Vector3 arrays in C#

Hi everyone, thanks for checking this out. I am totally mystified... trying to build an array which keeps the Vector3 position of a game object over time so it can be referenced when a triggering event occurs.

Here's the relevant code:

 using UnityEngine;
 using System.Collections;
 
 public class ThrowTrajectory : MonoBehaviour {
 
     public int arraySpaces = 20;
 
     public Transform rightHand;
 
     private Vector3[] rhPos;


     void Start () { //THIS WORKS PERFECTLY
 
         Vector3[] rhPos = new Vector3[arraySpaces];
   
         Vector3 v = new Vector3(0,0,0);
 
         //Fill the array with initial vectors
         for (int i = 0; i < arraySpaces; i++){
 
             rhPos[i] = v;
             Debug.Log ("rh " + i + " : " + rhPos[i]);
 
         }
     }
 
     void Update () {
 
         Vector3 v = rightHand.position; //This works
         Debug.Log (v); //This works

         for (int i = 0; i < arraySpaces; i++){

             rhPos[i] = v; //THIS DOESN'T WORK - null reference exception - WHY? It's the identical syntax to the Start method?????
             Debug.Log ("rh " + i + " : " + rhPos[i]);
 
         }        
     }
 }

Any ideas??