Download Word

HOMEWORK #12 (due start of class November 17)
My Messier object is ____________
(copyright D. McCarthy)
LEARNING GOALS:
1. Continue practicing Polya’s 4-step plan in “How to Solve It.”
2. Practice mathematical concepts relating to exponentials, significant figures, percentages.
3. Practice astronomical concepts relating to optical depth, tides, spacetime diagrams in the context of interstellar
dust, black-holes, quasars, the expanding universe.
4. Your second Fermi Problem!
TO RECEIVE FULL CREDIT:
1. Write your Messier name at the top of your homework.
2. If you submit multiple pages, staple them together.
3. To receive any credit on these problems, you must show how you derived your answer by writing
all the logical steps that led you to it.
4. All sentence responses must be typewritten and in complete sentences. You may handwrite any
arithmetic. Use good English grammar.
5. If you work more than three hours on this assignment, you should stop, record your work
here, and contact Dr. McCarthy.
------------------------------------------------------
I. Optional:
a. Want to be a better scientist? Here is an interesting article.
“Dow Chemical Executive Inspires Through STEM Education”
(V. Craig, FOXBusiness, November 14, 2014)
http://www.foxbusiness.com/business-leaders/2014/11/14/dow-chemical-exec-inspires-through-stem-education/
Do problems #1,4 and choose one from any of the other five problems.
II. Exponentials
#1. Show that for the exponential function (P = P0enr), the number of investment cycles to double the
original amount (P0) can be expressed as 69.3/r. Here P0 is the initial amount, “r” is the rate of growth in
percent in a specific time period, and “n” is the total number of those periods that have elapsed.
For example, "Moore's law" is the empirical observation that, over the history of modern computer
technology, the number of transistors in an integrated circuit doubles approximately every two years. Using
the above formula, how rapid is this growth per month?
#2. Interstellar dust scatters and absorbs light, making distant objects appear fainter and redder. This effect
is called “extinction.” It is wavelength dependent such that shorter wavelengths are attenuated more
strongly than longer wavelengths; thus, objects appear redder with increasing distance. The amount of
extinction is designated as A(), measured in magnitudes. In the direction of our Galactic Center A(V) ≈
30 mag and A(K) is ~10x less, where V refers to visual wavelengths (~500 nm) and K represents the “Kband” in the near-infrared at =2.2 m = 2200 nm.
We can express the effect of extinction as follows, where Bo() is an object’s original brightness at
wavelength , B() is the brightness we observe, and () is the “optical depth.” So, for =1, the final
brightness is 1/e = 36.7% of the original value.
B( = Bo()e-
1
Using your knowledge of magnitudes and brightness, show that A() = 1.086 (). Also, calculate the
optical depths to the center of our galaxy at both V and K wavelength bands.
III. Black holes and Tides
#3. The giant elliptical galaxy M87 has an accretion disk of material orbiting its central black hole. The
Hubble Space Telescope observed the spectrum of this material and, from the Doppler shifts of the spectral
lines, found that the material is orbiting at 550 km/sec at a distance of 40 pc from the center of that galaxy.
Use Kepler’s Third Law to calculate the black hole’s mass to two significant figures. Also, comment on the
assumptions inherent in this calculation and the effects these may have on the accuracy of the answers.
#4. “Tidal forces” are ubiquitous but often poorly appreciated or understood. As astronauts orbit the Earth
they experience “weightlessness” because all parts of their bodies are being pulled with the same
acceleration towards the center of the Earth. In truth, however, small gravitational forces exist across the
widths of their bodies such that the closest parts are pulled more strongly than the parts farthest away. This
difference in force across the width of an object is called a “tidal force,” the most obvious results of which
are the bi-daily water tides on Earth and the Moon’s synchronous orbit around Earth.
Suppose an object of width “l” is located a distance “r” from a gravitating object of mass “M.” From
physics you will learn that the gravitational acceleration is GM/r2. Derive the difference in gravitational
acceleration between distance “r” and distance “r+l” from the object. Show that as long as “l” is small
compared to “r” (i.e., r >>l), the result is (2GM/r3)l. Calculate this difference for the following two cases.
What would happen to each person?
a). A person of height l=170 cm located r=1000 km from a 1.5 MSun neutron star.
b). The same person a distance 1010 km (i.e., the width of a Solar System) from a 109 MSun black-hole as
could be present in the nucleus of a typical galaxy.
IV. Quasars and Superluminal Motion
#5. Quasars are the extremely luminous (~1012 LSun) centers of galaxies in the early Universe. As gas falls
towards the central supermassive black-hole (~109 MSun), the gas is compressed and heated to temperatures
~106 Kelvin, and its resulting brightness exceeds the emission from all stars in the galaxy. Before falling
completely into the black-hole, some of the material is ejected in a narrow “jet” that can move away at
speeds (v) like v/c ~0.9, where “c’ represents the speed of light.. From Earth, it can appear that the jet
moves faster than light, v ~10c (i.e., “superluminal motion”). Yet we know that no mass can reach the
speed of light. Using the diagram below for a jet of v/c=0.8, explain how superluminal motion is an illusion
caused by the jet’s motion at an acute angle towards our line-of-sight.
2
V. Spacetime Diagram and the Expanding Universe
#6. Our Universe is currently expanding at a rate of ~7% every billion years. Space itself is expanding and
pulling galaxies farther apart from each other. Pretend that tonight we record the light from a galaxy at a
distance of 100 million light-years. Complete the spacetime diagram below showing the trajectories of both
the light traveling from galaxy to Earth and the galaxy’s motion over time. Since Earth is our reference
point, you may assume that Earth is stationary at the origin but moving through time. Label the axes
numerically. Could we see the galaxy as it really is today; why or why not?
3
VI. Fermi Problem
#7. If the Sun were made of people, then it turns out the Earth would be incinerated. Compare the power
output (i.e., luminosity) per mass of the Sun and a person. (You may assume the power output of a person
is 100 W.) Discuss why the Stefan-Boltzmann Law implies that Earth’s temperature would become 10x
higher. Finally, discuss why this problem is not a fair comparison based on the fuel sources of a person and
a star.
4