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    Re: Tycho Brahe Mars oppositions
    From: George Huxtable
    Date: 2004 Nov 30, 19:18 +0000

    Michael Dorl offered his own calculations of when the Right-Ascensions of
    Sun and Mars differed by 180 degrees, about 400 years ago.
    In another message, he asked me about the table of Mars oppositions, of the
    same era, that I had quoted extracts from, in Meeus' "Astronomical Tables"-
    "Do you now how these times were calculated or how opposition is defined here?"
    Meeus defines his values as "the instant when the true heliocentric
    longitudes of the Earth and Mars, referred to the mean equinox of the date,
    are equal."
    I suggest that Michael, on one hand, and Meeus and Tycho, on the other
    hand, are referring to rather different quantities.
    Michael refers to RA (Right-Ascension) which is a measure of the positions
    of Sun, and Mars, projected on to the plane of the Earth's equator. It
    depends on the direction of the spin-axis of the Earth, in space. And the
    moment when two bodies have RAs 180 degrees apart, would depend on that
    tilt. But that isn't very appropriate when we are considering the dynamics
    of bodies that are revolving about the Sun, as Tycho was. The Earth's spin
    has nothing to do with that question, except to confuse the observations.
    All planets revolve around the Sun in the same direction, and (nearly) in
    the same plane, tilting no more than a few degrees. Tycho (and mainly
    Kepler) were trying to "stand back" from their observing point on a
    spinning Earth, and instead to uncover the planetary motions as they would
    seen from outside by a distant observer.
    A set of coordinates was needed for the Solar system, in which the planets
    rotate around the Sun. A plane had to be defined, to measure up and down
    from, and measure round in, and it was sensible to choose the plane of one
    of those planets as the reference plane. Living on the Earth, the plane of
    the Earth's orbit around the Sun was the natural choice. (We refer to this
    as "the plane of the ecliptic", because eclipses can occur when the Moon
    crosses it.) Angles measured up or down from that plane were named
    "ecliptic latitude", by analogy with our latitudes on Earth. Similarly,
    angles measured Eastwards around that plane were called "ecliptic
    longitudes". Those terms have caused much confusion ever since. Finally, a
    marker on that plane was needed, to measure longitudes from, similar to our
    use of Greenwich on Earth. It was convenient in many ways to take one of
    the two crossing-points of the planes of the Earth's equator and the
    ecliptic (the direction of the Sun in Spring), in that it allowed for easy
    conversion between the Earth-based equatorial system and the
    solar-system-based ecliptic system, because their zero was at the same
    point. However, it has the severe disadvanage that it shifts, slowly, as
    the Eath's spin-axis precesses.
    Not that Tycho and Kepler had to invent that system of ecliptic
    coordinates: that had been done by Greek astronomers two millennia earlier.
    All that stuff was no more than an explanation of why opposition of a
    planet (i.e. the moment when it lies in the opposite direction, 180 degrees
    away the Sun) needs to be defined in terms of ecliptic coordinates, not
    equatorial coordinates..
    Meeus puts it a slightly different way, referring to the "true heliocentric
    longitudes" of Earth and Mars, being equal. An astronomer referring to
    longitudes is always speaking in ecliptic coordinates. Meeus' use of
    "heliocentric" is because he is looking at the two planets from the point
    of view of an observer on the Sun. When Mars is in opposition to the Sun,
    then Mars, Earth, and Sun are in a straight line (or as nearly so as the
    2-degree tilt of Mars' orbit allows). In which case, seen from the Sun, the
    ecliptic longitudes of Earth and Mars will be equal.
    It's interesting that such oppositions, which happen roughly every two
    years 50 days, are the times when an observer on Mars might get a chance to
    see a "transit of Earth", across the face of the Sun, similar to our own
    occasional chance to see a transit of Venus. Not at every opposition: it
    depends on Mars' ecliptic latitude, that is, how near it is to the point
    where it's slightly tilted orbit crosses the ecliptic.
    When the position of a planet such as Mars is initially predicted, it's
    calculated, first, in terms of its own orbital plane around the Sun's
    centre (or, more precisely, about the centre of gravity of the solar
    system, which is the same thing, near as dammit). Then those positions are
    recalculated in terms of the Earth's ecliptic plane. Then the Earth's
    position, in the ecliptic, is calculated for that same instant. The Earth's
    ecliptic latitude (or the Sun's, from Earth) is, by definition, zero,
    except for some tiny perturbations. The distances from the Sun, for Earth
    and Mars, must be predicted, as well as the ecliptic angles.
    Now we have two vectors, in the same coordinate system, centred at the Sun,
    and it's only a matter of geometry to subtract them, to give the direction
    of Mars from Earth, in the same ecliptic coordinates. That gets put into an
    astronomical almanac.
    Then, taking account of the Earth's tilt, of 23 and-a-bit degrees, the
    angles are converted from ecliptic to equatorial coordinates,
    Right-Ascension and Declination, of Mars. Finally, knowing the Sidereal
    time, the GHA of Aries is obtained, and GHA Mars follows, which, with the
    declination, goes into the Nautical Almanac.
    The most modern computations differ in that they use XYZ Cartesian
    coordinates, and make numerical integrations along the paths of all
    solar-system bodies, as their various distances and attractions change, but
    fundamentally they are doing the same thing.
    What Michael Dorl has computed is the Right-Ascension and Declination of
    Mars, and he needs to take a step back to obtain the ecliptic coordinates,
    if the program he uses makes that possible. Otherwise, he needs to convert
    back again, from equatorial to ecliptic, which is quite a simple process.
    He can use (from Meeus, "Algorithms", equations 13.1 and 13.2)
    Tan ecliptic_long = (sin RA cos tilt + tan dec sin tilt) / cos RA
    The ecliptic long, derived from its tangent needs to be put into the same
    quadrant that the RA was in.
    Sin ecliptic_lat = sin RA cos tilt - cos dec sin tilt sin RA
    Where tilt is the obliquity of the ecliptic, being, to sufficient accuracy,
    23.439 degrees, + .00013 degrees for each year before the year 2000. For
    the year 1600, I make it to be 23.491 degrees.
    Kepler, using the Tycho observations, was obtaining the interval between
    Mars oppositions (2 years 50 days). From that interval, and knowing the
    Earth's period of revolution around the Sun (1 year), he could deduce the
    duration of a Martian year. This, with information from other planets, led
    to the Kepler law that stated that the revolution period, squared, was
    proportional to the size of the orbit, cubed.
    I hope that I've got all that astronomical stuff right. I'm not much of an
    astronomer, having picked up all I know as a by-product of investigating
    contact George Huxtable by email at george---.u-net.com, by phone at
    01865 820222 (from outside UK, +44 1865 820222), or by mail at 1 Sandy
    Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK.

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