The gravitomagnetic influence on Earth-orbiting spacecrafts and on the lunar orbit

Gravitomagnetic field is covariantly split in the {\it intrinsic} and {\it extrinsic} parts, which are generated by rotational and translational currents of matter respectively. The {\it intrinsic} component has been recently discovered in the LAGEOS spacecraft experiment. We discuss the method of detection of the {\it extrinsic} tidal component with the lunar laser ranging (LLR) technique. Analysis of the gauge residual freedom in the relativistic theory of three-body problem demonstrates that LLR is currently not capable to detect the {\it extrinsic} gravitomagnetic effects which are at the ranging level of few millimeters. Its detection requires further advances in the LLR technique that are coming in the next 5-10 years.