The fields already exist. The Earth basically follows the geometry of space due to the presence of the energy of the bodies contained within that space. The Sun being the major energy body, is the object that is cheifly responsible for the curvature of the space around it. It's akin to laying tracks. The Earth follows the track in space laid down by the Sun. The Sun does not radiate gravity, as it does light.
some abstracts re gravitons, from Scirus:
Local Photon and Graviton Mass and its Consequences
Authors: M. Tajmar, C.J. de Matos
http://arxiv.org/abs/gr-qc/0603032
Graviton emission from a higher-dimensional black hole
Authors: A. S. Cornell, Wade Naylor, Misao Sasaki
http://arxiv.org/abs/hep-th/0510009
Gravitomagnetic London Moment and the Graviton Mass inside a Superconductor
Authors: Clovis Jacinto de Matos, Martin Tajmar
http://arxiv.org/abs/cond-mat/0602591