In this section we plot the separated E(H - K)cs against the equivalent widths of various emission features from the spectra presented in Papers II, III and 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 7.2.
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 Figures 7.3, 7.7, 7.8, 7.9, 7.10, 7.12, 7.13 & 7.14: 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)cs, we therefore remove the point from our plots but for completeness list the object in Table 7.1. 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)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.
, Br11, Br18 and H
Our Br EW come from (Clark and Steele, 2000) and have an error of ~10%. Br18 EW
and Br11 EW are extracted from (Steele and Clark, 2001) and also have errors of
~10%. H data come from (Steele and Negueruela, 2002) and again have an error of
~10%.
We plot the E(H - K)cs against Br, Br18, Br11 and H in Figures 7.7, 7.8, 7.9 and 7.10.
There is an obvious correlation in each of the plots, re-enforced by the > 4.0
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 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 (Waters, 1986b), and the Poeckert & Marlborough model (PM)
(Poeckert and Marlborough, 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.
mm emission is confined to the early stars of the
sample, being seen in 19 of the 34 stars with spectral types determined in (Steele
et al., 1999) to be earlier than B2.5. It should be noted that 15 stars showed no Helium I
2.058m emission within the accuracy of our measurements. Whilst a reading of
absolutely zero is unlikely we believe that more accurate readings of these objects will
only reveal a gaussian distribution of the objects about zero. Our correlation test
will therefore remain uneffected. In Figure 7.11 we plot E(J - H)cs versus HeI
2.058m. We note that data in Hanson et al. 1996 shows the absorption lines for the HeI
2.0581m line to be negligible and so no correction has been made. Figure 7.11 has
rs = 0.38 and is therefore correlated at > 3.5 confidence level, although no linear
correlation seems to exist. This is likely due to the fact that the HeI 2.058m line is
extremely sensitive to changes in the UV continuum and optical depth (Clark and
Steele, 2000).
Figure 7.12 shows a plot of circumstellar excess against spectral type, we note that there
appears to be no correlation and that rs = 4 × 10-3 which gives a confidence level of < 1.
However, the shape of the distribution is similar to that seen in (Clark and Steele, 2000)
and (Steele and Clark, 2001) 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.
sin(i) confidence level. In an attempt to remove spectral
type dependence we also plot sin(i) versus E(H - K)cs, see Figure 7.14, where
sin(i) = v sin(i)/vcrit with vcrit taken from (Clark and Steele, 2000), ( see Porter 1996 for a
discussion of the merits of using sin(i) compared to v sin(i)). This plot exhibits a
smaller scatter than our v sin(i) plot, it is also correlated at confidence level of
> 4.0.
|
Plot r Sig. level Conf. level Stan. dev. Gradient Intercept ____
E(H - K)cs vs H
0.83 > 0.0005 > 4.0 ±0.07 0.006±0.000 -0.030±0.002
E(H - K)cs vs Br11 0.75 > 0.0005 > 4.0 ±0.07 0.023±0.000 -0.072±0.003
E(B - V )(H-K)
is vs E(B - V )is+cs 0.74 > 0.0005 > 4.0 ±0.25 0.814±0.007 0.168±0.003
E(H - K)cs vs Br18 0.70 > 0.0005 > 4.0 ±0.07 0.028±0.000 0.017±0.002
E(H - K)cs vs Br 0.69 > 0.0005 > 4.0 ±0.09 0.010±0.000 0.008±0.001
E(H - K)cs vs v sin(i) 0.58 > 0.0005 > 4.0 - - -
E(H - K)cs vs sin(i) 0.54 > 0.0005 > 4.0 - - -
E(B - V )is+cs vs sodium EW 0.56 > 0.0005 > 4.0 ±0.24 0.39 0.13
E(H - K)is vs sodium EW 0.47 > 0.0005 > 4.0 ±0.06 0.065±0.005 0.049±0.003
E(B - V )(H-K)
is vs sodium EW 0.45 > 0.0005 > 4.0 ±0.31 0.37 0.15
E(H - K)cs vs HeI EW 0.38 0.005 > 3.5 - - -
E(H - K)cs vs spectral type 4 × 10-3 - < 1 - - -
E(H - K)cs vs sodium EW -0.08 - < 1 - - -__________
Table 7.2: 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). Col 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. |
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