## 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:

## Renewable Energy 101

Renewable energy is energy that is generated from sunlight, rain, tides, geothermal heat and wind. These sources are naturally and constantly replenished, which is why they are deemed as renewable. The usage of renewable energy sources is very important when considering the sustainability of the existing energy usage of the world. While there is currently an abundance of non-renewable energy sources, such as nuclear fuels, these energy sources are depleting. In addition to being a non-renewable supply, the non-renewable energy sources release emissions into the air, which has an adverse effect on the environment.

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