Notes_H%20BP

Notes_H%20BP - H1 f and g Functions E and H are most...

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SD f and g Functions E and H are most definitely useful for “time” relationships But they are also useful in other ways. new expressions for , rv Begin with the elliptic case () ˆˆ cos sin sin cos ra E e e b E p vr a E E e b E E p =− + == + ±± ± ( ) sin 1 cos dE r Mn tE e E nE e E n dt a = = ± ± H1
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2 ˆˆ sin cos a n abn vE e E p rr =− + Substitute into v equation () 0 0 00 0 0 cos sin ˆ cos sin ˆ Ee E a pv r pr na p EE er v ra n ⎧⎫ =+ ⎪⎪ ⎨⎬ ⎩⎭ Substitute , ep into original expressions H2 ( )( ) 0 0 0 2 0 0 0 sin 11 c o s sin 1 1 cos a rE E r t t v rn a E r E E v r −− ⎡⎤ + + ⎢⎥ ⎣⎦ +
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Example: 211 rf r g v =+ Do same in terms of θ ; Do same in hyperbolic orbits H3
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f and g Relationships Any conic () 0 00 0 0 0 0 0 0 11 c o s s i n 1 1c o s s i n 1 o s ( ) rr r v p p rv r vr v pr r p p θθ μ ∗∗ ⎧⎫ ⎡⎤ =− + ⎨⎬ ⎣⎦ ⎩⎭ + i Elliptic Orbits Hyperbolic Orbits H4 3 0 0 0 0 0 0 0 0 c o s s i n sin 1 1 cos aa rE E r t t E E E E v r a a vE E r E E v rr r ⎪⎪ =− − + − − − − + − 3 0 0 0 0 0 0 0 0 1 cosh 1 sinh sinh 1 cosh 1 r HH r tt v r a a vH H r H H v r +
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Notes_H%20BP - H1 f and g Functions E and H are most...

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