16.522, Space Propulsion Lecture 2
Prof. Manuel Martinez-Sanchez Page 1 of 19
16.522, Space Propulsion
Prof. Manuel Martinez-Sanchez
Lecture 2: Mission Analysis for Low Thrust
1. Constant Power and Thrust: Prescribed Mission Time
Starting with a mass
0
M , and operating for a time t an electric thruster of jet speed
c, such as to accomplish an equivalent (force-free) velocity change of V? , the final
mass is
dv dM
M=-c
dt dt
dM
dv = -c
M
if c=constant (consistent with constant power and thrust), then
f
M
v=c ln
M
0
-Vc
f0
M=Me
?
(1)
and the propellant mass used
V
-
c
P0
M=M 1-e
?
??
??
??
(2)
The structural mass is comprised of a part Mso which is independent of power level,
plus a part Pα proportional to rated power P, whereα is the specific mass of the
powerplant and thruster system. In turn, the power can be expressed as the rate of
expenditure of jet kinetic energy, divided by the propulsive efficiency:
2
1
P= mc
2η
i
(3)
and, since m
i
is also a constant in this case,
p
m=M t.
i
Altogether, then,
2P
sso
M
M=M + c
2t
α
η
(4)
The payload mass is
Lfs
M=M-M.
Combining the above expressions,
16.522, Space Propulsion Lecture 2
Prof. Manuel Martinez-Sanchez Page 2 of 19
()
2
-Vc -VcsoL
oo
MM c
=e - - 1-e
MMt
??
α
η
(5)
Stuhlinger
[1]
introduced a “characteristic velocity”
ch
2t
v=
η
α
(6)
whose meaning, from the definition of αis that, if the powerplant mass above were
to be accelerated by converting all of the electrical energy generated during t, it
would then reach the velocity
ch
v .
Since other masses are also present,
ch
v must clearly represent an upper limit to the
achievable mission V? and is in any case a convenient yardstick for both V? and c.
Figure 1 shows the shape of the curves of
Lso
o
M+M
M
versus
ch
c/v with
ch
Vv? as a
parameter. The existence of an optimum c in each case is apparent from the figure.
This optimum c is seen to be near
ch
v hence greater than V? . If
V
c
?
is taken to be a
small quantity, expansion of the exponentials in (5) allows an approximate analytical
expression for the optimum c:
2
OPT ch
ch
11V
cv-V-
224v
?
?? (7)
Figure 1 also shows that, as anticipated, the maximum V? for which a positive
payload can be carried (with negligible
so
M ) is of the order of
ch
0.8 v . Even at this
high V? , Equation (7) is seen to still hold fairly well. To the same order of
approximation, the mass breakdown for the optimum c is as shown in Figure 2.
The effects of (constant) efficiency, powerplant specific mass and mission time are all
lumped into the parameter
ch
v . Equation (7) then shows that a high specific impulse
sp
I = c gis indicated when the powerplant is light and/or the mission is allowed a long
duration. Figure 2 then shows that, for a fixed V? , these same attributes tend to
give a high payload fraction and small (and comparable) structural and fuel fractions.
Of course the same breakdown trends can be realized by reducing V? for a fixed
ch
v .
This regime was called quite graphically the “trucking” regime by Loh
[2]
. At the
opposite end (short mission, heavy powerplant) we have a low
ch
v , hence low
optimum specific impulse, and, from Figure 2, small payload and large fuel fractions.
This is then the “sports car” regime
[2]
.
References:
Ref. [1]: Stuhlinger, E. Ion Propulsion For Space Flight. New York: Mc Graw-Hill Book Co., 1964.
Ref. [2]: Loh, W. H. Jet, Rocket, Nuclear, Ion and Electric Propulsion Theory and Design. New York:
Springer-Verlag, 1968.
16.522, Space Propulsion Lecture 2
Prof. Manuel Martinez-Sanchez Page 3 of 19
We have, so far, regarded the efficiency η as a constant, independent of the choice
of specific impulse. This is not, in general, a good assumption for electric thrusters
where the physics of the gas acceleration process can change significantly as the
power loading (hence the jet velocity) is increased. For each thruster family
(resistojets, arcjets, ion engines, MPD thrusters) and for each fuel and design, one
can typically establish a connection between η and c alone. Thus, as we will see in
detail later, η increases with c in both ion and MPD thrusters, whereas it typically
decays with c for arcjets (beyond a certain c). In general, then, one needs to return
to Equation (5) with ()=cηη in order to discover the best choice of c in each case. It
is instructive to consider in some detail the particular case of the ion engine, both
because of its own importance and because relatively simple and accurate laws can
be obtained in that case.
Ion engine losses can be fairly well characterized by a constant voltage drop per
accelerated ion. If this is called? φ, and singly charged ions are assumed, the energy
spent per ion is
()
2
ii
1 2mc + m = ion mass; e = electron charge? φ ,
of which only
2
i
12m c is useful.
The efficiency is then
2
2
i
c
=
2e
c+
m
η
?φ
(8)
We should also include a factor of
0
1η ∠ to account for power processing and other
losses. We then have
2
0 22
L
c
=
c+v
ηη (9)
16.522, Space Propulsion Lecture 2
Prof. Manuel Martinez-Sanchez Page 4 of 19
16.522, Space Propulsion Lecture 2
Prof. Manuel Martinez-Sanchez Page 5 of 19
were
L
v is a “loss velocity”, equal to the velocity to which one ion would be
accelerated by the voltage drop .? φ Notice how this simple expression already
indicates the importance of a high atomic mass propellant; ? φ is insensitive to
propellant choice, and so
L
v can be reduced if
i
m is large. Equation (9) also shows the
rapid loss of efficiency when c is reduced below
L
v .
Using (9), we can rewrite (5) as
16.522, Space Propulsion Lecture 2
Prof. Manuel Martinez-Sanchez Page 6 of 19
V22V
--
soLL
cc
2
ooch
MMc+v
=e - - 1-e
MMv
??
??
??
??
(10)
where the definition of
ch
v (Equation (6)) is now made using
0
η instead of η. Once
again, only approximate expressions for
ch
Vv? are feasible for the optimum c and
mass fractions. Normalizing all velocities by
ch
v :
L
ch ch ch
vcv
x; v=; =
vvv
?
≡δ (11)
we obtain
2
2
OPT
2
vv
x = 1+ - - +...
2
24 1+
δ
δ
(12)
3
22soL
2
oo
MAX
MM v
+=1-21+ v+v- +.
MM
12 1+
??
δ
??
δ??
(13)
3
P
22
o
M v1v
= - +...
M24
1+ 1+
??
??
δδ
??
(14)
For =0,δ and neglecting the last term included in each case, we recover the simple
expressions of Equation (7) and Figure 2. The main effect of the losses ()δ can be
seen to be:
(a) An increase of the optimum c, seeking to take advantage of the higher efficiency
thus obtained.
(b) A reduction of the maximum payload,
(c) A reduction of the fuel fraction.
Both these last effects indicate a higher structural fraction, due to the need to raise
rated power to compensate for the efficiency loss. It is worth noting also that the
losses are felt least in the “trucking” mode (high
ch
v , i.e. light engine or long
duration).
2. The Optimum: Thrust Profile
As was mentioned, there is no a priori reason to operate an electric thruster at a
constant thrust or specific impulse, even if the power is indeed fixed. We examine
here a simple case to illustrate this point, namely, one with a constant efficiency as
in the classical Stuhlinger optimization, but allowing F, m
i
and c to vary in time if this
is advantageous. Of course these variations are linked by the constancy of the
power:
16.522, Space Propulsion Lecture 2
Prof. Manuel Martinez-Sanchez Page 7 of 19
() () () ()
2
11
P= m t c t = F t c t
22ηη
i
(15)
Consider the rate of change of the inverse mass with time:
22
1
d
1dM mM
=- =
dt dtMM
??
??
??
i
(16a)
Multiplying and dividing by
2
22
F=mc,
i
22
22
1
d
FaM
==
dt 2 P
Mmc
??
??
??
η
i
(16b)
where a=FM is the acceleration due to thrust.
Integrating,
t
2
f0 0
11 1
-= adt
MM 2Pη
∫
(17)
On the other hand, the mission V? is
t
0
V= a dt?
∫
(18)
and is a prescribed quantity. We wish to select the function a(t) which will give a
maximum
f
M (Equation 17) while preserving this value of V? . The problem reduces
to finding the shape of a(t), whose square integrates to a minimum while its own
value has a fixed integral. The solution (which can be found by various mathematical
techniques, but is intuitively clear) is that a should be a constant.
Using this condition, (17) and (18) integrate immediately. Eliminating a between
these, we obtain
f
2
0 0
M 1
=
M MV
1+
2tP
?
η
(19)
The level of power is yet to be selected; it will determine the average specific
impulse, and it is to be expected that an optimum will also exist.
Using
fLs
M=M+M
and
16.522, Space Propulsion Lecture 2
Prof. Manuel Martinez-Sanchez Page 8 of 19
s
M
P=
α
and introducing the characteristic velocity (Equation 6), we rewrite (19) as
()()
sL
2
00
s0 ch
MM 1
=-1
MM
MM + Vv
??
??
?
??
(20)
and select the value of
s
0
M
M
that will maximize
L
0
M
M
. This is easily found to be
s
0chch
OPT
M VV
=1-
Mvv
?? ? ???
?? ? ?
?? ? ?
(21)
which, when used back in (21) gives
2
L
0ch
MAX
M V
=1-
Mv
?? ? ?
?
?? ? ?
?? ? ?
(22)
and then
P
0ch
OPT
M V
=
Mv
?? ?
??
??
(23)
These are, within the assumptions, exact expressions. They are to be compared to
the approximate expressions in Figure 2 or Equation (12)-(14) with
s0
0
M
=0, =0
M
δ
which were found to apply when c, and not a, was assumed constant. Clearly, the
difference is noticeable only for
ch
v= Vv? near unity (its highest value), and is
negligible for smaller values.
It is of some interest to inquire at this point how the jet velocity c should vary with
time in order to keep the acceleration constant.
We have
2
2
mc m a
a= = Mc= Mc
M2PM η
ii
where (16 a, b) have been used. Hence,
2P1
c=
aM
η
16.522, Space Propulsion Lecture 2
Prof. Manuel Martinez-Sanchez Page 9 of 19
and since, by (16b),
1
M
varies linearly with time, so will c. At the final time, when
f
M=M ,
2
schs
f
fsL
f
2M v M2P
c= = =
V
aM V M +M
M
t
ηαη
?
?
and, from (21), (22),
s
sL ch
M V
=
M+M v
?
so that
fch
c=v (24)
The rate of change of c follows from that of 1M(Equation 16) as
1
d
dc 2 P M
==a
dt a dt
??
??
η
??
(25)
so that, altogether, if t'represents some intermediate time, while t is the final time
(used in
ch
v ),
() ( )
ch
ct'=v -at-t' (26)
This varies between
ch
c=v - V? at t' = 0
and
ch
c=v at t' = t .
The approximate result
OPT ch
1
cv-V
2
?? found when c was constrained to remain
constant is therefore quite reasonable. Notice that (26) implies a constant absolute
velocity of the exhaust gas, at the value
()
abs ch
c=v-V-V0?
Alternative Derivation with Variable Specific Impulse
This is a more general treatment than that in pp. 15-19 of the Notes, and can be
extended to more complicated situations, like non-constant efficiency. It also
introduces some elements of Calculus of Variations, which is of general utility, and
has many commonalities with Optimal Control theory.
Decision
#2
τ 2 1 Stage 1
v
ch
16.522, Space Propulsion Lecture 2
Prof. Manuel Martinez-Sanchez Page 10 of 19
We wish to minimize
ft
0
F
V= dt
m
??
?
∫
??
??
(1)
subject to a given power (constant in time)
2
1mc 1Fc
p= =
22ηη
i
(2)
and to
dm
m=-
dt
i
(3)
write (1) as
ft
0
2P
V= dt
mc
η
?
∫
(4)
which eliminates thrust.
Next, treat (3) as a dynamic constraint, and append it to the cost through a time-
dependent Lagrange multiplier (t).λ
Define the Hamiltonian
2P dm
H= - m+
mc dt
η ??
λ
??
??
i
(5)
or, using (2),
2
2P 2P dm
H= - +
mc dtc
ηη??
λ
??
??
(6)
and minimize (unconstrained) the integral
ft
0
Hdt
∫
.To do this, perturb about the
optimum solution:
fftt
00
HH Hdm
Hdt = c + m+ dt = 0
dmcm dt
dt
??? ????
δ δδδ
∫∫
????
?? ??
????
?
??
??
()
d
m
dt
δ
Integrate last term by parts:
16.522, Space Propulsion Lecture 2
Prof. Manuel Martinez-Sanchez Page 11 of 19
f
f
t
t
0
0
HHdH H
c+ - m dt+ m =0
dm dmcmt
dt dt
????
?? ? ?
????
?? ??
?? ? ?
δδδ
∫
???? ??
??
?? ????
?? ??? ?
????
(17)
For optimality, we want to impose
H
=0
c
?
?
(18)
Hd H
=
dmmdt
dt
??
??
??
?
??
?
??
(19)
and at the ends, we say
( )
0
mt=m (fixed),
so
()
0
m=0,δ
and also
ffm(t) = m (fixed, since we minimize V? between given masses),
so, again,
()
ft
m=0δ .
Equation (18) gives in general (assuming
( )=cηη
)
23 2
2P 2 P 2P 2P
-+2 +- =0
mc cmc c c
ηη ?η??
λλ
??
?
??
or
12 c ln
-+ + - =0
mc m c
λ?η??
λ
??
?
??
(20)
For example, if
2
0 22
2
c
=,
c+v
ηη
16.522, Space Propulsion Lecture 2
Prof. Manuel Martinez-Sanchez Page 12 of 19
()
2
2
22
2
2v ln
=
c
cc+v
?η
?
Here we take the simple case where η = constant, so (20) gives
c=2 mλ (21)
From (19),
2
2P d
-+=0
dtmc
ηλ
(22)
and using (21),
3
dP
=
dt m
λη
λ
(24)
We also can substitute (21) into (3) to get
22
dm P
=-
dt 2m
η
λ
(25)
Divide (24) by (25):
d
=-2
dm m
λλ
(26)
which integrates to
2
m=Aλ (27)
(A = undetermined constant). The value of A can be easily related to the optimized
V? from Equation (4):
fftt
f2
00
2P P P
V = dt = dt = t
mc Am
ηηη
?
λ
∫∫
f
Pt
A=
V
η
∴
?
(28)
To complete the time integration, go back to (24):
1
2
33
22
dPP
== ;
dt
AA
λη η
λ
??
λ
??
λ
??
-1
2
3
2
P
d= dt
A
η
λλ
16.522, Space Propulsion Lecture 2
Prof. Manuel Martinez-Sanchez Page 13 of 19
and integrating,
1
2
3
2
P
2= t+B
2A
η
λ
2
3
2
PB
=t+
2
2A
??
η
λ
??
??
(29)
and from (27),
1
2
3
2
A
m=
PB
t+
2
2A
η
(30)
The constants A and B can now be related to m
0
and m
f
:
1
2
0
2A
m;
B
=
1
2
0
f
ff f
32 2
2
0
mA1
m= = =
Pt Pt m PtB1
++1+
2m2A 2A
2A
ηη η
(31)
Using now (28),
2
0f 0f 0
22
f
f
mPt mPt mV
==
2Pt2A
Pt
2
V
ηη?
η
η??
??
?
??
(32)
We now introduce the specific mass of the power/propulsion system
PP
M
,
P
α≡ and
the Characteristic Velocity
f
ch
2t
ν =,
η
α
to rewrite (32) as
2
0f 0
2
ch
pp
mPt m V
=
m ν2A
η ?
??
??
??
We also remember now that the final mass m
f
contains payload Lm ,
power/propulsion mass, and random dry mass
so
m , and write (31) as
0
Lsop 2
0
pp ch
m
m+m +m =
m V
1+
m ν
??
?
??
??
(33)
or
16.522, Space Propulsion Lecture 2
Prof. Manuel Martinez-Sanchez Page 14 of 19
pp
Lso
2
00
pp
0ch
m
m+m 1
=-1
mm
m
V
+
m ν
??
??
??
?
??
??
????
(34)
At this point we have the optimum time profiles, but we can do better by also
selecting the optimum power level, which amounts to selecting the optimum specific
impulse as well. We allow
pp
0
m
x=
m
to vary in (34) and maximize
Lso
0
m+m
=y
m
.
Using
ch
V
= ν
ν
?
(35)
We had
2
x
y= -x
x+ν
()
2
2
2
dy x +ν -x
=-1=0
dx
x+ν
()
2
22
ν =x+ν
( )
2
x=ν - ν = ν 1-ν
or
pp
0chch
OPT
m
VV
=1-
m νν
?? ??
??
?? ??
????
(36)
Putting
2
x=ν - ν intoy ,
()
2
2
OPT
ν - ν 1
y= -ν+ν = ν 1-ν -1
νν
??
??
??
()
2
OPT
y=1-ν
or
2
Lso
0ch
OPT
m+m V
=1-
m ν
?????
????
????
(37)
and then the propellant fraction follows from
ppp
LS0
000
mm
m+m
=1-
m
?
16.522, Space Propulsion Lecture 2
Prof. Manuel Martinez-Sanchez Page 15 of 19
p
0ch
OPT
m
V
=
m ν
??
?
??
??
(38)
Note these are exact formulas, no trailing terms. The specific impulse now follows
from (21):
2
00ff
2
00 f
mmVt2Pt 2Pt2A
c=2 m= = = 1+
mmVmmV 2Pt
???ηη
λ
??
??η
??
f
0f
2Pt t
c= + V
mV t
η
?
?
and if we use optimum power,
2
pp
ch
ch
0f chf
m
ν tVt
c= + V=ν 1- + V
mVt ν t
???
??
??
?
??
ch
f
t
c=ν -V1-
t
??
?
??
??
(39)
so that c increases linearly from
ch
V at t=0 to
ch
V-V? at
f
t .
Optimum Mission Time
So far, t has been a free parameter, and we have found the mission performance
(payload fraction) to improve with large t. But time has some costs associated with
it, so increasing t is not necessarily desirable. These “costs” of time including capital
immobilization, personnel costs during the long thrusting period, loss of opportunity,
etc. There are several simple analytical ways to penalize long t choices. We select
here one that maximizes the “Transportation Rate” M
L
/t. This makes most sense
when there is a sequence of identical flights, each delivering payload M
L
in time t,
but it can serve as a crude indicator even for a single mission.
We also assume constant thrust, constant power (for simplicity of operation), and we
include in the analysis the effect of a variable efficiency
()
0
22
L
=
1+c v
η
η ,
with
LLOSi
v=2ev m? , as in ion engines with constant ion cost
LOSS
?v .
We had for this case a payload ratio (optimized with respect to specific impulse)
16.522, Space Propulsion Lecture 2
Prof. Manuel Martinez-Sanchez Page 16 of 19
22
LL
ochchch
Mvvv
=1-2 1+ +
vv
??????
????
????
(1)
where
0
ch
2t
v=
η
α
(α=specific mass of the Power/Propulsion system)
The thrusting time t is buried in v
ch
. To bring it out more explicitly, let us define a
reference time
2
*
V
t=
2
0
α ?
η
(2)
which depends on specified quantities only, and then a non-dimensional time
*
t
t
τ≡ (3)
From the definition of v
ch
, then,
2
ch
*2
2t vt
== =
VtV
0
η ??
τ
??
?α ?
??
(4)
and so (1) becomes
2
LL
o
Mv2z1
=1- 1+ + ; z
MV
≡
ττ ?
τ
(5)
A normalized transportation rate is now defined as
LO
*
MM
tt
ψ≡ (6)
or, from (5),
2
32
2
12 z 1
=- 1+ +ψ
τττ
τ
(7)
To maximize ,ψ set =0
1
? ψ
??
?
??
τ
??
.
After some re-grouping, this leads to the equation for
OPT
τ :
22
(+2) +z =3+4zττ τ (8)
16.522, Space Propulsion Lecture 2
Prof. Manuel Martinez-Sanchez Page 17 of 19
A convenient way to view this
OPT
(z)τ dependence is to define an intermediate
parameter u=
2
+zτ ; we then obtain the parametric representation (for u≥2)
()
OPT
2
OPT
2u 2u-1
=
1+u
z= u -
?
τ
?
?
?
τ
?
(9)
A good analytical approximation (valid for z=0, asymptotic for z >>1) is
OPT
6
2+4z+
z+3
τ null (10)
In particular, for a constant-efficiency model (v
L
=0, z=0) we see that
OPT
=4τ ,
or
2
OPT
0
2V
t =
α ?
η
.
For other values of v
L
(or z), the results are shown in the attached graphs.
16.522, Space Propulsion Lecture 2
Prof. Manuel Martinez-Sanchez Page 18 of 19
22
(+2) +z =3+4zττ τ
6
4z + 2 + +...
z+3
τ null (z>>1)
2
4+5z +...τ null (z<<1)
16.522, Space Propulsion Lecture 2
Prof. Manuel Martinez-Sanchez Page 19 of 19