Can you adjust a f/7.7 spherical mirror to act like a f/8.0 parabolidal mirror by racking the focus farther out?

My 114-millimeter (4.5-inch) Newtonian reflector came with an f/7.7 spherical mirror. I can purchase a 114-mm f/8.0 paraboloidal mirror. Can I simply adjust for the 34 mm of added focal length by racking the focuser farther out, or should I extend the main tube? Either way, is the upgrade worth it?

Newtonian anatomy

This diagram illustrates the Newtonian reflector's optical components and some of the structures that support them.

When replacing a Newtonian primary mirror, it’s best to compensate for any change in focal length by moving the mirror cell. You always want the focal plane to end up as close to the side of the tube as your focuser and eyepieces will allow, because racking out the focuser can vignette the light beam. Remounting the focuser along the main tube is not a good idea either, because you’re apt to be left with an unsightly hole or patch hiding its old location. I suspect what you’re really wondering is whether the paraboloidal primary will yield better optical performance than the sphere. In your case, don’t count on it. Here is the formula for the wavefront error, e, when a spherical mirror is used as a Newtonian primary:

e = 22 D / F^3,

where D is the primary mirror’s diameter in inches and F is the focal ratio. The error of your 4.5-inch f/7.7 sphere works out to be 0.2 or 1/5 wave, so your spherical mirror already meets the Rayleigh 1/4-wave tolerance for optical performance. You may notice a slight improvement with the paraboloid, but not much.

— Roger W. Sinnott