Goal: given the location of asteroid in the sky (in terms of right ascension and declination), the date for which the location is known, and the proper motion of the asteroid, this procedure will determine the radius of the asteroid's orbit (under the assumptions operating below).
 

Assumptions:

1) the orbit of both earth and the asteroid are circular
2) the orbit of the asteroid is coplanar with the orbit of the earth
3) direction of the orbit revolution of the asteroid is the same as that of earth
4) the asteroid orbit is larger than that of the earth (a_a > a_e)
5) the sun is infinitely massive (this simplifies Kepler's 3rd law)
6) the asteroid is at least 90 degrees in direction away from the sun as seen from
    the earth, i.e.,  | _s - _a | >  90   
    (this assumption may be retracted in the next version)

definitions:

s, e, a     subscript symbols for sun, earth, asteroid

    celestial longitude (angle measured eastward along the ecliptic from the vernal
        equinox)

    absolute value of the angle between directions to sun and asteroid, as seen
        from earth; by definition  =  | _s - _a |

    angle between directions to earth and to sun, as seen from the asteroid

    angle (as seen from the asteroid) between direction to earth and direction of the
      asteroid's orbital motion; by definition,   = 90 -                                      (1)

     180 - the angle (as seen from earth) between direction to asteroid and direction
        ofearth's orbital motion; by definition,   =  - 90                                   (2)

other orbital parameters

a     orbital radius

P     orbital period
v     orbital speed;by definition,                                                        ₍₃₎
 
but Kepler's 3rd law requires that                                      (4)

therefore,                                                                              (5)

by the law of sines,                                                           (6)

 by the law of cosines,             a_a²  =  d² + a_e² - 2 d a_e cos                       (7)

and                                        a_e²  =   a_a² + d² - 2 d a_acos q                       (8)

by definition, angular speed =                              (9)
 

but , by (2), sin   =  sin (  - 90)  =  - cos                                               (10)
 

substituting  (5)  and  (10)  into  (9) gives,
 

                                                               (11)

now use (6) to solve for                 (12)

to obtain [by substituting (12) into (11)]

                                         (13)

now use (7) to solve for  d  via the quadratic formula,

                                                  (14)

then,

(15)

notice that the right side of the equation has units the of 

if numbers are now entered for M_s and G, and if a is now expressed in units of au's (1 au  =  a_e), then

              (16)

the equation that must then be solved for the unknown asteroid orbit radius x
(formerly called a_a ) (which will come out in units of au's) is

        (17)

The absolute value (abs) is necessary because only the magnitude of the proper motion is entered as w. Remember that assumption (4) above requires that x > 1.

 The most straightforward way to solve this equation is graphically.
 

To determine the orbit size of an asteroid, the following inputs are needed:

1) the celestial longitude of the asteroid on the date that the image was taken; if the
    right ascension and declination of the asteroid are known

    go to the NED Coordinate Calculator  to determine the celestial longitude of the
    asteroid     (when you get there, select "equatorial" as your input system and
    select "ecliptic" for your output system)

2) the celestial longitude of the sun on the date that the image was taken; the date
    that the image wastaken can be used to calculate the celestial longitude;

    to determine the sun's right ascension and declination from the date of the
    image, try using any planetarium program. e.g., TheSky
     (I am trying to find a web link for this step, but have been unsuccessful to date.)

    again go to the NED Coordinate Calculator to determine the celestial longitude of the
   sun from its rightascension and declination

3) now that both celestial longitudes are known, calculate  =  | _s - _a |

4) the proper motion of the asteroid in units of radians/sec

     if the proper motion is known in arcsec/sec, then

         (radians/sec)  =  (arcsec/sec) / 206265
 

To solve equation (17) for x , you can

1) program your graphing calculator

or

2) go to  Visual Math For Java for a Java applet that will solve the equation

PM - 1.991E-7*abs(sqrt((1-((sin(del))/x)^2)/x)+cos(del))/(cos(del)+sqrt((cos(del))^2-1+x^2))

where  PM stands for the proper motion 

and      del stands for the angle    (which will need to be converted to radians)

you can copy the above string and paste it into a dialog box in a Java applet; don't forget to replace PM and del (3 times) by numbers!