HW2 Soln - ASTR 204 Home Homework 2 You are a daring, young...

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Unformatted text preview: ASTR 204 Home Homework 2 You are a daring, young astronomer who works at the forefront of dark matter research. You point your telescope at a giant, spiral galaxy. Describe three possible experiments/observations that might allow you to determine whether that galaxy has a halo composed of dark matter. Dark matter, as its name indicates, is not luminous matter. It either emits no light at all, or is so faint that it is beyond the ability of our current instruments to detect directly. Therefore, the only way we can tell it is there is by looking for the effects it has upon luminous objects. If a spiral galaxy has a dark halo that extends well beyond the luminous disk, the dark matter's gravitational force will affect the motion of stars inside the luminous disk, the motions of dwarf galaxies orbiting beyond the luminous disk that are satellites of the spiral galaxy, and the path of light that passes near the galaxy's dark halo. Using these effects, the following are some examples of how one might detect the dark matter in the halo of a giant spiral galaxy. 1) Rotation curve. The stars in a spiral galaxy are rotating around the galaxy center as our planets do around the sun. It is the gravitational force of the galaxy that provides the acceleration needed for this motion. The larger the gravitational force is, the higher the rotation speed can be. So we can study the rotation velocity of the stars to detect the gravitational force and therefore the distribution of matter in this galaxy. As we know, the gravitational force decreases as the square of the distance from the source. In a galaxy where mass is distributed roughly as a sphere, the gravity at each point is proportional to the enclosed mass of the sphere inside this point. Therefore, gravitational force at each point in a galaxy is related to the enclosed mass and to the inverse square of the distance from the center of the galaxy. (In other words, a star at distance r from the center of the galaxy feels a gravitational force equal to the mass inside the star's orbit divided by r2.) Equating the gravity and the acceleration needed for a star rotating with a speed v at a distance r from the galaxy center, we get v2/r ~ M/r2, or v ~ sqrt(M/r) where M is the enclosed mass of the sphere of radius r. If that mass over volume (= density) in the galaxy is roughly constant, then M increases as the cube of r. In this case, the equation shows that the rotation velocity increases as r does. According to this equation, the rotation curve, the measured variation of v with r in a galaxy, demonstrates how mass changes with radius. As a result, the rotation curve tells is how much mass is enclosed at each radius and how that mass is distributed with radius. In the (erroneous) case where all the mass of the galaxy is distributed as the light is (no dark matter halo), then we'd expect that most of the mass of the galaxy would be in the brightest part of the galaxy -- the bulge. In such a case, and given the equation for the rotation curve above, we'd predict that, in the inner (brightest, most massive) part of the galaxy, the rotation speed will continue to increase with radius as the stars at each radius outward see more and more mass within their orbits. This is because the enclosed mass here increases faster than the inverse square of the distance decreases, leading to increased gravitational force as radius increases and therefore allowing faster speeds. At even larger radii (outside the bulge), however, stars at each increasing radius should experience less and less gravitational force because the amount of enclosed mass is increasing only slowly as we move further and further away from the bulge. At large radii then, the gravitational force is dominated by the inverse square of the distance, any fast moving stars cannot stay bound to the galaxy, and so the rotation curve must decline (as the inverse square root of the radius). However, in the case where the galaxy has a dark matter halo, the dark matter's gravity will change the behavior of the rotation curve. More specifically, if the dark matter is more extended than just concentrated in the bulge, the rotation curve should remain flat at very large radii. By comparing the observed rotation curve to the predictions above, you can tell whether the galaxy has a dark matter halo. Measurements show that the rotation curve of a spiral galaxy doesn't in fact decline at larger radii, but rather remains flat far beyond the visible boundary of the galaxy. This is the evidence that galaxies have dark matter halos extending much beyond their luminous edges. 2) Gravitational lensing. According to Einstein's Theory of General Relativity, gravitational force not only changes the way the objects move, but also the direction in which light travels. If there is a huge amount of mass in a certain volume, it will produce a strong gravitational force that will bend the space around it. Light traveling near this mass will follow the curvature that the mass makes in the space. In the case of the mass associated with a galaxy, this mass will act as a focusing lens and bend the light traveling towards us from an object like another galaxy or a quasar behind the lensing galaxy. As a result, we would see two effects on the image of the background object: a) The gravitational lens galaxy will distort the morphology of the background object. A point source like a quasar might then be observed as some arc structures, paired point sources, or, sometimes, a ring. (click here to see some extraordinary pictures of gravitationally lensed objects). b) The gravitational lens basically works as an amplifier, making the image of the background object appear far brighter than it actually is. Without this lensing effect, we might not detect those background sources that are too distant or otherwise faint to be naturally detected. Because the distortion introduced by lensing is mostly determined by the mass of the lens galaxy, we can use the observed Einstein angle, the angle over which the light from the background object is deflected by the foreground lens, to determine the mass of the lens. In particular, the Einstein angle is the size of the observed arc, ring, or separation on the sky and is proportional to the square root of the galaxy mass (ref. Longair 1998). By comparing the mass deduced by this way to the total visible mass in the galaxy, we can determine if the galaxy has a dark matter halo. The answer is yes from many gravitational lensing observations. It is important to keep in mind that this is distinct from microlensing. Microlensing involves a massive object, usually a Massive Compact Halo Object (MACHO), within our own galaxy passing in front of a background star. The same principles of Einstein's theory apply in both situations with only the mass of the lensing object changing. Above the massive object is an entire galaxy while with microlensing it is a brown dwarf, black hole, white dwarf, etc. The MACHO is small enough that we do not see any arcs, paired sources or rings but we still see the amplification of the background star. This amplification has a distinctive signature that allows astronomers to distinguish it from regular variable stars. When observing a giant spiral galaxy we would not be able to detect any microlensing events within this galaxy because they are rare enough that they will not affect the total light emitted by the galaxy. Also microlensing is only sensitive to MACHOs. If the dark matter was in the form of Weakly Interacting Massive Particles (WIMPs), we would not detect any dark matter with microlensing. 3) The effect on nearby satellite galaxies. The gravity of the galaxy will not only affect the motion of the stars inside of the galaxy, but also act upon nearby galaxies. As we know, most of the galaxies in the Universe are not isolated. They tend to form groups, clusters, and even larger scale structures. On smaller scales, many galaxy have nearby neighbors, including dwarf satellite galaxies orbiting around them. For example, our Milky Way galaxy is orbited by a number of dwarf galaxies, such as the Carina Dwarf, Draco Dwarf, Leo II, and the Large Magellanic Clouds. If we measure the speed at which these dwarf galaxies rotating around their parent galaxy, we can determine the mass of the central galaxy. (Note that the rotation speeds of the dwarf galaxies depend only on the mass of the parent galaxy, not on the masses of the dwarfs themselves.) On the larger scales of the galaxy groups, galaxies are also seen falling into each other. For instance, the two dominant members of our Local Group, the Milky Way and M31, are falling together at a speed of about 100 km/s (ref. Binney & Tremaine 1994) due to their mutual gravitational attraction. In this case, if we can determine the mass of the companion galaxy and its infalling speed, we can determine the mass of the first galaxy. This method may seem to be circular, but sometimes it is practical as not every galaxy has a well-measured rotation curve and or detectable background lensed images. From these dynamical methods, you can determine the mass of the galaxy you are observing and compare it with the visible mass of the galaxy. As it turns out, the dynamical mass is usually much larger than the visible mass of the galaxy, providing evidence for the existence of dark matter. Note: Although this problem specifically refers to a spiral galaxy, the 2nd and 3rd methods actually work on all types of galaxies (including elliptical and irregular galaxies). However, the rotation curve is a special feature of spiral galaxies. Other types of galaxies normally don't have stars systematically rotating around the center. Nevertheless, the motions of stars in elliptical and sometimes in irregular galaxies can be used to determine the amount and distribution of dark matter in such galaxies. Methods such as measuring the motion of galaxies within a galaxy cluster and the temperature of hot X-ray emitting gas are good at detecting dark matter but do not work in the specific case of observing a giant spiral galaxy. The motion of galaxies in a galaxy cluster works on the same basic physics as the motion of dwarf galaxies around a large spiral galaxy. The large velocity tells us that there must be a large halo of dark matter. The motion of the galaxies in a cluster depends on the mass of the entire cluster, not on the individual galaxies just as the motion of dwarf galaxies depends on the mass of the large spiral galaxy they orbit, not on the mass of the dwarf galaxies. So measuring the motion of galaxies in a galaxy cluster allows us to measure the dark matter of the entire cluster, not of an individual galaxy in the cluster. As for hot X-ray emitting gas, spirals galaxies do not have enough for this technique to work. In galaxy clusters the high temperature and density of the gas requires a large amount of matter to keep it bound to the cluster. Since the gas is very hot the gas atoms are moving very quickly and would fly out of the galaxy if there was no dark matter in the galaxy. Observing the hot gas using an X-ray telescope can be used to measure the amount of dark matter in galaxy clusters and some elliptical galaxies since they contain hot X-ray emitting gas. Spiral galaxies do not have very much hot gas and this method would not be viable. References [1] Binney, J. & Tremaine, S. 1994, Galactic Dynamics, Third printing, Princeton University Press, p605 [2] Longair, M. S. 1998, Galaxy Formation, Springer-Verlag Berlin Heidelberg New York, p95 ...
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This note was uploaded on 04/02/2008 for the course ASTR 204 taught by Professor Zabludoff during the Spring '08 term at University of Arizona- Tucson.

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