  Satellite tracking involves sending electromagnetic signals to Earth. Both
the orbit of the spacecraft and the electromagnetic signals themselves are
affected by the curvature of spacetime. The arrival time of the pulses is
compared to the ticks of local clocks to reconstruct the orbital path of the
satellite to high accuracy, and to implicitly measure general relativistic
effects. In particular, Schwarzschild space curvature (static) and
frame-dragging (stationary) due to the planet's spin affect the satellite's
orbit. The dominant relativistic effect on the path of the signal photons is
Shapiro delay due to static space curvature. We compute these effects for some
current and proposed space missions, using a Hamiltonian formulation in four
dimensions. For highly eccentric orbits, such as in the Juno mission and in the
Cassini Grand Finale, the relativistic effects have a kick-like nature, which
could be advantageous for detecting them if their signatures are properly
modeled as functions of time. Frame-dragging appears, in principle, measurable
by Juno and Cassini, though not by Galileo 5 and 6. Practical measurement would
require disentangling frame-dragging from the Newtonian 'foreground' such as
the gravitational quadrupole which has an impact on both the spacecraft's orbit
and the signal propagation. The foreground problem remains to be solved.
