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Distance measures
Moving outward from Earth we are introduced to the quantitative relation between the parallax of
a nearby star and its distance from us:
D = 1/p
where D is the distance in parsecs (1 parsec = 3.26 light years), and p is the parallax of the star,
defined to be one-half the angle that a star shifts when seen from opposite sides of Earth's orbit, in
arcseconds. You should be able to work with this equation to derive distances when given
measured parallaxes (and vice-versa).
Sample:
Using the equation d = 1/P, how many parsecs away is a star with a parallax of .4?
a. .4
b. 1
c. 2.5
d. 4
e. 5
1/p = 1/0.4
= 10/4
= 2.5
How does it work?
The Parsec
The closer a star is to us, the larger its angle of parallax will be. Astronomers have defined a
standard unit of distance to be the parsec (pc). One parsec is the distance to a point in space that
subtends a parallax angle of one arc second. This produces the simple but effective relationship:
distance, d (in parsecs) = 1 / parallax angle, p (in arcsec) or
d = 1/p
By measuring the parallax angle for a star, astronomers can then directly determine its distance.
In reality it is not quite so simple for several reasons. The angle of parallax for even the closest
stars is always < 1.0 arcsec so careful observation and precision is required. Corrections have to
be made for atmospheric refraction and the effects of "seeing". Also stars actually do appear to
move across the sky relative to other stars in a definite direction over time. This is called proper
motion and must be accounted for when determining parallaxes. If you study the parallax diagram
you will see that the greatest baseline ground-based astronomers can obtain is by observing the
star at times six months apart. This gives a baseline of two astronomical units and an angle of 2p.
How far are the stars?
Apart from the Sun, currently the closest star to us is a small red dwarf M5 class star, Proxima
Centauri. It has a parallax of 0.772". Therefore its distance is:
d = 1/p
so d = 1/0.772
d = 1.30 pc
This corresponds to a distance of 4.24 light years. All other stars are even further away. Sirius,
the brightest star in our night sky has a parallax of 0.375" and is 2.7 pc distant. Our Sun is about 8
kpc, i.e. 8,000 parsecs from the centre of our galaxy, the Milky Way. Its’ nearest large neighbor,
the spiral galaxy M 31 or the Andromeda Galaxy is 725 kpc.