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Thread: Is the Parallax measurement accurate? What about relative movement of Sun and Star?

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    Is the Parallax measurement accurate? What about relative movement of Sun and Star?

    Hello,

    I am sure this question has been asked before by many people, but it's only just struck me now.

    A Parsec is defined to be the distance subtended by a distance of 1 A.U and we take advantage of this effect to measure the distance to a star by measuring it's angular movement on the celestial background as the Earth moves around the Sun i.e as it moves 1 A.U.

    d (pc) = 1/p (arcseconds)

    But doesn't this assume that the Sun and star have the same relative motion? If the Sun orbits the Galactic centre at 200 km/s, then there is scope for considerable error if we have a rogue star. For the purposes of parallax distance measurement, do we just assume that on average stars move with the same relative velocity as the Sun. I think that would be a realistic assumption but it does introduce scope for a large error.

    Do you agree? Is there something obvious I have missed? (quite likely !)

    Thank you

    Edit: I think I figured it out. After 12 months, the Earth will be back in the "same place", and hence any angular movement due to relative motion can be discounted? Is that how it works?
    Last edited by Naz; 2018-Jun-21 at 03:50 PM. Reason: Might have figured it out?

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    Quote Originally Posted by Naz View Post
    ... Edit: I think I figured it out. After 12 months, the Earth will be back in the "same place", and hence any angular movement due to relative motion can be discounted? Is that how it works?
    Actually, what you do is make the measurements repeatedly over several years, so that you can find both the distance AND the relative motion.
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    Suppose that a star moves at 200 km/sec along our line of sight, and suppose further that it is only 10 parsecs away from the Sun. This is extreme example is a "worst case" scenario for the issue you've raised.

    How much will this motion ruin our measurement of the distance to the star? Let's find out.

    We make one measurement on Jan 1, 2018, and a second measurement on Jan 1, 2019. The first measurement occurs when the star is exactly 10 parsecs from the Sun = 3.08 x 10^(14) km. The second occurs one year later = 3.15 x 10^(7) seconds later. How far does the star move away from us during that year?

    distance moved away from Sun = 200 km/sec * 3.15 x 10^(7) seconds = 6.3 x 10^(9) km.

    That sounds like a lot, but how big a fraction of the original distance is it?

    fraction of original distance = 6.3 x 10^(9) km / 3.08 x 10^(14) km = 0.00002 = 0.002 percent

    Not very important.

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    I wouldn't be surprised to learn that the very distant quasars are used for reference as well in determining "proper motion" (determined apparent angular movement). Scientists determine the daily variations in the rotation of the Earth by using quasars to get their accuracy.

    [I missed StupendousMan's post. Quasars would seem to be overkill.]
    Last edited by George; 2018-Jun-21 at 04:58 PM.
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    When I was in college we did a computer programming exercise using matrix techniques to disentangle the sinusoidal parallax from the linear proper motion, and got printouts showing both. The numbers made it obvious that the target was Barnard's Star. The annual proper motion was about 20 times the amplitude of the parallax.

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    Quote Originally Posted by StupendousMan View Post

    fraction of original distance = 6.3 x 10^(9) km / 3.08 x 10^(14) km = 0.00002 = 0.002 percent

    Not very important.
    No it is very important, although your number is twice what it should be (you should calculate it over 6 months not a year). Compare with the parallax calculated the same way
    2,992◊10^8 / 3,08◊10^14 = 0.0000001 = 0.00001% = 0.1 ppm

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    Quote Originally Posted by glappkaeft View Post
    No it is very important, although your number is twice what it should be (you should calculate it over 6 months not a year). Compare with the parallax calculated the same way
    2,992◊10^8 / 3,08◊10^14 = 0.0000001 = 0.00001% = 0.1 ppm
    I think SupendousMan is addressing radial change but the OP is about proper motion.

    In terms of actual arcseconds, after 12 months, at 200kps the proper motion for a star at 32 parsecs will have an apparent motion of 4.2 arcseconds, thus 2.1 arcseconds of movement for a parallax motion (6 mos.) that is found to be 0.1 arcsecond (32 lyrs). So, yes, that proper motion is quite large!
    Last edited by George; 2018-Jun-21 at 06:48 PM.
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    Quote Originally Posted by Naz View Post
    Do you agree? Is there something obvious I have missed? (quite likely !)
    Yes, I think thereís something you missed, but Iím not sure I can explain it well. You know of course that as an object gets further away it appears to move more slowly compared to a fixed background. And stars are so far away that even if they are moving quickly, they appear to be essentially immobile against the background. It would be correct that parallax will not work for an object that is very close, unless you can do it relatively quickly, for example by comparing the images between two eyes.


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    Quote Originally Posted by Jens View Post
    Yes, I think there’s something you missed, but I’m not sure I can explain it well. You know of course that as an object gets further away it appears to move more slowly compared to a fixed background. And stars are so far away that even if they are moving quickly, they appear to be essentially immobile against the background. It would be correct that parallax will not work for an object that is very close, unless you can do it relatively quickly, for example by comparing the images between two eyes.


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    My bold. No, it doesn't matter how fast the proper motion is, provided you can keep it in the same field of reference stars for several exposures. The nearby star will follow a wavy path across the field, with that path being the resultant of moving around the center of a small ellipse, with that center moving at constant velocity. From the star's position we know the shape and orientation of the ellipse, so we need to find the size that fits the observed positions. If we had exact daily measurements for a nice clean curve it would be a slam dunk. With the usual set of data points at intervals of 6 months over a period of several years, along with uncertainties in the individual points, we use the matrix techniques to fit a curve to the points, and thus extract the size of the elliptical component.

    An extreme case is Groombridge 1830, which has an annual proper motion of about 7 arcseconds and a parallax amplitude of only about 0.1 arcsecond. The large proper motion is not an obstacle.

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    To amplify Hornblower's point - a key part of this is that we know the orientation and shape of the parallax ellipse for objects in a particular direction, so a long-enough time baseline to separate that from the linear proper motion gives the two components well.

    A more subtle gotcha happens for binary stars with orbital period close to one Earth year, where the center of light of the blended image (photocenter in the biz) has its own motion which might be in or out of phase with the parallax ellipse, resulting into a wrong parallax just from those data. With the latest Gaia data release, I've seen discussions of the fastest way to identify and correct these (multiple radial-velocity observations are good). The fraction of stars this applies to is tiny, but if your starting sampling has a few hundred million, some of these will show up.

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    Here is an image of the proper motion and parallax of a few stars:

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    I'm sorry I misunderstood the point of the OP. The diagram below shows the actual measurements of a nearby star made by the Hipparcos satellite over a period of a bit more than 3 years. As you can see, the magnitude of the proper motion over this period is larger than that of the parallax .... and yet it's very simple to disentangle the two effects, and still measure the parallax accurately.

    vega_motion.gif

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    Quote Originally Posted by ngc3314 View Post
    To amplify Hornblower's point - a key part of this is that we know the orientation and shape of the parallax ellipse for objects in a particular direction, so a long-enough time baseline to separate that from the linear proper motion gives the two components well.
    From tony's cool graphics, and Hornblower's example of a large ratio of proper motion to parallax, do any exhibit a closed or nearly closed ellipse or must the two motions be separated in almost every case, as you state, even for just one year?

    It looks like this ratio and the angle to the ecliptic is all one would need to generate the curve, I suppose.
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    Quote Originally Posted by George View Post
    From tony's cool graphics, and Hornblower's example of a large ratio of proper motion to parallax, do any exhibit a closed or nearly closed ellipse or must the two motions be separated in almost every case, as you state, even for just one year?

    It looks like this ratio and the angle to the ecliptic is all one would need to generate the curve, I suppose.
    If the star is on the ecliptic, the parallax motion will be back and forth along that line instead of around an ellipse. If the proper motion is also along the ecliptic, then the resultant will be motion along the ecliptic at a variable speed. The parallax can still be extracted from the resultant. If the proper motion is perpendicular to the ecliptic, then the resultant will be a sine wave.

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    You would not need a curve.
    After all, in precisely 12 months, parallax is precisely 0. So just make 3 measurements.
    2 of them 12 months apart. Precisely no parallax. Pure proper motion.
    And the third 6 months apart. Compute the fraction of proper motion that goes into the 6 months - the rest is parallax.
    What it depends on is the orientation of parallax, so you can choose season of observations according to location of star.

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    Quote Originally Posted by chornedsnorkack View Post
    You would not need a curve.
    After all, in precisely 12 months, parallax is precisely 0. So just make 3 measurements.
    2 of them 12 months apart. Precisely no parallax. Pure proper motion.
    And the third 6 months apart. Compute the fraction of proper motion that goes into the 6 months - the rest is parallax.
    What it depends on is the orientation of parallax, so you can choose season of observations according to location of star.
    That is exactly how I would do it if I had the luxury of unlimited telescope time for observing zillions of stars at just the right times for each star.

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    Quote Originally Posted by Hornblower View Post
    That is exactly how I would do it if I had the luxury of unlimited telescope time for observing zillions of stars at just the right times for each star.
    Making just 3 observations, not a whole curve, saves observation time.
    If in end of June, Sun in in Cancer, then observing a star in Cancer or Capricorn in July and 6 months later in January would show no parallax. Because the Earth would be closer and more distant, but the star would be in the same direction.
    Instead, in end of June, you can observe stars in Aries and Libra. These would show parallax when seen in end of December.

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    So what's the answer to the OP? How much "error" does the proper motion introduce into the measurements? Is it a lot or not? At first it seems like not a lot (tiny fraction of a percent) but then you all start putting up these curlicue graphs that show huge deviations.

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    Quote Originally Posted by CJSF View Post
    So what's the answer to the OP? How much "error" does the proper motion introduce into the measurements? Is it a lot or not? At first it seems like not a lot (tiny fraction of a percent) but then you all start putting up these curlicue graphs that show huge deviations.

    CJSF
    In a thought exercise with no uncertainty in the individual measurements, the proper motion does not introduce any error. None. Nada. Nyet. Zilch. Doing the math correctly extracts the parallax just fine. Those curlicues are not uncertainties. They are simply vector sums of an elliptical motion and a linear motion. In the real world, where the raw data do not make nice clean curlicues, the matrix techniques with observations over several years reduce the uncertainties of both components of the fitted curve.

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    Quote Originally Posted by chornedsnorkack View Post
    You would not need a curve.
    After all, in precisely 12 months, parallax is precisely 0. So just make 3 measurements.
    2 of them 12 months apart. Precisely no parallax. Pure proper motion.
    And the third 6 months apart. Compute the fraction of proper motion that goes into the 6 months - the rest is parallax.
    What it depends on is the orientation of parallax, so you can choose season of observations according to location of star.
    My bold. I think the original question was asking if the motion of the Sun made any difference. After one year, the Earth moves along with the Solar System far away from its year-ago location. Paralax isn't zero without taking that into consideration.
    Depending on whom you ask, everything is relative.

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    Quote Originally Posted by mkline55 View Post
    My bold. I think the original question was asking if the motion of the Sun made any difference. After one year, the Earth moves along with the Solar System far away from its year-ago location. Paralax isn't zero without taking that into consideration.
    Annual parallax is the displacement of the star's position from what it would be if viewed from the Sun. It will be zero when the star is at opposition to the Sun.

    If the motion is toward or away from the target star, by all means the amount of parallax will change from year to year, but over just a few years the change will be immeasurably slight. If the motion is purely lateral, that is the cause of the proper motion we have been discussing.

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