"The Noblest Problem in Astronomy": The importance of the distance to the Sun and the ways in which it was determined.
By Dr William Tobin, Department of Physics and Astronomy, University of Canterbury.
'The Noblest Problem in Astronomy': This article look at the importance of the distance to the Sun and the ways in which it was determined. Find out why was this question so crucial---so crucial that the British Admiralty went to the enormous expense of sending Captain Cook to the South Pacific as part of a campaign to it in 1769, and in 1874 some eighty expeditions spread out across the globe for the same purpose? Prepared specially for this website by Dr William Tobin, Department of Physics and Astronomy, University of Canterbury. |
|
|
The importance of the distance to the Sun
 |
| Sir George Airy |
In 1857 the English Astronomer Royal, Sir George Airy, described the determination of the average distance to the Sun as `the noblest problem in astronomy'. Why was this question so crucial---so crucial that the British Admiralty went to the enormous expense of sending Captain Cook to the South Pacific as part of a campaign to answer it in 1769, and in 1874 some eighty expeditions spread out across the globe for the same purpose?
By Airy's time and in Airy's view, the distance to the Sun was of key importance because it was the yardstick for measuring the solar system and the few stars that were close enough to be triangulated when the Earth was in diametrically opposite positions in its orbit.
In 1769, however, the underlying reason was primarily the practical one of determining longitudes on land at at sea. The Greenwich Observatory in England and the Bureau des Longitudes in France had been founded a century earlier to work on the problem.
top 
back 
|
|
Longitude is really the difference between the local time, which is easily determined from observations of the Sun, and the time at some reference meridian, such as Greenwich. (For example, in New Zealand we are almost 12 hours ahead of Greenwich time, and therfore almost 180 degrees to the east of the Greenwich meridian.) Prior to the advent of accurate marine chronometers towards the end of the 18th century, the clock that relayed the time in Greenwich was provided by motions of bodies within the solar system. Astronomers predicted these motions and tabulated them as a function of Greenwich time in almanacs. A distant geometer could observe the motion in the sky and read off the Greenwich time from the almanac. The motion used by surveyors on land were eclipses of Io, the innermost of Jupiter's moons. These delicate measures could not be made afloat, however. Mariners made less-accurate readings of the position of the Moon as it moved relative to the Sun and stars.
Accurate predictions of solar system motions were therefore crucial for
accurate navigation, and formed the principal work of astronomers until their
attention turned to astrophysics at the end of the 19th century. Celestial
mechanics, as it was and is called, needed understanding of all the forces and
distances in the solar system. The distance to the Sun was an important
element in this grand undertaking which reached its peak in the 19th
century, even though by that time accurate marine chronometers and the
electric telegraph were providing simpler, non-astronomical means of
transmitting the time---and hence longitude---to distant locations.
It is of interest to note that celestial mechanics continues to be
involved in the quest for more perfect navigation. The failure of
Newtonian dynamics to explain the motion of the planet Mercury was a
pointer towards Einstein's development of General Relativity,
and our ability to navigate is now so great that General Relativistic
corrections are needed in Global Positioning System satellites and
receivers.
top
back
|
|
Measuring the Sun's distance from triangulation
The oldest and most obvious method for determining to distance to the Sun
is by triangulation. Observers at two stations of known separation
form the base of the triangle. In theory, they might then
observe the angles towards the limb of the Sun taken as the triangle's apex.
With these measures, trigonometry would yield the distance to the Sun.
However, the Sun is far away and the accuracy would be poor. It is better
to triangulate a nearby object. Kepler and Newton had understood the law
describing how the periods of the planets' orbits reflect their relative
distances from the Sun and had drawn a
scale map of the solar system. It was only necessary set the scale by
measuring one interplanetary distance and all others, such as the
distance to the Sun, would follow.
Venus and Mars are the planets that come closest to Earth as they orbit
around the Sun. Mars is closest at opposition (i.e. when it is in the
opposite direction from the Sun).
Venus is closest at conjunction (i.e. when it lies between the Earth and
the Sun). Mars reaches opposition every 26 months while Venus is in
conjunction every 19 months. However, the alignments are
rarely perfect since the orbits of the Earth, Venus and Mars are inclined.
This has little effect on Martian oppositions, which are easily
visible in the night-time sky. Indeed, the first
reasonably-correct measurement of the solar distance came from
measurements of the opposition of Mars in 1672. In contrast, Venus at
conjunction is hidden in the glare of the daytime sky unless the alignment
is so good that the planet is betrayed as a dark spot passing in front of
the disc of the Sun in a transit. Such alignments are rare,
often occuring in pairs that recur less than once per century. There were
no transits of Venus in the 20th century; prior to that there were transits
in 1882, 1874, 1769 and 1761. The great interest of transits of Venus
was that Venus is then at nearly half the distance of Mars at its closest,
with the prospect of correspondingly more accurate triangulations.
top
back
|
|
 |
| Urbain Le Verrier |
Two further methods for determining the solar distance became possible
around the middle of the 19th century.
The first derived from celestial mechanics and perturbations in planetary
motions. If the only gravitational force acting on a planet were that of
the Sun, the planet would orbit in an ellipse. But a planet is also affected
by the gravitational pull of all the other planets in the solar system, and
this causes the planets to deviate from truly elliptical paths. In 1846
analysis of deviations in the orbit of Uranus led the French astronomer
Urbain Le Verrier to predict the existence and whereabouts of an eighth
planet, Neptune. Le Verrier went on to analyse the observed perturbations
in the orbits of all the objects in the solar system. His analysis equation
included over 500 terms! Between 1853 and 1861 he deduced a value for the
Sun's distance that was some 4% smaller than generally believed at the time.
It is interesting to note that these gravitational calculations did not
require knowledge of the Newtonian gravitational constant, G, which was
poorly-known in the 19th century. Newton's constant always appeared in
Le Verrier's equations multiplied by the mass of the Earth, and that
product, GM_Earth, had been measured with high accuracy because it can
be derived from the rate at which objects accelerate when dropped near
the Earth's surface.
top
back
|
|
 |
| Leon Foucault |
A second method involved the speed of light. In the mid-19th century this
quantity was derived from astronomical measurements, and in two different
ways. First, the 17th-century Danish astronomer Ole Roemer had calculated
the time taken by light to cross the Earth's orbit from delays and advances
in the times of Io's eclipses. The Earth's orbital diameter (twice the
distance to the Sun) divided by this 'light time' yielded the speed of
light. Second, the speed of light was related to the Earth's orbital
speed through an accurately-measured phenomenon called the aberration
of starlight, which is a small annual wobble in the position of stars caused
by the Earth's orbital motion in space. The aberration is equal to the ratio
of the orbital speed to the speed of light, and the orbital speed is simply
the orbit's circumference (2.pi times the distance to the Sun)
divided by the duration of the orbit, or one year. But if the speed of light
could measured by some laboratory experiment, both arguments could be
reversed to yield the distance to the Sun. This is what the French physicist
Leon Foucault did in 1862 at the behest of his boss at the Paris Observatory,
who was none other than Le Verrier. Foucault used a small, fast-spinning
mirror to convert the travel time of a luminous beam into a small but
measurable angular displacement, and obtained a value essentially in
agreement with Le Verrier's value derived from planetary perturbations.
top
back
|
|
Later developments
We know now that Le Verrier's and Foucault's results were essentially correct,
but at the time people were suspicious of Foucault's measuments.
Astronomers hoped the 1874 transit of Venus would resolve these discepancies,
especially since photography was to be used to record the phenomenon, but
in the event the uncertainty in the solar distance was not reduced and it
was recognised that it never would be with transits of Venus. Attention
turned to triangulation of Eros and other asteroids at opposition, which
though further than Venus or Mars, could be measured much more accurately
because they were small points of light rather than extended discs.
In the 1960s the navigational needs of interplanetary probes required further
improvements in our knowledge of the average distance to the Sun. It is
now one of astronomy's most accurately measured quantities and is
derived from observations of planetary perturbations made since 1910 by
the U.S. Naval Observatory combined with a fourth method, radar ranging
of planets and spacecraft. Its value is 149,597,870.66 km where the
uncertainty is less than 1 km.
top
back
|
|
For fuller details on Foucault's and Le Verrier's determinations, see
Chapter 13 in "The Life and Science of Leon Foucault: The Man Who Proved
the Earth Rotates" by William Tobin, Cambridge University Press,
Cambridge (2003).
William Tobin page
top
back
|
