SME
Orbital Mechanics · Problem type
Speed anywhere on an ellipse (vis-viva)
Use this for “On an orbit from 6778 to 42,164 km, how fast is the craft 20,000 km from Earth’s center?”
Diagram
a
=
rₚ + rₐ
2
=
6,778 + 42,164
2
a = 24,471 km
v
=
√(μ(
2
r
−
1
a
))
v
=
√(398,600
×
(
2
20,000
−
1
24,471
))
v = 4.855 km/s
That is the vis-viva equation, energy per kilogram kept along the orbit.
Fastest at perigee, slowest at apogee: energy is shared between speed and height.
This is an example. Type your own values to start a new problem.
a (km)
=
rₚ (km)
+
rₐ (km)
2
(
v (km/s)
)
2
=
μ
2
r (km)
−
1
a (km)
Formulas
a
=
rₚ + rₐ
2
24,471
=
6,778 + 42,164
2
v
2
=
398,600
×
(
2
r
−
1
a
)
4.855
2
=
398,600
×
(
2
20,000
−
1
24,471
)
Assumptions
•
Two bodies only: Earth as a point mass, μ = 398,600 km³/s²; radii from its center.
•
The orbit’s energy per kilogram, ½v² − μ ÷ r = −μ ÷ 2a, is the same all the way round.
•
r lies between the perigee radius rₚ and the apogee radius rₐ.
Step-by-step
WE KNOW
r = 20,000 km, rₚ = 6,778 km, rₐ = 42,164 km
FIND
semi-major axis (a), speed there (v)
Step 1 · Find semi-major axis (a)
Use a = (rₚ + rₐ) ÷ 2
The major axis runs from perigee to apogee through Earth’s center, so it is rₚ + rₐ.
a
=
6,778 + 42,164
2
a
=
48,942
2
a
=
24,471
km
Step 2 · Find speed there (v)
Use v² = 398,600 × (2 ÷ r − 1 ÷ a)
Vis-viva:
set
1
2
v
2
−
μ
r
equal
to
the
orbit’s
energy
−
μ
2
a
and
solve
for
v
.
v
=
√(398,600
×
(
2
r
−
1
a
))
v
=
√(398,600
×
(
2
20,000
−
1
24,471
))
v
2
=
398,600
×
5.9135 ×
10
-5
v
2
=
23.571
v
=
4.855
km/s
Check
Put the numbers back in. Both sides should match.
24,471
=
6,778 + 42,164
2
✓
4.855
2
=
398,600
×
(
2
20,000
−
1
24,471
)
✓
Topic check
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