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    Re: Polaris SHA
    From: Frank Reed
    Date: 2008 Sep 08, 04:03 -0400

    Bill, you wrote:
    "Why does the SHA of Polaris shift so dramatically through the month/year
    (as opposed to other stars?)"
    
    Fundamentally, this is a "coordinate singularity" effect. Polaris isn't
    moving faster. It just happens to be moving fast very close to the North
    Celestial Pole. It's a lot like someone walking past one of the geographic
    poles on Earth at a distance of fifty feet. You can walk through a dozen
    time zones in a few minutes.
    
    The largest change in the coordinates of the stars (and therefore the
    coordinates of Polaris) is due to precession. To picture this, imagine
    holding a nice celestial sphere in your hands. On this star globe, let's
    draw the stars and constellations only --no coordinates. Imagine that this
    celestial globe has the SHA/Dec coordinate grid as a separate transparent
    sphere wrapped around it. The coordinate sphere is separated by a small
    fraction of an inch from the star globe with little rollers holding it up at
    a few strategic points. We can now slide the coordinate sphere about while
    keep the stars fixed (or vice versa). Precession causes that coordinate
    sphere to roll about the star sphere, wobbling like a top. It starts with
    its coordinate pole near Polaris and then traces out a big circle in the sky
    over the course of thousands of years with a radius of 23.5 degrees, and as
    you know it will later pass not too far from the stars Thuban and Vega
    sometime in distant millennia (contrary to common textbook representations,
    it's not a simple circle since the Earth's polar inclination changes on a
    similar time scale). Every time the coordinate pole passes near a star, that
    star's SHA will pass rapidly through almost 180 degrees.
    
    Nutation is very similar to precession since the cause is fundamentally the
    same (with the Moon's gravity replacing the Sun's). In fact, it's fair to
    refer to nutation as a type of precession, or a variation in precession.
    
    The important thing to notice about nutation and precession is that they are
    strictly coordinate effects due to the changing orientation of the Earth.
    The angular distances among the star never change from precession and
    nutation alone. The coordinate sphere is "sliding around" on the star globe
    due to precession and nutation, but the star globe is not changed by them.
    
    Other motions of the stars actually change the stars positions on the star
    globe. Aberration is a temporary effect causes the star's to bunch together
    slightly in the direction towards which the Earth is travelling relative to
    the Sun and spread out slightly in the opposite directon (*see PS). To
    picture this in terms of a star globe, imagine that the surface of the star
    globe is slightly "rubbery". Find the spot on the star globe towards which
    the Earth is moving relative to the Sun (rather close to 90 degrees away
    from the Sun's position in the sky on that date). Now squeeze or drag the
    surface of the globe towards that direction. Notice that the entire sky is
    affected. Numerically the effect is largest along a great circle in the
    direction perpendicular to the Earth's velocity vector and amounts to about
    20 arcseconds (40 arcseconds back and forth over the course of a year).
    That's big enough to affect accurate sextant observations. If you find a
    sextant or navigation manual that tells you that you don't have to worry
    about annual aberration when observing angles between stars, it's not
    correct (at least not at the level of +/-0.3 minutes of arc). When you find
    tables of star-to-star angular distances that do not vary over the course of
    a year, something is missing.
    
    Another "back and forth" annual motion is due to the annual parallax of the
    nearby stars. This is smaller than 1 arcsecond for all known stars. Since
    it's an annual effect like aberration, it was extremely difficult in the
    18th and 19th centuries for astronomers to detect it and separate it from
    aberration. There were many false positives in the early detections of
    stellar parallax. Remarkably, in the 21st century, backyard astronomers with
    high-end digital camera/telescope combinations now routinely detect the
    annual parallax of the nearest stars (here's an example:
    http://www.richweb.f9.co.uk/astro/nearby_stars.htm). That's SUB-arcsecond
    accuracy with non-professional, albeit expensive, equipment! Note that
    navigators never have to worry about annual stellar parallax. Also notice
    that annual stellar parallax differs from annual stellar aberration in one
    important respect: parallax affects each star separately in inverse
    proportion to its distance from the Earth while aberration is a distortion
    of the entire celestial sphere.
    
    Finally, star positions are affected by proper motion. The stars are all
    moving about on different paths through the Galaxy, some fast, some slower.
    The star Arcturus is moving at an unusually high speed, and its position
    relative to the other stars changes rapidly enough that an observer with a
    sextant could see it in just a few years. Backyard astronomers (with
    high-end equipment) can detect proper motions in hundreds of stars with a
    year's worth of observations. Over the long-term, proper motion has the
    biggest effect on the sky we see. Stars nearer than a hundred lightyears or
    so (roughly half of the bright stars) will have moved noticeably over a few
    thousand years. Notice that stars with high annual parallax tend to have
    high apparent proper motions.
    
    By the way, checking for these various effects is a good test of navigation
    software. Much software forgets to include proper motion. The authors
    sometimes sell it on the basis that you "will never need to buy another
    almanac". True perhaps, but you may need to buy an upgrade to the software
    after ten years. I guess this is true in any case since very little software
    of any type is intended to run for much longer than ten years. Other
    navigational software products use a low-grade formula for precession. This
    is a case where that fast SHA change for Polaris comes in handy.
    
    So finally, this raises a question: what are the correct SHA and Dec for
    Polaris TODAY (September 8, 2008, 0800 UT to be specific) to the nearest
    tenth of a minute of arc for the sake of navigational calculations? What do
    various software products give for these values?
    
     -FER
    PS: Regarding aberration, if the Earth were moving much faster relative to
    the Sun, at a large fraction of the speed of light (!), the effect would be
    visually dramatic. The constellations would all appear pressed together
    directly ahead of us (towards the direction the Earth is moving), and the
    sky behind would be sparsely filled with stars.
    PPS: So why aren't the constellations compressed together in the direction
    towards which the Sun is moving? They are, but it's undetectable. Since the
    Sun's velocity isn't changing, this is a "fixed" aberration of the sky.
    
    
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