PaperIII_63

# PaperIII_63 - MATHEMATICAL TRIPOS Part III Wednesday 3 June...

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MATHEMATICAL TRIPOS Part III Wednesday, 3 June, 2015 1:30 pm to 4:30 pm PAPER 63 PLANETARY SYSTEM DYNAMICS Attempt no more than THREE questions. There are FOUR questions in total. The questions carry equal weight. STATIONERY REQUIREMENTS SPECIAL REQUIREMENTS Cover sheet None Treasury Tag Script paper You may not start to read the questions printed on the subsequent pages until instructed to do so by the Invigilator. 2 1 A nearly edge-on axisymmetric circumstellar disk is observed with a mid-plane inclined Id ≈ 87◦ to the sky-plane. An orthonormal astrocentric reference frame [ X , Ŷ , Z] is defined with X and Ŷ in the sky-plane, X pointing toward North and Z toward Earth. The disk’s ascending node through the sky-plane (in which disk material approaches us) lies at a position angle Ωd = 45◦ from North. Another orthonormal astrocentric reference frame [x, ŷ, z] is defined with x and ŷ in the disk mid-plane with x in the sky- plane in the direction of the disk’s ascending node, at which location material moves in the ŷ direction. Provide an annotated sketch of this geometry, and determine the transformation matrix T that converts between locations in the two reference frames X and x, respectively, such that X = T x. A planet is also observed to orbit the star with an inclination to the sky-plane of Ip 88◦ . The planet’s ascending node through the sky-plane lies at a position angle Ωp = 49◦ from North. The planet’s ascending node through the disk mid-plane (that in which motion is in the positive z direction) is in the direction x′ at an angle Ωm from the x direction. The orthonormal astrocentric reference frame [x′, ŷ′ , z′ ] has ŷ′ in the planet’s orbital plane oriented in the direction of motion at the ascending node. Provide an updated sketch of this geometry. There are two ways of applying successive rotations to convert between the [ X , Ŷ , Z] and [x′ , ŷ′ , z′ ] reference frames. Describe how this can be used to determine both Ωm and the inclination of the planet’s orbit to the disk mid-plane Im as a function of Id, Ip, Ωp and Ωd, defining any additional angles needed for the transformation. Hence, or otherwise, show that cos Im = cos Id cos Ip + sin Id sin Ip cos (Ωp − Ωd), sin Ip sin (Ωp − Ωd)

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tan Ωm = cos Id sin Ip cos (Ωp − Ωd) − sin Id cos Ip .
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