Geller et al. 10 5.3.1. Contribution from Undetected Binaries The combined RV dispersion defined in Equation 3 is also described by, σ$_{cb}$ = σ$_{c}$ + β (5) where σ$_{c}$ is the true RV dispersion of the cluster and β represents the contribution from undetected binaries within our sample. Therefore, in order to derive the true RV dispersion of the cluster we have performed a Monte Carlo analysis to determine this contribution from unde- tected binaries. Fig. 10.— The true cluster RV dispersion ( σ$_{c}$ ) plotted against the combined RV dispersion ( σ$_{cb}$ ) for a range of total binary frequencies. Each line corresponds to a different binary frequency in steps of 10%, with 0% at the far left and 100% at the far right. With the vertical gray rectangle, we plot the region included in the combined RV dispersion for M35 of 0 . 81 ± 0 . 08 km s − $^{1}$. The diagonal gray region covers the pos- sible lines within our extrapolated true binary frequency in M35 of 66% ± 8% (derived assuming the M35 binaries follow a Duquennoy & Mayor (1991) period distribution). Finally, we plot the resulting true RV dispersion in M35 of 0.65 ± 0.10 km s − 1 with the black point at the intersection of these two shaded regions. analyses for other star clusters, provided that the binary population is consistent with the Duquennoy & Mayor (1991) field binaries. 5.3.2. True Radial-Velocity Dispersion We first create a set of simulated binaries with or- bital parameters distributed according to the Galactic field solar-type binaries studied by Duquennoy & Mayor (1991). Specifically, these binaries have a log-normal pe- riod distribution centered on log( P [days] ) = 4.8 with σ = 2 . 3, and a Gaussian eccentricity distribution cen- tered on e = 0 . 3. For binaries with periods below the circularization period of 10.2 days (Meibom & Mathieu 2005), we set the eccentricity to zero. We use only solar- mass primary stars, and a distribution in secondary mass between 0.08 - 1 M ⊙ described by a Gaussian centered on M 2 = 0 . 23 M ⊙ with σ = 0 . 42 M ⊙ (Kroupa et al. 1990). Duquennoy & Mayor (1991) found this Gaussian to be the best fit to their solar-type field binaries, and this dis- tribution is also consistent with that of Goldberg et al. (2003) for their field binaries with primary masses > 0.67 M ⊙ . The orbital inclinations and phases of the binaries are chosen randomly. We then generate three RVs for these simulated binaries distributed in time according to the actual distribution of our first three observations for stars in M35. The majority of the SM stars in our sample have only three observations. To these RVs, we also add a random error generated from a Gaussian centered on zero and with σ = 0 . 5 km s − $^{1}$, the RV precision for narrow- lined stars in M35. We also add a random velocity offset generated from a Gaussian centered on zero with a stan- dard deviation equal to an adopted one-dimensional RV dispersion. To date, we have detected 55 binaries in M35 out of 360 cluster members. If we assume a similar com- pleteness as in Geller & Mathieu (in preparation), for NGC 188, then we can assume that we have detected 63% of the binaries with periods less than 10 4 days (and a negligible fraction of binaries with longer peri- ods). This correction results in a binary frequency of 24% ± 3% for P < 10 4 days. This binary frequency is consistent with that of solar-type stars in the Galac- tic field Duquennoy & Mayor (1991) out to the same period limit. If we assume the M35 binaries follow a Duquennoy & Mayor (1991) period distribution, then our binary frequency for P < 10 4 days implies a total bi- nary frequency of 66% ± 8%, with the inclusion of wider binaries currently beyond our detection limits. We then take this value for the total binary frequency and correct our combined RV dispersion for undetected binaries. To this sample, we add a number of simulated single stars to produce a desired binary frequency. We generate three RVs for each single star from a Gaussian described by our precision. To the mean RV for every single star we also add a random offset described by the assumed RV dispersion in the same manner as for the simulated bi- naries. We then keep only those simulated binaries (and single stars) whose first three RVs result in an e/i < 4, and whose mean RVs are within three standard devia- tions of the mean RV from a Gaussian fit to the simu- lated RV distribution. This cutoff in standard deviation reflects our membership criterion of P$_{RV}$ ≥ 50% for the M35 observations. These binaries would be undetected within the SM sample. In the filled gray areas in Figure 10 we show the re- gions defined by our M35 combined RV dispersion and the total binary frequency. At the intersection, we plot the derived true RV dispersion in M35 of σ$_{c}$ = 0 . 65 ± 0 . 10 km s − $^{1}$. Using a flat distribution in secondary mass (and mass ratio), as has been suggested by some studies (e.g., Mazeh et al. 1992, 2003), has a negligible effect on the derived true RV dispersion. This true RV dispersion is consistent with the projected velocity dispersion of 1.0 ± 0.15 km s − $^{1}$, derived by Leonard & Merritt (1989) us- ing the proper-motion data from McNamara & Sekiguchi (1986a). We then follow the equations given above to derive β for a range of binary frequencies and velocity dispersions, σ$_{c}$ . In Figure 10, we plot the true cluster RV dispersion ( σ$_{c}$ ) as a function of the combined RV dispersion ( σ$_{cb}$ ) for a range of total binary frequencies, where each line corresponds to a different binary frequency between 0% (far left) to 100% (far right) in steps of 10%. We can then use the results shown in Figure 10 to derive the true RV dispersion for M35. Furthermore, the results shown in this figure are also applicable to RV dispersion This is the first paper in a series studying the dynam- ical state of the young ( ∼ 150 Myr) open cluster M35 6. SUMMARY