Small angle formula (repeat) --- we started talking about this by imagining an
object at distance d that has an angular size a. Imagine
drawing a big circle through the body, with radius d. Then because
a is small, the body's real diameter will be nearly the same as the
arc of the big circle that passes through it, so we can say that
(diameter of body)/(2 x pi x d) = a / 360 degrees
because 2 x (pi) x d is the circumference of the big circle. This is
equivalent to expressing the angle a in radians and then
using the formula, that
(real diameter) = (distance) x (angular diameter
in radians).
You get from degrees to radians by remembering that 360 degrees
equals 2 (pi) radians. I went on to show some examples of using this small
angle formaula, the first one being to define a new unit of distance, the
parsec, which is defined by using the observed property of
parallax.
Parsec --- A parsec is defined to be the distance to an object that has an
(annual) parallax of one second of arc. In terms of the small angle
formula, 1 parsec = 1 AU / 1 arc second (expressed in radians). Remember, a
radian is 57.3 degrees, which is (57.3 x 60 x 60) arc seconds, or 206,265 arc
seconds, so 1 arc second = 1/206,265 of a radian. Then 1 parsec = 1 AU /
(1/206,265), or 206,265 AU. Since we know that 1 AU = 1.5 x 10^8 km, then 1
parsec = 3.09 x 10^13 km. It so happens that this is equal to 3.26 light
years, so a parsec is not too different a length from a light year, and we
tend to use either unit when talking about distances from now on. Notice that
this is a coincidence: these two units are defined in completely
different ways. We write parsec as pc.
Distances with parsecs --- Remember that as the distance to an object
increases its parallax decreases; in other words, parallax
is inversely proportional to distance. If distance is measured in parsecs,
this gets particularly simple: If an object has a parallax of 1 arc second,
it's distance must be 1 pc (by definition); if it has a parallax of 2 arc
seconds, it is twice as close, or at 0.5 pc; if it is 2 pc away, it's
parallax is 0.5 arc seconds. In other words,
distance in parsecs = 1/parallax in arc seconds .
A note on parallax --- The existence of
stellar parallax is a very important link in the chain of reasoning that lets
us find out distances throughout the universe, so that finding stars close
enough to show annual parallax is an important enterprise. The HIPARCOS
satellite recently measured many more such stars with parallaxes as small as 1
milli-arcsecond (one thousandth of an arc second). What would be the
corresponding distance in parsecs?
Sizes and distances to the Moon and Sun (early measurements) --- You should
treat this section mostly as illustrations of using the small angle formula to
measure distances from angular sizes and real sizes of bodies. I will expect
you to know that Aristarchus attempted these measurements, but I won't test
you on the details of the methods. You should, however, try to follow the
reasoning when it comes to solving the triangles involved. Aristarchus
(310-230 B.C.) used the timing of lunar eclipses to get a handle on the
relative sizes of the Earth and Moon (see handout). The Moon travels through
the Earth's shadow during a lunar eclipse in about 2.5 times the time it takes
to move its own diameter. The size of the Earth's shadow at the position of
the Moon is naturally related to the Earth's own diameter, being almost
exactly 1 Moon diameter smaller than the Earth's diameter. So Earth's
diameter is about (2.5 + 1) times the Moon's diameter, or, the Moon's diameter
is 0.29 times the Earth's diameter (modern value is 0.27). Knowing the real
size of the Moon and it's angular diamter (0.5 degrees) let's you work out the
distance to the Moon, from the small angle formula: size = distance times 0.5
degrees in radians. He then used this distance to the Moon to try to find out
the distance to the Sun (the Astronomical Unit). We saw his argument for
finding the relative distances to the Sun and Moon by observing the Moon at
first quarter and third quarter when the angle Earth-Moon_Sun is a right
angle. He estimated the angle Moon-Earth-Sun by timing the Moon's orbit
between quarters he got 87 degrees), and so could know the angle
Earth-Sun_Moon he got 3 degrees). Then the small angle formula gives the
Earth-Moon distance in terms of the Earth-Sun distance as Earth-Moon =
Earth-Sun times 3 degrees/57.3 degrees per radian. He got the scale about a
factor of 20 too small since the Moon-Earth-Sun angle is really much closer to
90 degrees, so Earth-Sun-Moon is smaller than 3 degrees.
Hipparchus (160 - 127 B.C.) --- made many contributions, but the only one that
was emphasised was the introduction of the system of Magnitudes to describe
the relative brightnesses of objects in the sky. He divided the visible
objects in the sky into "first magnitude", second magnitude" ... down to
"sixth magnitude", which were only just visible to the naked eye. Notice that
smaller numbers mean brighter. Modern masurement has shown
that this is a system that is related to the ratio of brightness of two
objects: second magnitude compares to first magnitude the same way sixth
magnitude compares to fifth, for example. In other words, a
difference in "magnitude" implies a ratio in brightness.
Modern definition below.
Magnitudes (modern) --- are the astronomical way of talking about the
brightness or "luminosity" (will be defined later) of objects. The magnitude
scale is logarithmic so that differences of magnitude represent ratios of
brightness, defined so that a difference of 5 magnitudes corresponds to a
brightness ratio of 100. The scale is also upside-down: smaller magnitudes
mean brighter objects.