Discovering the Stars
Since ancient times it has
been speculated that stars are objects like the sun which appear faint only
because they are so far away. However,
to demonstrate this we need to know how far away they are. Once the distance is known, the luminosity
(total amount of light radiated) can then be calculated using the inverse
square law for light (see Luminosity and Distance write up
below). How is it possible to determine
the distance of inaccessible objects like the stars? In principle, we should be
able to use the parallax technique. This ancient method for estimating distances on the earth involves
the apparent shift of position of an object against a distant background when
viewed from different positions. The
distance between viewing positions, the baseline, and the parallax
angle (derived from the apparent shift), can be used to construct
triangle, one side of which will be the desired distance. However, no baseline on earth, even one as
long as the earth’s diameter, is large enough to show stellar parallax. Assuming the heliocentric system (i.e., that
the Earth moves along an orbit around the sun), the maximum possible baseline
is the diameter of Earth’s orbit. The ancient Greeks recognized this, reasoning
that if the earth were moving around the sun, the nearer stars should shift
with the seasons against a background of fainter background stars. However, no such shift was ever observed,
so most ancient astronomers concluded that the earth was not moving. Without the distance determination, it was
not possible to say definitively that stars were objects with luminosities
comparable to the sun. Further, the
size of the universe was anybody’s guess.
§
(150 A.D.) Ptolemy, using Aristotle’s model of nested spheres,
Aristarchus’ distances to the moon and the sun, and Erastosthenes’ radius of
the Earth, set the lower boundary for the sphere of fixed stars to be 19,865
Earth radii. In modern numbers this
comes out to be about 80,000,000 miles.
§
(1590) Tycho calculated that the stars would have to be at
least 6,650 times farther away from the Earth than the sun for them not to show
parallax. This seemed like a
ridiculously large figure to him, and on this basis he also rejected the notion
of a moving Earth.
§
(1609) Galileo showed that there were many more stars
visible through the telescope than visible with the naked eye. If you assume that all stars are the same
luminosity, this implies great depth to the universe. It could also be argued that this implied a much larger universe
than previously supposed.
§
(1670) Cassini’s measurement of the parallax of Mars finally
established the true dimensions of the solar system. With the sun 93,000,000 miles from the Earth, Tycho’s earlier
estimate for the minimum star distance went from 6,650 to over 200,000! By this
time, however, the heliocentric system was firmly established and this large
number seemed quite believable.
§
(1718) Edmond Halley found that some stars had moved from
their positions plotted by Hipparchus 1500 years earlier. This movement of the stars is known as proper
motion. Since some stars had large proper motions while others had
small or none, this was taken as evidence that some stars are closer than
others; space had depth.
§
(1728) James Bradley discovered the aberration of light ;
the shifting of stellar position in the direction of earth’s motion. This was the first direct evidence that the
Earth was moving, but did not help in determining the distance to the stars.
§
(1785 – 1822) William Hershel began making systematic surveys of
the stars. With the assumption that all
stars have the same luminosity (hence, “dimness means farness”) he developed a disk shaped model of the
universe called the “galaxy”, since most of the stars lie close to the Milky
Way (galaxy in Greek). This model was also referred to as a “grindstone”
or “lens” from its apparent shape. The dimensions of the galaxy were given in a
unit Herschel christened the siriometer, the distance from the
solar system to the star Sirius. He
estimated the galaxy to be 1,000 siriometers in diameter and 100 siriometers thick. The actual distance to Sirius was not known
in Herschel’s day, but the modern figure is 48 trillion miles or 8 light
years. Thus, according to Herschel the
Milky way is 8,000 light years in diameter and 800 light years thick, roughly
one tenth the value of modern estimates.
§
(1785 – 1822) Hershel discovered that some double stars are true
star systems, with the stars revolving around a common center of gravity. Thus, they must be at approximately the same
distance from us, and, since most of these systems contain stars of unequal
brightness, this implies that not all stars have the same luminosity. This
realization caused Hershel to abandon his original grindstone model of the
galaxy.
§
(1785 – 1822) Herschel also catalogued “nebulae”, faint fuzzy patches,
some of which resolved into stars and some of which did not. The nature of these objects were subjects of
speculation. Some thought that they
might be distant versions of the Milky Way or island universes (a
term coined by Alexander von Humboldt in 1845). Others such as Pierre Laplace
viewed them as solar systems in formation.
§
Stellar parallax
measurements were attempted throughout the 18th century
without success. Two criteria were
developed for believing that stars were nearby: (1) brightness and (2) large
proper motion, preferably both.
§
(1838) Friedrich Bessel finally measured parallax in 1838
using a heliometer (split lens telescope). The star, 61 Cygni (selected because
it had the largest proper motion then measured), was found to have a parallax
of 0.3136”, translating to a distance of more than 60,000,000,000,000 (60
trillion) miles.
§
Since the number of
miles (or kilometers) to even the nearest stars is so immense, astronomers use
two more easily handled units: (1) the parsec (1/parallax angle)
and (2) the light year (the distance light travels through a
vacuum in one year, about 6 trillion miles). Using these units, 61 Cygni is
about 3.46 parsecs or 11 light years from our solar system.
§
(1838 – 1850) Parallax angles for other nearby stars were found by
Bessel and other observers. These included the bright stars Alpha Centuri (the
nearest at 1.3 parsecs or 4 light years), Vega and Altair. From the now known distances, it was quickly
determined that 61 Cygni was significantly less luminous than the
sun while Vega and Altair were more luminous. Only
Alpha Centuri had a luminosity nearly the same as the sun.
Luminosity and Distance
If we assume that space is completely transparent , then the
relationship between the apparent brightness of a star B (how
bright it appears in the night sky) and its luminosity L (its
intrinsic brightness, i.e., the total amount of light radiated) is given by:
(luminosity, knowing distance)
Where d is the distance to the star. We will always B, how bright
it looks to us. Thus, if we know the
distance d, then we can calculate the luminosity L
as above. Conversely, if we know the luminosity L we can
calculate the distance d.:
(distance, knowing luminosity)
Suppose we know that a distant star has the same luminosity as the
sun. Then we have:
Since
the sun and the star have the same luminosity,
Be
aware that the original assumption, that space is completely transparent, is
not valid in many significant cases due to obscuring matter in the form of interstellar
dust and gas. If such matter is
present, then the perceived brightness of a star will not depend solely on the
distance, and the calculations above will yield inaccurate results. In such a case the obscuring matter must be
accounted for before an accurate assessment of distance can be made.