Relating the Turn Angle to the Change in Longitude
Fundamentals of Orbital Mechanics
The hyperbolic trajectory near a planet has several unique characteristics. First, any
given hyperbolic trajectory is defined by the interplanetary velocity vectors outside of the
sphere of influence of the planet. Tracy Smith provided the radius, velocity magnitude,
and flight path angle for entry and exit of each planet. Utilizing this information, the
eccentricity of the hyperbolic orbit can be found using Equations (1) and (2), and
defining the angle β in Figure 10:
e2  1 
v 4 r 2 sin 2  β  2v 2 r sin 2  β 

μ
μ2
  90 o  
(1)
(2)

v



r
local horizon
Figure 1
The point in space in which the hyperbolic trajectory passes closest to the planet is
known as periapsis. An imaginary line, known as the line of apsides, may be drawn from
the center of the planet to the periapsis (see Figure 2). A point on the hyperbolic
trajectory can be referenced from the line of apsides using true anomaly (θ*). The true
anomaly of any point along the orbit can be found by using Equations (3), (4), and (5),
solving for θ*:
r
p
1  e cos  *
 
(3)
BMW 1.5-4
h  rv cos 
(4)
BMW 1.4-3
p
h2

(5)
BMW 1.6-1
The true anomalies for the initial and final conditions given by Tracy Smith are found
with these equations. Taking the limit of Equation (3) as r approaches infinity gives an
equation for θ∞*, the true anomaly when the vehicle is outside the sphere of influence of
the planet’s gravitational field. This is shown as Equation (6):
 
cos    
*
1
e
(6)
Take limit of BMW 1.5-4 as r approaches infinity
V
Initial line of
apsides
 * 
λ
Δω
V 
 *
ξ
final line of
apsides
Figure 2
Figure 9 above shows the relationships between all the angles of interest in this problem.
The θ∞* angles have already been explained, and the ξ and λ angles are defined by the dot
product of the V∞+ and V∞- vectors. Equation (7) shows how to get ξ:
 V  V 
  cos   
 V  V

1




(7)
The angle λ is simply 180- ξ, which can bee seen from Figure 9. Equations (8) shows
the relationship between the rest of the angles in Figure 9:
  180   *   *  
(8)
It may be hard to see how this works, but Figure 10 below is a manipulation of the angles
in Figure 9 to make it easier to see how the angles are related:
θ*-∞
λ
Δω
θ*+∞
Figure 10
The angle Δω is the difference in the line of apsides from the initial orbit to the final
orbit. Since θ* is measured from the line of apsides in any given orbit, θi* and θf*, the
initial and final true anomalies, are measured from two different lines of apsides. In
order to find the total longitudinal turn angle between the initial and final states, the
change in the line of apsides, Δω , must simply be added to the true anomalies at the
initial and final states. This is quantified in equation (9) and further clarified in Figure
11:
    i*   *f
(9)
Ri
θi*
Δω
θf*
Rf
Figure 11
In the simulation, the code keeps track of the longitude angle, θ, during flight. This angle
can be initialized at the beginning to be whatever is convenient. For this mission, θ = 0 is
the most convenient initial longitude angle. Once the desired turn angle from the initial
to the final conditions is determined, the longitude angle θ can be utilized to control the
vehicle to the desired exit state.