Download Measuring the Size of the Astronomical Unit (AU)

APS 1010 Laboratory
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Distances of the Earth, Venus, and Mercury
Measuring the Size of the Astronomical Unit (AU)
SYNOPSIS: In this lab, you will estimate the size of the orbits of Venus and Mercury, and measure the
distance that the Earth is from the Sun. You will also be able to derive the size of the Sun, and the size
of Venus as part of the exercise.
EQUIPMENT: Voyager II program in the SBO computer lab, ruler, calculator.
Even if one has accepted the idea of a Copernican universe where all the planets including the Earth
circle the Sun, the next logical question one might ask is, “Exactly how far away do they orbit?” This is
a non-trivial and very important question. The distance of the Earth to the Sun wasn’t determined
accurately until the 18th century, and the first measurements that gave values close to today’s accepted
value of 150,000,000 km were not made until the late 19th century. The distance of the astronomical
unit (AU) has direct bearing on the scale of the solar system. Even though Kepler’s Laws can be used
to infer the distances of the planets, the distances that one gets is in units of AUs, so without knowing
how big an AU is, we only know the relative distances to the other planets, not the exact distances.1 In
this lab, we will attempt to figure out the distances from the Sun to the Earth and the inferior planets.
(The inferior planets have orbits closer in towards the Sun than the Earth.) We will use the geometry of
similar triangles and trigonometry to calculate the orbital radii of Venus and Mercury, and to eventually
figure out the exact distance defined by the AU.
Part I. Maximum Elongations of Venus and Mercury
If we start out assuming the Copernican model of the solar system, we note that Venus and Mercury,
because they lie in orbits closer in towards the Sun than the Earth, never move very far away from the
Sun. In fact, as we watch the motion of Venus over the course of its orbit, we will see it move to a
maximum elongation from the Sun in one direction, and then reverse itself. It will then move to the
maximum elongation on the other side of the Sun.
If we were to look down upon the plane of the Solar System, the positions of maximum elongation can
be shown as in the diagram below. Notice that if you drew lines connecting the Earth, Venus, and the
Sun while Venus was at maximum elongation, you would get a right triangle with the Earth-Sun
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Measuring the AU not only tells us the distances to the other objects in our Solar System, but it is the starting
point for determining the distances to everything else in the universe! Trigonometric parallax is used to determine
the distances to the nearest stars, and for trigonometric parallax to work, you must know how far the Earth is from
the Sun.
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Distances of the Earth, Venus, and Mercury
distance DES as the hypotenuse, the Venus-Sun distance DVS as one leg of the triangle, and the EarthVenus distance as the other leg of the triangle. Note that by definition, DES is equal to 1 AU.
I.1
The maximum elongation can be expressed as an angle of separation between Venus
and the Sun in the sky. This angle is shown in the above figure as θmax. From the
above diagram you should convince yourself that if Venus was anywhere in its orbit
other than the maximum elongation, the angle between it and the Sun as seen from the
Earth would be smaller than θmax.
I.2
What is the maximum angle of elongation that an inferior planet can have? How big
would its orbit be? What is the minimum angle of elongation that an inferior planet can
have? (For this second question, you can give a qualitative answer as opposed to
giving numbers.)
From the figure above, and using basic trigonometry, we can derive the following relationship between
the angle of maximum elongation, θmax, the distance from the Earth to the Sun, DES, and the distance of
Venus to the Sun, DVS:
sin θmax = DVS / DES
I.3
(1)
Now load up Venus-Elongation simulation in Voyager II by going to the File menu at
the top left-hand corner of the menu-bar and go to Open Setting....
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Distances of the Earth, Venus, and Mercury
You will find a distorted projection of the sky on the screen that allows you to see much
of the sky at once. The Sun is up and Venus is the purple dot by it. The program will
start out a few months before maximum elongation. The time step in the Control
Panel should be in units of “7 days.” Stepping through the simulation one week at a
time with the “+” button means you advance to the same time of the day but a week
later. You should do this until you get close to the point of maximum elongation. You
may have to step backwards in time with the “-” button if you pass the point of
maximum elongation. Once you get close, you will also want to change the time step
to “1 day” in the Control Panel by holding down the mouse while the cursor is over
the “7 days” button and waiting for a pop-up menu that allows you to change the Time
Step. Once you do this, you can fine-tune your measurement of the maximum
elongation. To measure the angle of separation between Venus and the Sun at each
step, follow these instructions:
A. Put your cursor over the Sun and click once. The Angular Separation
display will now measure the separation of your cursor from the current position
of the Sun.
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Distances of the Earth, Venus, and Mercury
B. Now place the cursor over Venus but do not click. Now the Angular
Separation counter will tell you what the separation is between the Sun and
Venus in the sky in degrees (o) and arcminutes (′). (An arcminute is 1/60th of a
degree.) If you had clicked on Venus, the separation will be 0o00′ since the
separation will be measured from Venus! You will want to center the cursor on
Venus as accurately as possible before making your measurement.
C. As you step through the simulation using the “+” and “-” buttons, you will
have to move the cursor back to the Sun and reclick, before moving the cursor
back to Venus and reading off the Angular Separation counter. This is because
the Sun also moves from day to day. If you don’t reset the counter to the current
position of the Sun, your measurement of the separation will be with respect to a
past Sun position.
D. Once you are close to the maximum elongation point, switch to a smaller time
step, like “1 day,” and repeat steps A through C. This will allow you to get a
more accurate measurement of the angle.
What is the angle of maximum elongation for Venus?
measurements? What limits your accuracy?
How accurate were your
Using Equation (1), plug in the angle of maximum elongation that you’ve measured
along with DES = 1 AU and solve for DVS in units of AUs. (If you don’t know how to
take the sine of an angle that has degrees and arcminutes on your calculator, ask your
TA!) What is the distance from Venus to the Sun in AUs? How does it compare to the
value tabulated at the end of your textbook? How might you have improved your
measurements and arrived at a more accurate result?
I.4
Repeat the measurement of maximum elongation angle and calculation for Mercury, by
loading up the Mercury-Elongation simulation in Voyager II. Mercury will be appear
as an orange dot in this simulation. Calculate the distance to Mercury in AUs. Again
how does your answer compare with that in your textbook?
I.5
Extra Credit Part I: Turn on the Paths feature by clicking the button that is labeled
“Paths” in the Planet Panel:
On the Control Panel, reset the time step to “7 days.” Now step forwards and
backwards in time through the simulation again. You should see the Sun and Mercury
tracing out colored paths showing where they appear in the sky at different times of the
year. Trace their motions in the sky for several years; you will see that the Sun moves
in a regular pattern in the sky, whereas the path of Mercury is more complicated. Draw
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Distances of the Earth, Venus, and Mercury
the path of the Sun. Where is the Sun on this path during the equinoxes and the
solstices? Why would the Sun be where it is along the path at these dates? What is the
reason for the shape of the Sun’s path? Give a general description of why the motion of
Mercury is so peculiar.
I.6
Extra Credit Part II: You’ve deduced the distances of Venus and Mercury from the
Sun by measuring maximum elongation. What is another technique that you could
have used to figure out their distances from the Sun. What would be the sources of
error from this technique? How do you think the two techniques would compare as far
as accuracy?
Part II. Using Transits to Measure the Astronomical Unit
We have figured out the distances of Venus and Mercury from the Sun in units of AUs, but how large is
an AU? The first person we know of who attempted to get a rough idea of how far the Sun is was
Aristarchus of Samos in 275 BC. However his answer was considerably off because he was observing
with the naked eye, and his technique, although valid, required extremely precise measurements of the
angle between the first quarter Moon and the Sun. Aristarchus came up with the Sun being only 19
times farther away from the Earth than the Moon. This is considerably off from the true distance, which
is equal to 390 times the distance to the Moon.
The technique that was eventually used to accurately measure the distance to the Sun from the Earth was
by observing transits. A transit occurs when a planet closer in to the Sun passes between us and the
Sun, and is so aligned that when we observe it, it appears to cross the disk of the Sun. Because Mercury
and Venus, the only planets that can transit the Sun for observers on Earth, have slightly tilted orbits
with respect to the Earth’s orbit, a transit happens rarely. More often than not, the inferior planet will
pass above or below the Sun as seen from the Earth (see diagram below). Mercury in a closer in orbit
will transit the Sun only 14 times in the 20th century. The orbit of Venus relative to the orbit of the
Earth is such that transits of Venus against the Sun are even rarer. In fact, the last one occurred on
December 6, 1882, and the next won’t occur until 2004!
During a transit, one can actually observe the inferior planet traversing across the disk of the Sun.
Because of the brightness of the solar disk and the small relative size of the transiting planet, observing
transits was possible only after the invention of the telescope whereby one could use filters to cut down
the bright solar glare.
Depending on where you are on the Earth, the transit of Venus will appear to take place across different
parts of the Sun. In the exaggerated diagram below, the transit of Venus is observed by Observer A in
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Distances of the Earth, Venus, and Mercury
the northern hemisphere and Observer B in the southern hemisphere. Observer A sees the transit of
Venus across a different part of the Sun than Observer B because they are viewing it from different
places on the Earth. Venus appears to shift its position relative to the Sun because it is much closer to
the Earth. The same parallax effect occurs when you hold up your thumb and look at it with the left
versus the right eye. The position of your thumb will appear to shift relative to a background object
much further away.
Notice that in the above diagram, there are two similar triangles, one connecting up the Earth and Venus,
and the other connecting up the Sun and Venus. The base of one triangle is the separation of Observers
A and B on the Earth, SEarth. The base of the other triangle is the separation between the apparent
transit paths on the Sun, SSun. For two observers at different latitudes, one can calculate how far they
are separated from one another, SEarth. From similar triangles one can figure out what the corresponding
distance, SSun, is. From there one can estimate the actual size of the Sun, and using the small angle
approximation to figure out how far the Sun is from the Earth, and hence measure the AU.
For this lab, we will follow the same procedure, by simulating the upcoming transit of June 8, 2004.
Because the difference in transit paths due to the parallax effect is very small when measured from the
Earth, instead of measuring from the Earth’s surface, we will make our measurements from a pair of
imaginary spacecraft situated above and below the Earth’s poles:
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Distances of the Earth, Venus, and Mercury
The geometry will be similar to the first example, this time with the separation between the satellites as
Ssat and the separation of transits as seen on the Sun as Stransit. Using similar triangles, it is possible to
derive the following relationship between the transit separation on the Sun as a function of satellite
separation and Earth-Venus distance DEV and the Venus-Sun distance DVS:
Stransit / DVS = Ssat / D EV
(2)
For this exercise, you will assume that the separation between the two satellites is Ssat = 114,000 km,
and use the value for DVS in units of AUs that you found in I.4. Although you haven’t explicitly found
the Earth-Venus distance DEV, you can solve for it by using the relation:
DEV = DES - DVS
(3)
Remember that by definition, the distance of the Earth to the Sun, DES, is 1 AU!
II.1
Load up in Voyager II the Venus Transit 2004/Satellite 1 simulation. You will see a
negative image, i.e., the Sun will look black and the surrounding sky will be white.
Venus won’t appear with any visible phases because it is between us and the Sun and
we are seeing a New Venus (like a New Moon when the Moon is in the direction of the
Sun). Click on the ‘+’ button to advance the simulation. You will soon see the
silhouette of Venus cross the disk of the Sun. The tracing feature has been turned on so
you will see Venus’ path marked out as a line. After you run through the entire
simulation making sure the entire path of Venus across the Sun has been traced out,
print out the screen by going up to the File menu and releasing the mouse on “Print
Sky Chart...”
II.2
Load up in Voyager II the Venus Transit 2004/Satellite 2 simulation. Run the
simulation forward with the ‘+’ button until you trace out the entire transit path as seen
from Satellite 2. Now print out the screen again.
II.3
Overlay the two printouts over one another and use a ruler and measure the separation
between the two transit paths. Measure the diameter of the Sun on your printout as
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Distances of the Earth, Venus, and Mercury
accurately as possible with the ruler in centimeters. What is the transit separation and
the diameter of the Sun? What fraction of the Sun’s diameter is the separation between
the transit paths? That is, how many times bigger is the Sun’s diameter than the
separation between the two transits?
II.4
Now insert the value for Ssat, and the values for DEV and DVS in AUs into Equation
(2), and solve for Stransit, the physical separation between the transit paths on the Sun.
What are the units of your answer?
II.5
Using the ratio between Stransit and the Sun’s diameter from II.3, and the value for
Stransit from the previous question, what is the diameter of the Sun in kilometers?
II.6
Now we know the actual diameter of the Sun, we can use the small angle
approximation and our knowledge about the angle that the Sun subtends in the sky to
figure out how far away it is. The Sun appears to us to be 30 arcminutes across. Using
the value for the diameter of the Sun that you derived, what is the distance to the Sun in
kilometers? How far off is your calculated value compared to the accepted value of
150,000,000 kilometers?
II.7
Now that you know what 1 AU is, what are the distances of Venus and Mercury to the
Sun in units of kilometers?
II.8
Venus during a transit across the Sun appears as a small disk 1 arcminute across.
During a transit, the separation between the Earth and Venus is just the difference
between DE and DV . Using the small angle approximation again, what is the diameter
of Venus in kilometers? How does that compare with the diameter of the Earth?
II.9
Extra Credit Part III: We had mentioned earlier that Aristarchus was the first person
to attempt to figure out how far away the Sun was. Do some research at the library and
find out how exactly did he do this. What was he trying to measure and how did that
allow him to guess the distance to the Sun? Why did his initial method give an answer
that is so far off from today’s accepted value? Could we use his method today to get a
more accurate result than he did?