Tuesday February 19
The space between the stars is not empty. It contains both gas and solid matter. The gas (mainly hydrogen, as always) can be in a variety of phases from cold, relatively dense molecular material to hot, very diffuse fully ionized material. The solid material, commonly referred to as dust, is composed of a mixture of ices, graphite, silicates, and iron. The average properties of this Interstellar Medium (ISM for short) are as follows:
The advent of photography in the mid-1800s also led to the development of photographic spectroscopy. Spectroscopic studies of bright nebulae led to the realization that some of these nebulae (the Orion nebula, in particular) had emission-line spectra. This was happening in the same time-frame as the development of Kirchhoff's laws, so it was rapidly appreciated that the observations indicated that objects like the Orion nebula must be composed of hot (ionized) gas to produce the observed emission-line spectra.
An historical note is that the brightest spectral feature from the Orion nebula is a line at 5007 Angstroms that was unidentified for nearly a century after it was discovered. This line (along with many other lines seen from emission-line nebulae) is nearly impossible to produce in terrestrial conditions because it is due to an electronic transition (in oxygen, in this case) that is classically forbidden. The result of this is that for terrestrial conditions, oxygen ions in the upper level that leads to this line are always collisionally de-excited long before they have the time to radiate. Only in very low density conditions can such ions radiate these forbidden lines.
Although the existence of obscuring material in the ISM was obvious even in the late 1800s, it was generally assumed that this material was confined to the dense clouds that produce the absorption lanes and patches. The general interstellar space was assumed to be fully transparent. This position was shown to be untenable by work published in 1930 by Robert Trumpler.
Trumpler was interested in studying the properties of open star clusters. To do this, he needed to measure their distances. His method for doing so follows from the definition of the distance modulus:
Trumpler measured the apparent magnitudes of stars in clusters, and deduced their absolute magnitudes from their spectral types. This allowed him to calculate distances for his cluster sample. He then used the measured angular diameters (d) and these distances to calculate the absolute physical diameters (in parsecs or inches or whatever) as follows:
Trumpler found that the result of this procedure was that the supposed true physical diameter of open clusters was a function of their distance from us (in the sense that "more distant" clusters were physically larger). This is such obvious nonsense that it HAD to be wrong. Instead, what this test demonstrated was that Trumpler's distances were systematically overestimated, and the larger the distance the larger the overestimate.
The implication of this is that interstellar space is not transparent. This follows because such a non-transparency would cause the measured apparent magnitudes of the open cluster stars to be too faint, leading to an overestimate of their distance. Thus the distance modulus relation must be amended:
Here A must be non-negative.
If the opacity were uniform, one could express the absorption as a product of a constant times the distance:
The opacity isn't really uniform. One can attempt to determine an average value for the scale-constant "a", but variations of more than a factor of two for different lines of sight are found. Given that warning, a modern value for the scale-constant in the Galactic Plane is about:
The physical cause of the observed extinction is a combination of absorption and scattering of photons by dust grains. The typical grain size is about 1 micron. This means that the absorption and scattering functions of the dust are both strong functions of wavelength. In essence, this is because photons that are much longer in wavelength than the average grain size will not interact with the grains.
The detailed study of the extinction properties of dust is a matter of ongoing cutting-edge astronomical research. For our current purposes, it is sufficient to study a highly simplified version of the problem. In what follows we will assume:
Here the Q-term is where we hide all the effects that are complicated. The Q-term is called the efficiency factor, and will depend on (at least) the wavelength of the incident light, the distribution of true grain sizes and shapes, and the grain composition.
If one considers light propagating in some direction through a dusty medium, and can define a covering factor, and from there an optical depth, for the dust in terms of the path length, the effective cross section, and the dust density as follows:
In order to express this extinction as a correction to the distance modulus, one must recast it in logarithmic form as follows:
As noted above, this discussion is really a gross oversimplification of the real problem. First of all, extinction has a strong wavelength dependence. Second, grains are neither uniform nor spherical. One consequence of the wavelength dependence is that light travelling through a dusty medium suffers both extinction and reddening. The reddening is caused by the greater extinction of blue light compared to red light. Thus if one measures the color of a star, one is measuring the intrinsic color modified by some reddening term:
By measuring the observed colors of stars, and determining their true colors from their spectral types, one can derive estimates of the selective extinction along many lines of sight, and out to many distances. This leads to a general result for the total to selective extinction as follows:
The value of 3.2 is the rough average for sight-lines in the Galaxy. There is substantial variation from sight-line to sight-line. The causes for this variation remain something of a mystery, but are clearly related to variations in dust properties in different environments.
As noted above, grains are neither uniform nor spherical. The evidence that they cannot generally be spherical comes from the observed polarization of scattered light. Scattering only causes polarization if the scattering particles are non-spherical.
The dust, like the rest of the ISM is concentrated toward the Galactic plane. We are in the plane, and thus embedded in the dust. A question that turns out to be both crucial for extragalactic astronomy and very difficult to answer unambiguously is "What is the extinction in the directions of the Galactic Poles?"
One model for the distribution of dust is that of a uniform slab in the plane of the Galaxy. If this model holds then one can express the extinction as a function of Galactic latitude as the combination of the extinction toward the Galactic Poles and a term that increases with decreasing latitude:
One then takes some limiting magnitude (in some waveband), and counts the number of galaxies per unit area on the sky as a function of Galactic latitude. Under the assumption that the true distribution of galaxies is uniform on the sky, any modulation will be due entirely to extinction. The result of this test is an estimate that the vertical extinction in the B-band is quite high -- about half a magnitude.
There are other ways to test this, however. And they do not generally give such high values. The range in values obtained by reasonable methods is quite large:
The underlying reason for this substantial range in results is that the dust layer really isn't uniform. It is patchy on all scales. The existence of Dark Nebulae is one clear demonstration of this. These are regions of very high dust densities, and correspondingly high extinction values.
Another manifestation of the patchiness of dust is the existence of Reflection Nebulae. These are caused by light scattering off dust in the vicinity of hot stars. Because dust scatters blue light more effectively than red light, one generally needs hot stars to provide sufficient blue flux for scattering.
Dust grains absorb optical photons. The energy carried by those photons can't simply vanish. Instead, it must go to heat the dust grains. Grains are solid. Thus, if they are heated, they radiate a blackbody spectrum. For typical grain sizes of a micron or so, and the observed spectrum of the interstellar radiation field, one can derive typical grain temperatures in the ISM, and, from applying Wein's law, the typical peak in the resulting blackbody spectrum:
This turns out to be impossible to observe from the ground. Thus we did not get good data on the emission from dust until the IRAS satellite mission in the 1980s. This was the first satellite to map the sky in the mid- to far-IR (between 10 and 100 microns). It detected copious thermal emission from dust at a Temperature of around 20 K.
The observed emission properties of grains also have demonstrated that the assumption of standard grain size must be incorrect. This is because very small grains will be heated to higher temperatures by a given photon flux than will large grains. And the grain size distribution will also be modified by sputtering near hot stars. The combination of these two factors means that hot dust can have temperatures and blackbody peaks much different than the typical numbers quoted above:
Emission from dust isn't really a pure blackbody. There are clear spectral features due to solid-state emission bands in the mid-IR, and emission lines due to polycyclic aromatic hydrocarbons or PAHs. PAHs basically blur the line between a "small grain" and a "large molecule". These emission properties of grains allow us to determine the general chemical composition of the dust:
Gas makes up 99% of the mass in the ISM. But it is generally very difficult to observe at optical wavelengths. The most obvious optical signature of the gas phase ISM is the existence of emission-line nebulae (HII regions and Planetary nebulae). But these account for a miniscule fraction of the total gas mass.
A more general, but far more subtle tracer of the gas-phase ISM are optical absorption lines. These were first noted in the spectra of stars about a century ago. Typically one sees lines from neutral or singly ionized metals (NaI and CaII are the most common species seen). These lines are much narrower than the photospheric lines of the stars, and are generally at slightly different wavelengths (that is, the absorbing gas has a slightly different radial velocity than does the background star).
With the advent of UV spectroscopy in the 1970s it became possible to extract much more information from absorption lines studies. This is because one could now observe hydrogen-line absorption (Lyman-alpha in particular). One does not see the optical hydrogen lines (the Balmer lines) in absorption because the temperature of the ISM is too low for H to be excited to the second energy level in any quantity. Thus only the ground-state lines (the Lyman lines) are signficant and these are UV lines.
One result of the abundance studies from absorption lines is that the relative abundances of heavy elements in the gas phase of the ISM do not match the Solar abundance. In particular, the gas is depleted of the refractory elements. That is, the elements that form dust grains most easily are less common in the gas phase. When one includes the composition of the grains along with the gas, the overall ISM abundances are a fairly good match to the Solar abundance.
Another curious result of absorption line studies is that the average gas density within 100 pc or so of the Sun is substantially lower than the global average gas density of the ISM:
That is, the Sun is within a low-density region we call The Local Bubble. Such bubbles turn out to be not uncommon, and are created by energy input from supernovae.
You may recall the energy level diagram of hydrogen. Well, it turns out there is an additional way for hydrogen to emit. Consider the ground-state of hydrogen. Protons and electrons have angular momentum -- they spin. In a hydrogen atom, the spins can be aligned, or counter-aligned. Because the two particles have opposite charges, the system has slightly lower energy if the spins are opposite than if they are parallel. When a hydrogen atom goes from the parallel spin-state to the anti-parallel spin-state, it loses energy. That energy is emitted as a photon. But it's only a little bit of energy, so the photon is in the radio regime, with a wavelength of 21 cm.
This has the excellent property of being totally unaffected by dust. So we can observe neutral hydrogen throughout the entire Galaxy. The HI gas is orbiting the center of the Galaxy, so clouds of gas in different locations have different velocities along our line of sight. This means that we can use the Doppler effect to map out the distribution of HI gas in the Galaxy as if we had a top-down view. (The missing section is in the part of the Galaxy where all the HI is moving across our line of sight, so we have to way of figuring out the relative distance of the gas in that sector.)
The frequency of a 21 cm photon is low. Given that the microwave background radiation imposes a minimum temperature of T ~ 3K, this means
This means the intensity of the 21cm radiation can be expressed by the Rayleigh-Jeans approximation:
Where the T in the above expression is the brightness Temperature. This is related to the kinetic temperature of the gas, but should not generally be taken to equal the kinetic temperature. In fact, the brightness temperature is not generally the measurable quantity. The measurable quantity is the Antenna Temperature, which is related to the brightness temperature by a filling factor:
The Antenna Temperature is related to the measured intensity per beam. But if the angular size of the beam is larger than the angular size of the source, then the flux from the source is diluted. The filling factor expresses how much of the beam is filled by the source.
The brightness temperature is related to the Excitation Temperature as follows:
The excitation temperature is not equal to the kinetic temperature in general, but for conditions typical of interstellar hydrogen, they are equal. A typical value for the kinetic temperature of the HI in the Galaxy is about 100 K.
In most cases, the HI is optically thin. That is, when one looks along any given line of sight, one can see the emission from all of the HI in that direction out to the edge of the Galaxy. Thus there is a straightforward mapping between the measured intensity and the column density (the number of atoms per unit area) of the HI. This is why one can use the kinematic information from the observed pattern of Doppler shifts of a set of HI spectra along many lines of sight, and use them to reconstruct a "top-down" map of the HI distribution in the Galaxy.
Thursday February 21
Hot young stars are able to ionize the gas that surrounds them. This gas then produces an emission-line spectrum. Such regions are called HII regions. They only occur around hot stars because a substantial flux of UV photons are needed to ionize hydrogen:
The optical spectrum of an HII region is dominated by the Balmer lines of hydrogen, and low-ionization level forbidden lines of metals such as [OII], [OIII], [NII], [SII]. As noted last week, these lines are very prominant in HII region spectra, but were not identified until the mid-20th century as they are due to the radiative decay of excited states that have such long radiative lifetimes that they are *always* collisionally de-excited at terrestrial densities.
The IR spectrum of an HII region is likewise dominated by hydrogen emission lines, but the lines are due to higher-level series than the Balmer lines (n=2). There are also very bright metal lines in the IR, and there is continuum radiated by hot dust grains.
The radio spectrum of an HII region has both very high order hydrogen emission lines (n=100 and higher), and free-free radio continuum (also known as bremsstrahlung. This continuum is generated by electron-electron scattering. As a result, the shape of the spectrum depends on the gas temperature.
One can use the optical, IR, and radio spectral observations to determine the physical state of the gas in HII regions. Typical gas temperatures and densities are roughly T~10000 K and n~100 atoms per cc.
The hotter a star is, the larger the flux of ionizing photons. Thus hotter stars can create larger HII regions. One can calculate the expected size of an HII region around a star of a given temperature for typical ISM conditions. The essence of the calculation involves studying the balance between the ionization and recombination rates for the gas. If one has some source of ionizing flux in a gas, one can write down these rates as follows:
Here alpha is a fudge factor that includes all the hard stuff we want to ignore. If we take the region in question to be a sphere, and assume that the gas is Hydrogen, then we can procede as follows:
This size is called the Stromgren Sphere. Note that the results of such calculations only roughly describe actual HII region properties (because the ISM isn't uniform). But as a pair of examples:
The first evidence of molecular material in the ISM was due to observations of CH and CN ISM absorption lines in stellar spectra beginning in the 1930s. Molecular hydrogen absorption line studies began with the advent of UV astronomy in the 1970s.
It was realized early on that the relative amounts of molecular and atomic hydrogen along any given line of sight was clearly correlated with the observed optical extinction in the sense that the more dust there is along a line of sight, the more molecular gas there is.
This pointed out an intimate relationship between molecular gas and dust. By far the easiest way to make molecular hydrogen in the ISM is to do it on the surface of grains. Also, as grains absorb uv photons, they shield the molecular gas from radiation that would otherwise dissociate the molecules.
Direct observations of molecular hydrogen emission are essentially impossible as the gas is too cold. Molecular hydrogen has a very rich spectrum of electronic and vibrational features, but they can only be excited at gas temperatures of 100s or 1000s of Kelvin. At typical molecular gas temperatures (10s of K), molecular hydrogen has no emission features. This is because it is a symmetric molecule, and thus does not have rotational features.
As a result, the workhorse tracer of molecular material in the ISM is CO. CO is an asymmetric molecule, and thus has a rich set of mm-wavelength rotational features. It is also the 2nd most abundant molecule in the ISM (after H2).
Dense clouds are rich in molecular gas, and the denser and colder the cloud, the more complex are the molecules seen within it. There have been more than 100 different molecular species identified in cold clouds to date. Molecular cloud densities are much higher than average HI densities (up to 10^6 per cc), and the largest molecular clouds can have masses up to a million Solar masses.
Stars form due to the collapse of dense knots in molecular clouds. The Galaxy currently forms new stars at a rate of about 1 Solar mass per year. And we observe stars to form in clusters. Single star formation turns out to be physically very difficult to do. The basic reason for this was first explained in the 1920s by Sir James Jeans.
For a gas to collapse, its self-gravity must overcome its thermal pressure.
This means the ingredients that matter are the pressure and density of the gas, and gravity.
Jeans simply took these quantities, and asked "How can these three things be combined to produce something with the dimensions of mass?" The answer is:
Putting in numbers to give us a convenient expression, this becomes:
If n~10^6 and T~100, the Jeans mass is about 30,000 Solar masses. One needs n~10^12 and T~10 to have a Jeans mass of ~1 Solar mass.
Consider the case of a cloud core that is just barely Jeans unstable. It will begin to collapse, and it will thus begin to radiate away energy from the collapse. Because the gas is optically thin, it collapses without heating up. So the Jeans mass of the gas drops, and the cloud fragments. This process continues until the individual cloudlets are sufficiently dense that they become optically thick. At that point they begin to heat up, and fragmentation stops.
The above is a hand-wavy argument for how a massive collapsing core ends up producing a cluster of many stars rather than one supermassive star. But it does not give a receipe for making the IMF. This is because it leaves out at least two crucial pieces of physics:
Consider a collapsing cloud with some initial angular momentum. To make things easy on ourselves, we will assume that the cloud is spherical (it will actually collapse to form a disk. This will introduce numerical factors into the analysis below, but will not change the essential physics). From this, we can make the following argument:
Now consider the collapse of the cloud. At any given point, the material in the cloud will be accelerating due to the combination of gravitational and centripetal forces. Obviously, there is no centripetal force parallel to the rotation axis. Perpendicular to the rotation axis, one can write down the condition for rotation to by dynamically important as follows:
The only way for collapse to continue along the perpendicular is for the cloud to fragment, and each smaller cloud to undergo its own disk formation.
Again, we shall use simple scaling arguments to study the conditions under which magnetic fields should be dynamically important to the star formation process. Roughly speaking, this will be when the magnetic potential energy is comparable to the gravitational potential energy. The analysis is as follows:
We procede by equating this with the gravitational potential energy as follows:
For typical molecular cloud numbers, this implies that magnetic fields become important when they are at the 10s of microGauss level. It is unlikely to be a coincidence that this is approximately the measured magnetic field strength in the ISM.
Now consider that a molecular cloud isn't entirely neutral. There will be some residual ionization fraction even in very cold clouds. Thus the magnetic field is frozen to the gas by interactions with the residual charges. The magnetic flux will be conserved as the cloud collapses. The consequence of this can be seen by considering the following:
The magnetic and gravitational potential energies have the same radial scaling. So if magnetic fields are initially important, they stay that way.
This ignores an effect known as ambipolar diffusion. The essential point to understand is that the ionization fraction is very low. Most of the gas is neutral, so it will collapse "through" the magnetic field. The collapse will be slowed down by the magnetic pressure support of the ionized gas. But as this is a small fraction of the total, the collapse can still procede.
At any instant, the conditions in the cloud are quasi-equilibrium. So the virial theorem should be at least approximately correct at all times. The instantaneous total energy is half the instantaneous potential energy, which becomes more negative as the cloud collapses. Thus the cloud must radiate energy away. One can get a feel for this by considering the following (assuming a spherical cloud for simplicity):
Now consider the following hand-wavy arguments:
When a protostar begins to form in the core of a molecular cloud, it collapses and heats up. The luminosity of such objects can be quite high. A Solar-mass collapsing protostar can easily have a luminosity 100 times that of the sun. But such an object is much cooler than the Sun, and it is typically embedded in its host molecular cloud. How can such objects be observed?
Observations of protostars are a very recent development, driven by the advances in IR Array technology. This is because protostars are copious emitters of near-IR light, and because they are embedded in dense molecular material with enough dust to block all the optical photons.
Recall the Stefan-Boltzman equation:
Now, let's consider what happens when a roughly Solar mass protostar collapses. Whenever something undergoes gravitational collapse, it heats up, and thus radiates away the gravitational energy. For a ~Solar mass protostar, the envelope surrounding the collapsing core is mostly neutral, so it has a high opacity. This means the envelope will transport energy convectively, just as in the Sun. Because the envelope is convective, it will have a roughly constant temperature as it collapses.
The envelopes of both RGB and AGB stars are mostly neutral, even quite deep in the star. This means that the star is convective throughout (recall that only the outer 30% or so of the Sun is convective). A result of this is that material that has been processed via nuclear burning is convected up to the surface. This process is known as Dredge Up.
This process alters the amounts of CNO relative to one another, and generally increases them relative to Hydrogen. In AGB stars, dredge-up mainly increases the amount of carbon in the photosphere. Combined with the low surface temperatures of AGB stars, this leads to the formation of carbon-based molecules such as CO, CH, CN. The lines of these molecules are very pronounced in the red part of the spectrum of such Carbon Stars.
The atmospheres of AGB stars are only weakly bound to the star, and this leads to very high mass-loss rates:
Thus the dredged-up material gets dumped back into the ISM, increasing the relative abundance of carbon. This material can either be in the gas phase, or, more commonly, in the form of dust grains. Basically, soot. This is the main source of dust grains in the Galaxy.
The surface mass-loss from AGB stars isn't just a gradual, steady thing. It is driven by a series of thermal pulses. These pulses are due to runaway bursts of helium burning in the stellar interior. When the interior pulses, the energy eventually lifts a layer of the surface off the star, and ejects it into the ISM. This will continue until the entire envelope is ejected. A 1 Solar mass M-S star will end up ejecting something like 0.4 Solar masses of material in this manner.
At this point, the exposed surface of the star is, essentially, the stellar core. The surface is still extremely hot:
And the envelope is now diffuse extended gas, surrounding the star. Thus the gas becomes an emission-line nebula. Such objects are called Planetary Nebulae.
We can measure the expansion velocities from the Doppler shifts. And, for the closer examples, we can measure the proper exapansion by comparing current images with archival plate material from 100+ years ago. The expansion timescales of PNs are 1000-10000 years. They are imporant sources of CNO in the interstellar medium.
There are about 100 known SNR in the Galaxy. Most of these are sufficiently far away that they are only detected in the radio and the x-rays (because of optical extinction).
There are two classes of SNR: Filled and shell. In both, the radio spectrum is due to synchrotron radiation. The optical emission in shell SNRs is emission-line radiation (the same lines as one sees in HII regions, but at different relative strengths). In filled SNR, the optical emission is due to both emission-line radiation and optical synchrotron. A shell SNR is basically a blast wave propagating through the ISM. A filled SNR is also powered by energy from a central pulsar (recall the discussion of the Crab SNR from the compact objects unit). In either type, the expansion timescale is about 10^5 years.
Synchrotron Radiation is caused by relativistic electrons confined by a magnetic field. As the electrons are forced to spiral around the field-lines, they are constantly accelerated, and thus they radiate. The spectrum of the radiation is:
For a typical Galactic B-field strength, this requires electron energies of about a GeV to produce radio synchrotron. Very high energies, indeed.
Tuesday February 26
The existence of a very hot phase of the ISM was first suggested in the 1950s as a means of providing pressure to confine the cool ISM. The first observational evidence of this material was uv absorption line studies showing species such as OVI, NV, and CIV. The temperature required to excite such states is roughly a million Kelvin. But the density is very low - only about 10E-6 per cc. This gas is very extended. It has a much larger scale-height than the cool ISM, and thus does, in fact, act as a means of pressure confinement.
The source of this gas is heating due to Supernova explosions, The Local Bubble is a local example of this.
Cosmic Rays are high-energy particles. Below energies of about 10E8 eV we cannot detect them as they are swamped by Solar wind particles. Above 10E8 eV, the cosmic ray spectrum declines to higher energies. The most energetic examples have energies of ~10E20 eV.
About 90% of cosmic rays are protons. Another 9% are Helium nuclei. The remaining 1% are heavy nuclei and electrons. The moderate-energy cosmic rays are excited by Supernovae. The source of the most energetic cosmic rays is a mystery. They appear to be of extragalactic origin, as the Galactic magnectic field is too weak to confine them.
Stars form in clusters.
We observe this to be the case, and simple theoretical arguments (Jeans mass arguments that will be made in detail when we discuss the ISM) are able to explain this observation, at least in a general way.
Star clusters are essential tools in astrophysics because they provide samples of stars that are
It was first recognized in the mid-20th century that the youngest "field" stars (stars not clearly parts of "clusters") were not randomly distributed. That is, they tended to group together, forming structures called OB Associations.
These OB Associations are extended groups of young stars that do not form a gravitationally bound structure. Thus they will disperse with time due to the random motions of the stars in the association, and to gravitational interactions with surrounding material. The timescale for typical associations to disperse is on the order of ten million years. Thus, by the time the O and B stars have evolved the association is no longer recognizable as such.
The stars in an OB Association formed from a single molecular cloud core that was not only gravitationally bound, but unstable to gravitational collapse. Why is the subsequent association unbound?
This harkens back to the situation involving binary star systems in which one of the stars is massive enough to produce a supernova. Recall that one can make an energetics argument that if more than half of the mass of the binary is lost due to the supernova, the binary should become unbound. That turns out to be a very general, and very useful point. For any self-gravitating system, if more than half the mass is lost, the system becomes unbound.
Star formation is a very inefficient process, and young stars return a great deal of energy to the ISM surrounding them. Thus it appears that in many (if not most) cases, when a group of young stars forms, a sufficient amount of energy gets dumped into the surrounding medium that a majority of the initial mass is lost, and the resulting collection of stars is not gravitationally bound together.
As our ability to study star formation has improved, younger analogs of the OB Associations have been recognized. In these T Tauri Associations, the stars have not yet reached the main sequence, and are often still embedded in the molecular cloud out of which they are forming.
These are collections of 10s to 1000s of stars. Although they are technically gravitationally bound, they will disperse over time due to energy input from stellar evolution and external gravitational interactions. This leads to the consequence that the least well-populated open clusters are all very young. The oldest clusters (5-6 Gyr) are all massive because they are the only ones that can survive for that long.
Because of this the typical open cluster is something like 10E8 years old. The Pleiades is an excellent nearby example. A color-magnitude diagram of the Pleiades shows that the A stars are still on the main sequence.
The traditional limit of ~50 pc on direct trigonometric parallax led people to devise many ways of measuring distances to stars in the local disk. The advent of the Hipparchos catalog has been an enormous help in sorting out the disk distance scale (parallaxes out to ~1 kpc). But other techniques remain important. One of these techniques is the Moving Cluster Method, also known as statistical parallax, cluster parallax and kinematic parallax. The method takes advantage of the coherant motion of cluster stars with respect to the Sun, and the optical illusion that parallel lines meet at infinity. If one can measure the radial velocities (the easy part) and proper motions (the hard part) of many cluster stars, one can derive a geometric distance estimate.
This method was historically absolutely critical for setting the distance scale to star clusters, and thus for attaching absolute magnitudes to cluster stars for the purpose of studying the effects of stellar evolution. The method was so important historically because even the nearest open cluster (The Hyades) is distant enough that one cannot determine a good distance for it from ground-based trigonometric parallax.
All of the stars in the cluster should have nearly the same space velocity. If this is so, then the measured proper motion vectors of the stars in the cluster should all point away from a spot on the sky called the Convergence Point.
Once this Covergence Point is determined, one can use trigonometry to set up the following calculation for the individual distance estimates of the stars in the cluster:
Note that one determines a distance estimate for each star in the cluster (with a measurable proper motion), and the determines the cluster distance by taking the average of the individual distance estimates.
Because cluster stars are all the the same distance, one does not need to know what that distance is to plot the Color-Magnitude Diagram (CMD) for a star cluster. One simply plots the apparent magnitudes and observed colors of the stars. When one does so, one sees that most open cluster CMDs are dominated by Main Sequence stars. This is another way of saying that the clusters are relatively young.
One observation that astronomers in the early 20th century realized could be made, but that was demanding enough that it was not actually made until the 1980s is the detection of the sequence of binary stars above the MS of a cluster CMD. Consider the case of a binary composed of two equal-mass MS stars. Because they have the same mass, they will have the same temperature, and thus the same color. As a result, the binary will be displace straight up from the MS locus of the CMD, and the amount of the displacement will be as follows:
As photometric precision increased with the advent of CCD detectors, this binary displacement sequence was found in more and more clusters.
I have mentioned that the study of clusters is crucial to our understanding of stellar evolution. The basic reason for this is that the Absolute Magnitude and Temperature of the Main Sequence Turn-Off (MSTO) is a monotonic function of cluster age. Thus one can determine an excellent relative age sequence for open star clusters simply by determining their MSTO points. The more luminous and hot the stars at the MSTO the younger the cluster. This is a purely observational statement. In order to attach actual ages (in years or whatever) to clusters, one must use stellar evolution models.
Also known as Main-Sequence Fitting. Comparing the MS loci of two clusters tells us about their relative ages. It also tells us about their relative distances. One must be able to measure and correct for extinction to take advantage of this. And one must also know the distance to at least one cluster. This underscores the historic importance of statistical parallax. Using statistical parallax, astronomers determined distances to nearby clusters such as the Pleiades and the Hyades. They could then plot CMDs for these clusters using Absolute Magnitudes instead of apparent magnitudes.
If one has a set of good distances to disk open clusters, one can determine a good calibration for the absolute magnitude of the Main Sequence as a function of color (or temperature). And one can then apply that calibration to clusters of unknown distance (but KNOWN extinction!) to determine their distances. In practice, one can then simply slide the CMD of the new cluster up or down in magnitude until the MS overlaps that of the clusters of known distance. One thus determines the distance modulus of the new cluster:
This technique extends the range over which distances can be estimated from the limit of statistical parallax (which is basically set by the distance out to which we can measure significant proper motions) out to the limit of our ability to obtain good photometry of Main Sequence stars. But there is an effect that one must be careful of. If one plots the H-R diagram for local disk stars, and local halo stars, one finds that the halo stars populate a Main Sequence that is shifted to the blue of the disk Main Sequence at a given Absolute Magnitude (or shifted fainter at a given color). This leads to the name Sub-dwarfs for the halo MS stars. I will return to this issue shortly, in the context of the CMDs of Globular Clusters.
There are between 150 and 200 globular clusters in the Galaxy, where the range is due to our inability to detect clusters through the absorbing medium in the disk of the Galaxy. Globular clusters are both much more massive and much denser than open clusters:
The "rc" above is the core radius. That is the radius that encloses about half of the total mass of the cluster. Note that there are zero stars within 1 pc of the Sun. Thus the stellar densities of globular clusters are a factor of 1000 or more higher than is typical for the Disk.
Another feature of globular clusters is that they have very low luminosity, cool MSTOs. Typically, the hottest MS stars in a globular cluster are K stars. Thus globular clusters are much older than even the oldest open clusters in the disk.
A feature of globular cluster CMDs that was a puzzle for some time shows up when one attempts to plot the Absolute Magnitude CMD for both a globular and an open cluster. If one does one's best to account for the differences in distance and foreground reddening, one finds that the globular cluster MS lies below/to the left of the open cluster MS. That is, MS stars of a given temperature are less luminous in globular clusters than in open clusters. Or MS stars of a given luminosity are hotter in a globular cluster than in an open cluster. This is a real effect. And it is the result of the much lower relative abundance of heavy elements (or metallicity) in globular cluster stars than in open cluster stars.
Because metals are ionized at lower temperatures than hydrogen or helium, they are a very important source of electrons in stellar interiors. In low-metallicity stars, there are fewer electrons and thus the opacity of the interiors is lower. This causes stars of a given mass to be more compact and hotter than higher metallicity stars of the same mass.
Open cluster stars (and stars in the disk in general) have metallicities from about half that of the Sun up to a little higher than Solar. The most metal-rich globular clusters only reach up to about half Solar metallicity, and the most metal-poor have metallicities less than 1% that of the Sun.
One important constituent of globular clusters are the RR Lyrae Stars. These are intermediate-temperature Horizontal Branch stars. The middle of the HB cuts across the Instability Strip. Thus RR Lyrae are pulsational variables for the same basic physical reason as Cepheids. The difference is that Cepheids are young metal-rich stars (they are supergiants of several Solar masses), while RR Lyrae stars are old, metal-poor stars. Both classes of variable can be used for distance estimation, and as they probe different stellar histories, they are very good compliments to one another.
Notice the following curiosity: Globular clusters are very massive, dense star clusters. They are much more massive and dense than open star clusters. They are also much older than even the oldest the open star clusters. In other words, very massive, dense star clusters do not form in the Galaxy at the current time. They ONLY formed long ago (about 12 Gyr ago).
Why?
The beginning of the answer to this questions is in a consideration of the Jeans equation:
Note the temperature dependence. The hotter the gas, the higher the mass for gravitational instability. Globular cluster stars have very low metallicities. Just as the metals are the dominant source of opacity in stars, they are also the dominant mechanism for cooling in interstellar gas clouds. A gas of pure hydrogen can cool by line emission and free-free emission down to the temperature at which the gas becomes neutral. That is about 3000 K. Once the gas is neutral it is very difficult for it to cool further. If there is a small contamination of heavy elements, then line emission from the metals will dominate the cooling.
So if the metallicity of the gas is low, then it stays hotter. Thus the mass needed to initial collapse has to be higher, and gas density needs to be higher at a given mass. These factors combine to argue that star clusters forming from low-metallicity gas must be both more massive and denser than clusters that form from higher-metallicity gas.
One can ask what the "size" of a globular cluster is. The core radius, as noted above, encloses about half the total mass. But clusters don't exist in isolation. They are part of the Galaxy. Thus one can make a calculation of the maximum radius of the cluster subject to the gravitational field of the Galaxy. Stars that move farther from the cluster center than this tidal radius are likely to be stripped away by the Galaxy. For typical cluster properites and orbits, the tidal radii are in the range 10-100 pc. Thus a globular cluster extends far beyond the obvious concentration of stars.
The determination of cluster tidal radii was first done in the early 1960s. And it led nearly immediately to the realization that clusters should lose stars from their tidal limits when they passed closest to the Galactic Center. Searching for such tidal debris turns out to be a very demanding observational program. Thus, even though the idea was there as early as ~1965, it took until ~1995 for any real observational evidence of this tidal stripping to show up.
Thursday February 28
The standard Astronomy 101 story is that "Globular Clusters are part of the Galactic Halo". This is an oversimplification. In fact, there are very clearly two populations of globular clusters in the Galaxy. One really is a part of the Halo. These Halo clusters have the following general properties:
We do not have a sufficiently detailed understanding of the structure and evolution of stars to correctly model the evolutionary history of stars of a range of mass and chemical composition on purely theoretical grounds. Instead, we use a feedback process in which theoretical stellar models are compared to observed spectra of stars, and theoretical stellar evolution models are compared to the observed color-magnitude diagrams of clusters. The result of this process is the creation of sets of isochrones (lines of constant age) that can be compared with observed cluster CMDs.
One of the utilities of isochrone fitting is that it can be applied to determine both the age and composition of a cluster. This is because the abundance of heavy elements has a large effect on the colors of evolved stars. As it happens, most of the strong spectral features of neutral and low-ionization states of heavy elements are in the uv/blue part of the spectrum. Red giant are cool enough for these to be the dominant ionization states of the metals in their atmospheres. Thus the higher the heavy element abundance, the more of the blue flux is removed by spectral lines and the redder in color is the star.
When one compares the CMDs of globular clusters of a range in metallicity, the result is that the higher the metallicity of the cluster, the redder the RGB. Similar results are found when examining the Horizontal Branch stars in clusters (in old clusters, low metallicity results in blue HB stars, and high metallicity results in red HB stars). This is a much larger effect than that seen for the main sequence stars. When one takes high-dispersion spectra of globular cluster RGB stars, one can use these spectra to calibrate the relation between metallicity and RGB color.
As a point of definition, astronomers often use a metallicity scale that is referenced to the Solar metal abundance, and is logarithmic in form. Also, Iron is typically used as a proxy for all heavy elements. (This is actually not the best choice in many cases, but it is the convention). The resulting abundance scale is defined as follows:
Note that one can make a similar definition for any abundance ratio.
There are a number of ways one can measure the metallicity of a cluster. As one attempts to study more and more distant clusters, one is left with increasingly crude methods.
A final comment I want to make about Stellar Population studies in the context of clusters is that HST observations of clusters in the last decade have led to the clear detection of White Dwarf sequences in several clusters. This means that we have samples of White Dwarfs all at the same distance, all at the same age, and all with the same initial chemical composition. Our understanding of white dwarf physics and evolution is rapidly improving as a result of this.
There is a very large range in the degree of concentration of globular clusters. The concentration is typically defined as a ratio of the core radius to the tidal radius (different workers make somewhat different definitions, so one must take care when reading the papers). At the extreme low-concentration end of the scale, all of the clusters are from the Halo population. This doesn't have to mean that low concentration systems only formed in the Halo. It could be that such clusters formed as part of the Disk population as well. But they no longer exist because of the much larger Galactic tidal effects on the Disk clusters (a larger Galactic tidal effect being a result of the much smaller average radial distance of Disk globulars from the Galactic Center).
The term core radius has been loosely defined as the radius enclosing half the cluster mass. This is not the strict definition, but it will generally suffice for our purposes. When one examines the radial profiles of globular clusters, one typically finds an inner region with an essentially flat surface density of stars. That is, a core. The formal definition of the core radius is tied up with this observed core structure.
Some clusters do not show these core structures. Instead, they show density cusps. That is, the surface density of stars increases all the way down to the resolution limit of the observations.
It is also worth commenting on the tidal radius. If a globular cluster existed in isolation, then the surface density of stars would decrease in a smooth way with increasing radius, and there would be no break in the surface density profile. But clusters exist in the gravitational potential field of the Galaxy. And this imposes a cut-off radius on the cluster. Stars that move further than this distance from the cluster center are likely to be gravitationally stripped off by the Galaxy. Thus the observed cut-off radii of clusters are called the tidal radii.
If one considers long-enough timescales, globular clusters are clearly not in equilibrium. This comes back to the earlier discussion of tidal truncation by the Galaxy. Over time, globular clusters lose stars, and thus lose internal energy. As a result of this loss, the cluster will slowly contract. This is a potential problem for the analogy between a cluster and a star, because we describe stellar structure with the physics of thermodynamic equilibrium. If clusters are not equilibrium systems, is the analogy really a fair one?
The answer is "Yes.", and the reason is that the timescale on which clusters are non-equilibrium systems is very long. On the order of a billion years. But the time scale for clusters to reach internal equilibrium is much shorter than this. One can get a sense of this by thinking about the relaxation time for gravitational interaction between any two stars in a globular cluster.
If one considers a cluster with a number density of stars equal to n, and the typical velocity of a star with respect to the cluster center of v, one can define a radius r of that star's strong gravitational influence. As that star moves through the cluster, it sweeps out a volume V. If one wants to know how long it takes between significant interactions, one wants to examine the case in which nV=1. The mathematical details are thus as follows:
Now, one must consider what to use as the radius r. Given that we are treating gravitational interactions, this suggests that the radius be defined as the radius at which the gravitational potential energy between a pair of stars is equal to the kinetic energy of a given star:
One can then insert this into the relation for relaxation time, and procede with the algebra. The result of this calculation is as follows:
If one puts reasonable numbers into this equation (m of 1 Solar mass, n of 1000 stars per cubic parsec, and v of 10 km per second are what I used), one finds a value for the relaxation time of a few Gyr.
But this is the timescale for strong encounters. If one considers the cumulative effects of many weak encounters, the result is changed. Consider the effect on some star M as it passes by some other star m, with a close approach distance of some b, known as the impact parameter. The interaction will cause a small deflection in the trajectory of the star M. If that deflection really is small, then the result can be summarized as follows:
Note that for the impulse approximation to be valid, the deflection angle phi must be small. But this is identical to saying that the impact parameter must be much less than the strong collision radius (about 1 AU for the disk) calculated above.
Now consider a star moving through a field of stars (all of which are quite distant). Each of those stars will perturb the motion of the star by a little bit. The upshot of this is that after a certain time, the star will have "forgotten" its initial trajectory. This time is called the relaxation time of the system. We discussed the calculation of the relaxation time in class. Here I quote the result:
The lambda term comes from the integration about the impact paramter. For a large range of conditions ln(Lambda) is about 20. Putting in typical disk numbers to the above implies a disk relaxation time of about 1E13 years, or three to four orders of magnitude larger than the age of the Sun. So orbits in the disk are not significantly perturbed over a stellar lifetime.
The typical stellar density in the core of a globular cluster is much higher than in the disk. To see how important relaxation is, one usually compares the relaxation time with the core crossing time. The crossing time is just the time a star would take to cross the core of the cluster if it had a velocity equal to the velocity dispersion of the cluster. For globular clusters, the two time scales are about
So one can ignore relaxation for short time-scale problems. But it becomes important for things like studying the long-term evolution of globular clusters. Thus any internal inhomogeneities in a globular star cluster should smooth out on timescales much shorter than the cluster age, and one CAN treat the cluster as an equilibrium system on even million-year timescales.
This turns out to be a very important bit of information, because it means that one can use kinematic information from globular clusters to determine their masses. Thus one can study their observed mass-to-light ratios. The result of such studies is that globular clusters have M/L ~ 3 to 5 in Solar units. In other words, there is no evidence for the existence of dark matter associated with globular clusters. Globular clusters turn out to be the most massive stellar associations that do NOT show evidence for dark matter. We shall return to this point in the context of the Galaxy, and galaxies in general.
