NEO Tracking Calculations in the Celestial Reference Frame

 

  I redid the previous somewhat simplified NEO tracking problem using celestial coordinates for the sightings and geographic coordinates for the observer locations. For each pair of sightings one needs to find the observers' positions in the celestial frame by determining the Greenwich Mean Sidereal Time (GMST) for Earth's orientation at the given time and then rotate the position coordinates. One also has to convert the Right Ascension and Declination of the observations into celestial unit vectors for the parallax calculation to work properly. In the Mathcad function shown below the procedure goes as follows. First calculate the GMST using the Julian Date and convert hours to an angle. Then create a matrix to do the rotation about the Earth's celestial pole and use it to rotate the observer positions. Finally one calculates the distance along the line of sight to determine the position of the object.

 

 

  Suppose that on Jan 12, 2013 two pairs of sighting on an object are made from Sydney and Perth at 12:00 UT and 13:00 UT. The coordinates for positions and sightings are as indicated. The procedure gives the object's position in the celestial coordinate frame and one can convert the Cartesian coordinates to celestial angles and the range of the object. As was done previously, the RA is in hours and the range is given in Earth radii.

 

 

 The second sighting is done the same way and one again gets a speed of 0.8 R_E/hour for the object.

 

 

  If one needs more accuracy the JDPe2X procedure could be modified to take into account precession, nutation, the motion of the Earth about the Earth-Moon barycenter, etc. in order to get a better position for the observers. The rotation for the first two corrections can be found by accurately measuring the position of the Earth's celestial pole and checking the sidereal time for the Greenwich Meridian. By definition the Equinoctial point is determined by the intersection of the Equatorial and Ecliptic planes.

 

  Supplemental (Jan 13): The Earth-Moon barycenter is closer to a Galilean reference frame than the geocentric frame but since the intended goal is to track the NEO's motion relative to the Earth the geocentric frame will suffice. The position of the observers relative to the Earth's center is all that is necessary to measure the parallax and calculate the object's geocentric range.