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. Divide that by two and imagine how is the acceleration due to the Earth mass mod- much smaller the force has become. Now divide eled as a point and a comprises all the and the spacecraft Y-axis, yˆsc , points in a the original one newton force by 10,000, and perturbing accelerations due to perturbing forces. direction perpendicular to both zˆsc and p, try to imagine how much it would push your If this equation were solved numerically hence hand. That is the magnitude of the first order including models for forces due to lunar, solar force due to SRP acting on a spacecraft of mass and planetary gravitation, solid Earth tides and circa 1,500 kilograms. Now make that tiny other small forces, as well as SRP effects, and force 100 times smaller still (one millionth of a then we did the same calculation but ignoring Finally, the spacecraft X-axis, xˆ sc,is orthog- newton!) and that is the magnitude of the force NCF, the difference between these two trajecto- onal to the other two axes, completing the that needs to be modeled accurately for precise ries after one revolution around the Earth right-handed system. orbit determination. This is equivalent to the (about 12 hours for the GPS satellites) would be So, given the geocentric position vectors force of just two grains of salt pushing down on between 100 and 200 meters in the along-track of the satellite and the Sun, this procedure your hand! This is at the level where thermal re- direction. In practice, the process of orbit pre- yields the BFS axial unit vectors in terms radiation, albedo and subtle SRP effects become diction does involve estimating empirical para- of ECI vector components. important. meters, so even if SRP effects were ignored then Under these attitude constraints the BFS So, what would happen to your predicted the modeling would pick up some of the effect. X-Z plane nominally includes the Sun and is, orbit if you were to ignore NCF? The second However, from the above it is clear that over to first order, also a plane of symmetry for order differential equation of motion of the time SRP effects have a big influence on the the spacecraft (see Figure 2). As a result of satellite is: spacecraft trajectory. the attitude control algorithm, the profile of the spacecraft presented to the Sun over most
44 GPS World January 2002 www.gpsworld.com Innovation of its mission lifetime varies uniquely as a the ‘Probe-Sun’vector (see Figure 3). Some applied to Block II and Block IIA spacecraft). function of the so-called Earth-probe-Sun SRP models use the EPS angle as an inde- Optical Properties of Spacecraft Surface . The (EPS) angle. This is the angle between the pendent variable for computation of the force. interaction of solar radiation with the space- BFS Z-axis (which points to the Earth) and The only problem we may have to deal craft can be modeled using coefficients that with using the approach above is that it is describe how much radiation is absorbed, based on a nominal attitude algorithm, as how much is reflected and in what way. These opposed to some measurements that might can be expressed in various ways, but one p give us an estimate of the true attitude of the common system uses reflectivity and spec- spacecraft. While it might be possible to esti- ularity. See the “Terminology” sidebar for mate attitude variations within the orbit deter- details. Provided we know the material type, s EPS angle mination process, in practice it is difficult to standard measured values of the coefficients de-correlate other anomalous, un-mod- can be used. Once the surface has been r Earth eled forces from the true variations in atti- exposed to the space environment, these tude. Unpredictable attitude variations can optical properties may change slightly, tend- FIGURE 3 The Earth-probe-Sun (EPS) angle give rise to the so-called Y-bias. The more ing towards more diffuse reflection as the is the angle between the Earth and Sun predictable and deliberately inserted yaw surface becomes pitted by galactic cosmic directions as viewed from a spacecraft or bias also changes the physical attitude, albeit rays.However, caution is needed in decid- probe. by a very small amount (yaw biases are only ing whether these coefficients are changing or not – some secular changes in the space- craft dynamics are due to other causes, such Terminology as space vehicle out-gassing. Reflectivity ( ) – the proportion of radiation incident on a surface that is reflected, the reflected radiation being separated into diffuse (scattered) and specular (beamed) components. Spacecraft Description Specularity ( ) – the proportion of reflected radiation that is reflected specularly. Specular In the body-fixed frame, as each satellite reflection implies that the surface behaves like a perfect mirror. moves through its orbit, the Earth is con- Y-bias – a force acting along the spacecraft BFS Y-axis and believed to derive from NCF effects. tinually on the Z-axis and the Sun apparently A likely mechanism for the Y-bias is due to non-orthogonality of the solar panels with respect to rises and sets above the Z-Y plane. Apart the solar photon flux, as a result of attitude bias or variations. However, another possible contri- from eclipse seasons (which are discussed bution could come from heat dissipation effects of payload components. later), when the EPS angle varies from close to zero through to 180 degrees, this appar- ent movement of the Sun is always less than Normal Normal 180 degrees in any one orbit. In the BFS frame the only motion of the space vehicle is the Specularly rotation of the solar panels about the Y- reflected axis as they attempt to keep track of the Sun. light Incident For this reason it makes sense to model θ θ light the variations in NCF effects in the BFS frame, and therefore the spacecraft structure should be described in terms of BFS co-ordinates. Shear Shear θ SRP can be modeled a priori along the Z and Fdf Fsp X axes, provided that the profile of the space- craft surface components as seen from the Fdf = reaction force vector Fsp = reaction force vector Sun can be computed. One problematic issue due to diffusely reflected light due to specularly reflected light here is accounting for the shadowing of one component by another. Similarly, albedo Nominal attitude - no component Incorrect solar panel orientation effects are functionally related to the vari- of SRP acts along Y-axis resulting in the Y-bias ations in the spacecraft profile as seen from the Earth along the Z-axis. Thermal Properties. Modeling the ther- s/c Y-axis s/c Y- axis mal state of the satellite requires knowledge of the conductivity and emissivity of the space- craft materials. Thermal conduction is the process of heat transfer by molecular motion. The thermal conductivity of a material gives an indication of how well the heat energy is transferred and it depends on chemical com- Photon flux position, physical structure, state of the mate- rial and temperature (satellites in orbit endure Reflected radiation from a spacecraft may be separated into diffuse and specu- an extreme temperature range of 130K - 350K). lar components. If a spacecraft’s solar panels are not oriented precisely orthog- onal to the photon flux, an anomalous bias force is generated along the space- The emissivity of a body is the ratio of the radi- craft Y-axis. ation actually emitted by a surface and the radiation that would have been emitted from
46 GPS World January 2002 www.gpsworld.com Innovation a perfect blackbody (a perfect blackbody A GPS satellite’s attitude changes slowly, Eclipse Seasons absorbs all incident radiation, and thus has a therefore in orbit it is heated unevenly since During eclipse seasons, the satellites pass reflectivity of zero). only the Sun-facing side receives direct solar into the shadow of the Earth for a short peri- It is possible to measure these coefficients radiation and, due to the complex shape od (up to about one hour) in each 12-hour experimentally, and these would be used in of a real satellite, some parts are shadowed orbit. The shadow region is divided into two the process of spacecraft design and man- by other parts. This results in an uneven tem- parts, the umbra and the penumbra (see ufacture. Their values are assumed to be con- perature distribution and since the amount Figure 4). An eclipse season for a particular stant though they may change slightly with of energy radiated by a surface is dependent orbital plane lasts for between four and eight time as the surface of the satellite degrades on its temperature, this leads to anisotropic weeks and occurs twice per year. and this may lead to possible errors in the emission of radiation. As soon as the satellite passes into the models. Any heated surface which radiates loses penumbra two things happen. First, the SRP energy in the form of force is effectively turned off. However, TRR photons. This energy forces still act while the temperature of loss causes a reac- the spacecraft drops exponentially. The sec- tion force against the ond occurrence is that the attitude control radiating surface system (ACS) solar sensors lose sight of Sun (see “Theoretical Back- the Sun. Prior to 1994, this would cause the ground” sidebar). This spacecraft to yaw wildly about the BFS Z- force must be inte- axis. Since then, to make this movement Umbra grated over the whole more predictable, a yaw-bias has been applied Penumbra surface to calculate by the GPS control segment. Once the satel- the total force. To do lite emerges from the shadow crossing, it FIGURE 4 At certain times of the year, GPS satellites find them- this, the temperatures begins its recovery to attain nominal attitude selves in the Earth’s shadow for a short period during each orbit. During an eclipse, the Earth partially (while the satellite at each point on the once again. This so-called “midnight turn” is in the penumbra) or completely (while it is in the umbra) surface must be can take from zero to forty minutes, depend- blocks the Sun’s photons from reaching the satellite. known. ing on the satellite attitude at the start of the
Theoretical Background momentum per unit time per unit area is perpendicularly onto a surface as a function James Clerk Maxwell, the Scottish physicist, therefore n( )hn/c. And as first proposed of the solar irradiance parameter. first showed the theoretical basis for radia- by Isaac Newton, the rate of change of It can be shown that this leads to the fol- tion pressure in 1871. The Russian physicist, momentum of a body is equal to the applied lowing functions, which model the force Pyotr Nicholaievich Lebedev, demonstrated force. So, integrating over the electromag- acting on a spacecraft surface component experimental evidence in 1900. Ernest netic spectrum, we have for the magnitude due to SRP: Nichols and Gordon Hull in the United of the force: F = P{(1+ )cos + (2/3) (1 )} States also independently showed this in nˆ Fdue to absorbed = n( )h /cd 1901. Electromagnetic radiation possesses radiation F ˆ = P{(1 )sin )} momentum. This is described in Albert But P=A W cos / c Einstein’s special theory of relativity in that: n( )h d A=surface area of component is just the solar irradiance (W) in watts per W= solar irradiance square meter. Therefore, at 1 AU, the force =radiation angle of incidence where per unit area (pressure) due to absorbed =reflectivity of component radiation is = specularity of component E = energy of the particle F = W/c. c=vacuum speed of light p = momentum magnitude This expression gives the force acting on a unit area due to absorbed radiation falling One additional model is often used, and m0 = rest mass c = speed of light in a vacuum. this relates the temperature of a spacecraft surface component (typically a solar panel) For a photon m0 = 0, and hence E = cp. Specularly Einstein, building on ideas developed by reflected nˆ Incident to the force normal to the surface caused by Max Planck, also proposed a corpuscular rays rays the radiation of heat: theory of light in which each photon has an energy proportional to its frequency ( ) such that E = h ,where h is Planck’s con- where the additional parameters are stant. From which it is seen that p = h /c. θ θ So, the photon’s momentum is proportional = Stefan-Boltmann constant to its frequency. If such a photon is sˆ = emissivity absorbed by some surface, the momentum T = temperature is transferred to the body. Snell’s law predicts that the angle of The factor 2/3 arises in this equation due Let the average number of solar photons specularly reflected rays with to the diffuse nature of the heat radiation; a of frequency ,striking a unit surface area respect to the normal direction similar term arises in the SRP functions. per second at 1 AU, be n( ). The change in equals the angle of the incident rays.
48 GPS World January 2002 www.gpsworld.com Innovation maneuver. This is problematic in that the entific community, the empirical terms tend incident on any particular part of the space- attitude is unpredictable. In general, even to soak up any unmodeled forces that affect craft structure. The method we have adopt- with the post-processed International GPS the orbit, including the effect of satellite mass ed to solve this problem is to simulate the Service (IGS) orbits, the orbital precision for changes as orbit adjustment fuel is consumed. flux of electromagnetic radiation from the satellites in eclipse season is somewhat Hence the variation in the empirical terms Sun using a pixel array (see Figure 5). The degraded. These problems map into greater does not necessarily help us to understand pixel array is rotated around a computer sim- uncertainties for GPS applications. better the physical mechanisms that drive ulation of the spacecraft, using a body-fixed the true trajectories. Having said this, the coordinate system, in accordance with the Modeling Methods well known (but mysterious) Y-bias was first spacecraft attitude control algorithm. This To date, most modeling has concentrated discovered through empirical methods, and simulates the effect of the changing geom- on solar radiation pressure (as opposed this has spawned much useful research. etry of the EPS system. Any part of the space- to explicitly modeling TRR and albedo effects Because of this drive to improve the post- craft that would change its orientation in this as well). There are three main methods used processed precise orbits,little work has been system as a function of the Sun’s position can to calculate SRP models: done on the problem to develop high pre- be adjusted accordingly. The SRP interaction Analytical methods cision analytical modeling techniques that of the photon flux with the spacecraft is cal- Analytical methods with empirical scal- might aid design and operational applica- culated by projecting the pixel array onto the ing or augmentation tions. All the methods involving empirical spacecraft simulation at discrete points in Empirical methods. estimation rely upon a large data volume for the periodic EPS geometry. This process results In the purely analytical methods, the the- success. in a series of data points giving the acceler- oretical ideas presented in the sidebar are Finally, there are purely empirical approach- ation of the spacecraft due to SRP along its used to compute models based on the struc- es where the knowledge of the spacecraft BFS X and Z-axes as a function of the EPS tural, nominal attitude and optical proper- structure and attitude is effectively ignored. angle. The SRP model is formed by fitting a ties of the spacecraft alone. The early ana- These methods include a number of para- Fourier series to the data, using the EPS angle lytical models for GPS, termed the ROCK meters to which no particular physical mean- as the independent variable. The variations series (after Rockwell International), were ing can be attributed, and these effective- in the solar irradiance are modeled by tak- typical of this approach, which works well ly soak up any unmodeled force effects. ing a nominal value at one astronomical unit, provided that the spacecraft structure is quite Pixel Array Methods. As we try to model the and then scaling it based on the actual dis- simple. Later versions of the ROCK models forces on real spacecraft, the complexity of tance of the spacecraft from the Sun at any used basic thermal re-radiation modeling the spacecraft’s physical form causes diffi- point in its trajectory. to augment the SRP component of the force culties. The main problems that have to be With knowledge of the precise shape and model. The ROCK 42 model was adopted as overcome involve accounting for the chang- orientation of the spacecraft, the pixel array an International Earth Rotation Service stan- ing profile of the spacecraft in the course method can be used to calculate the dose dard in 1996. The main drawback of the of its orbit, and the way in which these vari- rate of radiation incident upon each com- approach used in computing the ROCK mod- ations change the amount of solar radiation ponent, and with values for the conductiv- els is that it becomes cumbersome when the spacecraft structure becomes complicated. δ Lockheed Martin, the designer and manu- r How do orbital errors map into position errors? facturer of the Block IIR satellites, has been In terms of absolute positioning, where data from only a single receiver is continually improving its NCF models, and being used, position errors can be of a similar order of magnitude to orbit the latest versions include thermal model- errors.However, if we are thinking about estimating a baseline ing for payload components. More recently, length between two points then the following “rule of thumb” we have developed a high precision approach (validated using IGS data) is a handy device: to analytical model computation based on If s is the baseline separation between two pixel array methods that can be used easi- r receivers, with s the uncertainty we are pre- ly with very complicated structures, and this pared to accept, and r is the distance to is described in the following section. the satellite then r, In post-processed orbit determination as given at left, is the applications, typically those carried out by ‘acceptable’uncertain- ty in the satellite the various agencies contributing to the IGS So, for example, if we position. orbit, empirical scaling or augmentation para- were looking for 20 millime- meters related to NCF can be estimated as ters of precision over a baseline part of heavily over-determined global net- of 100 kilometers, then we work analyses. These approaches are main- would require an orbital preci- ly characterized by the methods developed sion of around 4 meters. at the Centre for Orbit Determination in Europe and at the Jet Propulsion Laboratory. Such methods generally start with the ROCK s models and estimate parameters that cor- δs rect for the perceived deficiencies in the a priori analytical models. While these meth- For a given satellite orbit error, r, the error in the estimated baseline vector ods do result in very precise orbits that length, s, in relative GPS positioning, is roughly proportional to the ratio of base- line length to the distance to the satellite. are of enormous benefit to the global sci- www.gpsworld.com GPS World January 2002 49 Innovation although the variation in the flux of radi- based on all the available structural and atti- Sun ation from the Earth is much less pre- tude data for the spacecraft, and a number dictable compared to solar irradiance, due of carefully selected empirical parameters is to the effect of cloud cover. a powerful tool for calculating high precision Pixel array X orbits. However, the benefit of good a priori Discussion analytical modeling is that it enhances the In post-processed applications where a ability to understand the system, and hence, EPS angle precise orbit is needed, provided that a predict how it will function over time. This sufficiently dense tracking station network can, in turn be exploited to either improve Z is available (such as the IGS network) then the accuracy of predicted orbits, or, main- SRP effects can be accounted for empiri- taining a certain level of accuracy in the orbits, Earth cally. However, if the number of track- reduce the number of tracking stations required FIGURE 5 The effect of the Sun’s photons on a ing stations is reduced, simply applying to support the system. spacecraft may be simulated using a pixel the existing empirical methods to the com- Improving the accuracy of the orbit reaps array which is rotated around a computer putation for a new constellation does other benefits in that physical parameters, model of the spacecraft. not solve the problem, and this is borne such as the wet tropospheric propagation out by recent work in the International delay, also become more accurately deter- ity and emissivity finite element analysis GLONASS Experiment 1998 (IGEX98) cam- mined. These parameters are increasingly (FEA) can be used to calculate the tem- paign. At the very least, starting the esti- important in applications such as virtual RTK perature at each point. The principle of FEA mation process with a relevant analytical networks and interferometric synthetic aper- is based on the premise that an approximate model provides a hypothesis that can be ture radar. solution to any complex problem can be tested. reached by subdividing the problem into Where the requirement is to predict the satel- Conclusion smaller more manageable (finite) elements. lite orbital trajectory,the application of an ana- Solar radiation pressure, thermal re-radia- Using finite elements, solving complex par- lytical model is yet more important. Operational tion and albedo forces are tiny in magnitude, tial differential equations that describe heat systems are tracked by a relatively small num- and yet have a strong perturbing effect on transfer mechanisms can be reduced to a ber of ground stations and hence the number GPS satellite orbits. The modeling of these set of linear equations that can easily be of observations used to help predict the orbits forces is important at the stages of system solved using the standard techniques of is correspondingly small. The more accurate design, operation and scientific analysis. matrix algebra. So the principles above can the a priori SRP modeling, the more accu- Although much previous work has con- be used to develop a thermal analysis model, rate is the predicted trajectory, with associat- centrated on empirical methods (where a using FEA to determine the temperatures of ed improvements in system performance for large number of tracking stations are required), each part of the satellite in order to com- all real-time applications. newer high precision analytical techniques pute the force due to thermal re-radiation. This is not to dismiss in any way the role make it feasible to model non-conservative In a similar fashion these pixel array meth- of empirical methods in orbit determination. force effects more accurately at the design ods can be used to model albedo effects, The combination of an analytical model, and operational stages. This can be exploit- ed to either increase the length of time over which Further Reading For a discussion of solar radiation pressure model- a predicted orbit is valid, or For an introduction to GPS satellite orbit modeling, ing for the Block IIR satellites, see decrease the number of tracking sta- see “Solar Force Modeling of Block IIR Global tions required for the system, and thereby Positioning System Satellites” by H.F. Fliegel and T.E. “GPS Satellite Orbits” by G. Beutler, R. Weber, U. reduce running costs. Gallini in Journal of Spacecraft and Rockets, Vol. 33, Hugentobler, M. Rothacher, and A. Verdun in GPS for As Prof. Richard Feynman once said, “It Geodesy, 2nd edition, edited by P.J.G. Teunissen and A. No. 6, 1996, pp. 863-866. is only the principle of what you think will Kleusberg and published by Springer-Verlag, Berlin, For a discussion of solar radiation pressure sto- Heidelberg and New York, 1998. chastic modeling, see happen in a case you have not tried that is For a compendium of papers on GLONASS orbit “A New Solar Radiation Pressure Model for GPS worth knowing about. Knowledge is of no modeling, see Satellites” by T. A. Springer, G. Beutler and M. real value if all you can tell me is what hap- “Session 5: Orbit Determination” in Proceedings Rothacher in GPS Solutions, Vol. 2, No. 3, 1999, pp. pened yesterday. It is necessary to tell what of the International GLONASS Experiment, IGEX-98, 50-62. will happen tomorrow if you do something”. Workshop, Nashville, Tennessee, September 13-14, For further information on the use of finite ele- 1999, published by the International GPS Service ment modeling in GPS satellite orbit determination, “Innovation”is a regular Central Bureau, Jet Propulsion Laboratory, Pasadena, see column featuring California, 2000, pp. 155-258. An on-line version of “Thermal Force Modeling for Global Positioning discussions about recent the proceedings is available at System Satellites Using the Finite Element Method” by advances in GPS tech-
50 GPS World January 2002 www.gpsworld.com