On being an Orbital Mechanic

I spent yesterday implementing a digital orrey in my WebGL environment. (New furniture is nice) Just the major planetary barycenters, so far. All the numbers and equations come from this JPL document, and frankly I think I spent more time typing and rechecking all the numbers than actually coding the algorithm.

(Yes, I tried copying and pasting the table at the end, in various PDF readers, and invariably got a screen full of Zapf Dingbats.)

Then there was a good half-hour of staring at random messy lines, while I figured out that all the equations assumes angles in degrees rather than radians. Then I noticed that I had the whole solar system upside-down.

You know why Orbital Mechanics has a reputation for being hard? Because it's deliberately made so. Even professional astronomers can't remember the terms or actually do the math. They just use the on-line calculators like HORIZON, and have to look up what "argument of perihelion" means, just like everyone else.

There's a philosophical question hidden in there: when a Jargon becomes so dense and archaic that modern experts in the field don't understand it, what use does it have?

The other reason is the complete lack of a sensible reference frame. Orbital mechanics specified in "Heliocentric" co-ordinates don't actually have the Sun at the origin. (There are side-tables of where the sun is actually located) "Equatorial" co-ordinates are based on the Earth's position, (literally extending the plane of the equator out into space) which sounds easy, until you hear the word "precession", let alone "nutation", or "three body problem".

Earth is probably the worst place to define as a reference point, because it's locked in a three-way chaotic dance with the moon and sun. Yet pretty much all intra-solar-system co-ordinates are based on the "longitude of vernal eqinox" and the "ecliptic plane" from Earth. Both of which are constantly shifting.

OK, it makes the math simple and convenient for Earth-based astronomy. If you're doing it with pencil and paper. Which no-one does anymore. You can tell because all the math is presented as trigonometry, not vector/matrix forms. What's the difference? Quadrants. Inverse trig functions lead to quadrant ambiguity. Which a human must sanity-check along the way with their pencil.

Now I'm on to the task of plotting comets and asteroids, which is fun because their orbits are specified in a different format. Yup, different classes of objects get different co-ordinate formats. Why? No good reason.

Perhaps my biggest surprise in all of this is that Javascript seems quite capable of handing the precision required to crunch the numbers. I was all set to use the 'big.js" arbitrary precision library, (and still should implement that in parallel as a numerical check) but I'm not seeing any numeric instability yet. Granted, if the precision of the keplerian elements goes up another few decimal places then Javascript clearly won't have the mantissa to encode it, but for the moment it matches or exceeds the precision of the data.

Since I like to post code, here's my table of keplerian orbital elements. I've double-checked it, but a triple-check probably wouldn't go astray before you plan a space mission. And they'll slightly improve the numbers in a year or so, so they're always provisional.


var planets_ESAA2006_6K = {
source: 'http://iau-comm4.jpl.nasa.gov/keplerformulae/kepform.pdf',
mercury:{ 
a:['0.38709843','0.000000'], e:['0.20563661','0.00002123'], I:['7.00559432','-0.00590158'], 
L:['252.25166724','149472.67486623'], w:['77.45771895','0.15940013'], O:['48.33961819','-0.12214182'] },
venus:  { 
a:['0.72332102','-0.00000026'], e:['0.00676399','-0.00005107'], I:['3.39777545','0.0043494'], 
L:['181.97970850','58517.81560260'], w:['131.76755713','0.05679648'], O:['76.67261496','-0.27274174'] },
earthmoon:  { 
a:['1.00000018','-0.00000003'], e:['0.01673163','-0.00003661'], I:['-0.00054346','-0.01337178'], 
L:['100.46691572','35999.37306329'], w:['102.93005885','0.31795260'], O:['-5.11260389','-0.24123856'] },
mars:   { 
a:['1.52371243','0.00000097'], e:['0.09336511','0.00009149'], I:['1.85181869','-0.00724757'], 
L:['-4.56813164','19140.29934243'], w:['-23.91744784','0.45223625'], O:['49.71320984','-0.26852431'] },
jupiter:{ 
a:['5.20248019','0.00002864'], e:['0.04853590','0.00018026'], I:['1.29861416','-0.00322699'], 
L:['34.22479152','3034.90371757'], w:['14.27495244','0.18199196'], O:['100.29282654','0.13024619'],
b:'-0.00012452', c:'0.06064060', s:'-0.35635438', f:'38.35125000' },
saturn: { 
a:['9.54149883','-0.00003065'], e:['0.05550825','-0.00032044'], I:['2.49424102','0.00451969'], 
L:['50.07571329','1222.11494724'], w:['92.86136063','0.54179478'], O:['113.63998702','-0.25015002'],
b:'0.00025899', c:'-0.13434469', s:'0.87320147', f:'38.35125000' },
uranus: { 
a:['19.18797948','-0.00020455'], e:['0.04685740','-0.0001550'], I:['0.77298127','-0.000180155'], 
L:['314.20276625','428.49512595'], w:['172.43404441','0.09266985'], O:['73.96250215','0.05739699'],
b:'0.00058331', c:'-0.97731848', s:'0.17689245', f:'7.67025000' },
neptune:{ 
a:['30.06952752','0.00006447'], e:['0.00895439','0.00000818'], I:['1.77005520','0.00022400'], 
L:['304.22289287','218.46515314'], w:['46.68158724','0.01009938'], O:['131.78635853','-0.00606302'],
b:'-0.0041348', c:'0.68346318', s:'-0.10162547', f:'7.67025000' },
pluto:  { 
a:['39.48686035','0.00449751'], e:['0.24885238','0.00006106'], I:['17.14104260','0.00000501'], 
L:['238.96535011','145.18042903'], w:['224.09702598','-0.00968827'], O:['110.30167986','-0.00809981'],
b:'-0.01262724', c:'0', s:'0', f:'0' },
};