Homework 3 – MAE 215 – Introduction to Programming

In the early 17th century Johannes Kepler developed the equation, now often referred to as kepler’s

equation for the solution of the mean anomaly (M) of a planet in an elliptical orbit as a function of it’s

eccentricity (e) and it’s eccentric anomaly (E).

E – e*sin[E] = M

If you are interested in exactly what this equation represents please feel free to look deeper into orbital

mechanics. Curtis’s Orbital Mechanics for Engineers is an excellent place to start. For this assignement

however the important thing to notice is that given E and e, solving for M is quite trivial. Given M and

e, solving for E can prove quite difficult. In fact this problem is considered transcendental, meaning E

can not be represented as a finite set of algebraic terms of e and M. As such this problem is almost

always solved using some numerical method.

For this assignment I would like you create a function called kepler which solves the kepler equation

for E given e and M (your function should accept M and e as inputs, and return E). The actual solution

should be performed using the MATLAB built in function fzero() to solve the following form of

Kepler’s equation:

0 = E – e*sin[E] – M

We use this form because fzero solves for roots of the input function. Recall that fzero() is a function

function, and that it that it expects that the function argument which you pass it will only accept one

argument. This means that fzero(f(x)) is acceptable, but fzero(f(x,y)) is not. As such you might have to

use the shared workspace property of nested functions to your advantage to complete this assignment.

Please plot E vs. M for e of values [0 0.25 0.5 0.75 1]. Please include all 5 curves in 1 figure axis.

Please label axis and include a legend.

Last –6*

Homework 3 – MAE 215 – Introduction to Programming

In the early 17th century Johannes Kepler developed the equation, now often referred to as kepler’s

equation for the solution of the mean anomaly (M) of a planet in an elliptical orbit as a function of it’s

eccentricity (e) and it’s eccentric anomaly (E).

E – e*sin[E] = M

If you are interested in exactly what this equation represents please feel free to look deeper into orbital

mechanics. Curtis’s Orbital Mechanics for Engineers is an excellent place to start. For this assignement

however the important thing to notice is that given E and e, solving for M is quite trivial. Given M and

e, solving for E can prove quite difficult. In fact this problem is considered transcendental, meaning E

can not be represented as a finite set of algebraic terms of e and M. As such this problem is almost

always solved using some numerical method.

For this assignment I would like you create a function called kepler which solves the kepler equation

for E given e and M (your function should accept M and e as inputs, and return E). The actual solution

should be performed using the MATLAB built in function fzero() to solve the following form of

Kepler’s equation:

0 = E – e*sin[E] – M

We use this form because fzero solves for roots of the input function. Recall that fzero() is a function

function, and that it that it expects that the function argument which you pass it will only accept one

argument. This means that fzero(f(x)) is acceptable, but fzero(f(x,y)) is not. As such you might have to

use the shared workspace property of nested functions to your advantage to complete this assignment.

Please plot E vs. M for e of values [0 0.25 0.5 0.75 1]. Please include all 5 curves in 1 figure axis.

Please label axis and include a legend.

Last –6*