# Root of Kepler's Equation

The equation of eccentric anomaly is given by
\begin{equation}
\psi - e\sin \psi = \omega t \tag{1}
\end{equation}
Using simple iteration which can be written as
\begin{equation}
\psi_{n+1} = \omega t + e \sin \psi_n \tag{2}
\end{equation}
, in which I iterate over and over until the difference between $\psi_{n+1}$ and $\psi_n$ is less than $\varepsilon$. In this case I let $\varepsilon=1.D0$.

## Code

!========================================================
!   Purpose: Solve Kepler eccentric anomaly
!
!   Methods: Simple iteration (E=M+e*sinE)
!       
!       Date        Programer       Description of change
!       ====        =========       =====================
!      9/30/20        Morris        Original Code
!========================================================
PROGRAM kepler
    IMPLICIT NONE
    REAL(16), PARAMETER :: pi = 3.1415926535898D0
    REAL(16) :: phi, e, M, tmp
    INTEGER :: i

    open(1, file='kepler_2.dat', status='unknown')
    e = 0.2D0
    do i = 1, 100
        M = pi * dble(i) / 50.D0
        do 
            tmp = phi
            phi = M + e * sin(phi) 
            write(*, *) abs(tmp-phi)
            if (abs(tmp - phi) <= 1D-8) then
                EXIT
            end if
        end do
        write(1, '(3F20.17)') M, phi
    end do
END PROGRAM kepler

## Result
In this program, it tries to find $\psi$ as a function of $\omega t$. By letting $\omega t$ go through $0$ to $2\pi$. I can get the following relation. Futhermore, I can adjust different $e$ to get a different function.