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16.522, Space Propulsion
Lecture 3
Prof. Manuel MartinezSanchez
Page 1 of
9
16.522, Space Propulsion
Prof.
Manuel MartinezSanchez
Lecture 3: Approximate
∆
V for LowThrust Spiral Climb
Assume initial circular orbit, at
co
0
v=v =
r
µ
.
Thrust is applied tangentially.
Call
F
a=
M
.
By conservation of energy, assuming the orbit remains nearcircular
v
r
⎛⎞
µ
⎜⎟
⎝⎠
±
,
d
a
dt
2r
r
µµ
±
2
dr
a
dt
r
2r
±
3
2
1
2
r
dr
a dt
2
µ
±
When we integrate,
b
t
0
a dt = V,
∆
∫
and so
b
t
1
1
22
0
r
=V
µ∆
()
0b
V=

rr
t
∆
or
final
co
c
V=v v
∆
(1)
The result appears to be trivial, but it is not. Notice that the “velocity increment” V
∆
is actually equal to the decrease
in orbital velocity. The rocket is pushing forward,
but the velocity is decreasing. This is because in a r
2
force field, the kinetic energy is
equal in magnitude but of the opposite sign as the total energy (potential = 
2
×
kinetic).
If thrust were applied opposite the velocity (negative a), the definition of V
∆
would
be
b
t
0
(a) dt
∫
, so the result in general is
c
V= v
∆∆
(2)
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View Full Document16.522, Space Propulsion
Lecture 3
Prof. Manuel MartinezSanchez
Page 2 of
9
For simplicity, assume now
F
a =
M
= constant, which is actually optimal for many
situations. Equation (1) can be recast as
0
=
a
t
rr
µµ
and solving for r,
00
22
co
0
r=
=
at
at
1
v
r
⎡⎤
⎢⎥
⎛⎞
⎛
⎞
⎜⎟
⎜
⎟
⎝⎠
µ
⎣⎦
(3)
This shows how the radial distance “spirals out” in time. In principle, this says r
→ ∞
at
co
v
t=
a
, a crude indication of “escape”. But of course, the orbit is no longer “near
circular” when approaching escape, so this result is not precise. One can get some
improvement for the estimation of escape V
∆
as follows.
The radial
velocity r
i
can be calculated from (3) by differentiation. Notice that this is
in the nature of an iteration, since r
i
was implicitly ignored in the energy balance
which led to (3). We obtain
0
co
3
co
2a
r
v
at
v
i
(4)
The tangential component v=
±r
θ
θ
i
is still approximated as the orbital velocity, i.e.,
co
0c
o
at
r= =

a
v 1

v
θ
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 Fall '05
 ManuelMartinezSanchez
 Propulsion

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