TRAJECTORY SOLUTION Sample Solutions
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
is desired to launch the ASAT missile within a 24-hour window beginning
at GMT = 2200 hours, February 10, 2007.
first step is to
estimate the best launch opportunity within the 24-hour window by looking at
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
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.
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
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
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
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
Figure 3. Intercept as Viewed By a High-Altitude Observer in the
Equatorial Plane, from a Longitude of 30 deg.