4 Results

In this section we plot the separated $E(H-K)_{\rm cs}$ against the equivalent widths of various emission features from the spectra presented in Papers II-IV and test to see if the quantities are related by using a Spearman rank test which is non-parametric. Where we believe a linear correlation exists we calculate best fit lines using a least squares fit weighted to errors in the ordinate axis. For each of these linear cases we display the calculated equations above each plot and the standard deviation, in the ordinate direction, about the fitted line in the residuals section of the plot. These values are also recorded in Table 2.

We removed the underlying photospheric absorption by combining our spectra with values of corresponding B star absorption lines tabulated in Hanson et al. (1996) and Hanson et al. (1998). We note here two stars whose results do not conform with rest of the sample. When plotted in our preliminary results, specifically Fig. 2 (right panel), 5, 6, 8 & 9: BD $+37\ 03856$ has an anomalous spectral energy distribution (SED) for a Be star. Each of the other stars in our sample has a SED of the the form J>H>K or K>H>J, where JHK are fluxes, however the SED of this star is such that J>H<K. This star also exhibited the most extreme point on our plots having the most negative $E(H-K)_{\rm cs}$, we therefore remove the point from our plots but for completeness list the object in Table 1. A possible explanation for this SED is thermal emission from dust, although we note that the object is not in the IRAS point source catalogue.

BD $+57\ 00681$ also exhibits a large, negative $E(H-K)_{\rm cs}$. The random error on this object is small at <1% and it lies a long way from our calculated fit. We find no reason however to remove the point from the data set.

4.1 Br$\gamma$, Br11, Br18 and H$\alpha$

Figure 5: Circumstellar excess versus H$\alpha$ EW/Å (left panel) and Br$\gamma$ EW/Å (right panel). The fitted lines are least squares fits weighted to the ordinate axis errors

Figure 6: Circumstellar excess versus Br11 EW/Å (left panel) and Br18 EW/Å (right panel). The fitted lines are least squares fits weighted to the ordinate axis errors

Our Br$\gamma$ EW come from Paper II and have an error of $\sim$10%. Br18 EW and Br11 EW are extracted from Paper III and also have errors of $\sim$10%. H$\alpha$ data come from Paper V and again have an error of $\sim$10%.

We plot the $E(H-K)_{\rm cs}$ against Br$\gamma$, Br18, Br11 and H$\alpha$in Figs. 5 and 6. There is an obvious correlation in each of the plots, re-enforced by the >$4.5\sigma$confidence levels produced by the Spearman tests. We fit lines of least squares, weighted to the ordinate axis errors, to the data in order to ascertain any linear correlation.

It is worthy of note that van Kerkwijk et al. (1995) present similar results to ours regarding the relationship between line equivalent width and continuum excess, although they present H$\alpha$ versus J-L excess emission. In that study, as with our results there is a strong (apparently linear) correlation between the lines and continuum excess. This correlation is not surprising, as the hydrogen lines and the near-IR excess continuum are typically formed in the same regions of the disc. Van Kerkwijk et al. (1995) also show the line-excess continuum correlations for two popular models of the disc - Waters' disc model (1986a), and the Poeckert & Marlborough model (PM) (1978) - and find that neither can replicate the results particularly well. The PM model produces too little line emission for a given continuum excess, and the disc model produces too much line emission, unless a large density gradient is used (a radial density power law with an index larger than 3.5 seems to be necessary which appears to be inconsistent with the results obtained from IRAS data). Whilst there is a large scatter in the data, we present the linear best-fit relationship from our results which any new model of Be star discs should attempt to reproduce.

4.2 Helium I 2.058 $\mu$m

The HeI 2.058 $\mu$m emission is confined to the early stars of the sample, being seen in 19 of the 34 stars with spectral types determined in Paper I to be earlier than B2.5. In Fig. 7 we plot $E(J-H)_{\rm cs}$ versus HeI 2.058 $\mu$m. We note that data in Hanson et al. (1996) shows the absorption lines for the HeI 2.0581 $\mu$m line to be negligible and so no correction has been made. Figure 7 has r=0.38 and is therefore correlated at >$4\sigma$ confidence level, although no linear correlation seems to exist. This is likely due to the fact that the HeI 2.058 $\mu$m line is extremely sensitive to changes in the UV continuum and optical depth (Paper II).

4.3 Spectral type

Figure 8 shows a plot of circumstellar excess against spectral type, we note that there appears to be no linear correlation and that r=4 10^-3 which gives a confidence level of >$2\sigma$. However, the overall shape of the distribution is similar to that seen in Papers II and III for the strength of the Balmer series lines, with a broader range of excess around B1-B2. We note also that there is no correlation between luminosity class and circumstellar excess.

v $\mathsf {sin}$(i) & $\omega$ $\mathsf {sin}$(i)

Stellar rotation has been fundamentally linked with the generation of the Be phenomenon (e.g. Slettebak 1988, 1982). If rotation is the sole cause of the phenomenon then we would expect to see a strong correlation between rotational velocity and circumstellar excess. We therefore also plot $E(H-K)_{\rm cs}$ versus $v \sin (i)$: see Fig. 9, where velocity data are extracted from Papers I and II. Dougherty et al. (1994), Waters (1986b) and Gehrz et al. (1974) all find a similar result, that there is no correlation between $v \sin (i)$ and colour excess. However from Spearman rank tests we are able say that $v \sin (i)$ and $E(H-K)_{\rm cs}$ (see Fig. 9, left panel) are related at a >$4.5\sigma$ confidence level. In an attempt to remove spectral type dependence we also plot $\omega\sin(i)$ versus $E(H-K)_{\rm cs}$, see Fig. 9, right panel, where $\omega\sin(i) = v\sin(i)/v_{\rm crit}$ with $v_{\rm crit}$ taken from Paper II, (see Porter 1996 for a discussion of the merits of using $\omega\sin(i)$ compared to $v \sin (i)$). While this plot exhibits a smaller scatter than our $v \sin (i)$ plot it is still correlated at confidence level of >$4.5\sigma$.

Figure 7: Circumstellar excess versus HeI 2.058 $\mu$m EW/Å

Figure 8: Circumstellar excess versus spectral subtype

Figure 9: Circumstellar excess versus $v \sin (i)$ (left panel) and $\omega\sin(i)$ (right panel)

 

 

Table 2: We present our data table in order of descending Spearman rank coefficient, r (Col. 2), Col. 3, sig. level, are the one-tailed significance levels of the Spearman rank Correlation and are extracted from Wall (1996). Column 4 are the one-tailed confidence levels of our correlations and are extracted from Wall (1979) and Col. 5 is the standard deviation about the fitted least squares fits line in the ordinate direction. In Col. 6 we present the gradient of those fits and Col. 7 is the intercept

Plot	r	Sig. level	Conf. level	Stan. dev.	Gradient	Intercept
$E(H-K)_{\rm cs}$ vs. H$\alpha$	0.83	0.005	$>4.5\sigma$	$\pm$0.07	$0.006\pm0.000$	$-0.030\pm0.002$
$E(H-K)_{\rm cs}$ vs. Br11	0.75	0.005	$>4.5\sigma$	$\pm$0.07	$0.023\pm0.000$	$-0.072\pm0.003$
$E(B-V)_{{(H-K)}_{\rm is}}$ vs. $E(B-V)_{\rm is+cs}$	0.74	0.005	$>4.5\sigma$	$\pm$0.25	$0.814\pm0.007$	$0.168\pm0.003$
$E(H-K)_{\rm cs}$ vs. Br18	0.70	0.005	$>4.5\sigma$	$\pm$0.07	$0.028\pm0.000$	$0.017\pm0.002$
$E(H-K)_{\rm cs}$ vs. Br$\gamma$	0.69	0.005	$>4.5\sigma$	$\pm$0.09	$0.010\pm0.000$	$0.008\pm0.001$
$E(H-K)_{\rm cs}$ vs. $v \sin (i)$	0.58	0.005	$>4.5\sigma$	-	-	-
$E(H-K)_{\rm cs}$ vs. $\omega\sin(i)$	0.54	0.005	$>4.5\sigma$	-	-	-
$E(B-V)_{\rm is+cs}$ vs. sodium EW	0.56	0.005	$>4.5\sigma$	$\pm$0.24	0.39	0.13
$E(H-K)_{\rm is}$ vs. sodium EW	0.47	0.005	$>4.5\sigma$	$\pm$0.06	$0.065\pm0.005$	$0.049\pm0.003$
$E(B-V)_{{(H-K)}_{\rm is}}$ vs. sodium EW	0.45	0.005	$>4.5\sigma$	$\pm$0.31	0.37	0.15
$E(H-K)_{\rm cs}$ vs. HeI EW	0.38	0.37	$>4\sigma$	-	-	-
$E(H-K)_{\rm cs}$ vs. spectral type	4 10-3	-	$>2\sigma$	-	-	-
$E(H-K)_{\rm cs}$ vs. sodium EW	-0.08	-	$<1\sigma$	-	-	-