Parallax Method for Determining Star Distances

At first I thought that this would be an almost impossible task because even the nearest star, Alpha Centauri, has a parallax angle, p, of only 0.77 arc seconds and it would involve re-aiming the telescope by about 1.44 arc seconds after a six month delay even though the earth is rotating 15 arc seconds per second-therefore at what time would you aim the telescope?. However, the miracle of photography greatly reduces the difficulty of this task because there are always much more distant and even brighter stars and galaxies in the background. If these background objects are, say, a 1,000,000 times farther away than the star whose distance we are trying to measure, then their parallax angle will be 1 millionth of that star and, when we take a picture of these along with our star in, say, September and then later in March (6 months apart), we will see large displacements of the star with respect to the background object. Even so, we need a telescope that can actually observe these displacements and this requires, for visible light of 400 nm wavelength, an aperture of about 8 meters for a parallax of 0.0077 arc seconds of arc. From Earth, a star with parallax of 0.0077 arc seconds is a distance of only 440 light years which is also a very close star. We have to conclude that the parallax method is a very limited way of determining star distance. For this animation I have decided to express all angles as radians and distances as light years. There are 2pi radians in a complete 360 degree circle. More important, radians are much better for expressing the small angles involved in star distance measurements. If a star is 1 light year away and the planet orbit radius is r light years, then its radian angle will be r radians. For star distance that are n light years away, the parallax radian angle will be p=r/n. There is no need to convert through degrees, minutes, and seconds. It happens that Earth's orbit radius is 0.0000158 light years so a star one light year away will have a parallax of 15.8 microradians. For your convenience, 1 arc second is the same as 4.85 microradians so, from Earth orbit, Alpha Centauri has a radian angle of 3.73 microradians. Since the minimum resolved angle of a telescope of aperture, d, is wavelength, lambda, divided by d we can compute the aperture needed to resolve a microradian of arc as d=lambda/1e-6. For a wavelength of 400 nanometers, this is 0.4 meters and is just barely able to resolve the parallax of a star at a distance that is 3.73 times farther than Alpha Centauri or 3.73*4.367 ~ 16 light years. Because of the extremely small angles involved in measuring star distance, I cannot hope to show the actual angles. Therefore, the only thing the adjustable itmes will do is to print out the parallax angle in both radians and arc seconds, and the telescope aperture needed for the wavelength used to view the star. What is shown for the animation is an on-axis image of a very distant galaxy as well as that of the star at the focal plane of the telescope used to image the star. This image has the star above the galaxy for the first photograph and below the galaxy for the second photograph which occurs 6 months late. When the photographs are compared, by super-posing the galaxy images, after the second photograph is taken, the distance between the star images divided by twice the telescope focal length is the parallax angle, p.