SME
Propulsion · Problem type
Propellant and burn time at low thrust
Use this for “A 1000 kg probe with an 1800 s, 339.8 mN thruster needs Δv = 2 km/s. How much propellant, and how long a burn?”
Diagram
Δv
=
1,800
×
9.81
×
ln(
1,000
892.9
)
Δv = 2 km/s
This is an example. Type your own values to start a new problem.
Δv (km/s)
=
Iₛₚ (s)
×
g
×
ln(
m₀ (kg)
÷
mf (kg)
)
ṁ (kg/s)
=
F (mN)
÷
(
Iₛₚ (s)
×
g)
tb (days)
=
(
m₀ (kg)
−
mf (kg)
)
÷
ṁ (kg/s)
mₚ
Propellant mass
calculated
Formulas
Δv
=
Iₛₚ
×
9.81
×
ln(
m₀
mf
)
2,000
=
1,800
×
9.81
×
ln(
1,000
892.9157
)
mₚ
=
m₀
−
mf
107.0843 = 1,000 − 892.9157
ṁ
=
F
Iₛₚ × 9.81
1.9243 ×
10
-5
=
0.3398
1,800 × 9.81
tb
=
mₚ
ṁ
5,564,728.2984
=
107.0843
1.9243 ×
10
-5
Formulas are worked in m/s, N, s. The step-by-step shows the conversions.
Assumptions
•
Steady thrust F for the whole burn, so ṁ stays fixed and tb = mₚ ÷ ṁ.
•
The rocket equation’s ideal Δv, with g = 9.81 m/s² in vₑ = Iₛₚ × g.
•
A slow spiral needs more Δv than an impulsive transfer: 400 km to GEO is 4.594 km/s, not 3.854.
Step-by-step
WE KNOW
Δv = 2 km/s, Iₛₚ = 1,800 s, m₀ = 1,000 kg, F = 339.8 mN
FIND
final mass (mf), propellant mass (mₚ), propellant flow (ṁ), burn time (tb)
Step 1 · Convert to the formula’s units
The formulas don’t work directly in the units you chose, so convert first (to m/s, N, s).
Δv = 2 km/s = 2,000 m/s (1 km/s = 1000 m/s)
F = 339.8 mN = 0.3398 N (1 N = 1000 mN)
Step 2 · Find final mass (mf)
Use Δv = Iₛₚ × 9.81 × ln(m₀ ÷ mf)
Undo
the
log:
divide
m₀
by
e
to
the
power
Δv
vₑ
.
mf
=
m₀
e
Δv ÷ (Iₛₚ × 9.81)
mf
=
1,000
e
2,000 ÷ (1,800 × 9.81)
2,000
1,800 × 9.81
=
0.11326
e
0.11326
=
1.1199
mf
=
1,000
1.1199
mf = 892.9157 kg
Step 3 · Find propellant mass (mₚ)
Use mₚ = m₀ − mf
The propellant is the mass the burn leaves behind.
mₚ = 1,000 − 892.9157
mₚ = 107.0843 kg
Step 4 · Find propellant flow (ṁ)
Use ṁ = F ÷ (Iₛₚ × 9.81)
F
=
ṁvₑ,
so
the
flow
is
the
thrust
over
vₑ
=
Iₛₚ
×
g.
ṁ
=
0.3398
1,800 × 9.81
ṁ
=
0.3398
17,658
ṁ
=
1.9243 ×
10
-5
kg/s
Step 5 · Find burn time (tb)
Use tb = mₚ ÷ ṁ
At
a
steady
flow
the
tank
empties
in
mₚ
ṁ
seconds.
tb
=
107.0843
1.9243 ×
10
-5
tb = 5,564,728.2984 s
Step 6 · Convert the answers
Change each answer back to the unit you chose.
tb = 5,564,728.2984 s = 64.4066 days (1 day = 86400 s)
Check
Put the numbers back in. Both sides should match.
2,000
=
1,800
×
9.81
×
ln(
1,000
892.9157
)
✓
107.0843 = 1,000 − 892.9157 ✓
1.9243 ×
10
-5
=
0.3398
1,800 × 9.81
✓
5,564,728
=
107.084306
1.9243402 ×
10
-5
✓
Topic check
See what you know: 5 questions on this topic
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