THREE-DIMENSIONAL GRAVITATIONAL SWINGBYS
BPlane Targeting
Interplanetary Mission Design
Kate Davis
INTRODUCTION
The B-Plane is a planar coordinate system that allows targeting during a gravity assist. The B-Plane
can be thought of as a target attached to the central (flyby) body and can be used to target the desired
trajectory needed for the gravitational swingby. For this analysis of the B-Plane, assume that the incoming
~ ∞, in and V
~ ∞, out ) are known. These vectors will
and outgoing velocity vectors at the sphere of influence (V
have the same magnitude since the swingby simply reorients the velocity vector relative to the central body.
However, the magnitude of the spacecraft’s velocity relative to another body, such as the Sun, may change.
The B-Plane is defined to contain the focus of an idealized two-body trajectory, assumed to be a hyperbola. The B-Plane passes through the central body and is perpendicular to the incoming asymptote of this
~ ∞, in ). The intersection of the B-Plane and the trajectory plane defines a
hyperbola (i.e., perpendicular to V
~
line in space. The B-vector (B) is defined to lie along this line, starting on the focus and ending at the spot
where the incoming asymptote pierces the B-Plane. The geometry of the B-Plane is shown in Figure 1.
Target B-Plane
! to Incoming
Asymptote
!S
S/C Path
Central
Body
Trajectory
Plane
ψ
θ
!
T
!
R
B
!∞, in
V
Figure 1. The Geometry of the B-Plane
The target vector B in the B-Plane is often described by two components, B · T̂ and B · R̂. The radius of
periapse is also important to ensure that the spacecraft does not pass too closely to the surface of the flyby
body.
1
~
B
=
Ŝ
T̂
R̂
ĥ
rp
=
=
=
=
=
V∞
=
µ
θ
ψ
=
=
=
vector from the center of mass of the flyby body to the point where the spacecraft would
encounter the B-Plane if the central body were massless (where the zero-gravity extension
~ ∞, in penetrates the B-Plane)
of V
~ ∞, in
a unit vector in the direction of V
a unit vector parallel to the ecliptic plane and normal to Ŝ
a unit vector defined by Ŝ × T̂
a unit vector normal to the plane of the orbit
radius
of
approach
(periapse radius)
closest
~
~
V∞, in = V∞, out gravitational parameter of the flyby body
angle between the vectors T̂ and B̂
Turn angle (angle through which the V∞ vector is turned by the central body. In some
notations, it may also be called δ
Derivation
~ ∞, in .
The vector Ŝ may be computed from the vector V
Ŝ =
~ ∞, in
V
~ ∞, in |
|V
(1)
The quantity ĥ can be computed from the the incoming and outgoing velocity vectors.
~ ∞, in × V
~ ∞, out
V
ĥ = ~
~
V∞, in × V∞, out (2)
Next, the quantity B̂ can be computed using the orbit normal to the B-Plane and the normal to the orbital
plane (Ŝ and ĥ).
~
B
B̂ =
= Ŝ × ĥ
(3)
~
|B|
If k̂ is a unit vector in the z-direction of the ecliptic coordinate system (i.e., [0, 0, 1]), then T̂ can be calculated
using the normal to the ecliptic plane (k̂) and the normal to the B-Plane (Ŝ).
~S × ~k
T̂ = ~S × ~k
(4)
Next, R̂ may be calculated to complete the right-hand reference frame using Ŝ and T̂.
R̂ = Ŝ × T̂
The angle between the unit vectors T̂ and B̂ can be computed using the dot product.
θ = cos−1 T̂ · B̂
Note that if (B̂ · R̂ < 0) then θ = 2π − θ.
2
(5)
(6)
~ ∞, out .
The turn angle ψ is the angle through which the V∞ must turn to obtain the desired V


~ ∞, in · V
~ ∞, out
V

ψ = cos−1  ~ ∞, in V
~ ∞, out V
The radius of periapse can be found using the equation for planar gravity swingbys.1
1
µ
−1
rp = 2
V∞ cos ((π − ψ) /2)
(7)
(8)
Verify that the radius of periapse is greater than the radius of the planet, plus the desired margin for
~ can be computed using the radius of closest approach.1
mission constraints. Finally, the magnitude of the B
"
#1/2
2
2
V
r
µ
~ p
1+ ∞
−1
B = 2
µ
V∞
(9)
REFERENCES
[1] A. Sergeyevsky, G. Snyder, and R. Cunniff, “Interplanetary Mission Design Handbook, Volume 1, Part 2: Earth to
Mars Ballistic Mission Opportunities, 1990-2005,” Tech. Rep. JPL Publication 82-43, Jet Propulsion Laboratory,
1983.
3