Asteriod Orbits...

"Michael Stogden" wrote in message
...
Hi,

Does anyone have any links to sites or suggestions for books that describe
the method used to calculate orbits from observations?

It's just that I can't see how a series of observations of objects at
particular co-ordinates will enable you to determine what the orbit is - a
set of three observations will have an infinite number of orbits as you
probably cannot accurately determine the distance from the observer.

I suspect that it has something to do with angular velocity and the not
unreasonable assumption that the object is orbiting the sun.


Yes, the assumption is made that the orbit is an ellipse (or a parabola, for
a comet, at least for starters). The other assumption is that the object is
at a reasonable distance. In the Laplace method, the analysis first yields
the distance from the observer to be a root of a fifth order polynomial,
IIRC. Some of the roots are unphysical, such as inside the Earth, or
perhaps complex numbers. There are usually at most one, or two, reasonable
possibilities.

The best orbits come from three well-spaced observations over a few weeks
that yield a good value for the angular velocity vector and position vector
at the middle observation. These can be transformed into the orbital
elements via a lot of algebra and numerical slogging with simultaneous
equations.

Does Duffet-Smith come to mind as an author whose book covers this subject?
Also Danby. I'm sure there are many such. There is a rare book by Herget
that spells out a step by step procedure.

The Gauss method is more generally used these days. The method in effect
uses exact geometry but approximates the dynamics; in the Laplace method,
the dynamics are exact but the geometry is approximated.

--
Mike Dworetsky

(Remove "pants" spamblock to send e-mail)

Wasn't it Michael Stogden who wrote:
Hi,

Does anyone have any links to sites or suggestions for books that describe
the method used to calculate orbits from observations?

It's just that I can't see how a series of observations of objects at
particular co-ordinates will enable you to determine what the orbit is - a
set of three observations will have an infinite number of orbits as you
probably cannot accurately determine the distance from the observer.

I suspect that it has something to do with angular velocity and the not
unreasonable assumption that the object is orbiting the sun.

If we assume that the orbit is an ellipse with the Sun at one focus,
then the shape of the orbit has six degrees of freedom (eccentricity,
inclination, length of major axis, yaw, pitch and roll) and there's a
seventh degree of freedom in that the asteroid can be at different
points along that orbit. Each observation gives us two pieces of
information (Right Ascension and Declination) at a specific time. With
four observations, it's therefore possible to write down eight
simultaneous equations with seven unknowns, which can then be solved.
The equations are complicated by the fact that the observer is also
moving as the Earth moves along its orbit and rotates, so the maths gets
a little bit messy.

Consider the simpler case where there's no gravity, so the asteroid
moves in a straight line with constant velocity, and the observer is
stationary at the point x=0 y=0 z=0.

The parametric equations that describe the asteroid's motion would be
x=a*t+b y=c*t+d z=e*t+f
The observer observes the Altitude and Azimuth of the asteroid at times
t=0, t=1 and t=2
The Alt and Az values are related to the x,y,z position by
tan(Alt) = y/z tan(Az) = x/z
which we switch around to
y = z * tan(Alt) x = z * tan(Az)

Substituting "a*t+b" for "x", etc. we get six simultaneous equations
containing six unknown values which can be solved to give the orbit
parameters a to f.

c*0 + d = (e*0 + f)*tan(Alt0) a*0 + b = (e*0 + f)*tan(Az0)
c*1 + d = (e*1 + f)*tan(Alt1) a*1 + b = (e*1 + f)*tan(Az1)
c*2 + d = (e*2 + f)*tan(Alt2) a*2 + b = (e*2 + f)*tan(Az2)

--
Mike Williams
Gentleman of Leisure

"Mike Williams" wrote in message
...
Wasn't it Michael Stogden who wrote:
Hi,

Does anyone have any links to sites or suggestions for books that
describe
the method used to calculate orbits from observations?

It's just that I can't see how a series of observations of objects at
particular co-ordinates will enable you to determine what the orbit is -
a
set of three observations will have an infinite number of orbits as you
probably cannot accurately determine the distance from the observer.

I suspect that it has something to do with angular velocity and the not
unreasonable assumption that the object is orbiting the sun.

If we assume that the orbit is an ellipse with the Sun at one focus,
then the shape of the orbit has six degrees of freedom (eccentricity,
inclination, length of major axis, yaw, pitch and roll) and there's a
seventh degree of freedom in that the asteroid can be at different
points along that orbit. Each observation gives us two pieces of
information (Right Ascension and Declination) at a specific time. With
four observations, it's therefore possible to write down eight
simultaneous equations with seven unknowns, which can then be solved.
The equations are complicated by the fact that the observer is also
moving as the Earth moves along its orbit and rotates, so the maths gets
a little bit messy.


The orbit problem can be solved (with no room for observational error) using
three, not four, observations. From 3 RAs, 3 Decs, and times, you can
derive the values of (X,Y,Z) and (dXdt,dY/dt,dZ/dt) in ecliptic coordinates
at time t2 (the middle observation). This can then be transformed into
orbital elements, which include the position along the orbit at the time of
middle observation. The last of these can be transformed into
T(perihelion).

An obvious requirement is that you know the position of the Earth, and the
observer's position on it, with high accuracy.

Once you get extra observations, you can refine the orbit by a least-squares
adjustment.

Talk about "fun"--as a student, I had to calculate an asteroid orbit from 3
observations as a homework project, without using a computer. I was allowed
7-place trig tables and an electromechanical calculator. Took about 15
man-hours including all the checks. OK, well you try inverting a 3x3 matrix
by hand...without making a mistake...

--
Mike Dworetsky

(Remove "pants" spamblock to send e-mail)

JRS: In article , seen in
news:uk.sci.astronomy, Mike Dworetsky
posted at Wed, 25 Feb 2004 20:56:48 :-

Does Duffet-Smith come to mind as an author whose book covers this subject?

For a WWW search, Duffett-Smith should be better. The one book by him
that I own does not cover orbit determination.

--
© John Stockton, Surrey, UK. Turnpike v4.00 MIME. ©
Web URL:http://www.merlyn.demon.co.uk/ - FAQqish topics, acronyms & links;
some Astro stuff via astro.htm, gravity0.htm; quotes.htm; pascal.htm; &c, &c.
No Encoding. Quotes before replies. Snip well. Write clearly. Don't Mail News.

In article , Dr John Stockton
writes
s Duffet-Smith come to mind as an author whose book covers this subject?

For a WWW search, Duffett-Smith should be better. The one book by him
that I own does not cover orbit determination.


A description of the method and an implementation in BASIC is included
in Peter Duffett-Smith's 'Astronomy with your Personal Computer' (ISBN
0-521-38995-X) under the routines EFIT and PFIT.

'These routines (with their handling programs) calculate sets of orbital
elements consistent with three or more observations of the position of a
member of the Solar System made at different times. PFIT finds the
parabolic elements; EFIT finds the elliptical elements'.

--
David Entwistle