Centennial Total Solar Irradiance Variation

Abstract

Total Solar Irradiance (TSI) quantifies the solar energy received by the Earth and therefore is of direct relevance for a possible solar influence on climate change on Earth. We analyse the TSI space measurements from 1991 to 2021, and we derive a regression model that reproduces the measured daily TSI variations with a Root Mean Square Error (RMSE) of 0.17 W/m². The daily TSI regression model uses the MgII core to wing ratio as a facular brightening proxy and the Photometric Sunspot Index (PSI) as a measure of sunspot darkening. We reconstruct the annual mean TSI backwards to 1700 based on the Sunspot Number (SN), calibrated on the space measurements with an RMSE of 0.086 W/m². The analysis of the 11 year running mean TSI reconstruction confirms the existence of a 105 year Gleissberg cycle. The TSI level of the current grand minimum is only about 0.15 W/m² higher than the TSI level of the grand minimum in the beginning of the 18th century.

1. Introduction

The climate on Earth is determined by the balance between the incoming solar radiation—quantified by the Total Solar Irradiance (TSI)—and the outgoing terrestrial radiation. A change in TSI is a solar force of climate change on Earth; therefore, the TSI needs to be monitored as an Essential Climate Variable (ECV) [1].

The first measurement of TSI from space was made in 1969 [2], and continuous monitoring of the TSI with space radiometers started in 1978 [3]. In general, TSI radiometers measure at different absolute levels [4], and are subject to ageing due to solar exposure [5]. Several authors have proposed so-called TSI composite time series quantifying the long-term TSI variation as measured by the space instruments [5,6,7,8,9].

From the available TSI composites, it is now well-established that the TSI varies in phase with the 11 year sunspot cycle [10]. In particular, there is a short-term TSI decrease—referred to as sunspot darkening—when a sunspot characterised by a strong surface magnetic field occurs. There is also a longer-term TSI increase—referred to as facular brightening—caused by the facula, characterised by an intermediate-strength magnetic field, which form when a sunspot decays and which have a significantly longer lifetime than the original sunspot.

On top of the 11 year solar cycle TSI variation, there exists a longer-term variation of the ’quiet sun’ [11]. The TSI level observed during the 11 year cycle minima has long been a matter of speculation. Following [12], it was believed that the sun evolved from the so-called ’Maunder Minimum’ from about 1645 until 1715 when the 11 year cycle amplitude was minimal, to a so-called ’Grand Modern Maximum’ [13], where the 11 year cycle amplitude is supposed to be maximal. Centennial TSI reconstructions, such as the one of [14], used for the characterisation of solar climate forcing by the Intergovernmental Panel on Climate Change (IPCC), include a slow increase in the ’quiet sun’ TSI level from the Maunder Minimum to the Grand Modern Maximum by 1.25 W/m² over a period of about 300 years.

Table 1. List of various studies on the TSI increase since the Maunder Minimum.

Reference	TSI Increase Since Maunder Minimum
[15]	3.3 W/m2
[16]	1.3 W/m2
[14]	1.25 W/m2
[17]	6 W/m2
[18]	0.34 W/m2
[19]	0.93 W/m2

gives an overview of the TSI increase since the Maunder Minimum found by different studies.

Recently, the Sunspot Index and Long-term Solar Observations (SILSO) Sunspot Number (SN) have been revised [20,21]. Following the latest insights, the Grand Modern Maximum does not exist, and so the 300 year increase in the TSI level from the Maunder Minimum to the Grand Modern Maximum should also be revisited. Independently from the SN revision, from an analysis of the extended 2008–2009 solar minimum, ref. [18] came to the conclusion that the TSI increase from the Maunder Minimum to the present needs to be revised. In addition, the careful intercomparison of all available space radiometer TSI time series in [9] indicates no variation of the quiet sun TSI level over a 32 year period from 1984 to 2016 within a 95% uncertainty of ±0.17 W/m².

The goal of this paper is to reconstruct the centennial TSI variation back to 1700 based on the available TSI space measurements and the revised SN, in agreement with the insights from [18]. This new centennial TSI reconstruction is a paradigm shift [22] compared to the long-held belief based on [12] that there was a significant increase in the TSI, and hence solar climate change forcing, from the Maunder Minimum to the present. In Section 2, we review the available TSI space measurements and TSI regression models reproducing the sunspot darkening and facular brightening from observations of the solar surface magnetic field. In Section 3, we reconstruct the TSI variation back to 1700 based on the revised SN.

2. Materials and Methods

2.1. TSI Composite

The TSI quantifies the amount of solar energy that is received by the Earth [23]. The TSI is defined as the amount of solar power that reaches the Earth per unit surface perpendicular to the Sun–Earth direction at the mean Sun–Earth distance. The TSI is measured from space by electrical substitution cavity radiometers.

In [9], a TSI composite time series $TSI\left(t\right)$ was constructed by compositing the measurement time series $TS{I}_i\left(t\right)$ of individual TSI instruments with index i. In summary, the compositing technique consists of the following steps:

• The $TS{I}_i\left(t\right)$ are scaled with an instrument adjustment factor $a_i$, such that the adjusted time series $a_{i}TS{I}_i\left(t\right)$ have the same absolute level. The reference absolute level is the one from the Differential Absolute Radiometer (DIARAD) instrument, as part of the Solar Variability and Irradiance Monitor (SOVIM) experiment on the International Space Station (ISS) [24], with application of the new calibration procedure of [25]. The corresponding TSI level at solar minimum, also known as the quiet sun TSI level, is 1362.9 W/m².
• The adjusted time series $a_{i}TS{I}_i\left(t\right)$ are quality-controlled by intercomparison. Parts of individual time series which differ too much from the other time series—for details, see [9]—are removed, and the Total Irradiance Monitor (TIM) on the Solar Radiation and Climate Experiment (SORCE) satellite [26] is corrected for its apparent linear drift compared to the other TSI radiometers. This results in quality-controlled adjusted time series $a_{i}TS{I}_i^{qc}\left(t\right)$.
• For every day t, the composite $TSI\left(t\right)$ is calculated as the mean of the available time series $a_{i}TS{I}_i^{qc}\left(t\right)$:

  $$TSI\left(t\right)=\frac{{\sum }_i{a}_{i}TS{I}_i^{qc}\left(t\right)}{{\sum }_i}$$

  (1)

Here, we extend the work of [9] by adding the recent TSI time series measured by the TIM instruments on the Total Solar Irradiance Calibration Transfer Experiment (TCTE) [27] and on the Total and Spectral Solar Irradiance Sensor (TSIS) 1 [28] and by omitting the oldest and apparently less-reliable measurements on the Earth Radiation Budget (ERB)/Nimbus-7 experiment [3], on the Earth Radiation Budget Satellite (ERBS) [29] and the Active Cavity Radiometer Irradiance Monitor (ACRIM) 1 [30]. The 121 day running means of the resulting quality-controlled adjusted time series $a_{i}TS{I}_i^{qc}\left(t\right)$ are illustrated in Figure 1. As in [9], the period of 121 days averaging is chosen to remove the short-term solar variability, e.g., due to sunspot darkening linked to the average solar rotation period of 27 days [31], while highlighting the instrumental differences and the climate-relevant long-term solar variation linked to facular brightening.

The characteristics of the individual TSI time series are summarised in

Table 2. Characteristics of individual TSI instrument time series. The instrument acronyms are listed at the end of the paper.

Instrument	Reference	Start	End	ai
ACRIM 2	[30]	1991	2001	0.999220
SOVA 1	[32]	1992	1993	0.998723
SOVA 2	[32]	1992	1993	0.99744
DIARAD/VIRGO	[33]	1997	2017	0.997873
PMO-6/VIRGO	[33]	1997	2017	0.998241
ACRIM 3	[6]	2002	2013	1.001572
TIM/SORCE	[26,34]	2003	2017	1.0039663
SOVA-Picard	[25,35]	2010	2013	1.001152
PREMOS	[36]	2010	2013	1.001719
TIM/TCTE	[27]	2013	2019	1.001267
TIM/TSIS 1	[28]	2018	2021	1.001014

.

The resulting composite TSI timeseries $TS{I}_i\left(t\right)$ is illustrated in Figure 2. The daily mean TSI values are shown in purple, and the 121 day running mean values are shown in green.

To quantify the deviation between individual TSI time series and the composite, the Root Mean Square Differences (RMSD) between the composite and the longest individual TSI time series are given in

Table 3. RMSDs between longest individual TSI instrument time series and composite TSI time series.

Instrument	RMSD (W/m2)
ACRIM 2	0.16
DIARAD/VIRGO	0.10
PMO-6/VIRGO	0.12
TIM/SORCE	0.15

.

2.2. Daily TSI Regression Model

We validate our updated composite TSI series by two regression models using different proxies for the facular brightening and an estimate of sunspot darkening. The sunspot darkening estimate we use for both regression models is the Photometric Sunspot Index (PSI)—defined in [37]—obtained from [38].

For the first regression model, the facular brightening proxy we use is the F10.7 radio flux [39]. The F10.7 radio flux is a well-known, long-term solar activity indicator [40], and it has been successfully applied for TSI modelling [29]. The daily TSI $TSI\left(d\right)$ is modelled as a function of the daily F10.7 radio flux $F10.7\left(d\right)$ and the daily PSI $PSI\left(d\right)$ as follows:

$$TSI\left(d\right)=1360.92+0.0332\ast F10.7\left(d\right)-5.5\ast {10}^{-5}\ast F{10.7}^2\left(d\right)-14.6\ast PSI\left(d\right)$$

(2)

with an RMSE of 0.22 W/m² and a correlation coefficient of 0.89.

The 121 day running mean of the F10.7-based daily TSI regression model is shown as the light blue curve in Figure 2.

For the second regression model, the facular brightening proxy we use is the MgII core-to-wing ratio index from [41]. The MgII core-to-wing ratio is an excellent indicator of facular brightening [42] and is commonly used for TSI modelling [43]. The daily TSI $TSI\left(d\right)$ is modelled as a function of the daily MgII core-to-wing ratio $MgII\left(d\right)$ and the daily PSI $PSI\left(d\right)$ as follows:

$$TSI\left(d\right)=1362.68+5.09\ast MgII\left(d\right)-1.09\ast MgI{I}^2\left(d\right)-11.77\ast PSI\left(d\right)$$

(3)

with an RMSE of 0.17 W/m² and a correlation coefficient of 0.94.

The 121 day running mean of the daily MgII-based TSI regression model is shown as the orange curve in Figure 2.

The reference TSI minimum level of 1362.9 W/m² is shown as the yellow line in Figure 2.

3. Results

The daily mean TSI values discussed in the previous section are of interest for a detailed analysis and validation of the measured TSI values in terms of facular brightening and sunspot darkening. For climate change studies, the annual mean TSI values are more relevant, and those annual mean values will be studied in the current section.

The orange curve in Figure 3 shows the annual mean TSI values $TSI\left(y\right)$—obtained by averaging the TSI composite from the previous section, shown in Figure 2—extending from 1992 to 2020.

We consider three annual mean TSI regression models based on solar activity indicators with different lengths:

• The above-mentioned MgII index, extending from 1979 to 2020;
• The above-mentioned F10.7 radio flux, extending from 1948 to 2020;
• The SILSO SN, extending from 1700 to 2020.

For the first regression model, the yearly mean TSI $TSI\left(y\right)$ is modelled as a linear function of the yearly mean MgII index $MgII\left(y\right)$, as follows:

$$TSI\left(y\right)=1362.79+2.52\ast MgII\left(y\right)$$

(4)

The RMSE of the annual mean MgII-based model is 0.071 W/m².

The light blue curve in Figure 3 shows the annual mean MgII-based model.

For the second regression model, the yearly mean TSI $TSI\left(y\right)$ is modelled as a linear function of the yearly mean F10.7 radio flux $F10.7\left(y\right)$, as follows:

$$TSI\left(y\right)=1362.26+0.0099\ast F10.7\left(y\right)$$

(5)

The RMSE of the annual mean F10.7-based model is 0.081 W/m².

The green curve in Figure 3 shows the annual mean F10.7-based model.

For the third regression model, the yearly mean TSI $TSI\left(y\right)$ is modelled as a linear function of the yearly mean SN $SN\left(y\right)$, as follows:

$$TSI\left(y\right)=1362.88+0.0066\ast SN\left(y\right)$$

(6)

The RMSE of the annual mean SI based model is 0.086 W/m².

The purple curve in Figure 3 shows the annual mean SN-based model.

In order to assess the accuracy of the long-term TSI reconstruction from Figure 3, in Figure 4 we show the differences between the annual mean sunspot-based TSI model and the other TSI estimates:

• The purple curve shows the difference between the SN-based model and the F10.7-based model.
• The green curve shows the difference between the SN-based model and the MgII-based model.
• The light blue curve shows the difference between the SN-based model and the composite TSI observations.

All differences are contained within ±0.25 W/m², which we will adopt as an uncertainty estimate of the SN-based long-term TSI reconstruction.

Figure 5 shows the 11 year running mean of the annual mean TSI of Figure 3. The grand minima of this 11 year mean occur roughly in the beginning of each century, confirming the existence of the so-called Gleissberg cycle [44] with a period of around 100 years [45,46].

For a precise determination of the Gleissberg cycle period, we define the Root Mean Square Difference (RMSD) costfunction $RMSD\left(T^{\prime }\right)$:

$$RMSD\left(T^{\prime }\right)=\sqrt{\frac{{\sum }_{y=1916}^{2015}{\sum }_{k=1,2}{(TSI\left(y\right)-TSI(y-k{T}^{\prime }))}^2}{{\sum }_{y=1916}^{2015}{\sum }_{k=1,2}}}$$

(7)

where $RMSD\left(T^{\prime }\right)$ is the RMSD between the 11 year running mean TSI for the last 100 years and the time-shifted versions for the two earlier periods, as a function of the variable time shift $T^{\prime }$. The Gleissberg period T is obtained from a minimisation of this costfunction:

$$T=argmi{n}_{T^{\prime }}RMSD\left(T^{\prime }\right)=105+/-1 \mathrm{year}$$

(8)

Figure 6 shows the variation in the $RMSD\left(T^{\prime }\right)$ for T’ between 90 and 110 years.

The Gleissberg period that minimises the cost function is 105 ± 1 year.

The peak-to-peak variation in the 11 year running mean TSI in Figure 5 is of the order of 0.7 W/m². The corresponding variation of the solar forcing is 4 times smaller, so equal to 0.7/4 = 0.175 W/m². Factor 4 is the ratio of the surface of a sphere to the surface of a circle with the same radius. The centennial solar variability is 5.6% of the estimated 3.1 W/m² greenhouse gas radiative forcing in 2015 [47]. Over the last 50 years, the TSI varied by approximately −0.5 W/m² (a drop from 1363.7 W/m² to 1363.2 W/m², see Figure 5). Using the same factor 4 as earlier, the sun has caused a modest radiative cooling of −0.5/4 = −0.125 W/m² over the last 50 years, which was not sufficient to counteract the strong radiative warming from greenhouse gases.

Figure 7 shows the global temperature rise over the period of 1800–2020, as well as the greenhouse gas radiative forcing (green curve, left scale), the solar radiative forcing (blue curve, left scale) and the combined greenhouse gas and solar radiative forcing (ochre curve, left scale). For the combined greenhouse gas and solar forcing, the correlation with temperature change is 0.894, which is slightly higher than the correlation of 0.892 that is obtained when only the greenhouse gas radiative forcing is considered.

The long-term average of the TSI from 1700 to 2020 is 1363.4 W/m², indicated by the green line in Figure 5.

4. Discussion

Since [12], solar-climate research has been dominated by the idea that during the Maunder Minimum, the TSI was significantly lower than the current conditions, characterised by a Grand Modern Maximum [13] of solar activity, and that this lower TSI could be at least partially responsible for the lower temperatures during the so-called Little Ice Age (LIA) [48] from the 15th to the 19th century, where the temperatures in the Northern Hemisphere dropped by about 0.6 °C. For example, in [15], it is estimated that the TSI during the Maunder Minimum could be 3.3 W/m² lower then its mean value from 1980 to 1986. The theory of the Grand Modern Maximum had to be abandoned after the revision of the sunspot number [20] and after the occurrence of the low solar cycle 24 occurring between 2008 and 2019—see Figure 3. Therefore, the long-term TSI reconstruction needs to be revised.

A reconstruction of past TSI variations needs to be based on the analysis of existing TSI space measurements. We demonstrated in Section 2 that the daily composite TSI from 1991 to 2021 can be reconstructed with an RMSE as low as 0.17 W/m² and a correlation coefficient as high as 0.94 from a regression model based only on a facular brightening proxy and a sunspot darkening estimate. There is no evidence that other physical effects other than facular brightening and sunspot darkening, both linked to the magnetic field on the solar surface, are needed to explain observed TSI variations.

We can then endeavour the extrapolation of the TSI variations prior to their reliable measurement from space. On annual mean timescales, facular brightening and sunspot darkening are strongly correlated since the faculae result from the decay of sunspots on timescales shorter then 1 year, so that a single proxy for both can be used. In Section 3, we have used two facular brightening proxies—the MgII core-to-wing ratio and the F10.7 radio flux—and one sunspot darkening estimate—the SN—to reconstruct the measured annual mean TSI variation from 1992 to 2020, with RMSEs of 0.071 W/m², 0.081 W/m² and 0.086 W/m², respectively. Prior to the used TSI space observations, the annual TSI extrapolations using any of these proxies agree well during their period of overlap, giving confidence in the soundness of the extrapolation. From the comparison of the sunspot-based TSI model with the other TSI estimates during their period of overlap, the stability of the annual mean sunspot-based TSI reconstruction is estimated to be ±0.25 W/m². A TSI reconstruction similar to ours was used in [49] for an adequate reconstruction of global temperature change from 1850 to 2019, increasing the confidence in the validity of our TSI reconstruction.

The occurrence of grand solar minima and maxima [50,51] can be studied from the 11 year running mean TSI reconstruction shown in Figure 5. An RMSD analysis as a function of timeshift confirms the existence of a 105 year Gleissberg cycle, similar to the one found in [45,46]. The TSI levels during the earlier grand minima in the beginning of the 18th and the 19th centuries are comparable, around 1363.05 W/m², while the TSI levels during the later grand minima, in the beginning of the 20th and 21st centuries are also comparable, around 1363.2 W/m², only 0.15 W/m² higher than the earlier grand minima. Clearly, this small TSI level variation cannot explain the occurrence of the LIA.

The main contribution of our study is that, in opposition to earlier studies based on [12], we do not find a significant increase in TSI and hence solar influence on climate change between the Maunder Minimum and the present.

5. Conclusions

We have obtained a new TSI reconstruction from 1700 to 2020. It is based on a careful intercomparison and analysis of the TSI space measurements from 1991 to 2021 and an extrapolation back to 1700 based on the latest version of the annual mean SN. The daily mean TSI space measurements can be reconstructed with an RMSE of 0.71 W/m² and a correlation coefficient of 0.94 by a regression model using the MgII core-to-wing ratio facular brightening proxy and the PSI sunspot darkening estimate. The annual mean TSI model agrees with the TSI space measurements with an RMSE of 0.086 W/m² and has an estimated stability of ±0.25 W/m². The analysis of the 11 year running mean TSI reconstruction confirms the existence of a 105 year Gleissberg cycle with grand minima occurring in the beginning of each century. The TSI level of the latest grand minimum is only 0.15 W/m² higher than the TSI level of the earliest grand minimum.