This Article Presents the Rudiments of Constellation Design to Familiarize

This articlepresents the rudiments ofconstellation design to familiarize the reader with some of this field’smethods, issues, and concerns. Two exotic constellations are under consideration for various missions. The traveling-salesman problem The need for satellite constellationsis a natural outgrowth of the need for global communications and intelligence gathering. A single satellite can only observea spherical cap regionof the Earth (as illustrated in Figure l), called the in- stantaneous nadir-pointing coverage circle (also tational harmonics,produce an array of constel- referred to as a footprint) of the satellite. The lations with specific propertiesto support Various coverage circle’s area,A, is a function of altitude, mission constraints. Akey point is that constella- h, and the Earth’s radius,R (6,378.14 km), tion design isnot the mere repetitionof a satellite orbit as a template. A(h)= 2a2(1 - COS 0)= 2d2h/(R + h), (1) The aggregate of satellites andorbits presents a totally new systemsproblem with a combina- which is always smaller than the hemisphere’s torial growth in complexity for the most ele- area, 2d2.Hence, we can get a quick estimate mentary analysis questions. The fundamental of the lower boundfor the number, N, of satel- performance metric is constellation geometric- lites with altitude, h, required to provide instan- coverage statistics.Ergodic theory, the multires- taneous global coverageby dividing the Earth’s olution visual calculus, and dynamical systems surface byA(h): theory are some new approachesto constellation design problems. New infrastructures and new N > 4d2/ A(h) = 2 + 2Wh. (2) paradigms willfurther develop this technology. The importance of communications satellite Although Equation 2 is not a tight estimate, it constellations cannot be overstated. In one fell tells us right away that we must always have swoop, sucha constellation can provide an under-more than two satellites to provide simultane- developed region withouta modern communica- ous global coverage.Thus satellite constellations tion infrastructure with aninstant modem com- are essential to provide any kindof global satel- munications network.The social and economic lite service. implications of this technology are enormous. At first glance, it might seemthat constellation design is merelythe act of replicating multiple copies of a single satellitein slightly different or- 1521-9615/99/$10.000 1999 IEEE bits. Althoughthe end result might seemto support that idea, the design process is actually very 58 COMPUTINGIN SCIENCE & ENGINEERING Satellite I ~ ~~ ~ Figure 2. Elliptical orbit in the orbitplane. Basic orbital mechanics To intelligently discuss constellation design, let’s cover some basic orbital mechanics to describe the geomeuy and provide metricsto measure performance. I will touch upon only circular and el- Figure 1. The nadir-pointing coverage circle of a liptical orbits,as depictedin Figure 2. The semi- satellite on Earth. major axis, a, is half the distance of the axis through the ellipse’s longest part.The semiminor axis, b, is halfthe distance of the axis through the difficult. It is a much more organic process some-ellipse’s shortest part. The eccentricity, e = cia, what akinto the buildmg of a multicellular organ- measures the ellipse’s shape.For a circular orbit, ism. In its most basic form, each theof cells is an the eccenmcityis 0 because inthis case, the axes a identical satellite. For more complex missions, and b are equal, andc degenerates to 0. In the lim- satellites with different capabilities, such as differ-iting case whena approaches infinity witha finite entiated cellulargroupings, might be neceswy. Butb, c also approaches infinity, and we obtaina par- even inthe simplest case of identical satellites in abolic orbit with eccenmcity equalto 1. Thus, for similar orbits,the constellation performanceanaly- elliptical orbits, the eccentricity always falls be- sis can be extraordinarily complex dueto the com- tween 0 and l. The satellite’s locationon theel- binatorial effectsof multiple satellitesMung with liptical orbit is given by angleuin Figure 2 (called multiple locationson the ground. the true anomaly). The closest approachto Earth Even before the current interest in constella- is called perigee,the farthest is called apogee. tions, ground-station plannersfaced the difficult From Keplerian orbit theory, without pertur- problem of scheduling communications passesfor bations, elliptical orbits arealways confined to a multiple satellites.One way to see whythis is such plane centeredon Earth known as the orbit plane. a hard problem isto compare it to the well-known To provide space with coordinates,a fundamental traveling-salesman problem: Givena list of cities, plane must be chosento represent the q plane. what is the optimal path for a salesman to visit For Earth satellites, this plane is usually the each city once? Now suppose the salesman is a equator’s plane withthe x-axis pointed in the di- satellite and the path traversed is a fixed orbit. rection of the constellation Aries, also knownas What orbit, if it is even possible, providesthe op- the vernal equinox direction. Our orbit plane timum pathto visit the list of cities? Now increase might be inclined with respect to the equator, the number of satellites, each ain different orbit. and the angle between these two planesis simply What combination of orbits provides the opti- called the inclination, usually denoted by i. The mum coverageof allground stations?The once- orbit plane intersects the equatorial plane in a simple graph-theory problem has not only in- line known as the line of nodes. The line of nodes creased in size and combinatorial complexity, butintersects the orbit in theequatorial planein two the paths must now satisfythe constraint of a set points. The point where the satellite moves into of differential equations for artificial satellites. the +Z hemisphere is called the ascending node; Fortunately for most applications, designerstyp- the othernode is calledthe descending node. Q ically use certain well-understood satelliteorbits generally denotes the angle between the x-axis for constellation design. and the ascending nodes. Finally,the last para- JANUARY-FEBRUARY 1999 59 sooner than expected. This tends to make the satellite reachits ascending node sooner than it t‘ would withoutthe perturbation. Hence the node appears to move backward (seeEquation 3) and the effect is called nodal regression. In other words, the orbital plane is precessing due to J2 perturbation. For a space shuttle to orbit at 200 km altitude and 28.5 degrees (deg) inclination, the nodal regression is roughly -7 deg per day, which isquite significant. dWdt = -2.06474 x lOI4 cos(t) (1 - e2)-2 deglday. (3 1 A second effect dueto32 perturbation is called the precession of the argument of perigee. 32 causes the perigee to rotate around the orbit’s ~~ ~ normal vector, which is thealso orbit plane’s nor- Figure 3. The orbital inclination, ascending node, mal vector (see Equation4). The orbital period is and the argument of perigee. shortened because as the orbit moves from node to node, it reaches the nodal crossings fasterdue to the equatorial bulge’s gravitational forces: meter required to specify the orbit is the angle o, called the argtcment ofpengee, between the as- dddt = 1.03237 X 1014 (4- 5 sin2(t]) cending node and the perigee. These angles are (1 - e2)-2 deg/day. (4) indicated on theinclined orbit in Figure 3. The e, six parameters, a, i, Q, m, and Malled the . ..,. Qrblial &&o;gy:: ;:: .. .: classical orbital elements-completely charac- .. terize a conic orbit. Now that we have the basic tools for orbital mechanics, let’s examinesome of the space in- Perturbations dustry’s most useful orbits. For Earth-orbit design, we must consider three major perturbations: atmospheric drag,the Cirnrlm orbit. Circular orbits arethe most use- gravitation ofthe sun and moon, andthe Earth’s ful because of their symmetry andease of analysis. equatorial bulge. For typical constellations,av- They move with uniform speed, andwe can fre- erage orbit altitude is generally assumed to be quently estimate their coverage geometry with high enough to avoid the atmospheric drag that good, closed-form approximations. However, would causethe orbit to decay ifunmaintained. note that therate at which circularorbits cover Similarly, the altitude is assumedto be no higher ground is not uniform. In other words, a circular than thatof geosynchronous orbits (35,863 km) orbit’s projected velocityon the Fundis not uni- so that luni-solar perturbations are small enough form due to the Earth’s rotation. All points on and might be ignored during the design stage. Earth rotate at the same angular velocity around In actual satellite operation, these forces mustbe the North Pole. However, the actual velocityat modeled and included in the orbit calculations the equator is faster than that at a higher latitude. to the degree of accuracy that the mission re- This is because latitude circles have different radii, quires. But during typical constellation design, but the same angular rate. Because velocityis an- the main effect that must be included isthe so- gular rate times radius,the points at different lat- called32 term, named after the coefficient of a itude circles have different velocities. planet’s geopotential harmonic expansion. Three distinct cirmlar-orbit classes have emerged The32 term corresponds to the planet’s equa- in orbital nomenclature: the geosynchronousEarth torial bulge and

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