Assignment #7 – Transits of Venus & Mercury - Faculty Web Pages

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Name: ___________________________
Class: ___________________________
Date: ____________________________
Assignment #7 –
Transits of Venus & Mercury
On November 8, 2006, the planet Mercury did something very unusual and exciting: it passed directly
in front of the Sun. In astronomy, this is called a transit - when one astronomical object passes in front of
another one from our point of view here on Earth. When the objects transiting are the Moon and Sun we call
a transit an eclipse. The only two planets which can transit across the face of the Sun are, of course, Mercury
and Venus, since they are the only two planets between Earth and the Sun. Transits of Mercury and Venus
are not very common, since, as you have seen, the orbital planes of the planets are not exactly aligned, so it's
rare that the Earth lines up with the Sun and either Mercury or Venus. There was a transit of Venus on June 8,
2004, and one of Mercury in 2003. Prior to that, the last transit of Venus was in 1882, and it was a huge
event at the time!
The 2004 Transit of Venus, captured in a multiple exposure photo
(from the Sydney Morning Herald in Australia)
Besides being a beautiful event, the Transit of Venus in 1882 led to a crucial scientific calculation.
By carefully measuring the path of Venus across the face of the Sun, astronomers could calculate both the
diameter of the Sun and the value of the Astronomical Unit (the distance between the Earth and the Sun).
Until then, no one knew the exact value of either one, and all the distances to the planets were expressed as
multiples of an AU, even though no one knew exactly how long an AU actually was! The Transit of Venus
answered both questions. In this exercise we'll simulate the transit, using Stellarium, and use it to calculate
the diameter of the Sun, just as astronomers did in 1882! We'll do the same with the 2006 transit of Mercury.
In order to understand how we can calculate the diameter of the Sun from watching a planet pass in
front of it, look at Figure 1 on the next page. It is a side view of the transit of Venus, as it is happening, with
Earth on the left, Venus in the middle, and the Sun on the right. The relative distances of the planets are
known – The Earth is 1.00 AU from the Sun, and Venus is 0.72 AU from the Sun. Remember, even if we
don't know how long an AU is, at least we know that Venus is 0.72 times closer to the Sun than Earth. The
Greeks figured this out thousands of years ago using simple geometry, even though they had no idea how long
an AU actually was!

Figure 1
(from GSFC/NASA)
If the Earth is 1.00 AU from the Sun, and Venus is 0.72 AU from the Sun, then during transit, when
all three are in a straight line, Venus must be 0.28 AU from Earth (1.00 AU - 0.72 AU = 0.28 AU).
Two observers on Earth (Observers A and B in Figure 1) take pictures of the transit. They see two
different things, thanks to the idea of parallax. Because the two observers are at different points of the Earth,
they each see Venus take a slightly different path across the Sun. Observer A will see Venus transiting a bit
lower on the Sun than Observer B (the difference between the two paths of Venus across the Sun are greatly
exaggerated in the diagram). The larger the separation between Observer A and Observer B, the larger the
difference in what they see. If we carefully measure the difference between the position of Venus against the
Sun during the transit as observed by Observer A and the transit as observed by Observer B, we can use that
difference to calculate the diameter of the Sun. Let's do this with Stellarium. Everything we do with Venus,
is, of course, the same as what we will do later with Mercury.
Let's look at the transit of Venus from the North Pole. Start Stellarium. Change the Date & Time to
June 7, 2004, at 10:00 PM. Change your location to the North Pole by clicking as close to the top edge of
the map in the Location window, or, more accurately, by setting the Latitude to be 90° in the Location
window, if you didn't click exactly on the North Pole. Close the Location window. Notice that it is still light
out, even though it's 9 PM! Find and center the Sun, and then zoom in until it fills the screen. The Field of
View (FOV) should be about 0.7°. If you don't see the Sun, it probably means you haven't displayed the
Planets - do this by checking the Show Planets box in the Planets and Satellites section of the Sky sub-menu
in the View window. Turn off the Atmosphere (if it is on) by pressing the A key. Finally, press the Switch
Between Equatorial and Azimuthal Mount button in the bottom toolbar until the little telescope icon I slit
up, to make the plane of the Ecliptic horizontal. This last step is very important! The Switch Between
Equatorial and Azimuthal Mount button should be lit up.
Press L a few times to speed up the passage of time and watch for a few moments. At about 10:15
PM, you should see the tiny black circle appear on the lower left side of the Sun, blocking the light of the Sun.
That's Venus! Watch as Venus starts to transit across the Sun. This is what you would have seen if you had
looked at the Sun (with heavy sunglasses, of course!), on June 7 of 2004, from the North Pole! You can see
that a transit is really another word for an eclipse, except that now the object passing between us and the Sun
is not the Moon, but Venus. Stop the motion of Venus when it gets about halfway across the Sun. Now
carefully mark the position of the center of Venus (as best you can) on the screen with a piece of tape or a
Post-It note.
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Now, while Venus is sitting there in the middle of the Sun, open the Location window and change
your location to the South Pole by clicking as close as you can to the South Pole on the map, or by adjusting
the Latitude to 90° South. Close the Location Window. Notice that you will have to turn off the Ground
(by pressing the G key) to be able to see the Sun.
Did you see that Venus hopped to a slightly different position? The hop is small, but noticeable. The
hop is caused by parallax, which is due to the different latitudes of the two observing locations! We looked
at the Sun at the same moment from the North Pole, and then the South Pole. This, of course is impossible in
real life.
• Why is it impossible to see Venus from both the North Pole and the South Pole at the same time?
(Hint: look at the date and remember our “Seasons” assignment) ____________________________
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Get a ruler (marked in centimeters and millimeters, not inches!), and measure the size of the entire
Sun, in millimeters, on screen, from the left to the right edge. Enter this in Table 1 below as Total Sun Size.
Now measure the amount that Venus shifted when we moved from the North to the South Pole. Measure
from the point you marked with your Post-It note to the center of Venus' new position. Measure carefully -
Venus will have only shifted by a few millimeters! Enter this as Venus' position change in Table 1.
Table 1
Total Sun size (mm) Venus' position change (mm)
Now let's use this information to calculate the diameter of the Sun. First let's figure out how far
Venus moved from Observer A to Observer B, expressed as a fraction of the whole size of the Sun. To do
this, divide Venus' position change by the Total Sun size from Table 1. The result is the fraction of the
entire Sun that Venus moved. Call it p .
p=
Venus' position change
Total Sun size
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= ________________
This should be a pretty small number, since the difference between the two positions of Venus is
pretty small. The fraction of a Sun diameter that Venus moved between our two views is a small fraction of
the Sun's total diameter.
Now, if you look at Figure 1, you might notice (if you remember your geometry) that the triangles
formed by the two bases SO and ST and their common apex at Venus are similar triangles. This means that
that the ratio of any two “similar” parts (sides or angles) of the two triangles are the same. Ratio, of course,
just means “divided by.” Let's choose the two bases (SO and ST) and the two heights (0.28 AU and 0.72 AU)
as our two similar parts of the two triangles. So we can write:
S O
0 .28AU =
S T
0.72AU

or
S T = 2 . 571× S O
Equation 1
Now we know what SO is – it's the distance between our two observers, which in this case is just the
diameter of the Earth, since one observer is at the North Pole, and the other is at the South Pole! Look at
Figure 1 if you're confused about why this is. Look up the diameter of the Earth in your textbook (there's a
table of Planetary Data in one of the appendices in the back of the book). Remember, astronomers have
known this number since Eratosthenes first calculated it 2200 years ago!
S O = diameter of the Earth = ______________________________________ km
Now plug this value into Equation 1 to find S T - the separation of the two transit locations.
S T = ________________________________________
S T is the separation of the transit positions at the distance of the Sun – in other words, it's how far
apart the two images of Venus would be if we projected them in straight lines onto the surface of the Sun.
But we've already shown that the size of the change in Venus' position on the Sun is only a small
fraction of the whole Sun's diameter. So ST is only a correspondingly small fraction of the whole Sun
diameter. To find the whole Sun diameter, Ds, we must divide ST by the fraction s .
units!
D S = S T
p
Equation 2
Plug in your values of p and S T into Equation 2 and calculate D S . Don't forget to list the correct
D S = _________________________________
Congratulations! You've just calculated the diameter of the Sun, merely by watching Venus pass in
front of it and using some clever geometry! Pretty amazing!
We could also, if we wanted, use these same measurements to calculate the exact length of the
Astronomical Unit, or AU, in kilometers, but that would require a tiny bit of slightly advanced math
(trigonometry), so we'll leave that for another exercise!
Now let's do the same thing, only this time with the November 8, 2006 transit of Mercury. Change
the Date & Time to November 8, 2006, at 11:00 AM. Change your Location back to the North Pole.
Mercury is about to begin crossing the face of the Sun. Press L to speed up time and watch as Mercury
transits across the face of the Sun. Stop time when Mercury is halfway across the Sun. Notice that since
Mercury is much smaller than Venus, and is farther away than Venus, it shows up as a much smaller dot on
the face of the Sun. Again, make sure the Ecliptic plane is horizontal by checking to make sure the Switch
Between Equatorial and Azimuthal Mount button is lit-up in the toolbar.
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As you did with Venus, change your observing site between the North Pole and the South Pole and
measure the change in Mercury's position. It's much harder this time, because Mercury, being much closer to
the Sun than Venus, hops a much smaller amount when you switch observing locations. Do your best to use
the ruler to measure the small change in Mercury's position between the two observing locations and record it
and the Sun's total size in Table 2 below.
Table 2
Total Sun size (mm) Mercury's position change (mm)
Now we just do the same calculations for Mercury that we did for Venus. The only change is that
Mercury's distance from the Sun is 0.39 AU, not 0.72 AU (Mercury is, of course, closer to the Sun than
Venus!). So now, instead of Equation 1, we have
S O
0.61AU =
or
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S T
0.39AU
S T = 0 .639× S O
Equation 3
Do this calculation, using the same value for S O - the diameter of the Earth - that you used before.
S T = ________________________________________
Now calculate p the same way you calculated it before: by dividing Mercury's position change by the
Total Sun size.
p=
Mercury's position change
Total Sun size
= _____________________________
And finally, calculate D S (the diameter of the Sun), just like you did with Equation 2, only this time
using Mercury instead of Venus:
D S = S T
p
Equation 4
D S = _________________________________
Now we have two measurements of the diameter of the Sun. Hopefully they're the same – or at least
near each other! Take the average of your two values of DS by adding the two values and dividing by two.
Call this average value

= _____________________________________
Let's see how close our average value is to the real value of the diameter of the Sun by calculating our
percentage error. The percentage error is given by
6
100
Equation 5
Where D acc is the accepted value of the diameter of the Sun. Look this number up in your textbook,
just as you did for the Earth.
D acc =___________________________
Now, using Equation 5, calculate your percent error.
% error = ________________________________________________
Not too bad! If you made careful measurements your percentage error should be small. Careful
measurements are exactly what many astronomers made in 1882, to calculate the diameter of the Sun for the
first time. These astronomers, and people like you, using some clever intuition and high-school math, can
measure our entire Solar System, without ever leaving Earth. Events like transits are not only beautiful and
exciting events to watch in the sky, but provide crucial information that can help us understand our Solar
System in great detail.
Write a conclusion explaining what you learned in this exercise.
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