1.1. MARS 3
Fig. 1.2. Apparen t motio n o f Mars relativ e t o th e Eart h
Figure 1.2 make s it clea r why the pre-Copernican, Earth-centere d preju -
dice made i t s o difficult t o predic t th e motio n o f the planet s an d t o develo p
a "Newtonia n Theory. " Fo r a segmen t o f tim e o n thi s orbit , everythin g i s
regular. Indeed , startin g a t th e poin t wher e th e loo p intersect s itself , Mar s
starts o n it s lon g journey movin g awa y fro m th e Eart h unti l eventuall y i t
is five half-astronomica l unit s away . (Thi s positio n correspond s t o wher e
Earth an d Mar s ar e o n opposit e side s o f th e Sun. ) Th e interesting , coun -
terintuitive actio n start s whe n Mar s return s t o begi n it s clos e approac h t o
the Earth . First , i t quickl y swoop s i n a radica l plung e towar d th e Earth .
But rathe r tha n colliding , Mar s suddenl y reverses direction t o swoo p out
a motio n suggestin g tha t th e physics—fo r som e strang e reason—suddenl y
changes t o a law of repulsion rathe r tha n attraction . Finall y Mar s change s
direction onc e more s o that i t ca n repea t it s lon g two-year journey .
Imagine th e difficult y i n determinin g th e appropriat e forc e law— a la w
that resemble s som e form o f attraction fo r mos t o f the journey onl y to sud -
denly becom e a la w o f repulsio n whe n Mar s approache s Earth , an d the n
reverts bac k int o a la w o f attraction . Othe r tha n resortin g t o ba d joke s
about th e annoyanc e o f Earthling's politic s o r their behavior , ho w does on e
explain th e sudde n repulsio n o f Mar s whe n i t start s approachin g Earth ?
In othe r words , th e chang e o f variable s fro m a Earth-centere d t o a Sun -
centered syste m make s a considerable difference : withou t it , i t i s difficult t o
example t o pu t lif e int o thos e mandator y review s o f trigonometry. Th e trigonometri c ver -
sion just use s doubl e angl e formulae ; e.g. , (3cos(7r£) , 3sin(7rt)) 2(cos(27r£), 2sin(27rt)) =
(2, 0) + p(cos(7rt), sin(7rt) ) wher e p 3 4 cos(7r£).
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