PDyx Kyx MP2 bDyy Kyy j

To obtain a solution, the determinant of the coefficient matrix must vanish. Expansion produces a polynomial in b that can be solved for the roots of b, which in turn provide the growth factors and frequencies of vibration.

It is possible to obtain a closed-form solution of Equations 41 and 42 for the critical mass and resulting orbital frequency. The critical mass M is defined as that mass above which an instability will occur. At the threshold of instability, the real part of b = a goes to zero and b = iv.

Substituting into Equation 49, expanding the determinant and separating real and imaginary components produces the following equations:

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