Page 158 - Intro to Tensor Calculus

153

~
where C is a vector constant of integration. The triple scalar product formula gives us
d~ρ
~ ~ ~ 2 ~
~ ρ · (V × h)= h · (~ρ × )= h = GM~ρ · b e ρ + ~ρ · C
dt
or
2
h = GMρ + Cρ cos φ (1.5.135)
~
where φ is the angle between the vectors C and ~. From the equation (1.5.135) we find that
ρ
p
ρ = (1.5.136)
1+ cos φ
2
where p = h /GM and = C/GM. This result is known as Kepler’s first law and implies that when < 1
the mass m describes an elliptical orbit with the sun at one focus.
We present now an alternate derivation of equation (1.5.130) for later use. From the equation (1.5.128)
we have
2
d~ρ d ~ρ d d~ρ d~ρ GM d~ρ GM d
2 · = · = −2 ~ ρ · = − (~ρ · ~ρ) . (1.5.137)
dt dt 2 dt dt dt ρ 3 dt ρ 3 dt
1
Consider the equation (1.5.137) in spherical coordinates ρ, θ, φ. The tensor velocity components are V = dρ ,
dt
2
3
V = dθ , V = dφ and the physical components of velocity are given by V ρ = dρ , V θ = ρ dθ , V φ = ρ sin θ dφ
dt dt dt dt dt
so that the velocity can be written
d~ρ dρ dθ dφ
~
V = = b e ρ + ρ b e θ + ρ sin θ b e φ . (1.5.138)
dt dt dt dt
Substituting equation (1.5.138) into equation (1.5.137) gives the result
" #
2 2 2
d dρ 2 dθ 2 2 dφ GM d 2 2GM dρ d 1
+ ρ + ρ sin θ = − (ρ )= − =2GM
dt dt dt dt ρ 3 dt ρ 2 dt dt ρ
which can be integrated directly to give

2 2 2
dρ 2 dθ 2 2 dφ 2GM
+ ρ + ρ sin θ = − E (1.5.139)
dt dt dt ρ
where −E is a constant of integration. In the special case of a planar orbit we set θ = π constant so that
2
the equation (1.5.139) reduces to

2 2
dρ 2 dφ 2GM
+ ρ = − E
dt dt ρ
(1.5.140)
2 2
dρ dφ 2 dφ 2GM
+ ρ = − E.
dφ dt dt ρ
Also for this special case of planar motion we have
d~ρ 2 dφ
|~ρ × | = ρ = h. (1.5.141)
dt dt
By eliminating dφ from the equation (1.5.140) we obtain the result
dt
2
dρ 2 2GM 3 E 4
+ ρ = ρ − ρ . (1.5.142)
dφ h 2 h 2