SME
Orbital Mechanics · Problem type
Combined circularize-and-turn burn
Use this for “At apogee of a transfer orbit a craft moves at 1.618 km/s. One burn reaches a 3.075 km/s circle and turns the plane 28.5°. Find Δv.”
Diagram
v₂: 3.075 km/s at 28.5°, so v₂ = ⟨2.702, 1.467⟩.
−vₐ: 1.618 km/s at 180°, so −vₐ = ⟨−1.618, 0⟩.
Tip to tail: Δv = ⟨1.084, 1.467⟩
|Δv| = 1.824 km/s at 53.53°.
This is an example. Type your own values to start a new problem.
(
Δv (km/s)
)
2
=
(
vₐ (km/s)
)
2
+
(
v₂ (km/s)
)
2
−
2
×
vₐ (km/s)
×
v₂ (km/s)
×
cos(
Δi
°)
Δvₛₑₚ
Two separate burns
worked out (not typed)
Formulas
Δv
=
√(
vₐ
2
+
v₂
2
−
2
×
vₐ
×
v₂
×
cos(Δi))
1.8245
=
√(
1.618
2
+
3.075
2
−
2
×
1.618
×
3.075
×
cos(28.5°))
Δvₛₑₚ
=
v₂
−
vₐ
+
2
×
v₂
×
sin(Δi°
÷
2)
2.9708 = 3.075 − 1.618 + 2 × 3.075 × sin(28.5° ÷ 2)
Assumptions
•
One impulsive burn at apogee, which lies on the node line of the two planes.
•
vₐ and v₂ meet at the angle Δi, so Δv closes their triangle (law of cosines).
•
Turning where the craft is slowest, and in the same burn, costs least.
Step-by-step
WE KNOW
vₐ = 1.618 km/s, v₂ = 3.075 km/s, Δi = 28.5°
FIND
combined burn (Δv), two separate burns (Δvₛₑₚ)
Step 1 · Find combined burn (Δv)
Use Δv = √(vₐ² + v₂² − 2 × vₐ × v₂ × cos(Δi))
The
law
of
cosines
on
the
triangle
of
vₐ,
v₂
and
the
burn
between
their
tips.
Δv
=
√(
1.618
2
+
3.075
2
−
2
×
1.618
×
3.075
×
cos(28.5°))
Δv
=
√
2.6179 + 9.4556 − 2 × 1.618 × 3.075 × 0.87882
Δv
=
√
2.6179 + 9.4556 − 3.236 × 3.075 × 0.87882
Δv
=
√
2.6179 + 9.4556 − 9.9507 × 0.87882
Δv
=
√
2.6179 + 9.4556 − 8.7449
Δv
=
√
3.3286
Δv = 1.8245 km/s
Step 2 · Find two separate burns (Δvₛₑₚ)
Use Δvₛₑₚ = v₂ − vₐ + 2 × v₂ × sin(Δi° ÷ 2)
For
comparison:
circularize
first,
then
turn
the
plane
at
v₂
in
a
second
burn.
Δvₛₑₚ = 3.075 − 1.618 + 2 × 3.075 × sin(28.5° ÷ 2)
Δvₛₑₚ = 3.075 − 1.618 + 2 × 3.075 × sin(14.25°)
Δvₛₑₚ = 3.075 − 1.618 + 2 × 3.075 × 0.2462
Δvₛₑₚ = 3.075 − 1.618 + 6.15 × 0.2462
Δvₛₑₚ = 3.075 − 1.618 + 1.514
Δvₛₑₚ = 2.9708 km/s
Check
Put the numbers back in. Both sides should match.
1.8245
=
√(
1.618
2
+
3.075
2
−
2
×
1.618
×
3.075
×
cos(28.5°))
✓
2.9708 = 3.075 − 1.618 + 2 × 3.075 × sin(28.5° ÷ 2) ✓
Topic check
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