Explain what is happening to a satellite's orbit if the orbit is becoming more eccentric?

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In Physics, there is a parameter referred to as orbital eccentricity denoted by the symbol e. This parameter allows one to mathematically determine the extent to which an orbit deviates from a perfect circular orbit. Consider the possible values for e shown below:

e = 0 , circular orbit

0 < e < 1 , elliptic orbit

e = 1 , parabolic trajectory

e > 1 , hyperbolic trajectory

Upon a careful review of the derivation of e, one will find that there are a number of complex consequences associated with the value for e associated with a given system. One of the most important of these consequences involves the geometry or trajectory of the predicted orbit. This consequence will in turn potentially impact to varying extents on other planetary bodies as well as having direct influences on a planetary body itself.

Please refer to the provided resource for the derivation of e.