Determination of spiral orbits with constant tangential velocity - English

vT=r d
(θ) (4)
dt
Inserting the above equations (1) and (2) yields
vT=r 0( t
B
d
)
t0 dt (θ 0( t
A
) ) (5)
t 0
v T=r 0
t B
t 0
B A θ 0
v T= A r 0θ 0
t A−1
A (6)
t 0
A+ B−1
t
A+B (7)
t 0
In order to satisfy the required condition, namely the constancy of vT, the exponent of t must be
zero.
A+ B−1=0 (8)
Using this result, for a given exponent A, the exponent B can now be determined.
B=1−A (9)
In the above equations (1) and (2)
θ=θ 0( t
A
)
t 0
r=r 0( t
B
)
t 0
B can now be inserted
r=r 0( t
1− A
)
t 0
whereby the constraint ensures that r is not constant as well. Dissolved after time t this equation
gives
t=t 0( r
)
r0 (10)
(11)
(12)
1
1− A
(13)
and inserted into (1 / 10), finally yields the desired spiral
A
θ=θ 0( r 1−A ) ; A≠1 (14)
r 0
The resulting functions r(t) and θ(t) show for the following ranges different behaviors with
increasing t: