Gravitation and Cosmology Lecture 23: Solution of the Keppler problem Solution of the Keppler problem Perihelion precession 7KH HTXDWLRQ IRU WKH RUELW

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1 Solution of the Keppler problem Perihelion precession 7KH HTXDWLRQ IRU WKH RUELW RI D WHVW SDUWLFOH LQ WKH ILHOG RI D SRLQW JUDYLWDWLQJ PDVV 6FKZDU]VFKLOG PHWULF LV GU Gϕ =± *0 U ( P + - PU P U - %\ FKDQJLQJ YDULDEOHV WR X = ZH ILQG DQ HTXDWLRQ RI IRUP U GX Gϕ =± X + DX E + 0*X &OHDUO\ WKH H[SUHVVLRQ ZLWKLQ WKH VTXDUH URRW FDQ EH IDFWRUHG VLQFH LW LV FXELF LW KDV URRWV PRUHRYHU VLQFH WKH QXPEHU RI VLJQ FKDQJHV DPRQJ WKH FRHIILFLHQWV LV WKH URRWV DUH DOO UHDO ILQDOO\ WKH URRWV DUH DOO SRVLWLYH VLQFH LW VWDUWV RXW QHJDWLYH DW X DQG EHFRPHV SRVLWLYH IRU ODUJH X ZKHUHDV IRU X LW LV QHJDWLYH GHILQLWH JLYLQJ X + DX E + 0*X = X > X X X < α 0*X ZKHUH α= X < + X > $V WKH WHVW ERG\ SODQHW PRYHV DURXQG LWV RUELW WKH LQYHUVH UDGLXV X ZLOO UHPDLQ EHWZHHQ WKH OLPLWV X DQG X! H[SHFWHG WR EH FORVH WR WKHLU 1HZWRQLDQ YDOXHV EHFDXVH 0*X > LV YHU\ VPDOO 7KH DQJOH ϕ³phdvxuhg IURP X ³ZLOO LQFUHDVH IURP WR ZKDWHYHU YDOXH LW KDV ZKHQ X = X > 7KH HDVLHVW ZD\ WR FDOFXODWH WKH FKDQJH RI DQJOH XVHV WKH FKDQJH RI YDULDEOH X = X > X < FRVχ + X > + X < OHDGLQJ WR GX Gϕ = X > X < VLQχ Gχ Gϕ =± X > X < VLQχ X > + X < X > X < FRVχ LH Gχ Gϕ =± X > + X < 0* X > X < FRVχ $V WKH YDULDEOH χ FKDQJHV E\ WKH SODQHW PRYHV DURXQG RQFH LQ LWV RUELW 7KH 1HZWRQLDQ FDVH FRUUHVSRQGV WR UHSODFLQJ WKH ULJKW VLGH RI E\ XQLW\ JLYLQJ ϕ = χ = 2I FRXUVH X DQG X! PXVW EH UHSODFHG ZLWK WKHLU 1HZWRQLDQ YDOXHV DOVR +RZHYHU WKH FKDQJH LQ χ LQFOXGLQJ WKH FRUUHFWLRQV RI (LQVWHLQ V WKHRU\ RI JUDYLWDWLRQ LV! %

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4 Contour integration 1RZ ZH FRPH WR VLQJXODULWLHV 7KHVH DUH SRLQWV ZKHUH I] LV QRW DQDO\WLF 7KH\ FRPH LQ WKUHH IODYRUV R Poles near a simple pole a function behaves as f(z) = + g(z). Near a pole of order n the z z 0 R behavior is f(z) = + g(z) where n is a positive integer. n (z z 0 ) Essential singularities a function that has an infinite series of poles of orders 1, 2,... is said to have an essential singularity at z 0. An example is e 1 z = n=0 z n n!. Branch cuts these are like a continuous line of poles. For example, the function 1 f(z) = 0 du u z = log(1z) log(z) has a branch cut from 0 to 1 along the real axis. If we define f(z) to be real for negative real z, then obviously it is discontinuous across the real axis for 0 < Re(z) < 1. Another example is z 1 2. Whenever a function has a discontinuity across a line, it is impossible to define the derivative in the direction perpendicular to that line. So we cut the line of discontinuity out of the plane (that is the meaning of the dashed line in the figure above) and say the function is analytic in the cut plane. To specify such a function we have to specify its branch cuts and the direction we cut them out. For example, it would have been equally permissible in the above logarithmic case to run the cuts from 0 and from 0 along the real axis. That would define a different, but equally valid function. Generally we choose branch lines to suit our convenience. 3ROHV DQG HVVHQWLDO VLQJXODULWLHV DUH LVRODWHG³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±LLQWKH FRPSOH[ SODQH,Q WKLV FDVH ZH ZDQW D FRQWRXU WKDW LQFOXGHV WKH UHDO D[LV IURP ²5 WR 5 SOXV VRPH SDUW WKDW FORVHV WKH FRQWRXU DQG RQ ZKLFK WKH LQWHJUDO FDQ EH HDVLO\ HYDOXDWHG,Q WKLV FDVH WKH FKRLFH GLFWDWHG E\ H[SHULHQFH LV D VHPLFLUFOH LQ WKH XSSHU KDOI RI WKH FRPSOH[ SODQH DV VKRZQ EHORZ! (

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