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.

31 pages (including 11 postscript figures). Submitted to ApJ