ASTR001 HW 6
12) What is the Cosmological Principle, and how is it important to our understanding of the universe?
The Cosmological Principle states that the matter in the universe is uniformly distributed without a center or an edge. This idea is important to our understanding of the universe because it presumes that we do not live in a special place in the universe, and thus allows us to make models of how the universe as a whole behaves.
9) What are cosmic rays? Where do they come from?
Cosmic rays are electrons, protons, and neutrons that move at close to the speed of light and permeate interstellar space. We think that they come from supernova explosions.
4) How can we use orbital properties to learn about the mass of the galaxy? What have we learned?
If we know a star's orbital period and orbital radius, we can use Newton's version of Kepler's third law to determine the mass of the galaxy, with a minor warning: We get only the mass of the galaxy within the orbit of the star we examine. What we have learned from our studies to find the distribution of mass in the galaxy is that the stars in the galactic disk orbit at about the same speeds. This observation tells us that most of the galaxy's mass resides far from the center and is distributed throughout the halo.
7) How do the galaxy types found in clusters of galaxies differ from those in smaller groups and those of isolated galaxies?
Elliptical galaxies are much more common in large galaxy clusters than they are among isolated galaxies. Half of the large galaxies in large clusters are ellipticals, while among isolated galaxies they make up only about 15%.
10) How can we use variations in luminosity to set limits on the size of an active galactic nucleus? Give an example.
If we see an object brighten in 1 hour, we know that the object cannot be larger than 1 light-hour in size. If it were, light from the far side of the object would start arriving at our telescopes over an hour after the light from the front side and we would see the brightening take longer than an hour to occur.
45) Distances Between Galaxies. If you were to divide the present-day universe into cubes with sides 10 million light- years long, each cube would contain, on average, about one galaxy similar in size to the Milky Way. Now suppose you travel back in time, to an era when the average between galaxies was one quarter of its current value, corresponding to a cosmological redshift of z 5 3. How many galaxies similar in size to the Milky Way would you expect to find, on average, in cubes of that same size? In order to simplify the problem, assume that the total number of galaxies of each type has not changed between then and now. Based on your answer, would you expect collisions to be much more frequent at that time or only moderately more frequent?
If we traveled back in time to an era when the distance between galaxies was a quarter of what it is today, then we would find that the density of galaxies would be 4^3=64 times higher since the galaxies would be four times closer together in each of the three dimensions. In a cube that today contains one galaxy the size of the Milky Way, we would instead find 64 galaxies this size. With this higher density, we expect that the number of collisions between galaxies would be much higher as well.
7) What do we mean by inflation, and when do we think it occurred?
Inflation refers to the dramatic expansion of the universe that brought it from the size of an atom to the size of the solar system in a mere 10^(-36) second. We think that this occurred at the end of the GUT era when the strong force froze out of the GUT force, around 10^(-38) second into the universe's history.
12) What are ionization nebulae, and why are they found near hot, massive stars?
Ionization nebulae are colorful, wispy blobs of gas found near hot stars. They require the hot stars' ultraviolet light to raise the electrons in the atoms of the gas to high energy levels or to ionize the atoms completely. When the electrons fall back down to the lower energy levels, they emit photons, causing the nebulae to glow.
49) The Black Hole in M31. Measurements of star motions at the center of the Andromeda Galaxy, also known as M31, show that stars about 3 light-years from the center are orbiting at a speed of 400 km/s. These stars are suspected to be orbiting a supermassive black hole. Use the orbital velocity law from Mathematical Insight 21.2 to estimate the mass of this black hole.
Mathematical Insight 21.2 tells us that we can find the mass of the central black hole from the orbital speed and radius of the stars around it with the formula: M=rxv^2/G We will need our values of the radius and speed in meters and meters per second to use this formula. The speed is given as 400 kilometers per second, so it is easy to convert this into 4x10^5 meters per second. The radius requires us to recall that 1 light-year is 9.46x10^15 meters. Using this conversion factor, we see the 3-light-year orbital radius is equal to 2.84x10^16 meters. So plugging these values into the formula above, we discover that the mass of the central black hole in M31 is 6.8x10^37 kilograms or 34 million times the mass of the Sun.
54) Cluster Mass from Hot Gas. The gas temperature of the Coma Cluster of galaxies is about 9x10^7 K. What is the mass of this cluster within 15 million light-years of the cluster center?
Mathematical Insight 23.3 tells us that we can find the average speed of hydrogen nuclei at a given temperature with the equation: v(H)=140(m/s)xsqrt(T) This tells us that hydrogen nuclei in a gas at 9x10^7K have an average velocity of 1.3x10^6 meters per second. We can use this speed along with Newton's law of universal gravitation to find the mass of the cluster: M=rxv^2/G The radius is 15 million light-years or 1.4x10^23 meters. Plugging in the speed found above and this radius, we find that the mass of the cluster is 3.5x10^45 kilograms or 1.8x10^35M(sun).
40) The Color of an Elliptical Galaxy. Explain how the color of an old elliptical galaxy has changed during the last 10 billion years. How might you be able to use the galaxy's color to determine when it formed?
The color of an elliptical has changed as the population of stars, most of which formed at some early time, ages together. So similarly to a globular cluster, an elliptical's color gets redder as the high-mass stars evolve off the main sequence to become red giants.
4) Distinguish between the disk component and the halo component of a spiral galaxy. Which component includes cool gas and active star formation?
The disk component of spiral galaxies is a flat disk in which stars follow orderly, nearly circular orbits around the galactic center. The disk contains cool gas and active star formation. The spheroidal component of spiral galaxies has a round shape with stars orbiting at many different inclinations.
6) What is a starburst galaxy? What do starbursts tell us about the star-gas-star cycle within galaxies?
A starburst galaxy is a galaxy in the present-day universe that is forming stars at a prodigious rate. Starbursts can explain the ages of the stars in some of the small elliptical galaxies in the local group: The large number of supernovae after the first starburst stage created a galactic wind that blew most of the gas out of the galaxy, ending star formation until the gas could cool and reaccumulate within the galaxy for the next starburst. If ellipticals in general experienced a massive starburst episode early in their lives, that could explain the lack of ongoing star formation in ellipticals and why their stars are on average fairly old.
1) Define dark matter and dark energy, and clearly distinguish between them. What types of observations have led scientists to propose the existence of each of these unseen influences?
Dark matter is matter that gives off little or no light. Dark energy is the name given to whatever it is that is causing the universe's expansion to accelerate. While they have similar names, they are not similar in nature. Dark matter is massive, and behaves like something that gravitates, and is really dark in the sense that we do not see it. Dark energy is not matter and it is dark in the sense that we can't see it. Dark matter has been detected by carefully observing the gravitational effects on matter we can see in clusters of galaxies and in galaxies. Dark energy has been detected by observing the rate of expansion of the universe, from studies of white dwarf supernovae. (There is also less direct evidence for dark energy that goes beyond the scope of the course.)
11) Summarize each of the major links in the distance chain. Why are Cepheid variable stars so important? Why are white dwarf supernovae so useful, even though they are quite rare?
For planets, we can use radar ranging to determine distances. For the closest stars, we can use parallax to determine the distances directly. For more distant stars within our galaxy, we use main-sequence fitting, a technique that uses the known brightness of the various classes of stars to turn main-sequence stars into standard candles. This does not work to find the distance to other galaxies since most main-sequence stars are too faint to be seen in other galaxies. In this case, we use Cepheid variables, variable stars whose luminosities are related to the period of their brightness variations. These stars have been very important since they have allowed us to measure the distances to nearby galaxies and played a key role in Edwin Hubble's discoveries. Even Cepheid variables cannot tell us the distance to most galaxies, however, because we cannot see them from so far away. White dwarf supernovae are much brighter and all have nearly the same peak luminosity. Although such supernovae are rare, they are bright enough to see to the far reaches of the universe, making them an extremely powerful tool in determining distances.
50) Redshift and Hubble's Law. Imagine that you have obtained spectra for several galaxies and have measured the observed wavelength of a hydrogen emission line that has a rest wavelength of 656.3 nanometers. Here are your results: -->Galaxy 1: Observed wavelength of hydrogen line is 659.6 nanometers. -->Galaxy 2: Observed wavelength of hydrogen line is 664.7 nanometers. -->Galaxy 3: Observed wavelength of hydrogen line is 679.2 nanometers. a. Calculate the redshift, z, for each of these galaxies. b. From its redshift, calculate the speed at which each of the galaxies is moving away from us. Give your answers both in kilometers per second and as a fraction of the speed of light. c. Estimate the distance to each galaxy from Hubble's law. Assume that H0 5 22 km/s/Mly.
From the data we have provided for each galaxy, we can make a table of derived values. The lab (rest) wavelength is 656.3 nanometers for the hydrogen emission line of interest here. Note that the redshift z we compute is equivalent to the derived recession velocity divided by the speed of light. The speed of light is 300,000 kilometers per second. The Hubble constant we assume here is 24 km/s/Mega light-year, so the distance we quote is in Mega-light- years (Mlt-yr). Redshift Equation: z=(lambda(obs)-lambda(lab))/(lambda(lab)) Speed Equation: v=cz Distance Equation: d=v/(H(0)) Galaxy 1 (Wavelength 659.6) a) 0.0050 b) 1500 km/s c) 62.5 Mlt-yr Galaxy 2 (Wavelength 664.7) a) 0.0128 b) 3840 km/s c) 160 Mlt-yr Galaxy 3 (Wavelength 679.2) a) 0.0349 b) 10500 km/s c) 437.5 Mlt-yr
1) What do we mean by galaxy evolution? How do telescopic observations allow us to study galaxy evolution? How do theoretical models help us study galaxy formation?
Galaxy evolution is the study of how galaxies form and develop in our expanding universe. Telescopic observations allow us to observe the history of galaxies since the farther we look into the universe, the further we can see back in time. Theoretical modeling helps us study galaxy formation because we cannot see the galaxies before the first stars formed, and the time scales over which galaxies change and interact with each other are much longer than an astronomer's lifetime. Theoretical modeling allows us to test our ideas about how galaxies might have formed and changed over time as they grew more massive (or not) and interacted with other galaxies.
6) What is gravitational lensing? Why does it occur? How can we use it to estimate the masses of lensing objects?
Gravitational lensing is the bending of beams of light by massive objects. This bending occurs because masses distort spacetime, according to general relativity. Since more massive objects distort spacetime more and therefore bend the light more, we can use the amount of bending to calculate the mass of the object doing the lensing.
15) What is Olbers' paradox, and how is it resolved by the Big Bang theory?
Olbers' paradox notes that if the universe and the number of stars in the universe were infinite, then in every direction we looked, every line of sight would end up hitting a star eventually (like looking around in a dense forest: in every direction you see a tree trunk). However, the night sky is dark, so either the universe is finite in size or there are a finite number of stars. The resolution of this is the Big Bang: The universe has a finite age so we can see only out to the cosmological horizon. The observable universe is finite, resolving the paradox.
5) Why can't our current theories describe the conditions that existed in the universe during the Planck era?
Our current theories cannot describe the universe during the Planck era because we require a theory that links quantum mechanics and general relativity. We do not yet have such a theory, so we are currently unable to describe the universe in this era.
13) What is a radio galaxy? How can radio galaxies affect the gas surrounding them?
Radio galaxies are galaxies that emit unusually strong radio waves. Many radio galaxies have jets of plasma shooting out in opposite directions. The ends of the jets are radio lobes where the strong radio emission comes from. We think that the ultimate energy source is the galactic nucleus because that is where the jets originate.
3) Describe the basic characteristics of stars' orbits in the disk, halo, and bulge of our galaxy.
Stars in the disk of the galaxy have nearly circular orbits that are mostly in the plane of the galactic disk. The disk stars have vertical motions out of the plane, making them appear to bob up and down, but they never get "too far" from the disk. Orbits of stars in the bulge and the halo of the galaxy are much less orderly, traveling around the galactic center on elliptical orbits with more or less random orientations.
13) What do the large-scale structures of the universe look like? Explain why we think these structures reflect the density patterns of the early universe.
The large-scale structure of the universe shows galaxies arranged in huge chains and sheets that span millions of light-years. Between the chains and sheets are giant voids. This structure probably mirrors the original distribution of dark matter in the early universe. The more dense regions in the early universe would have gone on to collapse and form galaxies, clusters, and superclusters while the less dense regions would have gone on to form the voids.
14) What do we mean by the lookback time to a distant galaxy? Briefly explain why lookback times are less ambiguous than distances for discussing objects very far away.
The lookback time to a distant galaxy is the difference between the present age of the universe and the age of the universe when the light left the galaxy. The lookback time is less ambiguous than the distance to a galaxy because in the time that the light has been traveling to us, the universe has expanded, changing its distance, but the time that the photons have been traveling is definite.
17) Based on current evidence, what is the overall inventory of the mass-energy contents of the universe?
The overall inventory of mass-energy contents of the universe is 74% dark energy, 22% dark matter, and 4% ordinary matter (3.5% atoms, 0.5% stars). These percentages of dark matter and dark energy are good to plus or minus a few percent; the percentage of ordinary matter to the best accuracy is around 4.4%.
2) What are the three major types of galaxies, and how do their appearances differ?
The three major types of galaxies are spirals, ellipticals, and irregulars. Spiral galaxies, like our own, are flat white disks with yellowish bulges in the centers and contain cool gas interspersed with hotter ionized gas. Elliptical galaxies are redder, more rounded, and often football-shaped. They contain very little cool gas and often contain hot, ionized gas. Irregulars do not fit either of the other categories.
14) Describe and compare the four general patterns for the expansion of the universe: recollapsing, critical, coasting, and accelerating. Observationally, how can we decide which of the four general expansion models best describes the present-day universe?
There are four possible patterns for the expansion of the universe: • Recollapsing—We get a recollapsing universe if there is enough mass in the universe. If this were the case, eventually gravity will halt the expansion of the universe and reverse it. All of the matter in the universe will eventually be crushed back together again, re-creating the conditions of the Big Bang. • Critical—We get a critical universe if the density of the universe is exactly the critical density. In this case, the expansion will slow with time but never reverse. (It will halt after an infinite time.) • Coasting—A coasting universe occurs if the density of the universe is less than the critical density and there is no dark energy to accelerate the rate of expansion. The universe will continue expanding forever with little change in the rate of expansion. • Accelerating—An accelerating universe occurs if there is dark energy in the universe to exert a repulsive force. This will increase the rate of expansion of the universe over time. We can tell which of these models is the right one with white dwarf supernovae measurements. These supernovae are bright enough to be seen across vast distances and function as standard candles, letting us find their distances. Their redshifts tell us how much the universe has expanded since their light was emitted, so we can work out the rate of expansion of the universe in the past.
48) Background Radiation During Galaxy Formation. What was the peak wavelength of the background radiation at the time light left the most distant galaxies we can currently see? Assume those galaxies have a cosmological redshift of z 5 10. What is the temperature corresponding to that peak wavelength?
To answer this question, we will first use the equation given in Mathematical Insight 22.1 to calculate the peak wavelength of the microwave background at z = 7: Lambda(max)=1.1mm/(1+z)=1.1mm/(1+7)=0.14mm The peak wavelength of the microwave background at the time when the light left the most distant galaxies we can see was 0.14 mm. We can also find the temperature at that time with another equation from Mathematical Insight 22.1: T(universe)=2.73Kx(1+z)=2.73Kx(8)=21.8K The temperature of the universe at this time was around 21.8 K.
49) Mass of the Central Black Hole. Suppose you observe a star orbiting the galactic center at a speed of 1000 km/s in a circular orbit with a radius of 20 light-days. Calculate the mass of the object the star is orbiting.
To find the mass of the central black hole, we will use the orbital velocity law: M(r)= (rv^2)/G To use this law, we need to convert the radius and the speed of the star into meters and meters per second. The speed we find to be 1.0x10^6 meters per second. The radius is a bit more work, since it is given as 20 light-days. There should be about 365 light-days in a light-year, so we can convert and find that the orbital radius is also 2.5x10^(-2) light-year. A light-year is 9.46x10^15 meters, so the orbital radius is 5.2x10^14 meters. We therefore find that the mass of the central black hole is 7.8x10^36 kilograms, or about 3,900,000 times the mass of the Sun.
50) Mass of a Globular Cluster. Stars in the outskirts of a globular cluster are typically about 50 light-years from the cluster's center, which they orbit at speeds of about 10 km/s. Use these data to calculate the mass of a typical globular cluster.
To find the mass of the globular cluster, we will use the orbital velocity law: M(r)=rv^2/G To use this law, we need to convert the radius and the speed of the star into meters and meters per second. The speed is about 10,000 kilometers per second. Recalling that 1 light-year is 159.4610 meters the 50-light- year radius is equivalent to 4.7x10^17 meters. Plugging in the numbers, we find that the mass of the globular cluster is 7.1x10^35 kilograms or 350,000 times the mass of the Sun.
48) Mass of the Milky Way's Halo. The Large Magellanic Cloud orbits the Milky Way at a distance of roughly 160,000 light-years from the galactic center and a velocity of about 300 km/s. Use these values in the orbital velocity law (Mathematical Insight 19.1) to estimate the Milky Way's mass within 160,000 light-years from the center. (The value you obtain is a fairly rough estimate because the orbit of the Large Magellanic Cloud is not circular.)
To find the mass within 160,000 light-years of the center of the galaxy, we will use the orbital velocity law: M(r)= (rv^2)/G To use this law, we need to convert the radius and the speed of the Large Magellanic Cloud into meters and meters per second. We are told that the orbital radius is 160,000 light-years. Since Appendix A tells us that 1 light-year is 9.46x10^15 meters, this is the same as 1.51x10^21 meters. The speed is given as 300 kilometers per second, which we can easily convert to 3.0x10^5 meters per second. Plugging into the orbital velocity law, we learn that the approximate mass of the galaxy within 160,00 light-years of the center is 2.0x10^42 kilograms, or about 1 trillion times the mass of the Sun.
52) Mass from Orbital Velocities. Study the graph of orbital speeds for the spiral galaxy NGC 7541, which is shown in Figure 23.4. a. Use the orbital velocity law to determine the mass (in solar masses) of NGC 7541 enclosed within a radius of 30,000 light-years from its center. (Hint: 1 light-year 5 9.461x10^15 m.) b. Use the orbital velocity law to deter- mine the mass of NGC 7541 enclosed within a radius of 60,000 light-years from its center. c. Based on your answers to parts a and b, what can you conclude about the distribution of mass in this galaxy?
We can derive a general-purpose formula for this problem and for the next problem by using the mass/velocity/radius relation and plugging in 100 kilometers per second for the velocity and 10,000 light-years for the radius. Then, for this and the next problem, we need to do fewer computations: M(r)=rv^2/G=[r+((9.46x10^19m)/(10000 lt-yr))x(v^2x(1000m/1km))]/([6.67x10^(-11) m^3/(kgxs^2)]x[2x10^30kg/M(sun)])=7.1x10^5M(sun)x (r/10000lt-yr)x(v/100km/s)^2 For NGC7541 from Figure 23.4, the velocity at 30,000 light-years is 200 kilometers per second, and the velocity at 60,000 light-years is about 220 kilometers per second. Plugging into our handy formula, we can solve parts (a), (b), and (c). a) Within 30,000 light-years: M(r)=7.1x10^9M(sun)x[(30000 lt-yr/10000 lt-yr)x(200 km/s / 100 km/s)^2=8.5x10^10M(sun) b) Within 60,000 light-years: M(r)=7.1x10^9M(sun)x[(60000 lt-yr/10000 lt-yr)x(220 km/s / 100 km/s)^2=2.1x10^11M(sun) c) The mass at 30,000 light-years is about half the mass at 60,000 light-years (since the velocity curve is flat, the velocity isn't much different, and thus the mass enclosed increases proportionally to the radius). The mass is not concentrated in the center of the galaxy.
5) Briefly describe the three different ways of measuring the mass of a cluster of galaxies. Do the results from the different methods agree? What do they tell us about dark matter in galaxy clusters?
We can measure the masses of clusters of galaxies in three ways. The first method is to measure the speeds and positions of the galaxies in the cluster as they orbit. Applying Newton's law of gravitation to the speeds, we can deduce the mass of the cluster. The second method for finding the mass of a cluster is to measure the temperature of the gas between galaxies in the cluster and the approximate size of the cluster. The pressure of the gas must balance the pull of gravity from the mass in the cluster, so the temperature and size lead us to the mass. The final method of finding the mass of a galactic cluster is gravitational lensing. This technique measures the amount that a beam of light is bent as it passes near the cluster and uses Einstein's theory of general relativity to determine the mass of the cluster. The results from all three methods agree on the masses of the clusters, and these results tell us that the clusters hold substantial amounts of dark matter.
48) Cepheids in M100. Scientists using the Hubble Space Telescope have observed Cepheids in the galaxy M100. Here are the actual data for three Cepheids in M100: -->Cepheid 1: luminosity 5 3.9 3 1030 watts brightness 5 9.3 3 10-19 watt/m2 -->Cepheid 2: luminosity 5 1.2 3 1030 watts brightness 5 3.8 3 10-19 watt/m2 -->Cepheid 3: luminosity 5 2.5 3 1030 watts brightness 5 8.7 3 10-19 watt/m2 Compute the distance to M100 with data from each of the three Cepheids. Do all three distance computations agree? Based on your results, estimate the uncertainty in the distance you have found.
We can solve the luminosity-distance formula for the distance d: d=sqrt(luminosity/(4(pi)x(apparent brightness))) Substituting the given values, we find: • Cepheid 1: d= 5.8x10^23=6.1x10^7 light-years • Cepheid 2: 5.0x10^23=5.3x10^7 light-years • Cepheid 3: 4.8x10^23=5.1x10^7 light-years The results do not perfectly agree because of observational uncertainties. Taking the average, the distance is 5.6x10^7 light-years with a spread of +/-4.5 million light- years, or a spread of a little less than 10% of the total distance to the galaxy M100.
51) Estimating the Universe's Age. What would be your estimate of the age of the universe if you measured a value for Hubble's constant of H0 5 33 km/s/Mly? You can assume that the expansion rate has remained unchanged during the history of the universe.
We estimate the age of the universe to be 1/H(0). Before we can use the value given, we need to convert it: H(0)=33(km/s/10^6lt-yr)=33(km/s/10^6(3x10^5 km/s)x1yr)=1.1x10^(-10) (km/[(km/s)x1yr] Inverting this result leads to an estimated age of the universe of 9 billion years.
5) Describe some of the consequences of galaxy collisions. Why were collisions more common in the past?
We expect collisions between galaxies to be relatively common (while star-star collisions are rare) because the typical distance between galaxies is comparable in scale to the size of the galaxies themselves. However, the typical distance between stars is millions of times larger than the size of the stars, so collisions betweens stars are relatively rare. Galaxy collisions should have been even more common in the past then they are today because the density of galaxies was larger. Since, approximately, a similar number of galaxies existed in closer proximity to each other, they were more likely to encounter each other.
50) Temperature of the Universe. What will the temperature of the cosmic microwave background be when the average distances between galaxies are twice as large as they are today?
We know that a redshift of z, the average spacing between galaxies, was 1/(1 + z) what it is today. So for a spacing of two times what we see today, we can solve for z and find that z = -0.5. Plugging this into the formula for the temperature of the cosmic microwave background from Mathematical Insight 22.1: T(universe)=2.73K(1+z)=2.73K(0.5)=1.37K So the temperature of the microwave background will be 1.37 K at that time.
4) How do we measure the masses of elliptical galaxies? What do these masses lead us to conclude about dark matter in elliptical galaxies?
We measure the masses of elliptical galaxies by measuring how much the spectral absorption lines of stars in the galaxy are broadened in a spectrum that includes measurements of many stars at once. The broader the spectral line (composed of many lines from individual stars, blurred together), the faster the stars are moving relative to each other. Faster motion in the stars means that the galaxy is more massive. We can also use the orbital speeds and distance of globular star clusters to get the masses of elliptical galaxies. Like spiral galaxies, ellipticals also seem to contain far more matter than is accounted for by stars.
10) When we observe the cosmic microwave background, at what age are we seeing the universe? How long have the photons in the background been traveling through space? Explain.
When we observe the cosmic microwave background, we are seeing the universe as it was when it was 380,000 years old. The photons we see have been traveling for about 14 billion years, since not long after the Big Bang.
