Preliminary Orbit Determination

This model consists of a ground station at some position and velocity (Gnd), the position and speed of some object in the sky as seen from that ground station (Obs), and the corresponding orbit of that object (Orb).

The relationship between these three entities (Gnd, Obs, and Orb) is complicated by the shape, spin, and wobbles of the Earth. An observer needs to know the motion of their sky to relate positions in the sky to positions in an orbit at any given time. The more precisely the sky’s motion is known, the more precisely an object’s orbit can be inferred (or vice versa).

The word ‘coordinate’ means ‘relating to’. Coordinates in space and time relate to something measurable; the ground, the sun, or the stars, depending on what is easy/precise to measure in any given situation. This model simply transforms coordinates of one type to another.

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Infer orbit from a ground-based radar observation

Ground-based radar can measure the range, azimuth, elevation and time rates of change of a target at a single point in time. From this information a preliminary orbit can be determined. Following the SITE-TRACK example of Vallado:

  1. A ground tracking station is located at 39.007° geodetic latitude Gnd.«ctrl+g»f = 39.007 , 104.883°W longitude Gnd.«ctrl+g»l = -104.883 , and 2.187 km altitude Gnd.h =2.187 . In Vallado’s nomenclature, this model infers intermediate quantities rδ = 4964.5377 km (Gnd.xEQ ) and rK = 3994.2975 (Gnd.zEQ ), which in turn infers the ground station’s Earth-centered Earth-fixed (ECEF) coordinates (Gnd.ECEF ) to be {-1275.12, -4797.99, 3994.3} km.
  2. The tracking station observes the target’s range to be 604.68 km Obs.«ctrl+g»r = 604.68 , at 205.6° azimuth Obs.Az = 205.6  and 30.7° elevation Obs.El = 30.7 . The model infers the target’s ECEF coordinates relative to the ground station (Obs.ECEF ) to be {-354.538, -459.372, -170.054} km. By adding this position to the position found in step 1, the target’s ECEF coordinates relative to the center of mass of the Earth (Orb.ECEF ) are {-1629.66, -5257.36, 3824.24} km at the instant of observation. In the ECEF coordinate system the Earth’s surface is stationary and the stars rotate. To convert the ECEF coordinates to inertial coordinates for which the stars are stationary and the Earth rotates (as needed to represent an orbit), the velocity of the target is needed.
  3. The tracking station observes the target’s range to be increasing at 2.08 km/s Obs.«ctrl+g»r’ = 2.08 , azimuth to be increasing at 0.15°/s Obs.Az‘ = 0.15 , and elevation to be rising at 0.17°/s Obs.El’ = 0.17 . The model infers the target to be moving relative to the ground station at {-2.10271, -1.66481, 1.48497} km/s (Obs.ECEF). Because the ground is stationary in ECEF coordinates, the target’s velocity relative to the center of mass of the Earth (Orb.ECEF) is the same as relative to the ground station. If the ground station were moving (by changing Gnd.ECEF‘.X , for example) then the two velocities would differ.
  4. To transform the ECEF coordinates for which the ground is stationary and the stars rotate to inertial coordinates for which the stars are stationary and the Earth rotates, not only is velocity of the target needed, but also the time of the observation so as to determine which piece of the sky is overhead at that instant. As measured at the ground station, the observation is made on May 20, 1995 at 03:17:02.00 UT Gnd.JDUT1.M = 5  Gnd.JDUT1.D = 20  Gnd.JDUT1.Y = 1995 Gnd.JDUT1.h = 5  Gnd.JDUT1.m = 17  Gnd.JDUT1.s = 2 . The model converts the observer’s position and time into an angle between their local horizon and the stars, which is to say it converts solar time (relative to the Sun) to sidereal time (relative to the stars).
  5. Finally, we assume no difference between UTC and UT1 time Gnd.DUT1 = 0 and that the pole is exactly co-aligned with the geodetic system Gnd.xp = 0  Gnd.yp = 0 . The model infers that the target has an ECI position Orb.R  of
    {-5505.5015992927711, 56.455496432931874, 3821.88} km

    and ECI velocity Orb.V of

    {-2.2017912339421413, 1.1366604100046094, 1.48408} km/s

    corresponding to a suborbital trajectory Orb.2D for which the target reached 759 km apogee 12 minutes ago and has fallen to 324 km altitude at the time of observation, to reenter imminently.

Validation: SOFA formulation for the IAU 2006/2000A CIP using X,Y series

The ground station model uses the IAU SOFA library to convert between ECEF and ECI coordinates. To ensure that the ground station model carries out this conversion correctly, its output is compared to the example in section 5.6 of the SOFA documentation.

Gnd.xp = 0.0349282
Gnd.yp = 0.48331629999999998
Gnd.dxp = 0.000175
Gnd.dyp = -0.00022589999999999999
Gnd.DUT1 = -0.072073684999999998
Gnd.JDUTC.Y = 2007
Gnd.JDUTC.M = 4
Gnd.JDUTC.D = 5
Gnd.JDUTC.h = 12
Gnd.JDUTC.m = 0
Gnd.JDUTC.s = 0

The inputs above may be copied and pasted verbatim into the ground station model. The resulting ECEF->ECI transformation matrix is Gnd.ITRF/GCRF, which should display at the end of computation. The transformation matrix should be:

Ground station model:
+0.97310431769767702    +0.23036382623853102    -0.00070316348176937648 
-0.23036380045543875    +0.97310457063294209    +0.00011854536811676633 
+0.00071156016259418711 +4.6626402444386325e-05 +0.9999997457540245
SOFA documentation page 27, celestial to terrestrial matrix:
+0.973104317697536 +0.230363826239128 −0.000703163481769
−0.230363800456036 +0.973104570632801 +0.000118545368117
+0.000711560162594 +0.000046626402444 +0.999999745754024

The elements of both matrices agree to at least 11 significant figures.

See also

3D Orbit Calculator

Crouse, David F. “An overview of major terrestrial, celestial, and temporal coordinate systems for target tracking.” No. NRL/FR/5344-16-10. Naval Research Lab, Washington DC Surveillance Technology Branch (2016).