International Journal of Aerospace Engineering

Volume 2015 (2015), Article ID 928206, 14 pages

http://dx.doi.org/10.1155/2015/928206

## Modelling of Solar Radiation Pressure Effects: Parameter Analysis for the MICROSCOPE Mission

ZARM, University of Bremen, Am Fallturm, 28359 Bremen, Germany

Received 30 July 2015; Accepted 19 October 2015

Academic Editor: Paolo Tortora

Copyright © 2015 Meike List et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Modern scientific space missions pose high requirements on the accuracy of the prediction and the analysis of satellite motion. On the one hand, accurate orbit propagation models are needed for the design and the preparation of a mission. On the other hand, these models are needed for the mission data analysis itself, thus allowing for the identification of unexpected disturbances, couplings, and noises which may affect the scientific signals. We present a numerical approach for Solar Radiation Pressure modelling, which is one of the main contributors for nongravitational disturbances for Earth orbiting satellites. The here introduced modelling approach allows for the inclusion of detailed spacecraft geometries, optical surface properties, and the variation of these optical surface properties (material degradation) during the mission lifetime. By using the geometry definition, surface property definitions, and mission definition of the French MICROSCOPE mission we highlight the benefit of an accurate Solar Radiation Pressure modelling versus conventional methods such as the Cannonball model or a Wing-Box approach. Our analysis shows that the implementation of a detailed satellite geometry and the consideration of changing surface properties allow for the detection of systematics which are not detectable by conventional models.

#### 1. Introduction

The modelling and propagation of satellite motion are one of the central tasks in mission analysis. The main driver for the evolution of a satellite orbit is the gravitational field of the central attracting mass. While a spherical symmetric approach for the gravitational field delivers undisturbed Kepler orbits, more realistic approaches employ spherical harmonics to model the gravitational potential. Among others, these models implement the effect of Earth oblateness, zonal, and tesseral variations of the mass distribution. Consequently, the introduced corrections of the gravitational field can be interpreted as a gravitational disturbance of an ideal Kepler orbit.

However, besides these perturbations, nongravitational disturbance (NGD) effects have a large influence on satellite motion. The largest of these NGDs in low orbit altitudes is the atmospheric drag resulting from the resistance of residual atmosphere against the satellite body moving at high relative speed. For higher altitudes, where the influence of residual atmosphere can be neglected, the dominant NGDs result from interaction of the satellite surface with solar photons, causing a drag force known as the Solar Radiation Pressure (SRP). The magnitude of the SRP acting on the satellite depends on a wide range of parameters. The distance to the Sun and the position of the satellite with respect to Earth and Sun (regarding possible eclipses) define the intensity of the incoming radiation. The geometry of the satellite, the optical properties of the external surfaces, and the actual orientation with respect to the Sun largely influence the orientation and magnitude of the evolving SRP. According to this, any SRP model depends on an accurate implementation of the satellite orbit, the attitude, and the geometric/physical properties of the satellite structure. As a consequence, a high modelling effort has to be made in order to obtain precise results. However, if mission planning and analysis for the satellite mission at hand possess high requirements on orbit modelling precision, a sophisticated SRP model is needed.

It has been argued for quite some time that commonly used SRP models like the Cannonball and the Wing-Box model are not sufficient enough for an accurate SRP analysis [1, 2]. This is particularly true if the involved geometries differ considerably from a spherical shape or a standard bus and solar panel assembly. The high gain in modelling accuracy by means of a realistic implementation of the satellite geometry has also been demonstrated with an analysis of NGDs acting during the cruise phases of the ESA Rosetta spacecraft [3]. Here a nonphysical solar constant was measured resulting from a parametric fit of the measured contribution of SRP on the total acceleration. By means of a sophisticated SRP and thermal radiation pressure (TRP) (TRP results from photons emitted by the spacecraft itself) model this offset was explained as a nonmodeled TRP correlated with the acting SRP. Further examples for a successful implementation of enhanced SRP models are GNSS satellites, where navigation accuracy directly benefits from an improved SRP modelling approach [4–6].

The modelling effort for an accurate analysis of the SRP effect on a given satellite is considerably high. Consequently, a trade-off has to be made between the precision requirements for the specific mission, the effort that one is willing to take, and the possible gain with respect to an improvement of a precise implementation of NGDs. This paper intends to give an overview on the implications of accurate SRP modelling and the expected improvement of NGD implementation. The parameters for the subsequent SRP analysis are derived from the French space mission MICROSCOPE [7] which delivers a suitable test case with respect to the specified mission profile.

The MICROSCOPE mission requires a very high accuracy of the spacecraft attitude and attitude stability due to the specific mission specification. In order to realize the high performance of the differential acceleration measurement of the two test masses to test the Weak Equivalence Principle (EP) it is essential to ensure a very low disturbance level (forces and torques acting on the satellite). For this purpose MICROSCOPE will be operated in drag-free mode. Any disturbance will be compensated by forces and torques generated by a cold gas propulsion system in closed loop control. The input to the corresponding controller is given by the common mode acceleration signal of the differential accelerometer while the science signal is extracted from the differential acceleration signal. However, in spite of the drag-free control, the exact modelling of NGDs is still necessary due to couplings between the accelerometers and the satellite structure. As a consequence, external disturbance effects influence the scientific signals since the drag-free control forces and torques needed to compensate the NGDs introduce a disturbance translated by the coupling.

The actual requirements of MICROSCOPE are quite demanding. For the EP measurement sessions the residual acceleration of the spacecraft shall be less than . At the EP test frequency, the angular pointing stability should be better than *μ*rad, and the angular velocity stability is required to not exceed rads^{−1}, respectively [8].

The sun-synchronous polar MICROSCOPE orbit leads to a force vector due to SRP always directed to one side of the orbital plane. This leads to a linear acceleration normal to the orbital plane which is superimposed by an angular variation due to seasonal change of the angle between the orbital plane normal and the direction to the sun. In case of simulating a drag-free mission a detailed modelling of the corresponding SRP forces and torques is important to estimate the actual control forces of the Attitude and Orbit Control System (AOCS) which keep the spacecraft in the favored state. Considering MICROSCOPE, NGD effects due to SRP can easily reach several . Divided by the satellite’s mass (330 kg) this force induces disturbing accelerations of some ms^{−2} which is not negligible in a premission end-to-end simulation and for developing and implementing data analysis and data processing strategies.

Due to the high demands of the mission, the sun-synchronous orbital plane, and the LEO character of the orbit, MICROSCOPE is an ideal test case scenario for the analysis of the general benefit of accurate SRP modelling for space missions. After a general introduction of the SRP modelling method and the derivation of SRP characteristics for chosen orbit and mission examples we will use MICROSCOPE as a test case scenario for a detailed SRP analysis. By looking at different approaches for the implementation of the geometry of the satellite and applying surface degradation models we highlight the possible benefits and the involved costs of high accuracy SRP modelling.

#### 2. Orbit and Attitude Propagation

Since the evolving SRP magnitude and orientation depend on the position and the attitude of the spacecraft, a dynamic orbit simulation including the gravitational acceleration caused by the Earth’s gravitational field is necessary. The calculation of the gravitational influence and the integration of the equation of motion are realized within the framework of the generic simulation tool* High Performance Satellite Dynamics Simulator* (HPS) [9]. The HPS is a MATLAB/Simulink library which is developed at ZARM in cooperation with the DLR Institute of Space Systems, Bremen. The main focus of HPS is the propagation of satellite orbits and the computation of the satellite’s orientation, depending on specific initial conditions and the space environment. Furthermore, the coupled motion of up to eight on-board test masses (arranged pairwise in up to four accelerometers) can be computed in six degrees of freedom.

Coupling effects between the satellite and the test masses as well as among the test masses themselves are included in the implemented differential equation systems of each considered body. In the following the satellite’s equations of motion are shown exemplarily. The satellite motion is given by (1):Here (i) is the mass of the satellite, (ii) is the acceleration of the satellite relative to the ECI frame, (iii) is the gravitational acceleration, (iv) is the control force, (v) is the sum of all disturbance forces acting on the satellite, and (vi) is the force due to the coupling between the satellite and all considered test masses. The superscript indicates that all components of (1) are given in ECI coordinates.

The satellite’s rotation and the satellite’s attitude motion are computed by using (2) and (3):Here (i) is the angular velocity of the satellite relative to the ECI frame, (ii) is the moment of inertia matrix of the satellite, (iii) is the sum of the control torques applied for attitude control, (iv) are the disturbance torques acting on the satellite, (v) are the torques generated from the satellite-test mass coupling, (vi) is the quaternion representation of , and (vii) represents the Euler symmetric parameters. Here the superscript indicates a description of the equations’ components in the satellite body fixed frame.

It is obvious that the satellite’s motion is affected by the acceleration due to the Earth’s gravitational field which cannot be considered to be spherically symmetric. This is due to the nonuniform mass distribution of the Earth and it results in (i) perturbations of the pure Kepler orbit and in (ii) perturbations of the satellite’s attitude. But apart from this, for a complete orbit and attitude propagation simulation, one has to take into account nongravitational effects acting on the satellite, too. They force it to go astray from its purely gravitational orbit and induce undesired rotations. For many missions, one of the most prominent effects of these NGDs is the SRP which will be discussed in detail in the next sections.

#### 3. SRP Model

The disturbance forces and torques due to SRP originate from the interaction of the satellite’s surface with the photons emitted by the sun. It is assumed that each photon that hits the satellite is either absorbed or reflected in a specular or diffuse way, thus effectively changing the momentum of the satellite. As a consequence, the resulting force acting on an elemental area can be expressed as the sum of three individual contributions [10]:where (i) is the SRP, (ii) is the elemental area, (iii) and are the unit vector in Sun direction and the unit vector normal on the elemental area , respectively, and (iv) is the angle between and . Finally, (v) , , and are the coefficients of absorption, of specular reflection, and of diffuse reflection.

With (assuming a nontransparent material) the force due to SRP can be derived as follows:Hence, the computation of requires the modelling of (i)the satellite orbit because the magnitude of depends on the distance to the Sun,(ii)the satellite attitude in order to derive the correct incident angle between and ,(iii)the satellite geometry for defining appropriate values of , , , and .The propagation of the satellite orbit and its orientation is one of the basic tasks within the simulation software HPS and can be applied to any Earth orbiting satellite mission. Since most NGDs such as the SRP are surface-based effects, the propagation needs an input model for the satellite geometry specific to the actual mission. Due to the variations in satellite components, general dimensions, and external materials it is not possible to find a suitable standard model that can be used in a flexible way with respect to the variety of spacecraft geometries. This is one of the main setbacks of standard approaches such as the Cannonball or the Wing-Box model.

Instead of using a simplistic approach, where is calculated with respect to (i) an effective projected satellite surface area and to (ii) averaged optical surface properties, the focus of the HPS SRP interface lies on the capturing of the influence of details of the satellite geometry and the involved material parameters of each component of the satellite. In order to realize this, the HPS SRP approach is divided into two main steps. Before the actual SRP is calculated, the satellite’s surface is discretized in small elements and the different optical properties are assigned to the corresponding elements. For a complex geometry this is realized by means of a finite element (FE) preprocessor, where the meshes of the external surfaces are exported together with their respective optical property definitions. Subsequently, the HPS algorithm for SRP computation evaluates (6) which is the discrete form of (5) for each element that is illuminated by the Sun for the chosen vector :The overall force is then derived by computing the sum over all elements:By means of geometric criteria (see [11] for details), the algorithm determines automatically if an element is lit by the Sun and considers shadowing by other parts of the satellite as well. In combination with an eclipse model, the global SRP acting on the satellite is calculated with respect to a realistic illumination scenario.

In order to speed up the simulation process, resulting SRP magnitude and directions can be derived in a normalized form. Here a lookup table can be derived where parameters of the stored SRP values are the solar elevation and azimuth consequently defining the current sun angle. When the lookup table is computed in preprocessing, the results can be used to determine the dynamical evolution of the SRP during flight within an HPS simulation. For this, the normalized SRP values are converted to the actual SRP with respect to the current solar distance and orientation of the satellite as well as the eclipse condition.

#### 4. Parameter Analysis

In order to review the systematics of the SRP force model discussed here, a parameter analysis is performed. The implications of changes of the relevant input parameters such as orbital elements and geometrical and technical features with respect to the overall magnitude of the resulting disturbance force due to SRP are discussed in the following.

##### 4.1. Solar Radiation Pressure

Since the magnitude of the incident solar radiation does depend not only on the orbit of the satellite around the Earth but rather on the Earth’s orbit around the Sun, too, it is sensible to analyse the influence of the implications of the central body orbit during the year. The annual variation of the strength of is depicted in Figure 1. Exemplarily, the resulting SRP in for the CHAMP mission orbit (see Table 1 for details) is presented.