EORSAT #22808 and ndot2 values

Bjorn Gimle (bjorn.gimle@online.dextel.se)
31 Jul 1995 20:21:13 GMT

Bill Krosney recently asked me :

>> I've been following with interest the postings regarding the coming decay
>> of Cosmos 2264 (USSPACECOM #22808).

>> Of particular interest are the comments about "ndot2" values.  What
>> are these values?  Are they present in the TLE listing (i.e. are they the
>> drag term)?  Or is it a derived value from info contained within the TLE?

>> In any case, I would be interested in a brief explanation how one uses 
>> a ndot2 value along with other parms to deduce a probable re-entry date.
>> And what are "exceptional" values.

>> I understand that to some of the more experienced observers out there,
>> there may be something akin to a "gut feel" about these values.  But we
>> all have to start somewhere :-)

I ought to write a more thorough description on how I use elsets to
(try to) predict decays, but your question deserves at least a partial

First, I do not have a "gut feel" about those values, nor recognize
exceptional values by themselves. They depend so much on the area/mass
of the satellite, the geometry of itself and its tumbling axes, the
perigee and eccentricity of the orbit, current solar activity and its
effects on the atmospheric pressure and temperature. 
My comments, therefore, are mainly in relation to the values in recent 
elsets of the same (and similar) object(s), and in the case of #22808 
the previously decayed EORSATs. 

The naming of the fields of the elsets varies, but this is what
I have adopted:

Line 1, col. 21-32  Epoch = Julian Day and fractional portion of the day
             34-43  ndot2 = First Time Derivative of the Mean Motion/2
             45-52  ndotdot6 = Second Time Derivative of Mean Motion/6
             54-61  BSTAR = drag term if SGP4 model was used.
Line 2, col. 27-33  e = Eccentricity (decimal point assumed)
             53-63  n = MM = Anomalistic Mean Motion [Revs per day]

The SGP4 model uses BSTAR, and a "real world" atmospheric model to
compute the effect of drag, whereas SGP uses something like :
dN = dt * ( n + dt * ( ndot2 + dt * ndotdot6 ) ) , where 
dN is the number of anomalistic ( perigee-perigee ) revolutions
from Epoch until dt days later.
This would give the following quadratic formula for MM = n at Epoch+dt :
MM = n + dt * ( 2*ndot2 + dt * 3*ndotdot6 ), 
but SGP supposedly does not use the quadratic term, giving an 
(almost) linear decrease of the orbital period with time - 
clearly inappropriate for the accelerating decay of low, 
near-circular orbits, (but quite accurate for very eccentric orbits!)

There is a simple model for atmospheric pressure, P = C * exp(h/H),
where h is height, and H is called "Scale Height", and another for
the drag force F = A * D * rho * v * v 
The orbital velocity, v, and the effect of F on MM, vary with the
orbital radius, but this variation is small for a near-circular orbit
during the last few months of a satellite's life. Neglecting also the
effect on rho = atmospheric density of temperature and chemical
composition, the exponential model of pressure could be used for
density, at least to give better decay predictions than SGP.

With some similar linearisation of MM to orbital radius, 
to simplify integration, I arrived at :

MM(t)    = F - G * ( 1 + ln(H-t) )      , which gives :
ndot2(t) = 0.5 G / (H-t)

where t = day no. and F, G and H are constants, that I determine by a 
program I wrote, fitting them to actual elsets ( I actually use 
similar formulae for nodal period vs. revolution no., and Epoch times )

But there are some straightforward consequences of these formulae,
more useful for estimates :

The G value is (mainly) related to atmospheric scale height, i.e.
usually quite constant for all low, near-circular orbits ( about 0.1 )

The inverse of ndot2 is a linear function of time. The slope is 2/G
(about 20), and the intersection with the time axes gives H.

The value of F can be found from G, H, and the most recent elset.
This is the value that varies most with the area/mass of the object,
and once it is determined, you can find the estimated decay date
by solving for MM=16.55, which usually gives a time less than one
day before the 'H' value.

And, to simplify it even further :

"At any time, the remaining life time of a circular orbit, is about
 0.05 / ndot2  days."

For a highly eccentric orbit, it is instead (16-MM)/ndot2/2 days ?

Bjorn_gimle@lector.kth.se == bjorn.gimle@duesenberg.se ==
== bjorn.gimle@online.dextel.se ;  59.22371 N, 18.22857 E


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