1.1. MARS 5
orbit o f the plane t relativ e t o th e Eart h i n the usua l comple x variabl e for m
of z(t) = r(t)e
l0^\
a wa y t o determin e whethe r th e orbi t i s movin g i n a
clockwise or counter-clockwise manne r (relativ e t o the Earth ) i s to examin e
the sig n o f 0'(t) .
The sign of
6f{t)
i s the imaginar y par t o f (I n zp{t))' = -f- y +i6'. Bu t
since
, z'
P
m(aa - 4e(
2
-")^)
(ln Zp{t)) = TP=
a-2e(^ W '
L4
it follow s fro m th e for m o f th e numerato r tha t th e sig n o f 9
f
mus t chang e
periodically wheneve r aa 4.
The reaso n thi s aa 4 inequality mus t hol d fo r al l o f th e planet s tha t
are farthe r fro m th e Su n tha n th e Eart h i s Kepler's third law. Thi s la w
asserts tha t
a V = k (1.5)
where k i s a constant . Consequently , aa (^)
1'2
i s a decreasin g functio n
of a: remember , a i s th e distanc e o f th e plane t t o th e Sun . Thus , fo r a
planet sufficientl y fa r fro m th e Sun , w e must expec t it s orbi t t o experienc e
loops whe n expresse d relativ e t o th e Earth . Accordin g t o Eq . 1.4, th e loo p
occurs wheneve r th e distanc e betwee n th e Eart h an d th e plane t decrease s
toward a (local ) minima l value. Bu t becaus e those far-out planet s take fro m
decades to a couple of Earth centurie s to circle the Sun,
3
it follows that thei r
apparent orbit s mus t exhibi t man y loops .
A natura l relate d question , whic h i s neede d fo r late r purposes , i s t o
determine ho w fa r a plane t mus t b e beyon d th e Eart h s o that it s apparen t
orbit ha s a loop . Usin g th e unit s o f th e Earth , a = 2 , a = 2 , we have tha t
k 32 for Eq . 1.5. Thus , a
3a2
= 32, or the crucia l parameter ha s the valu e
aa = [32/a]2 . Becaus e apparen t loop s occu r whe n aa 4 , i t follow s tha t
these loop s occu r whe n [32/a] 2 4 , or whe n a 2 . Restate d i n words ,
the apparent motion of any planet that is farther from the Sun
than the Earth has a loop.
Of course, this assertion holds for al l bodies governed by Newton's equation :
this fac t play s a ke y rol e i n th e discussio n abou t th e ring s o f Satur n give n
in th e las t sectio n o f this chapter .
Notice how this simple argument just describe s a circular uniform motio n
relative t o anothe r circula r unifor m orbit . Th e importanc e o f this commen t
3While Venu s take s onl y abou t 22 4 Eart h day s t o circl e th e Sun , Jupite r take s 433 2
(about 11. 9 Eart h years) , Satur n 10,760 (abou t 29. 5 years) , Uranu s 30,68 5 (abou t 8 4
years), Neptun e 60,190 (abou t 165 years), an d Plut o 90,80 0 day s (abou t 24 9 years) .
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