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An Intrinsic Baldwin Effect in the H-beta Broad Emission Line in the Spectrum of NGC 5548

We investigate the possibility of an intrinsic Baldwin Effect (i.e.,nonlinear emission-line response to continuum variations) in the broad H-beta emission line of the active galaxy NGC 5548 using cross-correlation techniques to remove light travel-time effects from the data. We find a nonlinear relationship between the H-beta emission line and continuum fluxes that is in good agreement with theoretical predictions. We suggest that similar analysis of multiple lines might provide a useful diagnostic of physical conditions in the broad-line region.


Introduction
Correlations between the continuum and emission-line properties of active galactic nuclei (AGNs) afford potentially important probes of the structure and physical conditions of the broad-line region (BLR) and of the unobservable ionizing continuum in these sources. Probably the best-established correlation between continuum and emission-line properties is the "Baldwin Effect" (Baldwin 1977), an inverse, non-linear correlation between the equivalent width of the broad UV emission lines, most notably C iv λ1549, and the luminosity of the underlying continuum (e.g., Osmer, Porter, & Green 1994). The Baldwin relationship can be parameterized as a simple power law of the form with index α < ∼ 1. Kinney, Rivolo, & Koratkar (1990) and Pogge & Peterson (1992) investigated the effect of variability on this relationship in C iv and Lyα. The results can be summarized as follows: 1. There is a global or object-to-object relationship with slope α ≈ 0.83 for C iv and α ≈ 0.88 for Lyα.
2. There is a rather flatter intrinsic relationship with α ≈ 0.4 that describes how the line and continuum relationship changes due to variability. At least some of the scatter in the global relationship is attributable to variability and the flatter slope of the intrinsic relationship. Moreover, the scatter in this intrinsic relationship is largely attributable to light-travel time effects within the BLR (i.e., changes in the line flux follow those in the continuum flux with some measureable time delay).
While the Baldwin Effect is well-established in most of the strong UV lines, any global Baldwin Effect in the Balmer lines seems to be quite weak (e.g., Dietrich et al. 2002). However, to the best of our knowledge, the question of whether or not there is an intrinsic relationship in the Balmer lines has not been specifically addressed. Our goal in this paper is to remedy this. We adopt an approach similar to that of Pogge & Peterson (1992) in attempting to remove light travel-time effects when determining the slope of the relationship. We will make use of numerous well-sampled optical spectra of NGC 5548 obtained by the International AGN Watch in this analysis to show that an intrinsic Baldwin Effect is present in Hβ.

The Data
For this investigation, we used ten years of spectra from the International AGN Watch collaboration (Peterson et al. 2002 and references therein). These data are fairly inhomogeneous, with differences in spectral resolution, quality, and amounts of contaminating light from the host galaxy. In order to minimize the effects of these problems, we restricted our analysis to the high-quality homogeneous data obtained with the Ohio State CCD spectrograph on the Perkins 1.8-m telescope at the Lowell Observatory in Flagstaff, Arizona, between 1988 and1998. The optical continuum and Hβ emission-line fluxes for this subset are shown in Fig. 1. The continuum flux was measured in a 10Å wide bin at a wavelength of 5100Å in the rest frame of the galaxy, which has a redshift of z = 0.017. The continuum underneath the Hβ line was modeled by linear interpolation between two windows on either side of the Hβ line, at wavelengths of 4710Å and 5100Å in the rest frame of the galaxy. The Hβ flux was then measured by integrating over the flux above this continuum. The measurement uncertainties in these spectral measurements were estimated by examining the differences in fluxes measured on very short time scales (i.e. less than or equal to 2 days). We assumed that flux differences on such short timescales are stochastic errors rather than real differences. We estimated the fractional error in the integrated continuum and line fluxes to be 2.5%, consistent with previous estimates of the error in these types of calculations (e.g., Peterson et al. 2002).

Light Travel-Time Effects
In this investigation, it is important to take into account the effects of light-travel time. Continuum radiation emitted by the central continuum source (presumably an accretion disk) is absorbed and reprocessed into emission-line radiation by diffuse BLR gas. Due to light travel-time effects within the BLR, we observe a time delay of order days to weeks between variations in the continuum and the response of the emission lines. A simple linear model (Blandford & McKee 1982) to describe the emission-line response to continuum variations can be written as where C(t) and L(t) are the continuum and emission-line light curves, respectively, and τ is the time delay, and Ψ(τ ) is the transfer function. The transfer function depends on the size, geometry, and structure of the BLR. Under typical conditions, the centroid of the transfer function is essentially the mean response time of an emission line to changes in the continuum. The mean response time, or lag, for an emission line can be found by cross-correlation of the light curves.
In the case of NGC 5548, multiple years of monitoring optical continuum and emissionline variability has shown that the Hβ time lag changes over time. More precisely, the lag varies with the mean continuum level in a fashion that clearly suggests that the response is dominated by gas in which physical conditions are optimal for the Hβ line. Thus, the mean response time for a given line is a dynamic quantity and eq. (2) is an overly simplistic model of the line response. However, it seems to be sufficient to account for time-dependent changes in the lag by taking yearly averages (Peterson et al. 2002).
We characterize the time delay by the value of the centroid of the optical continuum/Hβ emission-line cross-correlation functions for each year. The cross-correlation centroids are taken from Table 8 of Peterson et al. (2002); values range from a low of 11.37 days in 1992 to a high of 26.44 days in 1998. For each point in the Hβ light curve, a corresponding lagcorrected continuum point was determined. The lag was subtracted from the time of each Hβ measurement to identify the time at which the optical continuum was best-correlated with the Hβ emission-line flux. The continuum flux at this time was determined by linear interpolation between the actual observations immediately preceding and following this time. Errors on interpolated continuum points were estimated by adding the errors of the preceding and following continuum points in quadrature, which probably slightly overestimates the actual uncertainty.

Fitting Procedure and Results
In Fig. 2, we show the total Hβ flux as a function of optical continuum flux, corrected for time delay as described in the previous section. The relationship is clearly non-linear. However, we have not yet accounted for constant spectral contaminants that have to be removed from the observed optical continuum and emission-line fluxes. The two major contaminants to the spectrum of importance to our investigation are (a) the contribution of the host galaxy to the optical continuum and (b) the Hβ narrow-line contribution to the total Hβ flux. Assuming that these are the only two necessary corrections, we can write eq. (1) in terms of the observables, namely the total continuum flux F λ (5100Å) (i.e., AGN plus host galaxy) and the total Hβ flux F (Hβ total ) (i.e., broad component plus narrow component) as We can approach eq. (3) in two ways, first by a complete non-linear least squares treatment, and second by fixing the host galaxy and narrow-component fluxes, which are wellconstrained by independent analyses, which reduces the problem to a linear least-squares exercise. We consider both of these in turn.

Non-Linear Least Squares Fit.
We employed a Marquardt algorithm as described by Press et al. (1989) to determine the constant components F (Hβ narrow ) and F galaxy , the scaling factor A, and the Baldwin Effect index α, all as defined in eq. (3). The values of these parameters are shown in Table 1, along with uncertainties estimated by bootstrap resampling. The best-fit locus is shown in Fig. 2.
Simple Power-Law Fit. We made use of the fact that two of the parameters in eq. (3) can be accurately determined by independent means. The host galaxy flux at 5100Å is estimated to be F galaxy = (3.37 ± 0.54) × 10 −15 ergs s −1 cm −2Å−1 (Romanishin et al. 1995) by using a model of the host-galaxy surface brightness distribution obtained by careful deconvolution of the nuclear point source and the host galaxy.
We used a value for the narrow Hβ line component of F (Hβ narrow ) = 6.7 × 10 −14 ergs s −1 cm −2 which was estimated by trial-and-error scaling of the [O iii] λ5007 emission-line profile and subtracting the scaled profile at the position of the Hβ narrow line. This works best on high-quality data obtained during a low-flux state when the narrow component of Hβ is relatively prominent. A plausible narrow-Hβ model was found by scaling the [O iii] λ5007 emission line by 0.12 (±0.01), as shown in Fig. 3. After subtraction of these constant components, eq. (3) reduces to a simple linear least squares problem The linear regression was performed using the BCES (bivariate correlated errors and intrinsic scatter) program described by Akritas & Bershady (1996). This technique accounts for measurement errors in both parameters. The results for BCES(Y X) regression are given in Table 1, again with uncertainties in the fitted parameters determined by bootstrap resampling. The best fit function is shown in Fig. 4.

Discussion
Both methods of fitting show a clear Baldwin Effect, with α ≈ 0.65 for the simple linearized fit and α ≈ 0.53 for the non-linear least-squares fit, in both cases rather steeper than the values found by Pogge & Peterson (1992) for C iv and Lyα. The fit shown in Fig. 4 raises concern about either the parameterization of the relationship or the estimate of the background components (narrow-line Hβ and host-galaxy); in other words, if the independent estimates of the background components are correct, then as shown in Fig.  4, most of the low-flux points fall below the best-fit line, indicating that the slope of the relationship must be different in this regime. While the non-linear least-squares fit (Fig. 2) does not have this problem at the low-flux end, the background estimates from the fit are in both cases much too high, although certainly the constraint on the host-galaxy flux is less stringent than that on the Hβ narrow component. To pursue this, if we adjust upwards from the Romanishin et al. (1995) estimate of the host-galaxy flux, we find the best overall fit (as measured by χ 2 ) is achieved for a value of 4.6 × 10 −15 ergs s −1 cm −2Å−1 (yielding α ≈ 0.52), which is, not surprisingly, close to the host-galaxy estimate we obtained (Table  1) by the non-linear least-squares fit to eq. (3), the only difference being whether or not the narrow-line contribution is a free parameter or held fixed. A galaxy contribution of 4.6 × 10 −15 ergs s −1 cm −2Å−1 exceeds the Romanishin et al. (1995) estimate by ∼ 2σ, but it is still (although only slightly) below the minimum published optical continuum flux (Peterson et al. 2002). However, during the 2002 observing season, NGC 5548 went into the lowest-flux state recorded over the last 30 years, reaching a minimum 5100Å flux of (3.94 ± 0.10) × 10 −15 ergs s −1 cm −2Å−1 on JD2452471 (Sergeev 2002, private communication). Given this result, we doubt that the estimate of Romanishin et al. (1995) significantly underestimates the hostgalaxy contribution; it seems more likely that our parameterization is oversimplified.
While Fig. 4 suggests that the relationship between the continuum and the Hβ emission line changes at low flux levels, it is also possible that the relationship simply varies somewhat with time. As all of the low-flux points below the best-fit lines are from a single epoch (the fourth year of this monitoring program, during the period JD 2448713 -2448848), we are unable to discriminate between flux dependence and time dependence. A complete analysis of the most recent faint-state data should clarify this.
Finally, we as show in Fig. 3, our best estimate of the Hβ narrow component, based on a profile decomposition, is 6.7 × 10 −14 ergs s −1 cm −2 . This is much lower than the non-linear least-squares result of 1.8 × 10 −13 ergs s −1 cm −2 (Table 1; also see Fig. 2). It is very unlikely that the Hβ narrow component based on profile decomposition is underestimated. However, the non-linear least-squares result may include other less-variable contaminants to the broad Hβ profile, and this is under investigation.
Depending on how we account for the constant components, we find possible power-law indices (eq. 3) in the range 0.52 < α < 0.65. In any case, we see clear evidence for an intrinsic Baldwin Effect in Hβ, i.e., the line flux does not increase linearly with the continuum flux. To explore the physical nature of this relationship, however, we need to relate the Hβ flux to the ionizing continuum flux rather than the optical continuum flux, since the former actually drives the emission-line variations. While the ionizing flux (λ < 912Å) is not directly observable, measurements closer to the ionizing continuum, at 1350Å, are available from the International Ultraviolet Explorer archive. Assuming a relationship of the form Peterson et al. (2002) find β ≈ 0.56, based on contemporaneous pairs of 1350Å (F UV ) and 5100Å (F opt ) continuum fluxes. However, by using the BCES regression, we found a slightly steeper slope, β = 0.67, which we will use in this discussion. We also note that in the case of NGC 5548, Galactic extinction is a small enough effect that it makes no difference in the slope β of eq. (5).
Our analysis thus far has been based only on optical data, and we have assumed specifically that F (Hβ) ∝ F α opt . Combining this with eq. (5). we can write the relationship between the Hβ emission line flux and the UV ionizing continuum as F (Hβ) ∝ F αβ UV , so that 0.35 < ∼ αβ < ∼ 0.44.

The Effect of Geometric Dilution
The emission-line light curve is a convolution of the continuum light curve with a transfer function, as shown in eq. (2). The effect of the transfer function is to shift and broaden the emission-line light curve in time relative to the continuum curve; the effect of making the BLR larger is not only to shift the cross-correlation lag to larger values, but also to smear out the response in time, reducing the apparent amplitude of the response at any given time. We therefore consider here how important an effect this geometrical dilution of the line response might be in this particular case. We attempt to estimate this by convolving the real optical continuum light curve with a transfer function for a thin spherical shell, since this is a rectangular function that will have a relatively large dilution of the line response through time-smearing effects, and repeating the analysis described above.
The effect of geometric dilution on the amplitude of variability of emission lines was explored by convolving the NGC 5548 continuum with thin-shell transfer functions of radius between 1 and 10 light days. We then used the original continuum data and the simulated Hβ emission to calculate the slope of the Baldwin Effect as discussed above. We found that the slope does decrease slightly with increasing geometric dilution; however, it does so at a very small rate. The difference in α is found to be only about 0.05 for rectangular transfer functions of half width 1 and 10 days. We tested transfer functions for other geometries, and found that, as expected, the effects of smearing were even smaller for narrower profile transfer functions.

Comparison with Theoretical Predictions
We can compare our empirical results on the intrinsic Baldwin Effect in Hβ with the theoretical predictions made in a series of photoionization calculations by Korista et al. (1997;see their Fig. 3g). The quantity Φ(H) is defined as the number of ionizing photons arriving at the inner face of a BLR cloud per square cm per sec. If the shape of the ionizing continuum is fixed, the emission-line spectrum will be a function of Φ(H), which is proportional to the photoionization rate, and of the particle density n (cm −3 ), which is proportional to the recombination rate, as well as chemical abundances and cloud column density. Korista et al. show the strength of various emission lines normalized by the continnum flux (i.e., the line equivalent widths) for various values of Φ(H) and n for different column densities and standard assumptions about elemental abundances. For NGC 5548, log Φ(H) = 20 at a distance of 12.6 light days from the central source (Korista & Goad 2001). The equivalent width used by Korista et al. is defined as the ratio of the Hβ flux to the continuum flux at 1215Å. Using the above relationships, the equivalent width should be Using our values of α and β, we find that we expect the equivalent width to be proportional to the UV flux to the −0.56 or −0.65 power, depending how we account for the constant components. In other words, an order of magnitude change in Φ(H) should change the equivalent width by around −0.60 dex. From Fig. 3g of Korista et al. (1997), one can see that for log Φ(H) ≈ 20 and for hydrogen particle densities n(H)(cm −3 ) in the range 11 < ∼ log n(H) < ∼ 13, the slope of the log W/ log Φ(H) relationship is consistent with our results. The grids 2 were investigated to find the slope of the relationship for varying values of column density and continuum shape with solar metallicity. Since the value of log Φ(H) ≈ 19.6 at a radius of 20 light days, where typically the Hβ response is maximized in NGC 5548, we determined the slope of the grid for varying values of the hydrogen particle density by linear interpolation along the grid. We found that regardless of the model, the slope of the log W/ log Φ(H) relationship reaches a maximum in the range −0.57 to −0.60 when the value of the hydrogen particle density is in the range 12 < ∼ log n(H) < ∼ 12.5.

Conclusions
We have investigated the relationship between the variable Hβ emission line and optical continuum of NGC 5548. We discovered that an intrinsic Baldwin Effect does exist; after for correcting for light travel-time effects, we find that the amplitude of broad Hβ variation is less than the amplitude of variation of the underlying optical continuum. By combining these results with the relationship between the amplitudes of variability of the optical and UV continuum, we determine the relationship between the Hβ emission-line flux and the UV continuum flux, and find that our result is in good agreement with the predictions of photoionization calculations, even accounting for smearing of the line response by light traveltime effects within the Hβ-emitting region. We suggest that similar analysis of multiple lines have potential as diagnostics of the physical nature of the BLR.  (Romanishin et al. 1995) in the upper panel, and the narrow-line Hβ contribution in the lower panel. Flux measurements are in the observer's reference frame and are uncorrected for Galactic extinction. The broad-line Hβ flux is obtained by subtracting an Hβ narrow-line contribution of 6.7 × 10 −14 ergs s −1 cm −2 from the total line measurement. The non-stellar continuum flux is obtained by subtracting a starlight contribution of 3.37 × 10 −15 ergs s −1 cm −2Å−1 from the optical continuum measurements. The solid line is the best fit of eq. (4) to these data, with slope α = 0.648 (Table 2).
c Parameter held fixed.