ResultsOblatenessTrendsAsimulationwasrunfor25orbits,giventhefollowinginitialorbitalelements:
Oncethesimulationswerecomplete,orbitalelementswerecalculatedfromthepositionandvelocitytimehistories.
Above:Changeininitialorbitduetooblatenessperturbation.Below:Animationofchangeinorbitovertime.
Secular Trends of Perturbations Explored NumericallyKyle Litzer CU Boulder ASEN 5050
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Above:Timehistoryofsemimajoraxis,eccentricity,radiusofperigee,andradiusofapogee.Below:Timehistoryofinclination,rightascensionoftheascendingnode,andargumentofperiapse.
Asseenabove,semimajoraxis,eccentricity,radiusofapogee,radiusofperigeeandinclinationareallunaffectedsecularlybyoblateness.Thereisauniquetradeoffwitnessedbetweenradiusatapogeeandperigee.Whenoneisataminimum,theotherisatamaximum.
Aspredictedanalytically,thetimehistoriesshowaseculartrendinbothrightascensionoftheascendingnode(RAAN)andduringtheapproximately65hoursimulationtimespan.RAANdecreasesbyapprox.7.5,whileargumentofperiapserisesbyapprox.13.
Asmentionedbefore,inhistextValladogivesanalyticalexpressionsfortheseculartrendsintheorbitalelementsduetotheoblatenessofEarth.ItisshownthattheonlyorbitalelementsaffectedinasecularwayaretheRAAN,theargumentofperiapseandthereference(initial)meananomaly.
Comparingtheseequationstothebehaviorwitnessedabove,thenetchangeinRAANandargumentofperiapseagreewiththeanalyticalexpressions.
DragTrendsAsimulationwasrunfor100orbits,giventhefollowinginitialorbitalelements:
Oncethesimulationswerecomplete,orbitalelementswerecalculatedfromthepositionandvelocitytimehistories.
Above:Changeininitialorbitduetodragperturbation.Below:Animationofchangeinorbitovertime.
Above:Timehistoryofsemimajoraxis,eccentricity,radiusofperigee,andradiusofapogee.Below:Timehistoryofinclination,rightascensionoftheascendingnode,andargumentofperiapse.
Above:Detailoftimehistoryofsemimajoraxis,eccentricity,radiusofperigee,andradiusofapogee.Below:Detailoftimehistoryofinclination,rightascensionoftheascendingnode,andargumentofperiapse.
Theperturbationsduetodragyieldquiteinterestingresults.Allsevenelementsplottedfollowadefiniteseculartrend.Allsixelementsasidefromargumentofperiapseshowadownwardtrend.ItalmostappearsasiftheorbitistryingtoalignitsperigeedirectiontobecomeparallelwithtotheEarth/Sundirection.Themagnitudeofchangeinradiusofapogeeversusperigeeshouldalsobenoted.Thenetchangeinapogeeisontheorderof30km,whilethenetchangeinperigeeisontheorderof200m.Thiscanbeconceptuallyverifiedifwecomparethelossofspeedduetodragwithacircularizingantivelocitydirectionburnperformedatperiapse.Sinceperiapseisthemostefficientplacetoperformorbitchangingmaneuvers(duetohighspeeds),smallerchangesinvelocityatperigeebecomelargerchangesintheorbit,especiallyatapogee.
Uponcloserinspection(referringtothezoomedinplots),itcanbeseenthattheelementsallremainrelativelyunchanging.Sincethisorbitiseccentric,andasareasonablylowperigeealtitudethatdipssignificantlyintotheEarthsatmosphere,theeffectsofdragareonlywitnessedduringshortspurtsoftimeneartheorbitsperigee.Curiously,theargumentofperiapsetimehistoryshowssharpspikeseverytimethespacecraftiswithintheatmosphere.Furtherinvestigationwouldneedtobeperformedtofindthereasonforsuchhighrateofchange.
SolarRadiationPressureTrends
Asimulationwasrunfor100orbits,giventhefollowinginitialorbitalelements:
Oncethesimulationswerecomplete,orbitalelementswerecalculatedfromthepositionandvelocitytimehistories.
Above:ChangeininitialorbitduetoSRPperturbation.Below:Animationofchangeinorbitovertime.
Above:Timehistoryofsemimajoraxis,eccentricity,radiusofperigee,andradiusofapogee.Below:Timehistoryofinclination,rightascensionoftheascendingnode,andargumentofperiapse.
Theonlyoftheorbitalelementsinduringthissimulationthatshowsnoseculartrendisthesemimajoraxis.Itisverypeculiarthateventhougheccentricity,radiusofapogeeandperigeechange,thesemimajoraxisremainsrelativelyconstant.Furtherinvestigationwouldbenecessarytounderstandthisrelationship.
ItcanalsobeseenthatthetrendsforRAANandargumentofperiapsemirrorthatoftheoblatenessperturbation.
Finally,itshouldbetakenintoconsiderationthatthemagnitudeofsecularchangeintheorbitalelementsduetoSRPismuchsmallerinmagnitudethanthesecularchangesduetoEarthsoblateness.