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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
astro-nav.

George

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