Stellar Parallax Theory
9/15/2015
Seed Question: Hold your finger up at arms length. Focus on your finger with
one eye. Note the objects in the background. What do you notice?
Now open and close alternating eyes. What do you notice about the position
of your finger compared to the background?
This is called parallax.
Exploration: Write small! If some stars are closer than other stars, we should be able to see
the shift in a nearby star’s position against the distant stellar background by using different lines
of sight. Imagine looking up at the night sky and alternating your open eyes. If you were
looking at a “nearby” star would you see a shift? Why not? Demo with artificial star in front
of star field.
The distance between your eyes is your own personal “baseline.” What do we need to do to the
baseline in order to try and see stellar parallax?
Imagine a nearby star that exhibits an annual parallax angle of θ = 1/3600 of a degree = 1
second of arc. The average distance from the Earth to the Sun is called 1 Astronomical
Unit, or 1 AU. How far away in AU’s would that nearby star be? Start by drawing this
diagram.
Distant Background Stars on
Celestial Sphere use color!
Not to Scale!
Parallax angle
(1 sec of arc)
1 AU
Sun
θ
d
Imaginary nearby
star
Earth’s Orbit
Label the lines of sight.
Where does the nearby star appear to be located when Earth is in the “top” position? Make a
tiny sketch here.
How about when Earth is in the “bottom” position? Make a tiny sketch here.
What is the baseline in AU’s?
22
What is the slope in terms of the rise and run of the right triangle with parallax angle θ?
Any pair of right triangles with the same angle θ in the same
orientation will have the same slope, so there should be a way to get
the slope just from the angle, without having to measure the rise and
the run. There is, and that nearly magical mathematical function is
called the tangent function :
Slope = 1 AU
d
tangent θ = slope = rise/run. For our parallax problem here, that means:
tan θ = 1 AU
d
d = 1 AU
tan θ
Solving for d…
Finally, put in our angle for θ and use your calculator (set to degrees!) to find
the actual number for d in AU’s.
d = 1 AU_______________
tan (1/3600 of a deg)
so
d = 206,300 AU’s
(to 4 sig figs)
Big Idea: Write small! Parallax is the apparent shift in a nearby object’s position against
a distant background due to a change in the observer’s location.
Stellar parallax is based on the Earth’s Orbit around the Sun where the Parallax angle θ
is half of the total shift:
Distant Background
Stars on Celestial Sphere
Not to Scale!
Parallax angle
1 AU
θ
d
Sun
Nearby Star
Earth’s Orbit
d = 1 AU
tan θ
Discussion: ?
where θ is in degrees
or
d = 206,300 AU’s where θ is in sec of arc
θ