Chapter 4.9, Problem 461E

In the following exercises, consider Kepler's equation regarding planetary orbits, $M=E-\epsilon \sin \left(E\right)$ , where M is the mean anomaly, E is eccentric anomaly, and $\epsilon$ measures eccentricity.

461. Use Newton’s method to solve for the eccentric anomaly E when the mean anomaly $M=\frac{3\pi }{2}$ and the eccentricity of the orbit $\epsilon =0.8$ ; round to three decimals.