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Thread: Present uncertainty in relative positions of Alpha and Proxima Centauri

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    Present uncertainty in relative positions of Alpha and Proxima Centauri

    What exactly are the state of the art uncertainties of the relative positions of Alpha and Proxima?
    I found a source from Kervella:
    http://userpages.irap.omp.eu/~bdintr...S_Kervella.pdf
    Is this the newest, or since replaced (such as by Gaia)?
    The quoted distances are:
    Proxima - 1,3008+-0,0006 pc
    Alpha AB - 1,3384+-0,0011 pc

    Are these uncertainties independent errors (to be added to each other, so the error in relative distance is 0,0017 pc) or including a common error of distance scale, so the error of relative distance is smaller?

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    Wikipedia gives about 600 AU uncertainty in the major axis of its 550,000 year orbit, which we've been observing for 100 years or so, which is about 1% of a light year. The values you cite are 3 to 10 times more accurate. I wonder if a more careful astrometric study using historic images would give significantly more accurate parameters for the orbit.
    Forming opinions as we speak

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    Quote Originally Posted by antoniseb View Post
    Wikipedia gives about 600 AU uncertainty in the major axis of its 550,000 year orbit, which we've been observing for 100 years or so, which is about 1% of a light year. The values you cite are 3 to 10 times more accurate. I wonder if a more careful astrometric study using historic images would give significantly more accurate parameters for the orbit.
    I donīt expect so.

    It is quite natural that the uncertainty in orbit would be bigger than the uncertainty in present position. Because orbital predictions include the uncertainty in present position, plus uncertainty in speed.
    Astrometry would give just two dimensions of the relative speed. Orbit needs the third, too, and that relies mainly on radial velocity.

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    Quote Originally Posted by chornedsnorkack View Post
    What exactly are the state of the art uncertainties of the relative positions of Alpha and Proxima?
    I found a source from Kervella:
    http://userpages.irap.omp.eu/~bdintr...S_Kervella.pdf
    Is this the newest, or since replaced (such as by Gaia)?
    The quoted distances are:
    Proxima - 1,3008+-0,0006 pc
    Alpha AB - 1,3384+-0,0011 pc

    Are these uncertainties independent errors (to be added to each other, so the error in relative distance is 0,0017 pc) or including a common error of distance scale, so the error of relative distance is smaller?
    Assuming you want to calculate the Alpha AB to Proxima distance, you can't just find the difference between the two distances above. Unless Proxima were to be exactly on the line of sight to AB, that is.

    What you have are the lengths of two sides of a triangle. If we know the angle between the two sides, the third (the Proxima to AB distance) can be calculated.

    That angle will have an uncertainty attached which will add to the uncertainty on the result.

    You could argue the angle is so small we can justifiably use the difference in line of sight distance as the result. If so, you don't simply add the uncertainties together.

    If you assume that all uncertainties are independent of each other, the uncertainty on the result is:

    SQRT(0.0006^2 + 0.0011^2) = 0.0013 pc.

    However I am sure that is a gross oversimplification because I don't know how much of the uncertainty is random and how much is common to both measurements.

    Also, if we did it properly with trigonometry, the small uncertainties on the two very long sides coupled with the uncertainty on the angle will have a large proportionate effect on the short side calculation result.

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    A general approximate overview of the system - that does NOT include the uncertainties:
    https://en.wikipedia.org/wiki/Alpha_...a_Centauri.png

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    Quote Originally Posted by chornedsnorkack View Post
    A general approximate overview of the system - that does NOT include the uncertainties:
    https://en.wikipedia.org/wiki/Alpha_...a_Centauri.png
    The angular separation is given as 2 degrees 11' 06".

    TBH I hadn't realised it was as big as that.

    But what is the uncertainty on it?

    Point number 1, the angular separation between A and B is several arcsec, way exceeding the probable precision of the angle measurement.

    SO for starters we need to know what this angular separation is in reference to. Is it the Proxima - AC-A or Proxima- AC-B angle, or some sort of average? The barycentre maybe?

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    Quote Originally Posted by kzb View Post
    The angular separation is given as 2 degrees 11' 06".

    TBH I hadn't realised it was as big as that.

    But what is the uncertainty on it?
    Very low.


    Point number 1, the angular separation between A and B is several arcsec, way exceeding the probable precision of the angle measurement.

    SO for starters we need to know what this angular separation is in reference to. Is it the Proxima - AC-A or Proxima- AC-B angle, or some sort of average? The barycentre maybe?
    If you click on more details you will see that it is sourced to "own work" by a unlisted wikipedia editor. If you want details search for a proper source or use simbad and do your own calculations.

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    The angular positions in the sky are known with greater precision than the distances, if I am not mistaken.

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    Quote Originally Posted by Hornblower View Post
    The angular positions in the sky are known with greater precision than the distances, if I am not mistaken.
    Yes but we need to know what the angular separation is between.

    A and B are several arcsec apart. The angular separation of A and B exceeds the precision of the measurement we are provided with. This is the difficulty.

    The first part of any uncertainty evaluation is to write down exactly what are all the calculation steps.

    At present I could take the angular separation of A and B as the uncertainty on the Proxima -> AB angular separation. That would make it several arcseconds and a lot larger than the precision with which angular separations can be measured in reality.

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    Quote Originally Posted by kzb View Post
    Yes but we need to know what the angular separation is between.

    A and B are several arcsec apart. The angular separation of A and B exceeds the precision of the measurement we are provided with. This is the difficulty.

    The first part of any uncertainty evaluation is to write down exactly what are all the calculation steps.

    At present I could take the angular separation of A and B as the uncertainty on the Proxima -> AB angular separation. That would make it several arcseconds and a lot larger than the precision with which angular separations can be measured in reality.
    As I see it any uncertainty in the AB separation is the tiny fraction of an arcsecond that would occur with any star in the same field. Since the orbit is well observed and determined we should have a similarly small uncertainty in the barycentric position of the pair. I don't see why you are seeing an uncertainty of several arcseconds.

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    The locations of bright fixed stars are known to precision of Hipparcos - 1 milliarcsecond.
    Alpha Centauri AB are not fixed stars. Uncertainty of true orbits (80 year orbit, 4 year Hipparcos observation arc) might make the location of Alpha Centauri more uncertain. And of course itīs 22 years since the 4 year observation arc of Hipparcos - any errors of proper motion have been growing since.
    The brightness of Proxima, above +11, is precisely the range in which Tycho coverage is partial. So Proxima may not be in Tycho.
    Still, I expect the uncertainty in Alpha-Sun-Proxima angle to be under 10 milliarcsec.
    Agreed?

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    Quote Originally Posted by chornedsnorkack View Post
    The locations of bright fixed stars are known to precision of Hipparcos - 1 milliarcsecond.
    Alpha Centauri AB are not fixed stars. Uncertainty of true orbits (80 year orbit, 4 year Hipparcos observation arc) might make the location of Alpha Centauri more uncertain. And of course itīs 22 years since the 4 year observation arc of Hipparcos - any errors of proper motion have been growing since.
    The brightness of Proxima, above +11, is precisely the range in which Tycho coverage is partial. So Proxima may not be in Tycho.
    Still, I expect the uncertainty in Alpha-Sun-Proxima angle to be under 10 milliarcsec.
    Agreed?
    Well no I don't agree really.

    If "Alpha" were a single star then I would agree with you.

    But it is not. It is two objects separated by several arcseconds.

    So what distance am I actually calculating here ?

    Is it the Proxima to Alpha Centauri A distance ?

    Is it the Proxima to Alpha Centauri B distance ?

    Is it the distance of Proxima to the mid-point of Alpha Centauri A and Alpha Centauri B ?

    Is it the distance of Proxima to the barycentre of Alpha Centauri A and B ?

    You see my problem ?

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    Quote Originally Posted by kzb View Post
    Well no I don't agree really.

    You see my problem ?
    Not really, just pick one that makes sense for what you want to do and find/calculate it.

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    Quote Originally Posted by glappkaeft View Post
    Not really, just pick one that makes sense for what you want to do and find/calculate it.
    But all of them are relatively imprecise:
    Hipparcos operated for 4 years, and shut down 22 years ago. So if there were any measurements of proper motion of Alpha, these had a short observation arc. Leading to an error in proper motion that has been accumulating since.
    But still... if the error in Alpha-Sun-Proxima angle is 10 milliarcsec, then what is the error in Alpha-A distance, where A is an empty place behind Proxima and Sun-A is equal distance to Sun-Alpha?
    The error from the angle error is 0,013 AU.
    The error from the error of Sun-Alpha distance is about 9 AU
    So thatīs the prevalent error, but still small compared to the error in distance to Sun.

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    Quote Originally Posted by chornedsnorkack View Post
    But all of them are relatively imprecise:
    Hipparcos operated for 4 years, and shut down 22 years ago. So if there were any measurements of proper motion of Alpha, these had a short observation arc. Leading to an error in proper motion that has been accumulating since.
    But still... if the error in Alpha-Sun-Proxima angle is 10 milliarcsec, then what is the error in Alpha-A distance, where A is an empty place behind Proxima and Sun-A is equal distance to Sun-Alpha?
    The error from the angle error is 0,013 AU.
    The error from the error of Sun-Alpha distance is about 9 AU
    So thatīs the prevalent error, but still small compared to the error in distance to Sun.
    The uncertainty in the angle we are provided with is not 10 mas. We are given the angle to a precision of 1 arcsecond: 2 degrees, 11 minutes, 6 arcseconds.

    Note it does not say 6.00 arcseconds which would indeed imply a precision of 10mas. We are given zero decimal places of arcseconds. It is reported with a precision of 1 arcsecond.

    It may well be that this number has been reported with greater precision than this, but I have not seen it.

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    Quote Originally Posted by glappkaeft View Post
    Not really, just pick one that makes sense for what you want to do and find/calculate it.
    It wasn't me that started this you know. I am actually trying to help believe it or not.

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    The next problem is finding the cosine of very small angles !

    The standard uncertainty of 2 degrees 11' 06":

    As reported, the arcsecond number could be between 5.5 and 6.5 arcseconds.

    The standard uncertainty (assuming all values between these limits are equally probable) is 0.5/SQRT(3) = 0.289 arcseconds

    0.289 arcseconds is 0.0000802 degrees.

    How do you find the cosine of that?

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    Quote Originally Posted by kzb View Post
    The uncertainty in the angle we are provided with is not 10 mas. We are given the angle to a precision of 1 arcsecond: 2 degrees, 11 minutes, 6 arcseconds.

    Note it does not say 6.00 arcseconds which would indeed imply a precision of 10mas. We are given zero decimal places of arcseconds. It is reported with a precision of 1 arcsecond.

    It may well be that this number has been reported with greater precision than this, but I have not seen it.
    Yes. I am combining sources which give the number with precision of 1 arcsecond and sources which imply that the number is known to precision better than 10 milliarcseconds, without specifying the value.

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    Quote Originally Posted by chornedsnorkack View Post
    Yes. I am combining sources which give the number with precision of 1 arcsecond and sources which imply that the number is known to precision better than 10 milliarcseconds, without specifying the value.
    SO, mathematically, how are you doing that? On the face of it, it sounds dodgy.

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    Quote Originally Posted by kzb View Post
    SO, mathematically, how are you doing that? On the face of it, it sounds dodgy.
    I donīt need the actual number. I just want to know whether someone as the number (or data from which the number can be derived) or whether nobody has the number.

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    Quote Originally Posted by kzb View Post
    The next problem is finding the cosine of very small angles !

    The standard uncertainty of 2 degrees 11' 06":

    As reported, the arcsecond number could be between 5.5 and 6.5 arcseconds.

    The standard uncertainty (assuming all values between these limits are equally probable) is 0.5/SQRT(3) = 0.289 arcseconds

    0.289 arcseconds is 0.0000802 degrees.

    How do you find the cosine of that?
    A trivial answer is "use the cosine function of your calculator," but I suspect you already know that . But that begs the question: What is your real question?

    If you're looking for an approximation that holds for small angles (and this is a very small angle, indeed), just use the first couple of terms in the series expansion for cosine -- 1 - x^2/2 -- where x is the angle in radians).

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    Quote Originally Posted by kzb View Post
    SO, mathematically, how are you doing that? On the face of it, it sounds dodgy.
    Of course it would be dodgy to declare, on the basis of what we have in this thread, that the angular separation of Proxima and Alpha is something like 2 degrees 11' 06.000" plus or minus .001". But let's put some things in perspective.

    Suppose the angular separation is uncertain by 1" for whatever reason. That is about 1.3 astronomical unit (AU) out of a transverse component over 10,000 AU as measured along a heliocentric circular arc from Proxima to the line of sight to Alpha. Converting the published distances along that line of sight to AU gives uncertainties on the order of 100 times as large in the radial component.

    We have measured positions of A and B that are good about a milliarcsecond or better. We also have excellent data on the shape and orientation of the relative orbit from increasingly accurate measurements over more than three complete revolutions since the mid-1700s. It should be possible in principle to reduce the uncertainty in the barycenter to far less than 1" with curve fitting techniques that have been known for a very long time.

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    Quote Originally Posted by Hornblower View Post
    We have measured positions of A and B that are good about a milliarcsecond or better.
    22 years ago, over an arc of 4 years. Which wonīt be repeated. Both A and B are too bright for Gaia to see.
    Quote Originally Posted by Hornblower View Post
    We also have excellent data on the shape and orientation of the relative orbit from increasingly accurate measurements over more than three complete revolutions since the mid-1700s. It should be possible in principle to reduce the uncertainty in the barycenter to far less than 1" with curve fitting techniques that have been known for a very long time.
    Those two centuries of observations were with errors much bigger than milliarcsecond.

    However, am I right in assuming that after Hipparcos is over and Gaia does not replace it, precision of 10 milliarcseconds continues to be available?
    Anyway, from the drawing: I named the point behind Proxima as A.
    The drawing quotes distance Proxima-A as 0,038 pc. Their numbers would give 0,0376 pc. Kervellaīs agree.
    Kervella gives precision for the component distances. 0,0006 and 0,0011. If they add as squares, the total uncertainty would be 0,0013 pc. For better following of commas, the distance comes as 37,6+-1,3 mpc
    Correct?

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    Quote Originally Posted by Geo Kaplan View Post
    A trivial answer is "use the cosine function of your calculator," but I suspect you already know that . But that begs the question: What is your real question?

    If you're looking for an approximation that holds for small angles (and this is a very small angle, indeed), just use the first couple of terms in the series expansion for cosine -- 1 - x^2/2 -- where x is the angle in radians).
    My calculator returns "1" exactly...not good enough

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    Quote Originally Posted by Hornblower View Post
    Of course it would be dodgy to declare, on the basis of what we have in this thread, that the angular separation of Proxima and Alpha is something like 2 degrees 11' 06.000" plus or minus .001". But let's put some things in perspective.

    Suppose the angular separation is uncertain by 1" for whatever reason. That is about 1.3 astronomical unit (AU) out of a transverse component over 10,000 AU as measured along a heliocentric circular arc from Proxima to the line of sight to Alpha. Converting the published distances along that line of sight to AU gives uncertainties on the order of 100 times as large in the radial component (2).

    We have measured positions of A and B that are good about a milliarcsecond or better. We also have excellent data on the shape and orientation of the relative orbit from increasingly accurate measurements over more than three complete revolutions since the mid-1700s. It should be possible in principle to reduce the uncertainty in the barycenter to far less than 1" with curve fitting techniques that have been known for a very long time (1).
    (1) I'm sure that is all true but I can't do that sat here with my Casio calculator.

    (2) To be honest I have no feeling for how the angular separation uncertainty would propagate through the calculation. We have to find the cosine of the angle so it may not be directly proportional to the (small) relative uncertainty in the angle itself. Or maybe it is, perhaps you can tell.

    At present I have not found a device that can even find the cosine of a very small angle reliably, never mind anything else.

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    Now, how does the 1,3 mpc (270 AU) uncertainty of one leg of right triangle map to uncertainty of the hypotenuse, seeing how the other leg has little uncertainty, dominated by the uncertainty of distance to Sun (under 10 AU)?
    My estimate would be 270 AU*38mpc/63 mpc=160 AU uncertainty of Alpha-Proxima distance. Yours?

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    Quote Originally Posted by chornedsnorkack View Post
    Now, how does the 1,3 mpc (270 AU) uncertainty of one leg of right triangle map to uncertainty of the hypotenuse, seeing how the other leg has little uncertainty, dominated by the uncertainty of distance to Sun (under 10 AU)?
    My estimate would be 270 AU*38mpc/63 mpc=160 AU uncertainty of Alpha-Proxima distance. Yours?
    Without actually doing the calculation I don't know. I have never done an uncertainty evaluation using cosines and trigonometry before.

    However this is the way I would do it:

    We have a triangle with sides a, b and c.

    We have values for a and b, and we are given an angle (theta) between them.

    We wish to calculate side c:

    c = SQRT(a^2+b^2-2ab.cos(theta))

    Now, the question is, how does uncertainty in theta affect the uncertainty in c ? Looking at the form of the equation I couldn't assume it is directly proportional.

    My idea for estimating this is to use the uncertainty of theta as the value of theta in the above equation.

    A variation on this is to repeat the calculation using theta plus its uncertainty and see what the difference is. This would show us the magnitude of the affect on c of a small error in theta.

    I lack the mathematical skills to do it any other way. But I am struggling to find accurate ways of finding the cosines.

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    Quote Originally Posted by kzb View Post
    Without actually doing the calculation I don't know. I have never done an uncertainty evaluation using cosines and trigonometry before.
    I think that finding uncertainty in distance does not actually need trigonometry. Just similar triangles.

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    Quote Originally Posted by chornedsnorkack View Post
    I think that finding uncertainty in distance does not actually need trigonometry. Just similar triangles.
    Tell us the exact calculation algorithm and then we can sort out how to propagate the uncertainties through it.

    I have some good news, and that is I believe the uncertainty dependence on the angular separation is less than unity. I have ascertained this by trying out different angles.

    This is good because if the per cent uncertainty in the angle is much smaller than the per cent uncertainty in the triangle side lengths (and it is going by what we know), we can probably neglect it. Because uncertainties square in the calculation, the small uncertainties tend to disappear into a rounding error of the total.

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    Quote Originally Posted by kzb View Post
    Without actually doing the calculation I don't know. I have never done an uncertainty evaluation using cosines and trigonometry before.

    However this is the way I would do it:

    We have a triangle with sides a, b and c.

    We have values for a and b, and we are given an angle (theta) between them.

    We wish to calculate side c:

    c = SQRT(a^2+b^2-2ab.cos(theta))

    Now, the question is, how does uncertainty in theta affect the uncertainty in c ? Looking at the form of the equation I couldn't assume it is directly proportional.

    My idea for estimating this is to use the uncertainty of theta as the value of theta in the above equation.

    A variation on this is to repeat the calculation using theta plus its uncertainty and see what the difference is. This would show us the magnitude of the affect on c of a small error in theta.

    I lack the mathematical skills to do it any other way. But I am struggling to find accurate ways of finding the cosines.
    I just crunched the equation in my trusty Texas Instruments calculator, which can do trig functions to ten figures. A 1" uncertainty in the angular separation had less than 1% of the effect of the published distance uncertainties.

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