Download The Measurement of the Astronomical Unit

The Measurement of the Astronomical Unit
The goal of this lab is to determine the distance between the Earth and the Sun, referred to as the
Astronomical Unit (AU), using a basic understanding of geometry and tools available in the lab. The
significance of this measurement cannot be overstated: the AU is the basis of the distance ladder. An
incorrect value of the AU would yield errors in parallax measurements to nearby stars and thereby errors
in all subsequent distance measurements. Using Kepler's 3rd Law and observations of the sidereal period
of the planets in our solar system, astronomers were able to create a scaled model of the solar system in
units of the AU. Errors in the AU would thus also result in an under/over-estimation of the size of our
local neighborhood. In this lab, you will use the size of the Doppler shift due to the Sun's rotation to
determine the rotational speed, and then, using the rotation period and the angular size of the Sun,
determine the value of the AU.
There are three steps :
Step 1: Determine the Angular Size of the Sun
The sun appears to move all the way around the Earth (360 degrees) in 24 hours. If we measure how long
(minutes and seconds) the Sun takes to move its own diameter along the sundial in our lab, we can
measure its angular diameter, in degrees.
In the lab you will see a lens attached to the window. The lens is used to focus the Sun's light and make a
sharper image on the easel. This is our "sundial".
One student should mark the right edge of the projected image of the Sun on the easel and AT THE
SAME TIME another student should note the exact time (including seconds).
Wait until the Sun has moved its entire diameter and as soon as it crosses the mark that you made
on the easel, record the new time.
We suggest each group get 3-5 measurements and then you can take an average and estimate your
uncertainty.
Analysis:
Using simple geometry we will can calculate the angular diameter, θ, of the Sun relating the
fraction of time it takes the sun to move its diameter, in degrees, to the time it takes the sun to
move a complete rotation.
Step 2: Determine the Rotational Speed of the Sun
The doppler shift allows us to measure radial velocity. We will make several spectral measures of the
NaD lines (at 5889 and 5896 Angstroms) from the East limb of the Sun and then the West limb of the
Sun. Then with the CCD (and attached microscope lens) we take careful measurements of the Doppler
shifts of these two lines relative to that of a Telluric absorption line, which arises from H2O in the Earth's
atmosphere. Because we do not know the orientation of the solar equator, we will take 8 measurements
around the edge in pairs (left, right; top, bottom; top right, bottom left; top left, bottom right).
The first step is to carefully align the CCD with the NaD lines. This is a slightly tedious process
and your TF will show you the tricks to aligning it painlessly!
First, align the CCD with the Sodium lamp. This will give you a chance to adjust the slit
width to get the sharpest lines possible without compromising brightness. You will need to
take a series of short exposures until you get a non-saturated, sharp image. If you need help
refer to the MaximDL help on taking images.
When you have everything aligned, remove the lamp and now allow sunlight to shine
through the slit! Take a test image to ensure that the table has not been bumped and that you
have the right exposure time and slit width. This may need to be adjusted for the Sun's
spectrum.
OK the hard part is over; everything is aligned so don't bump the CCD or lab bench!
We don't know the exact orientation of the equator of the Sun since the Sun's rotation axis is
tilted due to the Earth's axis tilt and the solar image is "reversed" after it bounces off the
mirrors and enters the spectrograph slit so we will take images at 8 locations of the Sun's
limb (left, right; top, bottom; top right, bottom left; top left, bottom right). The two
(opposite) locations which give the NaD line measurement with the biggest shift will be
closest to defining the equator of the Sun!
Take 2 spectra (i.e. record the CCD images) at each of the 8 locations on the limb. Take the
images in pairs - i.e. N-S (2 exposures for each N and S) and then E-W, etc. Reference the
online Solar spectrum and compare with the solar atlas to ensure that you include an H2O
line in your images!!
Analysis:
We need to derive the most accurate possible minimum positions of the NaD lines vs. the H2O telluric
line in order to measure what will be a small Doppler shift.
Open your images (spectra) in MaximDL.
Open a line information box and draw a horizontal box (~ 10pixels wide) across the middle
of your spectrum. It is VERY important that you start your box from the very left edge of the
image and that it spans the entire width of the image. Pull the box from top-left to bottomright. The data exported will be pixel number (starting with pixel 1 no matter where your box
starts) and number of counts/pixel.
Click export and save your file accordingly.
Do this for all of your spectra.
You will now have 8 Excel data sheets. Your TF will guide you through entering this data into the
Doppler Shift template. There are also instructions on the template to help you as well.
Once you enter all your data in the template you should have all the information needed to
determine a value for the rotational velocity of the Sun.
Step 3: Determine the Rotational Period of the Sun
The motion of the Sun in the sky tells us something about the rotation of the Earth, and the Earth's orbit
around the Sun. An important motion of the Sun itself is its rotation about its own axis. This can be
measured by using sunspots as tracers of that rotation, as Galileo first did in 1612.
Why is rotation period important? For one thing, it is believed that the solar rotation ultimately causes the
creation of sunspots themselves. Other stars (with similar, or cooler, temperature) that rotate more rapidly
than the Sun can have more and bigger spots. A primary goal for this section is to determine the period of
rotation of the Sun - that is, how many days it takes to rotate once around its own axis.
If we take data over a period of a few days to a few weeks, we can determine how much the sunspots
move and determine its apparent motion across the face of the Sun. If we make the assumption that the
spots are carried across the Sun by rotation, we can then infer the period with which the Sun rotates.
First, setup the heliostat so that the projected image of the sun is on the table (you will need the
table mirror setup in front of the spectrgraph slit).
Place a white sheet of paper on the table so that the sun's image fills the circle and adjust the focus
of the projected image by moving the table mirror (starting at the slit) away from the slit. Find the
spot where the image looks most crisp at the edges of the Sun.
Once in focus place the neutral density filter over the projected image of the sun so you can see the
sunspots easier.
When you think you located a sunspot, check to mark sure its a real spot and not just dust or a mark
on the table mirror. To do this, have one person use the motor controls in the lab to move the image
E or W. If you see the sunspot stay in the same spot on the Sun's disk as the image moves, then it's
real!
You can also check the Solar Heliospheric Observatory website to see real time data and
check if there are any current sunspots.
http://sohowww.nascom.nasa.gov/data/realtime/hmi_igr/512/
Another good site is the United States Air Force/National Oceanic and Atmospheric
Administration website that shows the Sun's active regions as well.
http://www.solar.ifa.hawaii.edu/ARMaps/
Now that you have located one (or more!) sunspots you can mark the edge of the sun in at least
three spots on the white paper. You also want to mark the center of your sunspot and then circle it.
If there is more than one spot, mark each sunspot with a different color. Make sure not to move the
paper!
Because there are so many mirrors
that the sun is being reflected off of
to get to our projected image on the
table, we cannot be sure that the
direction E-W is necessarily where
you think it would be, so we need to
check. Sunspots should move on
lines of solar latitude so by marking
the direction the spot is moving we
can determine E-W on the Sun. To
do this, we will have one person
unplug the heliostat motor controls
so the heliostat is no longer tracking
with the Earth's rotation. As the
image of the Sun begins to move
(DO NOT move the paper), another
person needs to BE READY to start
marking sunspot(s). As the
sunspot(s) begin to move, begin
marking the direction the spot is moving by continuing to mark the center of the spot as the Sun
moves. DO NOT move the paper! Once you can tell the direction the sunspot is moving (from
having made a few marks) you can stop marking and turn the motors back on. This tells us the
Terrestrial E-W line.
Also, make sure to record the date and time on the sheet. You will repeat this over a each lab
session over the period of the lab but make sure you are marking the SAME spot(s) each day! Each
day you start with a new blank paper. When we are done collecting data, we will overlay all the
data on the transparent grid to see how the sunspots moved!
NOTE: If there are no sunspots or it is cloudy and you are unable to get enough data, you will have
to use archival data. There are archival sunspot movies on each computer desktop. Open the movie
and resize the window to the same size as the transparency grid of the Sun. Find a sunspot that is
aligned on a line of latitude. You will then be able to follow the analysis below using the time from
the time stamp on the movie.
Analysis:
The first step is to overlay each day's data on top of each other (it helps to hold the papers up to a
light or the window) and rotate them so that the lines of sunspot motion are overlayed. You will
then want to overlay the transparency grid of the Sun. Make sure to note the direction the sunspot is
moving and rotate your marked paper to line it up on a line of latitude. Mark the sunspot (that was
circled) with a marker (labeled by date and time) on the transparent grid. NOTE: the E-W motion
of the sunspot that you marked by turning the heliostat motors off will not be the same
motion/direction that the sunspot moves over the period of your lab.
Once all the data is marked on the transparency, you can then note the direction the sunspot is
moving and begin to calculate the rotational period using simple ratios. We can now determine the
amount of time it takes to for the sunspot to move a X degrees of longitude and the full period, P,
will take a full rotation of 360 degrees.
Final calculations/results:
Now we have everything we need to calculate the Astronomical
Unit, AU. We can do this using simple trigonometry and the skinny
triangle approximation.
First, solve for the solar radius, V=2πR/P
Using the skinny triangle approximation, you can now
determine the AU.
Be sure to include an uncertainty estimate with your final
calculations.
How does this compare to the actual measurement of the AU=1.49597870 x 1013 cm?
Include a discussion of potential sources of error or uncertainty in your calculations.