What exactly does it take to figure out the orbital parameters or planets and what's the Math behind it? Can it be done using only a telescope or is it necessary to send a probe?
Planetary orbits were known to such precision PRIOR to the space age that even 100 years ago everyone knew that Mars precessed in a way that didn't fit the math of Newton's Laws and then Einstein figured it out with General Relativity which gives greater precision (and MUCH greater for faster-moving objects and very high gravitational fields).
Actually, a good place to start would be the Ptolemaic model and the Copernican model (circular model vs Kepler's elliptical model). Making an incorrect model accurately predict future positions of the planets using deferents and epicycles was quite an accomplishment! That's using naked eye observations and math. Even Kepler's model (using elliptical orbits) was created using a large database of naked eye observations and then creating mathematical formulas that would recreate those observations. HOWEVER.... Kepler never did really determine all of the planets' orbital parameters. He determined relationships. For example, Kepler never knew how far away any of the planets (including Earth) were from the Sun. He only knew their distance in Astronomical Units (in other words, how many times further away from the Sun a planet was compared to Earth's distance). It didn't take probes to determine that, though. It took the development of telescopes and for Venus to pass in between the Earth and the Sun. Because Venus's orbital plane is slightly different than Earth's, a pair of transits only occurs every 100+ years (with the transits 8 years apart). So, yes, it can be done just with telescopes and math.
It was the precession of the perihelion of Mercury, not Mars, in its orbit around the sun which was finally calculated using relativity to agree with observations. This was one of the key tests of relativity. http://physics.ucr.edu/~wudka/Physics7/Notes_www/node98.html The method of least squares was developed by Gauss to help him determine the orbit of the first asteroid discovered, Ceres, in 1801. http://en.wikipedia.org/wiki/Ceres_(dwarf_planet) http://en.wikipedia.org/wiki/Carl_Friedrich_Gauss The asteroid had been observed by others, but its track was lost when it went behind the sun. By using the new method, Gauss was able to use the previous observations to predict where Ceres could be found when it came into view again, without interference from the glare of the sun. Gauss' predictions were verified when astronomers found Ceres close to the position where Gauss predicted it would reappear. Another triumph for the 24-year old genius. Most orbits can be determined by telescopic observations of the body in question. For satellites in earth orbit, as few as three observations are required, which can be done with radar instead of a telescope.
Thank you all for your replies. Can you explain or link a resource explaining what measurements are made and how they are made and what calculations are then performed to determine the orbital parameters? Thanks in advance.
Time, elevation, azimuth, and range for each observation. Convert your spherical coordinates to Cartesian. Then transform your Cartesian to geocentric coordinates. From the geocentric coordinates of each observation, you can determine the velocity vector for the middle vector using the Gibbs method - provided each observation is at least 5 degrees apart. It's essentially solving simultaneous equations to find a velocity that would result in all three observations, but it's done using vectors. Converting from spherical to Cartesian is simple. Coordinate transformations are more involved and require knowing how to do matrix multiplication. The Gibbs method is very involved to use and even harder to follow how it was developed. The essence of the Gibbs method is comparing the area encompassed by a chord and the secant line joining the ends of the chord to the difference in the three radii. This solves for the eccentricity of the orbit (which is what the procedure was originally designed for). The remainder of the procedure is to determine the velocity vector (a process easier to understand if you've looked at orbital velocity hodographs) and numerous algebra steps (except with vectors) to isolate what you're solving for. It helps to understand vector algebra (cross products, dot products, etc). All three processes are detailed in Vallado's Fundamentals of Astrodynamics and Applications. There's probably a slew of other books that include the process. Vallado also details just about all of the other methods, including angles only, how to handle situations where separation is less than 5 degrees (initial observations of a newly discovered comet, for example), etc. About the only ones I'd try to include in a short reply would be the spherical to Cartesian conversion, but why bother when you need a real book to do the rest.
No probe necessary. Rather simply stated, Drocta, the basic idea is this;..., If we consider the mass of any planet is somewhat negligible compared to that of the sun, then all planets (asteroids, etc.) have the same ratio of orbital period squared to semi-major axis cubed. IOWs, T^2/R^3 is constant for all solar orbiting bodies, (T =period; R = semi-major axis of an elliptical orbit). That constant value for any solar orbiting object is given by 4(pi)^2 / GM, (where M = mass of sun), it is dependent only upon the value of solar mass. IOWs, that is Kepler's 3rd Law. Thus by measuring earth's period about the sun, and measuring its distance from the sun, you can calculate the solar mass. Then measuring the solar orbital period of any planet and plugging into the equation, you can calculate its semi-major axis. Now I can hear you say: But how can we get the Earth-Sun distance? It can be done (alluded to by Bob) simply by observing (measuring) the greatest elongation of Venus in the sky and using some trig. we can get the distance of Earth-Venus in terms of a fraction of the earth-sun distance...Then by bouncing a radar beam of Venus we can get its absolute distance, and calculate earth-sun distance....( the method described here: http://curious.astro.cornell.edu/question.php?number=400 ) Simplistic way, but workable. Creator
Hipparchus made the first stab at being able to predict the locations of the planets, moon and sun in 250 BCE, using the obsevation data from the Babylonians and Sumerians. Not too shabby a stab, either, because with the exception of some of the do-si-does that happen in retrograde motion, it was within 10 or so degrees. Close enough to get impressed into bronze with the Antikythera Mechanism in 80 BCE. Ptolemy, in 150 AD, closed another loophole, and things were good within a degree, and his theory lasted for 1500 years, until replaced by Kepler, who, by the way, also used eye-balled observations, good to within 1 minute of arc. The only real improvement since then has been Einstein's. The math is essentially spherical trig, perhaps embraced within matrix operations. Nothing that can't be done by hand, at need.