Notes_G%20BP

Notes_G%20BP - G1 Eccentric Anomaly E Additional variable...

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Additional variable defined for ellipse cos cos(180 ) cos cos cos cos aE a e r ae r ae r E a r e a θ =− =+ + = D OR G1 Eccentric Anomaly E 1 cos 1c o s pp r re e e = →= + cos ar E ae = cos r=a(1-e E)

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E obviously related to θ . How?? Previously cos cos ra E a e =− Identity 2 cos2 2cos 1 αα = 2 1 cos 2 E a e ⎛⎞ ⎜⎟ ⎝⎠ 2 2 cos cos (1 cos ) 2 E a e a e E + = () c o s aa e E aa e +− 2 2 cos )(1 cos ) 2 e E + Identity 2 1 2sin α 2 12 s i n c o s 2 E a e −= 2 2s i n c o s ( 1 c o s ) 2 E a e a e E = c o s e e + −+ 2 i n ( 1 ) ( c o s 1 ) 2 e E + 2 2 i n 2 )(cos 1) )(1 cos ) 2c o s 2 r ae E E r = G2 1 2 tan tan * θ 1+e E = 21 - e 2 Identity: 2 1c o s tan 21c o s E E E = +
Note: 1c o s e θ + cos e =− G3 At E = 90 D 222 2 22 2 (1 cos 90 ) ) ra e a bra e ba e = −= D 2 1 e =

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Equation of an ellipse All of these fundamental relationships are useful but one of the most important reasons to introduce E is to obtain a relation between position and time How? Various approaches – consider one CD = a cos E 22 2 1 (CD) (SD) 1 cos (SD) 1 xy ab aE + = + = + = G4 Kepler’s Equation Let x, y be point S on the ellipse Æ measured from center C SD?
Kepler’s Equation (Relation between position and time) Begin with some relationships that are already known 2 22 1c o s h p p r e d hr r dt μ θ = = + == ± Combine to eliminate r and h () 2 2 2 o s d dt p d p dt e = = + Rearrange

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Notes_G%20BP - G1 Eccentric Anomaly E Additional variable...

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