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Solved: III Tercil"e.\' 24 alld 25, filld milk .'\, by obtaillillg (/ matrix of [ /, "op

Elementary Linear Algebra with Applications | 9th Edition | ISBN: 9780132296540 | Authors: Bernard Kolman David Hill ISBN: 9780132296540 301

Solution for problem 24 Chapter 4.9

Elementary Linear Algebra with Applications | 9th Edition

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Elementary Linear Algebra with Applications | 9th Edition | ISBN: 9780132296540 | Authors: Bernard Kolman David Hill

Elementary Linear Algebra with Applications | 9th Edition

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Problem 24

III Tercil"e.\' 24 alld 25, filld milk .'\, by obtaillillg (/ matrix of [ /, "op fnrm 0 ~] l/t{l/ is e,!lIi,"ln,//o A(a ) A = [l 2 -nU - 2 0n (b) A = 2 3 6

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1 Astronomy Unit 4B Milky Way Galaxy Part 2: The Mass of the Galaxy- We can estimate the mass of the galaxy from the sun’s orbital motion Kepler’s 3 Law applied to the Galaxy is: 3 2 o M = galp o Here, a= the distance to the center of the galaxy, 8.5 kpc or 1.75 AU o The orbital period of the sun around the galaxy is the circumference of its orbit, 2pia, divided by its orbital speed of 220 km/s o It takes the sun 240 million years to orbit the galaxy once Kepler’s Law gives: o M =(gal.75 X 10 AU) / (2.4 X 10 yrs) ) M. = 1.0 X 10 M. 11 o The galaxy contains the equivalent of about 100 billion Suns Mass Beyond the Sun- Kepler’s law only estimates the mass of the Galaxy interior to the Sun’s orbit How much mass lies beyond the orbit of the Sun o Rotational velocities in the disk reach about 300 km/s beyond the sun and may stay that high out to at least 30 kpc o The velocities of distant clusters and small nearby galaxies that orbit the Milky Way suggest that velocities remain at 300 km/s out to 100 kpc Keplerian rotation curve: o More distant planets orbit slower than planets closer to the sun o The gravitational force exerted by the sun is greater on closer planets— these planets must move faster to maintain their orbits than more distant ones o Velocities decrease with distance The Dark Halo- A large, yet visible component where the disk and halo of the Milky Way Galaxy lie o Suggested because of the masses from the observed rotation curve of the Galaxy and form the rotation curves of other galaxies o The rotation curve implies that only 5% of the mass of the galaxy is visible 2 o The unseen mass of the Milky Way and other galaxies is called dark matter o White dwarfs are the major component (but can’t contribute more than 20% of the mass) Astronomers were surprised when they discovered that velocities did not decrease beyond the edge of the visible disk at the rotation curve of the galaxy o Rotational velocities increase beyond the sun’s position and beyond the visible edge of the galaxy o This behavior where stars move at constant velocities no matter where they are in the disk is true of other galaxies that have disks o For velocities to increase, a substantial amount of matter must lie behind the sun’s orbit o The additional mass needed to account for these high orbital speeds is huge (at least 2X10 M.)—20 times the estimated mass of visible matter in the galaxy Evolution of MWG 1. Large, Z=0 cloud of gas  No dust, no stars  H and He 2. Collapse under self-gravity  Globular clusters  In/out orbits reflect the inward collapse  Halo: old stars/ metal poor - First generation of stars - In/out orbits with no rotation - Inward motion of original material - Outlines original size and shape of the MWG 3. Starburst in the center  High concentration of gas  Burst of SF 4. Supernovae heating  Caused by the SF bursting  Temporarily stops the collapse  Enriches gas and dust—Z increases 5. Cooling and collapse to form in the disk 3  Forms the disk and bulge  Bulge: high density/lots of SF which leads to highest Z  Disk: Less SF which leads to lower Z - Young stars/ metal rich - Later generations - Flat/rotating implies collapse Galaxies and Distances Island Universes: As more powerful telescopes became available, observers could separate the fuzzy patches of light into star clusters and nebulae They consisted of two groups: 1. Consisted of H II regions and planetary nebulae 2. Spiral Nebulae- small symmetrical objects, many of which exhibited spiral structures  Astronomers needed to know their distances to understand their nature  This was possible after building the 100 inch Hooker reflector  This telescope was used to resolve a few of the nearest spiral nebulae into individual stars  Because these stars were so faint, they had to be very distant 4  Knowing the distance (determined by apparent magnitudes and periods of the Cepheids) , astronomers could identify other objects in Andromeda as bright as O and B stars, H II regions, and star clusters  The similarity between the Milky Way Galaxy and the spiral nebulae was complete; they were extragalactic and part of a real of galaxies, which could be different shapes and sizes Classification of Galaxies: The majority of galaxies fall into one of two types 1. Spirals  Show flat, prominent disks similar to that of the Milky Way  Most of the brightest visible galaxies are spiral  Have a nuclear bulge, a flat disk, a halo of old stars, and spiral arms  Look like two fried eggs placed back to back if viewed from the edge  Half of the spiral galaxies show bar-shaped concentration of stars centered on the nucleus with the arms emerging from the ends of the bar  The dust in the disk is visible as dark lanes against a bright nucleus  Disks are prominent due to H II regions and young stars in the arms  An older fainter population of stars and clusters are also present throughout the disk  The old stars in the bulges produce a yellowish-orange glow in contrast to the blue arms  Spiral galaxies are divided into subclasses according to: 1) the size or brightness of the nuclear bulge compared to the disk and 2) the patchiness or definition of the spiral arms in the disk  Spirals are designated as Sa, Sb, and Sc  Spiral galaxies tend to be large, bright, and massive 2. Ellipticals  Ellipsoidal-shaped swarms of stars  1 in 5 bright, visible galaxies are elliptical  When considering all galaxies, the faint ellipticals increase in numbers: ellipticals outnumber spirals 2 to 1  Shaped like melons 5  Some may be triaxial (3 axes in unequal length)  Labeled by the letter E followed by a number representing their degree of flattening  EO ellipticals are spherical and E7’s are the flattest  Ellipticals lack prominent internal structure  The concentration of stars decreases away from the center  The color-magnitude diagram resembles those of globular clusters  The majority of stars are old, Population II stars  The reddish B-V color arises from the many red giants  These galaxies have converted nearly all their gas and dust into stars  They rotate very slowly  Orbital velocities of individual stars not being equal in all 3 directions causes the observed flattening Order of Galaxies using Hubble’s Tuning Fork (picture on page 207): Elliptical Galaxies- 1. E0 2. E4 3. E7 S0 Branches into Normal Spirals or Barred Spirals- 1. Sa (normal) or SBa (barred) 2. Sb or SBb 3. Sc or SBc Hubble Classification: o Reflects how fast gas is converted into stars o Initial rotation determines:  No/little rotation - Fast collapse to the center - Elliptical galaxies - No disk - High density gas: high rate of star formation leads to using up the gas fast 6  Rotation - Slower collapse - Spiral galaxies - Low rotation: Sa which forms a large bulge - Higher rotation: Sc which forms a small bulge - Highest rotation: Irregular galaxy which has no bulge Galaxy Age and Distance: Light travel time vs. Distance o Light from galaxies takes time to reach us (travels at the speed of light, c)  We see all galaxies as they looked in the past  So, they appear younger than they really are o The farther away the galaxy:  The longer it takes its light to reach us o To view younger galaxies, we must photograph more distant galaxies  The greater the distance, the younger the galaxy Calculation example: A galaxy is at a distance of 3 billion pc o 1 pc= 3.26 ly Distance Scale/Indicator: 7 The Local Group (page 216): o The Milky Way galaxy itself is a member of at least 30 galaxies called the local group o The nearest clusters are 3 to 5 Mpc away o Two large spirals, the Andromeda nebula (M31) and the Milky Way galaxy, dominate the Local Group o They have satellites of small irregular and elliptical galaxies Hubble’s Law: o Hubble found that redial velocity increased with distance o Relationship between velocity and distance o Distance is proportional to velocity o The last “bridge” across the abyss of space o D represents distance, V represents the measured radial velocity, and HO represent the constant of proportionality o V= H X O o The constant of proportionality is the Hubble Constant and we must calibrate it from galaxies with known distances o Once we know H , tOe measurement of the radial velocity of a galaxy gives its distance o H hOs an average value of 71 km/s/Mpc Main Sequence Fitting: o Technique of comparing main sequences o Comparing absolute magnitudes of MS stars in the Hyades cluster with the apparent magnitudes of the MS star in the Pleiades cluster gives the distance modulus of the Pleiades o The distance modulus comes from the shifting in magnitude of the MS of the distant cluster until it lies over that of the Hyades o The vertical axis of the Hyades cluster is in absolute magnitude and the vertical axis of the more distant cluster is in apparent magnitude o The difference, m-M, is the distance modulus of the cluster 8 (Main Sequence Fitting) Superclusters (page 221): o An irregularly grouping of clustering of galaxies o The Local Group is a part of this o Consist of the large Virgo cluster o The largest clusters form a flat structure about 20-30 Mpc across and a few Mpc thick o The Local Group is at one edge of the supercluster o Many other smaller clusters and individual galaxies form a halo with a diameter of about 30 Mpc around this group o Masses are as high as 10 M. and they can be 100 Mpc long o Flat structures with thin bridges of galaxy clusters connecting each other o 2-D surveys can identify clusters of galaxies and long strings of superclusters Voids (page 222): o 2-D surveys of superclusters also reveal large, nearly empty areas 9 o The galaxies beyond or in front of dark areas tend to fill in regions that could really lack galaxies o To isolate voids, 3-D surveys must be used to study the distribution in space of many galaxies o Extensive surveys have shown hundreds of thousands of galaxies since 1989 have long filaments of galaxies and huge, galaxy-free voids o Filaments define boundaries or surfaces of low-density voids o The distribution of galaxies is sponge-like or foam-like, surrounding the empty space o Galaxies congregate along the perimeter of voids Walls: o The largest structures in the figure and in the universe o Long lines of galaxies o Largest walls are about 200 Mpc in length o They are called walls because they are so long but narrow and extend in and out o Evidence comes from redshift surveys of tiny areas of the sky Active Galaxies  Active galaxies (pg. 226) o Do not show absorption lines o Emission lines exhibit a much wider range of velocities o More violent, explosive activity o More rapid rotations o Differ from normal galaxies also because they have a continuum that originates 2 ways:  Thermal radiation- Come from hot gas and produces a smooth Planck-shaped spectrum.  The shape of the spectrum depends on the temperature of the gas  Synchrotron radiation—occurs when magnetic fields accelerate fast moving electrons, causing them to emit radiation as they spiral along field lines  The synchrotron spectrum is a smooth curve but with a broader distribution, covering a greater range of wavelengths 10  Active Galactic Nuclei or AGNs—the nuclear location of the intense emissions in active galaxies (pg. 226) o We measure the size of the emitting region at the center of AGNs by observing light variations o The environment around the source in AGNs must be able to produce a non-thermal continuum and sometimes emission lines in a very small volume o Seyfert galaxies—spiral AGNs  Most numerous type of AGNs  Contain an intense, variable compact, nonstellar emission from their nuclei (Just like radio galaxies and BL Lac objects)  Their emitting region is smaller than a few light-months across, which is less than the distance from the sun to the inner regions of the Oort Cloud  Continuum radiation is due to synchrotron radiation  Have small, bright nuclei that outshine their disks and bulges  Differ from continuum radiation because of they vary their light output, and when they do, their continuum varies, NOT the emission lines  Seyfert 1 Galaxies exhibit broad hydrogen and helium emission lines.  The broadest lines imply velocities of up to 4000km/s  Seyfert 2 Galaxies B have emission lines that are narrow and imply velocities of only 400 km/s.  Also exhibit an excess of infrared emission, which is absent in Seyfert 1 galaxies o BL Lacertae Objects- elliptical AGNs  Has a star-like image and exhibits erratic light variations  Their spectra do not show emission lines even though the observed continuum radiation appears to be due to synchrotron radiation  Have intense, variable, compact, nonstellar emission from their nuclei (Just like radio and Seyfert galaxies)  Have emitting regions of only a couple hundred astronomical units 11  Have active nuclei o Radio galaxies—the most powerful AGNs (pg. 228)  Made up of Double radio sources  Contain an intense, variable, compact, nonstellar emission from their nuclei (just like Seyfert galaxies and BL Lac objects)  Emit a million times more radio energy than any galaxy  Cyg A ( which is actually 2 colliding galaxies) is one of the most intense radio sources in the sky  Colliding galaxies can produce radio emission  These galaxies cane also generate a small radio source called a core which is less than 1 parsec across  The two lobes of the colliding galaxies are the most intense radio features of a radio galaxy  The largest known separation of radio lobes is almost 6 Mpc  Hypothesis says optical galaxies between the lobes explosively eject the blobs of gas now emitting radio waves.  A jet, or narrow beam stretches between the galaxies and the lobes, reinforcing the impression of an ejection (pg. 229)  Narrow jets connect the core with one or both lobes  Split into Narrow line radio galaxies and broad line radio galaxies (This is referring to the size of emission lines) o Quasars- galaxies that are star-like, compact radio sources with high redshifts (pg. 234)  They are the most distant and enigmatic AGNs  Have small radio diameters  Have intense continuum with broad emission lines, implying velocities of several thousands of kilometers per second for emitting gas clouds  A typical velocity for a gas cloud that is 1 pc from the center of a quasar is 5000km/s  They are bright nuclei of galaxies and their star-like appearance is due to their small nuclei outshining the starlight of the parent galaxy 12  Relativistic aberration- when an emitting cloud of particles move at near the same speed of light, radiating in the direction of its motion (pg. 229) o The greater the speed of the emitting particles, the narrower the cone of forward emitted radiation. o Implies that most of a jet’s emission is in the direction of its motion away from the nucleus.  Supermassive Black Holes (pg. 230) o These are the energy source of AGNs through gravitational expulsion. o Matter is accelerated inward toward a large mass concentration, explosively heated, and then funneled outward in opposite directions. o They are detected by observing the heated cloud of gas in AGNs that move rapidly around massive concentrations of matter, and rapid light variations that imply a compact source not much larger than our solar system. Sample Questions 1. When determining the mass of the MWG with the Sun’s orbital parameters, we are calculating the mass:  Of everything inside the Sun’s orbit 2. About how much of the mass of the Milky Way Galaxy is probably visible matter, such as stars, star clusters, and molecular clouds  5% Feedback: The rotation curve of the MWG shows high velocities beyond the visible boundaries of the galaxy - beyond the visible edge of the disk. Since the mass inside the orbiting object determines its orbital speed, high orbital speeds indicate large amounts of mass. The required mass to produce such high speeds represents about 95% of the mass we measure in the visible objects like stars, star clusters, molecular clouds etc. 3. The spherical distribution of the halo stars shows that the Milky Way Galaxy formed by collapse.  FALSE Feedback: The distribution of globular clusters indicates the extend of the MWG before it collapsed. Whenever a cloud of dust and gas 13 collapses, a disk of rotating gas and dust forms. See Figure 23.11; galaxy formation as the collapse of a cloud of dust and gas is the same as with stars. Both are due to the gravitational force pulling everything inward. 4. Which of the following came first in the evolution of the Milky Way Galaxy:  Collapse under self-gravity Feedback: The starburst phase occurs after the cloud collapses and concentrates a tremendous amount of matter at the center of the MWG. This concentration results in a burst of star formation because so much matter is concentrated in a small space; i.e., it has a high density, which encourages star formation because the inward-gravitational forces are strong. 5. Elliptical galaxies:  Contain mainly Population II objects with very little dust or gas Feedback: This is listed in Table 29.2 in the text. Elliptical galaxies do not show any current indications of star formation. Thus they cannot contain young objects - Pop 1 - since they have used up all the gas and dust already. None is left for star formation. Also elliptical galaxies use E for their classification E0 to E7 6. The most likely type of galaxy to form out of a cloud of gas that is rotating very fast:  Spiral- SC Feedback: Since spiral galaxies have disks, the clouds out of which they formed has high rotation to form a disk. Elliptical galaxies had very little rotation when they started to collapse, so they have not disk. Of the spiral galaxies, the one with the smallest nucleus must have had a faster initial rotation because the faster the rotation, the less material is allowed to fall into the nucleus. From Sc to Sa, rotation of initial cloud decreases, allowing more matter to accumulate in the nucleus. 7. One of the largest galaxies in the Local Group is:  The Andromeda Nebula 8. About how long has light taken to reach us from a galaxy 31 million parsecs away (1pc = 3.26 light years): 14  About 100 million years Feedback: The unit of distance called the light year also tells us how long it takes the light to reach us from distances measured in light years. A light year is the distance light (all electromagnetic radiation) travels in one year - about 9.5 trillion km (6 trillion miles). So the light from a star 1 light year away takes 1 year to reach us. Thus, we "see" the star as it looked one year ago. 1 parsec equals 3.26 light years. Thus, the light from a star 1 pc away takes 3.26 years to reach us. We see the star as it was 3.26 years ago. 31 million parsecs is how many light years 31 x 3.26 = 101 light years (about 100 light years). 9. Choose the correct arrangement of the following numbered list of distance indicators in order of increasing distance usage:  Parallaxes of stars, Luminosities and periods of Cepheids, Redshifts of galaxies Feedback: You simply have to memorize Figure 29.19. Note that astronomers use several techniques beyond 25 Mpc. 10. Measuring the luminosities and periods of Cepheids in star clusters of known distances in our galaxy is what step in the process astronomers use to determine the distance scale of the universe  Calibrate a distance indicator 11. The distance modulus of stars allowed astronomers to determine their:  Distance 12. The distance to the Hyades star cluster is the most fundamental parameter (...without it we can't even start measuring distances to anything) needed for our quest to determine the distances to the most distant galaxies in the universe:  FALSE Feedback: The most fundamental unit of distance is the astronomical unit 13. The largest known structures in the universe are called pencil beams:  FALSE Feedback: Pencil beam refers to a type of redshift survey.Astronomers look at a very tiny region of the sky and measure redshifts. An example of the results is shown in Figure 29.28. 15 14. When did galaxies first appear as elliptical galaxies:  About 6 billion years after the galaxies formed (about 8 billion years ago) Feedback: Spiral galaxies first appear about 5 billion years ago. Since the universe is 13.7 billion years old, spiral galaxies formed bout 9 billion years after the formation of galaxies. Elliptical galaxies first appear about 8 billion years ago. Since the universe is 13.7 billion years old, spiral galaxies formed bout 6 billion years after the formation of galaxies .So elliptical galaxies came first. 15. What objects appears first in the universe, when the universe was only 2-3 billion years old:  Quasars Feedback: Look at Figure 30.12. It plots z (time) as a function of the number of quasars (ellipticalAGNs) and QSO (spiral AGNs).It shows that the most active galaxies 2-3 billion years ago were the quasars. Radio galaxies, Seyfert, and BL Lacertae objects are very recent. 16. Active galaxies (AGNs) were most active (powerful) when the universe was how old  2 billion years ago 17. Why didn't astronomers initially recognize the spectral lines in the spectra of the star- like quasars:  The lines were Doppler shifted to wavelengths far different from their at-rest wavelengths. 18. Active galaxy (AGN) activity decrease in time because of the size for their central black holes decreases.  FALSE 19. The jets seen in active galaxies are probably produced by”  Supermassive black holes 16

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Textbook: Elementary Linear Algebra with Applications
Edition: 9
Author: Bernard Kolman David Hill
ISBN: 9780132296540

Elementary Linear Algebra with Applications was written by and is associated to the ISBN: 9780132296540. Since the solution to 24 from 4.9 chapter was answered, more than 266 students have viewed the full step-by-step answer. This full solution covers the following key subjects: . This expansive textbook survival guide covers 57 chapters, and 1519 solutions. The answer to “III Tercil"e.\' 24 alld 25, filld milk .'\, by obtaillillg (/ matrix of [ /, "op fnrm 0 ~] l/t{l/ is e,!lIi,"ln,//o A(a ) A = [l 2 -nU - 2 0n (b) A = 2 3 6” is broken down into a number of easy to follow steps, and 38 words. This textbook survival guide was created for the textbook: Elementary Linear Algebra with Applications, edition: 9. The full step-by-step solution to problem: 24 from chapter: 4.9 was answered by , our top Math solution expert on 01/30/18, 04:18PM.

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Solved: III Tercil"e.\' 24 alld 25, filld milk .'\, by obtaillillg (/ matrix of [ /, "op