Note: this is a work in progress! To do: provide version for your average non-astronomer, vs. a version for astronomers.
Information about the orbits, observations, etc. of various solar systems are available on this site, and some others, in the form of "pseudo-MPECs". The name is a tip of the hat to the MPECs (Minor Planet Electronic Circulars) distributed by the Minor Planet Center, the official clearinghouse for astrometric data. Pseudo-MPECs look approximately like the MPC's MPECs, but are generated using Find_Orb (software to determine the orbit of an object from observations).
The MPC's MPECs give data for asteroids, comets, and certain natural satellites. Pseudo-MPECs on this site and others also give data for these; they also cover artificial satellites and some planets.
Each pseudo-MPEC lists the astrometry (the raw observational data used to compute an orbit), the observing stations (which observatories gathered the data, and who did the observing, who analyzed the observations, and what equipment they used), orbital elements (a description of the shape and size and orientation of the orbit), residuals (how much the observations differ from the "computed" data), and, almost always, ephemerides (a table showing where the object is expected to be over a certain time span, or sometimes where it was before the observations were made).
The astrometry section gives a lot of information in a very compact, extremely cryptic form, as specified by the Minor Planet Center (MPC). Each line describes a particular measurement: it essentially says, "At this particular instant, we saw the object at this position in the sky, from this particular point on the earth." It may also tell you how bright the object was.
You can click here for the MPC's documentation of the astrometric format. Here's a quick overview, with an example :
Object name nnYYYY MM DD.ddddd Right Ascen Declination Mag BC Ref Cod ZC876D8 C2018 02 10.37451 09 24 00.32 +30 15 58.8 22.6 GUNEOCPI52
Each line is exactly 80 characters long, in order to make sure the data will fit on a standard IBM punched card, which gives you some idea of how long this format has been around. It also explains some of the cryptic nature of this format; back in the day, every byte was precious, and the 80-character limit was a hard one to break. Each line tells you the information about the object that was extracted from one digital image. (Or, for observations from about 1990 and earlier, from one old-fashioned photographic image.)
I'll hit the high points here; for details, see the aforementioned MPC documentation of the format.
Scanning across, you can see the name of the object, often in a cryptic packed format. A name such as 1999 XF11, for example, becomes J99X11F. Then the year, month, and day of the observation are given; the exact time is given in decimal days, rather than in hours, minutes, and seconds. The date/time is specified in Universal Time.
Next, the right ascension (RA) and declination are given; these are essentially the celestial equivalent of latitude and longitude, a way to specify the location of any point on the sky.
Often, but not always, the observer will measure and report the magnitude (brightness) of the object. In the above case, the object was at magnitude 22.6.
The last three columns (I52, in the above case) specify the observatory where the measurement was made. The MPC maintains a list of observatories; looking through the MPC list, you'll see that I52 is the Steward Observatory, Mt. Lemmon Station, plus some numbers which suitable software can convert to the exact position of that telescope. In rare cases, the three-character code may come from a file of observatories lacking "official" MPC codes.
In this section, information is given about each observatory. For each observatory, we start with its three-character observatory code, followed by the name of the observatory and its latitude/longitude. The observatory name is usually linked to its Web site; the latitude/longitude, to a map centered on that point. The names of the observers are given, and sometimes those of whoever measured the positions from the images. (Often, these are the same people, and only the observers are mentioned.) Finally, some data are given as to the telescope and camera that made the measurement.
This section packs in a lot of information about the trajectory of the object. I'll refer you to the Wikipedia article on orbital elements for a good explanation and diagrams of what the quantities mean. The section on Keplerian elements should be particularly helpful. I've also documented a few parameters below.
The format of the elements can vary a bit, but starts with lines such as
Orbital elements: 2007 LU19 Perihelion 2018 Jun 1.021489 +/- 18.4 TT = 0:30:56 (JD 2458270.521489)
identifying the object and telling you when it reaches perihelion (its closest point to the sun). If the object is actually orbiting the Earth, it'll say "perigee"; if it's orbiting Neptune, "perineptune", and so on.
You'll also see (usually) a plus-or-minus for the date/time. In the above case, it's plus-or-minus 18.4 days, meaning the object will probably arrive at perihelion between mid-May and mid-June, but might be a little outside that range. (If an object has been observed with good accuracy over a longer time span, that uncertainty may be mere seconds. For some objects observed only briefly and that are far away, it can be years. A similar wide range of uncertainties will be seen with the other quantities. Generally, as we get more observations, those uncertainties get small pretty quickly.)
The date is given first as 1.021489 (i.e., a little after the first of the month), but the ".021489" part is also shown in the more conventional hours/minutes/seconds form 0:30:56, just for convenience. You also get the time of perihelion in JD (Julian Day) form, which is easier for computer processing.
• MOID = Minimum Orbit Intersection Distance, the minimum distance between the orbit of the object and (usually) the Earth, in AU. This is only shown if it's less than one AU. Note that this is the minimum possible distance, assuming no perturbations. You can have orbits that intersect each other, but that doesn't necessarily guarantee that the objects will be at that intersection at the same time. You may also see MOIDs for other planets if they're small enough to be worth mentioning (usually under 0.1 AU).
• M = Mean anomaly, in degrees.
• n = mean motion, in degrees per day. If, for example, n=0.25, it'll take the object 360/0.25 days, or 1440 days, or a bit under four years, to complete an orbit.
• AMR = area/mass ratio. Artificial satellites are often small enough and light enough that they are "pushed" outward from the sun by solar radiation pressure. This offsets some fraction of the sun's gravity; for dust particles, it can actually cancel out the sun's gravity and the particles are repelled by the sun instead of being attracted to it. The magnitude of this effect is determined (mostly) by the ratio of the object's surface area to its mass. Hence the term "area/mass ratio", and the units of square meters per kilogram. (On rare occasions, it's possible to measure the area/mass ratio for asteroids. But almost always, the AMR is too low to be determined in such cases; it's effectively zero.)
• Peri = argument of perihelion (or perigee or perijove, etc.), in degrees. Often written with a lowercase omega ω.
• Node = longitude of the ascending node, in degrees. Often written with an uppercase Omega Ω.
• Incl. = inclination, the angle between the plane of the orbit of the object and the plane of the orbit of the earth. (At least, for heliocentric orbits. For geocentric orbits, it's the angle between the plane of the orbit of the object and the plane of the earth's equator.) Often written with a lowercase i or (rarely) a lowercase iota ι.
• e = eccentricity. This is zero for a circular orbit and one for a parabolic orbit. Heliocentric orbits almost always have e<1 or only slightly greater. (Orbits relative to the planets can have very high eccentricities.)
• a = semimajor axis. This is usually in AU for heliocentric orbits and in kilometers for geocentric ones, and is the average of q and Q.
• q = perihelion (or perigee or...) distance. This is usually in AU for heliocentric orbits and in kilometers for geocentric ones.
• Q = aphelion (or apogee or...) distance. This is usually in AU for heliocentric orbits and in kilometers for geocentric ones.
• H = absolute magnitude. This tells you the brightness of the object if it were one AU from the sun, and you were standing on the sun. (Which is not likely to happen and isn't useful in itself; but it's easy to use it to figure out how bright the object would be at other distances as seen from other places, such as the Earth.) You can also use absolute magnitude to estimate an object's diameter, at least approximately. (You can set a hard lower limit as to the object's size, but the real size depends on how shiny the object is.)
• G = slope parameter. This is used in computing the object's magnitude, and describes how quickly the object's brightness drops off as the illumination angle changes. Find_Orb usually just assumes G=0.15 at present, which is (usually) pretty close.
• U = uncertainty parameter. This is a quantity favored by the Minor Planet Center for describing the overall level of precision of orbital elements, as described here. In their use, 0 ≤ U ≤ 9, with 0=extremely well determined orbits and 9=only slightly determined orbits. Find_Orb extends the range to include negative U (happens if an object has been really well-observed for many years) and U values greater than 9, in cases where an object's orbit is extremely uncertain. You can click here for some mathematical details about this (search within the linked file for "The U value is calculated".)
Be advised that summarizing the "quality" of an orbit in a single parameter is not really possible. In general, a lower U parameter is better; but you can have a low U parameter and be utterly unable to find an object, or a high one and have its orbit actually be well determined, at least for current dates. In summary, I don't recommend paying much attention to it.
Given three observations, you can compute an orbit that fits those observations perfectly: positions computed from that orbit will be exactly equal to the measured positions. Given four or more observations, you can't fit them all to the orbit. (It's mathematically similar to the way you can draw a circle through any three points exactly, no matter where they are; but you can't draw a circle through any four (or more) points... though you may find a circle, or an orbit, that comes close, with only small differences.)
The differences between the observed positions and those computed using the best-fitting orbit are "residuals" or "residual errors".
For each observation listed in the pseudo-MPEC, residuals are shown in this section. The idea is that you've measured the coordinates of the object in the image and gotten a position on two axes, right ascension and declination (the celestial equivalent of latitude and longitude). The "residuals" are two numbers showing the difference between the observed right ascension in that image and the one computed from the orbit, and then the same thing for the declination.
With current technology (measurements made with a decent telescope and digital (CCD) camera), those differences will usually be under an arcsecond, and the numbers will be between -1 and +1. Some observers have better gear and technique and the numbers will be more closely clustered around zero. It was quite difficult to measure positions that accurately back in the days when they were measured from photographic plates, and values from -3 to +3 were fairly common. (Along with the occasional outright blunder, where perhaps the wrong position or time was entered and the errors are huge.)
Some of the residuals will be large enough that they don't seem particularly realistic, and it's safe to assume that the observation is bad. Such residuals are shown enclosed in parentheses () to indicate that the observation was rejected, and not used in computing the orbit.
The residuals are almost always shown in arcseconds. If they're really big and the number wouldn't fit in the space allocated for them, they may be shown in arcminutes with a ' added, or in degrees and followed by the letter d.
With a few exceptions, a pseudo-MPEC will show an ephemeris giving data as to where the object will be and how bright it will be over a particular time span. You usually see where it would be as seen from the center of the earth, though you may instead get where it would be as seen from a particular point on the earth, or as seen from the moon or, on rare occasions, from the surface of another planet.