How to account for the movement of stars during measurement of parallax?

Question

In my GCSE physics class today I was doing revision for my upcoming GCSEs, and we came to parallax measurements, as before my teacher explained that two measurements are taken six months apart as the Earth moves around the sun.

My question is, as my teacher couldn't answer, how is the movement of the star being measured, and that of the backdrop of more distant stars accounted for when taking measurements? Surely if the star has moved further or closer away during the six month period the measurement would be innacurate.

Answer 1

Yes, you often see sources that say a parallax is found from measurements taken 6 months apart. That is rarely the case, and if you think about it, not all stars are visible in the night sky 6 months apart (from the same location on Earth).

In fact a parallax measurement will consist of a series of measurements, possibly taken over more than 1 year. There is no formal requirement for any particular measurement interval, though of course the biggest parallax signal would be if you could get measurements 6 months apart.

In addition to parallax, you need to account for the "proper motion" of the star across the sky. To a first order approximation this is a linear change in right ascension and declination with time (though more complicated models can be used, and you have to account for the ellipticity of the Earth's orbit too). The position of the star in question (with respect to distant background stars) measured in multiple images at multiple times would then be fitted with a function that results in a determination of the parallax and the proper motions in both coordinates. Any motion of the "background stars" is compensated by averaging over all the faint stars in the field. They will all move in a systematic way due to the motion of the Sun in the Galaxy (this is part of the "proper motion"), but their random motions (if they are distant) will be very small and average out well.

A perfect visualisation is provided by HST images of Proxima Centauri which show its path across they sky and illustrates the cyclical motion due to parallax and the linear motion across the sky caused by its proper motion.

The motion of the star towards or away from the Sun could be accounted for from the measurement of the star's line of sight velocity. In practice, these motions are of order tens of km/s and this means it would take tens of thousands of years to change the distance by 1 light year.

Answer 2

You perform periodic measurements every 6 months. The measurements that are a year apart are only measuring the proper motion of the stars, this allows you to extract the parallax from the data.

Answer 3

In six months' time, that star is not going to change its distance enough to affect the amount of observed parallax. The distant background stars are, for all intents and purposes, a stationary background (but for best results you want to pick very distant stars and average over a bunch of them). The parallax causes a back and forth motion on the sky relative to the background stars, returning to the same spot each time the earth is back in the same part of its orbit. From year to year the size of this motion will not change. Over MILLIONS of years, yes, the parallax could gradually shrink or grow as a star gets farther or closer to us. Meanwhile, the proper motion is caused by the actual path of the star through the galaxy, which we can detect and is often about the same amount as the parallax. But it will not be in exactly the same direction as the parallax motion and it does not go back to its starting point each year -- it continues "forward" year after year. THAT'S what makes the star appear to move continuously in a particular direction. The image above shows the proper motion of the star moving from left to right, combined with an up and down motion that is cause by the parallax effect as the earth moves on its orbit each year. Highest peak to lowest trough takes six months, and peak to peak is one year.