Projects - Cengage Learning

Appendix C C-43
PROJECT
Transits of Venus and the Scale of the Solar System
Almost every High School child knows that the Sun is 93 million miles (or
150 million Kilometres) away from the Earth. Despite the incredible immensity
of this figure in comparison with everyday scales—or perhaps even because it
is so hard to grasp—astronomical data of this kind is accepted on trust by most
educated people. Very few pause to consider how it could be possible to measure
such a distance . . . and few are aware of the heroic efforts which attended
early attempts at measuring it. —David Sellers, The Transit of Venus & the
Quest for the Solar Parallax (Maga Velda Press, 2001)
A transit of Venus occurs when the planet Venus crosses a line of sight from
Earth to the Sun. For over four hundred years, transits of Venus have caused
excitement among astronomers, explorers, and the interested public. From the
seventeenth through the twenty-first century transits occurred in 1631, 1639,
1761, 1769, 1874, 1882, and 2004. The next transit, in 2012, will be the last of
the current century and will be visible from the western United States. Inspired
by a paper presented by Edmond Halley (1656–1742) to the Royal Society in
1716, expeditions set out to distant locations all over the Earth to observe the
transits of 1761 and 1769. Among famous explorers and mapmakers of the
eighteenth century, Charles Mason and Jeremiah Dixon observed the 1761
transit from South Africa, and Captain James Cook observed the 1769 transit
from Tahiti. When the data from the observations was processed, the value of
the solar parallax was determined to be between 8.5 and 8.9 seconds of arc.
The modern value is 8.794148 arc seconds.
Halley’s method used complicated tools from spherical trigonometry. In
this project, we will see how observations of a transit of Venus and some basic
plane trigonometry can be used to estimate the solar parallax and the distance
from Earth to the Sun. This project can be done as an individual or group
activity but may be more fun with a group.
Figure A shows the geometric relationship between the solar parallax,
angle a, the distance r e , from Earth to the Sun, and the radius R of the Earth.
The notation r e comes from thinking of the distance from Earth to the Sun as
the radius of the Earth’s orbit. Figure A is not drawn to scale. It greatly exaggerates
angle a, and r e is really about 25,000 times larger than R.
R
å
Earth
r e
Center of the Sun
Figure A
Exercise 1 Use the right triangle definition of the sine function to derive the
formula
r e
R
(1)
sin a
for the distance from Earth to the Sun in terms of the solar parallax and the
radius of Earth.