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Astronomy Homework #6
Name____________________________
DUE: April 15 (by 11 PM)
Your homework grade depends not only upon your getting the correct answer but also grammar,
spelling and punctuation, particularly in questions that require explanations. Numerical answers
to questions do not need to be written in complete sentences and you should show your work
where ever it is appropriate. Partial credit may be given for showing your work even if your
result is incorrect. You will also be graded on the use of significant figures, proper units of
measure and proper scientific notation. You may work with others in determining the answers to
the questions, but what you write should be in your own words – any homework assignments
that look too similar to that of other students will receive no credit. Unless otherwise noted, all
questions are worth 1 point. Homework can be turned in at the office Latham 121 during
business hours, during class, or on-line at e-Learning.
1. (9 points total) Radioactive Decay. People often wonder how scientists are able to determine
things such as the ages of rocks or the Earth. This is most frequently done by taking advantage
of a neat process in nature called radioactive decay. When a radioactive substance breaks down
it does so in a very systematic manner, and we can measure the rate of these break downs (or
decays) to determine how old an object is – basically the amount of radioactive stuff left over
can tell us how old the substance is. The rate of break down is often referred to as the half-life.
For example, Beryllium 10 (10Be) breaks down into Boron 10 (10B) with a half-life of 1.36
million years. This means that if you start out with a solid block of 10Be, then after 1.36 million
years, half of the block will still be 10Be and the other half will be 10B. After each increment of
the half-life, 1/2 of remaining radioactive material (10Be ) will become the decay product, in this
case 10B.
a. How much of a block that was originally all 10Be will still be 10Be and how much will be 10B
after a total of 5.44 million years?
b. (2 points) If you come across a block that is 1/8 10Be and 7/8 10B, can you unambiguously
determine its age? If so, what is the age? If not, why not?
Now what are you supposed to do if you come across a block that has 72% 10Be and 28% 10B, or
a block that is 0.23 million years old - how do you deal with these quantities to determine ages or
amounts of material? Well, I’ll tell you - you get to use some formulas (you knew those would
pop up somewhere in here didn’t you?) There are certain mathematical relations that help to
determine ages and abundances and they make use of the number e - yes, that’s a number. It’s
actually equal to something like 2.72. There is also a logarithmic function associated with this
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number (just like regular logarithms are associated with 10), but this one is called the natural
logarithm. It is usually denoted on calculators by an Ln key, which is often the same as the ex
key. Now you’ll finally get to use those keys!
First of all Ln works very much like Log does, so if you punch in Ln (24.5) you’ll get an answer
of 3.2. You can also take e3.2 which gives an answer of 24.5. Pretty easy, isn’t it?
c. What is the value of e -0.595 ?
d. What is the value of Ln(8.220)?
The formula for calculating how much radioactive material is left after a given time is
 / r
f e
where f = the fraction (percentage) of the original radioactive material that still exists, = time in
millions of years, r = decay rate (which is related to the half-life value). Since f is always less
than 1, it is a percentage of the material that remains (remember 0.35 = 35%). For 10Be the
value of r is 1.96, so if you wanted to find out how much 10Be remains after a certain amount of
time, the resulting formula would be
f e
  / 1 . 96
e. What fraction of a solid block of 10Be would remain after 1.01 million years?
f. If you start with a solid block of 10Be what would be the fraction that is still 10Be after 10.5
million years?
Now going the other way, let’s say you have a certain percentage or fraction of material that is
still radioactive and you want to determine how old it is. In that case the formula would be
 = - r Ln(f) = -1.96 Ln(f)
where the meaning of the terms is the same as before and r = 1.96 for the elements we are using
here.
g. So if you have a rock that is made up of 13.5% 10Be (and the rest is 10B), how old is it
(assuming it was all 10Be to begin with)?
h. What if you have a rock that is 5.62% 10Be - how old would it be (again assuming it was all
originally 10Be )?
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2. (11 points total) Astronomers have discovered quite a few planets around other stars in our
galaxy using Kepler’s Third Law (that thing just keeps on popping up, doesn’t it). Recently they
have used the data from the Kepler satellite to observe the passage of planets in front of their
star, which is known as a transit. By measuring the time between transits, it is possible to
determine the size of the planet’s orbit.
Follow the link at the course website to get the information about a planet that has transited a
star.
a. What is the spectral type of your star and its name?
b. What is the time for the orbit of the planet (this is in days)?
c. Convert the orbital period from days to years. What is the planet’s orbital period in years?
Use the table at the course website to get the mass for your star based upon its spectral type.
d. What is the mass for your star?
To determine the orbit size you can use the relationship
a3 = P2 M*
where a= average orbital distance in AU, P=orbital period in years, and M* = mass of the star in
solar masses. You have the values for the orbital period from part “c” and the mass of the star
from part “d”, so all you need to do is solve this formula for the average orbital distance, a.
e. What is the average orbital distance of the planet?
f. If this planet were in our solar system, where would it be located relative to the other planets
(between which planets would you find it)? Assume it has a circular orbit.
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Astronomers are particularly interested in planets that are found in the Habitable Zones (HZ) of
stars. This is the region around a star where a planet could have water on its surface in liquid
form. Planets that are in HZ are often considered as good places to look for life outside of our
solar system, since we view the presence of liquid water as a very important feature.
The size of a HZ depends upon the luminosity of the star. For the star you found in part “a”,
look up the value of its luminosity based upon its spectral type.
g. What is the luminosity of your star?
The extent of the HZ is given by the following relations –
Inner boundary (in AU) = 0.75 x L0.5
Outer boundary (in AU) = 1.55 x L0.5
Where L is the luminosity in terms of the Sun. For our solar system, L=1¸ so the HZ would
extend from 0.75 to 1.55 AU.
For the star system that you looked at, what are the boundaries of the HZ? Use the luminosity
from part “g” and the formulas provided above.
i. Inner HZ boundary =
Outer HZ boundary =
j. Is your planet’s orbital size that you found in part “e”, located within the star’s HZ?
k. (2 points) Do these result indicate that there is life on your planet? Why or why not?