Alternative Designs – Lunar Transit – Trajectory Alternatives
Section A-8.2.3
Bielliptic Transfer
While performing preliminary analysis on various methods for lunar transfer, we consider
a bielliptic transfer. For this analysis, we assume a two-body problem with point mass
central bodies, and we consider a coplanar transfer between a circular low-Earth orbit
(LEO) of 200 km altitude and a low lunar orbit (LLO) of 110 km altitude. We analyze
this problem employing the method of patched conics, which accounts for only Earth’s
gravity in Earth vicinity and only the Moon’s gravity in its vicinity.
Overview of Bielliptic Transfer. A bielliptic transfer involves three impulse burns for a
total transfer of 360o. Figure A-8.2.3-1 created in STK shows an example of the transfer
shape.
Fig. A-8.2.3-1: Example of Bielliptic Transfer
(Levi Brown)
Burn 1 accelerates the spacecraft from the initial parking orbit (periapsis of 1st transfer
arc) to some intermediate radius r (apoapsis of 1st transfer arc). Burn 2 accelerates the
spacecraft again, so the resultant periapsis of the 2nd transfer arc equals the final orbit
radius. For a lunar transfer, the final orbit radius is the Moon’s semi-major axis. Burn 3
decelerates the spacecraft to circularize the orbit at the final orbit radius.
Author: Levi Brown
Alternative Designs – Lunar Transit – Trajectory Alternatives
Section A-8.2.3
We perform an analysis of a bielliptic transfer and compare the results to a Hohmann
transfer. Per standard orbit mechanics, a three-burn bielliptic transfer actually costs less
than a two burn Hohmann transfer under certain conditions. The bielliptic transfer
reduces cost when the ratio between the final orbit and initial orbit radii exceeds 15.58.
(Howell, 2008) Because the spacecraft has a velocity near zero at radii far from Earth,
the change in velocity at the intermediate radius r requires a small amount of propellant.
We perform this analysis by running the Matlab code Bi_Elliptic.m. We define the
following constants.
μEarth = 398600.4418 km3/s2
μMoon = 4902.7854 km3/s2
rEarth = 6378.14 km
rMoon = 1737.1 km
aMoon = 384400 km
Finding ΔV1. We find the velocity of the spacecraft in a circular parking orbit.
𝜇𝐸𝑎𝑟𝑡ℎ
𝑉𝑝𝑎𝑟𝑘 = √
𝑟𝑝𝑎𝑟𝑘
(A-8.2.3-1)
We then find the conditions necessary to transfer to an intermediate radius. For the first
transfer arc, we define the periapsis and apoapsis radii as follows.
𝑟𝑝 = 𝑟𝑝𝑎𝑟𝑘
(A-8.2.3-2)
𝑟𝑎 = 𝑟
(A-8.2.3-3)
We calculate semi major axis (a), which is required to find the eccentricity for this orbit.
Author: Levi Brown
Alternative Designs – Lunar Transit – Trajectory Alternatives
𝑎𝑡1 = 1⁄2 (𝑟𝑝 + 𝑟𝑎 )
𝑒𝑡1 = 1 −
Section A-8.2.3
(A-8.2.3-4)
𝑟𝑝
𝑎
(A-8.2.3-5)
We find the time of flight by calculating half the orbit period for an elliptical orbit.
𝑡𝑓𝑙𝑖𝑔ℎ𝑡 = 1⁄2 ∗ 2𝜋√
𝑎3
𝜇𝐸𝑎𝑟𝑡ℎ
(A-8.2.3-6)
Applying the energy equation, we determine the departure velocity required to reach the
desired apoapsis.
𝑉𝑑𝑒𝑝𝑎𝑟𝑡1 = √2 (
𝜇𝐸𝑎𝑟𝑡ℎ 𝜇𝐸𝑎𝑟𝑡ℎ
−
)
𝑟𝑝𝑎𝑟𝑘
2𝑎𝑡1
(A-8.2.3-7)
Because we perform tangential burns for bielliptic transfers, we create vector diagrams as
shown in Fig. A-8.2.3-2 to describe the required ΔV.
∆V1
Vdepart1
Vpark
Fig. A-8.2.3-2: Vector Diagram for
1st Tangential Burn
(Levi Brown)
Based on the vector diagram in Fig. A-8.2.3-2, we calculate ∆V1 from the following
equation.
∆𝑉1 = 𝑉𝑑𝑒𝑝𝑎𝑟𝑡1 − 𝑉𝑝𝑎𝑟𝑘
(A-8.2.3-8)
Finding ΔV2. We next find the velocity at apoapsis on the first transfer arc. We
substitute the intermediate radius r for rpark in Eq. (A-8.2.3-7).
Author: Levi Brown
Alternative Designs – Lunar Transit – Trajectory Alternatives
𝜇𝐸𝑎𝑟𝑡ℎ 𝜇𝐸𝑎𝑟𝑡ℎ
𝑉𝑎𝑟𝑟𝑖𝑣𝑒1 = √2 (
−
)
𝑟
2𝑎𝑡1
Section A-8.2.3
(A-8.2.3-9)
We then calculate the conditions required for the second transfer arc. We define the
periapsis and apoapsis radii of the transfer arc by the following equations.
𝑟𝑝 = 𝑎𝑀𝑜𝑜𝑛
(A-8.2.3-10)
𝑟𝑎 = 𝑟
(A-8.2.3-11)
We find the semi-major axis and eccentricity of the second transfer arc, which is
necessary to calculate the velocity required to reach the final orbit radius.
𝑎𝑡2 = 1⁄2 (𝑟𝑝 + 𝑟𝑎 )
𝑒𝑡2 = 1 −
(A-8.2.3-12)
𝑟𝑝
𝑎
(A-8.2.3-13)
𝜇𝐸𝑎𝑟𝑡ℎ 𝜇𝐸𝑎𝑟𝑡ℎ
𝑉𝑑𝑒𝑝𝑎𝑟𝑡2 = √2 (
−
)
𝑟𝑎
2𝑎𝑡2
(A-8.2.3-14)
We calculate ∆V2 by referencing the vector diagram in Fig. A-8.2.3-3 and the following
equation.
Varrive1
Vdepart2
∆V2
Fig. A-8.2.3-3: Vector Diagram for
2nd Tangential Burn
(Levi Brown)
∆𝑉2 = 𝑉𝑑𝑒𝑝𝑎𝑟𝑡2 − 𝑉𝑎𝑟𝑟𝑖𝑣𝑒1
Author: Levi Brown
(A-8.2.3-15)
Alternative Designs – Lunar Transit – Trajectory Alternatives
Section A-8.2.3
Finding ΔV3. We find the velocity at periapsis of the second transfer arc.
𝑉𝑎𝑟𝑟𝑖𝑣𝑒2 = √2 (
𝜇𝐸𝑎𝑟𝑡ℎ 𝜇𝐸𝑎𝑟𝑡ℎ
−
)
𝑟𝑝
2𝑎𝑡2
(A-8.2.3-16)
We assume the spacecraft reaches the Moon’s vicinity as we reach the final orbit. We
incorporate the Moon’s gravity at this point as required per the method of patched conics.
We calculate the velocity of the Moon in its own orbit.
𝜇𝐸𝑎𝑟𝑡ℎ
𝑉𝑀𝑜𝑜𝑛 = √
𝑎𝑀𝑜𝑜𝑛
(A-8.2.3-17)
We compare the energy between the spacecraft and Moon’s orbit to determine the
direction of approach. The spacecraft possesses enough energy to travel far beyond
aMoon, thus it travels faster than the Moon. As we see in Fig. A-8.2.3-4 below, the
spacecraft approaches from behind. Consequently, we define 𝑉̅∞ as shown in Fig. A8.2.3-5.
Vm
E
M
Varrive2
Fig. A-8.2.3-4: Velocity Vectors at Final Orbit Arrival
(Levi Brown)
Author: Levi Brown
Alternative Designs – Lunar Transit – Trajectory Alternatives
Section A-8.2.3
Vm
Varrive2
V∞
Fig. A-8.2.3-5: Geocentric Vector Diagram
for Lunar Capture
(Levi Brown)
We then find 𝑉̅∞ .
𝑉∞ = 𝑉𝑎𝑟𝑟𝑖𝑣𝑒2 − 𝑉𝑚
(A-8.2.3-18)
As seen in Fig. A-8.2.3-6, the spacecraft approaches perilune of a hyperbolic orbit with
velocity 𝑉̅∞ . When the spacecraft reaches perilune, we perform ΔV3 to capture into a
circular low lunar orbit.
Perilune
M
Final
Capture
Orbit
Hyperbolic
Orbit Path
𝑉̅∞
.
Fig. A-8.2.3-6: Lunar Capture from Hyperbolic Orbit
(Levi Brown)
We calculate the orbit energy and define rp as the final orbit radius about the Moon (110
km altitude in this case). We then find the velocity at perilune on the hyperbolic orbit
and the capture velocity required.
Author: Levi Brown
Alternative Designs – Lunar Transit – Trajectory Alternatives
𝜀=
𝑉∞ 2
2
Section A-8.2.3
(A-8.2.3-19)
𝑉𝑝𝑒𝑟𝑖𝑙𝑢𝑛𝑒 = √2 (𝜀 −
𝜇𝑀𝑜𝑜𝑛
)
𝑟𝑝
𝜇𝑀𝑜𝑜𝑛
𝑉𝑐𝑎𝑝𝑡 = √
𝑟𝑝
(A-8.2.3-20)
(A-8.2.3-21)
We calculate ΔV3 referencing Fig. A-8.2.3-7.
∆𝑉3 = 𝑉𝑝 − 𝑉𝑐
(A-8.2.3-22)
Vp
ΔV3
Vc
Fig. A-8.2.3-7: Vector Diagram
for Lunar Capture
(Levi Brown)
Lastly we determine the overall total ΔVand total time of flight.
∆𝑉𝑡𝑜𝑡 = ∆𝑉1 + ∆𝑉2 + ∆𝑉3
(A-8.2.3-23)
𝑇𝑂𝐹𝑡𝑜𝑡 = 1⁄2 (𝑃𝑒𝑟𝑖𝑜𝑑1 + 𝑃𝑒𝑟𝑖𝑜𝑑2 )
(A-8.2.3-24)
Author: Levi Brown
Alternative Designs – Lunar Transit – Trajectory Alternatives
Section A-8.2.3
Results. As described above, we test different cases to determine the time of flight and
ΔV for a bielliptic transfer. We compare these results to a Hohmann transfer. We
arbitrarily select intermediate radii of 1 x 106 and 3.75 x 106 km beyond the moon’s orbit
radius of 384400 km as test cases. For these two intermediate radii, we test different
parking and capture orbits, as well. As we see in Table A-8.2.3-1, regardless of the test
conditions, a Hohmann transfer requires less ΔV with a much shorter time of flight.
Further investigation shows that increasing the intermediate radii further will result in a
smaller ΔV cost than a Hohmann transfer; however, the time of flight increases
dramatically.
We conclude that for a transfer employing chemical propulsion, a
Hohmann transfer is the most cost effective method.
Table A-8.2.3-1 Bielliptic Vs. Hohmann Transfer Result Comparison
Earth Parking Orbit Lunar Parking Orbit Intermediate radius
Parameter
km
km
km
ΔV (km/s)
200
110
1 x 106
TOF (days)
ΔV (km/s)
200
110
3.75 x 106
TOF (days)
ΔV (km/s)
36 x 104
110
1 x 106
TOF (days)
ΔV (km/s)
36 x 104
110
3.75 x 106
TOF (days)
ΔV (km/s)
200
500
1 x 106
TOF (days)
ΔV (km/s)
200
500
3.75 x 106
TOF (days)
ΔV (km/s)
36 x 104
500
1 x 106
TOF (days)
ΔV (km/s)
36 x 104
500
3.75 x 106
TOF (days)
Author: Levi Brown
Hohmann
Bielliptic
4.0
5
4.0
5
1.8
6
1.8
6
3.9
5
3.9
5
1.7
6
1.7
6
4.2
81
4.03
365
2.11
83
2.03
365
4.1
81
3.9
365
2.05
83
1.9
365