# NavList:

## A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding

**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 astro-nav. George ================================================================ 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. ================================================================