Kepler's Equation - Planet Orbit Simulation

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2016-01-21 22:36

This example demonstrates the elliptical orbit of planets around a star in Kepler's problem. Key math concepts include ellipses, vectors, and parametric equations. Construction highlights involve GeoGebra commands like Intersect, Vector, Sequence, CurveCartesian, Tangent, Point, UnitVector, Segment, Angle, Mirror, and Circle. Suitable for high school and above students to explore the intersection of celestial mechanics and analytical geometry.

GeoGebra Commands & Parameters

Intersect(M, r_1, (4.23825, 4.95)) → P
Vector(F, S) → r
Sequence(Intersect[M, y = k + α, (8.650787970620252, 8.091592653589792)], k, M((-π)), π, π / n) → lista1, {(2.3676026634416982, 1.808407346410207), (2.449702249060899, 1.9393070403097816), (2.5297247323846306, 2.070206734209356), (2.6080060986147893, 2.2011064281089308), (2.6848370299577757, 2.3320061220085053), (2.7604734760594787, 2.46290581590808), (2.8351446991867824, 2.5938055098076545), (2.9090596643290274, 2.724705203707229), (2.982412175772331, 2.8556048976068036), (3.0553851793494524, 2.986504591506378), (3.1281544914832486, 3.117404285405953), (3.200892166754764, 3.248303979305528), (3.2737696802010956, 3.3792036732051027), (3.3469610794056677, 3.5101033671046773), (3.4206462639082487, 3.6410030610042523), (3.4950145584483234, 3.7719027549038273), (3.5702687857377926, 3.902802448803402), (3.64663, 4.0337021427029764), (3.724343538406737, 4.164601836602551), (3.8036860874724825, 4.295501530502126), (3.8849754904758655, 4.426401224401701), (3.968583263639305, 4.557300918301276), (4.054951838660759, 4.68820061220085), (4.144618643036916, 4.819100306100425), (4.23825, 4.95), (4.336696642483753, 5.080899693899575), (4.441065712082371, 5.211799387799149), (4.552857613917095, 5.342699081698724), (4.674182600485173, 5.473598775598299), (4.808164874285426, 5.604498469497874), (4.959748271863796, 5.7353981633974485), (5.137505217425113, 5.866297857297023), (5.358408339568247, 5.997197551196598), (5.663425561441311, 6.128097245096173), (6.163388793632297, 6.258996938995748), (6.749761486238929, 6.389896632895322), (7.10873, 6.520796326794897), (7.353625392748443, 6.651696020694471), (7.5446176290380125, 6.782595714594047), (7.704584159483364, 6.913495408493621), (7.844358061123464, 7.044395102393196), (7.969923782476356, 7.1752947962927704), (8.08495, 7.306194490192345), (8.191861777169377, 7.43709418409192), (8.29235, 7.567993877991494), (8.38765, 7.698893571891069), (8.478696976804143, 7.829793265790643), (8.566215867907562, 7.960692959690218), (8.650787970620252, 8.091592653589792)}
CurveCartesian((a * (cos(θ + α) - e)), (a * sin(θ + α)), θ, 0, (2 * π)) → circonferenza
CurveCartesian((a * (cos(θ + α) - e)), (b * sin(θ + α)), θ, 0, (2 * π)) → ellisse
Sequence(((a * (cos(x(Element[lista1, i])) - e)), (b * sin(x(Element[lista1, i])))), i, 1, (2 * n)) → lista2, {(-3.03025477933577, 0.8387929755472369), (-3.1400826497369243, 0.7655928131275436), (-3.237153034525207, 0.689276997263195), (-3.3219768039635804, 0.6103495057588675), (-3.3949762276704902, 0.5292463728314193), (-3.4564978564100253, 0.44635149298752586), (-3.506822283980429, 0.3620087841940269), (-3.5461716149342046, 0.27653169127747423), (-3.574715033292627, 0.1902109182558022), (-3.592572872794273, 0.10332088258225572), (-3.5998194185167365, 0.01612530918828291), (-3.5964845980589435, -0.07111771858567294), (-3.582554654467923, -0.15815098841653455), (-3.5579718373530884, -0.24471342890732778), (-3.5226330960393994, -0.3305351903882906), (-3.4763877034169615, -0.4153322851977964), (-3.419033669658049, -0.4988005043242579), (-3.350312810689503, -0.5806081495337767), (-3.2699038723629243, -0.6603874472945648), (-3.177413963573412, -0.7377231645465), (-3.0723666634016, -0.8121386008935358), (-2.954186847984302, -0.8830764820040689), (-2.8221804050246044, -0.9498731603359478), (-2.6755071075329973, -1.0117224946558032), (-2.513145158542967, -1.067623403169223), (-2.333833333890631, -1.1163045774812292), (-2.1360130605319165, -1.1561005144960859), (-1.917711082655122, -1.1847622041848394), (-1.6763941708551329, -1.1991242700560139), (-1.408740929367117, -1.194500392818785), (-1.110311031350905, -1.1634748373016144), (-0.7751466491924559, -1.0931889598226665), (-0.3959746085831546, -0.9581838176761618), (0.02803605869066228, -0.697007527202591), (0.38566595022324557, -0.1434122180452047), (0.18622732198341652, 0.5397972800153613), (-0.2436864758404762, 0.881900734042959), (-0.6405237214720654, 1.0528940580795325), (-0.9910942093230891, 1.1430328716659222), (-1.3023153088260755, 1.1866331264846124), (-1.5807531513863995, 1.199944432536207), (-1.8313651254718464, 1.1919434820231798), (-2.0578405639079027, 1.1681342073985237), (-2.2629756788673623, 1.1321513899311033), (-2.44892535038933, 1.0865345230630672), (-2.617390049591666, 1.0331361455863752), (-2.769732805543203, 0.9733555665372081), (-2.9070530010742672, 0.9082843623325187)}
Sequence(((a * (cos(k + α) - e)), (a * sin(k + α))), k, 0, (2 * π), π / n) → lista3, {(-1.1292371140910977, -1.9438061388036416), (-0.8795469421281134, -1.8657297208875265), (-0.6421839217326413, -1.7557301501690221), (-0.42120939999311036, -1.6156895498007644), (-0.22040431224219414, -1.4480040532816427), (-0.043204489205002305, -1.2555428059483045), (0.10735813101722802, -1.0415988731029606), (0.2287073802486499, -0.8098328947530594), (0.3187669391905452, -0.5642104510458367), (0.3759958638201917, -0.3089342100925279), (0.3994149513703198, -0.04837201915179305), (0.3886234947599496, 0.2130178304478864), (0.3438061388036415, 0.47076288590890225), (0.26572972088752644, 0.7204530578718866), (0.15573015016902114, 0.9578160782673603), (0.015689549800763203, 1.178790600006891), (-0.15199594671835737, 1.379595687757806), (-0.3444571940516956, 1.5567955107949978), (-0.5584011268970395, 1.707358131017228), (-0.7901671052469406, 1.8287073802486498), (-1.0357895489541633, 1.9187669391905453), (-1.2910657899074702, 1.9759958638201915), (-1.5516279808482052, 1.9994149513703199), (-1.8130178304478846, 1.98862349475995), (-2.0707628859089002, 1.943806138803642), (-2.3204530578718865, 1.8657297208875268), (-2.557816078267357, 1.755730150169023), (-2.7787906000068867, 1.6156895498007666), (-2.9795956877578034, 1.4480040532816454), (-3.1567955107949968, 1.255542805948306), (-3.307358131017226, 1.0415988731029637), (-3.428707380248648, 0.8098328947530645), (-3.5187669391905443, 0.5642104510458402), (-3.575995863820191, 0.3089342100925317), (-3.5994149513703197, 0.048372019151798624), (-3.5886234947599505, -0.2130178304478791), (-3.5438061388036433, -0.4707628859088968), (-3.465729720887528, -0.7204530578718831), (-3.355730150169025, -0.9578160782673538), (-3.2156895498007687, -1.1787906000068837), (-3.048004053281648, -1.3795956877578006), (-2.855542805948309, -1.5567955107949942), (-2.641598873102967, -1.7073581310172243), (-2.409832894753068, -1.8287073802486462), (-2.164210451045844, -1.918766939190543), (-1.9089342100925353, -1.9759958638201907), (-1.6483720191518023, -1.9994149513703197), (-1.3869821695521247, -1.9886234947599508), (-1.1292371140911068, -1.9438061388036438)}
Tangent(S, ellisse) → Tangente
Point(Tangente) → V'
UnitVector(Vector[S, V']) → u
Segment(F, Translate[F, r]) → r_{modulo}
Angle(r_{modulo}, Tangente) → β
Vector(S, Translate[S, Vector[(L / ((r_{modulo} * sin(β))) * u)]]) → p
Mirror(F, C) → F'
Mirror(F', Tangente) → F''
Circle(F, F'') → d