2 CHAPTER 1. INTRODUCTION
difference doe s i t mak e i f th e Su n revolve s abou t th e Earth , o r th e Eart h
about the Sun? Afte r all , whichever occurs, one forms the center of a circular
motion fo r th e other . Wh y shoul d w e care whic h i s which?
•arth, z E(t) = 2e
2nit
Mars, z M(t) = 3e
nit
Fig. 1.1 . Sun-Earth-Mar s coordinate s i n half-astronomica l unit s
1.1.1 Motio n o f Mar s
To explain the kinds of difficulties tha t ar e introduced b y an Earth-centere d
prejudice, star t wit h th e Su n a s the cente r o f our sola r system . A simplifie d
story ha s Mars approximately 3/ 2 time s (actually , abou t 1.524 times ) a s fa r
from th e Su n a s the Earth , an d Mar s takes approximatel y tw o years (abou t
687 Earth days ) t o complet e it s journey abou t th e Sun .
To keep everything simple , eliminate fractions b y replacing the standar d
astronomical uni t (th e distance between the Earth an d the Sun) with what I
call "half-astronomical " units . I n th e ne w system , whic h i s depicted i n Fig .
1.1, the Eart h i s two units fro m th e Sun , an d Mar s i s three. Usin g comple x
variables, a reasonabl e descriptio n o f th e motio n o f th e Eart h i s give n b y
zE[t) = 2e
2nit
whil e that o f Mars i s z
M
= 3e*
il.
Finding th e orbi t o f Mars relativ e t o th e Eart h no w i s simple; it i s
z(t) = z M(t) - z E(t) = 3e
nit
- 2e
2nit.
(1.1)
To describe thi s orbit , ad d an d subtrac t th e distanc e t o th e Su n t o obtai n
z{t) = 3e nit - 2e 2nit - 2 + 2 = 2 + e nit[3 - 2e nit - 2~ nit]
= 2 + [3-4cos(7r£)]e
7r^.
^ '
According to Eq. 1.2, th e graph of this equation, a s given in Fig. 1.2, depict s
the surprisingl y complicate d orbi t o f Mar s whe n viewe d relativ e t o tha t o f
the Earth : i t i s a limacon wit h a nicely define d loop.
2
2In m y introductor y calculu s courses , I ofte n us e th e trigonometri c versio n o f thi s
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