For what eccentricity is the secondary focus (which is usually empty) located at the sun? What is the shape of this orbit?

The eccentricity, denoted by e, is defined as that fraction of an ellipse size which causes the focus to move away from the center. When an ellipse has an eccentricity of zero, this means that the focus and the center are in one same place. Thus the eccentricity of the secondary focus located at the sun is zero when located at the sun. Thus with an eccentricity of zero, the focus and center will be at same place therefore making the shape of the orbit to be a perfect circle.