gaussian orbit determination

Gaussian Orbit Determination Finding the Orbit of Ceres Aaron Eiben & Gregory Tewksbury

VERSION 8 NOTEBOOK UPDATED TO VERSION 9! VERSION 9 NOTEBOOK UPDATED TO VERSION 10! Version 8 functions have been commented out and replaced with their version 9 counterparts. Important! Mathematica 9 seems to deal with numerical precision differently than did version 8. While there is probably a very good reason for this, it kills this notebook by returning results that have no significant figures and are therefore useless for computation! Thus, the minimum precision is set to machine precision. This solves the sig. fig. problem and produces results that are identical to the version 8 notebook, but now Mathematica complains with each download of not-so-precise astronomical data and, for some reason, unit conversions. Therefore, these complaints will be turned off . . . $MinPrecision=$MachinePrecision; Off[Precision::precsm,SetPrecision::precsm] Load the PhysicalConstants package for easy unit conversions: EDIT: This package has been integrated!

{{10.31907452185567,2.664143042283759},{12.17044962463662,2.749211879943253},{14.40720484717656,2.978690498437773}} These angular coordinates must be converted into radians for later conversion into Cartesian coordinates. To accomplish this, we note that

[90,-90] =[0,]rad and [0,24]hours =[0,2]rad. Therefore,

->(90-) ( rad)/(180),

-> ( rad)/(12hours): Clear[DecRAConverter] DecRAConverter[{_,_}]:={(90-) Pi/180, Pi/12} Clear[ConvertedCeresData] Table[ConvertedCeresData[i]=DecRAConverter[CeresData[i]],{i,Nn}] {{1.3906945006298557,0.6974710174792517},{1.3583819093968796,0.7197419870992923},{1.3193431662023791,0.7798193489341522}} The same process is done for the Sun. These data must be obtained from a reference source, as the Sun will likely be below the horizon at the times our observations are made. Moreover, students should not be pointing a telescope anywhere near the Sun without proper solar filters and competent supervision! Clear[SunData] Table[SunData[i]=QuantityMagnitude[AstronomicalData["Sun",#,TimeZone->1]]&/@({#,NDate[i]}&/@{"Declination","RightAscension"}),{i,Nn}] {{-22.91901182368604,18.87260407392571},{-19.65550060681863,20.30877436827272},{-14.00105754545583,21.66277058728244}} Clear[ConvertedSunData] Table[ConvertedSunData[i]=DecRAConverter[SunData[i]],{i,Nn}] {{1.9708085444217282,4.940836192729485},{1.9138495285116251,5.316824696564857},{1.8151608797242965,5.671300077783963}} Additionally, we may look up the distances between the Earth and the Sun at the given times in astronomical units: Clear[EarthSunDistance]

(*Table[EarthSunDistance[i]=Convert[AstronomicalData["Sun",

{"Distance",NDate[i]},TimeZone 1]Meter,AU]/AU,{i,Nn}]*)

Table[EarthSunDistance[i]=QuantityMagnitude[UnitConvert[AstronomicalData["Sun",{"Distance",NDate[i]},TimeZone->1],"AU"]],{i,Nn}] {0.983191072677480,0.984468788380674,0.987625267479505}

With these data, we can construct the Suns geocentric positions (or

Earths heliocentric positions) in spherical coordinates:

Clear[SphericalSunPoints] Table[SphericalSunPoints[i]=Prepend[ConvertedSunData[i],EarthSunDistance[i]],{i,Nn}] {{0.983191072677480,1.9708085444217282,4.940836192729485},{0.984468788380674,1.9138495285116251,5.316824696564857},{0.987625267479505,1.8151608797242965,5.671300077783963}} Finally, we must define the vernal point (the position of the Sun at the vernal equinox) for later reference: VernalPoint={0,0}; ConvertedVernalPoint=DecRAConverter[VernalPoint] {/2,0} These can then be converted into Cartesian coordinates. Mathematica 9 can do this transformation with one simple commandunfortunately, I am running version eight . . . EDIT: I am now running version 9! However, for some reason, CoordinateTransform cannot handle the given solar points: CoordinateTransform["Spherical" -> "Cartesian",SphericalSunPoints[1]] CoordinateTransform::bdpt: Evaluation point {0.983191072677480,1.9708085444217282,4.940836192729485} is incompatible with the coordinate assumptions of the specified coordinate chart. >> CoordinateTransform[Spherical->Cartesian,{0.983191072677480,1.9708085444217282,4.940836192729485}] Nevertheless, the function can obtain the symbolic mapping without issue,

which can then be applied to the spherical solar coordinates. That could have been done by hand (and was in version 8)! Clear[CartesianSunPoints] (*Table[CartesianSunPoints[i]={#[[1]] Sin[#[[2]]]Cos[#[[3]]],#[[1]] Sin[#[[2]]]Sin[#[[3]]],#[[1]] Cos[#[[2]]]}&[SphericalSunPoints[i]],{i,Nn}]*) Table[CartesianSunPoints[i]=CoordinateTransform["Spherical" ->

"Cartesian",{r,,}]/.Thread[{r,,}->SphericalSunPoints[i]]

,{i,Nn}] {{0.2050811956804456,-0.8820467445378197,-0.3828837016917895},{0.5268722599196133,-0.7628439164108162,-0.3311398137009807},{0.7844193879680317,-0.5504496040065058,-0.2389458645653943}} The same is done for the vernal point: (*CartesianVernalUnit={Sin[#[[1]]]Cos[#[[2]]],Sin[#[[1]]]Sin[#[[2]]],Cos[#[[1]]]}&[ConvertedVernalPoint]*) CartesianVernalUnit=CoordinateTransform["Spherical" -> "Cartesian",Prepend[ConvertedVernalPoint,1]] {1,0,0} Let us plot these points: Graphics3D[ {{PointSize[Large], {Darker[Yellow], Point[CartesianSunPoints/@Range[Nn]]}, {Lighter[Blue,0.5], Point[Origin]}}, {Arrowheads[Tiny], Arrow[{CartesianSunPoints[#],Origin}&/@Range[Nn]]}}, Axes->True, AxesLabel->{x,y,z}]

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Earth (the pale blue dot) sits at the origin. The Sun (yellow dots) moves around it in the ecliptic plane. Arrows have been drawn from the Suns to the Earth in anticipation of a shift to a heliocentric reference frame. We now wish to create unit vectors in the direction of Ceres. This is achieved via the same spherical to Cartesian coordinate transformation as before: Clear[CartesianCeresUnits] (*Table[CartesianCeresUnits[i]={Sin[#[[1]]]Cos[#[[2]]],Sin[#[[1]]]Sin[#[[2]]],Cos[#[[1]]]}&[ConvertedCeresData[i]],{i,Nn}]*) Table[CartesianCeresUnits[i]=CoordinateTransform["Spherical" -> "Cartesian",Prepend[ConvertedCeresData[i],1]],{i,Nn}] {{0.7540716704866178,0.6318927497373350,0.1791297535277473},{0.7350749930171924,0.6443752027585923,0.2108206648092110},{0.6886796860832983,0.6810382048269793,0.2488116828893994}} Plotting these vectors produces the following: Graphics3D[ {{PointSize[Large],

{Darker[Yellow], Point[CartesianSunPoints/@Range[Nn]]}, {Lighter[Blue,0.5], Point[Origin]}}, {Arrowheads[Tiny], Arrow[{CartesianSunPoints[#],Origin}&/@Range[Nn]], Arrow[{Origin,CartesianCeresUnits[#]}&/@Range[Nn]]}}, Axes->True, AxesLabel->{x,y,z}]

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We now shift into a heliocentric frame of reference: Clear[CartesianEarthPoints] Table[CartesianEarthPoints[i]=Origin-CartesianSunPoints[i],{i,Nn}] {{-0.2050811956804456,0.8820467445378197,0.3828837016917895

},{-0.5268722599196133,0.7628439164108162,0.3311398137009807},{-0.7844193879680317,0.5504496040065058,0.2389458645653943}} and plot: Graphics3D[ {{PointSize[Large], {Darker[Yellow], Point[Origin]}, {Lighter[Blue,0.5], Point[CartesianEarthPoints/@Range[Nn]]}}, {Arrowheads[Tiny], Arrow[{Origin,CartesianEarthPoints[#]}&/@Range[Nn]], Arrow[Table[(#+CartesianEarthPoints[i])&/@{Origin,CartesianCeresUnits[i]},{i,Nn}]] }}, Axes->True, AxesLabel->{x,y,z}]

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At this point, we have positions of Earth relative to the Sun and lines of sight from Earth to Ceres at three different times. Following the description

of Gauss method as described by Jonathan Tennenbaum and Bruce

Director [1], we can approximate the positions of Ceres along the sight lines at the specified times. From these positions, one can obtain a

reasonably accurate approximation of Ceres orbit. Let us begin this

process by first writing parametric vector equations for Earths positions

and Ceres lines of sight:

Clear[EarthPointEquations] Table[EarthPointEquations[{i,e_}]=e CartesianEarthPoints[i],{i,Nn}] {{-0.2050811956804456 e,0.8820467445378197 e,0.3828837016917895 e},{-0.5268722599196133 e,0.7628439164108162 e,0.3311398137009807 e},{-0.7844193879680317 e,0.5504496040065058 e,0.2389458645653943 e}} Clear[CeresVectorEquations] Table[CeresVectorEquations[{i,c_}]=CartesianEarthPoints[i]+c CartesianCeresUnits[i],{i,Nn}] {{-0.2050811956804456+0.7540716704866178 c,0.8820467445378197+0.6318927497373350 c,0.3828837016917895+0.1791297535277473 c},{-0.5268722599196133+0.7350749930171924 c,0.7628439164108162+0.6443752027585923 c,0.3311398137009807+0.2108206648092110 c},{-0.7844193879680317+0.6886796860832983 c,0.5504496040065058+0.6810382048269793 c,0.2389458645653943+0.2488116828893994 c}}

Gauss method for determining Ceres positions hinges upon the ratios of

areas of triangles formed by the dwarf planets positions and the Sun. In

particular, we want to know T12/T13, (T23/T13).

As Ceres positions are unknown, these triangular areas are unknown;

however, they can be approximated by the areas of the orbital arcs between the unknown points, which are proportional to the elapsed times between observations:

T12/T13S12/S13=t12/t13, T23/T13S23/S13=(t23/t13).

We define these ratios: Clear[NSr] Table[NSr[i]=Nt[i]/Nt[13],{i,{3,1}}] {0.49991015828155564895,0.50008984171844435105}

and proceed with Gauss method. The first step is to find points F1 and F3

along the first and third Earth position vectors such that F1/E1=t23/t13, F3/E3=(t12/t13): Clear[FPoints] Evaluate[FPoints/@{1,3}]=EarthPointEquations/@{{1,NSr[1]},{3,NSr[3]}} {{-0.1025590226872634,0.4411026168641874,0.1914762497756191},{-0.3921392203982197,0.2751753486649119,0.1194514649756094}} These vectors are then combined to form a point F: FPoints[0]=Total[FPoints/@{1,3}] {-0.4946982430854831,0.7162779655290993,0.3109277147512285} Graphics3D[ {Point[FPoints/@{1,3}], Point[FPoints[0]], {PointSize[Large], {Darker[Yellow],

Point[Origin]}, {Lighter[Blue,0.5], Point[CartesianEarthPoints/@Range[Nn]]}}, {Arrowheads[Tiny], Arrow[{Origin,CartesianEarthPoints[#]}&/@Range[Nn]], Arrow[Table[(#+CartesianEarthPoints[i])&/@{Origin,CartesianCeresUnits[i]},{i,Nn}]], {Dashed, Arrow[{FPoints[#],FPoints[0]}&/@{1,3}]}}}, Axes->True, AxesLabel->{x,y,z}]

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The first and third Ceres lines of sight (L1 and L3) are shifted to point F; from them, we derive a plane Q that passes through F: Clear[ShiftedCeresVectorEquations] Table[ShiftedCeresVectorEquations[{i,sc_}]=FPoints[0]+sc CartesianCeresUnits[i],{i,{1,3}}] {{-0.4946982430854831+0.7540716704866178 sc,0.7162779655290993+0.6318927497373350 sc,0.3109277147512285+0.1791297535277473 sc},{-0.4946982430854831+0.6886796860832983

sc,0.7162779655290993+0.6810382048269793 sc,0.3109277147512285+0.2488116828893994 sc}} QNormal=Normalize[Cross@@CartesianCeresUnits/@{1,3}] {0.3283092197645911,-0.5988619051150029,0.7304638764645345} Qz=z/.Simplify[Solve[QNormal.({x,y,z}-FPoints[0])==0,z][[1]]] -0.4986476801296103-0.4494530535221219 x+0.819837810479433 y Show[{ Graphics3D[ {Point[FPoints/@{1,3}], Point[FPoints[0]], {PointSize[Large], {Darker[Yellow], Point[Origin]}, {Lighter[Blue,0.5], Point[CartesianEarthPoints/@Range[Nn]]}}, {Arrowheads[Tiny], Arrow[{Origin,CartesianEarthPoints[#]}&/@Range[Nn]], Arrow[Table[(#+CartesianEarthPoints[i])&/@{Origin,CartesianCeresUnits[i]},{i,Nn}]], {Dashed, Arrow[{FPoints[#],FPoints[0]}&/@{1,3}], Arrow[Table[ShiftedCeresVectorEquations[{i,#}]&/@{0,1},{i,{1,3}}]]}}}], ParametricPlot3D[{x,y,Qz},{x,-0.5,0.3},{y,0.7,1.4}, PlotStyle->{{Opacity[0.25],Brown}},Mesh->None]}, Axes->True, AxesLabel->{x,y,z}]

By construction, the intersection of L2 with plane Q is our first approximation of the second position of Ceres, P2: P2Solution=Solve[CeresVectorEquations[{2,c}]=={x,y,Qz}][[1]] {c->2.509839169621373,x->1.318047750164093,y->2.380122040227045} CartesianCeresPoints[2]=CeresVectorEquations[{2,c}]/.P2Solution {1.318047750164093,2.380122040227045,0.8602657760047565} CeresSunDistance[2]=Norm[CartesianCeresPoints[2]] 2.853469467762402 Show[{ Graphics3D[

{Point[FPoints/@{1,3}], Point[FPoints[0]], {PointSize[Large], {Darker[Yellow], Point[Origin]}, {Lighter[Blue,0.5], Point[CartesianEarthPoints/@Range[Nn]]}, {Gray, Point[CartesianCeresPoints[2]]}}, {Arrowheads[Tiny], Arrow[{Origin,CartesianEarthPoints[#]}&/@Range[Nn]], Arrow[Table[(#+CartesianEarthPoints[i])&/@{Origin,CartesianCeresUnits[i]},{i,{1,3}}]], Arrow[CeresVectorEquations[{2,#}]&/@{0,c}/.P2Solution], {Dashed, Arrow[{FPoints[#],FPoints[0]}&/@{1,3}], Arrow[Table[ShiftedCeresVectorEquations[{i,#}]&/@{0,1},{i,{1,3}}]]}}}], ParametricPlot3D[{x,y,Qz},{x,-0.5,0.3},{y,0.7,1.4}, PlotStyle->{{Opacity[0.25],Brown}},Mesh->None]}, Axes->True, AxesLabel->{x,y,z}]

At this time we have found a first approximation of Ceres second position.

Before proceeding farther, let us deal with the fact that the shallow angle between the observed line of sight L2 and the constructed plane Q means that any small error in Q turns into a gigantic error in P2. The former error stems from our use of orbital sectors to approximate triangular areas. Fortunately Gauss, being a master of geometry, formulated a correction factor which goes as follows: G=1+22 t12t23/r2

3 , r2 being the second distance between Ceres and the Sun. Initially, one might be put off by the fact that G, which is used to calculate r2, depends upon r2. However, we already have one estimate of r2. If we insert this into G, then we may calculate a better approximation, which can then be used to recalculate G, and so on. Evidently, Gauss abhorred this kind of brute force mechanism and developed methods of rapidly converging upon the true G and r2, but he did not have Mathematica to carry out his calculations for himwe do: NG=1+22 Nt[3]Nt[1]/CeresSunDistance[2]3

1.002537833595109918 The first correction factor is quite small, but we can do much better. The following loop will iteratively calculate r2 and G until they converge on the limits of machine precision: Distances={CeresSunDistance[2]}; GaussFactors={NG}; time=AbsoluteTime[]; Catch[Do[ Clear[NSr]; Table[NSr[i]=NG Nt[i]/Nt[13],{i,{3,1}}]; Clear[FPoints]; Evaluate[FPoints/@{1,3}]=EarthPointEquations/@{{1,NSr[1]},{3,NSr[3]}}; FPoints[0]=Total[FPoints/@{1,3}]; Clear[ShiftedCeresVectorEquations]; Table[ShiftedCeresVectorEquations[{i,sc_}]=FPoints[0]+sc CartesianCeresUnits[i],{i,{1,3}}]; QNormal=Normalize[Cross@@CartesianCeresUnits/@{1,3}]; Qz=z/.Simplify[Solve[QNormal.({x,y,z}-FPoints[0])==0,z][[1]]]; P2Solution=Solve[CeresVectorEquations[{2,c}]=={x,y,Qz}][[1]]; CartesianCeresPoints[2]=CeresVectorEquations[{2,c}]/.P2Solution; AppendTo[Distances,CeresSunDistance[2]=Norm[CartesianCeresPoints[2]]]; AppendTo[GaussFactors,NG=1+22 Nt[3]Nt[1]/CeresSunDistance[2]3]; If[Differences[Distances[[-2;;-1]]][[1]]==0,Throw[Null],Null], {100}]] (AbsoluteTime[]-time)Second Length[Rest[Distances]] {CeresSunDistance[2],NG} 0.03802430000000000 Second 16 {2.750890747152010,1.002832453601296655}

Within sixteen very short steps, we converge onto a value of r2 that is as

precise as our machine can handle! We are now ready to calculate Ceres

other positions. Again, by construction, Ceres second position (P2) lies

within the plain Q, which was generated by the first and third line of sight vectors after they had been shifted to the point F. Therefore, we can express P2 as a linear combination of these vectors. The coefficients of this expansion are as follows: Clear[QpScalars] Table[QpScalars[i]=Qp[i]/.Solve[Total[Qp[#] CartesianCeresUnits[#]&/@{1,3}]==CartesianCeresPoints[2]-FPoints[0]][[1]],{i,{1,3}}] {1.082298159096267,1.33248380866884} Show[{ Graphics3D[ {Point[FPoints/@{1,3}], Point[FPoints[0]], Point[Table[ShiftedCeresVectorEquations[{i,QpScalars[i]}],{i,{1,3}}]], {PointSize[Large], {Darker[Yellow], Point[Origin]}, {Lighter[Blue,0.5], Point[CartesianEarthPoints/@Range[Nn]]}, {Gray, Point[CartesianCeresPoints[2]]}}, {Arrowheads[Tiny], Arrow[{Origin,CartesianEarthPoints[#]}&/@Range[Nn]], Arrow[Table[(#+CartesianEarthPoints[i])&/@{Origin,CartesianCeresUnits[i]},{i,{1,3}}]], Arrow[CeresVectorEquations[{2,#}]&/@{0,c}/.P2Solution], {Dashed, Arrow[{FPoints[#],FPoints[0]}&/@{1,3}], Arrow[Table[ShiftedCeresVectorEquations[{i,#}]&/@{0,QpScala

rs[i]},{i,{1,3}}]], Arrow[Table[{ShiftedCeresVectorEquations[{i,QpScalars[i]}],CartesianCeresPoints[2]},{i,{1,3}}]]}}}]}, Axes->True, AxesLabel->{x,y,z}]

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It is now a simple matter of working backward and scaling the line of sight vectors by these coefficients divided by the approximate triangular area ratios: Table[CartesianCeresPoints[i]=CartesianEarthPoints[i]+CartesianCeresUnits[i]QpScalars[i]/NSr[i],{i,{1,3}}] {{1.422276912041750,2.245731165718013,0.7694626810473485},{1.046034831422242,2.360593391870084,0.9002669460230068}} Norm[CartesianCeresPoints[#]]&/@Range[Nn] {2.767354856307513,2.750890747152010,2.734423267234036} Show[{ Graphics3D[ {Point[FPoints/@{1,3}], Point[FPoints[0]],

Point[Table[ShiftedCeresVectorEquations[{i,QpScalars[i]}],{i,{1,3}}]], {PointSize[Large], {Darker[Yellow], Point[Origin]}, {Lighter[Blue,0.5], Point[CartesianEarthPoints/@Range[Nn]]}, {Gray, Point[CartesianCeresPoints/@Range[Nn]]}}, {Arrowheads[Tiny], Arrow[{Origin,CartesianEarthPoints[#]}&/@Range[Nn]], Arrow[Table[(#+CartesianEarthPoints[i])&/@{Origin,CartesianCeresUnits[i]QpScalars[i]/NSr[i]},{i,{1,3}}]], Arrow[CeresVectorEquations[{2,#}]&/@{0,c}/.P2Solution], {Dashed, Arrow[{FPoints[#],FPoints[0]}&/@{1,3}], Arrow[Table[ShiftedCeresVectorEquations[{i,#}]&/@{0,QpScalars[i]},{i,{1,3}}]], Arrow[Table[{ShiftedCeresVectorEquations[{i,QpScalars[i]}],CartesianCeresPoints[2]},{i,{1,3}}]]}}}]}, Axes->True, AxesLabel->{x,y,z}]

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Show[{ Graphics3D[ {{PointSize[Large], {Darker[Yellow], Point[Origin]}, {Lighter[Blue,0.5], Point[CartesianEarthPoints/@Range[Nn]]}, {Gray, Point[CartesianCeresPoints/@Range[Nn]]}}, {Arrowheads[Tiny], Arrow[{Origin,CartesianEarthPoints[#]}&/@Range[Nn]], Arrow[{Origin,CartesianCeresPoints[#]}&/@Range[Nn]]}}]}, Axes->True, AxesLabel->{x,y,z}]

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With that, the three positions of Ceres have been obtained! We are now

ready to calculate Ceres orbit. Let us begin by learning how to rotate our

points in Mathematica: EarthOrbitNormal=Normalize[Cross@@CartesianEarthPoints/@{1,3}] EarthOrbitPlane=EarthOrbitNormal.{x,y,z}==0 EarthOrbitPlaneZ=z/.Solve[EarthOrbitPlane,z][[1]] EarthInclinationAxis=Normalize[Cross[{0,0,1},EarthOrbitNormal]] EarthRotationMatrix=RotationMatrix[{EarthOrbitNormal,{0,0,1}}] {5.133007170822883*10-6,-0.3981872408585286,0.917304159584554} 5.133007170822883*10-6 x-0.3981872408585286 y+0.917304159584554 z==0 1.090150948898890 (-5.133007170822883*10-6 x+0.3981872408585286 y)

{0.999999999916912,0.00001289093834179405,0} {{0.999999999986258,1.066026979829785*10-6,-5.133007170822883*10-6},{1.066026979829785*10-6,0.917304159598296,0.3981872408585286},{5.133007170822883*10-6,-0.3981872408585286,0.917304159584554}} CeresOrbitNormal=Normalize[Cross@@CartesianCeresPoints/@{1,3}] CeresOrbitPlane=CeresOrbitNormal.{x,y,z}==0 CeresOrbitPlaneZ=z/.Solve[CeresOrbitPlane,z][[1]] CeresInclinationAxis=Normalize[Cross[{0,0,1},CeresOrbitNormal]] CeresRotationMatrix=RotationMatrix[{CeresOrbitNormal,{0,0,1}}] {0.1811690257373910,-0.4195076106284812,0.8894892628571450} 0.1811690257373910 x-0.4195076106284812 y+0.8894892628571450 z==0 1.124240664567306 (-0.1811690257373910 x+0.4195076106284812 y) {0.918048006825590,0.3964692386094553,0} {{0.982629054034949,0.04022345434874737,-0.1811690257373910},{0.04022345434874737,0.906860208822196,0.4195076106284812},{0.1811690257373910,-0.4195076106284812,0.8894892628571450}} Show[{ Graphics3D[ {{PointSize[Large], {Darker[Yellow], Point[Origin]}, {Lighter[Blue,0.5], Point[(CeresRotationMatrix.CartesianEarthPoints[#])&/@Range[Nn]]}, {Gray, Point[(CeresRotationMatrix.CartesianCeresPoints[#])&/@Range[Nn]]}}, {Arrowheads[Tiny], Arrow[{Origin,CeresRotationMatrix.CartesianEarthPoints[#]}&/@Range[Nn]],

Arrow[{Origin,CeresRotationMatrix.CartesianCeresPoints[#]}&/@Range[Nn]], {Dashed, Arrow[{Origin,CeresRotationMatrix.EarthOrbitNormal}], Arrow[{Origin,CeresRotationMatrix.CeresOrbitNormal}]}}}], ParametricPlot3D[{x,y,0},{x,-0.8,1.4},{y,0,2.6}, PlotStyle->{Gray,Opacity[0.25]},Mesh->None]}, Axes->True, AxesLabel->{x,y,z}]

Clear[HorizontalCeresPoints] Table[HorizontalCeresPoints[i]=Chop[CeresRotationMatrix.CartesianCeresPoints[i]],{i,Nn}] {{1.348498877412522,2.416568575138297,0},{1.157410896897420,2.495556033935053,0},{0.959714951907322,2.560472186038424,0}} CeresOrbitVertical=Chop[CeresRotationMatrix.CeresOrbitNorma

l] {0,0,1.000000000000000} RotatedVernalUnit=CeresRotationMatrix.CartesianVernalUnit {0.982629054034949,0.04022345434874737,0.1811690257373910} At this point, we have obtained a heliocentric coordinate system in which

Ceres orbital plane is horizontal. As the planets orbit is an ellipse, it can

be represented as a conic sectionthe intersection of a plane U with a cone. To find this intersection, we first create a right cone that is normal to the plane whose vertex is located at the Sun: Show[{ Graphics3D[ {{PointSize[Large], {Darker[Yellow], Point[Origin]}, {Gray, Point[HorizontalCeresPoints/@Range[Nn]]}}, {Arrowheads[Tiny], Arrow[{Origin,HorizontalCeresPoints[#]}&/@Range[Nn]], {Dashed, Arrow[{Origin,4CeresOrbitVertical}]}}}], ParametricPlot3D[{x,y,0},{x,-3,3},{y,-3,3}, PlotStyle->{Gray,Opacity[0.25]},Mesh->None], ParametricPlot3D[{x,y, x2 y2 },{x,-3,3},{y,-3,3}, PlotStyle->{Gray,Opacity[0.25]},Mesh->None]}, Axes->True, AxesLabel->{x,y,z}]

and then project the known Ceres positions onto this cone: Clear[CeresConeHeights] Table[CeresConeHeights[{i,r_}]=HorizontalCeresPoints[i]+r CeresOrbitVertical,{i,Nn}] {{1.348498877412522,2.416568575138297,1.000000000000000 r},{1.157410896897420,2.495556033935053,1.000000000000000 r},{0.959714951907322,2.560472186038424,1.000000000000000 r}} Clear[CeresConePointEquations] Table[CeresConePointEquations[{i,r_}]={x,y, x2 y2 }==CeresConeHeights[{i,r}],{i,Nn}] {{x,y, x2 y2 }=={1.348498877412522,2.416568575138297,1.000000000000000 r},{x,y, x2 y2 }=={1.157410896897420,2.495556033935053,1.000000000000000 r},{x,y, x2 y2 }=={0.959714951907322,2.560472186038424,1.00000

0000000000 r}} Clear[CeresConePointSolutions] Table[CeresConePointSolutions[i]=Solve[CeresConePointEquations[{i,r}]][[1]],{i,Nn}] {{r->2.767354856307513,x->1.348498877412522,y->2.416568575138297},{r->2.750890747152010,x->1.157410896897420,y->2.495556033935053},{r->2.734423267234036,x->0.959714951907322,y->2.560472186038424}} Clear[CeresConePoints] Table[CeresConePoints[i]=CeresConeHeights[{i,r/.CeresConePointSolutions[i]}],{i,Nn}] {{1.348498877412522,2.416568575138297,2.767354856307513},{1.157410896897420,2.495556033935053,2.750890747152010},{0.959714951907322,2.560472186038424,2.734423267234036}} Show[{ Graphics3D[ {Point[CeresConePoints/@Range[Nn]], {PointSize[Large], {Darker[Yellow], Point[Origin]}, {Gray, Point[HorizontalCeresPoints/@Range[Nn]]}}, {Arrowheads[Tiny], Arrow[{Origin,HorizontalCeresPoints[#]}&/@Range[Nn]], {Dashed, Arrow[{Origin,4CeresOrbitVertical}], Arrow[Table[CeresConeHeights[{i,#}]&/@{0,r/.CeresConePointSolutions[i]},{i,Nn}]]}}}], ParametricPlot3D[{x,y,0},{x,-3,3},{y,-3,3}, PlotStyle->{Gray,Opacity[0.25]},Mesh->None], ParametricPlot3D[{x,y, x2 y2 },{x,-3,3},{y,-3,3}, PlotStyle->{Gray,Opacity[0.25]},Mesh->None]}, Axes->True, AxesLabel->{x,y,z}]

One could determine the intersecting plane from these projected points on the cone. However, because these points are close to being aligned, this method is prone to large error. Gauss instead opted to determine a fourth

point V along the cones axis. The height of this point is equal to the

semi-orbital parameter, h (otherwise known as the semilatus rectum),

which appears in the equation form of Keplers second law:

S= t h \[DoubleLongRightArrow] h=(S/( t))2 Unfortunately, exact knowledge of the orbital sector S is unknown until we have the orbit. In the meantime, we can approximate it as the triangular area made by two known positions and the Sun, corrected by the G factor:

h((G T)/( t))2.

As before, we shall reflexively use this parameter to determine the orbit, which we then use to determine the parameter, followed by the orbit, etc. Finding the area of the T13 is straightforward: NT[13]=Norm[Cross@@HorizontalCeresPoints/@{1,3}]/2 0.5667884373246028 as is the point V: VPoint={0,0,Nh=((NG NT[13])/( Nt[13]))2} {0,0,2.739591956314323} Show[{ Graphics3D[ {Point[VPoint], Point[CeresConePoints/@Range[Nn]], {PointSize[Large], {Darker[Yellow], Point[Origin]}, {Gray, Point[HorizontalCeresPoints/@Range[Nn]]}}, {Arrowheads[Tiny], Arrow[{Origin,HorizontalCeresPoints[#]}&/@Range[Nn]], {Dashed, Arrow[{Origin,4CeresOrbitVertical}], Arrow[Table[CeresConeHeights[{i,#}]&/@{0,r/.CeresConePointSolutions[i]},{i,Nn}]]}}}], ParametricPlot3D[{x,y,0},{x,-3,3},{y,-3,3}, PlotStyle->{Gray,Opacity[0.25]},Mesh->None], ParametricPlot3D[{x,y, x2 y2 },{x,-3,3},{y,-3,3}, PlotStyle->{Gray,Opacity[0.25]},Mesh->None]}, Axes->True, AxesLabel->{x,y,z}]

We now determine the plane U following the usual method: Clear[CeresUPlanePoints] Table[CeresUPlanePoints[i]=CeresConePoints[i]-VPoint,{i,Nn}] {{1.348498877412522,2.416568575138297,0.02776289999318930},{1.157410896897420,2.495556033935053,0.01129879083768650},{0.959714951907322,2.560472186038424,-0.005168689080287674}} UPlaneNormal=Normalize[Cross@@CeresUPlanePoints/@{1,3}] UPlane=Simplify[UPlaneNormal.({x,y,z}-VPoint)==0] UPlaneZ=z/.Solve[UPlane,z][[1]] UPlaneInclinationAxis=Normalize[Cross[{0,0,1},UPlaneNormal]] {-0.07349653070039586,0.02956023706000831,0.996857287860184} 37.1579744157960+1.000000000000000 x==0.402199080396173

y+13.56332439586587 z -0.07372823732689031 (-37.15797441579597-1.000000000000000 x+0.402199080396173 y) {-0.3731488411179069,-0.927771492541328,0} Show[{ Graphics3D[ {Point[VPoint], Point[CeresConePoints/@Range[Nn]], {PointSize[Large], {Darker[Yellow], Point[Origin]}, {Gray, Point[HorizontalCeresPoints/@Range[Nn]]}}, {Arrowheads[Tiny], Arrow[{Origin,HorizontalCeresPoints[#]}&/@Range[Nn]], Arrow[Table[{#,#+CeresUPlanePoints[i]}&[VPoint],{i,{1,3}}]], {Dashed, Arrow[{Origin,4CeresOrbitVertical}], Arrow[Table[CeresConeHeights[{i,#}]&/@{0,r/.CeresConePointSolutions[i]},{i,Nn}]]}}}], ParametricPlot3D[{x,y,0},{x,-3,3},{y,-3,3}, PlotStyle->{Gray,Opacity[0.25]},Mesh->None], ParametricPlot3D[{x,y, x2 y2 },{x,-3,3},{y,-3,3}, PlotStyle->{Opacity[0.25]},Mesh->None], ParametricPlot3D[{x,y,UPlaneZ},{x,-3,3},{y,-3,3}, PlotStyle->{Brown,Opacity[0.25]},Mesh->None]}, Axes->True, AxesLabel->{x,y,z}]

Almost there! We must now find the intersection of the plane U with the cone. This task is simple, given our parametric representations of each. We are also better off if we can represent this curve in polar coordinates, which is nothing that a simple substitution cannot fix: UElipseEquation=FullSimplify[{x,y, x2 y2 }=={x,y,UPlaneZ}/.{x->r Cos[],y->r Sin[]},r>0] {r Cos[],r Sin[],r}=={r Cos[],r Sin[],2.73959195631432+0.0737282373268903 r Cos[]-0.02965342925210608 r Sin[]} UElipseSolution=FullSimplify[Solve[UElipseEquation,r]][[1]] {r->-(37.15797441579597/(-13.56332439586587+1.000000000000000 Cos[]-0.4021990803961730 Sin[]))} ParametricUElipse=FullSimplify[{x,y,UPlaneZ}/.{x->r Cos[],y->r Sin[]}/.UElipseSolution]//Chop

{2.73959195631432/(-0.07372823732689031+1.00000000000000 Sec[]+0.029653429252106 Tan[]),-(37.15797441579597/(-0.4021990803961730+1.00000000000000 Cot[]-13.56332439586587 Csc[])),-(37.15797441579597/(-13.56332439586587+1.000000000000000 Cos[]-0.4021990803961730 Sin[]))} ParametricCeresOrbit=FullSimplify[{x,y,0}/.{x->r Cos[],y->r Sin[]}/.UElipseSolution] {2.73959195631432/(-0.07372823732689031+1.00000000000000 Sec[]+0.029653429252106 Tan[]),-(37.15797441579597/(-0.4021990803961730+1.00000000000000 Cot[]-13.56332439586587 Csc[])),0} Show[{ Graphics3D[ {Point[VPoint], Point[CeresConePoints/@Range[Nn]], {PointSize[Large], {Darker[Yellow], Point[Origin]}, {Gray, Point[HorizontalCeresPoints/@Range[Nn]]}}, {Arrowheads[Tiny], Arrow[{Origin,HorizontalCeresPoints[#]}&/@Range[Nn]], Arrow[Table[{#,#+CeresUPlanePoints[i]}&[VPoint],{i,{1,3}}]], {Dashed, Arrow[{Origin,4CeresOrbitVertical}], Arrow[Table[CeresConeHeights[{i,#}]&/@{0,r/.CeresConePointSolutions[i]},{i,Nn}]]}}}], ParametricPlot3D[{x,y,0},{x,-3,3},{y,-3,3}, PlotStyle->{Gray,Opacity[0.25]},Mesh->None], ParametricPlot3D[{x,y, x2 y2 },{x,-3,3},{y,-3,3}, PlotStyle->{Opacity[0.25]},Mesh->None], ParametricPlot3D[{x,y,UPlaneZ},{x,-3,3},{y,-3,3}, PlotStyle->{Brown,Opacity[0.25]},Mesh->None], ParametricPlot3D[ParametricUElipse,{,0,2Pi}, PlotStyle->{{Black}}], ParametricPlot3D[ParametricCeresOrbit,{,0,2Pi}, PlotStyle->{{Black}}]}, Axes->True,

AxesLabel->{x,y,z}]

Success! Now, let us see the orbit by itself: Show[{ Graphics3D[ {{PointSize[Large], {Darker[Yellow], Point[Origin]}, {Gray, Point[HorizontalCeresPoints/@Range[Nn]]}}, {Arrowheads[Tiny], Arrow[{Origin,HorizontalCeresPoints[#]}&/@Range[Nn]]}}], ParametricPlot3D[{x,y,0},{x,-3,3},{y,-3,3}, PlotStyle->{Gray,Opacity[0.25]},Mesh->None], ParametricPlot3D[ParametricCeresOrbit,{,0,2Pi}, PlotStyle->{{Black}}]}, Axes->True, AxesLabel->{x,y,z}]

Wonderful, but we are not finished just yet! Remember that this orbit was determined using an approximate orbital parameter under the condition that we would use the orbit to correct the parameter and so on. We shall use this orbit to calculate the area of the orbital sector S13, which will require two things: 1 and 3 corresponding to the second and third positions of Ceres, the orbit as a polar function r(), which can be integrated between 1 and 3. The latter is easy enough to obtain: PolarCeresOrbit=Norm[ParametricCeresOrbit] \[Sqrt](1380.715062684948/Abs[-0.4021990803961730+1.00000000000000 Cot[]-13.56332439586587 Csc[]]2+7.505364087102141/Abs[-0.07372823732689031+1.00000000000000 Sec[]+0.029653429252106 Tan[]]2) PolarPlot[PolarCeresOrbit,{,0,2Pi}]

2 1 1 2 3

2

1

1

2

However, finding the relevant angles is a different story: Clear[Equations] Table[Equations[i]=Thread[ParametricCeresOrbit==HorizontalCeresPoints[i]],{i,Nn}] {{2.73959195631432/(-0.07372823732689031+1.00000000000000 Sec[]+0.029653429252106 Tan[])==1.348498877412522,-(37.15797441579597/(-0.4021990803961730+1.00000000000000 Cot[]-13.56332439586587 Csc[]))==2.416568575138297,True},{2.73959195631432/(-0.07372823732689031+1.00000000000000 Sec[]+0.029653429252106 Tan[])==1.157410896897420,-(37.15797441579597/(-0.4021990803961730+1.00000000000000 Cot[]-13.56332439586587 Csc[]))==2.495556033935053,True},{2.73959195631432/(-0.07372823732689031+1.00000000000000 Sec[]+0.029653429252106 Tan[])==0.959714951907322,-(37.15797441579597/(-0.402199080

3961730+1.00000000000000 Cot[]-13.56332439586587 Csc[]))==2.560472186038424,True}} Solve[Equations[#],]&/@Range[Nn] {{},{},{}} Nothing! If the orbit that we have derived does not contain the points from which it was derived, then it is possible that something has gone quite wrong. Therefore, let us investigate this with care by solving the x and y components separately: Clear[Solutions] Table[Solutions[i]=Quiet[Solve[Equations[i][[#]],]&/@Range[2]],{i,Nn}] {{{{->-1.089983125354243},{->1.06181491881519}},{{->1.06181491881519}}},{{{->-1.160838069013211},{->1.13654043023253}},{{->1.13650337153816}}},{{{->-1.232436082815825},{->1.21218389626726}},{{->1.21218389626725}}}} We see that each x component has two solutions, while the y components only has one. This should not present an issue, as simultaneously solving x and y parts ought to whittle the solutions down to a single number. Indeed, it does appear that each set of solutions do share a number in common, but let us examine this result with a bit more precision: Chop[Differences[/.Rest[Flatten[Solutions[#],1]]]&/@Range[Nn]] {{0},{-0.00003705869437043291},{0}} The discrepancies within the angles corresponding to the x and y components of the first and third positions are negligible, as should be expected since these were the points used to calculate the orbit. However, the difference between the second position angles is quite noticeable! Minimizing this quantity will be a goal ours whilst fine-tuning the orbit. In the meantime, let us take solace in the fact that we now know which angles

correspond to Ceres first and third locations:

Clear[N] Table[N[i]=Mean[/.Rest[Flatten[Solutions[i],1]]],{i,Nn}] {1.061814918815193,1.136521900885348,1.212183896267255} Clear[ParametricCeresPoints] Table[ParametricCeresPoints[i]=ParametricCeresOrbit/.{->N[i]},{i,Nn}]

{{1.348498877412516,2.416568575138300,0},{1.157458847478015,2.495573796395142,0},{0.959714951907323,2.560472186038423,0}} Chop[Table[HorizontalCeresPoints[i]-ParametricCeresPoints[i],{i,Nn}]] {{0,0,0},{-0.00004795058059542299,-0.00001776246008829907,0},{0,0,0}} Again, our final orbit should render all of these zero! Let us remind ourselves of the original approximation of the sectoral area from which we calculated h: NG NT[13] 0.5683938392750762 and let us obtain the area within our current orbit. For this, we shall use NIntegrate, as it is much faster than Integrate and practically produces the same result: time=AbsoluteTime[]; NS[13]=NIntegrate[1/2 PolarCeresOrbit2,{,N[1],N[3]}] (AbsoluteTime[]-time)Second 0.568939 0.07605430000000000 Second (*time=AbsoluteTime[]; NIS[13]=Integrate[1/2PolarCeresOrbit^2,{,N[1],N[3]}] (AbsoluteTime[]-time)Second*) Differences[{NS[13],NIS[13]}] {-0.568939+NIS[13]} Our new semi-orbital parameter is as follows: (NS[13]/( Nt[13]))2 2.74485 Compare it to the original: Nh 2.739591956314323 With this, we embark upon another iterative process la the determination of P2: SemiorbitalParameters={Nh}; SectoralAreas={NS[13]}; time=AbsoluteTime[]; Catch[Do[ AppendTo[SemiorbitalParameters,Nh=(NS[13]/( Nt[13]))2]; VPoint={0,0,Nh};

Clear[CeresUPlanePoints]; Table[CeresUPlanePoints[i]=CeresConePoints[i]-VPoint,{i,Nn}]; UPlaneNormal=Normalize[Cross@@CeresUPlanePoints/@{1,3}]; UPlane=Simplify[UPlaneNormal.({x,y,z}-VPoint)==0]; UPlaneZ=z/.Solve[UPlane,z][[1]]; UElipseEquation=FullSimplify[{x,y, x2 y2 }=={x,y,UPlaneZ}/.{x->r Cos[],y->r Sin[]},r>0]; UElipseSolution=FullSimplify[Solve[UElipseEquation,r]][[1]]; ParametricUElipse=FullSimplify[{x,y,UPlaneZ}/.{x->r Cos[],y->r Sin[]}/.UElipseSolution]; ParametricCeresOrbit=FullSimplify[{x,y,0}/.{x->r Cos[],y->r Sin[]}/.UElipseSolution]; PolarCeresOrbit=Norm[ParametricCeresOrbit]; Clear[Equations]; Table[Equations[i]=Thread[ParametricCeresOrbit==HorizontalCeresPoints[i]],{i,Nn}]; Clear[Solutions]; Table[Solutions[i]=Quiet[Solve[Equations[i][[#]],]&/@Range[2]],{i,Nn}]; Clear[N]; Table[N[i]=Mean[/.Rest[Flatten[Solutions[i],1]]],{i,Nn}]; Clear[ParametricCeresPoints]; Table[ParametricCeresPoints[i]=ParametricCeresOrbit/.{->N[i]},{i,Nn}]; AppendTo[SectoralAreas,NS[13]=NIntegrate[1/2 PolarCeresOrbit2,{,N[1],N[3]}]]; If[Chop[Differences[SectoralAreas[[-2;;-1]]][[1]],10-15]==0,Throw[Null],Null], {100}]] (AbsoluteTime[]-time)Second Length[Rest[SectoralAreas]] {Nh,NS[13]}

1.2358553000000000 Second 6 {2.74481,0.568935} This process has converged upon a value for h, and so let us see whether or not our new orbit is consistent with our original P2: Chop[Differences[/.Rest[Flatten[Solutions[#],1]]]&/@Range[Nn]] {{0},{-0.0000205673},{0}} Chop[Table[HorizontalCeresPoints[i]-ParametricCeresPoints[i],{i,Nn}]] {{0,0,0},{-0.0000266121,-9.8577*10-6,0},{0,0,0}} We are clearly moving in the right direction, but we have not reached our destination just yet. Nevertheless, this is a good point to pause and

calculate Ceres orbital elements for comparison to the known values:

AstronomicalData["Ceres","OrbitRules"] {SemimajorAxis->4.137811914496967*1011,Eccentricity->0.07976017327382710,Inclination->10.58671160799540,PeriapsisArgument->73.15073411326651,AscendingNodeLongitude->80.40695914792410,PeriapsisLongitude->153.5576932611906,Periapsis->3.807779319222182*1011,Apoapsis->4.467844509771751*1011} (*{KnownCeresSemimajorAxis, KnownCeresEccentricity, KnownCeresInclination, KnownCeresPeriapsisArgument, KnownCeresAscendingNodeLongitude, KnownCeresPeriapsisLongitude, KnownCeresPeriapsis, KnownCeresApoapsis}= {Convert["SemimajorAxis"Meter,AU]/AU, "Eccentricity", "Inclination", "PeriapsisArgument", "AscendingNodeLongitude", "PeriapsisLongitude", Convert["Periapsis"Meter,AU]/AU, Convert["Apoapsis"Meter,AU]/AU}/.AstronomicalData["Ceres","OrbitRules"]*)

{KnownCeresSemimajorAxis, KnownCeresEccentricity, KnownCeresInclination, KnownCeresPeriapsisArgument, KnownCeresAscendingNodeLongitude, KnownCeresPeriapsisLongitude, KnownCeresPeriapsis, KnownCeresApoapsis}= {QuantityMagnitude[UnitConvert[Quantity["SemimajorAxis","Meter"],"AU"]], "Eccentricity", "Inclination", "PeriapsisArgument", "AscendingNodeLongitude", "PeriapsisLongitude", QuantityMagnitude[UnitConvert[Quantity["Periapsis","Meter"],"AU"]], QuantityMagnitude[UnitConvert[Quantity["Apoapsis","Meter"],"AU"]]}/.AstronomicalData["Ceres","OrbitRules"]; Obtaining the apsides is easy enough: {{CeresApoapsis,{CeresApoapsis}},{CeresPeriapsis,{CeresPeriapsis}}}={Maximize[PolarCeresOrbit,],Minimize[PolarCeresOrbit,]} {{2.98201,{->-0.406376}},{2.54256,{->2.73522}}} Let us verify that these points occur opposite of each other: Abs[-Abs[Differences[/.{{CeresApoapsis},{CeresPeriapsis}}][[1]]]] 1.37838*10-8 and compare our results to the known values: 100Abs[CeresApoapsis-KnownCeresApoapsis]/KnownCeresApoapsis 0.152676 100Abs[CeresPeriapsis-KnownCeresPeriapsis]/KnownCeresPeriapsis 0.109258 We are off by between one and two tenths of a percent on eachnot bad! From these values, we derive the semi-major axis (a):

CeresSemimajorAxis=(CeresApoapsis+CeresPeriapsis)/2 2.76229 which we also compare to the known quantity: 100 Abs[CeresSemimajorAxis-KnownCeresSemimajorAxis]/KnownCeresSemimajorAxis 0.132698 This, of course, is as accurate as the previous two quantities. From it and

Keplers third law, we can obtain Ceres orbital period:

KnownCeresOrbitPeriod=QuantityMagnitude[AstronomicalData["Ceres","OrbitPeriodYears"]] 4.600194590273435 CeresOrbitPeriod=CeresSemimajorAxis3/2 4.59095 100 Abs[CeresOrbitPeriod-KnownCeresOrbitPeriod]/KnownCeresOrbitPeriod 0.200866 The orbit's eccentricity (e) follows from the semi-major axis and the semi-orbital parameter: CeresEccentricity= 1 Nh CeresSemimajorAxis 0.0795442 Compared to the known value: 100 Abs[CeresEccentricity-KnownCeresEccentricity]/KnownCeresEccentricity 0.270807 Errors, while still small, seem to be mounting. Nevertheless, it is clear that

the ellipse which we have found very closely matches that of Ceres orbit.

Now we must determine how that ellipse is oriented in space, beginning

with Ceres orbital inclination (i, in degrees) with respect to the ecliptic

plane; i.e. the plane of Earths orbit:

CeresInclination=(180/)ArcCos[EarthOrbitNormal.CeresOrbitNormal] 10.58742174583762 100Abs[CeresInclination-KnownCeresInclination]/KnownCeresInclination 0.006707822679146446 This value is quite good! However, that is because it was essentially known as soon as we had two (,) data points for both Ceres and the Sun. The

longitude of the ascending node (, the angle between Ceres inclination

axis and the vernal unit) might not be so accurate: CeresAscendingNode=Normalize[Cross[EarthOrbitNormal,CeresOrbitNormal]] {0.1667222217538919,0.904465824243575,0.3926133894161495} CeresAscendingNodeLongitude=(180/)ArcCos[CeresAscendingNode.CartesianVernalUnit] 80.40270353352374 100 Abs[CeresAscendingNodeLongitude-KnownCeresAscendingNodeLongitude]/KnownCeresAscendingNodeLongitude 0.005292594627951980 I stand corrected! However, the argument of periapsis (, the angle between the inclination axis the ellipse's major axis in the direction of periapsis) does not come out quite as nicely: HorizontalCeresAscendingNode=Chop[CeresRotationMatrix.CeresAscendingNode] {0.1290774535881167,0.991634514816425,0} CeresPeriapsisUnit=Normalize[ParametricCeresOrbit /. CeresPeriapsis] {-0.918559,0.395283,0.} CeresPeriapsisArgument=(180/)ArcCos[HorizontalCeresAscendin

gNode.CeresPeriapsisUnit] 74.1326 100Abs[CeresPeriapsisArgument-KnownCeresPeriapsisArgument]/KnownCeresPeriapsisArgument 1.3423

We finally achieve an error greater than one percent! However, a moments

thought leads me to question whether or not the known periapsis

argument is the correct periapsis argument! Remember, our raw data

comes from 1801, and it is quite possible that Ceres orbit has precessed

throughout the last two centuries. In fact, precession should be expected,

given Ceres proximity to mighty Jupiter! Unfortunately, Mathematica is of

no help in providing Ceres orbital elements from 1801:

AstronomicalData["Ceres","OrbitRules",NDate[2]] AstronomicalData::notsubprop: {1801, 1, 22, 19, 20, 21.7} is not a known subproperty of AstronomicalData. Use subproperty "Properties" for a list of subproperties. >> AstronomicalData[Ceres,OrbitRules,{1801,1,22,19,20,21.7}] Thus, I shall either have to look for reference values elsewhere, or use more modern observations. In the meantime, a quick jaunt over to the Internet reveals that for Ceres: Precession of perihelion 54.070272 arcsec / yr Precession of the ascending node -59.170034 arcsec / yr From this, we immediately see that the ascending node has been precessing as well, which means that if precession accounts for the discrepancy in the argument of periapsis, then we ought to see a similar error in the longitude of the ascending node, which we do not! Moreover,

Ceres periapsis has evidently precessed by an amount much greater than

that of our discrepancy: (*Convert[(2013-1801)Year 54.070272ArcSecond/Year,Degree]*) CeresPeriapsisPrecession=(180/)QuantityMagnitude[UnitConvert[Quantity[(2013-1801) 54.070272,"ArcSecond"],"Radian"]] 3.18414 CeresPeriapsisArgument-KnownCeresPeriapsisArgument 0.981906 Thus it seems that accounting for precession will only increase our error! Perhaps we shall simply have to accept the fact that our orbit is not yet perfect. Ignoring this, we can now easily determine the longitude of the

periapsis ( ):

CeresPeriapsisLongitude=CeresAscendingNodeLongitude+CeresPeriapsisArgument 154.535 100Abs[CeresPeriapsisLongitude-KnownCeresPeriapsisLongitude]/KnownCeresPeriapsisLongitude 0.636666 This quantity is more successful than the last, but it does not tell us anything that we did not already know. In that case, perhaps it can be used as a replacement for the error-ridden periapsis argument! There is one final element that we need to make this orbit complete: the mean anomaly at epoch (M0). The mean anomaly MeanAnomalyEquation=M==(2)/P t+M0;

is derived from Keplers second law and establishes the relationship

between time and position along the orbit. Although M varies from 0 to 2 over the course of one orbital period, it is not an angle, unlike the eccentric and true anomalies (E and , respectively). The mean anomaly at epoch is

simply a known reference that determines the systems initial condition.

The true and eccentric anomalies are related:

TrueEccentricEquation=Tan[/2]==1 e1 e Tan[Ee/2];

as are the eccentric and mean anomalies, via Keplers equation:

KeplerEquation=M==Ee-e Sin[Ee]; Combining these expressions with one for the mean anomaly above, we find that (*EccentricAnomaly=Ee/.FullSimplify[Quiet[Solve[TrueEccentricEquation,Ee][[1]]]]*) EccentricAnomaly=Ee/.FullSimplify[Quiet[Solve[{TrueEccentricEquation,C[1]==0},Ee][[1]]]] 2 ArcTan[Tan[/2]/

1 e1 e

] Refine[Eliminate[{MeanAnomalyEquation,KeplerEquation},M],P>0] Ee==M0+(2 t)/P+e Sin[Ee] MeanAnomalyAtEpoch=Expand[Solve[Refine[Eliminate[{MeanAnomalyEquation,KeplerEquation},M],P>0]/.Ee->EccentricAnomaly,M0][[1]]] {M0->-((2 t)/P)+2 ArcTan[Tan[/2]/

1 e1 e

]-e Sin[2

ArcTan[Tan[/2]/1 e1 e

]]} Inserting the known values for the first observation gives us the quantity which we seek: CeresMeanAnomalyAtEpoch=M0/.MeanAnomalyAtEpoch/.{P->CeresOrbitPeriod,e->CeresEccentricity,t->Nt[1],->N[1]} 136.501 Therefore CeresMeanAnomaly=M/.Expand[Solve[MeanAnomalyEquation,M][[1]]]/.{M0->CeresMeanAnomalyAtEpoch,P->CeresOrbitPeriod}

136.501 +1.3686 t With this expression (and its rather convoluted relationship with the true anomaly), we can calculate the position of Ceres along its orbit for a given time t.

References [1] http://www.schillerinstitute.org/fid_97-01/982_orbit _ceres.pdf

Future Work Before proceeding, let us review what we have done so far. First, we derived the second position of Ceres from our three original observations. This involved an approximation of the ratios of sectoral areas which were iteratively corrected by use of the factor G(r2). With the second position in hand, the first and third positions were obtained using the latest ratio approximation. From these points, we then derived a first approximation of

Ceres orbit, which was also adjusted iteratively using the semi-orbital

parameter h(S13). This is where we currently stand. Now that we have a

preliminary orbit, we need not approximate the ratios of sectoral areas in

the first step! Instead, we can obtain the ratios from our orbit, calculate P2, P1, and P3, recalculate the orbit, and so on until the area ratios converge and all three points lie on the same path. Let the finale begin! S= t h \[DoubleLongRightArrow] h=(S/( t))2 (*Distances={CeresSunDistance[2]}; SemiorbitalParameters={Nh}; SectoralAreas={NS[13]}; time=AbsoluteTime[]; Catch[Do[ Clear[NSr]; Table[NSr[i]= Nt[i]Sqrt[Nh]/NS[13],{i,{3,1}}]; Clear[FPoints]; Evaluate[FPoints/@{1,3}]=EarthPointEquations/@{{1,NSr[1]},{3,NSr[3]}}; FPoints[0]=Total[FPoints/@{1,3}]; Clear[ShiftedCeresVectorEquations]; Table[ShiftedCeresVectorEquations[{i,sc_}]=FPoints[0]+sc CartesianCeresUnits[i],{i,{1,3}}];

QNormal=Normalize[Cross@@CartesianCeresUnits/@{1,3}];

Qz=z/.Simplify[Solve[QNormal.({x,y,z}-FPoints[0]) 0,z][[1]]

];

P2Solution=Solve[CeresVectorEquations[{2,c}] {x,y,Qz}][[1]]

; CartesianCeresPoints[2]=CeresVectorEquations[{2,c}]/.P2Solution; AppendTo[Distances,CeresSunDistance[2]=Norm[CartesianCeresPoints[2]]]; Clear[QpScalars]; If[Solve[Total[Qp[#]

CartesianCeresUnits[#]&/@{1,3}] CartesianCeresPoints[2]-FPo

ints[0]] {},Throw[Null],Null];

Table[QpScalars[i]=Qp[i]/.Solve[Total[Qp[#]

CartesianCeresUnits[#]&/@{1,3}] CartesianCeresPoints[2]-FPo

ints[0]][[1]],{i,{1,3}}]; Table[CartesianCeresPoints[i]=CartesianEarthPoints[i]+CartesianCeresUnits[i]QpScalars[i]/NSr[i],{i,{1,3}}]; CeresOrbitNormal=Normalize[Cross@@CartesianCeresPoints/@{1,3}];

CeresOrbitPlane=CeresOrbitNormal.{x,y,z} 0;

CeresOrbitPlaneZ=z/.Solve[CeresOrbitPlane,z][[1]]; CeresRotationMatrix=RotationMatrix[{CeresOrbitNormal,{0,0,1}}]; Clear[HorizontalCeresPoints]; Table[HorizontalCeresPoints[i]=Chop[CeresRotationMatrix.CartesianCeresPoints[i]],{i,Nn}]; CeresOrbitVertical=Chop[CeresRotationMatrix.CeresOrbitNorma

l]; Clear[CeresConeHeights]; Table[CeresConeHeights[{i,r_}]=HorizontalCeresPoints[i]+r CeresOrbitVertical,{i,Nn}]; Clear[CeresConePointEquations];

Table[CeresConePointEquations[{i,r_}]={x,y,Sqrt[x^2+y^2]} C

eresConeHeights[{i,r}],{i,Nn}]; Clear[CeresConePointSolutions]; Table[CeresConePointSolutions[i]=Solve[CeresConePointEquations[{i,r}]][[1]],{i,Nn}]; Clear[CeresConePoints]; Table[CeresConePoints[i]=CeresConeHeights[{i,r/.CeresConePointSolutions[i]}],{i,Nn}]; AppendTo[SemiorbitalParameters,Nh=(NS[13]/( Nt[13]))^2]; VPoint={0,0,Nh}; Clear[CeresUPlanePoints]; Table[CeresUPlanePoints[i]=CeresConePoints[i]-VPoint,{i,Nn}]; UPlaneNormal=Normalize[Cross@@CeresUPlanePoints/@{1,3}];

UPlane=Simplify[UPlaneNormal.({x,y,z}-VPoint) 0];

UPlaneZ=z/.Solve[UPlane,z][[1]];

UElipseEquation=FullSimplify[{x,y,Sqrt[x^2+y^2]} {x,y,UPlan

eZ}/.{x r Cos[],y r Sin[]},r>0];

UElipseSolution=FullSimplify[Solve[UElipseEquation,r]][[1]];

ParametricUElipse=FullSimplify[{x,y,UPlaneZ}/.{x r

Cos[],y r Sin[]}/.UElipseSolution];

ParametricCeresOrbit=FullSimplify[{x,y,0}/.{x r Cos[],y r

Sin[]}/.UElipseSolution]; PolarCeresOrbit=Norm[ParametricCeresOrbit]; Clear[Equations];

Table[Equations[i]=Thread[ParametricCeresOrbit HorizontalC

eresPoints[i]],{i,Nn}]; Clear[Solutions]; Table[Solutions[i]=Quiet[Solve[Equations[i][[#]],]&/@Range[2]],{i,Nn}]; Clear[N]; Table[N[i]=Mean[/.Rest[Flatten[Solutions[i],1]]],{i,Nn}]; Clear[ParametricCeresPoints];

Table[ParametricCeresPoints[i]=ParametricCeresOrbit/.{ N[

i]},{i,Nn}]; AppendTo[SectoralAreas,NS[13]=NIntegrate[1/2PolarCeresOrbit^2,{,N[1],N[3]}]]; Print["NS[13]="ToString[NS[13]]]; Print[PolarPlot[PolarCeresOrbit,{,0,2Pi}]];

If[Chop[Differences[SectoralAreas[[-2;;-1]]][[1]],10^-15] 0

,Throw[Null],Null];, {100}]] (AbsoluteTime[]-time)Second Length[Rest[SectoralAreas]] "{CeresSunDistance[2],SemiorbitalParameters,SectoralAreas}" {CeresSunDistance[2],SemiorbitalParameters,SectoralAreas}*) (*Show[{ Graphics3D[ {Point[VPoint], Point[CeresConePoints/@Range[Nn]], {PointSize[Large], {Darker[Yellow], Point[Origin]},

{Gray, Point[HorizontalCeresPoints/@Range[Nn]]}}, {Arrowheads[Tiny], Arrow[{Origin,HorizontalCeresPoints[#]}&/@Range[Nn]], Arrow[Table[{#,#+CeresUPlanePoints[i]}&[VPoint],{i,{1,3}}]], {Dashed, Arrow[{Origin,4CeresOrbitVertical}], Arrow[Table[CeresConeHeights[{i,#}]&/@{0,r/.CeresConePointSolutions[i]},{i,Nn}]]}}}], ParametricPlot3D[{x,y,0},{x,-3,3},{y,-3,3},

PlotStyle {Gray,Opacity[0.25]},Mesh None],

ParametricPlot3D[{x,y,Sqrt[x^2+y^2]},{x,-3,3},{y,-3,3},

PlotStyle {Opacity[0.25]},Mesh None],

ParametricPlot3D[{x,y,UPlaneZ},{x,-3,3},{y,-3,3},

PlotStyle {Brown,Opacity[0.25]},Mesh None],

ParametricPlot3D[ParametricUElipse,{,0,2Pi},

PlotStyle {{Black}}],

ParametricPlot3D[ParametricCeresOrbit,{,0,2Pi},

PlotStyle {{Black}}]},

Axes True,

AxesLabel {x,y,z}]*)

(*Show[{ Graphics3D[ {{PointSize[Large],

{Darker[Yellow], Point[Origin]}, {Gray, Point[HorizontalCeresPoints/@Range[Nn]]}}, {Arrowheads[Tiny], Arrow[{Origin,HorizontalCeresPoints[#]}&/@Range[Nn]]}}], ParametricPlot3D[{x,y,0},{x,-3,3},{y,-3,3},

PlotStyle {Gray,Opacity[0.25]},Mesh None],

ParametricPlot3D[ParametricCeresOrbit,{,0,2Pi},

PlotStyle {{Black}}]},

Axes True,

AxesLabel {x,y,z}]*)

(*PolarPlot[PolarCeresOrbit,{,0,2Pi}]*)

Off[Precision::precsm,SetPrecision::precsm]These angular coordinates must be converted into radians for later conversion into Cartesian coordinates. To accomplish this, we note that [90,-90] =[0,]rad and [0,24]hours =[0,2]rad. Therefore, CoordinateTransform::bdpt: Evaluation point {0.983191072677480,1.9708085444217282,4.940836192729485} is incompatible with the coordinate assumptions of the specified coordinate chart. >>and proceed with Gauss method. The first step is to find points F1 and F3 along the first and third Earth position vectors such thatThe first and third Ceres lines of sight (L1 and L3) are shifted to point F; from them, we derive a plane Q that passes through F:By construction, the intersection of L2 with plane Q is our first approximation of the second position of Ceres, P2:At this time we have found a first approximation of Ceres second position. Before proceeding farther, let us deal with the fact that the shallow angle between the observed line of sight L2 and the constructed plane Q means that any small error in Q turns into a gigantic error in P2. The former error stems from our use of orbital sectors to approximate triangular areas. Fortunately Gauss, being a master of geometry, formulated a correction factor which goes as follows:r2 being the second distance between Ceres and the Sun. Initially, one might be put off by the fact that G, which is used to calculate r2, depends upon r2. However, we already have one estimate of r2. If we insert this into G, then we may calculate a better approximation, which can then be used to recalculate G, and so on. Evidently, Gauss abhorred this kind of brute force mechanism and developed methods of rapidly converging upon the true G and r2, but he did not have Mathematica to carry out his calculations for himwe do:The first correction factor is quite small, but we can do much better. The following loop will iteratively calculate r2 and G until they converge on the limits of machine precision:Within sixteen very short steps, we converge onto a value of r2 that is as precise as our machine can handle! We are now ready to calculate Ceres other positions. Again, by construction, Ceres second position (P2) lies within the plain Q, which was generated by the first and third line of sight vectors after they had been shifted to the point F. Therefore, we can express P2 as a linear combination of these vectors. The coefficients of this expansion are as follows:Finding the area of the T13 is straightforward:Wonderful, but we are not finished just yet! Remember that this orbit was determined using an approximate orbital parameter under the condition that we would use the orbit to correct the parameter and so on. We shall use this orbit to calculate the area of the orbital sector S13, which will require two things:1 and 3 corresponding to the second and third positions of Ceres,the orbit as a polar function r(), which can be integrated between 1 and 3.With this, we embark upon another iterative process la the determination of P2:This process has converged upon a value for h, and so let us see whether or not our new orbit is consistent with our original P2:This value is quite good! However, that is because it was essentially known as soon as we had two (,) data points for both Ceres and the Sun. The longitude of the ascending node (, the angle between Ceres inclination axis and the vernal unit) might not be so accurate: AstronomicalData::notsubprop: {1801, 1, 22, 19, 20, 21.7} is not a known subproperty of AstronomicalData. Use subproperty "Properties" for a list of subproperties. >>Precession of perihelion 54.070272 arcsec / yrPrecession of the ascending node -59.170034 arcsec / yrThere is one final element that we need to make this orbit complete: the mean anomaly at epoch (M0). The mean anomalyis derived from Keplers second law and establishes the relationship between time and position along the orbit. Although M varies from 0 to 2 over the course of one orbital period, it is not an angle, unlike the eccentric and true anomalies (E and , respectively). The mean anomaly at epoch is simply a known reference that determines the systems initial condition.Before proceeding, let us review what we have done so far. First, we derived the second position of Ceres from our three original observations. This involved an approximation of the ratios of sectoral areas which were iteratively corrected by use of the factor G(r2). With the second position in hand, the first and third positions were obtained using the latest ratio approximation. From these points, we then derived a first approximation of Ceres orbit, which was also adjusted iteratively using the semi-orbital parameter h(S13). This is where we currently stand. Now that we have a preliminary orbit, we need not approximate the ratios of sectoral areas in the first step! Instead, we can obtain the ratios from our orbit, calculate P2, P1, and P3, recalculate the orbit, and so on until the area ratios converge and all three points lie on the same path. Let the finale begin!

gaussian orbit determination

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