wavelength is shorter) than when it is receding from you (lower frequency, longer wavelength).

The Effect is explained by the fact that if a source is approaching you while emitting a wave of fixed wavelength, each successive crest of the wave starts from a position a bit closer to you than the preceding one. But the crests still travel at the same velocity. This means that the distance between successive crests (the wavelength) traveling in your direction will be smaller than if the source were stationary. The stream of crests moving in your direction is therefore more "compressed" than for a source at rest. You will observe a signal with shorter wavelength (higher frequency) than that from the same source at rest. Conversely, if the source moves away from you, the stream of crests is "stretched", and you will observe a longer wavelength (lower frequency). "Doppler radar" makes use of this phenomenon to measure the motion of air streams in the atmosphere or of automobiles on the highway (often in the hands of the state police).

Applied to astronomical spectra, the Doppler Effect is particularly valuable, since it allows us to measure the motion of distant objects toward us or away from us, what astronomers call the "radial velocity." By measuring the exact wavelengths of spectral emission or absorption lines and comparing them to the expected values for atoms which are at rest, we can estimate the motion of objects along our line of sight (i.e. radially). The method does not depend on the distance of the object. Since the change in wavelength is toward the blue end of the spectrum if the object is approaching and toward the red end if it is receding, astronomers speak of "blue shifts" and "red shifts." The cosmic red shift, observed in the spectra of distant galaxies, is the principal evidence that the universe is expanding.

The animation below shows how the Doppler Effect can be used to detect the motion of objects in orbit around one another. This is the technique used to discover the existence of exoplanets in orbit around other stars. Hundreds of exoplanetary systems have been discovered with this method to date.

DopplerEffect

Effect of Doppler shift on spectral lines



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Last modified February 2014 by rwo

Text copyright © 1978-2014 Robert W. O'Connell. All rights reserved. Illustrations of blackbody radiation, the 3 types of spectra, and the solar spectrum copyright © 2000 Harcourt, Inc. Doppler animation from Dick McCray, University of Colorado. These notes are intended for the private, noncommercial use of students enrolled in Astronomy 1210 at the University of Virginia.