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Is it possible to calculate $B^*$ term in TLE data using only a few or more velocity and position vectors for a satellite?

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David Vallado also thinks it is an impossible task, but for different reasons. A footnote in his Fundamentals of Astrodynamics and Applications (page 106 in the 4th edition) says:

Be aware that the value of B* is always modified. It’s really an arbitrary free parameter in differential correction. Chapter 10 will introduce how to estimate a drag parameter. The estimated value of B* may be completely unrelated to drag effects in the presence of satellite maneuvers, significant solar pressure and atmospheric perturbations, large third-body effects from the Sun or Moon, or large deflections caused by mismodeling of the Earth’s gravitational field. B* can even appear as a negative number!

I agree with this and with the quote posted in the other answer. What I would add is that since B-star is (used as) an arbitrary free parameter, there is exactly one way to calculate what it ought to be: have the same software NORAD uses to compute TLEs, and the same input data they use to compute them, and run the process the same way they do to make TLEs in the first place. Which is to say, it is impossible unless you happen to be the person at NORAD whose job is making TLEs. That person can do it easily, but they're not sharing all of their knowledge and tools, so no one else can reproduce it exactly. If you happen to be engaged in official business of the U.S. Government, it is possible to receive the differential corrector software, and use it to modify an existing TLE to incorporate new measurements. However, you still can't duplicate the original generation, because even the arduous process of requesting permission for the tools through your contracting officer does not also give you access to all the sources of input data they use routinely.

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Dwight E. Andersen claims that its an impossible task in his thesis "Computing Norad Mean Elements From a State Vector" (1994)

An extension of this [calculating other TLE elements than drag and mean motion] would be to compute, from just the state, the mean elements and the drag terms and B* in SGP4, SDP4, SGP8, and SDP8. Future research on satellite drag and methods of estimating the drag would be valuable. Since the drag term is a function of the physical geometry as well as atmospheric conditions, some aspects of the satellites' physical characteristics would be needed.

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