C*398 - How it Decayed

Fri, 15 Dec 1995 02:05:49 -0500

For kicks I took the final elsets of Cosmos 398 and looked at them to get
a profile of the satellite's decay.

    EPOCH    APOGEE       V(A)      DA/DT   HEIGHT   RHO
    325.562  6818.24      7.57              440.24  1.65E-19
    332.266  6771.71      7.60      -6.94   393.71  2.28E-18
    334.885  6745.89      7.63      -9.86   367.89  9.78E-18
    339.168  6690.71      7.67     -12.88   312.71  2.19E-16
    342.312  6635.26      7.72     -17.64   257.26  4.99E-15
    344.695  6534.85      7.80     -42.14   156.85  1.43E-12

    EPOCH    PERIGEE      V(P)      DP/DT   HEIGHT   RHO
    325.562  6538.15      7.89              160.15  1.19E-12
    332.266  6535.12      7.88      -0.45   157.12  1.41E-12
    334.885  6533.23      7.87      -0.72   155.23  1.57E-12
    339.168  6528.40      7.86      -1.13   150.40  2.06E-12
    342.312  6523.34      7.85      -1.61   145.34  2.74E-12
    344.695  6501.54      7.84      -9.15   123.54  9.37E-12

V(A) -  Velocity at apogee (km/sec)
V(P) -  Velocity at perigee (km/sec)
DA/DT - Rate of change in apogee (km/day)
DP/DT - Rate of change in perigee (km/day)
HEIGHT - Satellite's height above a 6378-km spherical Earth
PERIGEE, APOGEE - Expressed in km, geocentric
RHO - Atmospheric density.  I obtained actual measurements taken in
      December 94 between 120 and 200 km height, and used a best-fit
      exponential curve to generate the density numbers

The final elset at 344.695 shows that C*398 was very close to cir-
culization.  It might be tempting to think that because DA/DT is
increasing so much that the apogee somehow shoots past the perigee.
Not true.  The real story is in the convergence of V(A) and V(P).  
Once they converge, the orbit is fully "circularized."  I use the 
quotes because at this point the orbit is not circular; it is 
actually a spiral without eccentricity.  Apogee and perigee no longer 

Once C*398 was in its spiral orbit, decay occurred very quickly.
The reason for this is a combination of the non-eccentric orbit and
the roughly 120 km orbit height.  120 km is a lethal height for 

The form of the orbit is a spiral because the drag is acting on the 
satellite at all points in the orbit.  For every step forward in the 
orbit, C*398 is slowed incrementally by drag and must drop a bit.  
At this new lower level, rho is higher and C*398 drops even more.  
The classic drag equation

         F   =  1/2 (rho)(V^2)AC
          d                     D

shows that the drag force is proportional to atmospheric density and to
the square of the velocity.  But rho varies exponentially with height;
so then does the drag force.  A is the cross-sectional area; C-sub-D
is the coefficient of drag.  CD is typically between 2 and 10 for
a supersonic object in a thin atmosphere.

Once C*398 got into a low-height non-eccentric spiral orbit, the 
ever-increasing and relentless drag force increased to the point that
the satellite broke up, either from frictional overheating or from
aerodynamic structural failure.   

Other factors ignored here are what make precise decay prediction
impossible.  Rho can vary by a factor of ten at heights above 300 km
or so due to solar flux and geomagnetic activity.  At 120 km there
are wave disturbances of very short period that vary rho by 20%.
One of these random waves can accelerate decay quite a bit.  Aero-
dynamic lift can offset the drag force to some degree and delay decay.
Jim Varney          |  121^ 23' 54" W,  38^ 27' 28" N   |     Sacramento, CA
Civil Engineer      |            Elev. 20 ft.           |jvarney@quiknet.com