1) What are the practical limitations on measuring distances using parallax?
We saw on Page 2 that as the stars became more distant, the size of the parallax angle p gets smaller. At what point is it too small to measure? That depends on the telescope you are using to make the observation. Observations made using ground-based telescopes suffer from blurring caused by the Earth's atmosphere. Typically, the blurred image of a star may be about 1 arcsec in diameter. However, the closest star to the sun, alpha Centauri, has p = 0.73 arcsec -- even the nearest stars have a parallax shift that is comparable to the size of the dot the star makes on a photograph! The parallax angles are very small and very difficult to manage. The General Catalog of Trigonometric Stellar Parallaxes lists parallaxes for over 8000 stars, most of which are very near the sun.Space-based telescopes like the Hubble Space Telescope produce smaller star images, and thus enable parallaxes to be measured for more distant stars. However, observing time on the HST is very valuable, and only a few important stars have had their parallaxes measured using HST. The Hipparcos mission, a smaller satellite dedicated to measuring parallaxes, produced a catalog containing parallax measurements for over 100,000 stars. Still, only the nearest of the roughly 100 billion stars in our Galaxy have been measured. NASA and ESA are each planning more advanced satellites that will measure parallaxes to more stars with higher accuracy (SIM and GAIA).
2) On Page 2, why aren't the three stars "in sync"?
Because the stars are located in different directions in space. This picture shows where the stars are located relative to the Earth's orbit. The triangle for each star shows the star at the leftmost point in its annual wobble. For each star, this occurs at different places in the Earth's orbit around the sun, hence, at different times of the year. Star A is at the leftmost point in its wobble in January. Star B is leftmost in September. Star C is leftmost in May. Watch the stars again as they wobble to verify this for yourself.
3) Do all stars wobble back and forth in a straight line?
No, we made the animations that way to simplify the activity. In reality, only stars that are exactly in the plane of the Earth's orbit around the sun (the ecliptic) make straight-line, back-and-forth wobbles. Stars located exactly perpendicular to the plane appear to make circles. Stars located between these extremes appear to make ellipses -- rounder ellipses closer to the poles and flatter ellipses closer to the plane.
4) Mathematically, where does the parallax equation come from?
The diagram shows that p is half of the total shift the star makes relative to the background stars over the course of one year. Because it is a right triangle, we could use the trigonometry tangent function to solve for the Sun-star distance d in a.u. provided we measure the parallax angle p:
In practice, the angle p is very small -- the size of the Earth's orbit is tiny compared to the distance between the Sun and the star in the top panel of the movie, and the triangle is actually very long and skinny. This means that we can use the Small Angle Formula (instead of tangent) to solve for d after we measure p. The resulting relationship is given by:
This gives p = 206,265 a.u. / d, where the units of d are a.u.. In practice, this yields very large numbers, since p is so small. Rather than deal with large, unwieldy numbers, astronomers decided to define a new unit called parsecs (from parallax second): 1 pc = 206,265 a.u.. Then, the equation simplifies to p = 1 pc / d. Solving for d, we get d = 1 / p -- the units of d are pc, provided that the angle p is measured in arcsec.