# Quasi-optimum Trajectory for ASAT Missile

The problem is to determine the direct-ascent trajectory of a Chinese ASAT missile, launched from the Xichang test site, so that it intercepts a particular satellite at the maximum possible latitude.  The altitude of the target satellite's circular orbit is approximately 467 nautical miles, and the orbital inclination is 98.79 degrees (i.e., a retrograde orbit).  A specific requirement is that the missile and satellite inertial velocities be co-linear at the intercept point.

At Greenwich Mean Time (GMT) = 1430.25 hours, February 10, 2007, the target satellite's mean orbital elements are:

Perigee altitude = 863 km
Apogee altitude = 865 km
Right Ascension = 15 deg
Inclination = 98.79 deg
Argument of Perigee = 30 deg
True Anomaly = 120 deg

It is desired to launch the ASAT missile within a 24-hour window beginning at GMT = 2200 hours, February 10, 2007.

The first step is to estimate the best launch opportunity within the 24-hour window by looking at the satellite's ground track.  A Fortran program is used to generate the ground track and to identify the satellite's point of closest approach (PCA) to the launch site.  Figure 1 shows a segment of the satellite's ground track including the orbit (red color) where the satellite comes closest to the launch site at Xichang.  The PCA occurs approximately 8 hours and 41 minutes into the 24-hour window.

Figure 1.  Segment of Satellite Ground Track Containing Orbit
Where Satellite
Passes Closest to Launch Site

The intercept must occur when the satellite latitude is increasing in order to satisfy the co-linear velocity requirement and to achieve intercept at the maximum possible latitude.

The interceptor is a two-stage missile with a launch mass of 14,699 kg and payload (warhead) mass of 612 kg.  The launch thrust-to-weight ratio is 1.89, and the first stage will burn for 90 sec, consuming 9983 kg of propellant before its empty mass of 1361 kg is separated.
The second stage's initial thrust-to-weight ratio is 1.02 and it will burn for 188 sec, consuming 2414 kg of propellant before its empty mass of 329 kg is separated.  The 612 kg warhead will then coast to the intercept point.

A Fortran trajectory optimization program is used to determine the interceptor missile's quasi-optimum launch time, launch azimuth, steering profile, and time of intercept so as to satisfy the mission requirements.  The steering profile (missile attitude history) is characterized by five parameters, making a total of eight parameters that must be determined by the trajectory optimization program.

The trajectory optimization program determines that the missile interceptor should be launched 7 minutes and 51 sec before the satellite ground track comes nearest to the launch site.  The quasi-optimum launch azimuth is determined to be -10.83 deg.  On the quasi-optimum trajectory, the interceptor missile's first stage accelerates the missile to a speed of 1978 m/s*, achieving an altitude of 55 km with a flight path angle of 50.1 deg.  The second stage accelerates the missile to a speed of 3744 m/s, achieving an altitude of 384.7 km with a flight path angle of 45.72 deg.  After second-stage burnout and separation, the warhead coasts for 374.3 sec to intercept the target satellite at a latitude of 38.2 deg.  At the intercept point (altitude = 875.9 km), the warhead is very near its apogee, with a speed of 2455 m/s.  The target satellite hits the warhead from behind, with a speed of 7508 m/s, making the relative speed of impact equal to 5053 m/s.  The total flight time of the interceptor missile is 652.3 sec, covering a ground range of 1218 km.

* The speeds and flight path angles mentioned here are those
observed in the rotating earth reference frame.

Figure 2 illustrates the quasi-optimum intercept trajectory.

Figure 2.  Quasi-Optimum Intercept Trajectory

Figure 3 is an illustration of the intercept, to scale, showing the earth perimeter and Greenwich meridian.  The blue line is a segment of the target satellite orbit, and the red line is the interceptor trajectory.

Figure 3.  Intercept as Viewed By a High-Altitude Observer in the
Equatorial Plane, from a Longitude of 30 deg.