4 X-ray luminosity functions

The statistical analysis was performed with the ASURV package version 1.2 (see Feigelson et al. 1985; Isobe et al. 1986; LaValley et al. 1992). The ASURV software is particularly well suited for the study of data sets with censored points, i.e. non-detections. We exclude photons observed during the large X-ray flares presented by SNH00, i.e. for flaring stars only their quiescent radiation is taken into account.

XLF are frequently employed to characterize a stellar population. Our special interest here is to compare the XLF of the different stellar groups with respect to the following issues: (i) Are the luminosity functions of cTTS and wTTS different, (ii) how does the X-ray luminosity evolve with stellar age, (iii) how does it depend on the spectral types of the stars and their binary character.

A substantial number of stars are in the field of more than one pointed PSPC observation (see Table 8). However, every star should appear only once in the XLF. Therefore, we represent each star by its error weighted mean luminosity from all observations in which it was detected. If a star was observed in more than one observation, but not detected in any of them, we use the mean upper limit of all non-detections of this star as an estimate for its luminosity limit.

In Sect. 4.3 we will justify our assumption that the X-ray luminosity can be distributed equally among all stars in unresolved multiple systems. Therefore, if not specified otherwise, we have divided the mean X-ray luminosity by the number of components in the stellar system.

  
4.1 cTTS and wTTS in RASS and pointed PSPC data

When studying the X-ray emission of TTS in Taurus-Auriga observed during the RASS, N95 found that the wTTS are X-ray brighter than the cTTS. This is in contrast to findings in various other star forming regions (see e.g. Feigelson et al. 1993; Casanova et al. 1995; Grosso et al. 2000). This discrepancy is not yet understood. A possible explanation is that the XLF of the wTTS in Taurus-Auriga is uncomplete towards the low-luminosity end, because wTTS are not easily identified due to the lack of pronounced spectral features. In particular, many wTTS have been discovered with the EO. Therefore, even the pre-ROSAT sample studied in N95 could be biased towards X-ray bright wTTS.

Our analysis of a large set of pointed ROSAT observations allows to extend the sensitivity limit substantially with respect to the RASS. In Fig. 3 we compare the XLF of TTS derived from the pointed observations described in this paper to the results from the RASS.

Figure 3: XLF of TTS in Taurus-Auriga derived from the RASS and from pointed ROSAT PSPC observations. Shown are all cTTS and wTTS in Taurus-Auriga. The inset in the lower left shows the typical error bar.

The XLF of the RASS have been newly compiled with respect to the analysis by N95 to include all TTS discovered since then, i.e. the sample consists of all TTS from N95 plus those listed in König et al. (2001) (including both EO and ROSAT discovered TTS). N95 did include EO discovered but no ROSAT discovered TTS.

The comparison with the RASS data clearly demonstrates the better sensitivity of the pointed observations. The XLF computed from the PSPC pointings extends by $\sim$1-2 orders of magnitude further into the low luminosity regime. We reproduce the result first found by N95: in Taurus-Auriga the wTTS are clearly X-ray brighter than the cTTS.

It was noted by Feigelson et al. (1993) that the XLF can change, if the stars included in the sample were found by different methods, e.g. H$\alpha$ versus X-ray surveys. In order to overcome this bias we have computed XLF where we exclude all X-ray discovered TTS. Figure 4 shows the Kaplan-Meier Estimator (KME) for three subsets of wTTS in Taurus-Auriga: ROSAT discovered wTTS, EO discovered wTTS, and all other wTTS.

Figure 4: XLF of wTTS in Taurus-Auriga derived from pointed ROSAT PSPC observations. The three different distributions are ROSAT discovered wTTS (solid line), EO discovered wTTS (dashed line), and wTTS discovered by other means (dotted line). The inset in the lower left shows the typical error bar. All distributions are similar indicating that the inclusion of X-ray discovered TTS does not introduce a selection bias into the sample of wTTS.

The XLF of these groups do not differ significantly from each other. Therefore, we are led to conclude that the difference in the distributions of cTTS and wTTS is not due to an X-ray selection bias.

The differences to the $\rho$Oph and ChaI star forming regions (Feigelson et al. 1993; Casanova et al. 1995; Grosso et al. 2000) could also be caused by the difference in spatial extension between these two young clusters and the Taurus-Auriga region: The latter is widely dispersed, and, hence, its members may constitute a larger spread in age as compared to the more complex $\rho$Oph and ChaI regions in which the stars are probably more coeval. We can check this by selecting TTS from the central parts of the star forming region, and comparing the resulting XLF with that of the total sample. We have chosen the PSPC observations ROR 200001-0p and 200001-1p pointed on the L1495E cloud. These pointings are centered on the largest concentration of molecular material corresponding to a particular young part of the Taurus complex. In Fig. 5 we show the XLF for wTTS and cTTS in that region.

Figure 5: XLF of wTTS and cTTS in L1495E derived from a $\sim$30 ksec pointed ROSAT PSPC observation: dotted line - cTTS; dashed line - all wTTS; solid line - wTTS except those discovered by ROSAT. The inset in the lower left shows the typical error bar. The distributions of cTTS and wTTS are again different indicating that the discrepancy between the X-ray luminosities of cTTS and wTTS in Taurus-Auriga is not due to the spatial extension of the sample.

A third distribution consists of all wTTS in L1495E which have not been discovered by ROSAT. The general shape of the XLF in L1495E is the same as that for the complete Taurus-Auriga area: wTTS are X-ray brighter than cTTS. This is also evident from the data in Strom & Strom (1994), an earlier analysis of these pointings in L1495E. We conclude that the X-ray luminosity does not depend on the spatial location within the Taurus region.

The difference in the XLF of wTTS and cTTS does also not depend on our choice of roughly 10 Å as boundary between cTTS and wTTS. It is clear that one should use the H$\alpha$ flux instead of the equivalent width as boundary (hence, we classify SUAur as cTTS) because the equivalent width depends on the underlying continuum which varies with spectral type. Martín (1997) suggested three different equivalent width boundaries for three spectral type regimes chosen such as to exclude that the H$\alpha$emission is due to chromospheric activity. Adopting these criteria only a few TTS change classification, but the difference in the XLF remains.

In Sect. 2.3 the conversion from count rates to luminosities by use of hardness ratios was explained. Using hardness ratios allows to indirectly take account of the extinction in the absence of actual A_V measurements. However, HR1 is only sensitive to comparatively low extinctions. The extinction should generally be higher for the cTTS than for the wTTS due to the denser circumstellar environment of the former ones, and if not treated properly may lead to wrong estimates for the luminosities.

We have, therefore, applied an alternative way of deriving X-ray luminosities for the TTS in Taurus-Auriga making use of the available A_V data. In this approach the X-ray flux was computed with standard EXSAS tools assuming a 1 keV RS-model with absorbing column density $N_{\rm H}$ derived from A_V according to Paresce (1984). Similar values for $N_{\rm H}$ are obtained when using the conversion given by Ryter (1996). Stars for which A_V is $\leq$0.05 mag have been assigned a standard value of $N_{\rm H} = 10^{18}~{\rm cm}^{-2}$.

While for individual stars the $L_{\rm x}$ derived by the two methods show typical deviations of $\sim$50%, the statistical distribution of X-ray luminosities is unaffected by the specific choice of CECF, and the previously discussed differences between the XLF of cTTS and wTTS remain.

  
4.2 Dependence on spectral type

In the previous subsection, no distinction was drawn between stars of different spectral type, mass or other stellar parameters. This is justified for young, very low-mass stars which follow fully convective tracks. It is believed that for stars on the MS activity is governed by the relative size of radiative core and convective envelope. This should also apply to TTS once they have reached the radiative part of their PMS evolution. Therefore, to obtain homogeneous samples, stars with different interior structure, i.e. different mass, should be treated separately. As argued in Sect. 3 it is not possible to obtain reliable values for the individual masses and ages of the stars. As an approximation we distinguish the stars by their spectral type. But note, that while for stars on their Hayashi tracks this description is acceptable, for stars on radiative tracks a given spectral type represents a mass range rather than a single value for the mass.

Each subsample is subdivided in three spectral type bins: G, K, and M stars. The mean X-ray luminosities for the different stellar groups and spectral types are listed in Table 9.

 

 

Table 9: Mean X-ray luminosities $\langle \log{L_{\rm x}} \rangle$for cTTS, wTTS, and Pleiades and Hyades. The columns labeled "N'' and " $N_{\rm lim}$'' give the number of stars and number of upper limits within the sample. The second column provides a description of the sample: "C'' - cTTS, "W'' - wTTS, "s'' - single star, "b1'' - binary star assuming that all X-ray emission comes from one component, "b2'' - binary star assuming equal X-ray emission from both components.

Region		Spectral Type G			Spectral Type K			Spectral Type M
		N	$N_{\rm lim}$	$\log{L_{\rm x}}$	N	$N_{\rm lim}$	$\log{L_{\rm x}}$	N	$N_{\rm lim}$	$\log{L_{\rm x}}$
TTS	C	2	(1)	$29.60 \pm 0.66$	22	(9)	$28.93 \pm 0.16$	61	(30)	$28.54 \pm 0.14$
TTS	W	15	(0)	$30.02 \pm 0.17$	36	(5)	$29.78 \pm 0.10$	34	(9)	$29.20 \pm 0.10$
Pleiades		41	(18)	$28.98 \pm 0.12$	112	(41)	$28.94 \pm 0.06$	65	(29)	$28.80 \pm 0.07$
Hyades		22	(2)	$28.97 \pm 0.05$	54	(6)	$28.52 \pm 0.11$	99	(38)	$27.99 \pm 0.10$
TTS	s	-	-	-	34	(11)	$29.44 \pm 0.14$	60	(28)	$28.70 \pm 0.14$
TTS	b2	-	-	-	17	(3)	$29.47 \pm 0.18$	29	(20)	$28.85 \pm 0.18$
TTS	b1	-	-	-	17	(3)	$29.77 \pm 0.18$	29	(10)	$29.15 \pm 0.18$
Pleiades	s	25	(13)	$28.98 \pm 0.15$	84	(38)	$28.83 \pm 0.09$	60	(29)	$28.78 \pm 0.08$
Pleiades	b2	16	(5)	$29.03 \pm 0.16$	27	(3)	$29.00 \pm 0.08$	5	(0)	$28.93 \pm 0.08$
Pleiades	b1	16	(5)	$29.33 \pm 0.16$	27	(3)	$29.30 \pm 0.08$	5	(0)	$29.23 \pm 0.08$
Hyades	s	12	(1)	$28.97 \pm 0.06$	36	(5)	$28.41 \pm 0.15$	89	(35)	$27.95 \pm 0.10$
Hyades	b2	10	(1)	$28.96 \pm 0.07$	18	(1)	$28.75 \pm 0.14$	9	(3)	$28.47 \pm 0.19$
Hyades	b1	10	(1)	$29.26 \pm 0.07$	18	(1)	$29.05 \pm 0.14$	9	(3)	$28.77 \pm 0.19$

For all spectral types the wTTS distribution shows the largest values of $\langle \log{L_{\rm x}} \rangle$, and the Hyades have the lowest $\langle \log{L_{\rm x}} \rangle$. Note, that the group of cTTS of spectral type G is represented by only two stars. But for the other subsamples the statistics are significant. Since in most cases the spectral type (or B-V) is known only for the primary in multiples, we exclude the secondaries from this part of the analysis, except the few cases where the spectral types of all components are known (see Tables 2-7).

In Fig. 6 we provide a comparison of the XLF of TTS, Pleiads, and Hyads.

Figure 6: XLF for TTS in Taurus-Auriga, for the Pleiades, and the Hyades. The distributions are shown for different spectral types, corresponding to different values of B-V or effective temperature or mass for the MS stars. a) G stars, b) K stars, and c) M stars.

Throughout all examined spectral types the wTTS clearly represent the brightest class among the X-ray objects studied here, and Hyads show the weakest X-ray emission. For the M stars, where the mass range is comparatively small, the decline of $L_{\rm x}$ from TTS over Pleiades to the Hyades can be interpreted as an age effect. G and K type stars represent a larger spread in the mass distribution such that the influences of mass and age may not easily be disentangled. However, the difference between $\langle L_{\rm x} \rangle$ of Pleiades and Hyades decreases towards earlier types indicating that age and not mass is the dominant effect.

The distributions of cTTS and Pleiads intersect each other because of the much shallower slope of the XLF of cTTS, i.e. larger spread in luminosities. This effect may be caused by our assumption of uniform distance for all stars in a given sample: in contrast to the strongly clustered Pleiades region the TTS in Taurus-Auriga may be subject to a larger distance spread that translates to an apparent spread in $L_{\rm x}$.

Luminosity differences between various stars may generally be due to differences in emitting area. In order to eliminate this effect the X-ray to bolometric luminosity ratio, ${\log{(L_{\rm x}/L_{\rm bol})}}$, is often used to characterize the X-ray emission. We have examined the relation between the effective temperature and ${\log{(L_{\rm x}/L_{\rm bol})}}$. $L_{\rm bol}$ of Pleiads and Hyads was computed from the V magnitude and B-V (needed to determine the bolometric correction) given in the Open Cluster Data Base. The effective temperatures of Pleiades and Hyades stars were obtained from B - V. We have assumed negligible absorption to both star clusters. In Fig. 7 all late-type stars (spectral type later than F or ${\log{T_{\rm eff}}}$ < 3.78) are plotted.

Figure 7: Relation between X-ray to bolometric luminosity ratio, ${\log{(L_{\rm x}/L_{\rm bol})}}$, and effective temperature, ${\log{T_{\rm eff}}}$. From top to bottom: TTS, Pleiades, and Hyades. Only stars with spectral type later than F are considered. The plotting symbols have been scaled to the projected rotational velocity of the stars. Upper limits to $L_{\rm x}$ are indicated by arrows.

Figure 7 shows that within the TTS sample, which is characterized by a decline of $\log{L_{\rm x}}$ with spectral type, ${\log{(L_{\rm x}/L_{\rm bol})}}$ does not depend on effective temperature. Pleiades and Hyades, however, demonstrate a clear anticorrelation between ${\log{(L_{\rm x}/L_{\rm bol})}}$ and ${\log{T_{\rm eff}}}$ (see also e.g. Micela et al. 1999). The fact that we do not see such a trend in the TTS sample may be due to the large age spread among the TTS. Note, that in Fig. 7 only stars with known projected rotational velocity are shown. The plotting symbols have been scaled to ${v \sin{i}}$. With few exceptions the fastest rotators are situated close to the upper envelope, indicating a connection between the activity level and the rotation rate (see also Sect. 5).

  
4.3 Single and binary stars

All XLF presented above may rely to some degree on our assumption that all components in multiple systems emit X-rays (at the same level). In order to check this hypothesis we have studied the XLF of single and binary stars separately. Again we have constructed separate XLF for G, K, and M type stars. In Fig. 8 we show these XLF for TTS, Pleiades and Hyades stars. For comparison we display also the XLF for binaries derived without taking account of the binary character, i.e. XLF in which each binary has been regarded as a single star with the observed X-ray luminosity (dashed in Fig. 8). Henceforth, these distributions are termed "b1'', in contrast to the distributions "b2'' for which equal partition of $L_{\rm x}$ onto the components was assumed (dotted in Fig. 8). As before, binary components with unknown spectral type are not considered.

The mean and median of $\log{L_{\rm x}}$ for all compiled distributions are listed in Table 9. Obviously, throughout all examined groups of stars the distributions "b1'' are shifted towards higher luminosities with respect to the distributions "b2''. We have performed two-sample tests within ASURV to quantify the differences. The results are summarized in Table 10.

 

 

Table 10: Results of two-sample tests performed with ASURV to distinguish between the XLF of single and binary stars. For each group (TTS, Pleiads, and Hyads) and each spectral type we have compared three distributions: s - single stars, b1 - binary stars with only one X-ray emitter, b2 - binary stars assuming that both components emit equal amounts of X-rays. The probabilities given are for the null-hypothesis that the compared pair of XLF is drawn from the same parent distribution. We have applied Gehan's generalized Wilcoxon test (GW), the logrank test, and the Peto & Prentice generalized Wilcoxon test.

Sample	size (ul.)	Prob	Prob	Prob
		GW HV	log rank	P & Pren.
TTS K stars
s-b2	34 (11)-17 (3)	0.948	0.852	0.948
s-b1	34 (11)-17 (3)	0.073	0.165	0.084
TTS M stars
s-b2	60 (28)-29 (10)	0.238	0.471	0.275
s-b1	60 (28)-29 (10)	0.006	0.051	0.010
Pleiads G stars
s-b2	25 (13)-16 (5)	0.844	0.953	0.789
s-b1	25 (13)-16 (5)	0.085	0.103	0.089
Pleiads K stars
s-b2	84 (38)-27 (3)	0.825	0.286	0.688
s-b1	84 (38)-27 (3)	0.002	0.001	0.004
Pleiads M stars
s-b2	60 (29)-5 (0)	0.710	0.294	0.665
s-b1	60 (29)-5 (0)	0.002	0.001	0.009
Hyads G stars
s-b2	12 (1)-10 (1)	0.657	0.711	0.620
s-b1	12 (1)-10 (1)	0.003	0.005	0.005
Hyads K stars
s-b2	36 (5)-18 (1)	0.134	0.095	0.150
s-b1	36 (5)-18 (1)	0.000	0.000	0.001
Hyads M stars
s-b2	89 (35)-9 (3)	0.059	0.217	0.083
s-b1	89 (35)-9 (3)	0.002	0.022	0.005

The comparison between "s'' and "b2'' shows in most cases a high probability that the distributions are similar. Only for the Hyades K and M stars the probability that the distributions of singles and "b2'' are different is $\sim$90%. All samples "s'' and "b1'', on the contrary, have high probability for different underlying parent distributions.

The XLF of Hyades stars have first been examined by Pye et al. (1994) on the basis of ROSAT observations. Their finding that Hyades dK binaries are X-ray brighter than single Hyads of the same spectral type were confirmed by Stern et al. (1995) on a larger sample. Our analysis shows that the comparison depends sensitively on the way in which binary stars are treated. The effect is strongly reduced if it is assumed that both components in binaries emit X-rays ("b2'') with respect to distributions of type "b1'' examined by Pye et al. (1994) and Stern et al. (1995).