Math 1060 – 003
Transit of Venus
Summer 2012
Camacho
Measuring the Astronomical Unit
Using the Transit of Venus!
History
In the mid 1700’s, the transit of Venus was successfully used by astronomers of the time to
approximate the distance between the earth and the sun, otherwise known as the astronomical unit.
Transits of Venus had been observed previously, but there world had changed significantly since the last
observation. The most significant contributing factors were the following:
1. By the mid 1700’s, there were several imperial powers (English, Spanish, French,
Portuguese, etc.) who ruled over land in multiple continents. This made it possible to
coordinate an international effort and view the transit of Venus from several different
places on the earth. Navigation techniques of the time made it possible to calculate exact
distances between different points on the earth.
2. Lenses had been developed and perfected for approximately a century, which made it
possible to accurately view the transit of Venus and take careful measurements without
staring directly into the sun.
3.
A multitude of astrological data had been collected and analyzed by that point in time.
Johannes Kepler used this data to calculate the relative distance between celestial objects
(though not absolute distances). In 1716 Edmond Halley was able to use the data to predict
the exact date and time of the next transit of Venus.
In this document, we will let D represent the astronomical unit (Earth to Sun distance). We will let R
represent the radius of the sun, and V represents the Venus to Sun distance.
V
D
R
Math 1060 – 003
Transit of Venus
Summer 2012
Camacho
Sun’s Radius to Distance Ratio
The ancient Greeks were expert geometers. They thought about most quantities in terms of
ratios, that is, one quantity divided by another. One particular ratio they were able to calculate long ago
was the ratio of the Sun’s radius to its distance from the earth.
On a sunny day, one can hold a coin (whose radius is known) near a wall and measure the radius
of the coin’s shadow. By moving the coin far enough away from the wall the shadow should eventually
shrink to a single point. When this happens, we can let d represent the distance between the wall and
the coin, and let r represent the radius of the coin. One can easily use similar triangles to show that the
ratio of the coins radius to its distance from the wall, is equal to the ratio of the Sun’s distance to its
radius. In mathematical notation this is:
R r

D d
Sun
Wall
Coin
R
r
d
D
The Greeks conducted this experiment hundreds of years ago but their calculation was far from
the value that is accepted today. During the renaissance, astronomers took far more precise
measurements and were able to determine the ratio to be approximately:
Math 1060 – 003
Transit of Venus
Summer 2012
Camacho
Interplanetary Ratios
There was an incredible amount of astrological data collected during the renaissance which
allowed many mathematicians, physicists, and astronomers of the time to figure out all of the relative
distances between planets within our solar system. Relative means that they could determine the ratio
between distances, but not absolute distances. For example, astronomers could determine that Mars is
1.5 times further from the sun than the Earth is. However, they knew neither Earth’s absolute distance
or Mars’ absolute distance from the sun.
To calculate these ratios one could use astrological data as well as trigonometry. It was,
observed, for example, that Venus would rise higher and higher in the sky for a few months, then
remain stationary for a short period of time, and finally begin to descend in the sky for a few months.
From the simple diagram below, it is easy to see that during this short stationary period, the Earth, Sun,
and Venus form a right triangle. By measuring the angle between the Sun and Venus, as viewed from
the earth, one can easily calculate the ratio between Earth’s distance from the Sun and Venus’ distance
from the Sun. That is:
V

D
Using the data available at the time, Johannes Kepler was able to calculate the ratio to be approximately
Math 1060 – 003
Transit of Venus
Summer 2012
Camacho
Transit of Venus and the Astronomical Unit
Several imperial nations of the 1700’s were alerted 40 years in advance of the next transit of
Venus. They went to great lengths to prepare for its happening, as it is a very rare astrological event.
Several observatories were set up on different points of the earth. Astronomers used their instruments
to carefully measure the location on the sun in which Venus crossed. From the different vantage points
on the earth, Venus appeared to cross over different points on the sun, as in the diagram below:
Astronomers needed to carefully measure the relative distance between the two crossing paths.
That is, the ratio of the path distance to the radius of the sun, R. Let’s call this ratio f. Additionally, they
needed to carefully measure the distance between the two points on earth, let’s call this distance d.
V
D V
fR
d
D V
V
One can then use the known ratios from the previous section to rewrite this diagram as follows:
Math 1060 – 003
Transit of Venus
Summer 2012
Camacho
0.7 D
0.3D
fD
215
d
0.3D
0.7 D
Now, one can use the fact that the two triangles in the diagram are similar. Therefore, the ratio of all
corresponding side lengths must be equal. Thus we could say that
Solving this equation for D yields
Recall that and are both measurable quantities. The distance can be measured since it is the
distance between two points on the earth, and the ratio was able to be measured during the transit of
Venus by viewing the two different spots on which Venus passed over the sun from the two vantage
points. Therefore, one can plug the known values into this equation and calculate D. The value for D
that is accepted now-a-days is approximately 93,000,000 miles.