Mission Design for the CubeSat OUFTI-1

CHAPTER 55.3 Orbit perturbationsThe Keplerian orbit, considering only the earth gravitational force and thesatellite fictitious centrifugal force, provides an excellent reference but, for amore accurate study, we need to take into account some minor effects thatmake deviate the nominal orbit.We classify these variations of orbital elements in three main categories:• the secular variations: they are a linear variation of the element. Theireffect cumulates in time and therefore they are the cause of changing shapeand orientation of the orbit.• the long-period variations: they are those with a period greater than theorbital period.• the short-period variations: they have a period less than the orbital period.They can usually be neglected.In the sequel, three main effects will be considered: the earth’s oblateness, theatmospheric drag and the solar radiation pressure.5.3.1 The earth’s oblatenessThe gravitational potential in the Keplerian theory corresponds to that of anuniform sphere or, equivalently, to that of a punctual mass:V = − µ r(5.14)Unluckily, the earth isn’t a perfect sphere and its mass isn’t uniformly distributed:therefore some secondary effects are produced. To take them intoaccount, a more accurate model is necessary. We introduce, besides the radialcoordinate r representing the distance from the center of the earth, the latitudeλ and the longitude φ. The complete expression of the earth gavitationalpotential becomes:( ∞XnX n(C nmcos (mλ) S mnsin (mλ)) P nmsin (φ)#)V (r, φ, λ) = − µ r1 −n=2" Re nJ nP nsin (φ) +rm=1The coefficient C nm et S nm are constant while P nm sin (φ) are the associatedLegendre functions.The gravitational potential can be so expressed as a sum of infinite terms thatcan be classified into three groups (fig.5.13): Re• if m = 0 the potential depends only on the latitude. This effect, calledzonal harmonics, takes into account the earth oblateness. Often we callsC m0 = J m .rGalli Stefania 40 University of Liège