How do I make planet move slower the further away they are? [duplicate]

Question

I got this code from somewhere else, I want to implement Kepler's third law to make the simulation look more realistic, the simulation is a 2D satellite orbiting around a planet, the satellite(Planet) script here is :

using UnityEngine;
using System.Collections;
using System.Collections.Generic;

public class Planet : MonoBehaviour {
private float alpha; //will use for cos & sin (as angle)
private float distance;//distance from center
private float Rotation;//rotation angle
private float currentTime;//Time start when Instantiated
//input values
public float speed;
public Vector2 RotationValues;
public Vector2 distanceValues;
public Vector2 sizeValues;


void Start () {
    //Initialize with Random rotation & distance & size

    //Random Rotation between -2 & 2
    Rotation = Random.Range(RotationValues.x, RotationValues.y);
    //Random distance  between -1 & 2
    distance = Random.Range(distanceValues.x, distanceValues.y);
    //Random Size between 0.2f & 0.7f
    var Size = Random.Range(sizeValues.x, sizeValues.y);
    //Set localScale with random size
    transform.localScale = new Vector3(Size, Size, Size);

    //calculating current time by subtraction Time.time and lastTime
    currentTime = Time.time - Generator.lastTime;

    //GetComponent<MeshRenderer>().material.color = new Color(Random.value, Random.value, Random.value); //Random Color
}

void Update () {
    //Timer
    currentTime += Time.deltaTime;

    var X =  (distance * Mathf.Cos(currentTime)); //X
    var Y =  (distance * Mathf.Sin(currentTime)); //Y
    var R = (Rotation * Mathf.Cos(currentTime)); //Rotation
    //give earth from generator class
    var Earth = Generator.Earth;
    transform.position = Earth.transform.position + (new Vector3(X, R, Y));// RotateAround Earth

}
}

also I have this other script which generates the planets:

using UnityEngine;
using System.Collections;

public class Generator : MonoBehaviour {
public GameObject prefab;
public static GameObject Earth;
public static float lastTime;
public int numerOfSatellites;


void Start() {
    Earth = GameObject.FindGameObjectWithTag("MainPlanet");//Find Earth
    for (int i = 1; i <= numerOfSatellites; i++) {
        lastTime = Time.time;
        GameObject sat = Instantiate(prefab, Earth.transform.position, Quaternion.identity);
        Physics2D.IgnoreCollision(sat.GetComponent<CircleCollider2D>(), Earth.GetComponent<CircleCollider2D>());
    } }
}

I know that the main part here is

void Update () {

    //Timer
    currentTime += Time.deltaTime;

    var X =  (distance * Mathf.Cos(currentTime)); //X
    var Y =  (distance * Mathf.Sin(currentTime)); //Y
    var R = (Rotation * Mathf.Cos(currentTime)); //Rotation

    //give earth from generator class
    var Earth = Generator.Earth;
    transform.position = Earth.transform.position + (new Vector3(X, R, Y));// RotateAround Earth

}

The main problem is that whatever distance the planets have from the mainPlanet (called earth) they move at the same angular velocity, how would I change this?

Those are the basic 4 steps of every orbit, the problem here is that they all align in the center, which shouldn't be possible because of the random distances they have

Answer 1

Kepler's third law states that the square of the period of revolution is proportional to the cube of the distance

or equivalently

Now, the function has a period of .

You can change the period by changing . If you want it to satisfy Kepler law, you have introduce a dependence on the distance

so that

where C is a constant that physically should make the argument of the cosine dimensionless. You do not need to worry about it in the code though, so you can set it to 1.

In conclusion, just change your update function in this way

void Update () {

   //Timer
   currentTime += Time.deltaTime;

   var prefactor = Mathf.Pow(distance, -2/3);
   var X =  (distance * Mathf.Cos(prefactor * currentTime)); //X
   var Y =  (distance * Mathf.Sin(prefactor * currentTime)); //Y
   var R =  (Rotation * Mathf.Cos(prefactor * currentTime)); //Rotation

   //give earth from generator class
   var Earth = Generator.Earth;
   transform.position = Earth.transform.position + (new Vector3(X, R, Y));// RotateAround Earth

}

Answer 2

You have a fixed angular velocity for each planet.

The easiest way to fix this without changing much of your current code is to modify the code so that the angular change per Update() is inversely proportional to the orbital radius of the planet.

Let's assume Earth is at radius 2. Let's assume Jupiter is at radius 10. 10/2 = 5, thus you need to divide the angular change for Jupiter by 5 to get it to have roughly the same linear velocity as earth; this works because the formula for circumference, c = 2πr, is linear. Of course you may not want the same linear velocity; but this at least gives you a basis to work from, and you can then (randomly) adjust velocity of each heavenly body, from there.

The more difficult but also more rigourous way would be to use physics to correctly update the position of each planet on each FixedUpdate(), using vectors. This ranges from quite simple to pretty complex, depending on how much of the physics you want to explicitly control, for example, going as far as calculating mass per body may not be necessary, but adjusting the motion vector each update according to the gravity force being applied, would seem to be a sane requirement.