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HOMEWORK #5 (due start of class September 29)
My Messier object is ____________
(copyright D. McCarthy)
LEARNING GOALS:
1. Continue practicing Polya’s 4-step plan in “How to Solve It.”
2. Practice concepts relating to parallax, stellar magnitudes, luminosity, and brightness.
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.
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1. Optional:
a. Want to be a better problem solver? Here are interesting articles.
“13 things mentally strong people don’t do” (Amy Morin, 2013)
http://www.amymorinlcsw.com/mentally-strong-people/
“Hit the reset button in your brain” (D.J. Levitin, New York Times, Aug. 9, 2014)
http://www.nytimes.com/2014/08/10/opinion/sunday/hit-the-reset-button-in-your-brain.html?_r=0
Try to solve at least one problem in each of the three categories below.
2. Parallax
In recent years radio astronomers have tracked the angular position of the supermassive black hole (aka,
Sgr A*) located at the center of our Milky Way galaxy with respect to a very distant, fixed background of
quasars. That location seems to move because of the Sun’s orbital motion in our galaxy. Independently,
the Sun’s orbital speed in the galaxy has been determined to be 220 km/sec.
a. From the graphical data below, determine the distance to our Galactic Center using the small angle
equation and a parallax technique.
b. In terms of the width of our Solar System (e.g., diameter of the Oort Cloud is ~105 AU), how long does
it take for our Solar System to move its own diameter?
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c. The European Space Agency’s “Gaia” mission
Europe’s Gaia Mission is a currently operating space project designed to produce a three-dimensional map
of our Milky Way galaxy, providing unprecedented positional measurements for about one billion stars in
our galaxy - about 1% of our galaxy’s stars. This instrument will measure parallaxes as small as ~25
micro-arcseconds (25 x 10-6 arcsec = 25 as). Use the parallax equation we discussed in class to determine
the distance to an object of that parallax.
What if? About 20 solar-type stars are known within a radius of 10 parsecs around the Sun. Assuming
stars are distributed uniformly around us, how many solar-type stars would you expect to find within
Gaia’s search volume? Does this calculation seem reasonable? Why or why not?
3. Apparent and Absolute Magnitudes
Read the following articles to understand the concepts of how astronomers measure an object’s brightness.
http://www.skyandtelescope.com/astronomy-resources/the-stellar-magnitude-system/
http://en.wikipedia.org/wiki/Apparent_magnitude
a. In the following Hertzsprung-Russell Diagram every dot corresponds to a different star in our galaxy.
What is the total range in absolute magnitude and also in brightness? Using your knowledge of
magnitudes, describe why a star with color index of B-V=1.5 would appear red to the human eye. [HINT:
In the above Wiki article you’ll read that “B” stands for blue visual magnitude and “V” stands for yellow
visual magnitude.]
4. Luminosity and Brightness
The term “luminosity” refers to an object’s intrinsic brightness. However, its “apparent brightness” can
be much different depending on its distance from us. For example, imagine a 100-Watt light bulb. Its
luminosity is always 100 Watts and will not change no matter where you place it. However, you will
perceive that its apparent brightness changes when you move it closer or farther. Similarly, two identical
stars could have different apparent brightness because they are at different distances from us.
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The two images below illustrate how light moves away from a source (S). Each example shows a specific
portion of an expanding sphere or bubble of light as it moves outward. The original light spreads out over
a wider and taller surface area as distance increases. For example, the left hand picture shows nine light
rays hitting the surface area marked (A) at distance ‘r’ but, when the light has traveled three times farther
(3r), only one ray hits the same sized surface, so the source would appear nine times fainter there.
Therefore, the apparent brightness of the light decreases as the square of the distance away. We call this
phenomenon the “inverse-square law of light” but light is not special in this regard; sound behaves the
same way. The principle can be expressed as follows:
apparent brightness = luminosity
4(distance)2
Optional: A useful, sometimes funny, video to illustrate the inverse-square law:
http://www.youtube.com/watch?v=JW3tT0L2gpc&feature=related (Butter gun!!)
a. Pretend that one of your two favorite stars appears 100 times fainter than the other. You also know from
other data that both stars are fundamentally the same type of star with the same luminosity (i.e., a socalled “standard candle”). The parallax of the brighter star indicates its distance is 100 light-years. How
far away is the fainter star? Be sure to show your work.
b. Pretend you are comparing three stars in the same constellation. In each question below show your
work and explain your reasoning. As you proceed include a diagram showing the stars’ relative positions.
1. Your favorite star is 100 times more luminous than star #2 but both appear equally bright. How
many times farther away is your favorite star?
2. Stars #2 and #3 have the same luminosity, yet star #2 appears 100 times fainter than star #3. How
many times farther away is star #2?
3. The parallax of star #3 indicates its distance is 100 light-years. What are the distances to all three
stars?
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