The total solar irradiance (TSI) is the amount of solar radiative energy impinging on the Earth's upper atmosphere. As can be seen on Figure 1 below, the TSI is observed to vary in time on a variety of timescales, including a prominent variation in phase with the solar magnetic activity cycle, with yearly averages going from 1365.5 Watt per square meter at solar minimum, up to of 1366.6 at maximum. Superposed on this slow trend are fluctuations about the means of about +/- 1 Watt per square meter on timescales of a few days. Interestingly, the Sun is slightly brighter at solar maximum, even though sunspots are darker than the rest of the solar photosphere. This is because at solar maximum, a great many magnetized structures other than sunspots appear on the solar surface and many of them, such as faculae and active elements of the network, are brighter than the photosphere. They collectively end up slightly overcompensating for the overall irradiance deficit associated with the larger but less numerous sunspots.
Recent observations indicate that the primary driver of TSI changes is the varying photospheric coverage of these different types of solar magnetic structures, although contributions from long-timescale variations associated with a deep-seated physical process, such as cycle-mediated small changes in the efficiency of convective energy transport, cannot be ruled out entirely as yet.
A Fragmentation Model
Our goal has been to produce a physical model for TSI variations, i.e., a model based not on empirical correlations between various classes of surface magnetic structures, but rather on a (simplified) physical model linking these structures. Our starting point, observationally well-supported, is that large structures such as sunspots fragment and decay away, releasing in the process smaller magnetic structures over the solar photosphere. We model this as a stochastic fragmentation process, complemented by boundary erosion. We inject spots and active regions on a model solar surface, according to observed emerging sunspot area data. Under the action of fragmentation and erosion, this then yields a time-evolving size distribution of magnetic structures, which can be convolved with an semi-empirical photospheric emissivity contrast curve (brightness deficit/excess as a function of size), to produce an evolving TSI. We also account, albeit statistically, for spot emergences occurring on the backside of the sun.
The model involves a number of adjustable parameters, which are determined by simultaneously fitting the 1978-2007 TSI and spot area time series. This is a complex multimodal, multi-objective and partly stochastic optimization problem, which we tackle with the genetic algorithm PIKAIA. The orange/red curves on Figure 1 is one such best-fitting solution, which indeed is quite similar to the data, although not in all detail of course given the stochastic nature of the fragmentation process and of the backside emergences.
Figure 2: Time series of 81-day-boxcar-smoothed TSI (top panel) and similarly smoothed sunspot areas (bottom panel) over 1978-2007. The black lines are the observations, and the red bands indicate the +/- one-sigma range about the mean constructed from 1000 realizations of our best-fit model. The fits are quite good through most of the time interval. The gradual, excess downtrend in observed TSI starting around 2003 may be an artefact associated with instrumental drift, or part of a true, long-timescale dimming trend in luminosity. This remains currently under investigation.
The red bands Figure 2 shows the +/- one sigma range covered by our best fit model runs for 1000 distinct realizations of backside emergences and fragmentation sequences, for both TSI (top panel) and sunspot areas (bottom panel). Observations are shown as a solid black line. Careful examination reveals a systematic tendency for the observed TSI to exceed our modelled TSI during the rising phases of the activity cycle, even though the spot areas are well-fitted. We take this as evidence for source of bright magnetic features other than sunspot decay. One obvious candidate are the bright faculae that are observed to appear before the first spots of a given cycle.
TSI Reconstruction Back to 1874
Having thus fixed the adjustable parameters in our model by best-fitting the 1978-2007 time interval, we can use it to reconstruct the irradiance variations since the beginning of the Royal Greenwich Observatory sunspot area dataset, namely 1874. The result of this exercice is shown on Figure 3 below. As expected for a constant quiet sun background irradiance, the irradiance variations follow rather closely the variations in emergence rates, itself closely correlated to the variations in the solar cycle amplitude, as measured in sunspot number.
Figure 3: Temporal variation of the total solar irradiance from 1874 to the present, as reconstructed by our physical model. The orange curve shows daily values for a single representative solution, and the thick red curve mean +/- one sigma for 1000 realization of the best fit model.
Source : University of Montreal
