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Astrophysics
Assignment #1, Spring 2012
“Don’t panic.”
∼Douglas Adams
1: Pizzas and the Number of Stars on the Sky [Fermi Problem Exercise]
The classic Fermi problem is “How many piano tuners are in the city?” I have done with with my introductory
physics class every year since I become a professor, and we have never failed to be within 1 or 2 of the number
listed in the phonebook! Let’s practice on another problem. I’ll lead you through one, then set you off on your
own on a second. Remember: (1) There are many different ways to estimate and your logic may not be the
same as mine or your classmates; as a general rule we hope to get close to the same answer even if we arrive by
different paths. (2)there are no right or wrong answers — only answers that are correct based on
the assumptions you made.
[a] How much pizza is consumed on campus each semester? First we are going to write down a formula for the
answer to this question, then plug in your number estimates. At each stage give the quantity of interest a name
(A, χ, etc), define its units, and record your estimate of its value.
a1 ◮ Fermi estimates are based on numbers you know without referring to your phone or the source of all
knowledge (Wikipedia/Google). In this case, start with your own experience: how many pizzas do you consume
and in what time? How could you make this part of the estimate better?
a2 ◮ How long is a semester?
a3 ◮ How many students are on campus?
a4 ◮ Combine your above 3 numbers to get the number of pizzas consumed on campus per semester.
[b] Sitting in your backyard, you count 75 stars in the area of the Big Dipper, an area about the size of your
outstretched hand. How many total naked eye stars are visible on the entire sky?
HINT: The statement about your hand is a way of measuring angular size on the sky.
2: Space Astrometry Missions [∼C&O 3.5]
a ◮ The Hipparcos Space Astrometry Mission was able to measure parallax angles down to nearly 0.001′′. To
get a sense of that level of resolution, how far from a dime would you need to be to observe it subtending an
angle of 0.001′′ ? The diameter of a dime is approximately 1.9 cm.
b ◮ Assume that grass grows at the rate of 5 cm per week. (i) How much does the grass grow in 1 second? (ii)
How far from the grass would you need to be to see it grow at an angular rate of 0.000004′′ (4 microarcseconds
≡4 µas) per second? 4 µas is the estimated angular resolution of the proposed NASA astrometric mission, SIM
(<http://sim.jpl.nasa.gov>).
3: Connected Properties of Stars [∼C&O 3.9]
Consider the star Dschubba (δ Sco), the center star in the head of the constellation Scorpius. Assume that it is
a spherical blackbody with a surface temperature of 28, 000 K and a radius of 5.16 × 109 m, located at a distance
of 123 pc from Earth. Determine the following:
a ◮ Luminosity b ◮ Absolute magnitude c ◮ Apparent magnitude d ◮ Distance modulus e ◮ Radiant flux at
the surface of the star f ◮ Radiant flux at Earth (compare to the flux from the Sun) g ◮ Peak wavelength of
emission
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4: Moving Stars [∼C&O 5.1]
Barnard’s Star, named after the American astronomer E.E. Barnard (1857-1923) is an orange star in the constellation Ophiuchus. It has the largest known proper motion (motion perpendicular to the line of sight:
µ = 10.3577′′/yr) and the fourth largest parallax angle (p = 0.54901′′). Only the stars in the triple system
α Centauri have larger parallax angles. In the spectrum of Barnard’s star, the Hα absorption line is observed to
have a wavelength of 656.034 nm when measured from Earth.
a ◮ Determine the radial velocity of Barnard’s Star.
b ◮ Determine the transverse velocity fo Barnard’s Star.
c ◮ Calculate the speed of Barnard’s star through space.
5: Right from the Fish’s Mouth [∼C&O 8.16]
The blue-white star Fomalhaut (“the fish’s mouth” in Arabic) is in the southern constellation of Pisces Austrinus.
Fomalhaut has an apparent visual magnitude of mV = 1.19. Use the HR Diagram in Figure 8.16 of the text to
determine the distance to this star.
6: Lexical Analysis
Develop your own mnemonic(s) for the spectral type sequence, OBAFGKM. We will collect all the entries on
the class website for future reference.
7: The Hipparcos Catalog [Data Archive Exercise]
Learning to work with data is a big part of modern astrophysics. You are fortunate enough to live in a golden
age, when much of this data is accessible online. We will work repeatedly with data archives in this class, and
learn how to manipulate data and analyze it. Here in the early exercises, I will guide you step by step.
Archiving and making data accessible is a huge part of what our national agencies (NASA and the NSF) do
for us. One that we will use repeatedly in this class is the NASA HEASARC Archive. You can find it here:
<http://heasarc.gsfc.nasa.gov/docs/archive.html>
Hipparcos was a space-astrometry mission that measured the parallax angle of 2.5 million stars. In this exercise,
we will query the Hipparcos catalog, edit the data into a format suitable for input to a spreadsheet, then use the
data to study the distribution of absolute magnitudes M for approximately the 300 brightest stars in our sky.
a ◮ Go to the HEASARC Archive, and click on the Browse link partway down the page. This will open up a
page with many listed missions. If you are searching for a single object, you can work from this page directly,
but we want to query the entire Hipparcos catalog. Partway down the page, under the header “Popular Catalog
Choices” you will see Hipparcos Main listed. Click on the name! This will open up a short description of the
Hipparcos mission. Right at the top is a link that says HIPPARCOS Catalog — click on it!
b ◮ You should now be on the main catalog page. This page describes the catalog, provides a reference to the
literature (important, if you ever use this data in a report or publication) and a description of all the fields in
the catalog data. In the upper left is the link we want to generate our own custom data. Click on Browse this
table.
c ◮ This is the Query Page, where you construct your request for the data you are interested in. You can select
specific data, you can limit the values of the data you want to see, and you can ask that the data be sorted
when it is returned. In this case we want all the stars that have apparent magnitudes brighter than m = 3.6.
We want to calculate the absolute magnitude, so need the distance. If you browse your choices, you will see that
distance is not listed! But have no fear — the reason we are using Hipparcos is because it measured parallax
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angles, which can be used to compute distances! In the “View” column, check the following boxes (the data we
would like returned): name, ra, dec, parallax, vmag, parallax error. In the “Sort” column, sort by vmag. In the
“Query Terms” box, enter < 3.6 on the vmag row1.
d ◮ At the bottom of the page, there are a series of choices about the output. In the “Output Format” choose
Pure Text; I find this is the best choice to make the data easy to manipulate into a format that works with
spreadsheets, but you may find others work better for you. Feel free to experiment! Click Start Search.
e ◮ A pure text webpage should open up with your data. Save this to a text file. Keeping data organized
can be a massive chore, so it pays to be diligent about where you keep files and what you name them. Choose
an appropriate name, and keep the file handy! We will come back to this data again in the future! I used
hipparcos3.60maglimit.txt
f ◮ Open up the file in a text editor. You’ll notice that the NASA system used the symbol | to separate data
fields. For spreadsheets a comma is a better separator. Do a search and replace on the entire file to change | to
comma. Save the new file with the same name as your previous filename but the extension *.csv
g ◮ Now open the data up in a spreadsheet. We are going to use the spreadsheet to first compute the distance
d in parsecs to each star, and then to compute the absolute magnitude M . The Hipparcos mission measured
parallax angles in milli-arcseconds over a baseline of 1 AU. Remember the definition of parsec: the distance d
at which 1 AU subtends an angle of 1′′ . Since the parallax angle from our query is reported in milli-arcseconds,
the distance in parsecs is given by
d(pc) = 1000/parallax
Compute a column in your spreadsheet for the distance in parsecs.
h ◮ With the apparent magnitude m and the distance d in parsecs, you can calculate the absolute magnitude
M from the distance modulus formula:
m − M = 5 · log(d/10pc)
Compute a column in your spreadsheet for the absolute magnitude.
i ◮ Turn in with this problem: (1) Printout of the first 10 rows of your spreadsheet. (2) Histogram of the
apparent magnitudes m (the vmag from the catalog). (3) Histogram of your calculated distances in parsecs. (4)
Histogram of your calculated absolute magnitudes M .
1 this
is where we are selecting stars with m < 3.6; remember that this is stars brighter than 3.6!
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