SME
Orbital Mechanics · Problem type
Gravity-gradient torque
Use this for “A satellite at 500 km has Iz = 1200 and Iᵧ = 800 kg·m², its z axis tilted 10° from the local vertical. Find the gravity-gradient torque.”
Chart
T(10) ≈ 0.0002514
This is an example. Type your own values to start a new problem.
r (km)
=
R
E
+
z (km)
Tₘₐₓ (N·m)
=
3μ
2
×
(
r (km)
)
3
×
|
Iz (kg·m²)
−
Iᵧ (kg·m²)
|
T (N·m)
=
Tₘₐₓ (N·m)
×
sin(2
×
θ
°)
Formulas
r
=
6,378
+
z
6,878 = 6,378 + 500
Tₘₐₓ
=
3
×
3.986 ×
10
14
÷
(2
×
(
r
× 1,000)
3
)
×
|Iz
−
Iᵧ|
7.3502 ×
10
-4
=
3
×
3.986 ×
10
14
÷
(2
×
(6,878 × 1,000)
3
)
×
|1,200
−
800|
T
=
Tₘₐₓ
×
sin(2
×
θ
°)
2.5139 ×
10
-4
=
7.3502 ×
10
-4
×
sin(2
×
10°)
Assumptions
•
Circular orbit; μ = 3.986 × 10¹⁴ m³/s² and Earth’s radius 6378 km, with r in meters in Tₘₐₓ.
•
θ is the tilt of the Iz axis from the local vertical, in one plane.
•
The torque pulls the long axis (the smaller inertia) toward the vertical; it peaks at 45°.
Step-by-step
WE KNOW
z = 500 km, Iz = 1,200 kg·m², Iᵧ = 800 kg·m², θ = 10°
FIND
orbit radius (r), largest torque (Tₘₐₓ), torque (T)
Step 1 · Find orbit radius (r)
Use r = 6,378 + z
Add
Earth’s
radius
to
the
altitude:
r
is
from
Earth’s
center.
r
=
6,378
+
500
r
=
6,878
km
Step 2 · Find largest torque (Tₘₐₓ)
Use Tₘₐₓ = 3 × 3.986 × 10¹⁴ ÷ (2 × (r × 1,000)³) × |Iz − Iᵧ|
Gravity
falls
off
as
1
r
2
,
so
the
near
end
of
the
body
is
pulled
harder;
the
difference
in
inertia
sets
the
twist.
Tₘₐₓ
=
3
×
3.986 ×
10
14
÷
(2
×
(6,878 × 1,000)
3
)
×
|1,200
−
800|
r
=
6,878
km
=
6.878 ×
10
6
m
|1,200 − 800| = 400
Tₘₐₓ
=
3
×
3.986 ×
10
14
÷
(2
×
(6.878 ×
10
6
)
3
)
×
400
Tₘₐₓ
=
1.1958 ×
10
15
6.5075 ×
10
20
×
400
Tₘₐₓ
=
7.3502 ×
10
-4
N·m
Step 3 · Find torque (T)
Use T = Tₘₐₓ × sin(2 × θ°)
The twist grows from 0 at no tilt to its largest at 45°, then falls to 0 at 90°.
T
=
7.3502 ×
10
-4
×
sin(2
×
10°)
T
=
(7.3502
×
10
-4
)
×
sin(20°)
T
=
(7.3502
×
10
-4
)
×
0.342
T
=
2.5139 ×
10
-4
N·m
Check
Put the numbers back in. Both sides should match.
6,878 = 6,378 + 500 ✓
7.3502 ×
10
-4
=
3
×
3.986 ×
10
14
÷
(2
×
(6,878 × 1,000)
3
)
×
|1,200
−
800|
✓
2.5139 ×
10
-4
=
7.3502 ×
10
-4
×
sin(2
×
10°)
✓
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