Download HOMEWORK #1

HOMEWORK #7 (due start of class October 13)
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 stellar magnitudes, small angle equation, luminosity, and brightness.
3. Learn about Kepler’s Third Law and apply it in different astronomical situations.
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 some interesting articles.
“How curiosity changes the brain to enhance learning” (October 2, 2014, Science Daily)
http://www.sciencedaily.com/releases/2014/10/141002123631.htm
“How to get unstuck: The psychology of writer’s block” (Poets & Writers, D. Cass, Jan. 2010)
http://www.pw.org/content/how_to_get_unstuck_the_psychology_of_writer_s_block?article_page=4
b. Observe the lightcurve of Algol (Persei - the “Demon Star”)
Algol is the most famous eclipsing binary star system. The two stars orbit each other every 2 days 20
hours 49 minutes and periodically eclipse each other from the perspective of Earth. On the evenings of
October 9 and 12, you can observe the eclipse by naked eye. The two stars will appear as a single object
that varies in brightness from an apparent magnitude of +2.1 to +3.4 (3.3x in brightness). A full eclipse
lasts a total of ten hours as the companion star passes in front of the primary star.
At 11:47 pm on October 9 (and 8:36 pm on the 12th), Algol will be in full eclipse and at minimum
brightness. Over the next five hours it will brighten back to normal as the companion star moves out of
direct alignment with the “primary” star. You can watch that transition and even record it by noting how
bright Algol appears with respect to surrounding stars.
Below are some related articles:
http://stars.astro.illinois.edu/sow/algol.html
http://www.aavso.org/vsots_betaper
Below is a link to a finding chart to help you locate Algol. Look towards the northeast to find the
constellations of Cassiopeia and Perseus. The chart includes the apparent magnitudes of stars near Algol
so you can compare and estimate Algol’s changing magnitude from time to time.
http://www.arksky.org/ref_guides/images/sherrod-algol.jpg
2. Kepler’s Third Law
Kepler was the first person to realize that the orbital period (P) of a planet (mass M1) around the Sun
(mass M2) is related to the semi-major axis (a) of the planet’s orbit. From observational data he showed
that the square of the period is proportional to the cube of the semi-major axis: P a3. Later in your
physics education, you will understand how Newton used his Law of Gravity to explain Kepler’s Law and
to derive the exact equation
P2 = 42a3 / G(M1 + M2).
If we adopt units of solar masses for “M1 and M2,” years for “P,” and AU for “a,” this equation becomes
P2 = a3 / (M1 + M2).
Astronomers use this fundamental equation to determine the masses of objects throughout the Universe:
Stars, planets, black holes, galaxies, etc. Here is some optional background reading:
http://www.as.utexas.edu/astronomy/education/fall10/scalo/secure/301.F10.2.KeplerToNewton.pdf
http://burro.cwru.edu/Academics/Astr201/Chap04b.pdf
A fun simulator about binary stars and eclipsing binary stars:
http://astro.unl.edu/naap/ebs/animations/ebs.html
Do problems #1,2,6 and choose one of #3,4,5.
#1. Consider a planet orbiting the Sun. If the mass of the planet (say, Earth) were to double but the
planet stay at the same orbital distance from the Sun, then the planet would take
a) more than twice as long to orbit the Sun.
b) exactly twice as long to orbit the Sun.
c) the same amount of time to orbit the Sun.
d) exactly half as long to orbit the Sun.
e) less than half as long to orbit the Sun.
#2. Stellar Masses
Imagine you observe a binary star system with a period of 1 yr and a separation of 1 AU. According to
Kepler’s Third Law, the total mass of both stars (M1 + M2) is 1 MSun.
Part a. Suppose that instead of observing a period of 1 yr, you observe a period of 0.5 yr with the same
orbit. What total mass of the system would you measure?
Part b. Suppose one star’s spectrum indicates this star is identical to our Sun. What is the mass of the
other star?
Part c. Note that the orbital speed is twice as fast as the Earth moves in the same orbit around our Sun.
How can this be?
#3. The Kepler-21b exoplanet system
http://iopscience.iop.org/0004-637X/746/2/123/pdf/apj_746_2_123.pdf
The figure below shows the lightcurve of the star HD 179070 as an orbiting exoplanet transits in front of
the star every 2.785755 days and eclipses some of the star’s brightness. From the star’s spectrum, we
know the star has a mass of 1.3 MSun and a radius of 1.9 RSun. Based on the relative amount of light lost
during the eclipse, determine the radius of the planet compared to Earth. Based on the duration of the
eclipse, determine the planet’s orbital speed. Based on Kepler’s Third Law, calculate the planet’s mass.
You may assume the planet moves in a circular orbit and transits across the star’s diameter as seen from
Earth.
Show all your work even though you may not be satisfied with your final answer. We will discuss the
details afterwards and learn an important lesson about assumptions and errors.
#4. The Algol eclipsing binary system
From Earth, Algol looks like a single star because the two stars are close together and one star (the
“primary”) is much more luminous than the other. The right-hand lightcurve above shows how Algol
changes its brightness with time over one full orbital cycle. The left-hand schematic shows how the two
stars eclipse each other from our perspective and cause the lightcurve. The deeper eclipse occurs when the
red “secondary” star moves in front of the “primary” star.
Let’s make a few assumptions to help us calculate an approximate value for the total mass of both stars.
(1) Assume the two stars have the same diameter. Thus the secondary must move twice the primary’s
diameter to produce the deep eclipse: Once to fully eclipse it and another to uncover it. (2) Assume the
secondary’s orbit is circular.
By studying the spectrum of primary star, we know its mass and radius are 3.6 MSun and 3.2 RSun relative
to our Sun. Use the lightcurve to determine the speed of the companion star and its orbital period (P).
Calculate the semi-major axis (a) and total mass via Kepler’s Third Law. What is the mass of the
companion star?
Show all your work even though you may not be satisfied with your final answer. We will discuss the
details afterwards and learn an important lesson about assumptions and errors.
#5. A mystery object at our galactic center
Pretend this morning’s television news shows the following figure about a newly discovered star orbiting
an invisible object at coordinates (0,0) that is located at a distance of 8 kpc. The orbital period is 15.56
years. Use Kepler’s Third Law to calculate the mass of the invisible object. Explain your reasoning and
show your quantitative analysis.
Part a. The orbital semi-major axis is approximately ____ arcsec = ____ AU.
Part b. The mass of the object at coordinates (0,0) is ____ solar masses.
3. Luminosity of stars
The following problem is much simpler than it may seem at first. First, read the problem and be sure you
understand it. Second, develop a plan for solving it using the techniques we have used in class.
#6. We know that every star’s luminosity depends on its radius and temperature according to the equation
L = 4R2T4, where is the Stefan-Boltzmann constant. Ironically, the red giant star Arcturus is half the
temperature of our Sun yet 100x more luminous. Determine the radius of Arcturus compared to the Sun
and use your answer in a paragraph to explain this irony.