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I want to see what the stars looked like up to 15,000 years into the past and future.

I've chosen 15,000 years in order to see a full precession cycle. To make things easier, I would be satisfied with a relatively low precision of 60 arcseconds. I want to calculate the equatorial positions of bright stars, and show the path of the ecliptic. (I have no need to calculate the positions of the Sun or planets.)

My source data is the Yale Bright Star Catalog, revision 5, for J2000, which includes proper motion data. My core reference is Astronomical Algorithms by Jean Meeus (1991).

Precession. Is the algorithm in Meeus (equations 20.3 on page 126) suitable for +/- 15,000 years? The non-linear terms get up to about 20 degrees. Is that a problem?

Would it be reasonable to update Meeus with this paper from Capitaine et al 2003 (page 572, equations 21)? It has a remark on accuracy: "The following series with a 0.1 µas level of precision matches the canonical 4-rotation series to sub-microarcsecond accuracy over 4 centuries." Those equations (21) are for accumulated amounts (subscript A for accumulated). Is it OK to simply drop the constant terms in these expressions? Meeus has no constant terms in his algorithm.

Proper motion. The YBS catalog has proper motion in declination and right ascension. At what point would a simple linear treatment of proper motion start to be of dubious quality, given that I can tolerate a full 60 arcseconds of error? (I mean by this using the simple 2D proper motion across the sky, with no 3D velocity vector.)

Example: for the year -3200, the max proper motion for all of the stars in the YBS, between J2000 and that date, is +10°12' (using YBS data).

The ecliptic. I would be satisfied with simply seeing the path of the ecliptic, without trying to calculate the Sun's actual position. I'm assuming that all I really need is the value for the obliquity of the ecliptic. Meeus (page 135, equation 21.3) has an expression for the obliquity of the ecliptic, but he says that it shouldn't be used outside of 10,000 years from J2000. Does anyone know of a replacement for this expression, which is valid for a longer period?

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    $\begingroup$ The NOVAS software is the most precise software that can readily do this: aa.usno.navy.mil/software/novas_info $\endgroup$ Jan 28 at 22:06
  • $\begingroup$ aa.usno.navy.mil/downloads/novas/NOVAS_C3.1_Guide.pdf - its docs, like the Ephemeris itself, doesn't seem to have much explicit info regarding accuracy over millenia... I do notice two things. 1) its precession docs point to the Capitaine paper mentioned above. 2) Its proper motion algo is in 3 dimensions, not 2. $\endgroup$
    – John
    Jan 29 at 1:50

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This is not a full answer but more a sketch how I would try to address the issue (but too long for a comment):

If you use (or transform) all data you consider to the same datum, then you can consider the proper motion data linear over the time.

As for an error estimate, you will have to use the errors in proper motion which are given for each catalogue. Especially for the distant stars, the relative error may be big, but the absolute error and thus effect will not be big.

For better accuracy one should nowadays use the data from GAIA though as it has way better accuracy than previous catalogues, with an error given for proper motion as low as (quote from Eggle et al (2019)):

Uncertainties in the parallaxes range between 0.04 milliarcseconds (mas) for sources with Gmag<15 and 0.7 mas at Gmag=20. The corresponding un- certainties in the proper motion components start at 0.06 mas/yr for Gmag<15 and reach 1.2 mas/yr for Gmag=20. The Gaia astrometric catalog is, thus, a reference of unprecedented accuracy well suited to identify systematic errors in other star catalogs

Taking these error estimates, and linearily extrapolating you end up for stars of magnitude 15 or brighter with a time span of $60" / 0.00006"/yr = 1$ million years in order for the error to exceed 60 arcseconds. Thus you would be fine to analyse the look of the night sky with the GAIA data and their intrinsic accuracy.

If you use other data sources, do the analysis similarily. This website lists 1.5" / century for the PPM catalogue. So you have only 4000 years with the PPM catalogue until you reach with linear extrapolation a possible error in excess of 60". I don't have data for the YBS, but it's probably in this range.

After applying the time-propagation of the proper motion, thus re-calculated the places of each star, you then will have to do the transform of coordinates due to precession, nutation, obliquity of ecliptic etc which influence orientation of Earth's equatorial coordinate system with respect to the stellar reference frame.

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Precession and Obliquity. Almost all software will not do a good job of computing long-term precession over millenia. This includes NOVAS. As described here , the standard algorithm adopted by the IAU (and used by the Astronomical Almanac) is simply not designed to handle more than a few centuries from J2000. Outside of that range, the polynomial expressions diverge fairly rapidly from precise numerical integrations.

The good news is that this paper from 2011 by Vondrák, Capitaine, and Wallace describes a robust algorithm for calculating precession (and obliquity) on millenial time scales.

Proper motion. For time scales of +/- 15,000 years, you will see proper motion in the sky on the order of tens of degrees. This means that simple 2D proper motion across the sky will not be robust. It would be much better to track the 3D motion of the stars. The algorithm is described by Kaplan et al 1989. The algorithm needs data for parallax and radial velocity. The Yale Bright Star catalog has only partial coverage for that data. More precise data is found in GAIA, but GAIA seems to suffer from the defect of not including bright stars with magnitude less than about 3.

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