PARALLAX

 motion: TOC for Knowledge Concepts, Exercises, and Solutions

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Ancient societies understood how to use parallax to measure distances; the Greeks
were particularly adept at it, for example.  Even today, parallax remains one of
the most important methods to measure distances in the cosmos.

\         /        A brief review: if you want to find the distance (r)
\       /         between us and some star, the apparent change of angle
\     /          when you change vantage points from (a) to (b) will be
\   /           twice the angle (x).  If the distance between the
\ /            vantage points is twice the distance (l), then you can
*             use the trigonometric relation
/|\                       l
/x|x\            tan(x) = --- or   r = l/tan(x) to get the distance.
/  |  \                     r
/   |   \
/    |    \        This technique is called parallax.
/     |r    \       A simple example: hold your thumb out in front of you.
/      |      \      close one eye.  Line your thumb up with a distant
/       |       \     object, like a clock on a wall or a lightpost.
(a)---------------(b)   Now close that eye and open the other eye.  Your thumb
l       l         appears to have moved to one side!  Measure that angle
(remember that an outstretched finger width is about
2.5 degrees of arc) - that's 2(x).  Now measure the distance between your eyes -
that's 2(l).  Consult the diagram above, and get the answer (r).

Astronomers maximize (l), the baseline, by using Earth's orbit around the Sun.
The radius of Earth's orbit is one astronomical unit (AU), or 150 million
kilometers - so if you look at a star six months apart, (l) = 1 AU.

Even with this large baseline, however, the distances to even the nearest stars
are less than a thousandth of a degree.  Astronomers thus use units called
an arcminute (1/60 of a degree) or an arcsecond (1/3600 of a degree) more often.
This leads to a useful unit of length: if you use Earth's orbit as a baseline,
and measure one arcsecond of parallactic angle to an object, that object is one
"parsec" (pc) away.  So we can calculate:

1 pc = (1.5 * 10^8 km) * tan(1 arcsecond)

= (1.5 * 10^8 km) * (57.3 degrees/radian * 3600 arcseconds/degree)

= (1.5 * 10^8 km) * (2.06 * 10^5) = 3.1 * 10^13 km = 3.1 * 10^16 m

The distance from the Sun to Alpha Centauri, for example, is about 1.3 pc.
When we discuss even bigger distances, we use the standard metric prefixes:

1 kiloparsec (kpc) =       1000 pc = 3.1 * 10^16 km = 3.1 * 10^19 m
1 Megaparsec (Mpc) =    1000000 pc = 3.1 * 10^19 km = 3.1 * 10^22 m
1 Gigaparsec (Gpc) = 1000000000 pc = 3.1 * 10^22 km = 3.1 * 10^25 m

Notice that a parsec is about 3.26 light-years.  Astronomers tend to use parsecs
more often than light-years; generally, though, astronomical distances can be
conveniently expressed using either unit of length.

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