The theory of the discrete nature of light began a conceptual revolution in twentieth-century physics and astrophysics. It was used by Niels Bohr to formulate a new model for the atom that can be used to understand how light is created and destroyed inside atoms in distance stars. Using what they know about the properties of electromagnetic radiation, the atom's structure, the interaction between matter and energy, and spectrum analysis, astronomers can study the Sun and stars by means of the radiation they emit. Moreover, as we develop even greater understanding of the nature of radiation and its interaction with matter, we can explore more deeply the dim sources of radiation--galaxies and clusters of galaxies--in the outer reaches of the cosmos, almost back to the beginning of time.
One of the most perplexing problems for early twentieth-century physicists was why atoms emit a discrete pattern of spectral lines, that is to say, photons with only selected wavelengths. To understand why this was a problem, let us back up a few years to Maxwell's electromagnetic radiation theory which was advanced in the latter half of the nineteenth century. Maxwell's concept of a field provided a framework whereby electricity, magnetism, light, and even gravity could be seen theoretically as related ideas through "action-at-a-distance" and energy. His concept of light as being an electromagnetic wave composed of moving electric and magnetic fields which carry energy and that such waves are emitted by fluctuations in electric currents was a bold unification of seemingly unrelated phenomena.
Experimental confirmation was provided by the German physicist Henrich Hertz (1857-1894), who undertook in 1887 to show that oscillating electric currents send out electromagnetic waves possessing all the properties of light except visibility. Although Hertz's electromagnetic waves were microwaves, his experiment was overwhelmingly successful in convincing those who doubted Maxwell's theory that it was correct. And in so doing, he confirmed that electromagnetic waves are generated by accelerating electric charges. Consequently, there developed a theory for the emission of light as being due to the oscillatory motion of electric charges located in the atoms of the radiating source.
This theory had many early successes and from it was sought an
explanation of the various colors, photons of different energies,
emitted by radiating sources. The discovery by Thompson in 1897
of the electron as a negatively charged constituent of atoms seemed
to provide the needed accelerated charge for producing electromagnetic
waves. If the electron moved in a circular orbit inside the atom,
for example, then in Newtonian mechanics that is accelerated motion
since the direction of motion is constantly changing. Hence the
accelerating electron should emitted electromagnetic waves. But,
since electromagnetic waves carry away energy, this means that
the electron should lose energy and consequently spiral in an
ever tighter circle about the nucleus. This tightening spiral
motion should cause the emission of waves with a continuous range
of wavelengths, resulting in the electron spiraling ever closer
to the nucleus until its energy is gone and it is pulled into
the nucleus. Rutherford was to later say that, "I was perfectly
aware when I put forward the theory of the nuclear atom that according
to classical theory the electron ought to fall into the nucleus...."
This theory seemingly accounted for continuous spectra, but did
nothing to account for the discrete nature of emission spectra.
Also in this model, why did the atom sometimes emit electromagnetic
waves and not at other times? The answer to the dilemma lay in
the concept of the photon by Planck and Einstein, which Bohr was
to incorporate with Rutherford's nuclear atom (see Sections 4.2
and 5.1).
12.1.2. Electron Orbits and Energy Levels
In 1913, by which time the structure of the atom was reasonably well understood, Bohr proposed a theory for the orbital structure of the electron in hydrogen. He envisioned a hydrogen atom as being like, in some respects, a miniature planetary system extending about a tiny nucleus composed of the one proton. Orbiting the nucleus is the single electron, like a tiny planet moving in roughly a circular orbit held by the electric force of attraction between the proton and electron rather than Newton's gravity. However, and most importantly, unlike the gravitationally bound Solar System, Bohr made the following hypotheses concerning the behavior of an electron in the hydrogen atom:
In Bohr's theory, orbits representing higher levels of energy are increasingly farther from the nucleus. And, it is only when an electron changes from one of its permitted orbits, which is higher in energy to another of lower energy, that a photon of electromagnetic energy is emitted. In addition, Bohr indicated that the electron normally resides in the lowest-energy orbit, which is the one closest to the nucleus. The diameter of this first orbit corresponds to the normal size of the hydrogen atom, about 10-8 cm in diameter (Figure 5.1).
Think of the permitted orbits in hydrogen as being analogous to the steps of a ladder with the lowest-energy orbit being the ground. An electron in a higher-energy orbit, like a rubber ball on one of the steps of the ladder, is only partially stable. If the ladder is bumped, the ball bounces from one step to another down the ladder until it reaches the ground. In like manner, the electron resides in a higher-energy orbit only temporarily before finding its way back to the lowest-energy orbit or the ground state. In the ground state, the electron is stable indefinitely, as is the rubber ball when it is lying on the ground. But if the electron normally exists in the ground state, how does it gain the energy necessary to exist in one its higher-energy orbits?
When a hydrogen atom absorbs energy, it is said to be excited, and the single electron in the atom appears in one of the outer orbits, which have successively higher energies than the lowest orbit (ground state). The electron's change (up or down) from one permitted orbit to another is called an electron transition. A hydrogen atom in a gas may acquire the internal energy that excites its electron by
As representative of these three processes, let us consider the excitation of hydrogen by photon absorption. Of all the photons encountering an atom, only those possessing an amount of energy equal to the energy difference between a higher-energy orbit and the one in which the electron is located will be absorbed with the photon's energy being used to excite the atom. For example for hydrogen, it takes 10.2 electron volts (eV), or 1.63 x 10E-11 erg, of energy to raise an electron from the ground state to the next higher-energy level. Photons with energies below 10.2 eV can not be absorbed, and consequently the electron can not be excited. Photons with energies in excess of 10.2 eV cannot raise the electron to the second energy level, but they may, if they have the right amount of energy, excite the electron to even higher-energy levels.
How long does an excited atom remain that way? For an excited hydrogen atom as in Figure 12.2, in about a hundred-millionth of a second it rids itself of any energy in excess of that of the lowest-energy orbit by emitting the excess energy as one or more photons. Like the rubber ball bouncing down the steps of the ladder, the electron drops in succession into one or maybe several lower-energy orbits on its way to the ground state. In each energy level in which the electron appears, other than the ground state where it can reside indefinitely, the electron remains again about 10E-8 s. With each downward transition, a photon of electromagnetic radiation is emitted in which the photon contains the energy difference between the two orbits involved in the transition. The greater the energy difference, the greater is the amount of resident energy in the photon, and consequently, the shorter is the photon's wavelength.
In addition to the model of the atom that represents it by its electron orbits, we can make a model using the energy of each allowed electron orbit. Such an energy-level model for hydrogen appears in Figure 12.3. The number of energy levels corresponds in a one-to-one fashion with the number of electron orbits. The distance between successive electron orbits increases with higher orbit numbers, but the differences in energy between successive levels grows smaller as the orbit numbers increase. A comparison of Figures 12.2 and 12.3 illustrates this point.
Suppose a hydrogen atom is excited such that its electron is in the third energy level. The electron remains there about 10E-8 s and then may de-excite directly to the ground state, emitting one photon. The photon would have a wavelength of 1026 A, which corresponds to the energy difference between these two energy levels. Alternatively, the electron may de-excite to the second energy level, emitting a photon with a wavelength of 6562 A and then de-excite about 10E-8 s later from the second energy level to the ground state, emitting a photon with a wavelength of 1216 A. The total energy emitted in both cases is the same, but the wavelengths of the photons that result and hence the spectral lines differ.
The hydrogen-line spectrum in the visible region, known as the Balmer series, is prominent in the absorption spectra of most stars. It arises from electron transitions originating on the second energy level of hydrogen. In the same way, all possible transitions from the ground state are known as the Lyman series, which is in the ultraviolet part of the electromagnetic spectrum. Those transitions from the third level up to higher energy levels constitute the Paschen series, which is in the infrared region and so on for the remaining series, whose lines appear in the far infrared on out to the microwave region (see Table 12.1).
For hydrogen, each series of spectral lines comes to a limit toward shorter wavelengths. The uppermost energy levels, representing the electron's highest energy orbits, crowd together toward a series limit, which represents the point beyond which the proton can no longer bind the electron. Once an electron has an energy beyond the series limit, it leaves an atom, that is, the atom becomes ionized. An ionized hydrogen atom cannot absorb or reradiate energy in the form of discrete lines until it captures another electron. Protons capture electrons because of their electrical force of attraction between themselves being positively charged particles and the negatively charged electron. Figure 12.4 shows the Balmer series in the absorption spectrum of an actual star. Note that it converges toward its series limit at approximately 3646 A.
12.2.3. Spectra of Other Elements
In the Bohr atom, besides limits on the size of electron orbits, there is a limit to the number of electrons that may occupy a given orbit. These allowed orbits with a prescribed number of electrons in them are called electron shells (Figure 5.1).
In general, as one goes through the periodic table, electrons are added to balance the number of protons in the nucleus by filling the shells from the one closest to the nucleus outward. In hydrogen there is one electron in the innermost shell, which has room for a maximum of two electrons. Helium's two electrons fill, or close, the shell, so that for the element lithium the third electron must start a new shell, which is the next innermost. In the second shell there is room for only 8 electrons; in the third, 18 electrons; in the fourth, 32; and so on.
Considering the 92 naturally occurring elements, we find that there are 92 distinct configurations of electron orbits, that is, each element has a unique set of energy levels. Consequently, the wavelength of the spectral lines originating from electron transitions between various energy levels is also unique for each element--a clear fingerprint of the element (Figure 12.5).
The amount of energy needed to ionize an atom varies from one element to the next depending on the number and "position" of the electrons. For example, to remove the outermost electron from helium takes five times as much energy as it does to do the same for sodium. Also, for a given element each additional ionization takes more energy to free an electron from an inner orbit than from an outer one because the inner one is more tightly bound to the nucleus. Thus, with carbon, for example, it takes more than twice as much energy to remove the second electron than it does to remove the first electron, 4 times more for the third electron than for the first; almost 6 times more for the fourth electron, 35 times more for the fifth electron, and 44 times more for the innermost sixth electron.
Multiple ionization of carbon brings a corresponding readjustment
of the energy levels because of the altered electrical attraction
between the positive nucleus and the reduced number of electrons.
Altering the energy corresponding to each allowed orbit produces
different spectral lines with each succeeding ionization of the
carbon atom. Consequently, we see not only a different set of
wavelengths in absorption or emission spectra for different unionized
elements, but also for the same element after each ionization.
That is, the spectrum of singly ionized carbon differs from that
of neutral carbon, the spectrum of doubly ionized carbon differs
from that of singly ionized carbon, that of triply ionized carbon
differs from that of doubly ionized carbon, and so on.
12.2.4. Bohr's Conceptual Scheme and Reality
Typical reactions by nonscientists to Bohr's conceptual scheme is to wonder whether it is about reality, since his theory is quite remote from their everyday experience. For a moment, let us consider Bohr's theory as a process in science. The photon concept by Planck and Einstein was developed earlier on the basis of phenomena other than the emission of discrete wavelengths by atoms. What Bohr did was to combine Einstein's photon concept and that of Rutherford's nuclear atom in a bolder conceptual scheme to show that the internal behavior and structure of atoms can account for the observed emission (or absorption) spectra of atoms. Although additions have been made to Bohr's theory over time to broaden it into a mechanics of the atom called quantum mechanics, the basic concepts are still those of Bohr.
There exists a vast collection of data on emission and absorption spectra of atoms, ions, and molecules. Although Bohr's theory may provide a framework for understanding all of this data, why does one say that Bohr's theory "explains" this data? Let us remind ourselves that as discussed in Chapter 1, explanations are achieved in science when scientists are able to fit new physical phenomena into the body of science using a mixture of existing concepts and a minimum of new ones, all of which are consistent with their preconceptions about how nature operates. If the process of accounting for these new phenomena includes mathematical laws and principles, as does Bohr's theory, then all the better for we have a sound basis for making predictions about new but similar phenomena. In that sense, Bohr's theory is an explanation of nature.
Fine, but is Bohr's theory about reality? Does the "real" world really look like Bohr's miniature planetary system? That we even question its reality, is because Bohr's theory is based on a model that is unlike any mechanism with which we have had immediate experience. Note that we did not question the reality of Newton's gravitational model of the Solar System, yet with the exception of falling bodies we have no immediate experience with one body orbiting about another body because of something called gravity. Yes, we can build mechanical models, strings, gears, etc., of orbiting bodies, but we can not do laboratory experiments in which gravity is solely responsible for orbital motion. Why then is Newton's theory anymore about reality than is Bohr's theory? The answer is that the business of science is to make the unfamiliar familiar by bringing the unfamiliar into the structure of explained experience by an extension of existing conceptual schemes or the assertion of new conceptual schemes. After so doing, the human mind with time and repeated exposure makes the unfamiliar familiar and accepts the unfamiliar as "real" to the extent that "real" has any meaning in science (read Section 1.3.5 again). Newton's theory is familiar to us and is therefore about reality, while Bohr's theory is unfamiliar and whether or not it is about reality depends on your experiences with it. Given sufficient exposure to Bohr's theory, in time it will in your mind, as in the minds of physical scientists, be seen to capture reality.
At the same time physicists were working on the structure of
the atom and how atoms produce discrete emission spectra, work
had been underway for sometime on continuous spectra. Continuous
spectra were instrumental in the development of the photon concept
as mentioned in Section 4.2. From those considerations came fundamental
tools for astronomers in their pursuit of understanding of the
nature of stars.
12.3.1. Blackbodies
All material objects radiate and absorb electromagnetic radiation; the wavelength region and the amount of energy depends generally on the object's temperature and physical state. Varying the temperature in laboratory experiments and from theory, physicists in the nineteenth century analyzed how various bodies emit and absorb radiation. From this work they developed the concept of an idealized radiator called a blackbody.
Blackbody Concept: A blackbody is an imaginary body that, when cool, absorbs all wavelengths of radiant energy falling on its surface so that it is black in color; when hot, the blackbody emits energy with 100 percent efficiency. (Real matter is generally less than 100 percent efficient when it radiates.)
At room temperature, lampblack (a finely powdered black soot) is very close to being a blackbody because it absorbs almost all the radiation incident upon it and reflects very little.
For our purposes, the most important feature of blackbodies is the way in which their emitted radiant energy is spread out in wavelength, or the spectral energy distribution. Physicists have found that the distribution of energy depends only on a blackbody's temperature and not on its chemical composition or any other aspect of it. Figure 12.6 illustrates this point; note how the amount of radiant energy emitted by a blackbody varies with wavelength in a very recognizable way, even for different temperatures. The emission of radiant energy (or the brightness at each wavelength) covers a continuous range of wavelengths so that the spectrum of a blackbody is a continuous spectrum.
12.3.2. Planck's Law, Wien's Law, and the Stefan-Boltzmann Law
In 1900, the German physicist Max Planck derived a mathematical expression, now called Planck's law, that describes the distribution of brightness in the spectrum of a blackbody. There are two other distinguishing characteristics of the spectrum of blackbody radiation in Figure 12.6:
The significance of the blackbody-radiation laws--Planck's, the Stefan-Boltzmann, and Wien's--is that when bodies emit electromagnetic radiation like that of a blackbody, they do so because they are hot. Fortunately, the radiation emitted by stars tends to be much like that emitted by a blackbody. Thus the blackbody-radiation laws are powerful diagnostic tools for measuring the temperatures of stars as thermal sources of radiation. We shall use this fact in our study of the Sun and stars in this and the following chapters. For the study of bodies that emit radiation not because they are hot (called nonthermal sources of radiation) but because of some selective physical processes, the blackbody-radiation laws are of no use. Some everyday examples of thermal sources of radiation are an incandescent light bulb, the burner on an electric stove, and the flame of a cutting torch. Examples of nonthermal sources are a fluorescent light, lightning, and a television screen. However, stars are the examples of thermal sources in which we are most interested.
Let us use the blackbody radiation laws and our knowledge of
the absorption of radiation by atoms and ions to consider the
outer layers of stars, such as the Sun. For it is these outer
layers that are the sources of the radiation we receive from stars.
12.4.1. The Sun's Outer Layers
The Sun has a mass roughly 300,000 times that of the Earth and it emits by Earth standards an immense quantity of electromagnetic energy. Only about half a billionth of the Sun's radiation is actually intercepted by the Earth as it passes out through the Solar System.
How is this radiant energy outflow measured? Starting in 1980 with the launch of the Solar Maximum Mission spacecraft, astronomers could directly measure radiation falling on a unit area just outside the Earth's atmosphere within a certain time, a quantity known historically as the solar constant (Table 12.2). The name is now known to be somewhat of a misnomer inasmuch as spacecraft measurements revealed that the solar constant varies by as much as 0.1 to 0.3 percent in the time span of a week or two. However, if we ignore these small variations and their consequences for the moment, we can proceed to find the Sun's rate of emission of radiant energy over all wavelengths, or its luminosity. (Astronomers use the symbol . for the Sun, thus the solar luminosity is written L., mass as M., radius as R., and surface temperature as T..) If we multiply the average solar constant by the surface area of a sphere whose radius is the Earth's mean distance from the Sun (1 AU), we obtain the rate at which solar radiation flows out from the Sun in all directions. This number must also be the rate at which electromagnetic energy is radiated away from the Sun's surface or its luminosity, whose value works out to be about 4 x 1033 erg/s. Approximately 40 percent of this energy is in the visible part of the spectrum, 50 percent in the infrared region, and the remaining 10 percent in the ultraviolet.
This flood of radiant energy (the solar luminosity) comes from what appears to be the surface of the Sun. It is not in reality a distinct surface, like that of the Earth, but a layer of gas several hundred kilometers in thickness called the photosphere (Figure 12.7a). Dotted here and there with sunspots, as shown in Figure 12.7a, the photosphere is actually only the lowest visible level of a much more extensive solar atmosphere. Lying above the photosphere is a transparent, tenuous layer, the chromosphere, which is several thousand kilometers thick. This is topped by an even more rarefied layer, the corona, which extends millions of kilometers out from the Sun in all directions (Figure 12.7c). These three regions, which gradually merge into one another, can be distinguished from each other by their different physical characteristics.
Why do we see only the photosphere in visible light and not the
chromosphere and corona? The reason is that visible light coming
from the chromosphere and corona is usually too weak to be seen
against the much brighter photosphere. But they are visible outside
the Sun's limb (edge) during a total eclipse of the Sun, when
the Moon covers the photosphere. The gases of the chromosphere
and corona are transparent in the visible part of the spectrum,
and photons from the photosphere pass directly through these layers.
The chromosphere and corona can be observed directly in short-wavelength
regions of the electromagnetic spectrum (ultraviolet and X-ray)
or in the long-wavelength radio region. This is because the photosphere
is not very bright in either of these wavelength regions in comparison
with the chromosphere and corona. In part the chromosphere and
corona are brighter than the photosphere in the short wavelength
regions because the chromosphere and the corona are hotter than
the photosphere as we shall discuss in Chapter 17.
12.4.2. The Source of the Sun's Luminosity
Below the photosphere the solar gases become opaque and hide the Sun's interior from our view. Although the interior of the Sun cannot be studied directly, astronomers have devised mathematical models of the Sun's internal structure by which these hidden regions can be studied, as we shall discuss in Chapter 19. From such studies we find that the Sun's luminosity is the result of hydrogen fusion--the conversion of four hydrogen nuclei into one helium nucleus--occurring close to the Sun's center. This energy, when first released in the deep interior, is chiefly in the form of gamma-ray and X-ray photons. As these photons work their way toward the surface, various atoms and ions absorb and reemit them, which tends to shift the wavelengths of photons from short values toward longer ones.
The emerging photons finally reach those layers lying some 100,000 km below the photosphere, a region known as the hydrogen convection zone. Through this region the movement of energy is like that in a heated room, where cold, heavier air descends to be reheated near the floor and then rises, carrying heat toward the ceiling. In the Sun, hot gases bring thermal energy up from the bottom of the convection zone to its top lying just below the photosphere, and from there cool gases return to the bottom of the convection zone to start the cycle again.
At the surface most of the radiant energy that left the deep
interior hundreds of thousands of years earlier is now in the
visible part of the spectrum--the ordinary sunlight that we observe
here on Earth about 8 minutes after it leaves the Sun.
12.4.3. Spectrum of the Solar Photosphere
The spectrum of the visible solar disk is a continuous band of colors from red to violet crossed by many absorption lines that are easily seen in Figure 12.8. That is, the photospheric spectrum is an absorption spectrum. In 1814, German physicist, Joseph von Fraunhofer (1787-1826), mapped nearly 600 of the most prominent lines. He designated the strongest absorption lines by capital letters, beginning with A in the red and going to K in the violet. Since the photospheric spectrum is an absorption spectrum, we can interpret these lines according to Kirchhoff's third law of spectral analysis (Section 5.1): The radiation coming up from the interior of the Sun has a continuous spectrum; as the radiation passes through the photosphere, certain wavelengths are absorbed by atoms and ions of different chemical species in the photosphere's cooler layers and in the adjoining low chromosphere, causing the observed dark lines. The uninterrupted-wavelength regions, between absorption lines, are those of continuous radiation that passes into space without being absorbed.
12.4.4. Temperature of the Solar Photosphere
The temperature of the solar photosphere, or the Sun's surface temperature, is an important property of the Sun or for that matter any star. For it is a measure of the rate at which radiation is emitted by the star, i.e. the star's luminosity. To find the Sun's surface temperature, we utilize the three methodologies implied in the Stefan-Boltzmann law, Wien's law, and Planck's law. As discussed above, these laws characterize the radiation emitted by blackbodies, but because the Sun is not precisely a blackbody, the temperatures derived from these laws differ slightly; they yield an approximate value of 6000 K. We know that stars can not be precisely blackbodies because the spectra of the radiation from their photospheres would have to be continuous, but simple observation shows us that the spectra of stars are absorption spectra.
One means of demonstrating how a value for temperature is derived from Planck's law is shown in Figure 12.9, in which the amount of energy in the solar spectrum measured at a number of wavelengths is compared with that emitted by a blackbody. In such a comparison, we can see that the 6000 K blackbody energy curve is a reasonable approximation to the way energy is distributed in the Sun's continuous spectrum.
The important question is whether this 6000 K surface temperature
means the photosphere has but a single temperature (i.e., is constant
throughout the entire photosphere) or the surface temperature
is actually an average over the thickness of the photosphere.
To answer that question let us consider what we actually observe
in the case of the Sun.
12.4.5. Temperature Decline Through the Solar Photosphere
The Sun's limb (edge) looks sharp-edged to the naked eye because the layers responsible for emission of white light are too thin to be resolved. They are several hundred kilometers thick, whereas the typical resolution size for 1 second of arc seeing corresponds to about 750 km. Either to the naked eye or in large solar telescopes, such as the one in Figure 12.10, the Sun's limb looks darker than the center of the disk. Why is this so, since a shiny aluminum sphere does not appear darker near its edge? At the Sun's edge we are viewing a succession of photospheric layers obliquely, seeing light that comes only from the highest layers of the photosphere. Because the higher layers emit less radiation, we infer from the Stefan-Boltzmann law that they must be cooler, as is evident from the blackbody energy curves in Figure 12.9. Radiation visible to us from the center of the Sun's disk, however, comes from deeper, hotter layers and is more intense (Figure 12.11). Thus it is obvious that the temperature declines outward through the photosphere. From this fact astronomers can determine the decrease in temperature and density through the photosphere and use these data, for example, to determine the abundance of the chemical elements.
12.4.6. Chemical Composition of the Solar Photosphere
By measuring the wavelengths of absorption lines in the photospheric spectrum, astronomers have identified in the Sun nearly 70 of the 92 naturally occurring elements and about 20 molecules. The identifications are made by comparing wavelengths of lines in the solar spectrum with those obtained from laboratory analyses of spectra of elements. However, the presence of absorption lines of particular elements only confirms that the element is present, but does not provide the element's abundance directly. The few elements for which absorption lines are missing from the photospheric spectra, mostly the heavier ones, are probably also present in the Sun's atmosphere, but they are either not abundant enough to be detected spectroscopically or their spectral lines are not in the visible or ultraviolet regions, the only regions thoroughly explored.
In order to determine the abundance of elements in the photosphere, some of which are given in Table 12.3, we must combine our theoretical knowledge of the probability that an atom will absorb radiation at the wavelength in question at a specified temperature with measurements of the line's darkness and width. In the next section, we shall discuss such measurements for other stars besides the Sun.
Rotation is a common characteristic of most objects in the Universe. Like the planets (except Venus and Uranus), the Sun rotates counterclockwise as seen from north of the orbital plane of the planets. It does not rotate as a solid body, as does the Earth and other terrestrial planets, which is not surprising because the Sun is wholly gaseous. For the atmosphere, primarily the photosphere, the period of rotation progressively increases from 25 days at the solar equator to about 36 or 37 days at the poles. This behavior, called differential rotation, is similar to that which we found for the Jovian planets Jupiter and Saturn. The differential rotation of the Sun has taken on renewed importance in the last decade after it was realized that the interaction between rotation and convective currents below the Sun's surface generates the magnetic fields that are responsible for the host of observed surface activity, such as sunspots.
To find how long it takes the visible layers of the Sun to complete one rotation we measure the travel time of sunspots as they are carried across the disk (Figure 12.12). Another method, applicable to all solar latitudes (sunspots rarely appear beyond 40 degrees on either side of the solar equator), measures the difference in Doppler shift for spectral lines in the radiation from opposite limbs of the Sun (Figure 12.13). A Doppler shift is present because the eastern limb of the Sun rotates toward and the western limb away from the Earth. The velocity relative to the Sun's center is found to be about 2 km/s at the equator. Dividing the distance traveled in one rotation--that is, the circumference--by the velocity gives the time for one complete rotation at the equator, approximately 25 days.
The differential rotation of the photosphere probably does not extend much deeper than the bottom of the hydrogen convection zone. However, evidence suggests that the layers under the photosphere may rotate faster than the photospheric layers do. An additional fascinating possibility derived from long-term studies of sunspots and Doppler shifts is that the rotation of the photospheric surface is not uniform in time. The surface appears to speed up and slow down by several percent or a few tens of meters per second over the 11-year sunspot cycle discussed in Chapter 17.
The stars of our night sky are ones that are reasonable close to the Sun in our Galaxy, which is but one galaxy among billions of galaxies in the Universe. The population of stars in our Galaxy, like that in other galaxies, is measured in the hundreds of billions. Starlight is visible light coming from the photospheres of those stars. Some of the techniques used to study the solar photosphere can also be applied to the photospheres of other stars. The result of such studies is the belief that the photospheric structure of all stars are in general the same. But the same can not be said regarding the existence of chromospheres and coronas. Like the Sun, if a chromosphere and a corona exist in other stars, then we would expect that ultraviolet and X-ray or radio radiation can provide us evidence for these two regions. We will consider that evidence in Chapter 17.
Spectra of the radiation from the photospheres of stars are also absorption spectra with the exception that they do not possess identically the same pattern of absorption lines as does the Sun's spectrum. By applying either Planck's, Wien's, or the Stefan-Boltzmann law to the spectral energy distribution of a particular star, we can determine its surface temperature, just as we did for the Sun. The result of which is evidence that there exists a variation in surface temperatures among stars from some 3000 to 50,000 K.
Differences in absorption lines observed in the spectra of stars, to be discussed in the next chapter, stem from the manner in which atoms of various elements absorb and reradiate energy under the temperature and density conditions found in a star's photosphere. By far the most important variable affecting stellar spectra is temperature, the primary factor in Figure 12.14a, the consequence of which is the observation that strengths of absorption lines due to neutral, singly-ionized and doubly-ionized atoms should vary for stars of different temperatures. The large range in temperature among stars accounts for which absorption lines are and which ones are not present in stellar spectra.
Astronomers can obtain even more detailed information about stars'
physical and chemical properties than that found from the surface
temperature. For example, from the shape, width, and strength
of absorption lines in spectra, astronomers can infer the variation
in temperature and density down through the star's photosphere.
(A type of study on which we will not elaborated, other than
to say it can be done.) You can see how the intensity varies
across a part of a typical spectrogram in Figure 12.14b. The
absorption lines are the dips below the continuous background
spectrum, whereas the tiny wiggles are made by the photographic
emulsion's graininess. Atomic processes, large-scale motions,
and chemical compositions also affect stellar spectra, but they
do it in a much more subtle fashion than does temperature.
12.5.2. Processes That Affect Stellar Spectra
In actual practice it is not easy to sort out all the processes that affect the shape of an absorption line. Absorption lines are broadened by several atomic processes. For example, the absorption lines in the spectra of some stars are narrow because of a lower density in their photospheres than is true for other stars. As another example, we know that light from a radiating source that is in a magnetic field has a spectrum in which each absorption line is split into three or more closely spaced components--a phenomenon called the Zeeman effect after the Dutch physicist, Pieter Zeeman (1865-1943), who discovered it in 1886. A few stars are found to have their absorption lines broadened by the Zeeman effect corresponding to large-scale magnetic fields whose strengths are several thousand gauss. By comparison, the magnetic field throughout most of the Sun's photosphere is a few gauss, although there are small regions where the magnetic field is much stronger. Zeeman broadening of stellar absorption lines is illustrated in Figure 12.15a.
Large-scale motions in the outer layers of a star--such as random
thermal motions, streams of gas, and rotation of the star--all
broaden absorption lines through the Doppler effect. From these
Doppler-effect studies, astronomers know that stars do not all
rotate at the same rate. Rates of rotation for the some very
hot stars are several hundred kilometers per second compared with
a few kilometers per second for cooler stars like the Sun. This
difference is shown schematically in the widths of the absorption
lines in Figure 12.15b.
12.5.3. Chemical Composition of Stars
Analyzing absorption lines to determine the chemical composition of a star's atmosphere is an important part of astronomy. The intensity of an absorption line depends on the fraction of neutral or ionized atoms of an element capable of absorbing that particular wavelength of radiation. This fraction depends on the star's temperature, the atmospheric-gas density, and the abundance of the element. Astronomers generally have estimates of the first two factors from a star's luminosity and spectral type; the uncertain factor is the abundance of the element. From judicious choices for abundance, along with the estimates of temperature and density, theoretical absorption-line strengths can be calculated until they agree with observed line intensities (Figure 12.14) element by element. By so doing, astronomers determine abundances for the elements in a star's photosphere and refine the values of its atmospheric temperature and density.
Abundance studies for several hundred stars give values much like those for the Sun (Table 12.2). But some exceptions appear in the abundances both of all elements heavier than hydrogen and helium and of certain elements in some stars (Figure 15.12). The heavy-element abundances, all elements besides hydrogen and helium, appear to vary from 0.01 to about 4 percent by mass among all stars so far studied.
Having learned how astronomers analyze the light from stars, we can now proceed to a more complete study of the properties of stars in the next several chapters.