Tuesday September 25
The first application of spectroscopy to astrophysics was work by Wollaston (1804) and von Fraunhofer (1811). Both used a slit-plate and prism system to project the Solar spectrum onto a wall, and study it visually. von Fraunhofer did the most extensive analysis of the observations, and thus his name is still attached to the work.
Both noted that the Solar spectrum is best described as a continuum with occasional sharp dark features (or Absorption Lines). von Fraunhofer labelled a number of the stronger features with capital letters, starting with A at the red end, and proceeding blue-ward. This convention is still in use (the "G-band", Ca H+K lines, and so forth).
There was no explanation of the von Fraunhofer lines at the time. Some 30 years after von Fraunhofer's initial work, chemists began studying the spectra of substances by vaporising them in flames. They realized that the patterns of emission lines they produced were each due to specific elements. Experiments by a number of workers over the next several years ended in a synthesis of rules by Kirchhoff in 1859:
Wollaston and von Fraunhofer did their work by projecting the Solar spectrum on a wall, and looking at it. This works for the Sun because it is bright. But no other star is sufficiently bright to use this technique. However, the early 1800s saw the invention of photography, and one of the earliest applications of photography was to astronomy. The existence of an integrating detector (a photographic emulsion) made it possible to obtain useful spectra of other stars by the mid 1800s.
Many observatories began collecting stellar spectra. The most extensive collection was built at the Harvard College Observatory under the direction of Edward Pickering. By the late 1800s, Pickering and his assistants had accumulated spectra of more than 200,000 stars.
It was realized very early-on that the Solar spectrum was typical. Most stellar spectra were continua with absorption features. But the features detected, and their relative strengths varied greatly from star to star. With an enormous collection of data showing complex and totally mysterious variation, the basic rule of science is to pick some feature in the data to sort on. It usually doesn't matter very much what you pick, so long as it really does give you a means to sort by.
Draper decided to sort spectra based on the relative strength of the hydrogen absorption features. He chose hydrogen because he could see those lines in nearly all of the high-quality spectra he had available. In the original scheme, a star was assigned a letter between A and P depending on the strength of the star's hydrogen absorption lines. The stars with the strongest hydrogen absorption were type A, and the weakest were type P.
As it turned out, the strength of hydrogen absorption features in stellar spectra is not monotonic with any underlying physical property. But it does track stellar surface temperature. Extensive observational and experimental work from 1870-1900 led to a lot of the letters being dropped, and the rest re-organized into a sequence of stellar temperature, going from the hottest stars to the coolest as follows:
This sequence is further divided decimally, so that within any given letter there are ten subtypes running from 0 to 9, such that (say) an A0 star is hotter than an A1 star, but cooler than a B9 star.
The classification of the Sun is G2.
Notice something about this sequence - The strongest Balmer lines are found in neither the hottest, nor the coolest stars. This is because the hottest stars (O stars) are so hot (T ~ 25000 K) that the hydrogen in their atmospheres is all ionized, so there isn't any Balmer line formation. And the coolest stars (M stars) are so cool (T ~ 3000 K) that all the hydrogen stays in its ground state. There aren't any photons energetic enough to excite hydrogen, so once again, there isn't any Balmer line formation.
In the last decade, two new spectral classes have been added at the cool end of the sequence (L and T). Stars of these classes turn out to be very common, but they were not discovered until the 1990s. This is due to a couple of technological advances. First, as these stars are very cool, they have relatively low luminosities so even very nearby ones are quite faint. Second, again because they are cool, most of their flux is radiated in the infrared. Thus to detect these stars it was necessary to build 8-10m class telescopes, and then build IR array cameras and spectrographs for them.
While all this spectral classification was going on, a mostly distinct group of people were sorting out the structure of atoms. The modern starting point for this was Ernest Rutherford's work (Rutherford's graduate student's work, to be more honest). The experiment in question studied the deflection of "alpha rays" by a thin gold foil. Most of the alpha rays were essentially undeflected by the foil, but the occasional one experienced very large-angle scattering. The implication of this experiment is that most of the mass of atoms is contained in a very small volume (now called the atomic nucleus), and most of the volume of an atom is essentially empty space.
Further experiments on the charge state of atoms led to the modern picture of a nucleus containing nearly all the mass, and all the positive charge, and a surrounding set of electrons carrying the negative charge. This led to a very difficult problem: Why are atoms stable?
The physics of the late 19th century led to the conclusion that electrons should fall into nuclei due to electrostatic attraction. And that the atom should radiate like mad as the electron was accelerated into the nucleus. In fact, because the electron is charged, even if it were to just orbit the nucleus, it should radiate like mad. None of the above actually occur.
The patterns of lines formed by most substances are very complex. So much so that they defy any simple algebraic description. That of hydrogen turns out to be the exception. In fact, the visible lines of hydrogen obey a very simple algebraic rule, as first pointed out in 1885 by a Swiss school teacher named Balmer:
Here R is the Rydberg Constant. This formula turned out to predict the wavelengths of the visible transitions of hydrogen perfectly, given the experimental equipment of the day. But it is only function-fitting. Balmer did not explain why the formula worked, he simply pointed out that it did work.
The Bohr model for hydrogen is, in some sense, the most succesful ad hoc theory in modern science. Bohr postulated that the electron in a hydrogen atom is confined to exist only in certain specific orbital states. And that these states corresponded to allowed energy levels. Thus for an electron to move from some higher energy state n to some lower energy state m, the atom must emit a photon of energy En - Em. Conversely, the electron can move from m to n only by gaining a total energy of En - Em (this can be accomplished either by absorbing a photon or by collision).
Now, if this was all Bohr had done, no one would have paid his idea much mind. But he was able to make a succesful model of the hydrogen spectrum by taking this sketch of an idea, and adding the restriction that (somehow or another), the orbital angular momentum of the electron is not free to have any value. It must, instead, be quantized as follows:
Bohr then combined this restriction with an otherwise classical treatment of the hydrogen atom. He considered the energetics of the electron as follows:
Bohr went on to equate the electrostatic force between the electron and the proton with the centrifugal force that would keep the electron in a stable orbit. This led to a simple statement of the total energy of the system:
Now Bohr introduced his quantized angular momentum condition, and used it to evaluate the radii of the allowed states as follows:
This expression could be put back into the energy equation to give the following result:
Note that, formally, the m in all the above is the reduced mass, not the mass of the electron. The notion of reduced mass will return later in a very different circumstance, so it is worth defining. The explicit statement of reduced mass is as follows:
If one substitutes n=1 in the energy relationship above, one is solving for the binding energy of a hydrogen atom. The result is E1=-13.6 eV. Note that if n=infinity, E=0 and the atom is unbound.
By evaluating the resulting energy differences between any two allowed states, one can construct an Energy Level Diagram for hydrogen. This is, in effect, a prediction of the spectral line properties of the hydrogen atom. And the reason why Bohr's model is still discussed in classes like this is that his prediction matched the observed spectrum of hydrogen. In particular, it extended the work of Balmer. One can use the definition of the Rydberg constant, and Bohr's result for the energies of allowed states to derive the following general result:
The Balmer series is just the series for m=2. That is, the visible lines of hydrogen do not correspond to transitions into the ground state, but rather into the 1st excited state. The ground-state transitions, also called the Lyman series produces lines in the UV. Higher order transitions create further series in the infra red and radio.
It is worth noting at this point that the inverse of the Rydberg constant is thus the wavelength of an ionizing photon (m=1,n=infinity). This wavelength is about 91.2 nm.
What Bohr accomplished was a description of hydrogen using a hybrid model that was mainly classical physics, but required a key quantum assumption. Bohr's model did not address the deeper question of "Why is J quantized?". The full treatment of quantum mechanics is the subject of an entire course in its own right. Quantum issues will rear up now and again in this course, but we will generally deal with them in semi-classical ways.
Louis deBroglie made a fundamental contribution to the evolution of quantum mechanics with his extension of the notion of wave-particle duality from photons to massive particles. That is, any massive object can be described in wave-mechanical terms with the wavelength following from the momentum of the particle. Thus one can make wave-mechanical treatments of things like electrons and atoms. This insight is was led to the development of the wave equation of Erwin Schroedinger, and the quantum description of the microscopic world in terms of probability amplitudes, rather than deterministic events.
The line strength for a given line from a gas depends on both temperature and the composition of the gas. This is a generalization of the argument already made that Balmer line strength can be very low in a gas of pure hydrogen if the temperature is either so low that the hydrogen is all in the ground state, or so high that the hydrogen is all ionized.
We define a state population as the number of atoms in that state per unit volume in a gas. Excitation processes alter state populations. The two principle mechanisms of excitation are
Note that temperature has two roles here. Higher temperatures means both more energetic and more frequent collisions.
For some atom, one can describe the relative populations of any two atomic states in terms of the Boltzmann Distribution:
Here the n's represent the state populations of the i'th and j'th level. The g's are statistical weights. These are typically small integers that represent the available number of substates in a given atomic state. As an example, the ground state of hydrogen has two available sub-states, and the first excited state has eight available sub-states. Thus in the case of i=1, j=2 for hydrogen, the ratio of the statistical weights would be 4.
The Boltzmann distribution is defined for a system in thermodynamic equilibrium. However, even in non-equilibrium systems, one can still determine the population ratio of any two states, and use that to derive an Excitation Temperature for that pair of states.
Consider the limiting behavior of the Boltzmann distribution (and take j > i, for simplicity). As T goes to 0, the population ratio also goes to 0. That is, at low temperature all the atoms are in the lower energy state. As T goes to infinity, the population ratio converges on the ratio of the statistical weights. That is, the electons distribute themselves "at random" according to the available phase-space.
The above limit argument ignores another important issue. If the temperature is sufficiently high, the atoms begin to ionize. Thus a proper study of the population ratios must also include the effects of ionization.
A gas in thermodynamic equilibrium is (in most instances, at least) also in Ionization Equilibrium. In this state, there will be a balance amongst:
Here n(Xr) represents the population of ionization state r of element X, Ei represents the ionization energy required to go from r to r+1, ne is the electron density, and the g's are the statistical weights again.
The T dependence in the power-law part can be explained at the hand-waving level as following from the larger number of states available at higher temperatures. The 1/ne dependence, at a similar level of detail, follows from the increase in recombination at higher electron densities.
As is the case for the Boltzmann distribution, one can use the Saha equation in non-equilibrium conditions. In this case, the ratio of the ionization states can be used to derive an Ionization Temperature for that pair of ionization states.
Now consider the case of a gas of pure hydrogen. This turns out to be a useful astrophysical case, as stars are mostly hydrogen. Helium makes up most of the rest of the gas in a star, and helium has a sufficiently high ionization potential that it is neutral in all but the hottest stars. If one ignores the contributions from metals (in astronomy everything but Hydrogen and Helium is a metal), then the number of free electrons is equal to the number of free protons. In this case the Saha equation reduces to the following:
Notice that one cannot explicitly solve this equation for the ionization fraction. One still needs (or need to assume) a value of the electron density to reach an explicit solution for the ionization fraction in such a case.
Astronomers and Atomic Physicists use a special notation for ionization states. In this notation, a Roman numeral I indicates a neutral species, II indicates a singly-ionized species and so forth. Thus, as an example, FeXVII indicates sixteen-times ionized Iron.
Thursday September 27
We have already discussed the non-monotonic behavior of the Balmer lines of hydrogen as a function of temperature. This behavior is also seen with all other sets of lines from all other atoms, ions, and molecules, and is seen for exactly the same reasons (with the exception that it is dissociation rather than ionization that causes the high-temperature cut-off of molecular lines). The important variable is that the range of temperature over which a given line is strong differs from element to element. Thus the collective behavior of a large set of spectral line diagnostics makes an excellent thermometer.
We discussed the use of blackbody temperatures last week. Such temperatures are only accurate to a few hundred degrees for a typical star. However, a good spectral type classification for a star can yield a temperature that is good to tens of degrees. Without belaboring the details, I now review the basic sorts of line diagnostics that apply as a function of spectral type.
M Stars (T < ~ 3500 K):
Around 1910 Ejnar Hertzsprung and Henry Norris Russell independently plotted up the spectral types and absolute magnitudes of stars in the Solar neighborhood. That is, they plotted stellar temperature against stellar luminosity. Such a plot (and its variations) is now known as a Hertzsprung-Russell Diagram or HR Diagram for short.
The key and immediate realization is that such a diagram is NOT uniformly populated. There are particular combinations of (L,T) that are allowed. In fact, most stars fall along a sequence that runs from high luminosity and high temperature to low luminosity and low temperature. Because most stars follow this sequence, it is called The Main Sequence.
This means that, if a star is a Main Sequence star, a measure of its temperature (from its spectral type) is sufficient to determine its luminosity. Recall that two stars with the same luminosity and temperature must also have the same radius (from the Stefan-Boltzmann relation).
Pushing this idea a bit further, the existence of the Main Sequence also means that most stars form a One-parameter Family. This gave people like Russell real hope that it was possible to construct realistic models of stellar structure.
There are stars "above" the MS. These stars are overluminous for their temperatures. Again arguing from the Stefan-Boltmann relation, this means these stars must be physically larger than MS stars. Thus they are labelled Giants. In particular, as most such stars are relatively cool, they are called Red Giants.
A small number of extremely luminous stars, with a wide range of temperatures is also seen in the HR diagram. These stars are called Supergiants. Given these terms, Main Sequence stars are therefore often referred to as Dwarfs or Main-Sequence Dwarfs.
There are stars "below" the MS. These stars are underluminous for their temperatures. This means these stars must be physically smaller than MS stars. As these stars are relatively hot, they are called White Dwarfs. Despite the confusing terminology, do not confuse White Dwarfs and Main-Sequence Dwarfs. Note that White Dwarfs are about 10 magnitudes less luminous than MS stars of the same temperature. One can use the definition of the magnitude and the Stefan-Boltmann relation to show that this means they must have R ~ 0.01 R(MS). In other words, a White Dwarf with the same temperature as the Sun is about the size of the Earth.
If one consideres the H-R diagram, one may note that stars of a given temperature can have one of several very different luminosities. That is, for instance, a star with a surface temperature of 9000 K could be a white dwarf, a main-sequence star, or a supergiant.
Is it possible to distinguish between these various alternatives? Yes, it is. One can use stellar spectra to distinguish the Luminosity Class of a star. The Luminosity Classification system runs from Supergiants (classes Ia and Ib) through Main Sequence stars (class V). Thus the Sun has a full spectral classification of G2V.
I want to stress that the Luminosity Class of a star is a purely spectral designation. One does NOT need to know the distance of the star a priori to assign it a Luminosity Class. This is a very important point that has some interesting consequences. It is also an opportunity to discuss some good physics. The physical underpinnings of Luminosity Classification will take some time, so I defer it until next week. I close today with a discussion of the observational implications of the ability to assign luminosity classes to stars of (a priori) unknown distance.
Consider a cluster of stars of unknown distance. Even though the distance is unknown, all the stars in that cluster are at the same distance. Therefore one can plot up a Color-Magnitude Diagram for the cluster. A CM diagram is a kind of HR diagram, in which one plots apparent magnitude (in some bandpass) on the y-axis and a color on the x-axis. The traditional combination is V and (B-V), but that is not essential at the moment.
What you see in such a diagram depends on the history of the cluster in question. As an example, consider CM diagram of the Pleiades. Most of the stars in the Pleiades fall on the Main Sequence. One can therefore estimate the distance of the Pleiades by measuring the difference in magnitudes between the apparent magnitude of Pleiades stars at a given temperature and the Absolute magnitude of Main Sequence stars of that temperature from the HR diagram. This difference in magnitudes is just the Distance Modulus to the cluster.
Now consider the case of the observation of a single star. Here one does not have the aid of having many stars at the same distance. But if one can assign a spectral type and Spectral luminosity class to the star, then one can estimate its luminosity just from its spectrum. One can then measure the apparent magnitude of the star, and use the inferred luminosity to estimate the distance to the star. This method of distance estimation is known as Spectroscopic Parallax (even though one is not measuring a parallax for the star!).
I shall begin today by covering a topic that ought rightly have been covered in the previous unit. Compton scattering is important both astrophysically and historically. The historical import is that Compton used the photon model of light to predict the scattering behavior of photons off electrons, and then did the experiments to confirm his prediction. His experiment was the first to confirm that photons carried momentum, and thus was of enormous importance in convincing physicists that Einstein's explanation of the photoelectric effect was valid.
The idea is a variation on the classic scattering problem from introductory physics, in which a particle with some initial linear momentum collides with a second particle at rest. One can then use the laws of conservation of energy and momentum to study the behavior of the system after the collision.
The key difference in Compton scattering is that the photon must always travel at the speed of light. The scattering event gives the electron some kinetic energy, and thus the photon must end up with a longer wavelength (lower energy) than it started with. Given initial conditions as follows:
one can see that the final configuration will have conditions as follows:
Adopting theta as the scattering angle of the photon (and phi as the scattering angle of the electron), one find the following form for the wavelength shift from Compton scattering:
Note that this transfer of momentum from photons to electons is the underlying cause of radiation pressure.
Last week I pointed out that the definition of Luminosity Class for stars was a purely spectroscopic one. Hertzsprung was the first person to consider the possibility that there might be a spectrscopic tracer of luminosity. That this was the case was clearly demonstrated in the MKK Stellar Atlas, published in 1943. At a given spectral type, more luminous stars have sharper spectral features. To understand why this is so, we need to discuss spectral line formation in a bit more detail.
Consider the following isolated spectral line:
One defines the depth of the line, as a function of wavelength, as follows:
We further define, the Equivalenth Width of the line as follows:
In other words, the area of the line is just Fc W. Note that in this case we are dealing with an optically thin line. That is F(lambda) > 0 at all wavelengths.
The opacity of the line varies with wavelength, and reaches a maximum at the line center. Opacity is a measure of how far a photon can travel without being absorbed. When the opacity is high, the distance is small. This is a hand-wavy justification for the following statement: The line core forms highest in a stellar atmosphere, while the wings form lower. Both the temperature and density of the atmosphere increase with increasing depth, so this means the line core forms in cooler, lower density material than the line wings:
With this ideas in mind, we now consider mechanisms that lead to line broadening. Understand that the arguments presented last week would, on the surface, lead to the expectation that a spectral line should be a delta-function. Clearly, this is incorrect. And it is so for a number of reasons:
Consider the case of an isolated "motionless" atom. The Heisenberg uncertainty principle can used to derive the following:
Repeating the process, holding E_i constant, and adding the two components then gives the following:
Where we have taken state f to the be ground state (hence the infinite lifetime). For the case of excitation of hydrogen to the first excited state, this calculation yields an expected natural line-width of
Very narrow, indeed!
Now consider a gas of atoms of some mass, m in thermal equilibrium. In such a gas, the velocities of the atoms will follow the Maxwell-Boltzmann distribution function (or Maxwellian, for short):
I state without proof that this distribution gives a "most probable velocity" of
Given this, a photon that is absorbed or emitted by an atom in the case is Doppler shifted as follows:
Taking H-alpha, and a temperature roughly that of the Solar photosphere yields
Coherant streaming motions (from large-scale convection, as an example) can also contribute to line-broadening. If there is some characteristic turbulent velocity, it will combine with the atomic-level Doppler broadening to produce a broadening profile approximately as follows:
Stars rotate. If we observe a star with some characteristic rotational velocity, and we happen to be looking perpendicular to its rotational axis, we will see the line profiles broadened due to the Doppler effect on a global scale: One limb of the star is rotating toward us at the rotational velocity, and the other is rotating away.
Tuesday October 2
Again, consider a gas of atoms of some mass m, in thermal equilibrium. But now consider that the orbital structure of any given atom will be perturbed by those of its neighboring atoms. If one considers an atom, moving through the gas, one can define a mean time between collisions in terms of the mean free path and the characteristic velocity (as defined above) as follows:
If you consider the atom moving through a gas of some density, then it is easy to show that the mean free path ought to depend on the density and the interaction cross section appropriate to the type of atom in question:
Taken with the Maxwellian most-probable velocity, given above, one can thus write down the mean time between collisions as follows:
One can use the same argument for the dependence of the line width on the mean time between collisions as was used for the mean state lifetime in the case of natural broadening. This gives an expected line width as follows:
In other words, the line width is directly proportional to the density. The denser the atmosphere, the broader the lines. The implication is that Supergiants have much lower atmospheric densities than do Main Sequence stars, and they therefore have sharper spectral features.
1. Use Bohr's method to work out the expected energy levels for CVI, Hydrogen-like Carbon.
Bohr began his study of hydrogen by considering the total energy of a single electron bound to a nucleus. For CVI:
Where the factor of 6 reflects the nuclear charge Carbon. From here Bohr equated the electrostatic and centripital force on the electron, and used the result to simplify his expression for the electron energy:
Next comes the quantization condition on the electron angular momentum. One can use this to evaluate an expression for the allowed radii as follows:
This expression can be introduced into the energy equation and solved to yield the desired result:
Note that the m above is strictly the reduced mass. However as the carbon nucleus is more than an order of magnitude more massive than a single proton, the reduced mass is equal to the electron mass to at least five decimal places.
2. What is the relative density of the G2I star compared to the Sun, assuming the supergiant has a mass of 10 Solar masses?
Both the Sun (G2V) and the supergiant (G2I) are G2 stars so they have the same effective temperatures. Inspection of the HR diagram shows a G2I star to have an absolute magnitude of about -5. The Sun has M~+5. This means the G2I star has a luminosity of about 1e4 Solar luminosities. Given these, the result follows:
Notice that this supports the contention that pressure broadening is what causes MS stars to have broader lines than supergiants. In fact, if 10 Solar masses is a reasonable number for the supergiant (and, trust me for now, it is), one can see that the global average density ratio derived above is actually an overestimate of the atmospheric density ratio: The mass is only ten times as large, but the radius is ~100 times as large. Therefore the gravitational force at the photosphere is a factor of ~1000 smaller for the supergiant.
3. Consider a gas of simple two-level hydrogen atoms, with statistical weights of g1=2 and g2=8. What are the expected population ratios for T=5000K and T=10000K?
One can derive the energies for the first two levels of hydrogen from Bohr's formula. They are E1=-13.6 eV and E2=-3.4 eV. Begin with the Boltzmann distribution:
Taking care to use the correct units, one can then put the relevent numbers into the equation and evaluate the population ratios:
A star with T=5000 K is a K star. K stars have very weak Balmer lines because only one atom in 10e10 is above the ground state. A factor of two increase in temperature causes a factor of 10e5 increase in excitation. As hydrogen accounts for 90% of the atoms in a star, this means that stars with T=10000 (A stars!) will have very strong Balmer lines.
4. At what temperature is the average kinetic energy of a gas of pure hydrogen atoms equal to the ionization energy of hydrogen?
One begins with the thermodynamic relation equating temperature with average kinetic velocity, sets the resulting energy equal to the hydrogen ionization energy, and solves for the temperature as follows:
This is the temperature at which an average collision is energetic enough to ionize hydrogen. But the Balmer lines in stellar spectra become rapidly weaker for T > 10000 K, about an order of magnitude lower than the temperature just derived. In other words, ionization is not caused by average collisions, it is caused by collisions with atoms on the high energy tail of a distribution with an average energy that is well below the ionization energy.
