[e_engineer]

Remember when a planet is orbiting a star; it also has a gravitational influence on that star as well. So it’s not just a planet orbiting a star, it's both orbiting a common center of mass.

Agreed.

So as the planet becomes more massive, the further the center of mass moves from the center of the star (causing the star to wobble).

Agreed.

Also the period of an orbit is directly proportional to its distance from the star, closer is shorter.

Agreed.

So the massive close-in higher elliptical planets are far easier to detect.

If you change this to read: "So the massive close-in higher elliptical planets are far easier to detect.", Agreed.

Imagine a Jupiter mass planet in a circular orbit with perihelion = 1 AU. If you could change it's ellipticity (sp?) but keep the perihelion (closest approach) at 1 AU, does it become easier or harder to detect based on the stars radial velocity?

Since the maximum excursion of the center of mass would be the same in both cases, the elliptical orbit would be (at best) equally detectable to the circular orbit.

If we assume that both orbits are in the same plane with the observer on earth, the elliptical orbit becomes more difficult to detect if it's semi-major axis is perpendicular to the line of sight to earth.

[RadioAstronomer]

Imagine a Jupiter mass planet in a circular orbit with perihelion = 1 AU. If you could change it's ellipticity (sp?) but keep the perihelion (closest approach) at 1 AU, does it become easier or harder to detect based on the stars radial velocity?

Hmmm.... Think of it this way. If the planet is highly elliptical, the star will also exibit a elliptical motion around the center of mass (greater wobble). If the orbit plane is exactly in line with our line of sight of the star, I agree detection would be much more difficult. However, if we are looking at it face on, the motions of the star becomes much more apparent.