1 Introduction

The solar irradiance (SI) is the primary source of energy reaching the Earth-atmosphere system. Nevertheless, neither its direct or indirect influence is able to explain the global warming over the past century (Foukal et al. 2006; Foukal 2012), and certainly not over the past 35 years (changes in solar activity and climate have been going in opposite directions), which limits the role of solar irradiance variation in twentieth-century global warming. On longer time scales, records of the solar radiative output become essential when addressing the question of solar variability and its link with past climate changes, particularly during the Holocene period. However, solar irradiance measurements are only available for the last three decades, which calls for the use of models over longer time scales. These total solar irradiance (TSI) reconstructions (Vieira et al. 2011) rely, among others, on solar-activity indices such as the sunspot number (Lean 2000; Foster 2004), geomagnetic indices such as the aa index (Lockwood and Stamper 1999; Wenzler et al. 2006), cosmogenic isotopes such as the 14C or 10Be (Solanki et al. 2004; Steinhilber, Beer, and Fröhlich 2009; Roth and Joos 2013), or solar models such as the flux-transport models (Wang, Lean, and Sheeley 2005). The TSI is a crucial input for all climate models. The actual absolute value of TSI is still a matter of debate. Claude Pouillet (1790 – 1868), a French physicist conducted the first measurements of this fundamental astrophysical quantity between 1837 and 1838. “Solar constant” was in fact the name he gave to the amount of incoming solar electromagnetic radiation per unit area, at a Sun–Earth distance of one astronomical unit (AU). The first estimate of the solar constant was 1228 W m−2 (Pouillet 1838), very close to the current estimate. The solar constant is in fact not constant. Charles Greeley Abbot (1872 – 1973) claimed that his data showed variations in the solar constant and therefore the TSI, but he was not able to demonstrate this (Abbot, Fowle, and Aldrich 1923; Abbot, Aldrich, and Hoover 1942). Indeed, the Earth’s atmosphere affects ground-based measurements, which makes the determination of the TSI difficult. In particular, the solar spectral flux distribution (when transiting the Earth’s atmosphere) is modified by absorption and scattering processes. Therefore, TSI measurements from space are required. Indeed, TSI variations were undetectable until spacecraft observations began in 1978. These observations showed that the TSI variations are within Abbot’s data accuracy. The first long-term TSI measurements outside the Earth’s atmosphere were conducted with the sensors of the Earth radiation budget (ERB) experiment on the Nimbus 7 spacecraft (from Latin for “dark cloud”). Results provided a mean TSI value of 1370.56 W m−2 from 1978 to 1987, and a range of 5.95 W m−2 (Hickey et al. 1980, 1988). The Active Cavity Radiometer Irradiance Monitor (ACRIM) on the Solar Maximum Mission (SMM) gave a TSI variability of ± 0.05 % for a mean value of 1368.31 W m−2 throughout the first 153 days of observations (Willson et al. 1981).

During a solar cycle, TSI variations are relatively small. Nevertheless, these changes have been proposed as a possible driver of climate change (Willson and Hudson 1991; Fröhlich and Lean 1998). Based on measurements collected from various spaceborne instruments over the last 35 years, the measured TSI absolute value (for the mean annual TSI representative at solar minimum) has incrementally declined from 1371 W m−2 in 1978 to about 1361 W m−2 in 2013. The Total Irradiance Monitor (TIM) instrument onboard the Solar Radiation and Climate Experiment (SORCE) measured a TSI of about 1361 W m−2 (Kopp, Lawrence, and Rottman 2004; Kopp and Lean 2011). Results were further confirmed by the PREcision MOnitor Sensor (PREMOS) instrument (Schmutz et al. 2009, 2013). This measurement is lower than the mean TSI value of 1366.6±1.3 W m−2, computed from the TSI composite of the Royal Meteorological Institute of Belgium (RMIB) from 17 December 1984 to 24 December 2008. The RMIB model gives an 11-year TSI modulation.

Initially, TSI measurements were performed from the ground and from sounding rockets. The quality of these data may not be sufficient to converge to the TSI absolute value (effects of the atmosphere). Space-borne observations highlight other difficulties (behavior in space and on-orbit degradation). Space is a harsh environment for radiometers, with many physical interactions leading to potentially severe degradation of the performance (optical, thermal, and electrical). Investigation and analysis of the degradation of space radiometers are crucial parts of achieving the scientific goals of all such instruments. The TSI absolute value depends on each radiometer (ground calibration, behavior in space, space calibration, mastery of degradation, uncertainties budget). The TSI absolute value remains difficult to obtain.

The problem is different for the TSI variations. TSI variations were too small to be detected with the technology available before the spacecraft era. The instruments currently in space yield a TSI variation that is substantially identical. The total solar output is now measured to vary (over the last three 11-year sunspot cycles) by approximately ± 0.05 %. The TSI is in phase with solar activity because the Sun is faculae-dominated, which is not the case for every Sun-like star. The variations in TSI represent the integrated change across ultraviolet (UV), visible, and near-infrared regions. In the UV, the amplitude of the variations is much higher, with relative changes of 1 to 20 % observed in the UV band (Cebula and Deland 1998; Deland and Cebula 2012). In the visible and near-infrared bands, the amplitude of the variations rarely exceeds 0.5 % during a solar cycle. Spectral solar irradiance (SSI) variability influences the Earth’s atmosphere (Ermolli et al. 2012). SSI changes are more difficult to observe during the 11-year cycle. Indeed, the degradation of the instruments limits this observation, particularly in the UV (BenMoussa et al. 2013). The Solar Diameter Imager and Surface Mapper (SODISM) onboard the Picard space mission provides wide-field images of the chromosphere and photosphere of the Sun in five narrow-bandpasses centered at 215.0, 393.37, 535.7, 607.1, and 782.2 nm. TSI variations have been successfully modeled by using solar surface magnetic features (Domingo et al. 2009). A comparison between the normalized TSI and the SSI time series can be made over the whole time span of the Picard mission.

The SOlar VAriability Picard (SOVAP) radiometer measures one of the essential climate variable (TSI) onboard Picard. In this paper, we report on the TSI measurements of SOVAP (absolute value and variations) during the years 2011 and 2012.

2 SOVAP Flight Operations

SOVAP (Conscience et al. 2011) is a DIfferential Absolute RADiometer (DIARAD) developed at RMIB (Crommelynck and Domingo 1984). DIARAD-type radiometers are the first side-by-side absolute cavity radiometers operating in space. SOVAP radiometric core is formed by two blackened cavities constructed side by side on a common heat sink. A heat-flux transducer (thermo-electric sensor) is mounted between each cavity and the heat sink (Figure 1). The difference between the two balanced sensor outputs (open-circuit voltage V) gives an instantaneous differential temperature measurement (ΔT) in which the common part of the thermal surrounding radiation seen by the two cavities is eliminated. By symmetrical construction, good insulation, and proper selection of thermo-optical materials, thermal asymmetry is minimized. Both channels are equipped with a shutter in front of them by which sunlight can be occulted (closed shutter) or allowed into the cavity (open shutter). In the open-shutter phase, the solar flux (SI) flows into the cavity through a precision aperture (mirror) and is absorbed. The cavity is cylindrical (tube) and is covered with diffuse black paint. In addition to solar radiative power, electrical power (P) can be dissipated by a resistor mounted in the cavity (heating resistor). During the design of the cavities, special attention has been paid to obtain corresponding spatial locations and distributions for the solar and the electrical heating power flowing directly into the heat-flux sensor. SOVAP-DIARAD has a double-differential design, allowing differential left-right and differential open-closed measurements. We describe the operation modes we used and analyze here in the sections below. They differed depending on the left- and right-channel shutter operations.

Figure 1
figure 1

Schematic representation of the radiometer.

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2.1 Nominal Right-Channel Measurements – Auto 3

Nominal right-channel measurements were made in the instrument’s Auto 3 mode. During this mode, the left shutter is closed, while the right shutter sequentially opens and closes. This mode corresponds to a double-differential operation. It has a sampling period of three minutes and an instrument noise level of 0.1 W m−2. This mode was used to measure the absolute level of the TSI and to verify the short-term stability of the right channel. This mode was used intermittently from 22 July 2010 until 28 August 2010. It was abandoned due to a failure of the right shutter mechanism, which left it permanently open.

2.2 Nominal Left-Channel Measurements – Auto 2

The Auto 2 mode corresponds to open and closed states of the left shutter while the right shutter is closed (similarly to the Auto 3 mode). This mode was used to measure the TSI absolute level as well as the TSI variability from 28 August 2010 to 27 October 2010. This mode was used during about half an hour every month (at the beginning of the mission). Therefore, the left cavity was deliberately kept less exposed to solar radiation to minimize the degradation due to solar UV radiation. In early 2013, this mode was used again because we were near the end of the lifetime of the spacecraft. This channel (with low aging) provides the TSI measurement with the lowest uncertainty (Table 4).

2.3 Continuously Open Right-Channel Measurements – Rad 10

During the so-called Rad 10 mode, the right shutter is permanently open and the left shutter is kept closed. This new mode was used since the first occultation period on 18 November 2010 (after an update of the flight software). Indeed, an abnormal increase of the temperature of the shutters was observed from the beginning of the mission (Figure 2). This rise of temperature caused a failure of the right-shutter mechanism, leading to the adoption of this new mode of operation. The Rad 10 mode has the advantage of increasing the time resolution at the cost of a decrease in the accuracy and stability of the measurements. SOVAP measurements are organized in packets of 90 seconds (90 s), containing nine frames of 10-second measurements each. TSI acquisitions are made at seconds 20, 40, 70, 80, and 90. During nominal modes (Auto 2 and Auto 3), only the acquisition at second 90 is used. Indeed, all other acquisitions correspond to a transient state during which thermal equilibrium between the two cavities is not reached. In the Rad 10 measurements mode, this equilibrium is observed. Therefore, all TSI measurements at seconds 20, 40, 70, 80, and 90 can be used. Since early 2013, we have stopped using this mode to focus on nominal measurements with the left channel.

Figure 2
figure 2

The temperature of the shutters (red circles: left; black circles: right) has significantly varied since the beginning of the mission. We can observe the temperature drop of the right shutter at its permanent opening. SOVAP is particularly vulnerable because its shutters and radiators are exposed to unshielded solar radiation. SOVAP has suffered substantial degradation due to a combination of solar irradiation and contamination.

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3 SOVAP TSI Equation and Measurement Uncertainty

To extract the maximum amount of information about the TSI variation it is necessary to minimize the instrumental errors of the available radiometers as much and as objectively as possible. Minimization of the instrumental errors requires a careful instrument design, pre-flight calibration, flight data processing, and detailed data analysis including aging corrections. To derive the most accurate TSI measurements, it is important to understand the error sources of the measuring instrument to apply appropriate corrections and to quantify the remaining uncertainties for the temperature and the electrical power measurements. These corrections are used to deliver the final scientific TSI value. To determine the TSI from SOVAP we established the instrumental equation we present below.

3.1 SOVAP TSI Equation – Auto 2 and Auto 3 Modes

The TSI (Auto 3 mode, for example) is defined as the solar irradiance at the mean Earth–Sun distance according to

$$ { \mathit{TSI}_{\mathrm{r}}=\mathit{SI}_{\mathrm{r}}\frac {z^{2}}{(1~\mathrm{AU})^{2}} \frac{1}{(1-\frac{\partial z}{\partial t}\frac{1}{c})^{2}}}, $$
(1)

where TSI r is the total solar irradiance for the right cavity at 1 AU (W m−2), SI r is the solar irradiance obtained using the right cavity (W m−2), z is the distance between the spacecraft and the Sun (km), 1 AU is one astronomical unit (km), c is the speed of light (299 792.458 km per second), and t is the time (second). A small uncertainty on the TSI measurement associated with the Doppler effect (Fröhlich et al. 1997) is about ± 50 ppm (according to the spacecraft velocity of about ± 7.5 km s−1, in the worst case).

SI r is derived from the measurement of the electrical power supplied to the active cavity (right) of the radiometer to maintain the active cavity with the same heat sink flux in the open and closed states (Equation (3)) according to

$$\begin{aligned} \mathit{SI}_{\mathrm{r}} =& C_{5} \frac{((\overline{P_{\mathrm {cl},\mathrm{r}}}-P_{\mathrm{o},\mathrm{r}})-\Delta P_{\mathrm {o},\mathrm{r}}+C_{1}+C_{6})(1-C_{2})}{\alpha_{\mathrm{eff},\mathrm {r}}A_{\mathrm{r}} (1+\Delta_{\mathrm{r}}+\Delta'_{\mathrm{r}}+\Delta ''_{\mathrm{r}}) \cos\theta} +C_{3}+C_{4} , \end{aligned}$$
(2)
$$\begin{aligned} \overline{P_{\mathrm{cl},\mathrm{r}}} =&\frac{P_{\mathrm{cl},\mathrm {r},t1}+P_{\mathrm{cl},\mathrm{r},t3}}{2} , \end{aligned}$$
(3)

where P cl,r,t1 is the electrical power (W) fed to the right cavity in closed state of the shutter at the time t1, P cl,r,t3 is the electrical power (W) fed to the right cavity in closed state of the shutter at the time t3, P o,r is the electrical power (W) fed to the right cavity in open state of the shutter at the time t2. The definition of the parameters of Equation (2) is provided in Table 1. Each parameter is described in more detail in the instrumental paper (Conscience et al. 2011).

Table 1 Definition of the radiometer parameters.
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The thermo-electrical non-equivalence between the effects of the electrical (heating resistor) or solar flux heating of the bottom of the cavity defines ΔP o,r. It is characterized by a different topological distribution of the energy in the radiometer cavity (Figure 3).

Figure 3
figure 3

Demonstration of the thermo-electrical non-equivalence. For the same voltage at the output of the sensors, we do not obtain the same temperature distribution at the surface of the sensors. In the closed phase, the mean temperature of the sensor (\(\overline {T_{\mathrm{cl},\mathrm{r}}}\)) is 21.0789 °C (left). In the open phase, the mean temperature of the sensor (\(\overline {T_{\mathrm{o},\mathrm{r}}}\)) is 21.0808 °C (right). The temperature of the heat sink (T cold) is 20 °C. The temperature difference is small, but has a significant impact on the determination of the TSI. The temperature gradient between the faces of the sensor (thermo-electric generator) is near 1 °C.

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Each cavity is constituted from a thermoelectric sensor, a tube, and the rear side of the precision aperture. Between each cavity and the heat sink, a sensor and a heating resistor are mounted. The equilibrium between the output voltage of the two sensors is maintained by controlling the electrical power in one of the two cavities (right channel) using an analog-proportional integral servo system. The sensor voltage V(ΔT) (in Volt) for each cavity may be written as

$$ { {{V(\Delta T)}=N\alpha_{\mathrm{s}}\Delta T} }, $$
(4)

where N is the number of series-connected elements of the sensor (70), α s is the Seebeck coefficient (2.07×10−4 V K−1) of the sensor, and ΔT is the temperature gradient (between hot and cold sides) of the sensor (∼ 1 Kelvin).

Auto 3 mode (nominal right channel) corresponds to a double-differential operation (this is similar to the Auto 2 mode). One differential operation is effective (between the two cavities). Another differential operation represents a non-equivalence (the same cavity in open and closed phase) according to

$$ { {{V_{\mathrm{cl},\mathrm{r}}(\Delta T_{\mathrm{cl},\mathrm{r}})}\neq {V_{\mathrm{o},\mathrm{r}}(\Delta T_{\mathrm{o},\mathrm{r}})}} }, $$
(5)

where V cl,r(ΔT cl,r) is the sensor voltage in the closed state of the shutter (V) and V o,r(ΔT o,r) is the sensor voltage in the open-shutter phase (V). ΔT cl,r is the mean temperature of the sensor in the closed state of the shutter (in Kelvin), and ΔT o,r is the mean temperature of the sensor in the open-shutter phase (K). In the closed state of the shutter, the electrical power is dissipated on a disk of 12.5 mm diameter (left side). In the open-shutter state, the solar flux (SI) is absorbed on a circular spot of 10 mm diameter (right side) and an additional electrical power dissipated on a disk of 12.5 mm diameter (right resistor). The thermo-electrical configuration is given in Figure 3.

3.2 Typical Parameter Values

Typical values are provided in Table 2. The method for determining the parameter ΔP o is given below.

Table 2 Typical values of the SOVAP radiometer.
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The parameter ΔP o,r was determined from a theoretical analysis (by symmetrical construction we tend to have ΔP o,r=ΔP o,l). The thermal analysis includes effects of passive heat losses from radiation and conduction (Figure 4) on the mechanical part (ceramic sensor, silver tube, contacts).

Figure 4
figure 4

(Left) Shape of the temperature gradient in the cylindrical cavity (tube). The electrical power is dissipated on a diameter of 12.5 mm of the sensor (closed phase). (Right) Shape of the temperature gradient in the sensor (closed phase).

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We established the differential equations relating temperature in the sensor as a function of the radial coordinate x (in meters). The equations come from an energy balance on a differential ring element of the sensor. This can be expressed mathematically by

$$\begin{aligned} & x^{2} \frac{\partial^2{T_{\mathrm{o},\mathrm{r}}}}{\partial x^2} + x \frac{\partial{T_{\mathrm{o},\mathrm{r}}}}{\partial x} + \frac {x^{2}}{\lambda t} \biggl( \varepsilon\sigma \bigl(T_{\mathrm{t}}^{4} - T_{\mathrm{o},\mathrm{r}}^4 \bigr) + \frac{P(\mathit{SI}_{\mathrm {r}})}{A_{\mathrm{r}}} + \frac{P_{\mathrm{o},\mathrm{r}}}{A_{\mathrm {heater}}} \biggr) \\ &{\quad} {} + \frac{x^{2}}{\lambda t} \bigl( R_{\mathrm{p}} (T_{\mathrm{cold}} - {T_{\mathrm{o},\mathrm{r}}}) + R_{\mathrm{c}} (T_{\mathrm{t}} - {T_{\mathrm{o},\mathrm{r}}}) \bigr) = 0 , \end{aligned}$$
(6)
$$\begin{aligned} & x^{2} \frac{\partial^2{T_{\mathrm{cl},\mathrm{r}}}}{\partial x^2} + x \frac{\partial{T_{\mathrm{cl},\mathrm{r}}}}{\partial x} + \frac {x^{2}}{\lambda t} \biggl( \varepsilon\sigma \bigl(T_{\mathrm{t}}^{4} - T_{\mathrm{cl},\mathrm{r}}^4 \bigr) + \frac{\Delta P_{\mathrm {o},\mathrm{r}}}{A_{\mathrm{heater}}} + \frac{P_{\mathrm{cl},\mathrm {r}}}{A_{\mathrm{heater}}} \biggr) \\ &{\quad} {} + \frac{x^{2}}{\lambda t} \bigl( R_{\mathrm{p}} (T_{\mathrm{cold}} - {T_{\mathrm{cl},\mathrm{r}}}) + R_{\mathrm{c}} (T_{\mathrm{t}} - {T_{\mathrm{cl},\mathrm{r}}}) \bigr) = 0 , \end{aligned}$$
(7)

where T o,r is the temperature of the sensor hot plate (SHP) in the open phase (in Kelvin), T cl,r is the temperature of the SHP in the closed phase, T t is the temperature of the tube, T cold is the temperature of the heat sink (or sensor cold plate), \(\overline{T_{\mathrm{o},\mathrm{r}}}\) is the mean temperature of the SHP in the open phase with \(T_{\mathrm{hot},\mathrm{o},\mathrm{r}} = \overline {T_{\mathrm{o},\mathrm{r}}}\), \(\overline{T_{\mathrm{cl},\mathrm{r}}}\) is the mean temperature of the SHP in the closed phase with \(T_{\mathrm{hot},\mathrm{cl},\mathrm{r}} = \overline {T_{\mathrm{cl},\mathrm{r}}}\), ε is the equivalent emissivity between the SHP and the tube (∼ 0.8), σ is the Stefan–Boltzmann constant (5.6704×10−8 W m−2 K−4), λ is the thermal conductivity of the SHP (∼ 40 W m−1 K−1), t is the thickness of the SHP (m), P(SI r) represents the power consumption that has a solar origin, A r is the SHP area illuminated by the Sun, A heater is the heater area (∼ 1.227×10−4 m2), R p is the thermal coupling between the SHP and the heat sink (∼ 200 W m−2 K−1), and R c is the thermal coupling between the SHP and the tube (∼ 1000 W m−2 K−1).

From the model (when ΔP o,r=0), we obtained the temperature variations within the sensors (Figure 5). We can see the slight differences related to the state of the radiometer (ΔT cl,r≠ΔT o,r). With an equivalent energy balance, the voltage control system must compensate the slight temperature difference between the two cavities (or two states) according to the following equation:

$$ { V_{\mathrm{cl},\mathrm{r}}(\Delta T_{\mathrm{cl},\mathrm{r}}=\overline {T_{\mathrm{cl},\mathrm{r}}} -T_{\mathrm{cold}})}={V_{\mathrm{o},\mathrm {r}}(\Delta T_{\mathrm{o},\mathrm{r}}=\overline{T_{\mathrm{o},\mathrm {r}}}-T_{\mathrm{cold}}) }, $$
(8)

where \(\overline{V_{\mathrm{o},\mathrm{r}}}\) is the mean voltage of the SHP in the open phase (in V), \(\overline{V_{\mathrm{cl},\mathrm{r}}}\) is the mean voltage of the SHP in the closed phase (in V).

Figure 5
figure 5

Hot-side temperature of the sensor (open and closed phases). These two curves show the non-equivalence of the radiometer in the open and closed phases.

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If we add a small amount of power (ΔP o,r), we can balance the voltage of both sensors (Table 3). The thermo-electric non-equivalence ΔP o,r is about 0.3 mW. This result is achieved with an uncertainty of about 30 % (thickness, thermal conductivity, emissivity, contacts, thermoelectric parameters, and modeling).

Table 3 Thermo-electrical analysis: V cl,r=V o,r.
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3.3 Absolute-Accuracy Synthesis

The uncertainty for the absolute accuracy is given in Table 4. The TSI uncertainty from SOVAP is smaller than ± 1.52 W m−2 (k=1, or one sigma) for the right channel (Auto 3), and ± 1.39 W m−2 (k=1) for the left channel (Auto 2).

Table 4 Uncertainties of the TSI measurements.
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3.4 Resolution

The resolution is better than 0.01 W m−2 (for Auto 2 and Auto 3 modes). The DIfferential Absolute Radiometer has been making measurements of the total solar irradiance as part of the Variability of Irradiance and Gravity Oscillations (VIRGO) experiment on the Solar and Heliospheric Observatory (SOHO). SOVAP-DIARAD has the same heritage as VIRGO-DIARAD (Dewitte, Crommelynck, and Joukoff 2004). VIRGO-DIARAD has a repeatability of 0.1 W m−2, the same as SOVAP-DIARAD (for Auto 2 and Auto 3 modes).

4 TSI Absolute Value from SOVAP Nominal Modes

First light for the SOVAP radiometer (Auto 3) was on 22 July 2010. Applying the TSI equation (see Section 3.1) to the SOVAP measurements (Auto 3) provides a TSI value of 1363.0±0.08 W m−2 (1σ on the set of measurements), with an uncertainty of ± 1.52 W m−2 (k=1). Applying the TSI equation to the SOVAP measurements (Auto 2), provides a TSI value of 1361.2±0.12 W m−2 (1σ on the set of measurements), with an uncertainty of ± 1.39 W m−2 (k=1). The measurements of SOVAP in summer 2010 yielded a mean TSI value of 1362.1 W m−2 with an uncertainty of ± 2.4 W m−2 (k=1) for both cavities. From this new approach, we obtained values that are lower than those previously provided with instruments of the same type (Crommelynck, Domingo, and Fröhlich 1991; Crommelynck et al. 1996; Dewitte, Helizon, and Wilson 2001; Dewitte, Crommelynck, and Joukoff 2004; Mekaoui et al. 2010). The SOVAP mean TSI is therefore approximately 1 W m−2 higher than that of the simultaneously flying radiometers (TIM, PREMOS, and ACRIM 3). The SOVAP TSI (Auto 2 measurements, or left channel) is consistent with the PREMOS result (Schmutz et al. 2013) when using the Total solar irradiance Radiometer Facility (TRF) calibration. Recent characterizations have revealed that the World Radiometric Reference (WRR) (Fröhlich et al. 1995) is 0.34 % (∼ 4.6 W m−2) higher than the SI scale (Fehlmann et al. 2012). Before its flight, PREMOS was also compared with the TRF at the Laboratory for Atmospheric and Space Physics (LASP). Using the TRF calibration, PREMOS gives a TSI absolute value of about 1360.9 W m−2 (Schmutz et al. 2013). ACRIM 3 data ( http://acrim.com/ ) were further reevaluated after TRF comparisons. The new TSI value from ACRIM 3 is now within the uncertainties of TIM measurements. This made it interesting to perform a new SOVAP calibration. In particular, a better quantification of the inherent uncertainties associated with thermo-electrical non-equivalence of the cavities was investigated. The stray-light contribution is only a minor parameter for the SOVAP TSI determination. Indeed, multiple contributions such as the bias due to launch, vacuum, outgassing, contamination, radiations, as well as thermo-mechanical and thermo-electrical behavior can be better estimated.

5 SOVAP TSI Time Series

Owing to a failure of its mechanism, the right shutter of SOVAP has remained permanently open since November 2010. The instrument was therefore set to a new mode of functioning and a new instrument equation was required to accommodate this situation. In this new mode (Rad 10), the radiometer is no longer differential (see Equations (2) and (3)) and both channels have been used to determine the TSI. During the Rad 10 mode, the right shutter is open and the left shutter is closed. SI is derived from measuring the difference between the electrical power supplied to the closed cavity and the electrical power supplied to the open cavity according to

$$ { \mathit{SI}_{\mathrm{r}}=C_{5}\frac{({(P_{\mathrm{cl},\mathrm {l}}}-P_{\mathrm{o},\mathrm{r}})-\Delta P_{\mathrm{o},\mathrm {r}}-K_{\mathrm{Rad}10}+C_{1})(1-C_{2})}{\alpha_{\mathrm{eff},\mathrm {r}} A_{\mathrm{r}} (1+\Delta_{\mathrm{r}}+\Delta'_{\mathrm{r}}+\Delta ''_{\mathrm{r}}) \cos\theta} + C_{3}+C_{4} }, $$
(9)

where SI r is the solar irradiance for the right cavity of the radiometer, and K Rad10 is the thermo-electrical non-equivalence between the right and the left cavities of the radiometer (obtained during space calibration).

K Rad10 parameter changes over time (Figure 6). It highlights the difference between the two calibration channels (ground calibration).

Figure 6
figure 6

Evolution of the parameter K Rad10 over time.

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Applying the TSI equation (9) to SOVAP measurements provides a TSI value of 1362.9±0.39 W m−2 (1σ on the set of Rad 10 measurements during two years) and a range of 2.4 W m−2. The uncertainty of these measurements depends on several parameters (see Table 4). The uncertainty in the determination of the parameter K Rad10 is estimated at ± 0.6 W m−2 (Figure 6). For the Rad 10 mode, the mean quadratic uncertainty (k=1) is equal to ± 1.63 W m−2. The SOVAP TSI variability time-series starting in December 2010 is shown in Figure 7. The holes in the data correspond to periods during which SOVAP was switched off. The first switch-off was due to a spacecraft anomaly (in January 2011). This series of measurements stopped at the beginning of 2013.

Figure 7
figure 7

(Top) SOVAP TSI (Rad 10 mode) from December 2010. (Bottom) Daily mean sunspot number ( http://sidc.oma.be/sunspot-data/ ) for the same period. Data provided by the Solar Influences Data analysis Center.

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The instruments currently in space yield a TSI variation that is substantially identical (Figure 8). The SOVAP radiometer operating in a degraded mode also provides the TSI variation; it shows the robustness of the RMIB design.

Figure 8
figure 8

Total solar irradiance variability time-series (average over six hours) for three space radiometers: TIM ( http://lasp.colorado.edu/home/sorce/data/tsi-data/ ), PREMOS (Schmutz et al. 2013), and SOVAP.

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6 Conclusion

To determine the TSI from SOVAP we established a new instrumental equation. A new parameter was integrated from a theoretical analysis that highlighted the thermo-electrical non-equivalence of the radiometric cavity. From this new approach, we obtained values that are lower than those previously provided with the same type of instruments. The SOVAP right channel yields a TSI value of 1363.0 W m−2 (summer 2010) with an uncertainty of ± 1.52 W m−2 (k=1). The SOVAP left channel measured a TSI value of 1361.2 W m−2 (summer 2010) with an uncertainty of ± 1.39 W m−2 (k=1). This latter measurement is consistent with the PREMOS result (using the total solar irradiance radiometer facility calibration). Based on SOVAP data, we obtained that the TSI input at the top of the Earth’s atmosphere at a distance of 1 AU from the Sun is 1362.1 W m−2 (from the mean value of the left and right channels) with an uncertainty of ± 2.4 W m−2 (k=1). We based the thermo-electrical non-equivalence ΔP o of the radiometer on a theoretical model. Accurate estimates of the non-equivalence of the effects of the radiative and electrical heating can be obtained using the pre-fligth characterizations. Owing to a failure of its mechanism, the right shutter of SOVAP has remained permanently open since November 2010 up to the beginning of 2013. The instrument was therefore set to a new mode of functioning. From this non-differential mode operation, we were able to give the TSI variations. The SOVAP right channel (in non-differential mode) yields a mean TSI value of 1362.9 W m−2 with an uncertainty of ± 1.63 W m−2, and continuing the solar data record (from the end of 2010 to the beginning of 2013). During this period, the instruments in space (TIM, PREMOS, and ACRIM 3) yielded a TSI variation that is substantially identical.