  The technique of radial velocity (RV) has produced spectacular discoveries of
short-period Jovian mass objects around a fraction (5 to 10%) of nearby G
stars. Although we expect Jovian planets to be located in long-period orbits of
decades or longer (if our solar system is any guide), detecting such planets
with RV technique is difficult due to smaller velocity amplitudes and the
limited temporal baseline (5-10 yr) of current searches relative to the
expected orbital periods. In this paper, we develop an analytical understanding
of the sensitivity of RV technique in the regime where the the orbital period
is larger than the total baseline of the survey. Moreover, we focus on the
importance of the orbital phase in this ``long-period'' regime, and develop a
Least Squares detection technique based on the amplitude and phase of the
fitted signal.
  To illustrate the benefits of this amplitude-phase analysis, we compare it to
existing techniques. Previous authors (e.g. Nelson & Angel 1998) have explored
the sensitivity of an amplitude-only analysis using Monte Carlo simulations.
Others have supplemented this by using the slope of the linear component of the
fitted sinusoid in addition (e.g. Walker et al. 1995; Cumming et al. 1999). In
this paper, we illustrate the benefits of Least Squares over periodogram
analysis, and demonstrate the superiority of an amplitude-phase technique over
previous analyses.
