3 Results of the analysis

Figure 3: Combined PCU0 and HEXTE cluster 0 spectra of the $\chi$-states from 09.10.1997 (down, artificially offset for better discrimination) and 30.03.2000 (top). Theses spectra show extreme power law slope ( $\Gamma =3.41\pm 0.06$ (09.10.1997), $\Gamma =2.65\pm 0.05$ (30.03.2000) and reflection ( $R=7.43\pm 1.12$ (09.10.1997), $R=0.35\pm 0.23$ (30.03.2000) behavior. Note the obvious reflection hump for the 09.10.1997 spectra at $\sim$10 keV.

The spectra of all 139 datasets have been fitted with the above-described DISKBB+REFSCH model. Two example model spectra are shown in Fig. 3. These spectra have the extreme values of $\Gamma$ and R, revealing the spectral differences between different $\chi$-state observations.

The best fit parameters for the $\chi$-states of GRS 1915+105 are shown in Fig. 4 and Table 2, respectively. The upper two panels in Fig. 4 show the strong variability of the source in the 1.5-12 keV (RXTE All Sky Monitor (ASM)) and 2.25 GHz (GBI) bands from JD 2450300-2451900. Irregular outburst and relatively quiet phases alternate in X-ray and radio without any obvious coupling.

The lower six panels of Fig. 4 show the PCU0 count rate and the fit parameters of the $\chi$-state observations. The red and black data points mark two groups of observations with different $\Gamma$( $K_{\rm po}$) behavior (see Fig. 5).

Between the $\chi$-states, the PCU0 count rate varies, with some exceptions, by a factor of 2 at most. Only during the long continuous $\chi$-state at ${\rm JD} \sim 2450400{-}2450600$ and at JD $\sim$2451750 recurrent variability in the X-ray count rate is observed. The model parameters of DISKBB+REFSCH are variable. The power law slope, $\Gamma$, varies between 2.4 and 3.5 with a long-term periodicity of $\sim$590 days (Rau & Greiner 2002, in preparation). No correlation of $\Gamma$ and PCU0 count rate is seen, except around ${\rm JD} \sim 2450500$ and ${\rm JD} \sim 2451750$. The power law normalization behaves similarly to the slope.

The reflection component, R, is variable between different $\chi$-state observations and shows a long-term variability similar to the power law slope. It varies between 0 and 10 (upper limit of our model) with rather large uncertainties when R>4. Except for five observations when R<1, this rules out an isotropically sandwiching corona above the entire accretion disk.

Figure 4: 1.5-12 keV ASM count rate (1st panel from top) and 2.25 GHz GBI flux (2nd) from November 1996 to September 2000. ( GBI was off-line some time before ${\rm JD} \sim 2450400$ and following ${\rm JD} \sim 2451660$. The lower panels show the 3-20 keV PCU0 count rate (3rd), the power law slope, $\Gamma$, (4th) and power law normalization, $K_{\rm po}$, (5th), the reflection amplitude, R, (6th), the accretion disk temperature, $T_{\rm bb}$, (7th) and the inner disk radius (determined from the disk normalization), $r_{\rm i}$, (8th) for all analysed $\chi$-states of GRS 1915+105. Error bars are 1$\sigma$ for one parameter of interest. The observations of the two different branches in the $\Gamma$( $K_{\rm po}$) behavior are marked with red (steep) and black (flat) as in Fig. 5.

 

 

Table 2: Fit results of the RXTE spectra using the DISKBB+REFSCH model (The complete table is available at CDS). (1): ID of observation ( I=10402-01, J=20187-02, K=20402-01, L=30182-01, M=30402-01, N=30703-01, O=40703-01, P=50703-01), (2): exposure time of observation, (3): 3-20 keV PCU0 count rate, (4): 20-190 keV HEXTE cluster 0 count rate, (5): accretion disk temperature, (6): accretion disk normalization ( $K_{\rm bb}=(\frac{r_{\rm i}}{D/10~{\rm kpc}})^2\cos\theta$), (7): power law slope, (8): reflection amplitude, (9): power law normalization (photons/keV/cm²/s at 1 keV), (10): ionization parameter, (11): factor for normalization of PCU0 and HEXTE cluster 0, (12): reduced $\chi ^2$. All errors are 1$\sigma$ for each parameter of interest.

Obs-ID(1)	GD	JD	Exposure(2)	PCU0(3)	HEXTE0(4)	$T_{\rm bb}$(5)	$K_{\rm bb}$(6)	$\Gamma$(7)	R(8)
		(-2400000)	[s]	[cts/s]	[cts/s]	[keV]
K-02-01	14.11.1996	50401.12	2688	995	70	2.60 +4.81-0.49	1.67 +4.90-1.60	2.89 +0.05-0.07	1.37 +0.55-0.86
K-06-00	11.12.1996	50428.86	8656	587	60	2.50 +0.35-0.30	2.52 +1.46-1.45	2.69 +0.02-0.02	2.40 +0.40-0.44
K-07-00	19.12.1996	50436.74	8992	598	61	2.21 +0.36-0.21	3.42 +3.25-2.32	2.70 +0.01-0.03	2.25 +0.20-0.56
.	.	.	.	.	.	.	.	.	.
.	.	.	.	.	.	.	.	.	.
P-28-00	21.09.2000	51808.70	2384	706	69	2.70 +1.39-0.73	1.14 +8.85-1.01	2.84 +0.03-0.06	3.11 +1.42-2.64
P-28-01	21.09.2000	51808.77	2400	651	66	3.15 +3.59-1.05	0.38 +0.21-0.37	2.83 +0.08-0.04	2.74 +0.22-1.29
P-28-02	21.09.2000	51808.83	2336	645	44	2.00 +1.72-0.28	6.65 +23.43-6.47	2.87 +0.05-0.03	3.24 +0.24-2.29

Obs-ID	$K_{\rm po}$(9)	$\xi$(10)	c(11)	$\chi ^2$(12)
	[Pho/keV/cm2/s]
K-02-01	43.5 +2.0-2.6	4999 +1-3521	0.82	0.69
K-06-00	15.6 +1.2-0.3	5000 +0-4421	0.78	1.00
K-07-00	16.3 +1.1-0.4	996 +4004-338	0.79	1.01
.	.	.	.	.
.	.	.	.	.
P-28-00	34.1 +0.7-5.0	2902 +2098-2902	0.82	0.94
P-28-01	29.8 +0.1-2.1	462 +4538-461	0.81	0.84
P-28-02	27.2 +2.8-0.8	34 +2900-34	0.52	0.77

The significance of the accretion disk component varies through the $\chi$-states. For some observations the disk is more or less absent (e.g. JD 50400-50600; relatively low disk temperature and large inner disk radius (small normalization)), for other observations the disk component provides a non-negligible contribution to the X-ray flux (e.g. JD 51500-51600; high temperature and small inner radius). In order to be consistent and to fit all $\chi$-state observations with the same model, we included the DISKBB component in all of our fits, although it could be excluded for several observations.

Figure 5: Power law slope, $\Gamma$, as a function of power law normalization, $K_{\rm po}$. Each RXTE observation is represented by one data point. The upper branch presents the red points in Fig. 4 and the lower branch the black points, respectively. The dotted lines represent the fitted correlation functions (see text).

The accretion disk component shows a variable disk temperature of 1-4 keV. Sometimes, large uncertainties due to the small contribution of the DISKBB component to the total flux are seen. The inner disk radius, which can be determined from the disk normalization, varies between 1 and 20 km. For a non-rotating black hole of mass 14 $M_{\odot}$ (as measured for GRS 1915+105, Greiner et al. 2001b) the Schwarzschild radius is $\sim$40 km. It is known that the DISKBB model underestimates the inner disk radius by a factor of 1.7-3 due to Doppler blurring and gravitational redshift (Merloni et al. 2000). Also the neglect of comptonization in the surface layers of the disk leads to unphysical values when using then DISKBB model (Zdziarski et al. 2001). But even a maximally rotating black hole (where the inner disk radius reaches the Schwarzschild radius) cannot account for the majority of the parameter values. This problem has to be kept in mind when discussing the absolute values of the parameters.

Another free physical parameter of the REFSCH model is the ionization parameter, $\xi$. It has huge uncertainties because no Fe K$\alpha$line could be fitted but has a negligible influence on the hard X-ray continuum and our model parameters. Therefore we will not plot or discuss $\xi$ further. Theoretically, the reflection in the hard spectrum should depend on the ionization parameter because higher ionization means lower absorption and therefore higher reflection probability. But the REFSCH model includes a simple 1-ionization zone model only, which is very unlikely to be present in the disk and does not show any dependence of reflection and ionization at all.

3.1 The power law component

The model fits reveal an increasing power law normalization, $K_{\rm po}$, with a steepening power law component (Fig. 5) suggesting a pivoting behavior. Two branches with different slopes are seen in the correlation. No correlation of the power law slope with the X-ray count rate in ASM and/or PCU is observed.

The strength of the correlation can be tested using a Spearman rank-order correlation test (Press et al. 1992). Both branches show strong correlations (steeper: $r_{\rm S}=0.85$, flatter: $r_{\rm S}=0.87$). (Note, this statistical method does not take into account the particular uncertainties of the data points.)

The best descriptions of the correlations are functions of the type

$$\Gamma=u\cdot \log K + v$$ (1)

with ${u}=0.57\pm0.02$ and ${v}=2.0\pm0.1$ for the lower branch and ${u}=0.81\pm0.07$ and ${v}=2.0\pm0.1$ for the upper branch.

The pivoting has been tested in more detail by schematically plotting the obtained power law spectra. The two different branches consequently show different pivoting behavior (Fig. 6). Whereas the upper branch from Fig. 5 shows a pivoting energy of 4-8 keV (gray), the lower branch pivots at around 20-30 keV (black).

Figure 6: Schematic power law spectra for the upper (top) and lower (bottom) branch of Fig. 5, the red and black points from Fig. 4, respectively, artificially offset by a factor of 100. The pivoting happens at different energies (4-8 keV & 20-30 keV).

3.2 Reflection

Due to the known uncertainties of the PCA below 6 keV (Jahoda et al. 1996) and the low energy resolution, stringent conclusions about the behavior of an iron line in the RXTE spectra at 6.4-7 keV not can be drawn. For the analysis of the reflection, therefore, only the continuum radiation can be used, manifesting itself in the reflection hump from 10-30 keV.

Figure 7: R($\Gamma$)-correlation for GRS 1915+105 (crosses) and three X-ray binaries (Cyg X-1 = circles, GX 339-4 = triangles and GS 1354-644 = squares; from Gilfanov et al. 2000). The solid line represents the best fitting model function and the dotted lines the upper and lower limits. Overplotted are the 1, 2 and 3 $\sigma$ confidence contours for three different observations (from left: N-23-01, K-49-01, O-40-03). Note the data points (JD 2451497-2451513) at $\Gamma \sim 3.3{-}3.5$ and $R\sim 0{-}2$ behaving remarkably different.

The reflection amplitude in GRS 1915+105 as a function of $\Gamma$ is shown in Fig. 7 together with that of three black hole candidates (Cyg X-1, GX 339-4 and GS 1354-644; Gilfanov et al. 2000). Although R shows large uncertainties for the $\chi$-states of GRS 1915+105, a similar correlation is seen as in the other X-ray binaries. The steeper the power law component, the higher is the reflected fraction of photons. Figure 7 contains the confidence intervalls of R for three observations to clarify the influence of the model on the R($\Gamma$)-correlation. Although the contours are elongated similiar to the correlation (higher $\Gamma$has higher R), they are intrinsically steeper compared to the overall correlation. Thus, the existence of the R($\Gamma$)-correlation in the $\chi$-states of GRS 1915+105 is no artifact of the model.

A group of observations at high $\Gamma$ (3.3-3.5) and small R(0-2) behaves remarkably differently. These points belong to five datasets from JD 2451497 and JD 2451513 showing very high PCU0 count rates (1300-2100 cts/s) and are short duration $\chi$-states between different high variability states.

The correlation of R and $\Gamma$ is tested using a Spearman rank-order correlation test. The correlation is distinct ( $r_{\rm s}=0.61$) but not strong.

For a quantitative description of the correlation, a phenomenological function was fitted to the data. The best-fitting model is a power law

$$R= u \cdot \Gamma^v,$$ (2)

with ${u}=0.05\pm0.007$ and ${v}=3.7\pm0.4$ (solid line in Fig. 7).

3.3 X-ray-radio-correlation

No obvious correlation between the soft X-ray component and the radio emission is seen. Neither the soft X-ray flux, the temperature of the accretion disk nor the inner disk radius show a correlation with the 15 GHz radio flux from RT and/or GBI .

An unexpected result is found when plotting $\Gamma$ vs. $F_{\rm R}$(Fig. 8). The power law slope correlates positively with the radio flux at 2.25 GHz and 15 GHz. Observations with high radio emission show a softer X-ray spectrum (steeper power law component). Note that no bimodality in the radio emission exists. Instead, a continuous spread is observed. A strict separation of radio loud and radio quiet $\chi$-states, as done before (e.g. Muno et al. 2001; Trudolyubov 2001) seems therefore unsubstantiated.

The correlation of $\Gamma$ and F_R is most obvious for simultaneous ( $\Delta t=0$ hrs) observations. Datasets with radio observations $\pm$5 hrs offset also show the correlation (Fig. 8 upper panel) but with significant scattering. Usually, no statement about the variability state before and after the RXTE exposure can be made, because the source may have had several state alterations. Thus, non-simultaneity is the likely reason for the strong scattering of the correlation for $\Delta t=5$ hrs in comparison to $\Delta t=0$ hrs.

Figure 8: $\Gamma$($F_{\rm R}$)-correlation for GBI (left) and RT (right). Upper panels: correlation for $\Delta t=5$ hrs (time between RXTE and radio observation). As for GBI and for RT the correlation is visible but the data points show a large spread. Lower panel: correlation for $\Delta t=0$ hrs. The dotted lines represent the correlation functions.

To determine the strength of the correlation a Spearman rank-order correlation test was made for the RT data with $\Delta t=0$ h. The correlation is strong ( $r_{\rm s} = 0.83$). No test was made for GBI due to the small number of simultaneous GBI/RXTE datasets. The next step was to fit a phenomenological function (dotted line in Fig. 8) to the correlation. The best model functions are of the form

$$\Gamma=u\cdot \log F_{\rm R} + v,$$ (3)

with ${u}=0.54\pm0.02$ and ${v}=3.94\pm0.03$ for RT and ${u}=0.32\pm0.05$ and ${v}=3.3\pm0.1$ using GBI data.

Besides the $F_{\rm R}$($\Gamma$)-correlation we also investigated the relation between $F_{\rm R}$ and the X-ray flux of the power law component. This introduces some freedom as to at which lower energy the power law is "chosen''. Figure 9 shows the X-ray flux from 20-200 keV and 1-200 keV, respectively, over the radio flux.

Figure 9: 20-200 keV (left) and 1-200 keV (right) X-ray flux shown over the 15 GHz radio flux ( RT). The 20-200 keV flux has a negative correlation with the radio flux, whereas the 1-200 keV flux does not show such a correlation.

With increasing radio emission, the X-ray flux in the power law component between 20-200 keV decreases (neglecting reflection). The total 1-200 keV X-ray flux in the power law component is more or less constant. No correlation with the radio flux is seen. This is because the flux in the 1-20 keV range by far dominates, thus washing out the correlation. Muno et al. (1999) did not find a correlation of $F_{\rm X}$(50-100 keV) with the 15 GHz radio flux. But they used a model consisting only of DISKBB and a broken power law (BKNPO) for the X-ray spectra and RT data with $\pm$12 hrs offset.

A Spearman rank-order correlation test for $F_{\rm X}$(20-200) gives $r_{\rm s}=-0.75$, thus showing a strong anti-correlation. For $F_{\rm X}$(1-200 keV) the test results in $r_{\rm s}=-0.09$ and ratifies the lack of a correlation between the 1-200 keV non-thermal X-ray flux and $F_{\rm R}$.

   

Table 3: Observed correlation of the $\chi$-states of GRS 1915+105.

correlation between:		type(1)	ref.
$\Gamma$	$K_{\rm po}$	+
$\Gamma$	$F_{\rm R}$	+
$\Gamma$	R	+
$F_{\rm X}$(20-200 keV)	$F_{\rm R}$	-
$\nu_{\rm QPO}$(0.5-10 Hz)(2)	$F_{\rm R}$	-	Muno et al. (2001)
$\nu_{\rm QPO}$(0.5-10 Hz)	$T_{\rm bb}$	+	Muno et al. (1999)

(1): direction of correlation ("+" = positive, "-" = negative),
(2): frequency of the 0.5-10 Hz QPO.