3 Results

Table 3 gives the parameters for the four main two-step IMF cases we have computed with the MC approach using Eqs. (1) to (3) and $\tau = 0.10$. $N_{\rm max} = 10$ was chosen to represent the relatively wide cluster size distribution in Mot98. $N_{\rm ave}$ ($\approx$5) is the total number of stars $N_{\rm stars}$ plus the total number of brown dwarfs $N_{\rm bd}$ divided by the total number of clusters (100000) in the ensemble. A fixed N = 5 provides a good comparison with the variable N cases. $M_{\rm min} = 0.01~M_{\odot}$ and $0.08~M_{\odot}$ were used in order to explore differences between cases where brown dwarfs do and do not participate equally with stars in the pairing process (see also Sterzik & Durisen 1999). Choosing $M_{\rm min} = 0.01~M_{\odot}$ yields about 20 to 25% brown dwarfs for the SMS in Eq. (3).

 

 

Table 3: Ensembles of 100000 clusters calculated with the MC approach

N-type	N or $N_{\rm max}$	$M_{\rm min}$	$N_{\rm stars}$	$N_{\rm bd}$	$N_{\rm ave}$
fixed	5	0.08	500000	0	5
	5	0.01	394252	105748	5
variable	10	0.08	480406	0	4.80
	10	0.01	441325	87830	5.30

 

 

Table 4: Distribution of cluster sizes and average cluster masses for $N_{\rm max} = 10$

N	$M_{\rm min} = 0.08$	$M_{\rm min} = 0.01$
1	6827   0.86   0.86	5117   0.92   0.92
2	15846   0.88   0.44	11491   0.91   0.45
3	17893   1.03   0.34	14772   1.05   0.35
4	14124   1.40   0.35	13952   1.29   0.32
5	10054   1.91   0.36	11556   1.68   0.36
6	7685   2.56   0.44	9539   2.20   0.37
7	6840   3.33   0.48	8533   2.88   0.41
8	6494   4.00   0.50	8037   3.55   0.44
9	6870   4.62   0.51	8139   4.13   0.46
10	7367   5.29   0.53	8864   4.66   0.47

Table 4 shows the distribution of cluster sizes for $N_{\rm max} = 10$. The first number in each entry is the number of clusters of size N; the second number is the average $M_{\rm tot}$ for clusters of this size in the ensemble; and the third entry is the average stellar mass for clusters of this size. The modal cluster size is only N = 3 in both cases, but the cluster size distribution is somewhat more uniform with brown dwarfs. Both cluster size distributions tend to level off to a roughly constant value as $N \longrightarrow N_{\rm max}$. This means that imposition of an $N_{\rm max}$ is somewhat artificial, but $N_{\rm max} = 10$ is roughly consistent with the range of apparent cluster sizes observed by Mot98. Note that the $f_{\rm c}(M_{\rm tot})$ in Eq. (2) predicts an overall average cluster mass of 2.236 $M_{\odot}$.

3.1 Comparison of the resultant ${\mathsfsl f}_{\mathsf i}{\mathsfsl {(M)}}$

Figure 2 shows the IMF's for the four cases in Table 3. Recall from Fig. 1 that parameters were optimized to give a decent f_i(M) for the variable N case without brown dwarfs. Use of a fixed N = 5 degrades the agreement of the IMF with KTG93. The f_i for fixed N has too shallow a slope for moderate to high masses. For large $M_{\rm tot}$, disallowing N > 5 forces the creation of more high-mass stars at the expense of smaller masses. Introducing brown dwarfs does not degrade the variable N IMF as seriously as imposition of a fixed N. The deviation consists of a slightly shallower slope at high masses. We do not attempt to fit any particular form of the IMF in the brown dwarf region, because this is still uncertain observationally (Marcy et al. 2000).

Figure 2: The resultant two-step fi for the four cases in Table 3. The solid curves are for variable N; the dotted curves for fixed N=5. The heavy curves have no brown dwarfs; the lighter curves extending to lower masses are cases which include brown dwarfs. All histograms are fractions of stars or brown dwarfs in each logarithmic mass interval of 0.05 dex

3.2 Binary fractions

Table 5 gives the overall binary fractions as a function of mass and spectral type along the main sequence for the four cases from Table 3. Tabulated are the number of MC binaries in the ensemble whose primaries lie in the stated mass range divided by the total of these binaries plus all single stars in the mass range. Thus a star in the mass range which is the secondary mass of a binary is not included in the BF statistic. This is consistent with the way that observational results are presented in DM91, FM92, and L97. The fixed N=5 cases are denoted by a "5''; the variable N cases by "varN''. Cases run with brown dwarfs have the additional label "bd''. The mass boundaries in Table 5 are based on the one Gigayear isochrone of Yi et al. (2001) for solar composition. Allen (1976) is used to calibrate the conversion from effective temperature to spectral type. The boundary between G and $\ge$F is set a bit higher than the G0 mass of 1.14 $M_{\odot}$ in Yi et al. because DM91 extend their statistics for solar-type stars into late F spectral types. The M stars are split near the mid-M's because there is a steep gradient of BF in our results across the M star mass range while the observational studies of M star binary frequency (FM92, L97) are biased towards early-M types.

Figure 3: The BF(M)'s for the four cases in Tables 3 and 5. The solid curves are for variable N; the dotted curves for fixed N=5. The heavy curves have no brown dwarfs; the lighter curves are the cases which include brown dwarfs. For comparison, the data points from DM91, FM92, and L97 are plotted as cross-hairs

 

 

Table 5: Binary fractions as a function of spectral class

sp-type	masses	5	5bd	varN	varNbd
L	<0.08	-	2.2(-4)	-	2.6(-4)
late M	0.08-0.27	0.11	0.12	0.04	0.05
early M	0.27-0.47	0.35	0.45	0.31	0.35
K	0.47-0.84	0.54	0.61	0.48	0.51
G	0.84-1.20	0.73	0.77	0.64	0.67
$\ge$ F	>1.20	0.92	0.93	0.86	0.87

Figure 3 illustrates the BF(M)'s for stars in more detail. The variable N ensembles fit the BF data on M and G stars rather well. The introduction of brown dwarfs to either fixed N or variable N cases tends to increase the binary fractions of the stars by amounts ranging from 10's% down to $\sim$1% as mass increases. This is simply because brown dwarfs are rarely binary primaries in our ensembles, and the stellar mass objects must take up the slack. The effect is larger for fixed N. In fact, the solar-type BF for the fixed N case becomes uncomfortably large, especially with brown dwarfs. At first glance, the larger BF for all masses in the fixed N cases may seem inconsistent, because, with $N_{\rm ave} \approx$5 (see Table 3) for all ensembles, the total number of binaries should be similar. The systematically larger BF(M) is possible, however, because the f_i(M) are also significantly different between the fixed N and variable N cases (see Fig. 2). The total number of binaries remains 100000 in all cases, as required.

3.3 Relative importance of incomplete biasing, a mix of cluster sizes, and the two-step process

Our approach differs in several respects from McDC93. They used a one-step process with complete dynamical biasing and compared results for different fixed cluster sizes. For our variable N cases, we consider a mix of cluster sizes together with incomplete biasing and two-step mass selection. To disentangle the relative importance of these effects for each case in Tables 3 and 5, we have used the identical sets of stars to re-select binaries in two ways: 1) We use the same clusters selected by the two-step process but assume only M₁M₂ binaries result (i.e., we use weights of 1.0, 0.0, and 0.0 for the mass pairings in Table 1). We refer to this as "complete'' biasing statistics in Table 6. 2) We use the same stars and the same distribution of cluster sizes but randomize the star/cluster assignments and assign binaries by assuming complete dynamical biasing. This reproduces what McDC93 would obtain by choosing stars randomly from the same f_i and creating the same distribution of cluster sizes, but without imposing a cluster total mass constraint.

 

 

Table 6: Comparison with binary fractions from complete biasing and a one-step IMF

sp-type	statistics	5	5bd	varN	varNbd
early M	two-step	0.35	0.45	0.31	0.35
	complete	0.35	0.46	0.32	0.35
	McDC93	0.47	0.47	0.22	0.25
G	two-step	0.73	0.77	0.64	0.67
	complete	0.79	0.84	0.66	0.70
	McDC93	0.94	0.94	0.85	0.86

Table 6 compares the BF's from the three approaches. We single out the early-M stars and G stars because these are the ones for which some observational data are published. A plot showing the effect on BF(M) in more detail is given in Fig. 4 for the varN case. The effect of going from incomplete to complete biasing alone is an increase in the BF's of less than about 1-2% and 4-9% for early-M stars and G stars, respectively, typically within the uncertainties in the observed BF's. However, use of full McDC93 statistics results in dramatic changes. For varN, it shifts the BF for early-M stars downward and the BF for G stars upward by about 30% each! The agreement with observations is this case is seriously degraded, especially for the G stars. We conclude from Table 6 that the acceptable character of our two-step BF(M) is due almost entirely to the imposition of the $M_{\rm tot}$ constraint in the two-step process, not to the mixture of cluster sizes nor to the relaxation of "complete'' biasing.

Figure 4: A comparison of BF(M)'s for the varN case using our "two-step'' approach with incomplete biasing (heavy solid curve) and with "complete'' biasing (thin solid curve). The dotted curve "McDC93'' is obtained from the same cluster size and stellar mass distributions by choosing stars in a one-step process without an $M_{\rm tot}$ constraint using complete biasing. The cross-hairs have the same meaning as in Fig. 3

3.4 The secondary mass distribution

Another important characteristic of binary stars is the distribution of their mass ratios, $q = M_{\rm s}/M_{\rm p}$, where $M_{\rm p}$ is the mass of the primary and $M_{\rm s}$ is the mass of the secondary defined so that $M_{\rm p}$ is always $\ge M_{\rm s}$. The observed q-distributions for M stars in FM92 and L97 are flat near q = 1 and drop smoothly to zero for mid-q values. For G stars, DM91 indicate a strong tendency for the distribution to rise smoothly to a peak at about q = 0.25. McDC93 have already shown that, for fixed N, choosing stellar masses from an IMF in a one-step process tends to produce such systematics in the q-distribution as a function of primary mass. As $M_{\rm p}$ increases, the q-distribution becomes more and more peaked toward low q. Figure 5, for the varN case from Tables 3 and 5, shows similar trends in the q-distributions for either our two-step or the McDC93 one-step approach. The shapes are perhaps marginally better in the two-step case, because the peak in the G star distribution shifts to lower q compared to McDC93.

Figure 5: Histograms of q-distributions for the varN case. Three spectral class bins from Table 5 are shown, early M, G, and $\ge$F. Distributions are all normalized to unity for the q = 0.95 to 1.00 bin in order to emphasize comparison of the distribution shapes. The heavy curves are for the two-step process; the thin curves are for the equivalent McDC93 method of selecting cluster members

As illustrated in Fig. 6, the introduction of brown dwarfs and a fixed cluster size, both tend to make the G star q-distribution more peaked at low q. Unfortunately, as shown in Table 5, this comes at the cost of making the G star BF higher and hence less acceptable.

Figure 6: Histograms of the G star q-distribution for the four cases in Tables 3 and 5. The solid lines are for variable N; the dotted lines for fixed N=5. The heavy lines have no brown dwarfs; the lighter lines are the cases which include brown dwarfs. For comparison, the data from DM91 are plotted as cross-hairs. All distributions are normalized to the same total number of binaries as for varN

3.5 Varying input parameters

In this subsection, we report how changes in each major parameter affects the quality of results when they are varied one at a time about the values used for the varN calculation. For reference, the standard varN value of the parameter is given in parentheses at the start of each paragraph. Tolerance $\tau$ (0.10). The low and variable efficiency of gas utilization by star formation makes it difficult to know what value of $\tau$ to use. Increasing $\tau$ from 0.1 to 0.25 weakens the $M_{\rm tot}$ constraint, and distributions tend to shift somewhat toward one-step results. The f_i looks more like $f_{\rm s}$ and has a distinct deficit of stars above about 5 $M_{\odot}$ compared with the KTG93 IMF. The BF's for early-M, K, and G stars all increase by about 6 or 7%. When $\tau$ is decreased from 0.10 to 0.05, the IMF is slightly closer to KTG93, especially near a solar mass. Although the BF's do not change to within a percent when $\tau$ is reduced, the q-distribution for G stars becomes slightly closer to the DM91 distribution. Maximum Cluster Size $N_{\rm max}$ (10). Changing $N_{\rm max}$ from 10 to 15 has a similar but greater effect on f_i as an increase in $\tau$. It effectively weakens the $M_{\rm tot}$ constraint and causes $f_{i} \rightarrow f_{\rm s}$. For $N_{\rm max} = 15$, there is a severe deficit of stars more massive than a few $M_{\odot}$ compared to KTG93. On the other hand, when $N_{\rm max}$ is decreased to 7, the $M_{\rm tot}$ constraint strengthens to produce a substantial excess of high-mass stars over KTG93. Although the G star BF fluctuates nonmonotonically by a few percent as $N_{\rm max}$ increases from 7 to 15, the BF of early-M stars systematically decreases from 0.38 to 0.27. The q-distribution of the G stars looks more like the DM91 distribution as $N_{\rm max}$ increases. CMS Power-Law Index $\alpha_{\rm c}$ (1.5). As $\alpha_{\rm c}$ increases from 1.1 to 2.0, f_i goes from a severe overabundance to a severe deficit relative to KTG93 for stars of a solar mass or above, while BF's increase systematically. The early-M star BF increases from 0.20 to 0.48, while the G star BF rises more moderately from 0.57 to 0.70. The resemblence of the G star q-distribution to DM91 improves as $\alpha_{\rm c}$ goes up. Upper Mass Limit for $M_{\rm tot}$ (10 $M_{\odot}$). Everything is degraded by decreasing the upper limit for $M_{\rm tot}$ from 10 to 5 $M_{\odot}$. The BF for G stars climbs to 0.83, the IMF is a very poor match to KTG93, and the G star q-distribution has a broad, low peak near q = 0.4. When the upper limit for $M_{\rm tot}$ is increased from 10 to 15 $M_{\odot}$, the $M_{\rm tot}$ constraint forces a large excess of massive stars over KTG93, and the BF's for early-M, K, and G stars drop by 8 to 11%. The q-distribution for the G stars becomes a poorer fit to DM91. Upper Mass Limit for the Stars (10 $M_{\odot}$). In Sect. 2.3, the upper mass limits for the clusters and stars are taken to be the same. If the stellar upper mass limit is reduced from 10 to 5 $M_{\odot}$, there is little impact on any results, except that there is now a larger excess of stars between 2 and 5 $M_{\odot}$ over KTG93. Changes in the BF's are only at the 1 to 3% level. SMS Peak $\mu$ (log₁₀0.20). The parameter $\mu$ is defined so that it represents the peak, in solar units, that the lognormal $f_{\rm s}$ has when plotted in a logM histogram. In fact, for our standard choice of all other parameters, the f_i exhibits essentially this same peak. For $\mu = \log_{10}0.30$, the relatively strong resulting deficit of low-mass stars is compensated by a large excess in higher-mass stars. The IMF is a rather poor match to KTG93. For $\mu = \log_{10}0.15$, f_i stays closer (within about 20%) to the KTG93 over a much wider range of masses. The $\mu = \log_{10}0.30$ case tends to drive the early-M, K, and G star BF's down by roughly 10%, while $\mu = \log_{10}0.15$ increases them by 5 to 7%. The G star q-distribution is more like DM91 for $\mu = \log_{10}0.15$. SMS Width $\sigma (\log_{10}0.30$). Setting $\sigma$ to $\log_{10}0.25$ effectively makes the lognormal distribution in (3) broader and produces an excess of high-mass stars over KTG93. The M star and G star BF's go up and down, respectively, by a few percent each; and the G star q-distribution becomes somewhat more like DM91. A $\sigma$ change in the other direction, to $\log_{10}0.40$, which narrows the distribution, has quantitatively similar effects in the opposite direction, except that the M star BF is essentially the same; and the IMF does not differ as much from KTG93. Overall, there is a tendency for any change which produces fewer high-mass stars in the IMF to increase the BF for G stars while making their q-distribution look more like the DM91 distribution. One interesting exception is a decrease in $\tau$. As the $M_{\rm tot}$ constraint is strengthened by decreasing $\tau$ below 0.10, the IMF and G star BF remain essentially the same, while the q-distribution improves slightly. In effect, the degree of dynamical biasing remains nearly the same, while the two-step nature of the mass selection is enhanced. The tolerance $\tau$ seems unique in this respect. In a preliminary report on this work, Durisen et al. (2000) presented results for a "best case'' choice of parameters. We have made a similar effort here by making several minor tweaks to the varN parameters as follows: $\tau = 0.05$, $N_{\rm max} = 12$, $\alpha_{\rm c} = 1.7$, and $\mu = \log_{10}$0.18. The upper mass cutoff for the CMS was increased to 15 $M_{\odot}$, and the upper mass limit for the SMS was reduced to 7.5 $M_{\odot}$. We put "best case'' in quotes to indicate there has been no systematic optimization. The IMF and q-distributions are not markedly changed; but, as shown in Fig. 7, the observations of BF(M) can be matched well by these minor changes.

Figure 7: The BF(M) for the "best case'' using our two-step approach with incomplete biasing (heavy solid curve). The dotted curve "McDC93'' is obtained from the same cluster size and stellar mass distributions by choosing stars in a one-step process without an $M_{\rm tot}$ constraint. The cross-hairs have the same meaning as in Fig. 3

3.6 Effects of the ${\mathsfsl f}_{\mathsf c}{\mathsfsl {(M}}_{\mathsf {tot}}{\mathsfsl )}$ and ${\mathsfsl f}_{\mathsf s}{\mathsfsl {(M)}}$ functional forms

We test the sensitivity of our two-step IMF results to the shapes chosen for the CMS and SMS by computing three additional ensembles. We refer to them by names which designate first the shape of the CMS and then the shape of the SMS. In this notation, the varN ensemble becomes "Power/Log'', meaning that the CMS is a power law and the SMS a lognormal. These designations are further abbreviated in obvious ways as needed. For the new ensembles with variable N, the $\tau$, the $N_{\rm max}$, and the mass limits of the distributions are kept the same as for the varN case; and all power laws have the same index $\alpha = 1.5$. When a lognormal is used for the SMS, it is identical to (3). However, this distribution peaks at too low a mass and is too narrow for a proper CMS. So, when a lognormal is used for the CMS, we choose $\sigma = \log_{10}$0.38 and $\mu = \log_{10}$0.90. These values produce a reasonable fit to the lognormal obtained by Klessen et al. (1998) from simulations of gravitational interactions and turbulence in a molecular cloud, after it is scaled to give a peak cluster mass at 0.9 $M_{\odot}$.

Figure 8: The resultant two-step fi for the different choices of CMS and SMS in Table 7. The heavy solid line is varN, the standard power-law CMS with a lognormal SMS. The thin solid curve is LogPow; the heavy dotted, PowPow; the light dotted LogLog. All histograms are fractions of stars in each logarithmic mass interval of 0.05 dex

Figure 8 compares the two-step IMF's from the four ensembles. For reference, remember, from Fig. 1, that the standard Power/Log case does a reasonably good job of reproducing the KTG93 IMF. The Log/Power IMF actually fits the piecewise power-law KTG93 better at low masses but is slightly worse than Power/Log at the high-mass end. The Log/Log and Power/Power cases exhibit stronger, possibly unacceptable deviations. Power/Power fits KTG93 well enough at low and moderate masses, but it severely overproduces stars more massive than about 2 $M_{\odot}$. The Log/Log case is very similar to Power/Log at low and moderate masses, but underproduces massive stars. It might be possible to improve any one of these cases by varying parameters. We have not done so here, because there is little justification yet for any particular choices.

 

 

Table 7: Binary fractions as a function of spectral class for different CMS and SMS choices

sp-type	PowLog	LogPow	PowPow	LogLog
late M	0.04	0.04	0.04	0.03
early M	0.31	0.48	0.41	0.38
K	0.48	0.66	0.54	0.63
G	0.64	0.75	0.65	0.77
$\ge$ F	0.86	0.81	0.81	0.88

Despite the range of mass spectra, the two-step mass selection results in a similar BF pattern for these cases, as shown in Table 7. The range of BF's for each spectral type in Table 7 is similar to the range in Table 5. Use of a lognormal for the CMS does about as much damage to the G star BF as imposition of a fixed N = 5. Tests of these ensembles for the effects of complete biasing and the use of one-step McDC93 statistics yield results similar to those in Fig. 4. The form of BF(M) is strongly influenced by the $M_{\rm tot}$ constraint in the two-step mass selection. Compared with McDC93, two-step mass selection tends to increase the M star BF and decrease the G star BF.

Figure 9: Histograms of the G star q-distribution for the four cases in Table 7. The heavy solid line is varN, the standard power-law CMS with a lognormal SMS. The thin solid curve is LogPow; the heavy dashed, PowPow; the heavy dotted LogLog. For comparison, the data from DM91 are plotted as cross-hairs. For comparison purposes, all distributions are normalized to the same total number of binaries as for varN

Figure 9 illustrates the most important qualitative as well as quantitative change that results from varying the shapes of the mass spectra. All the cases other than Power/Log have G star q-distributions that are much more like the DM91 data. In the Log/Log and Log/Power cases, this comes at the expense of a G star BF of $\sim$0.76 which borders on unacceptable; but, in the Power/Power case, the BF's are as good as they are for the varN Power/Log standard. On the other hand, the Power/Power ensemble has far too many high-mass stars in its IMF. There is no one case we have identified which simultaneously satisfies all criteria for "goodness'', but the Power/Power case does demonstrate that good BF's and a good G star q-distribution can be obtained at the same time, something which was not evident in Sect. 3.5, when we restricted ourselves to Power/Log.