Sky & Telescope - December 2023 - 75

From Parallax . . .
How did we first start understanding
how far away celestial objects
are in the local universe? Quite
straightforwardly, in fact - expanding
on a familiar phenomenon,
stereoscopic vision.
If you're standing, say, in a field
of sunflowers and looking at a
distant copse, you'll notice that if
you close one eye, open it, then
close the other eye that the image
of the clump of trees doesn't move
very much against the more distant,
stationary landscape. But now
crouch down in the field and do
the same with a nearby sunflower.
You'll notice the sunflower appears
to move way more (than the copse)
against the background. The apparent
shift of a foreground object
against a backdrop farther away is
known as parallax.
This phenomenon happens
because our eyes are side by side
- they give us the perspective we
need to gauge distances. But it only
works for relatively nearby objects
- our eyes are too close together
for the effect to be useful for larger
distances. In other words, the baseline
of our two eyes is very short.
But imagine if we could extend
the distance between our eyes.
Obviously we can't do that, but we
do have a very useful tool to hand:
Earth's orbit around the Sun. If we
substitute telescopes for our eyes
and point them at a celestial target
at six-month intervals, we have a
baseline that's equivalent to twice
the distance of Earth from the Sun.
Now we're talking!
. . . to Parsec
To measure parallax, we need
Earth orbiting the Sun (check), a
target (pick your favorite object),
distant background stars (there are
many!), and a telescope with which
to make the measurements. Follow
along with the diagram at left.
Observations of a nearby object
taken six months apart will show
that its position shifts against
faraway background stars (that
appear stationary) by a certain
angle, the stellar parallax. Half that
angle helps us measure the star's
distance from the Sun using a very
simple equation.
The diagram shows a right
triangle with adjacent sides formed
by the distance between Earth and
the Sun as well as the distance
between the Sun and the target
star. The parallax is the angle
between them. So, applying highschool
trigonometry we have:
tan(p) =
1 a.u.
d
(Remember that? The tangent of
an angle equals the opposite side
divided by the adjacent side.)
Rearranging yields:
Star
d =
1 a.u.
tan(p)
For very small angles, the formula
simplifies to:
d = 1/p
If an object's position on the sky
appears to shift by an angle, p, of
1 arcsecond when observed over
a baseline of 1 a.u., then the object
lies at a distance, d, of 206,265 a.u.,
or 3.26 light-years. And voilà, we
have the definition and value for the
parsec - which is a portmanteau
for parallax of one second.
The table lists parallaxes (in
milliarcseconds) and distances (in
Sirius
Pollux
Castor
Aldebaran
Betelgeuse
Rigel
p (mas) d (pc) d (l-y)
379
96.5
63.3
48.9
6.55
3.78
2.66 8.68
10.4
15.8
20.4
153
265
33.9
51.5
66.5
499
864
both parsecs and light-years) for a
selection of stars. Note that we're
indeed dealing with very small
angles - and that the smaller the
angle, the farther the object.
So long as we can measure the
parallactic angle of an object, we
can derive its distance. And, until
we can find a way to stretch out our
cosmic tape measure, our parallax
estimates will have to do! ¢
The subdivisions of degrees we call minutes of arc (or arcminutes) and
seconds of arc (or arcseconds) date back to the ancient Babylonian
astronomers and are units for measuring celestial angles. Nothing to do
with the minutes and seconds that we use to keep time, they nevertheless
break down in a similar fashion:
1 degree (°) =
60 arcminutes (′)
1 arcminute =
60 arcseconds (″)
So...
1° = 3,600″
sk yand tele scope .o r g * DECEMBER 2023 75
https://skyandtelescope.org/
Sky & Telescope - December 2023

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