The simplest way to measure the distance to an object via
parallax is to make simultaneous measurements from two
locations on Earth.
However, as we saw last time, this method only works for
objects which are relatively nearby:
Object maximum possible parallax
(arcseconds)

Mars 34
Jupiter 4
Neptune 0.6
Proxima Centauri 0.00007

In short, simultaneous twolocation measurements will
only work for bodies within our own solar system.
We need to find some larger baseline to measure the parallax
to other stars....
A longer baseline: the orbit of the Earth around the Sun
Astronomers need a VERY long baseline in order to produce
a parallax angle which is large enough to detect via
conventional imaging.
Fortunately, there is a convenient baseline just waiting
to be used  if we are willing to discard the requirement
that measurements be simultaneous
(more on this later).
This longer baseline is the radius of the Earth's
orbit around the Sun:
Note the convention used only in this situation:
the quoted "heliocentric parallax angle" π (pi) is always
HALF the apparent angular shift.
So, if we measure a parallax halfangle π (pi) to a star,
we can calculate its distance very simply:
R
L = 
tan(π)
where
6
R = 149.6 x 10 km
= 1 Astronomical Unit (AU)
Small angles and peculiar units make life easy
Now, for small angles  and the angles are always small for
stars  we can use the small angle approximations:
tan(π) ~= sin(π) ~= π
as long as we measure the angle in radians. Even if we don't
use radians, it still remains true that cutting a small angle
in half will also cut its tangent in half
π 1
tan(  ) ~=  tan( π )
2 2
and, in general, there is a linear relationship between
the tiny angle π and its tangent, regardless of the units.
Take a look for yourself:
angle angle tan(angle)
(degrees) (radians)

1 0.017453 0.017455
0.5 0.008727 0.008727
0.1 0.0017453 0.0017453
0.01 0.00017453 0.00017453

Okay, fine. Why am I belaboring this point?
Because astronomers have chosen a set of units
for parallax calculations which look strange,
but turn out to simplify the actual work.
The relationships between these units depend on
the fact that there is a simple linear relationship
between a tiny parallax angle, in ANY units, and
the tangent of that angle.
 angles are measured in arcseconds (")
 Recall that 1 arcsecond = 1/3600 degree. Astronomers
use arcseconds because parallax values for nearby
stars are just a bit less than one arcsecond.
You may also see angles quoted in mas, which
stand for milliarcseconds:
1 mas = 1/1000 arcsecond.
 distances are measured in parsecs (pc)
 A parsec is defined as the distance at which a star
will have a heliocentric parallax halfangle of 1 arcsecond.
Q: How many meters are in one parsec?
How many light years are in one parsec?
Given these units, and the linear relationship between
a small angle and its tangent,
we can calculate the distance to a star (in pc)
very simply if we know its parallax halfangle in arcseconds:
1
distance (pc) = 
π
Give it a try: the first star to have its parallax measured
accurately was 61 Cygni. Way back in 1838, the German
astronomer Friedrich Bessel announced that its parallax
halfangle was 0.314 arcseconds.
Q: Based on Bessel's measurement,
what is the distance to 61 Cygni?
Some observational difficulties
Actually, 1838 isn't all that long ago.
Astronomers had been looking through telescopes since
the time of Galileo, in the early 1600s.
Why did it take over 200 years for someone to measure
the parallax to another star?
There are several reasons:
 The parallax shifts are always small.
Really small. Smaller than the apparent
size of stars as seen from the Earth's surface.
Starlight is refracted by air as it passes through the
Earth's atmosphere. It encounters layers with
a range of temperatures and pressures, and, what's worse,
all these layers are constantly in motion.
As a result of all this refraction, astronomers on the ground
perceive stars to be little blurry spots. The typical
size of a "seeing disk" is around 1 arcsecond.
In order to measure the parallax of a star, we must
determine its position  and that of several reference stars
in the same field  to a very small fraction of this seeing
disk.
That's not an easy task.
 All stars in a field exhibit parallax
In practice, astronomers usually measure the shift of one star
in an image relative to other stars in the same image;
differential measurements can be made much more precisely
than absolute ones.
However, as the Earth moves from one side of the Sun to the other,
we will see ALL the stars in the field shift, not only the
star of interest.
In other words, we'd like to see this:
But we instead some something like this:
The only hope is to pick out a set of reference stars which
happen to be much farther away than the target star.
Distant stars will shift by a much smaller angle  perhaps
small enough to be imperceptible. If we measure the position
of the nearby target star relative to those distant ones,
we might be able to detect its shift.
In practice, astronomers perform a rather complicated series
of computations:
 pick a set of reference stars
 estimate the distance to each reference star, using
some method other than parallax
 measure the shift of the target relative to the references
 correct the position of each reference for its own parallactic
shift
 recalculate the shift of the target, relative to these
corrected reference positions
How can we estimate the distance to the reference stars, if we
haven't even determined the distance to the single target star yet?
A good question. It requires a decadeslong series of
iterations:
 make a simple assumption, like "all stars are the
same intrinsic luminosity, so apparent brightness is
simply related to distance."
 use apparent brightness of reference stars to estimate their
distances
 calculate the distance to the target star
 do this for hundreds of nearby target stars
 based on these measurements of distance to nearby stars,
draw some general conclusions about the true luminosities
of stars
 use these conclusions to make improved rules for estimating
the distances to reference stars
 go to step 2
 Heliocentric parallax measurements are not simultaneous
We have to wait months for the Earth to move a significant
distance in its orbit around the Sun.
During that time, the Sun is moving through space at a
decent clip: very roughly 12 km/s relative to nearby stars
in the disk of the Milky Way.
Other stars are moving, too, by similar speeds.
Yes, most of the stars in the local neighborhood are orbiting
around the center of the galaxy, at a speed of roughly
200 km/s. If all stars moved with EXACTLY the same velocity,
we would see no relative motion as we all circled the
Milky Way together. However, each star has some individual
"peculiar velocity" relative to the average rotational velocity,
and that is the velocity discussed here.
There relative motions of the Sun and nearby stars produces
proper motion: a gradual drift in the position of each
star relative to all the others around it. These proper motions
are in almost all cases so small that it takes thousands of years
for the visual appearance of constellations to change:
Thanks to Anna Jangren at Wesleyan University
However, the proper motions can be signficant at the level
of precision required for parallax measurements.
Consider these observations of Vega over a period of three years,
made by the
Hipparcos satellite.
The bottom line
The best large set of parallax measurements come from
the
Hipparcos satellite,
which measured the position and brightness of relatively
bright (brighter than tenth magnitude or so) stars over the entire
sky during the period 19891993.
A large team of scientists turned its millions of raw measurements
into a consistent catalog of distances and luminosities.
The precision of the Hipparcos measurements of parallax
is about 0.001 arcsecond.
You might think that such precision would allow us to
measure distances to stars as far away as 1000 pc.
However, as we shall see,
it turns out that the true range for accurate
distances is quite a bit smaller.
Copyright © Michael Richmond.
This work is licensed under a Creative Commons License.
