How do I solve Kepler's equation for E using fzero?

I am required to do an assignment in which I need to do just that and I can't figure it out. Equation is:

M=E-e*sin(E)

M, e, and E are all variables.

I need to use fzero to create a script that will solve it for E when I input M and e. It should look something like Kepler(1,2) in the command window for example. Any help would be great thanks alot

[Walter Roberson]

See also http://www.mathworks.co.uk/matlabcentral/answers/87063-plotting-e-e-sin-e-m-as-e-vs-m-when-given-e-as-5-different-values-0-0-25-0-5-0-75-1

[Walter Roberson]

Is abs(e) greater than Pi/2 or less than Pi/2 ?

For e >= Pi/2, there are multiple solutions for E. For any given positive integer N, if P > (N+1/2)*Pi then there are at least N real-valued solutions for some values of E (there may be 1 or 2 fewer solutions for other values of E; the number of solutions will be periodic with E -- a larger E will just push the solutions further positive but they will still exist.)

[James Tursa]

@Walter: 0 <= e < 1

[Walter Roberson]

We don't know that, James. We are only told that e is a variable.

[James Tursa]

@Walter: It is true that OP did not tell you that, but I know that Kepler's equation is used for elliptical orbits. So I was just filling in a detail that OP did not mention.

https://en.wikipedia.org/wiki/Kepler%27s_equation

Answer 1

[James Tursa]

To use fzero, you first need to form an equation with 0 on one side. So this equation:

M = E - e * sin(E)

becomes this equation:

0 = E - e * sin(E) - M

Then set up your known values:

e = something
M = something

Then form a function handle for the equation that you want to be 0 as a function of what you want to solve for, which is E in this case:

kepler_equation = @(E) E - e * sin(E) - M

Then it is just a matter of calling fzero with this function handle and an initial guess. E.g., you could use M as the initial guess. I will leave that part for you to figure out. Look at the doc for fzero to see how to call it.

If you are creating a function that outputs E given e and M, then the outline of that function would be this in a file called Kepler.m:

% E = Kepler(e,M) solves Kepler's equation, E and M are in radians
function E = Kepler(e,M)
% form the function handle here  <-- you insert this code
% call fzero here  <-- you insert this code
return
end

The code you insert above will be based on the advice I gave you above.