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Innovation Large Changes to the Spacecraft Orbit Innovation large changes to the spacecraft orbit. See Modeling Photon Pressure the sidebar “How big are these radiation forces?” to get a “ball-park” feel for how big these forces are, and what effect they have on the The Key To High-Precision orbit. Overall force modeling is used in three areas of GNSS: design, operation and sci- GPS Satellite Orbits entific analysis. At the design stage we need to understand Marek Ziebart, Paul Cross, and Sima Adhya University College London how changes in the spacecraft structure (e.g. size of the solar panels, materials used for “Photons have mass?! I didn’t even know they were Catholic.” — Anonymous component shielding) affect the orbit dynam- Actually photons have no mass, but that does not mean they cannot affect GPS satellite ics. These systems are designed to pro- orbits. GPS satellites operate in a harsh, radiation-filled environment 20,000 kilometers above vide global coverage, and this entails being the surface of the Earth. Solar radiation pressure — the force due to the impact of solar able to predict how the orbits change over photons and the related effects of anisotropic thermal re-radiation and albedo are all tiny a number of years. Moreover, as the orbit forces and yet they have a strong perturbing effect on the GPS satellite orbits. Predicting how decays over time fuel must then be expend- GPS satellites will move in space relies upon understanding and modeling these effects, and ed in firing thrusters to bring the satellite the accuracy of these predicted orbits underpins the entire system for positioning, velocity trajectory back within its design thresh- determination, and a host of other applications. This month’s Innovation column exam- old. Only a limited amount of fuel can be ines the significance of these forces and how they can be modeled. carried, and therefore, optimizing the cost Marek Ziebart is a research fellow at the Department of Geomatic Engineering, University of launching the satellite (which is strong- College London (UCL). His main research areas are orbit determination problems, non- ly mass-dependent) against the mass of fuel conservative force modeling and space-borne GPS attitude control. Dr. Ziebart co-authored available at the start of the spacecraft’s oper- the article "Over the Silk Road: Bringing Satellite Imagery Down to Earth" in the April ational lifetime requires detailed under- 2001 issue of GPS World. Paul Cross is the Leica Professor and head of the Department of standing of how the orbit is going to evolve. Geomatic Engineering at UCL. He has been involved in GPS research since about 1980, con- The operational stage requires orbit pre- centrating mainly on high precision applications in engineering and geophysics. Dr. Cross is diction. The better we understand all the a member of the GPS World Editorial Advisory Board. Sima Adhya has degrees in Natural physical mechanisms affecting the satellite’s Science and Space Science from Cambridge and University College London respectively. She trajectory, the better we can predict the orbit. is currently reading for a Ph.D. in satellite geodesy and astrodynamics at UCL. Her princi- In determining the broadcast ephemerides, pal research interests are high precision orbit determination and GNSS system design. the Master Control Station estimates the val- ues of two parameters related to SRP for each spacecraft. Even in space-based augmentation systems such as WAAS (Wide Area Augmentation System), the modeling of the tiny SRP forces is impor- tant. All real-time GPS applications rely fundamentally upon the accuracy of the predicted orbit. Scientific analysis involves post-pro- cessing of the received signals and is A fundamental component of the Global are more problematic (see Figure 1): used in measuring geodynamic phenomena Po sitioning System is the calculation of a Solar radiation pressure (SRP), that is, highly accurate predicted orbit for each of the force produced by the impact of elec- Acceleration (meters the constellation spacecraft on a regular tromagnetic radiation from the Sun striking Force per second squared) basis. In general, the parameters used to the spacecraft. Earth gravity modeled as a point mass 6.1 x 10-1 describe these orbits are not stable and need Albedo, the force due to electromag- Earth gravity oblateness modeled by 1.0 x 10-4 the J2 coefficient to be constantly updated. Being able to pre- netic radiation reflected by the Earth which Lunar gravity 3.9 x 10-6 dict changes, and also being able to calcu- is supplemented by thermal radiation emit- Solar gravity 1.0 x 10-6 late the orbit after the fact, relies upon a com- ted by the Earth, and Summed effect of Earth gravity field, 2.2 x 10-7 bination of range observations and force Thermal re-radiation (TRR) forces, coefficients 2,1 to 4,4 modeling. A whole range of different forces, caused by anisotropic radiation of heat from Solar radiation pressure 7.2 x 10-8 with varying magnitudes, act on the GPS the spacecraft. Summed effect of Earth gravity field, 5.9 x 10-9 satellites. Tab le 1 lists these forces in decreas- These are non-conservative forces (NCF). coefficients 5,0 to 8,8 ing order of magnitude along with crude esti- This means that they change the energy state Albedo (or Earthshine) 1.5 x 10-9 mates of the accelerations they cause. of the spacecraft, and for this reason they Thermal re-radiation 1.4 x 10-9 There are also many smaller forces, such are important agents in changing the evo- Solid Earth tide, raised by the Moon 1.3 x 10-9 as drag effects, that are too small to worry lution of the orbit’s parameters in both Solid Earth tide, raised by the Sun 4.5 x 10-10 about for current techniques. the short and the long term. Venus gravity 1.1 x 10-10 While the modeling of gravitational forces Despite having tiny magnitudes, SRP and is well understood, the following forces the other non-conservative forces cause TABLE 1 Forces acting on GPS satellites www.gpsworld.com GPS World January 2002 43 Innovation ing through one square meter at one astronomical unit (AU, approximately Antenna support equal to the semimajor axis of the Earth’s sub-assembly Apogee SRP orbit) from the Sun. Antenna boresight engine Spacecraft attitude — this governs direction which parts of the spacecraft are illu- minated by the Sun, and which parts are in shadow,which parts are heating Albedo up, and which parts are cooling down. Optical parameters of spacecraft Central bus Solar Thermal re-radiation surface materials — typically these are panel the reflectivity and specularity coeffi- FIGURE 2 A GLONASS IIv satellite illustrates FIGURE 1 GPS satellite orbits are per- cients of each spacecraft surface com- turbed by the impact of photons in the spacecraft body-fixed system. ponent; see the sidebar “Terminology” solar radiation as well as sunlight (next page) for definitions. reflected from the Earth and also by the emission of photons in satellite Spacecraft structural details – the size over the entire solar cycle. heat dissipation. and shape of the structure, which parts Spacecraft Attitude. The spacecraft attitude are static and which parts can move. has two design constraints that can be used such as post-glacial rebound, plate tecton- Thermal conductivity and emissivity of in a model. Firstly, the antenna boresight is ic motion and volcano magma chamber infla- structural elements. constrained to point at the geocenter in order tion. Such ultra-high precision applications How can we assign values to these para- to distribute evenly the GPS signal over of GPS observables require positional accu- meters or create models for them? the hemisphere that is visible to the satel- racy of a few millimeters over length scales Solar Irradiance. The total solar irradiance lite.Secondly, the solar panels are orient- of 100-1000 kilometers. To this end, enor- (TSI) is measured directly in the space envi- ed to point continually towards the Sun. The mous effort has been expended in improv- ronment by a number of sensors on probes spacecraft body-fixed system (BFS) is then ing the accuracy of GPS orbits by trying to such as SOHO (Solar and Heliospheric related to the instantaneous geocentric posi- improve the accuracy of the force modeling. Observatory) or ERBS (Earth Radiation tion vector of the satellite center of mass as So, how can we deal with the problem? Budget Satellite). Because of this, we know follows: that the solar irradiance does vary as a f Let Data and Parameters unction of the solar cycle (the solar cycle r = Earth-centered inertial (ECI) position Somehow we need to build up a model of lasts approximately 11 years) over a range vector of the spacecraft center of mass at the non-conservative forces that are affect- of 1.7 watts per meter squared, with an time t ing an orbit. The following parameters are approximate mid-range value of 1369 watts s = ECI position vector of the Sun at important in this process: per meter squared. This variation introduces time t Solar irradiance — the amount of solar only very small changes to the SRP force p = vector from the spacecraft (or probe) electromagnetic radiation (in watts) pass- calculated using TSI, of the order of 0.1% to the Sun. Then, How big are these radiation forces? What effect do they have? p = s – r. The spacecraft Z-axis, zˆsc, points along Go to the kitchen and pick up 100 grams of –r, hence something. The force that you feel pushing where down on your hand is about one newton at sea level.
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