The Day the World Discovered the Sun (33 page)

This book requires no specialized mathematical or astronomical training to explore the human drama of the 1761 and 1769 Venus transit voyages. However, as a supplement to the discussion, the present appendix answers readers' curiosity about the specific methods the astronomers mentioned in this book used to calculate the solar distance. In so many words:
How, specifically, can one use the Venus transit to find the distance to the sun? And why wasn't there an easier way?

The answer to the second question sets the stage for the first.

Every astronomical object in the sky appears as if projected onto a flat screen. Nothing immediately apparent about any star or planet might give the casual observer hints about its distance.

The primary reason ancient astronomers distinguished between stars and planets was that stars remained fixed in their positions with respect to one another, as observed night after night. Planets (from the Greek for “wanderer”), however, moved across the familiar stellar tapestry over the course of weeks and months.

From the time of the ancient Mayan, Chinese, Greek, and other early civilizations, astronomers, typically also serving as astrologers, devised elaborate theories to explain planets' wanderings—and what those wanderings might portend for kings or great events of the day.

After the Polish astronomer Nicolaus Copernicus (1473–1543) committed the ultimate heresy of replacing the earth with the sun as the center of the solar system, the German mathematician Johannes Kepler (1571–1630)
made quantitative sense of Copernicus's framework. From a lifetime of studying detailed planetary observations by Danish astronomer Tycho Brahe (1546–1601) and others, Kepler ultimately derived three basic laws of planetary motion, which remain in use to this day.

Kepler's third law states that the square of the time a planet takes to complete one orbit of the sun is proportional to the cube of that planet's distance from the sun. In a simple equation:

Where
P
is the planet's orbital period (the length of time, measured in earth years, the planet takes to complete one orbit) and
a
is the planet's average distance from the sun, measured in fractions of the earth-sun distance (astronomical units or AU). From well before Kepler's day, detailed charts of Venus's motions established that the planet completes one orbital period, one Venusian “year,” every 0.615 Earth years.

Multiply 0.615 by itself and take the cube-root of the result to find Venus's distance from the sun as 0.72 AU.
¹
But what, in practical distance units such as miles, is an AU?

Here is where astronomy remained stuck for more than a century.

Clever attempts to leverage precision measurements of Mars's and Mercury's orbits brought some astronomers in the late seventeenth and early eighteenth centuries close to answering the question.
²
But planets move slowly across the sky, and tracking their motions with respect to background stars was necessarily imprecise. Discovering the sun's distance required greater precision.

It's useful now to introduce an important term: the “solar parallax,” not the solar distance, is actually the quantity astronomers sought. Solar parallax is an angular measurement, representing one-half of the angular size of the earth as seen from the sun. To use the analogy of a circle, the solar parallax is like the angular “radius” of the earth, as subtended from a distance of 1 AU. Fortunately, converting between solar parallax and solar
distance is relatively easy. The distance to the sun is just the radius of the earth divided by the solar parallax. In real numbers, 92,956,000 miles = 3,963.2 miles ÷ 8.79414 arc seconds. (To do this on a calculator, an extra factor of 206,265 is needed to convert arc seconds to “radians,” the natural unit of angular measurement.)

In 1716, soon to be Astronomer Royal Edmund Halley published an astronomical call to arms, revealing that for a brief window in June 1761 and again in June 1769, the planet Venus would be moving across a kind of interplanetary yardstick—the face of the sun.

Astronomers at different locations across earth could then time the duration of the Venus transit and compare answers (along with exact measurements of the observers' latitude) to triangulate the sun's distance.

Halley's method did
not
call for the observers to know their longitudes. And given how difficult longitude was to determine at the time, whether at sea or on land, Halley's method seemed to be a quick and easy route to the solar parallax, such a crucial number in science.

However, one of Halley's protégés, the French astronomer Joseph-Nicolas Delisle, examined the English method in closer detail and found it wanting. In 1761, for instance, the difference between the shortest and longest transit times would be just 13 minutes, making crucial the accuracy of each Venus transit duration measurement down to the second. It would also require the weather's cooperation for five or more hours. And the location of the longest transit time would be in the Indian ocean, notorious for its changeable weather conditions.

Delisle thus developed a supplementary method of discovering solar parallax from the Venus transit. Delisle's technique required an observer to mark the exact local time for just one of the four points of contact between Venus and the sun. (Those four points are the planet's outer and inner limbs touching the sun's edge on entry and exit—external and internal points of ingress and egress, respectively.) Delisle's method did only require observation of a single moment in the transit, thus making stations of multiple observers more likely to have at least one overlapping data point
even in very uncooperative weather. Crucially, however, Delisle's method also required knowing both latitude and longitude of the observing station.

The story in the present book describes the compromise Halley-Delisle method that astronomers used in the 1761 and 1769 Venus transit voyages: measure latitude, longitude and as much of the Venus transit as possible. Astronomers after the fact would then use both Halley's and Delisle's methods to discover solar parallax, ideally enabling them to cross-check their results as well.
³

For the purposes of the present appendix, we'll also consider two approaches to deriving solar parallax or distance. Neither is strictly the Halley or Delisle approach. Rather the first is much simpler—although far less accurate—than the other.

In both approaches, the basic idea is the same. Venus's silhouette follows different paths across the solar disk depending on where on the earth one observes the Venus transit. The greater the path length across the sun, the longer Venus will take to cross that path length. Observe Venus crossing the sun from two widely separated locations on earth, and the difference between the transit times will ultimately depend on three factors: the exact locations on earth where the astronomers observed the transit, the physical distance to Venus, and the distance to the sun.

Because of Kepler's third law, the proportional distance between Venus and the sun was well established. And if the observers independently determine their latitude and longitude, then the only remaining free variable is the distance to the sun.

The simpler approach, then, only requires some high school mathematics:
⁴

(relative sizes and distances not to scale)

Define the physical distance between two sets of Venus transit observers as
b
, and the distance between the earth and Venus as
p
. So by trigonometric definition—opposite over adjacent—the tangent of the apparent angular difference between the transit of Venus as seen by one observer compared to the other observer is:

But the angle
θ
here is small, so tan(
θ
) ≈
θ
. Therefore,

Multiply both sides of the equation by the ratio
, where
a
is the distance between the earth and the sun, and equation (3) becomes:

And
is already a known quantity,
, courtesy of Kepler's third law.