Impact Analysis of Solar Irradiance Change on Precision Orbit Determination of Navigation Satellites

ZHANG Yan，WANG Xiaoya,3*，XIKewei，SHAO Fan

1.Astro-Geodynamics Research Center，Shanghai Astronomical Observatory，Chinese Academy of Sciences，Shanghai 200030，P.R.China；2.University of Chinese Academy of Sciences，Beijing 100049，P.R.China；3.Shanghai Key Laboratory of Space Navigation and Positioning Techniques，Shanghai Astronomical Observatory，Chinese Academy of Sciences，Shanghai200030，P.R.China

Abstract:Solar radiation pressure is the main driving force and error source for precision orbit determination of navigation satellites.It is proportional to the solar irradiance，which is the“sun constant”.In regular calculation，the“solar constant”is regard as a constant.However，due to the existence of sunspots，flares，etc.，the solar constant is not fixed，the change in the year is about 1%.To investigate the variation of solar irradiance，we use interpolation and average segment modeling of total solar irradiance data of SORCE，establishing variance solar radiation pressure（VARSRP）model and average solar radiation pressure（AVESRP）model based on the built solar pressure model（SRPM）（constant model）.According to observation data of global positioning system（GPS）and Beidou system（BDS）in 2015 and comparing the solar pressure acceleration of VARSRP，AVESRP and SRPM，the magnitude of change can reach 10-10 m/s2.In addition，according to the satellite precise orbit determination，for GPS satellites，the results of VARSRP and AVESRP are slightly smaller than those of the SRPM model，and the improvement is between 0.1 to 0.5 mm.For geosynchronous orbit（GEO）satellites of BDS，The AVESRP and VARSRP have an improvement of 3.5 mm and 4.0 mm，respectively，based on overlapping arc，and SLR check results show the AVESRP model and the VARSRP model is improved by 2.3 mm and 3.5 mm，respectively.Moreover，the change of inclined geosynchronous orbit（IGSO）satellites and medium earth orbit（MEO）satellites is relatively small，and the improvement is smaller than 0.5 mm.

Key words:solar pressure acceleration;total solar irradiance;precise orbit determination;global positioning system（GPS);Beidou system（BDS)

0 Introduction

Solar radiation pressure is an important research topic in the precision orbit determination of navigation satellites.At present，the widely used solar radiation pressure models include three types in the precision orbit determination of navigation satellites：Analytical solar radiation pressure（SRP）models，semi-empirical and semi-analytical SRP models，and empirical SRP models.The analytical models and the semi-analytical and semi-empirical models need to use the solar constant while calculating the solar radiation force.The solar constant is the receiving solar radiation flux density which is perpendicular to the solar ray surface in the upper limit of the Earth’s atmosphere，with an unit of W/m2.In general，the solar constant is treated as a constant among the SRP models，and its value varies with the times.At present，most scientists choose the value of 1 368 W/m2.In fact，however，the solar constant is not a constant.In 1983，Climate Change[1]published the Nimbus-7 satellite measurements：The solar constant is changing.The valley values of most solar constant changes are corresponding to the peak of sunspots day by day.This is because the temperature of sunspots is lower than that of the surface of the photosphere，and solar radiation will decrease naturally with the increase of sunspots.This view is at variance with the most cognitions of scientists.It is generally believed that the activity of sunspots will increase and the solar radiation will increase.Foukal et al.[2]solved this problem.They thought that sunspots reducing solar radiation is a short-term behavior.When sunspots increase，the flares also increase.As a result of the increase of the flares，the solar radiation will increase，and the solar radiation increased by the flares is larger than the decrease due to the sunspots.So solar radiation is generally increased.Therefore，the solar constant shows a trend of synchronization with the sunspot in an 11-year cycle，so more and more scientists and research institutions have chosen solar irradiance or total solar irradiance（TSI），which is a more rigorous statement，instead of the term of solar constant.

In this paper，we use the measured solar irradiance data obtained by space observation to analyze the variation characteristics of solar irradiance.For global positioning system（GPS）and Beidou sysem（BDS）navigation satellites，we analyze the effects of solar irradiance change on solar pressure acceleration and its orbit determination accuracy.This provides a basis for considering the effect of global navigation satellite system（GNSS）high precision orbit determination.

1 Solar Irradiance and SRP

1.1 Current status of solar irradiance research

The solar irradiance is the receiving solar radiation flux density，which is perpendicular to the solar ray surface in the upper limit of the Earth’s atmosphere at the average distance（D=1 AU=1.496×108km），and the unit is W/m2.Solar irradiance is a relatively stable value：It is based on changes of the activity of sunspots，including all forms of solar radiation.It is constrained by solar activities and it has various temporal scale characteristics.In early days，the measurement of solar irradiance was based on the known mass of water placed in the sun for a period of time.The temperature rising process was measured with a thermometer，and the sunlight intensity could be calculated because the specific heat capacity of the water was known[3].Claude Pouillet and John Herschel designed different devices according to this principle and measured the value which was about 680 W/m2[3].Abbot[4]measured the solar irradiance based on ground-based measurements at a high-altitude observation for many years.The measurement result was 1 302—1 465 W/m2.

The Nimbus-7 meteorological satellite was successfully launched and it began to accurately measure the solar irradiance，thus the misunderstanding of the“solar constant”was changed.In 2003，the University of Colorado Boulder’s Laboratory for Atmospheric and Space Physics used the total irradiance monitor（TIM）[5]radiometer to measure the solar irradiance value every day or every 6 h，as shown in Fig.1.It can be seen that there is a solar activity cycle of about 11 years.Table 1 gives the TSI mean values for satellite measurements from different ages[6].The mean value of solar and heliospheric observatory（SOHO）is 1 365 W/m2，and that of TIM is 1 361 W/m2.According to Kopp et al.[7]，this is because the instruments used to measure solar irradiance on satellites are not different from those used on the ground in the past.In addition，most of the previous instruments were calibrated with world radiometric reference（WRR）before launch，and the current instruments are based on the system international（SI）.The WRR itself is relatively higher than the SI irradiance standard，which results in low values[6，19].

Fig.1 Solar irradiance time series measured by TIM/SORCE during 25 Feb.,2003 to 5 Dec.,2018

Table 1 The mean TSI of satellite measur ements in different ages

It can be seen that the values of different solar irradiance instruments are basically the same，the solar irradiance is not constant，but there is a solar activity cycle change，and the variation range is about 0.1%.For the solar irradiance data of SORCE based on the work of the predecessors，the accuracy has been improved by±0.035%[20].

1.2 Solar irradiance theory derivation

The solar radiation power comes from the gravitational self-polymerization energy of the sun，that is，the gravitational potential energy of the sun[21]

where M is the solar mass，G the universal gravitational constant，and R the solar radius.Considering the two-body problem of the sun and the earth，time and space are the basic forms of the existence of moving matter，and the basic properties inherent in the object.Time and space are inseparable from the moving matter，so time factor is introduced in the physical quantity for the two-body problem[22-23]

where H is the Hubble coefficient of the sun.After Eq.（2）is substituted to Eq.（1），it is obtained as

The solar radiation power is the differential of Ωversus time t

According to the relationship between solar irradiance and solar radiation power

whereΦ0is the solar irradiance，r the distance between the earth and the sun.We choose 1 AU.

According to astronomy constants，G=6.674 184×10-11N·m2/kg2，1 AU=1.495 978 70×10-11m，H≈6×10-16s-1，M=1.988 47×10-30kg，R=6.959 97×1011m. And we get Φ0=1 456.841 W/m2.

Comparing the above results with the measured values of the satellites，we can see that the theoretical calculations are large.This is probably because among the solar radiation calculation，the sun is not a simple point source，but a group of beam arrays，so the theory does not fully reflect the true situation of solar radiation，which means that the theory of solar radiation needs to be developed before it can be consistent with actual observations.

The theory of solar irradiance is also inferred to from radiation propagation theory with regard to solar radiation power. According to Stefan-Boltzmann’s law，the relationship between the radiant energy ETper unit area of the sun and the temperature T is

whereσis the Stefan-Boltzmann constant.The total radiant energy E of the sun is

where S is the surface area of the sun.When solar radiation is isotropically radiated outward in the sphere space，the solar radiation received in a unit area from the sun with the distance of r is the solar irradianceΦ0

Organize the above formulas

where r is the distance between the earth and the sun.We choose 1 AU.

According to astronomy constants，σ=5.67×10-8W/m2·K4，1 AU=1.495 978 70×1011m，T=5 780 K，S=6.087 35×1018m2.And we getΦ0=1 369.815 W/m2.

1.3 SRP models

According to the principle of the SRP model，the satellite can be decomposed into a plurality of components having different shapes and sizes.For the solar radiation force of the differential element of the satellite component，it can be expressed as[24]

whereλis the eclipse factor of the satellite；Φ0the solar irradiance at 1 AU；c the speed of light；rssthe distance from the satellite to the sun；d A the differential area element of the satellite；are the the normal vector of the element area and the direction vector of the satellite to the sun，respectively；θis the angle between the normal of the area element and the direction of the satellite to the sun；μ the specular reflection coefficient，andνthe reflectivity.Then the solar radiation pressure acceleration is[24]

ROCK 4 and ROCK 42 are the first analytical SRP models established by satellite manufacturers Rockwell International and IBM for GPS Block I and Block II/IIA satellites，respectively.Fliegel et al.[25]have made thermal radiation corrections for the ROCK model to establish T 10 and T 20 models，then Fliegel et al.[26]used the detailed physical parameters provided by satellite manufacturer Martin Marietta to establish the T 30 model for the Block IIR satellite.For the ROCK 4，ROCK 42，T 10，T 20 and T 30 models，the solar irradiance（formerly known as the solar constant）was taken to be 1 368 W/m2.

In recent years，Ziebart et al.[27，28]from University College London have established accurate models for non-conservative forces for handling different types of low-orbit and high-orbit satellites.They further considered the refined three-dimensional model of the satellite and used the ray tracing method to calculate the SRP of each component of the satellite comparing with T 20 and T 30 model.In this model，the solar irradiance is 1 368 W/m2.Since the analytical model cannot fully describe the SRP，Rodriguez-Solano et al.[29]established a semi-empirical and semi-analytical Adjustable Box-Wing model on the basis of Box-Wing by adjusting the optical characteristics of the solar panel and the illuminated panel.In this model，the solar irradiance is 1 367 W/m2.

Froideval[30]mentioned that the solar constant was not a constant，and it was a changing sequence of values.He took a solar constant equal to 1 367.2 W/m2.Zhao[24]analyzed several factors affecting the accuracy of the SRP models.In the SRP model he established，the solar constant was 1 368 W/m2.

2 Solar Irradiance Variance Characteristics and Its Influence on Precision Orbit Determination

In this paper，we use the solar irradiance data of SORCE from the Colorado University to analyze the variance characteristics，and the sampling interval is 6 h.We choose linear interpolation method and segmentation averaging method to calculate the instantaneous solar irradiance and the average solar irradiance of time period，respectively.According to the physical modeling approach of SRP model，we established a variation of solar irradiance SRP model，named as VARSRP，and a segmental average solar irradiance SRP model，named as AVESRP.Then，we take the constant solar irradiance model SRPM[13]to compare the SRP accelerations and the orbit determination accuracy，and investigate its influence on the orbit determination of navigation satellites.

2.1 Solar irr adiance var iance characteristics investigation

Fig.2 shows the difference sequence between Physikalisch-Meteorologische Observatorium Davos（PMOD）Virgo data（ranging from 30 January，1996 to 4 February，2002，with a sampling interval of 1 h），the Colorado SORCE data（ranging from 25 February，2003 to 11 October，2018，with a sampling interval of 6 h）.The constant value is Φ0=1 368 W/m2.The solar irradiance data of Virgo maintain at-4—1 W/m2，and the data of SORCE is-12—-5 W/m2.According to the statistical results，the average values of SORCE data changes is-7.093 07 W/m2，the standard deviation is 0.416 21，and the maximum change value is-11.018 90 W/m2；the average value of Virgo data changes is-1.283 68 W/m2.The standard deviation is0.605 38 and the maximum change-3.322 02 W/m2.There is a certain gap between the use of the constant calculation and the measured data among the results，so this element should be considered.

Fig.2 The difference sequence between the PMOD Virgo data and the Colorado SORCE data with the currently used constant solar irradiance values

Fig.3 shows difference sequence of the neighbor values between PMOD Virgo data（ranging from 30 January，1996 to 4 February，2002，with an interval of 1 h）and the Colorado SORCE data（ranging from 25 February，2003 to 11 October，2018，with an interval of 6 h），which presents the differences between forward and backward values in these two data sources.The difference between the solar irradiance values is maintained at-1—1 W/m2.According to the statistical results，the average value of SORCE data changes is-4.608 57e-05 W/m2，the standard deviation is 0.041 60，and the maximum change value is 1.104 50 W/m2；the average value of Virgo data change is 1.669 94e-04 W/m2，the standard deviation is 0.029 84，and the maximum change value is 0.226 51 W/m2.This confirms that the change in solar irradiance is relatively stable，and the variance range is about 0.1%.At the same time，this also provides a theoretical basis for the selection of interpolation methods for the next solar irradiance modeling.

2.2 Analysis on precision orbit determination for GPS

GPS is the first research object in this study，and it has high precision orbit[31].We use the global GPS data from Day of Year（DOY）060 to 149 in 2015，the constant solar irradiance SRP model SRPM（Φ0=1 368 W/m2），the variance solar irradiance SRP model VARSRP and the segmental average solar irradiance SRP model AVESRP to perform SRP accelerations and precise orbit determination.Fig.4 shows the difference accelerations of VARSRP and AVESRP comparing with the constant solar irradiance SRP model SRPM（regarding PRN01 satellite as an example）.The SRP acceleration differences of VARSRP and AVESRP relative to SRPM can reach 10-10m/s2.It means that the error caused by the change in solar irradiance is a factor should be considered when the perturbation magnitude of 10-10m/s2is considered.

Figs.5，6 show the differences in acceleration between different types of GPS satellites using VARSRP，AVESRP model and SRPM model.The VARSRP and AVESRP models are slightly different，but they are more consistent and show common features.The difference between the acceleration of most GPS satellites is up to 10-10m/s2，and some even reach to 10-9m/s2，but the difference in average acceleration is relatively small，basically at 10-13m/s2magnitude，and mean square error at 10-11m/s2.

Fig.5 Acceleration difference between AVESRP and SRPM

Fig.6 Acceleration difference between VARSRP and SRPM

Fig.7 illustrates the 3D-RMS values of GPS satellite orbit determination for three different SRP models（SRPM，VARSRP，and AVESRP）.The 3D-RMS is obtained from comparing the orbit results with the IGS precision orbit.We choose 30 GPS satellites including IIF，IIR-M，IIR-A and IIR-B to calculate the orbital results.It can be seen from Fig.7 that for most GPS satellites，the AVESRP model and the VARSRP model are slightly improved comparing with SRPM model.

Fig.7 3D-RMS of GPS orbit for the SRPM,VARSRP,and AVESRP

In order to reflect the change of orbital accuracy for the three models of GPS satellites，Table 2 gathers statistics for orbital accuracy of GPS satellites.It implies that the three models have little difference in the orbit determination accuracy of GPS satellites.AVESRP model has 10%（3 satellites）and 17%（5 satellites）satellites unchanged and 70%（21 satellites）and 73%（22 satellites）slightly improved comparing with the SRPM model，and the improvement is about 0.1—0.5 mm.In addition，the variance of solar irradiance has no obvious regularity for the orbit determination accuracy for different types of GPS navigation satellites.

2.3 Precision orbit determination analysis for BDS

BDS is the important research object in this study，and its orbit accuracy is not up to the level of GPS，so more deep research is needed[32].We used the multi-GNSS experiment（MGEX）network data from DOY 001 to DOY 093 in 2015 for the precision orbit determination.During this period，there are 50—60 stations containing BDS data，and the orbit determination strategy adopted non-difference phase and pseudorange observations with 3 d an arc.Then，three models SRPM（Φ0=1 368 W/m2），VARSRP（interpolationΦ0），and VARSRP（segment averageΦ0）are used to calculate the SRP acceleration and orbit determination accuracy，respectively.We select GEO（C04）IGSO（C06）and MEO（C14）which are representative of different types of BDSs.Fig.8 shows the difference accelerations of VARSRP and AVESRP comparing with the constant solar irradiance SRP model SRPM in three directions，which are along，cross and radial.We can see that the difference reaches 10-10m/s2.

Fig.9 shows the statistical results of the acceleration difference of VARSRP and AVESRP models relative to the SRPM model.It can be seen that the difference can reach to 10-10m/s2，and some even reach to 10-9—10-8m/s2，but the average is relatively small，basically at 10-11m/s2magnitude，and the mean square error is around 10-11m/s2.Moreover，the normal of the orbital plane changes greatly，and some reach 10-9m/s2.This is greater than the impact of GPS satellites.In addition，in the acceleration changes of the three types of satellites，the GEO satellites change the most；while in the radial direction，along-track and cross-track，the cross-track has the greatest influence.From the view of SRP acceleration for BDS，the solar irradiance should be a necessary factor for noticing.

Table 2 3D-RMS of GPSsatellites for thr ee models

Fig.10 shows the 3D RMS statistics analysis for three types of BDS satellites about orbital overlapping arc.Fig.11（a）shows all BDS satellites orbit determination accuracy differences of the AVESRP and VARSRP relative to SRPM.Table 3 gives the 3D-RMS values of the overlapping arc for the three models of BDS satellites.Fig.11（b）shows the RMS value of the BDS orbital residual.It can be seen that the accuracy of orbit determination with AVESRP and VARSRP is more obvious for GEO satellites，and the improvements are 3.5 mm and 4.0 mm，respectively.The VARSRP is slightly better than the AVESRP model，especially for the accuracy of the C05 satellite.But both models perform poorly for C03，and the reasons for this need to be further studied.For IGSO and MEO satellites，the accuracy of the three models is not much different.This may be due to the higher orbit of GEO satellites，which is more affected by changes in solar irradiance.In addition，according to the statistics of the daily orbital accuracy of SRPM for all satellites of the BDS，the accuracy of the orbit determination of 67%and 69%is improved，respectively.From the view of precision orbit determination，solar irradiance is an important element for BDS.

Fig.8 Differences between of VARSRP,AVESRP and constant solar irradiance SRP model in three directions including along trace a(top),orbital surface normal c(middle)and radial r(bottom)(C04,C06,C14)

Fig.9 Acceleration differences of VARSRP and SRPM(top),AVESRP and SRPM(bottom)

In order to more reliably evaluate the effects of BDS three different solar irradiances SRP models on the orbit determination，this paper uses the normal point（NP）data of the satellite laser ranging（SLR）from the international laser ranging service（ILRS）to check the orbit determination accuracy externally.There were C01，C08，C10，and C11 satellites with laser observations in 2015.The full rate data are used as reference and supplement to check results.Since the C08 and C11 data are less，we choose C01 and C10 to gather statistics.The results of C01 and C10 SLR check are given in Fig.12.It can be seen that the orbital accuracy of C01 using the AVESRP model and the VARSRP model is improved by 2.3 mm and 3.5 mm，respectively，and C10 has been improved by 0.2 mm and 0.4 mm，respectively.

Fig.10 Average 3D-RMS values of BDS satellites overlapping arc

Fig.11 Average 3D-RMS values of satellites overlapping arc differences and the RMS values for BDS orbital residuals

Table 3 3D-RMSvalues of the overlapping arc for the three models of BDSsatellite

Fig.12 SLR check results for BDS satellites

3 Conclusions

This paper firstly analyzes the physical mechanism that the“solar constant”is not constant.We start with the research status and theoretical introduction of solar irradiance，then use the spatial measurement data to analyze the variance characteristics of solar irradiance.Next，as for GPS and BDS navigation satellites，we use measured data to construct different types of SRP models including SRPM（Φ0=1 368 W/m2），VARSRP and AVESRP，and we use these three models to calculate and compare SRP acceleration and orbit determination accuracy.The results show that the SRP acceleration differences of VARSRP and AVESRP relative to SRPM can reach 10-10m/s2，and some satellites even reach 10-8m/s2，but the difference in average acceleration is relatively small，at the magnitude of 10-11m/s2，and the mean square error is around 10-11m/s2.In addition，the changes are relatively large in cross-trace，and some can reach 10-9m/s2.

The three models have little difference in the orbit accuracy of GPS satellites in totally.Compared with the SRPM model，the AVESRP and VARSRP have 10%（3 satellites）and 17%（5 satellites）unchanged respectively，70%（21 satellites）and 73%（22 satellites）have a slight improvement，and the increase range is 0.1—0.5 mm.Moreover，there is no obvious regularity for the orbit determination accuracy of different types of GPS navigation satellites.For the GEO satellites of BDS，the accuracy of orbit by AVESRP and VARSRP is obviously improved.But for IGSO and MEO satellites，the accuracy differences of the three models are small.This may be due to the higher orbit of GEO satellites，which is more affected by changes in solar irradiance.The statistics of the BDS daily orbital accuracy for all satellites of AVESRP and VARSRP models have 67%and 69%improved.

In conclusion，when we consider the magnitude of 10-10m/s2，the error caused by the change of solar irradiance is a factor that must be considered.In addition，high-precision navigation satellites precise orbit determination，especially for satellites with accuracy requirements are better than 2 cm，need to consider the influence of the solar radiation force caused by the change in solar irradiance.It is recommended that the solar irradiance obtained from satellite measurements should be used to model solar radiation pressure.