QUESTION:
How far away is the farthest sun from our earth and how do you estimate the
distance?

ANSWER from Sean W. J. Colgan on 30 October 1995: 
This is actually a very fundamental question we don't know the answer to,
yet.  All the stars in the sky are distant suns of course. All the stars we
can see with our unaided eye are in our Milky Way galaxy, which is pancake
shaped and contains several billion stars. The most distant star in our
galaxy is roughly 50,000 light years away. Galaxies tend to be found in
clusters - ours is called the Local cluster - and clusters of galaxies group
themselves into superclusters. The most distant suns are in superclusters at
the edge of the universe and the distance to them depends on how old the
universe is.

The reason the age of the universe comes in is that the universe is
expanding. It has been expanding since it began with the Big Bang, somewhere
between 8 and 15 billion years ago. The most distant suns are in those
galaxies which are expanding away from us at nearly the speed of light, so
that they are 8 to 15 billion light years away! We can only measure the
distance to these stars by measuring the velocities of the galaxies which
contain them - we can't really see individual stars at that distance - and
multiplying by the age of the universe.

Astronomers usually turn this around by trying to measure the distances and
velocities of galaxies which are much closer to us than the edge of the
universe, and from this information trying to find the age of the universe.
This is one of the projects the Hubble Space Telescope was built to try and
solve. To find the distance to objects far away, we start with nearby stars
and work our way farther out.  We find the distance to stars close to the
earth using a method just like surveyors: we observe the position of the star
and then wait six months for the earth to move to the other side of the
sun. We observe the position of the star again. Nearby stars will move
relative to distant background stars just like nearby objects move relative
to distant objects as one drives along in a car. From this we get the
distance to nearby stars. We also classify stars based on their colors, and
other features we can observe about them.  We then assume that distant stars
of a given class are the same brightness as nearby stars of the same type,
which we know the distance to.  By comparing the brightnesses, we then get
the distance to stars which are further away. While doing this, a class of
stars has been discovered which vary in brightness. Lots of stars vary in
brightness, but one group in particular is very useful for finding distances:
Cepheid Variables have a cyclical period of brightness variation and the
length of the period is related to how bright they really are! So all we have
to do is measure the period of a Cepheid and how bright the star appears to
us and we know how far away it is. This is the method the Hubble Space
telescope astronomers are using to find the age of the universe. So within a
couple years we should know how old the universe is, and hence, how far away
the most distant suns are.