MODELING SOLAR SPECTRAL IRRADIANCE AND TOTAL MAGNETIC FLUX USING SUNSPOT AREAS
DORA G. PREMINGER and STEPHEN R. WALTON San Fernando Observatory, California State University, Northridge,
CA 91330-8268, U.S.A.
(e-mails: [email protected]; [email protected])
(Received 9 August 2005; accepted 20 January 2006)
Abstract. We show that daily sunspot areas can be used in a simple, single parameter model to recon- struct daily variations in several other solar parameters, including solar spectral irradiance and total magnetic flux. The model assumes that changes in any given parameter can be treated mathematically as the response of the system to the emergence of a sunspot. Using cotemporal observational data, we compute the finite impulse response (FIR) function that describes that response in detail, and show that the response function has been approximately stationary over the time period for which data exist. For each parameter, the impulse response function describes the physical evolution of that part of a solar active region that is the source of the measured variability. We show that the impulse response functions are relatively narrow functions, no more than 3 years wide overall. Each exhibits a pre-active, active, and post-active region component; the active region component dominates the variability of most of the parameters studied.
1. Introduction
Measurements of the sun’s energy output over the last 25 years have shown that the solar irradiance varies, and that this variability is wavelength dependent, the magnitude of the changes being larger at shorter wavelengths (Lean, 1991; Solanki and Unruh, 1998). It has been shown that the solar irradiance is modulated by variable magnetic activity that causes dark sunspots and bright faculae to appear on the solar surface (Lean et al., 1997; Fr¨ohlich and Lean, 1998; Lean et al., 1998;
Preminger, Walton, and Chapman, 2002).
Since solar radiation is the earth’s primary source of energy, changes in solar irradiance are bound to have some effect on the earth’s climate. Variability in the ultraviolet portion of the spectrum is thought to play a particularly important role in climate forcing because it affects the amount of ozone in the stratosphere (Haigh, 1999; Rind, Lean, and Healy, 1999). Changes in the sun’s magnetic flux may also impact the earth’s climate indirectly by modulating the heliospheric magnetic field which in turn affects geomagnetic activity and the flux of cosmic rays on earth (Cane et al., 1999; Lockwood, Stamper, and Wild, 1999; Svensmark, 2000).
Climate models that aim to estimate the sensitivity of the climate to solar variability need to know the magnitude of that variability over historic time scales. Important
parameters for climate studies include the total solar irradiance, the irradiance in different portions of the spectrum, and the solar magnetic flux. Solar models based on sunspot data are needed to reconstruct the history of this variability, since the only direct solar data available over a long period of time is a catalog of observations of sunspots on the solar disk. Sunspot areas have been recorded regularly since 1874, and prior to that sunspot numbers were counted.
Several models of the total and spectral solar irradiance already exist. Fox (2004) offers an excellent overview of irradiance models and points out that the method- ology common to all these models is the separation of the irradiance signal into two or more components, which are related to typical solar surface features. One of the components must be sensitive to the presence of dark sunspots on the solar disk, while at least one more component must be sensitive to bright features such as faculae or plage. The main difficulty encountered by these models is the limited availability of historical data that measure the bright component. The digitization of the Mount Wilson CaIIK-line spectroheliograms have recently provided infor- mation on plage areas from 1915 to 1984 (Foukal, 1996), but no information on the brightness of these features. The bright component is therefore usually modeled on the sunspot component. While some models assume a linear relationship between the sunspot component and the bright component (Foukal and Lean, 1990; Lean, Beer, and Bradley, 1995), and some assume a quadratic relationship (Solanki and Fligge, 1999), all models generally assume that the relationship is instantaneous.
However, studies such as that by Bachmann et al. (2004) have shown that there is a temporal offset between sunspot indices and indices sensitive to bright features such as the MgIIindex. Another difficulty encountered by these models is that many spectral irradiance measurements are not photospheric in origin, but originate much higher up in the solar atmosphere. It is therefore not clear that a simple relationship exists between sunspots, which are photospheric features, and brightness variations in, for example, the Lyman-alpha irradiance.
Recent models of the total magnetic flux on the sun generally take into account the temporal evolution of the magnetic flux in active regions. In simulations such as those by Worden and Harvey (2000) and Wang, Sheely, and Lean (2002), bipolar magnetic regions (BMRs) were identified on magnetograms and allowed to evolve by surface transport processes such as differential rotation and meridional flow.
These models are reasonably successful, but require detailed information about the locations and fluxes of newly emerging BMRs. Models based on sunspot data alone generally assume that there is some relationship between total solar magnetic flux and the emergence and evolution of sunspots. The actual nature of this relationship is not yet well established. Thus, authors such as Solanki, Sch¨ussler, and Fligge (2002) and Wang, Lean, and Sheely (2005), who make different assumptions about the details of this relationship, obtain different results when modeling the sun’s magnetic flux.
In Preminger and Walton (2005) (henceforth Paper I), we presented a new model, the finite impulse response (FIR) model, for reconstructing total solar irradiance
from sunspot areas. In the present work, we show that the same modeling technique can be successfully used to reconstruct other measures of solar variability, such as solar spectral irradiance and total magnetic flux variations, from sunspot areas.
The FIR model is different from other models in two ways. Firstly, it is based on sunspot area only, and needs no other proxies. Secondly, the FIR function is an empirical function, computed from existing data, that represents the detailed relationship between sunspot area and any other given solar irradiance measure.
The FIR model does not require the relationship to be a simple, instantaneous one, but instead allows it to be a smooth function over time. The method requires that daily data be available for the parameter in question for at least one solar cycle.
Changes in the parameter are modeled as a response to the emergence of a sunspot, and the FIR function that describes that response is evaluated from co-temporal data. The FIR function gives insight into the physical evolution of that part of a solar active region that is the source of the measured variability. The FIR function also has an important practical application, since if we assume that the function does not change with time, we can use it to reconstruct historical variations of the given parameter during epochs when direct observations do not exist.
2. Data
The sunspot areas used in this work, As, are the daily total sunspot disk areas from 1874 to the present. Asis computed from the database of individual sunspot areas, provided by the Royal Greenwich Observatory, the US Air Force and the National Oceanic and Atmospheric Administration.
The other parameters of solar variability that we propose to model have been observed on a daily basis for at least one solar cycle and include the following:
S: The composite total solar irradiance at 1 AU, in W m−2, measured from space- craft and compiled from 1978 to the present by Fr¨ohlich and Lean (1998) and Fr¨ohlich (2004).
F10.7: The solar radio flux adjusted to 1 AU, in solar flux units (1 s.f.u. = 10−22W m−2 Hz−1). The integrated emission from the solar disk at 2800 MHz (10.7 cm wavelength), measured from 1947 to the present.
|B|: The average unsigned magnetic flux from Kitt Peak (1977 to present), in Gauss.
The absolute value of the line of sight magnetic field strength observed with 1 arcsec pixel size, averaged over the full disk.
CaK: The CaIIK Emission Index, the equivalent width, in ˚A, of a 1 ˚A band centered on the calciumIIK-line core at 3934 ˚A, obtained at Sac Peak since 1984.
Mgii: The MgIIindex, the ratio of core-to-wing intensity in the solar MgIIfeature at∼280 nm in the disk-integrated solar spectrum, measured since 1978 and compiled into a composite by Viereck and Puga (1999).
He1083: The equivalent width of the helium 1083 nm solar absorption line averaged over the solar disk, in m ˚A, measured from 1975 to the present.
Lyα: The Lyman-alpha irradiance, in photons cm−2s−1. The full-disk integrated flux in the Lyman-alpha solar emission line at 121.6 nm, measured since 1977 by various satellites and compiled into a composite data set by Woods et al. (2000).
Ap: Daily calcium plage areas from Mount Wilson Observatory, corrected for fore- shortening, in fractions of the solar hemisphere. The data are available from 1915 to 1984 and were digitized as described in Foukal (1996).
K: The relative change in the solar intensity in a 1 nm bandpass centered on the CaIIK-line, as computed from photometric images taken at the San Fernando Observatory. Preminger, Walton, and Chapman (2002) showed that this parameter is highly correlated with the 11-year cyclical change in S and can be considered a good proxy for irradiance variability caused by spectral lines formed in the lower chromosphere and upper photosphere.
K is a dimensionless parameter, expressed in ppm relative to the quiet sun.
UV200: Daily ultraviolet irradiance data from the NOAA-9 SBUV/2 instrument, averaged between 200 and 205 nm, in mW m−2 nm−1, calibrated by DeLand, Cebula, and Hilsenrath (2004).
Details of the sources of these data sets are given in the Acknowledgements section.
3. Method
Let X be one of the observed parameters we wish to model. Suppose X varies from its minimum level, Xmin, by an amountX,
X(t) = X(t) − Xmin.
Assume thatX is related to Asby convolution with a FIR function, hX(t):
X(t) = As(t)∗hX(t). (1)
Thus, we modelX as the response and Asas the stimulus of a physical system and assume that the properties of the system are linear and time invariant (Bracewell, 1965). We can regard this assumption as being justified if we can find a function hX(t) such that Equation (1) is true and hX(t) is non-zero over a time interval which is short compared to the length of the data sets used to compute it. We set out to find hX(t) from cotemporal data forX and As.
In principle, the calculation of hX(t) is straightforward, and can be done using Fourier transforms. Let HX(ω) be the Fourier transform of hX(t). Then
HX(ω) ≡ F(hX(t))= F(X)/F(As) (2)
and
hX(t)= F−1(HX(ω)) (3)
whereF denotes the Fourier transform.
In practice, the calculation of hX(t) is more complex because the data are noisy, and some effort is required to extract the signal. hX(t) is evaluated by averaging over many data points and filtering out high-frequency noise. For each parameter X , all available observational data are used to calculate hX(t). This includes over two solar cycles’ worth of data for most parameters. However, less than two complete cycles of data are available for CaK andK, while the ultraviolet data U V 200 are currently only available for solar cycle 22.
Gaps in all data sets are interpolated with a shape-preserving piecewise cubic interpolation. N different data sequences are selected from each X (and from the parallel As observations) by slightly staggering the start and end dates of the se- quences. N depends on the length of the observational data set X , and is about 2000 on average, although it is of order 15 000 for F10.7 and Ap, since these are such extended databases. Each data sequence is chosen to be an integral number of solar cycles to avoid introducing boundary problems when computing the fast Fourier transform. Where possible, each data sequence includes about 8000 daily values, approximately two solar cycles’ worth. For CaK,Kand U V 200 each data sequence includes about one solar cycle’s worth of data, approximately 4000 daily values. For the i th sequence, HX,i(ω) is computed from Equation (2). Each HX,i(ω) is transformed to give hX,i(t), using Equation (3). The hX,i(t) are averaged together to obtain the mean hX(t) and the standard deviation of the mean. High-frequency noise in hX(t) is then removed by multiplication in the frequency domain with a cosine-tapered window function, which gradually attenuates the power at fre- quencies higher thanω (14 days)−1and sets the power to zero for frequencies ω (7 days)−1.
Note that in Paper I we used slightly less data to evaluate the FIR for S, hS(t), than in the current work. While there is no significant difference in the hS(t) obtained, a longer data set generally improves the signal-to-noise ratio of any hX(t).
4. Results
The resultant hX(t) for all parameters X are shown in Figure 1, each scaled to a maximum of 1.0. The actual numerical values of the maxima are listed in Table II, and show the maximum change in parameter X that occurs in response to the
Figure 1. The finite impulse response functions, hX(t). The functions have been scaled to a maximum of 1.0 and their zero levels have been shifted vertically.
emergence of a hypothetical sunspot whose time signature is that of aδ function.
The standard deviations of the means are too small to see on the scale of the figure.
If the functions in Figure 1 are smoothed it becomes apparent that for|t| 400 days the hX(t) are dominated by noise that is more or less symmetrical about zero.
The effective width of each hX(t) is therefore much shorter than the length of the data sequences used to compute it. This is a robust result, since the data sequences are long enough that h(t) is well defined for|t| 2000 days. We are therefore justified in claiming that each hX(t) represents a FIR and before reconstructing
X, we further simplify the hX(t) by multiplying each one in the time domain by a cosine-tapered window function, which gradually attenuates h(t) starting at
|t| ≈ 225 days and sets h(t) to zero for |t| 450 days. This step is equivalent to smoothing each hX(t) in the frequency domain.
Since each hX(t) is assumed to be a stationary function, for each parameter X we can reconstruct the expected variation in that parameter,Xrec, for all times for which Asis known:
Xrec(t)= As(t)∗hX(t)
A least squares fit to the observations Xobs is needed to recover the value of the parameter in the absence of sunspots, X0:
X (t)= aXXrec(t)+ X0
where X0and aX are the fit parameters. From a fit to the same portion of X used in the computation of hX(t) we obtain the coefficients shown in Table I. Note that for most of the parameters aX is unity, so we recoverXobsto within an additive constant when we use the portion of h(t) for|t| 400 days. For |B| and Mgii, aX
is a bit greater than unity; it would be closer to 1 if we used the portion of hX(t) for|t| 500 days. However, doing so does not improve the regression coefficient for the fit. We therefore chose to do all the computations alike for consistency and simplicity.
Figures 2–4 show the observed and reconstructed irradiance parameters as well as the residuals (X− Xrec ≡ aXXrec+ X0). Assessing how well the reconstruction of X agrees with observed values is the best way to estimate the uncertainty in the hX(t) used in the model. The portion of the hX(t) that we use is the portion that results in the best match of the model to the data. The regression coefficients in Table I show that the model accounts for about 90% of the variability for many of the parameters, 94% of the variability for F10.7, but only 75% of the variability for S and Ap. In the case of Ap, the fit is particularly poor for solar cycle 19 and the residual is generally noisy, probably in part because we are using sunspot disk areas, which are foreshortened, to model hemispheric plage areas. The foreshortened plage areas are unavailable, but in fact the FIR function computed for the data will automatically contain a correction for the difference. In general, because the hX(t) have been smoothed, Xrec is always slightly smoother than X and this affects all the regression coefficients, but especially that for S since this parameter has very strong, sharp dips due to sunspots. Figure 5 shows sample scatter plots of X versus Asand X versus Xrecfor F10.7, Mgii, and K, and demonstrates that the degree of linear correlation between X and Xrecis much higher than that between X and As.
TABLE I Coefficients of fits to X .
X X0 aX R2
S 1365.5 1.02 0.75
F10.7 64.4 1.007 0.94
|B| 7.0 1.23 0.80
CaK 0.089 0.96 0.84
Mgii 0.264 1.14 0.89
He1083 45.5 1.04 0.87
Lyα 3.6 1.03 0.89
Ap 0.0034 1.08 0.75
K 1031.7 1.08 0.87
U V 200 8.6 0.96 0.87
Figure 2. Observations (dotted curves) and model reconstructions (solid black curves) of parameters X . The (shifted) residuals are also shown (solid grey curves).
A vertical cut-off is visible at the low end of some of the Xrec, for example,K,rec
in Figure 5. This is probably an indication that the smallest magnetic features are missing in the data for As so their signal does not show up in Xrec even though these features are contributing a small component to X . This implies that the fit parameters X0are most likely somewhat higher than the true minimum value of X . F10.7 and Apare the only observational data sets that span many solar cycles.
The reconstruction for F10.7 is very good over a time period that spans 58 years.
The reconstruction for Apis fairly good, but it underestimates plage area in cycle 19 by a significant amount and slightly overestimates it in cycle 21. We note that the reconstructions are somewhat better in cycles 21 and 22 than in cycle 23 for most parameters. In cycle 23 the residuals for S, CaK, Mgii and F10.7 are slightly positive, the residual for He1083 is significantly positive and the residual for Lyα is positive for the first half of the cycle. Some of these discrepancies between the
Figure 3. Observations (dotted curves) and model reconstructions (solid black curves) of parameters X . The (shifted) residuals are also shown (solid grey curves).
Figure 4. Observations (dotted curves) and model reconstructions (solid black curves) of parameters X . The (shifted) residuals are also shown (solid grey curves).
Figure 5. The correlation between X and As, compared to that between X and Xrec, for selected X .
model and the observations could be related to the observations themselves, since few of these observations are obtained with the same instrument for the entire time interval shown. Alternatively, the discrepancies could mean that the hX(t) are not strictly stationary functions. However, the residuals are small enough that it is reasonable to conclude that they are approximately stationary. Thus, since the hX(t) exist and are relatively short, well-defined functions that reproduce most of the variability in X over an extended period of time, we are led to the conclusion that the assumption underlying the FIR model is valid.
5. Analysis of the hX(t)
The FIR, hX(t), represents the average response of parameter X to a sunspot.
Changes in X must correspond to changes in the physical conditions in that part of the solar atmosphere where X is produced. If we assume that evolving active regions associated with sunspots are responsible for the changing physical condi- tions, we can interpret the different hX(t) as showing the time evolution of active regions at different heights in the atmosphere. The FIRs presented here explain the relative temporal characteristics of full-disk solar observations made at different wavelengths, such as those noted by Donnelly (1987) and Bachmann et al. (2004).
Figure 6. Portions of the hX(t), close to t= 0. The functions have been scaled to a maximum of 1.0 and their zero levels have been shifted vertically.
All the solar parameters we examine in this work reach their peak later in any given solar cycle than does sunspot area.
Figure 6 shows the portion of the hX(t) close to t = 0. Vertical reference lines are drawn every 27 days, about one solar rotation apart. The strong peaks of hX(t) are caused by rotational modulation of spatially concentrated active regions. Note that narrow, shallow dips and peaks are probably a “ringing” phenomenon caused by the removal of high-frequency information.
In Paper I we analyzed in detail the form of hS(t). It differs from the other FIRs in that the strongest peak is a negative one at t= 0, when the emergent sunspot first transits disk center and contributes a large negative signal toS. This dip is flanked by two peaks at t = ±6 days, which occur when the spot is near the limb. From this we concluded that there are bright regions associated with the sunspot whose contributions toS outweigh the contributions of the newly emergent sunspot when the active region is near the limb. The FIRs for all other parameters, with the exception of U V 200, show a positive peak at t = 0 and do not show any negative peaks. Therefore, the entire active region always contributes positively to these parameters. hU V 200(t) is exceptional in that it shows a significant negative peak at t= −14 days, a feature which is difficult to explain, since this occurs half a solar rotation before the eruption of the sunspot. Bearing in mind that hU V 200(t) is one of the noisiest of all the FIRs, we anticipate that as more data become available for U V 200 we will be able to tell whether or not this puzzling feature is real.
At t≈ 27 days (+1 solar rotation) all the hX(t) show a broad peak, indicating that the sunspot has decayed away and the active region is now predominantly a bright region, which contributes positively to all X. The 27-day peak is the strongest peak in hK(t) and hU V 200. However, at t = 27 days the signature of the active region is already faint for|B| and even more so for F10.7. For these two parameters, the FIRs decay quickly and show no strong peaks beyond t≈54 days.
A comparison of hF10.7 and h|B|(t) shows clearly that F10.7, which is formed in the corona, is highly correlated with the presence of strong magnetic fields. This is consistent with the results of Schmahl and Kundu (1995), who show that the rotationally modulated component of the microwave emission is produced mostly by gyroresonance emission above sunspots. hF10.7 has a particularly high signal- to-noise ratio and it is interesting to note that while the peak in this FIR at t = 0 is a single peak, the peaks at t= 27, 54, 81, and 108 days are all double. We have seen that the t = 0 peak is dominated by the sunspot, but that the later peaks are dominated by bright regions. This means that a bright active region contributes more to the F10.7 radiation when at the limb than when at disk center.
In addition to the rotational peaks at positive t, several of the FIRs show positive rotational peaks for t < 0 days, i.e., before the appearance of the sunspot. This is especially noticeable in the FIRs for Lyα, He1083, and Mgii. These features, visible one and two rotations before the sunspot signal, imply that at this early time there is a bright region visible at these wavelengths indicating the impending eruption of a sunspot.
A comparison of the forms of hS(t) and h|B|(t) leads to the conclusion that dark sunspots with large, concentrated magnetic fields decay, on average, within one solar rotation of their emergence. However, the FIRs for Lyα, He1083, Mgii, and CaK make it clear that the bright active region associated with the spot persists much longer and has a pronounced effect on the chromosphere, where these spectral lines are formed. On average, a new active region that includes a sunspot affects conditions in the solar atmosphere for as many as six solar rotations after the sunspot
disappears. The similarity of the hX(t), which describe conditions at a large range of heights in the solar atmosphere, confirms that active regions are coherent structures stretching from the photosphere to the corona. Because all the hX(t) have been similarly smoothed, it is not clear from this data whether the width of the rotational peaks in the hX(t) is a function of the height of formation of X . It is clear, however, that the 27-day peak, whose average FWHM is about 8.4 days, is always slightly wider than the 0-day peak, whose FWHM averages about 6.7 days.
The envelope of the peaks represents the approximate rate at which the part of the active region responsible for parameter X decays. Figure 7 is a plot of log of peak height versus time for a selection of the FIRs. In this figure, peak height is normalized to 1 at t = 27 days. It reveals that, in general, the decay is not strictly exponential. Decay rate during the first solar rotation after the formation of a sunspot is different than for subsequent rotations. For h|B|and hF10.7, decay is significantly more rapid for the first solar rotation than subsequently. Sunspots and bright regions both contribute a positive signal to these two parameters, but the dominant influence is that of the sunspots. The rapid initial decay rate of these parameters reflects the fact that the rate of decay of sunspots is significantly higher than that of bright active regions. Decay becomes slower and is approximately exponential after t ≈ 27 days, when the sunspot has already disappeared. For hK (and hU V 200, not shown), the signal actually increases during the first solar rotation after formation of a sunspot. For these two irradiance parameters, the contributions of sunspots and bright regions compete with each other, sunspots contributing negatively and bright regions contributing a positive signal, but the dominant contribution is that of the
Figure 7. Peak height as a function of time for selected hX(t).
bright regions. The initial increase in these parameters during the first solar rotation after sunspot eruption may simply be due to the rapid decay of the sunspot, but it may also reflect growth of the bright region associated with the sunspot. The parameter Ap measures the area of bright regions, so we examined the rate of decay of the peaks of hAp to try to determine whether the bright regions actually grow during the time when the sunspot is decaying. We find that hApdecays during the first solar rotation, but somewhat more slowly than during subsequent rotations.
Taken together, these results imply that while bright regions and sunspots do appear simultaneously and then decay away at different rates, it may also be the case that sunspots decay into bright regions, thus boosting the bright region signal over a time period of about one solar rotation. For other parameters such as Mgii, He1083, and L yα we find that the FIRs decay a bit faster during the first one or two solar rotations than subsequently. Using peaks from 27 t 162 days and assuming hX(t)∼ e−t/τ during this time we estimateτ for the component of each FIR that is not sunspot related. Table II lists the decay rates.τ is longest for He1083, Lyα,
K, and U V 200, somewhat shorter for Mgii and CaK, and shortest for|B|, Apand F10.7. τ for S is similar to that for Lyα and He1083. Although the decay rate of the active region after t = 27 days is rather similar for all the parameters, there is some indication that decay rate may be slower at greater heights in the solar atmosphere, with the obvious exception of F10.7.
We can also examine the hX(t) in Figure 1 for long-term components. When the hX(t) are smoothed it becomes apparent that the hX(t) are positive on average for|t| 400 days. The positive values of hX(t) for 200 t 400 days are small
TABLE II
Characteristics of the FIRs, hX(t).
Contributions (%)
X Max(hX(t)) τX (days) Pre-active Active Post-active
S 2.38e−05 108 49 21 30
F10.7 0.0030 57 16 80 4
|B| 2.09e−04 65 24 60 16
CaK 1.27e−07 77 22 64 14
Mgii 1.90e−07 88 19 77 4
He1083 2.79e−04 114 18 73 9
Lyα 1.66e−05 103 14 72 14
Ap 3.72e−07 69 30 67 3
K 0.059 98 13 82 5
U V 200 6.47e−06 117 9 60 31
and do not show clear signs of rotational modulation. We will call this component of hX(t) the post-active region component. It appears reasonable to conclude that the post-active region component arises from the remnants of concentrated active regions that have undergone diffusion and decay to become “enhanced” or “active”
magnetic network (Martin, 1988). The low-amplitude but consistently positive val- ues of hX(t) in the range−400 t < −100 days imply that there are also small, bright, magnetic regions that appear spread out over the solar disk starting about 1 year prior to the emergence of a sunspot. Let us call this component of hX(t) the pre-active region component. As we suggested in Paper I, this component may tentatively be identified with ephemeral regions, since ephemeral regions belonging to a given solar cycle start to emerge at least 1 year prior to the sunspots of that cycle (Harvey and Martin, 1973). Note that since hX(t) describes the variability in X that is correlated with sunspots, the pre-active region component must be a different aspect of the same dynamo that generates sunspots on the solar surface.
In Table II we compare the relative contributions to hX(t) of its active, pre-active and post-active region components. Each contribution is also the relative contribu- tion of that component to the total solar cycle variability of X . The active region component dominates the total variability for all X except S, for which the negative contribution of sunspots almost (but not quite) cancels out the positive contribution of bright active regions. The active region contribution is about 80% for F10.7 and forK, about 70% for Mgii, He1083, Lyα and Ap, and about 60% for|B|, CaK and U V 200. ForK this result is the same as that obtained by Walton, Preminger, and Chapman (2003). Worden and Harvey (2000) found that the amount of magnetic flux emerging in active regions is about half of the total flux, a fraction which is comparable to the∼60% we estimate here. For most of the parameters, the pre- active region component is the second most significant and the post-active region component is relatively small. However, in the case of Lyα, the pre- and post-active region components are of similar magnitude, while in the case of U V 200, the post-active region component is the second largest, contributing about 30% of the total variability. The post-active region component for Apis very low because the Apdata exclude most of the area covered by bright network. For S, the pre- and post-active region components account for 49% and 30% of the total variability respectively.
6. Summary and Discussion
In this work, we use our recently developed FIR model, introduced in Paper I, to reconstruct the variability of several solar parameters,X, from a single input parameter, As. This model does not assume a simple, instantaneous relationship betweenX and As. Instead, mathematically theX are treated as the responses of the system to the perturbation As. Using cotemporal observational data, we compute the FIR functions, hX(t), that describe these responses. The hX(t) are
empirical functions, and we show that they have been approximately stationary over the time period for which data exists, so they can be used in a simple model to reconstructX from As.
The models are quite successful at reproducing observations of X and in addi- tion offer insights into physical processes on the sun. hX(t) can be thought of as representing the physical evolution of that part of an active region that is responsible for producing the variation in parameter X . The following are the most important implications of the FIRs:
1. hX(t) is a robust measure of how the variability in solar parameter X is correlated with sunspot activity. For the parameters discussed here, hX(t) is a relatively narrow function, no more than 3 years wide.
2. A comparison of the forms of hS(t) and h|B|(t) shows that dark sunspots with large, concentrated magnetic fields decay within one solar rotation after their emergence. All the FIRs, taken together, show that there are bright active regions tightly associated with the sunspot whose contribution to S is positive and, integrated over time, larger than that of the sunspot. hS(t) shows that at least some of the bright regions appear at the same time as sunspots, and it is possible that their long-lasting influence on the chromosphere is simply a result of the fact that their decay rate is significantly slower than that of sunspots. It is also possible that dark sunspots themselves decay into bright regions, which is in fact widely believed to be the case (Foukal, 2004). The FIR for plage area, hAp, shows a slow initial decay rate for bright regions, during the first solar rotation after the formation of an active region. This is the time period when the sunspot is decaying rapidly, and it does suggest that the bright regions are being boosted at this time by the decaying sunspot. Thus, the tight relationship between sunspots and bright active regions seen in the FIRs suggests that a causal connection exists between the two phenomena; however, they may simply have a common underlying cause. It is also possible that there is a common cause and a causal connection. In this case, we would not expect the correlation between the decay of sunspots and the growth of bright active regions to be a simple one, and indeed the study by Chapman, Hoffer, and Walton (2005) found that the growth of facular area within an active region was not clearly related to the rate of decay of the region’s sunpots.
3. The peaks related to bright regions in the FIR for hF10.7are double, indicating that a bright active region contributes more to the F10.7 radiation when at the limb than when at disk center. This implies that bright active region radiation is directional in the F10.7 continuum.
4. The FIRs for Lyα, He1083, Mgii, and CaK have up to seven well-defined rotational peaks that occur after the emergence of a sunspot, making it clear that the whole active region has a pronounced, long-lasting effect on the chro- mosphere where these spectral lines are formed. The fact that the signature
of an active region at a particular longitude is so long-lived means that a significant fraction of the magnetic field responsible for the active region must decay in situ, or at least at a fixed longitude, rather than dispersing. The similarity of the hX(t), which describe conditions at a large range of heights in the solar atmosphere, implies that active regions are coherent structures stretching from the photosphere to the corona.
5. The FIRs for Lyα, He1083, and Mgii show peaks one and two rotations before the sunspot signal, implying that at this early time there is a bright region visible at these wavelengths indicating the impending eruption of a sunspot. This might be visible evidence of the coalescence of magnetic flux to form sunspots (Martin, 1988).
6. We identify three different components for each FIR. First there is a low-amplitude, positive, “pre-active region” component; small, bright, mag- netic regions appear spread out over the solar disk starting about 1 year prior to the emergence of a sunspot. Then there is the active-region component, which includes the contributions of sunspots and large bright active regions, and lasts up to∼200 days. Lastly, there is a broad, positive, low-amplitude,
“post-active region” component, which lasts another∼200 days. We sug- gest that the “pre-active region” component may plausibly be identified with ephemeral regions (Preminger and Walton, 2005), while we identify the
“post-active region” component with bright, “enhanced” magnetic network that is formed from the decay and diffusion of large active regions. We es- timate the contribution to the variability in X of the different components of hX(t). We find that the dominant contributor is usually, but not always, the active region component. Notable exceptions occur for the parameters S, U V 200, and Lyα.
A potentially useful application of this method is the historical reconstruction of irradiance parameters, which are important for global climate models. All the parameters discussed here can be reconstructed back to 1874, when the sunspot area database begins. In order to extrapolate the models this far back it must be assumed that the hX(t) have been invariant over the last 100 years. This work is in preparation.
Acknowledgements
The sunspot area data used in this work is available at:
http://science.nasa.gov/ssl/pad/solar/greenwch.htm. The com- posite total solar irradiance data used is version d40 60 0412 from PMOD/WRC, Davos, Switzerland. It includes unpublished data from the VIRGO experiment on the cooperative ESA/NASA Mission SoHO and is available at ftp://ftp.pmodwrc.ch/pub/data/irradiance/composite/.
The 10.7 cm solar radio data is a product of the National Research Coun- cil of Canada and is available from the National Geophysical Data Center archive at http://www.ngdc.noaa.gov/stp/SOLAR/ftpsolarradio.html.
The integrated solar magnetic flux and the equivalent width of the he- lium 1083 solar spectral line used here are from NSO/Kitt Peak and are produced cooperatively by NSF/NOAO, NASA/GSFC and NOAA/SEC.
These data are available at http://nsokp.nso.edu/dataarch.html. The SBUV Mg II core-to-wing index was obtained from the anonymous ftp server of the Space Environment Center, Boulder, CO, National Oceanic and Atmospheric Administration (NOAA), US Dept. of Commerce, and is available at http://www.sec.noaa.gov/ftpmenu/sbuv.html. The complete NOAA-9 SBUV/2 ultraviolet irradiance data set is available at http://ozone.sesda.com/solar/. The Lyman-alpha irradiance time series composite is available at http://lasp.colorado.edu/solstice/data.html.
The daily CaK emission index is from Sacramento Peak Observatory of the U.S. Air Force Phillips Laboratory and these data are available at http://www.ngdc.noaa.gov/stp/SOLAR/ftpcalcium.html. We grate- fully acknowledge the help of A. Cookson in preparing the manuscript, and the continuing support of CSUN for the program at the San Fernando Observatory.
This research was supported by NASA grant NAG5-12905 and NSF grant ATM-9912132.
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