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We can often read in the scientific and also more casual reader literature and articles about the angular resolution of various telescopes and other optical equipment, be it ground based or onboard space probes. They would often list their angular resolution, or in other words, their ability to resolve or distinguish small, distant objects with today's digital era mostly on a per sensor pixel basis.

            parallax

                Finding a star's distance from its parallax. Trigonometric Parallax method determines distance to star by measuring
                its slight shift in apparent position as seen from opposite ends of Earth's orbit. (Source: Measuring the Universe)

What I'm interested in is, is the precision in measurements of parallax and with it our ability to determine distance of observed objects directly analogous to mentioned radial resolution and how could it be calculated using data on angular resolution of a telescope alone, if we assume both ground based and in space observatories have more or less the same perihelion to aphelion distance (i.e. the space observatory is in Earth's orbit).

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2 Answers 2

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Parallax measurement in practice is not as is explained above using the popular diagram you have used. The parallax causes the star to prescribe an ellipse in the sky, the semi-major axis of whose is equal to the parallactic angle.

The telescopes generally measure the shift in co-ordinates of star(RA and Dec) and then translate the information to that of the ellipse prescribed and determine the parallactic angle. Now, the ability of telescopes to note this change in RA and Dec is based on their least count, obviously which is dependent on their angular resolution.

parallactic ellipse

Hence, distance measurements using parallax method have their range based on the angular resolution of the telescope being used.

Image source: http://documents.stsci.edu/hst/fgs/documents/handbooks/ihb_cycle16/c01_intro4.html

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    $\begingroup$ Where TGM in that diagram means what? $\endgroup$
    – Donald.McLean
    Jan 7, 2014 at 15:49
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    $\begingroup$ @Donald.McLean It's Hubble Space Telescope's Two-Gyro Mode. Image taken from this page : documents.stsci.edu/hst/fgs/documents/handbooks/ihb_cycle16/… They were demonstrating the porperties of the instrument, but gave a useful diagram for illustration. $\endgroup$
    – Cheeku
    Jan 7, 2014 at 23:15
  • $\begingroup$ Ahh. I'm familiar with Two Gyro Mode, but without any context for the diagram, I just didn't recognize it. Perhaps you could edit your answer to add some explanation? $\endgroup$
    – Donald.McLean
    Jan 8, 2014 at 14:30
  • $\begingroup$ StackExchange websites tend to be informative and descriptive, but they often drift. With all humility, I am against this, since the only thing I wanted to show in my diagram is the parallactic ellipse. The description of TGM, in my opinion, is not relevant. I am sorry and I hope you understand. $\endgroup$
    – Cheeku
    Jan 8, 2014 at 14:49
  • $\begingroup$ That's fine, but your answer should, at the very least, include proper attribution of the source of the picture. $\endgroup$
    – Donald.McLean
    Jan 8, 2014 at 15:26
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The probably most advanced system for determination of parallaxes is AGIS as used for Gaia. It's able to go far beyond the angular resolution of the telescopes. Angular resolution is just one parameter.

Actually it's just necessary to determine the luminosity centroids of the stars, almost independent of the resolution of the telescopes. That's mainly a statistical challenge, and it is especially dependent of the image noise, the brightness of the star and the number of observations.

The calibration of the telescope and the determination of the luminosity centroids can be done in one solution algorithm.

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