SME
Orbital Mechanics · Problem type
Sphere of influence
Use this for “Earth orbits 1.496 × 10⁸ km from the Sun and has 3.003 × 10⁻⁶ of its mass. How far out does Earth’s pull take over from the Sun’s?”
This is an example. Type your own values to start a new problem.
rSOI (km)
=
a (km)
×
(
)
0.4
Formulas
rSOI
=
a
×
(m/M)
0.4
924,599.6536
=
1.496 ×
10
8
×
(3.003 ×
10
-6
)
0.4
Assumptions
•
Laplace’s sphere of influence: inside it, the craft’s orbit is worked round the planet alone.
•
M is the central body’s mass (the Sun, or the planet for a moon); m is far smaller, and its orbit is near circular.
•
Patched conics switch from one body’s pull to the other’s at rSOI.
Step-by-step
WE KNOW
a = 1.496 × 10⁸ km, m/M = 3.003 × 10⁻⁶
FIND
sphere of influence radius (rSOI)
Step 1 · Find sphere of influence radius (rSOI)
Use rSOI = a × (m/M)0.4
Laplace’s
radius:
the
orbit
radius
times
the
mass
ratio
to
the
power
2
5
=
0.4.
rSOI
=
1.496 ×
10
8
×
(3.003 ×
10
-6
)
0.4
rSOI
=
(1.496
×
10
8
)
×
0.00618
rSOI = 924,599.7 km
Check
Put the numbers back in. Both sides should match.
924,599.7
=
1.496 ×
10
8
×
(3.003 ×
10
-6
)
0.4
✓
Topic check
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