SME
Orbital Mechanics
Two-body problem
Use this for “How fast does a satellite circle Earth 400 km up, how long is one lap, and what speed escapes from there?”
Diagram
Orbital
speed:
v
=
√(
μ
r
)
=
√(
398,600
6,778
)
=
7.669
km/s
Period:
T
=
2π√(
r
3
μ
)
=
2π
×
√(
6,
778
3
398,600
)
=
5,550
s
(92.6
min)
Gravity
is
the
centripetal
force:
μ
r
2
=
v
2
r
,
so
the
satellite’s
own
mass
cancels.
It orbits 400 km above the surface (r − R, with R = 6,378 km).
This is an example. Type your own values to start a new problem.
r (km)
=
R
+
z (km)
v (km/s)
=
μ
÷
r (km)
T (min)
=
2π
(
r (km)
)
3
÷
μ
vesc
Escape speed
calculated · km/s
Formulas
r
=
6,378
+
z
6,778 = 6,378 + 400
v
=
√(
398,600
r
)
7.6686
=
√(
398,600
6,778
)
T
=
2π
×
√(
r
3
398,600
)
5,553.459
=
2π
×
√(
6,
778
3
398,600
)
vesc
=
√(2
×
398,600
r
)
10.8451
=
√(2
×
398,600
6,778
)
Formulas are worked in s. The step-by-step shows the conversions.
Assumptions
•
Earth is a point mass with μ = 398,600 km³/s²; its equatorial radius is 6378 km.
•
A circular orbit with no air drag; the satellite’s mass is negligible beside Earth’s.
•
Escape: the kinetic energy just cancels the potential energy, ½vesc² = μ ÷ r.
Step-by-step
WE KNOW
z = 400 km
FIND
orbit radius (r), circular speed (v), period (T), escape speed (vesc)
Step 1 · Find orbit radius (r)
Use r = 6,378 + z
Orbit radii are measured from Earth’s center: add its radius to the altitude.
r
=
6,378
+
400
r
=
6,778
km
Step 2 · Find circular speed (v)
Use v = √(398,600 ÷ r)
Gravity
is
the
centripetal
force:
μ
r
2
=
v
2
r
,
so
v
2
=
μ
r
.
v
=
√(
398,600
6,778
)
v
=
√
58.81
v
=
7.6686
km/s
Step 3 · Find period (T)
Use T = 2π × √(r³ ÷ 398,600)
One
lap
is
the
circumference
2πr
at
the
speed
√(
μ
r
).
T
=
2π
×
√(
6,
778
3
398,600
)
T
=
5,553.459
s
Step 4 · Find escape speed (vesc)
Use vesc = √(2 × 398,600 ÷ r)
Set
1
2
v
2
=
μ
r
:
the
total
energy
is
zero,
just
enough
to
leave
for
good.
vesc
=
√(2
×
398,600
6,778
)
vesc
=
√(
797,200
6,778
)
vesc
=
√
117.62
vesc = 10.8451 km/s
Step 5 · Convert the answers
Change each answer back to the unit you chose.
T = 5,553.459 s = 92.5576 min (1 min = 60 s)
Check
Put the numbers back in. Both sides should match.
6,778 = 6,378 + 400 ✓
7.6686
=
√(
398,600
6,778
)
✓
5,553.459
=
2π
×
√(
6,
778
3
398,600
)
✓
10.8451
=
√(2
×
398,600
6,778
)
✓
Topic check
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