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Astronomy and Astrophysics
How to get state vector in 4-observation method of Gauss
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[QUOTE="Jenab2, post: 6054607, member: 107288"] Okay, after some research I've answered my own question. The reference material that I used is [I]The Determination of Orbits[/I] by A.D. Dubyago, especially chapter six. I used the method presented by Dubyago, which is a form of the method of Gauss, in my program ORBIT4. Actually, I followed Dubyago only until I had the range information. After that, I went my own way with a simpler route of march toward the orbital elements, which I understood better. There is no need to have [I]a priori[/I] range information. The method of Gauss ferrets it out by successive approximations. [URL='https://my.cloudme.com/jenab6/ORBIT4']ORBIT4[/URL] is available for free download from [URL]https://my.cloudme.com/jenab6/ORBIT4[/URL] ORBIT4 is designed to treat elliptical (two-body) orbits around the [B]sun.[/B] It is written in Prime Programming Langauge (PPL) for the HP Prime Calculator or its emulators. The input, in this case for the asteroid [B]1 Ceres,[/B] is in the program (inline code), and looks like this: // Data for time 1 L1:= {2457204.625, 0.155228396, −1.004732775, 0.00003295786, HMS→(20°46′57.02″), HMS→(−27°41′33.9″)}; // Data for time 2 L2:= {2457214.625, 0.319493277, −0.965116604, 0.0000311269, HMS→(20°39′57.10″), HMS→(−28°47′21.5″)}; // Data for time 3 L3:= {2457224.625, 0.4747795623, −0.8983801739, 0.00002841127, HMS→(20°31′22.81″), HMS→(−29°49′22.7″)}; // Data for time 4 L4:= {2457234.625, 0.616702829, −0.8063620175, 0.00002486325, HMS→(20°22′06.57″), HMS→(−30°41′57.3″)}; The first number (after the open curly bracket) is the time of observation in Julian Date. The observations should be separated in time by somewhere between 0.5% to 1.0% of the object's orbital period. The observation interval should be reasonably near the opposition of the asteroid with the Sun, but it should not span an apside of the asteroid's orbit. The 2nd number is the X component of the Earth's position in heliocentric ecliptic coordinates. The 3rd number is the Y component. The 4th number is the Z component. These can be obtained from [URL='https://ssd.jpl.nasa.gov/horizons.cgi']JPL Horizons[/URL]. The 5th number, inside the HMS operand parentheses, is the asteroid's geocentric right ascension in HH°MM'SS.SS" format. Right ascension should be accurate to 0.01 seconds of time. The 6th number, also inside HMS operand parantheses, is the asteroid's geocentric declination in degrees, arcminutes, arcseconds format. Declination should be accurate to 0.1 arcsec. If observational data aren't available, then test data for the RA & DEC of known asteroids or planets can be obtained from [URL='https://ssd.jpl.nasa.gov/horizons.cgi']JPL Horizons[/URL]. The output for the data (shown in the PPL code above) follows: ORBIT4 by David Sims Method of Gauss with four observed positions to find the Keplerian orbital elements. User provides input by adjusting inline data. r₁ 2.75 (initial guess) r₄ 2.75 (initial guess) Successive approximations r₁ 2.90652064 r₄ 2.92071388 r₁ 2.93008666 r₄ 2.94292188 r₁ 2.93307059 r₄ 2.94572742 r₁ 2.93344165 r₄ 2.94607627 r₁ 2.93348769 r₄ 2.94611956 r₁ 2.9334934 r₄ 2.94612492 r₁ 2.93349411 r₄ 2.94612559 r₁ 2.9334942 r₄ 2.94612567 r₁ 2.93349421 r₄ 2.94612568 r₁ 2.93349421 r₄ 2.94612568 Heliocentric distances in AU at t₁ & t₄ r₁ 2.93349421 r₄ 2.94612568 Geocentric distances in AU at t₁ & t₄ ρ₁ 2.00460681 ρ₄ 1.94781669 HEC positions in AU at t₁ & t₄ x₁ 1.33687069 y₁ −2.2463475 z₁ −1.33119794 x₄ 1.58994315 y₄ −2.10289848 z₄ −1.31512559 Aberration corrections to time Aρ₁ 0.011577643 days Aρ₄ 0.011249651 days Epoch of state vector & obliquity t₀ 2457219.61 JD ε₀ 0.409057547 radians HEC state vector x₀ 1.46520344 AU y₀ −2.52458426 AU z₀ −0.349479243 AU Vx₀ 14610.4367 m/s Vy₀ 7967.42879 m/s Vz₀ −2442.63758 m/s Heliocentric distance r 2.93980995 AU Sun−relative speed v 16819.9661 m/s True anomaly 147.669798° Ecc. anomaly 145.259666° Mean anomaly 142.77737° Period of orbit 1681.12408 days Orbital elements a 2.76694735 AU e 0.076026341 i 10.5918141° Ω 80.3183813° ω 72.6265868° T 2456552.87 JD T+P 2458234 JD For comparison, here is the official NASA/JPL orbit for Ceres: a 2.768008676 AU e 0.075773357 i 10.59221734° Ω 80.32683297° ω 72.66267214° T 2456552.644 JD P 1682.091 days The [URL='https://jenab6.livejournal.com/77958.html']source code[/URL] can be found on my LiveJournal website. [/QUOTE]
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How to get state vector in 4-observation method of Gauss
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