Determining orbits from R.A., Dec, & time

[bohr]

Hi all,

First post so here goes:

Anybody know of a reference(s) for determining the orbital equations of a celestial object from it's right acsension, declination, and time? I would like to write out some equations for a few asteroids I imaged with my telescope. Thanks.

 

[enigma]

Hi bohr,

Welcome to the forums!

If you don't know the distance to the asteroid, you'll need to take several different observations at different times. The calculations are far too intense to be done by hand (you need to root solve for an 8th order polynomial and do tons of matrix calculations), and the accuracy is not very good even if you've got very accurate data.

I can talk you through the steps if you want. How good are you at linear algebra and do you have access to math software like MATLAB (or similar)?

 

[chroot]

The Minor Planet Center can provide the orbital elements for the asteroids.

- Warren

 

[bohr]

Hi,

Thanks for the quick replies.

enigma: I am experienced in Mathematica and have the math background. What I am looking for is preferably a source book to go from i.e., a celestial mechanics text, but none have any that deal with multiple observations and r.a. and dec coordinates. When you say 8th order polynomials where did you learn that from. I have also read a book published by the US air force for determining orbits, but they used ground based radar to determine their orbits. Again, no r.a. or dec.

chroot: I know that the minor planet center has this info, but it would be a fun experiment to try to solve these on my own.

Thanks for the replies.

 

[enigma]

The book I have been using for my orbital dynamics and space nav classes is:

Fundamentals of Astrodynamics and Applications by David Vallado.

The method you're looking for is called the Angles-Only Gauss method. It's section 7.3 in Vallado.

If you do go find the book (second edition, anyway), note that there is a mistake in the algorithm listed for the method (computing the f and g functions in the last step doesn't work, so you can disregard and solve for the orbital elements from the obtained position and velocity vectors using a different algorithm). Also, the RA & Dec are topocentric, not geocentric (but I don't think that will make much of a difference for asteroid positions).

The major thing is that the results return the position and velocity in an Earth inertial coordinate system, so you'll need to find the position and velocity of the Earth relative to the Sun at your observation time to find the heliocentric position and velocity of the asteroid to convert to the orbital elements.

 

[bohr]

You rock! Off to the library in the morning. I'll note the mistake too and I'll post my results here when I get them. I'll need somebody to check my work. Thanks!

 

[enigma]

No problem.

If you do find either a formula to find the exact position of the Earth relative to the Sun, or a position and exact time let me know. Seems like it'd be a good thing to have stored somewhere.

I guess you could do a first cut approximation and just say you're at (-1,0,0)AU at the vernal equinox, but that's just assuming perfectly circular, zero inclination, etc. etc.

 

[Jenab]

The book I have been using for my orbital dynamics and space nav classes is:

Fundamentals of Astrodynamics and Applications by David Vallado.

The method you're looking for is called the Angles-Only Gauss method. It's section 7.3 in Vallado.

And in Chapter 5 of The Determination of Orbits by A.D. Dubyago.

These are preliminary orbits that you get from the Method of Gauss on three angle-only observations. To improve the orbit, you use a generalized Method of Gauss on a Taylor series having coefficients of the partial derivatives of each orbital element with respect to right ascension and again to declination. Into the resulting system of equations you stick in more observations of the object in RA and DEC. The idea is to converge on successively better values of the orbital elements.

Jerry Abbott

 

[BobG]

Also, the RA & Dec are topocentric, not geocentric (but I don't think that will make much of a difference for asteroid positions).

The major thing is that the results return the position and velocity in an Earth inertial coordinate system, so you'll need to find the position and velocity of the Earth relative to the Sun at your observation time to find the heliocentric position and velocity of the asteroid to convert to the orbital elements.

I think that comment needs a little clarification. If you're using right ascension and declination from a catalog, it's in geocentric spherical coordinates. If you're talking about the observations you make with your telescope, it's topocentric spherical (but then, technically, you're using azimuth - elevation, not right ascension - declination).

The "Astronomical Almanac" will have the Earth-Sun position data you need. It's an annual publication that costs around $55 or so (more for non-US distribution). Unfortunately, the almanac for 2004 is already out of print, since just about anyone intereseted in it probably bought it last summer or fall. You can still find the data you need (including orbital elements for asteroids) from the US Naval Observatory's web page under "Data Services": http://aa.usno.navy.mil/

 

[Jenab]

I think that comment needs a little clarification. If you're using right ascension and declination from a catalog, it's in geocentric spherical coordinates. If you're talking about the observations you make with your telescope, it's topocentric spherical (but then, technically, you're using azimuth - elevation, not right ascension - declination).

The "Astronomical Almanac" will have the Earth-Sun position data you need. It's an annual publication that costs around $55 or so (more for non-US distribution). Unfortunately, the almanac for 2004 is already out of print, since just about anyone intereseted in it probably bought it last summer or fall. You can still find the data you need (including orbital elements for asteroids) from the US Naval Observatory's web page under "Data Services": http://aa.usno.navy.mil/

I forgot the part where you have to correct your own position for being on Earth's surface.

r(you relative to sun) = r(EMB relative to sun) + R(Earth's center relative to EMB) + R(you relative to Earth's center)

 

[Jenab]

Orbit determination by 3 observations with Gauss method

Hi all,

First post so here goes:

Anybody know of a reference(s) for determining the orbital equations of a celestial object from it's right acsension, declination, and time? I would like to write out some equations for a few asteroids I imaged with my telescope. Thanks.

I got a copy of THE DETERMINATION OF ORBITS by Dubyago and studied up on how to do this. I put the procedure in greatly compressed form in a new thread:

https://www.physicsforums.com/showthread.php?t=36657

 

[remcook]

so you'll need to find the position and velocity of the Earth relative to the Sun at your observation time to find the heliocentric position and velocity of the asteroid to convert to the orbital elements.

for position...

http://www.gb.nrao.edu/~rfisher/Ephemerides/ephem_descr.html#get
http://pan-starrs.ifa.hawaii.edu/project/people/kaiser/imcat_doxygen/jpleph_8cpp-source.html

not sure if it all works (if you get all the info), but you can try.

haven't tried these sources. DE200 is on the university computer here.

 

[remcook]

http://ssd.jpl.nasa.gov/horizons.html