JonahThePoo - 10 months ago 31

iOS Question

I am writing codes to find a solution (E) to Kepler's equation:

E - e*sin(E) = M

and all angles are expressed in radians,

M = 3.52821,

e = 0.016714

and theses are the steps:

- First guess, put E = Eo = M
- Find the value of O = E - e*sin(E) - M
- If |O| <= 0.000006, go to step 6

If |O| > 0.000006, proceed with step 4 - Find delta_E = O/(1-e*sin(E))
- Take new value E1 = E - delta_E, go to step 2
- The present value of E is the solution, correct within 0.000006 of the true value

However, I don't know how to write codes (swift) of those steps, please help me.

Answer

First of all some corrections in your question which you apparently forgot to mention and I had to look for here.

```
E0 = M
E = E1 on next iteration if solution not found
```

Regarding technical terms, `E`

here is called Eccentric Anomaly and `M`

is called mean Anomaly. Where as `eps`

is precision diameter. Also, E=e according to the shared article

Also in swift, we use camel case naming convention for variables and constants but here I have tried to use your names so that you could understand the code.

Now back to business, Following methods will do it for you using recursion:

```
func solveKeplersEquationForParamas(M:Double)->Void{
let E:Double = M
let eps:Double = 0.000006
let AbsoluteValueOfO:Double = getAbsoluteValueOfO(E, M: M,eps: eps)
print(NSString(format:"Answer is:%f", AbsoluteValueOfO))
}
func getAbsoluteValueOfO(E:Double,M:Double,eps:Double) -> Double {
var SinOFE: Double = Double(sin(E))
SinOFE = E*SinOFE
var E1 = E;
let O = E - SinOFE - M
var AbsoluteValueOfO = fabs(O)
if AbsoluteValueOfO > eps {
print("Solution not found. go to step 4");
let denom = 1-E*sin(E)
let absDenom = fabs(denom)
if absDenom>0{
let deltaE = O/denom
E1 = E-deltaE
AbsoluteValueOfO = getAbsoluteValueOfO(E1, M: M, eps: eps)
}
else{
print("Denom became 0, Can't divide by zero Denom is:%f",denom);
return AbsoluteValueOfO
}
}
else if AbsoluteValueOfO < eps || AbsoluteValueOfO == eps{
print("Solution found. Returning the value");
print(NSString(format:"Value of E is:%f", E1))
}
return AbsoluteValueOfO
}
```

Running this in playground like:

```
solveKeplersEquationForParamas(3.094763)
```

Playground output:

**NOTE:** This is the Swift solution to the steps you have mentioned. Responsibility for any error in steps lies on you.