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CHAPTER 1. INTRODUCTION
even imagin e ho w Newton's law s of attraction coul d hav e bee n developed .
Incidentally, i t i s eas y t o observ e thi s retrograd e behavio r o f Mars . O f
course, th e chang e i n distanc e betwee n Eart h an d Mar s canno t b e detecte d
by th e untraine d nake d eye , bu t th e chang e i n direction—wher e Mar s ap -
pears t o b e movin g i n on e direction , stop s an d move s backwards , an d the n
stops agai n t o retur n t o it s origina l direction—i s quit e apparen t ove r th e
span o f severa l nights . Durin g thos e period s whe n Mar s approache s Eart h
to star t it s dippin g behavior , eve n a casua l observe r ca n notic e ho w a t a
fixed time eac h night th e positio n o f Mars swings to define , ove r a period of
days, a compresse d "Z. "
Fig. 1.3. Apparen t orbi t o f a planet 9 times farthe r fro m th e Su n
While the apparen t motio n o f Mars offers surprisin g behavior, th e orbit s
of the planets farther fro m th e Su n adopt a much more complicated appear -
ance wit h th e severa l loop s a s indicate d i n Fig . 1.3. Thi s figure depict s th e
apparent behavio r o f a plane t nin e A U awa y fro m th e Sun : a distanc e tha t
is a bi t shor t o f Saturn' s actua l orbit . Rathe r tha n developin g a compli -
cated versio n o f th e abov e description , a differen t elementar y approac h i s
described next .
1.1.2 Th e "fa r out " planet s
Consider th e circula r orbi t o f a far-ou t planet—Mars , Saturn , o r beyond
given b y zp(t)
aeanlt
wher e th e valu e o f a 3 defines th e distanc e fro m
the Su n i n ou r half-astronomica l units : th e a value s ar e discusse d below .
After expressin g thi s
z(t) = z P(t) - z E(t) = ae anlt - 2e 2m\ (1.3)
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