21ST CENTURY ASTR.:STARS..(LL)-PACKAGE
6th Edition
ISBN: 9780393448450
Author: Kay
Publisher: NORTON
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Chapter 20, Problem 16QP
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A galaxy's rotation curve is a measure of the orbital speed of stars as a function of distance
from the galaxy's centre. The fact that rotation curves are primarily flat at large galactocen-
tric distances (vrot(r) ~ constant) is the most common example of why astronomer's believe
dark matter exists. Let's work out why!
Assuming that each star in a given galaxy has a circular orbit, we know that the accelera-
tion due to gravity felt by each star is due to the mass enclosed within its orbital radius r and
equal to v?/r. Here, ve is the circular orbit velocity of the star. (a) Show that the expected
relationship between ve and r due to the stellar halo (p(r) xr-3.5) does not produce a flat
rotation curve. (b) Show that a p(r) ∞ r¯² density profile successfully produces a flat ro-
tation curve and must therefore be the general profile that dark matter follows in our galaxy.
How astronomers determine the distance of a galaxy? Explain.
Problem 1: (
Compute the Oort constants A and B for Keplerian rotation: 0(R) = O, (R/Ro)
terms of O, and Ro. If Keplerian rotation described the rotation of the Milky Way near the
Sun, what would the numerical value of A and B (in kms-1 kpc-1) be and how does this
-0.5
in
compare to the observed values?
Chapter 20 Solutions
21ST CENTURY ASTR.:STARS..(LL)-PACKAGE
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- Assume that dark matter is uniformly distributed throughout the Milky Way, not just in the outer halo but also throughout the bulge and in the disk, where the solar system lives. How much dark matter would you expect there to be inside the solar system? Would you expect that to be easily detectable? Hint: For the radius of the Milky Way’s dark matter halo, use R=300,000 light-years; for the solar system’s radius, use 100 AU; and start by calculating the ratio of the two volumes.arrow_forwardStars form in the Milky Way at a rate of about 1 solar mass per year. At this rate, how long would it take for all the interstellar gas in the Milky Way to be turned into stars if there were no fresh gas coming in from outside? How does this compare to the estimated age of the universe, 14 billion years? What do you conclude from this?arrow_forward1. The current (critical) density of our universe is pe = 10-26kg/m³. Assume the universe is filled with cubes with equal size that each contain one person of m = 100kg. What would the length of the side of such a cube have to be in order to give the correct critical density? How many hydrogen atoms would you need in a box of 1 m³ to reach the critical density? The matter we know, which consists mostly of hydrogen, constitutes only 4.8% of the current critical energy density of our universe. So how many hydrogen atoms are actually in a box of 1 m3 in our universe? Deep space is very empty and a much better vacuum than we can obtain on earth in a laboratory.arrow_forward
- Another commonly calculated velocity in galactic dynamics is the escape velocity vesc, that is the minimum velocity a star must have in order to escape the gravitational field of the galaxy. (a) Starting from the work required to move a body over a distance dr against f show that the escape velocity from a point mass galaxy is vse = 2GM/r where r is your initial distance. (b) Since we know galaxies aren't actually point-masses, also show that vesc from r for a galaxy with a p(r) x r-² density profile is vse = 2v²(1+ ln(R/r)). Here you must assume that R is a cutoff radius at which the mass density is zero. (c) The largest velocity measured for any star in the solar neighbourhood, at r=8 kpc, is 440 km/s. Assuming that this star is still bound to the galaxy, find the lower limit (in kiloparsecs), to the cutoff radius R and a lower limit (in solar units) to the mass of the galaxy. Note the solar rotation velocity is 220 km/s.arrow_forwardAn important part of the lifecycle of galaxies like the Milky Way is the self regulation of formation of future generations of stars. Which statement best describes this process? A) Massive stars explode as Supernovae, heating nearby gas which then can't form stars, and even forcing the gas out of the galaxy in asuperbubble. B) Low mass stars like our Sun explode as Supernovae, heating nearby gas which then can't form stars, and even forcing the gas out the galaxy in asuperbubble. C) Stars fuse new elements in their cores which mix with nearby gas clouds, preventing the collapse of the clouds and hence stopping new starformation. D) The stars lock up material in their cores (like White Dwarf and Neutron Stars) meaning they can act as gravitational seeds for future starformation.arrow_forwardAnother commonly calculated velocity in galactic dynamics is the escape velocity vesc, that is the minimum velocity a star must have in order to escape the gravitational field of the galaxy. (a) Starting from the work required to move a body over a distance dr against f show that the escape velocity from a point mass galaxy is vsc = 2GM/r where r is your initial distance. (b) Since we know galaxies aren't actually point-masses, also show that vesc from r for a galaxy with a p(r) xr¯² density profile is vese that R is a cutoff radius at which the mass density is zero. = 2v(1+ ln(R/r)). Here you must assume (c) The largest velocity measured for any star in the solar neighbourhood, at r=8 kpc, is 440 km/s. Assuming that this star is still bound to the galaxy, find the lower limit (in kiloparsecs), to the cutoff radius R and a lower limit (in solar units) to the mass of the galaxy. Note the solar rotation velocity is 220 km/s.arrow_forward
- The Tully-Fischer method relies on being able to relate the mass of a galaxy to its rotation velocity. Stars in the outer-most regions of the Milky Way galaxy, located at a distance of 50 kpc from the galactic centre, are observed to orbit at a speed vrot determine the mass in the Milky Way that lies interior to 50 kpc. Express your answer in units of the Solar mass. 250 km s-1. Using Kepler's 3rd Law,arrow_forwardOur galaxy is approximately 100,000 light years in diameter and 2,000 light years thick through the plane of the galaxy. If we were to compare the ratio of the diameter galaxy and its thickness to the ratio of the diameter of a CD and its thickness (CD has a diameter of 12 cm and thickness of 0.6 mm), what would be the factor differentiating those ratios? Put differently, if the galaxy were scaled down to the diameter of a CD, how many times thicker or thinner would the galaxy be than the CD? (For example if it would be twice as thick, you would answer 2 and if it were twice as thin you would answer 0.5 (aka 1/2))arrow_forwardThe Kormendy relation for ellipticals can be written as He = 20.2+ 3.0 log R. where R. is the half-light radius (in kpc) and 4e is the surface brightness (in magnitudes per square arc second) at R.. An elliptical galaxy obeying this relation will have a total luminosity Lo R for some index 7. What is the correct value of n? O a. n=-6/5 O b. n= 4/5 T23D Oc n= 16/5 O d. n cannot be determined with the information we have.arrow_forward
- An observational survey of distant galaxies is undertaken that involves measuring their distances using cepheid variables and red-shifts using spectroscopy. Explain how cepheid variables can be used to measure the distances to galaxies. A spectral line is observed whose wavelength in the laboratory is de length of this spectral line observed in each galaxy, Xo, is listed in the table, along with the distance, d, to the galaxy. Determine the red-shift and the recession velocity of each galaxy and tabulate your results by making a copy of the table and filling in the blank spaces. Sketch a Hubble diagram using your results and determine the value of the Hubble constant Ho in units of km s-1 Mpc. 650 nm. The wave- Galaxy 1 652.69 Galaxy 2 Galaxy 3 Galaxy 4 Galaxy 5 653.01 do (nm) d (Mpc) 658.54 662.18 681.63 17 19 54 77 200 v (km s-1)arrow_forwardAs we discussed, clouds are made of a great many small drops. Really - a great many. Imagine a liquid cloud that fills a volume of 1 km3. The clouds contains 100 drops per cubic centimeter; for the sake of argument assume that each is 10 microns (micrometers) in radius. A. How many drops does the cloud contain? Compare this to a big number - say, the number of stars in the galaxy. B. What mass of water does the cloud contain? Compare this to something big - elephants, trucks, that sort of thing. C. What fraction of the cloud volume is filled with condensed water? One way to approach this is to compare the density of the suspended liquid water to the density of the surrounding air. D. How many 1 mm drizzle drops could you make from all the cloud drops? E. How much energy was released when this water condensed from vapor to liquid? If the water condensed in 20 minutes (a reasonable lifetime for a small cloud), what was the (energy per time)? powerarrow_forwardAnalyzing the spectrum of a distant galaxy, you discover evidence that a type la supernova is occurring in that galaxy. A type la supernova has a peak luminosity of about 1010 solar luminosities (1 solar luminosity = 3.8e26 Watts). Looking at an image of the galaxy, you estimate that here on earth your telescope only sees a brightness of 8.45E-10 Watts/m². Using this information and the brightness equation, how distant is the galaxy in which the supernova is occurring? Give your answer in It yrs.arrow_forward
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