About every 10 minutes mark an additional x at the tip of the shadow. Do this until about 1 pm. By then, one of the x's should have touched the outline of your circle. This will be the X on the right
Now, use your string to draw two arcs. Start with the X that lies to your right. Using a 'radius' a little more than half the distance to the first X on the circle, make an arc that bows away from the x and intersects your circle. Repeat this procedure for the left X. The end result should look like this: