Page 157 - Intro to Tensor Calculus
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where p =2q and the angle φ is known as the true anomaly associated with the orbit. The quantity p is
called the semi-parameter of the conic section. (Note that when φ = π ,then ρ = p.) A more general form
2
of the above equation is
p 1
ρ = or u = = A[1 + cos(φ − φ 0 )], (1.5.130)
1+ cos(φ − φ 0 ) ρ
where φ 0 is an arbitrary starting anomaly. An additional symbol a, known as the semi-major axes of an
elliptical orbit can be introduced where q, p, , a are related by
p 2
= q = a(1 − ) or p = a(1 − ). (1.5.131)
1+
To show that the equation (1.5.128) produces a conic section for the motion of mass m with respect to
mass M we will show that one form of the solution of equation (1.5.128) is given by the equation (1.5.129).
To verify this we use the following vector identities:
~ ρ × b e ρ =0
2
d d~ρ d ~ρ
ρ
~ ρ × =~ ×
dt dt dt 2
(1.5.132)
d b e ρ
b e ρ · =0
dt
d b e ρ d b e ρ
b e ρ × b e ρ × = − .
dt dt
From the equation (1.5.128) we find that
2
d d~ρ d ~ρ GM
~
~ ρ × = ~ρ × = − ~ ρ × b e ρ = 0 (1.5.133)
dt dt dt 2 ρ 2
so that an integration of equation (1.5.133) produces
d~ρ
~
~ ρ × = h = constant. (1.5.134)
dt
~
~
~
The quantity H = ~ρ × mV = ~ρ × m d~ ρ is the angular momentum of the mass m so that the quantity h
dt
~
represents the angular momentum per unit mass. The equation (1.5.134) tells us that h is a constant for our
~
two body system. Note that because h is constant we have
~
d dV GM d~ρ
~
~
~
V × h = × h = − b e ρ × ~ρ ×
dt dt ρ 2 dt
GM d b e ρ dρ
= − b e ρ × [~ρb e ρ × (ρ + b e ρ )]
ρ 2 dt dt
GM d b e ρ 2 d b e ρ
= − 2 b e ρ × ( b e ρ × )ρ = GM
ρ dt dt
and consequently an integration produces
~
~
~
V × h = GM b e ρ + C