Download Refer to the class notes which describe how Aristarchus figured out

1. Aristarchus and the relative Earth-Moon and Earth-Sun distances. (20
points)
Refer to the class notes which describe how Aristarchus figured out the ratio of
the Earth-Moon distance to the Earth-Sun distance. To do this he had to measure
the angle between the Moon and the Sun when the Moon was exactly a quarter
moon. What if he had measured an angle of 45 degrees, instead of 87 degrees?
What would he have concluded about the ratio of the Earth-Moon to Earth-Sun
distance? Draw an accurate diagram showing the geometry of this measurement,
using a protractor. Then measure the relative distances with a ruler on your
drawing, or use your knowledge of trigonometry to derive the exact result.
The precision of the Hipparcos measurements of parallax is about 0.002
arcsecond. That means that we can rely on Hipparcos measurements to
determine distances out to 100 parsecs or a bit further.
Homework
1. Way back in 1838, Bessel determined the parallax to 61 Cygni to be
0.314 arcseconds. How good was his measurement? You can check it
against a more modern measurement by the Hipparcos satellite.
a. The star 61 Cygni appears in the Hipparcos catalog as star number
104214. Use the Hipparcos catalog search site to find the star's
parallax, as measured by Hipparcos; type this number into the
"Hipparcos identifier" box down near the bottom of the form.
b. What is the percentage difference between the Hipparcos angle
and Bessel's angle?
c. Use the Hipparcos parallax angle to calculate the distance to the
star.
d. What is the percentage difference between the Hipparcos distance
and Bessel's distance?
2. Computing the uncertainty in distance to a star based on its parallax can
be a tricky thing. Consider the case of Joe and the star Rigel. Joe
measures a parallax half-angle of π = 0.004 arcsec to Rigel. He
estimates the uncertainty in his measurement to be +/- 0.003 arcsec. In
other words, the parallax could be as large as 0.007 arcsec, or as small as
0.001 arcsec.
a. What is the distance to Rigel, if Joe's measurement is exactly
correct?
b. What is the distance to Rigel, if the error in Joe's measurement is
+0.003 arcsec (i.e. if Joe's measurement is at the lower end of its
estimated uncertainty)?
c. What is the distance to Rigel, if the error in Joe's measurement is 0.003 arcsec (i.e. if Joe's measurement is at the upper end of its
estimated uncertainty)?
Bonus! What is the actual distance to Rigel? How well do we know it?
Name: ________________________
DUE 10/25
ID number: ________________________
Homework 9. Finding Distance With Parallax
When your point of view changes, nearby objects appear to shift with respect to
more distant ones. This is called parallax, and it's a basic tool for measuring
astronomical distances. The same idea can be used to measure distances to
objects on Earth.
For this assignment, you need to measure small angles with an accuracy of
about 0.1°. An ordinary protractor is not that accurate, but you can easily make
something which is. You need a plastic ruler marked in centimeters, a length of
thread, and a paperclip. Tie one end of the thread to the paperclip. Starting at
the knot, measure out 57 cm of thread, and mark the thread at that point. Place
the ruler face-down and lay the thread across the middle, with the mark at the
bottom edge of the ruler. Put a piece of tape on the back of the ruler to hold the
thread in place. Done!
To use this contraption, take the ruler in one hand and the paperclip in the
other. Hold the paperclip to your cheek below your eye, and extend your other
arm until the thread is taut. Look toward the objects you want to measure and
hold the ruler at a right angle to your line of sight. With the ruler 57 cm from
your eye, an angle of 1° corresponds to a separation of 1 cm. Thus, when you
sight past the ruler, two objects which appear to be separated by 1 cm have an
angular separation of 1°.
This diagram is an overhead view showing the geometry of a parallax
measurement. Such a measurement requires observations from two different
places separated by a known distance. This distance, the baseline, is
represented by the symbol b. Pick a fairly nearby target which you can view in
front of a background much further away (for example, you might use the pole
of a streetlight as your target, with the side of the valley as a background). For
the first observation, line the target up with some definite landmark in the
background (for example, a rock on the side of the valley). Now move to your
second observation point, and use the ruler-and-thread to measure the
angle
between your target and the background landmark. The distance Rto
your target is
The key to this assignment is not just to make a measurement -- you will also
have to make some choices, and explain why you made those choices. At every
step, your choices affect the accuracy of your result, so think carefully before
choosing.
1. Describe the target you chose and the background landmark you will use for
your measurement.
2. Pick a value for the baseline distance b, and explain your choice. Ease of
measurement is a valid reason for picking a certain baseline, but accuracy is
also a consideration. If your baseline is too small, it will be hard to measure the
angle
accurately. (Note that it does not matter what units you use for b;
you will automatically get R in the same units!)
3. Now record the angle
side of the page to find R.
you measured, and use the equation on the other
4. Finally, try to guess the size of any errors you might have made in
measuring b and
. Which part of the assignment was most difficult? How
would you improve your measurements?
Joshua E. Barnes ([email protected])
Last modified: October 19, 1999
http://www.ifa.hawaii.edu/~barnes/ast110_99/homework/hw9.html