What is the best way to orbit objects in a scientific-like elliptical orbit?

It's quite possible that I'm in way over my head, but I decided to simulate 67 moons orbiting Jupiter.

I found lots of great data, but it's not fully complete for a Newtonian simulation.

I also found an interesting elliptical example and converted it into C#:

 //x, y: center of the ellipse
 //a, b: semimajor and semiminor axes
 
 using UnityEngine;
 using System.Collections;
 
 public class ElipseOrbit : MonoBehaviour {
 
     public float a;
     public float b;
     public float x;
     public float y;
 
     private float alpha;
     private float X;
     private float Y;
     private float mult = 0.005f;
 
     void Update () {
         alpha += 10;
         float multed = alpha * mult;
         X = x + (a * Mathf.Cos(multed));
         Y = y + (b * Mathf.Sin(multed));
         this.gameObject.transform.position = new Vector3(X, 0, Y);
     }
 
     // Use this for initialization
     void Start () {
     
     }
 }

However, the results were a very equalized elliptical orbit (not very Newtonian-like).

RotateAround is too round for this project.

Should I be searching for a sat tracking script and adapt from there? I'm still digging, but being able to showcase 67 moon orbits around: amazing right?

UPDATE: While I don't know all the data for Jupiter's moons, I'll focus on what we DO know first, returning to physics and the Galilean moons. That's a better start. More updates if/as they happen.