Simulation of Interplanetary Trajectories Using Forward Euler Numerical Integration

Abstract Knowledge of interplanetary trajectory mechanics is the reason that humans have been

able to realize such monumental missions in space exploration. For instance, NASA’s and ESA’s

joint mission, Ulysses, was the first mission to survey the space environment above and below

the poles of our star. The spacecraft utilized a gravity assist maneuver of unparalleled complexity

at Jupiter to shift its trajectory out of the plane of the ecliptic and into its desired solar polar orbit.

This pioneering in utilization of interplanetary trajectory mechanics exposed, for the first time,

the three-dimensional character of galactic cosmic radiation, energetic particles produced in solar

storms, and solar wind. (Missions to our Solar System, 2013) The Ulysses mission would not

have been able to achieve its solar polar orbit without utilizing the momentum of Jupiter. Even

modern-day capabilities of rocket power would not have enabled Ulysses to achieve such a

peculiar orbit; the efforts of conventional rocket technology alone would have been negligible in

comparison to transcendental powers of planetary flyby assistance.

The advantages of planetary flyby are clear; a spacecraft can essentially attain “free”

changes to its velocity and trajectory direction; moreover, if the spacecraft’s trajectory is timed

and positioned “perfectly”, it can strategically use its change in energy to realize such

monumental missions as Ulysses. Thus, the understanding of the sensitivity of interplanetary

trajectories to certain sphere of influence entry parameters is crucial. In this study, the mechanics

of interplanetary trajectories are taken a closer look at through numerical analysis using Euler’s

Method of numerical integration of ordinary differential equations.

The objectives of this study are to establish the validity of Euler’s Method of numerical

integration of ordinary differential equations with application to the gravitationally influenced

motion of interplanetary trajectories, and to study the effects of the spacecraft’s velocity and

position relative to its body of attraction upon its entry into the body’s sphere of influence on its

interplanetary trajectory. Of course, this numerical study is conducted with observance only to

values of the spacecraft’s velocity upon entry into its attracting body’s sphere of influence that

are sufficiently large enough to situate the spacecraft into a hyperbolic path about the body, even

for small values of the spacecraft’s position relative to the body at periapsis, such that the

spacecraft will maintain the trajectory it realized upon its exiting of the body’s sphere of

influence. “A spacecraft that enters a planet’s sphere of influence and does not impact the planet

or go into orbit around it will continue on its hyperbolic trajectory through periapsis and exit the

sphere of influence.” (Curtis, 2014)

Newton’s Second Law Newton’s Second Law can be defined as

𝐹 =𝑔𝑚1𝑚2

𝑟2

For the applications of this study, it is necessary to start by defining Newton’s Second Law in

vector form

𝐹1⃗⃗ ⃗ =

𝐺𝑚1𝑚2

𝑟2�̂�

Where

𝑟 = √𝑥2 + 𝑦2

The components can be rewritten as

(𝐹𝑥 , 𝐹𝑦) = (𝐺𝑚1𝑚2

𝑥2 + 𝑦2

−𝑥

√𝑥2 + 𝑦2,𝐺𝑚1𝑚2

𝑥2 + 𝑦2

−𝑦

√𝑥2 + 𝑦2)

This yields two second-order equations that may be implemented computationally.

𝑚1𝑥′′ = −

𝐺𝑚1𝑚2𝑥

(𝑥2 + 𝑦2)3/2

𝑚1𝑦′′ = −

𝐺𝑚1𝑚2𝑦

(𝑥2 + 𝑦2)3/2

Forward Euler’s Method of Numerical Integration of Ordinary Differential Equations Forward Euler’s Method of numerical integration of ordinary differential equations may be

shown as follows

𝜔0 = 𝑦0

𝜔𝑖+1 = 𝜔𝑖 + ℎ𝑓(𝑡𝑖, 𝜔𝑖)

ℎ = 𝑡𝑖+1 − 𝑡𝑖

(Sauer, 2012)

Implementation of Forward Euler’s Method in Gravitationally Influenced Hyperbolic Trajectory of Spacecraft As the motion of the spacecraft along its hyperbolic trajectory is what is trying to be simulated, it

is necessary to define the explicit equations of the spacecraft’s acceleration in both the x-

direction and y-direction

𝑥′′ = −𝜇𝑚𝑆𝐶𝑥

(𝑥2 + 𝑦2)3/2

𝑦′′ = −𝜇𝑚𝑆𝐶𝑦

(𝑥2 + 𝑦2)3/2

These equations can be implemented in a computational context as follows

𝑡𝑖+1 = 𝑡𝑖 + ℎ

𝑣𝑥𝑖+1= 𝑣𝑥𝑖

+ ℎ𝑥′′(𝑥𝑖, 𝑥𝑖)

𝑣𝑦𝑖+1= 𝑣𝑦𝑖

+ ℎ𝑦′′(𝑥𝑖, 𝑦𝑖)

𝑥𝑖+1 = 𝑥𝑖 + ℎ𝑣𝑥𝑖

𝑦𝑖+1 = 𝑦𝑖 + ℎ𝑣𝑦𝑖

𝑟𝑖+1 = √𝑥𝑖+12 + 𝑦𝑖+1

2

It is also necessary to impose some restrictions in this simulation: the simulation is restricted

within the attracting body’s sphere of influence

𝑟 ≤ 𝑟𝑆𝑂𝐼

Moreover, the spacecraft must not crash into the surface of the attracting body

𝑟 > 𝑟𝑆𝑢𝑟𝑓

Implementation in MATLAB The iterative equations shown above may be implemented in MATLAB, or other

languages. It is necessary to specify a value for the time step, h. For this implementation in

MATLAB, “h” is now referred to as “dt”. This time step, dt, can be thought of as the resolution

of the simulation. As the value of the time step, dt, decreases, the results of simulation become

more accurate. Of course, one is limited in how small a time step can be chosen, as computation

time increases significantly.

It is also necessary to implement the imposed restrictions of the context of this study.

That is that the spacecraft must not crash into the surface of the planet, and the simulation must

stop once the spacecraft has exited the attracting body’s sphere of influence. It is also, necessary

to define the “known”. Of the knowns to be specified, there are the standard gravitational

parameter of the attracting body (𝜇), the radius of the attracting body’s sphere of influence

(𝑟𝑆𝑂𝐼), and the attracting body’s surface radius (𝑟𝑆𝑢𝑟𝑓).

For simplification in this simulation’s context, the attracting body is held stationary, so

that we may analyze the sensitivity of the spacecraft’s trajectory in accordance to its sphere of

influence entry parameters. The sphere of influence entry parameters, 𝑟𝐸𝑥 𝑎𝑛𝑑 𝑣𝐸𝑦

, are specified,

such that the entry velocity only has magnitude in the y-direction and the spacecraft’s position

coincides with the sphere of influence’s boundary. In this study, the simulation is initialized in

the coordinate systems fourth quadrant, such that the position of the attracting body is centered at

the origin. To summarize

𝑟𝐸𝑥> 0 𝑎𝑛𝑑 |𝑣𝐸| = 𝑣𝐸𝑦

Moreover, since it is desired that the spacecraft’s entry position coincide with attracting

body’s sphere of influence, it should be noted that, for the context of this simulation that

𝑟𝐸𝑦= √𝑟𝑆𝑂𝐼

2 − 𝑟𝐸𝑥2

In addition, to insure that this study only investigates hyperbolic trajectories, it was

determined through this numerical analysis that for all entry x-positions from zero to the sphere

of influence’s boundary that valid entry velocities are distinguished from the minimum value

min (𝑣𝐸𝑦) = 5

𝑘𝑚

𝑠. In this study a range of entry velocities are tested 5

𝑘𝑚

𝑠≤ 𝑣𝐸𝑦

≤ 20𝑘𝑚

𝑠.

The Forward Euler’s Method of Numerical Integration of Ordinary Differential Equations

may be implemented in a “while” loop in MATLAB as follows

%% Numerical Integration dt=10; %[s] Time step %Forward Euler Method of Numerical Ordinary Differential Equation Integration i=1; while r<=RSOIE & r>RE t(i+1)=t(i)+dt; %[s] %Next step's time vx(i+1)=vx(i)+dt*dvxdt(x(i),y(i)); %[km/s] Next step's x-velocity vy(i+1)=vy(i)+dt*dvydt(x(i),y(i)); %[km/s] Next step's y-velocity x(i+1)=x(i)+dt*dxdt(vx(i)); %[km] Next step's x-position y(i+1)=y(i)+dt*dydt(vy(i)); %[km] Next step's y-position R(i+1)=sqrt(x(i+1).^2+y(i+1).^2); %[km] Next step's distance from

Earth V(i+1)=sqrt(vx(i+1).^2+vy(i+1).^2); %[km/s] Next step's velocity psi(i+1)=atan2(y(i+1),x(i+1)); r=R(i+1); %[km] Position of spacecraft in next step i=i+1; %Change index end

Results

For the sake of numerical analysis of this situation, it was chosen for familiarity, that the

attracting body be Earth.

Velocity Recovery Theory dictates that an object’s hyperbolic trajectory about an attracting body will recover its

velocity between its entry and exit of the attracting body’s sphere of influence. Numerically this

was simulated with a 10km/s entry velocity and a 20,000km entry x-position. We can see that,

over the course of the spacecraft’s trajectory, it completely recovers its initial velocity, in

accordance to theory. It can also be seen that the spacecraft’s maximum velocity will occur at

periapsis. Furthermore, it can be seen why gravitational assists are considered impulsive

maneuvers.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

x 105

10

10.5

11

11.5

12

12.5

Time [s]

VH

yperb

olic

[km

/s]

Hyperbolic Path Velocity versus Time VH(t) with V

E=10km/s and r

Ex

=20000km

Anomaly with Respect to the Attracting Body It can be seen, by that act of the satellite’s hyperbolic trajectory, that its anomaly with respect to

the attracting body is changed significantly. To be clear the anomaly in this simulation is the

angle of which the spacecraft is elevated from the positive x-axis.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

x 105

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

Angluar Position versus Time for VE=10 km/s and r

Ex

=20000 km

Time [s]

[

rad]

Entry Position’s Effect on Hyperbolic Trajectory It is evident that, as the spacecraft’s entry x-position decreases, its path deflection increases.

Notice that the satellite’s path never coincides with Earth’s radius because of the conditions

implemented in previous section’s MATLAB code. It is evident in the second plot of this

section, that the satellite’s path deflection becomes negligible as its entry x-position is moved out

toward Earth’s sphere of influence boundary.

-7 -6 -5 -4 -3 -2 -1 0 1 2

x 104

-1

0

1

2

3

4

5

6

7

x 104

rx [km]

r y [

km

]

Hyperbolic Trajectory for various rE

x

with VE=10km/s

Radius of Earth

Sphere of Influence

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

x 106

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1x 10

6

rx [km]

r y [

km

]

Hyperbolic Trajectory with VE = 10 km/s

Radius of Earth

Sphere of Influence

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

x 106

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1x 10

6

rx [km]

r y [

km

]

Hyperbolic Trajectory with VE = 5 km/s

Radius of Earth

Sphere of Influence

Entry Position’s Effect on Velocity at Periapsis It can be seen that as the spacecraft’s entry x-position decreases, its velocity at periapsis

increases. One should notice that the velocity at periapsis asymptotically approaches the

satellites entry velocity; again, because Earth’s gravitational influence becomes “negligible” at

its sphere of influence boundary.

1 2 3 4 5 6 7 8 9 10

x 104

10

10.5

11

11.5

12

12.5

13

13.5

14

14.5

15

Velocity at Periapsis verus Entry Position VP(r

Ex

) with VE=10km/s

rE

x

[km]

VP [

km

/s]

Hyperbolic Trajectory as a Function of Entry Position As one might predict, as the spacecraft’s entry x-position decreases, its hyperbolic path

deflection or exiting angle of trajectory relative to its entry angle of trajectory, increases. One

might be able to better understand the meaning behind the designation of an attracting body’s

sphere of influence. It can be seen from the plot that the spacecraft’s path deflection

asymptotically approaches zero. At Earth’s sphere of influence boundary the deflection of the

spacecraft’s path is “nearly’ zero; it is with this situation that astronautical engineers have

specified the boundary for which the attracting body’s gravitational pull on the spacecraft is

“negligible”.

1 2 3 4 5 6 7 8 9 10

x 104

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Path Rotation versus Entry Position (rE

x

) with VE=10km/s

rE

x

[km]

[

rad]

Entry Velocity’s Effect on Hyperbolic Trajectory It can be seen that as the spacecraft’s entry velocity increases, its path deflection decreases. As

the spacecraft’s entry velocity increases, the attracting body has less time to pull the satellite in.

Notice, in this particular plot, the regime of this study was broken; that is, the minimum entry

velocity being analyzed is 3.5 km/s. Notice that, because of this, the minimum entry velocity’s

path was rotated by more than 90 degrees, surely not practical with such a close approach to

Earth’s surface.

-20 -15 -10 -5 0

x 104

0

2

4

6

8

10

x 104

rx [km]

r y [

km

]

Hyperbolic Trajectory for various VE with r

Ex

=20000km

Radius of Earth

Sphere of Influence

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

x 106

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1x 10

6

rx [km]

r y [

km

]

Hyperbolic Trajectory for various Entry Velocities from 5km/s (magenta) to 20km/s (blue) with rE

x

=20000km

Radius of Earth

Sphere of Influence

Entry Velocity’s Effect on Periapsis Distance From the plot, it can be see that as the satellite’s entry velocity decreases, the satellite’s distance

of closest approach, periapsis, decreases. Note that the distance at periapsis asymptotically

approaches the spacecraft’s entry x-position. Obviously, it would be much more efficient in

terms of fuel costs to choose a greater entry x-position than to choose a higher entry velocity.

5 10 15 200.8

1

1.2

1.4

1.6

1.8

2x 10

4Periapsis Distance versus Entry Velocity r

P(V

E) with r

Ex

=20000km

VE [km/s]

r Periapsis [

km

]

Conclusion The goals of this study, to investigate entry positon’s and entry velocity’s effects on the

satellite’s hyperbolic trajectory, were realized. Indeed, the main conclusions were that decreasing

the entry velocity and decreasing the perpendicular distance from the attracting body at entry

both increased the angle by which the satellite’s hyperbolic path is rotated. To further validate

the findings of this study, it is evident from the plots that the spacecraft’s entry velocity has a

greater effect on its trajectory rotation than its perpendicular distance at entry, in accordance to

the definition of centripetal acceleration

𝑎𝑅𝑎𝑑𝑖𝑎𝑙 =𝑣𝑇𝑎𝑛𝑔𝑒𝑛𝑡𝑖𝑎𝑙

2

𝑟𝑃𝑒𝑟𝑝𝑒𝑛𝑑𝑖𝑐𝑢𝑙𝑎𝑟

It can be seen from the plots and theory, that increasing the entry velocity will increase the

spacecraft’s periapsis distance more than increasing the entry x-position will increase the

satellite’s periapsis velocity.

Future Development In the future, this code will be altered such that it will enable the analysis of gravity assist for

which the velocity of the attracting body will be nonzero. It may also be of interest to factor into

the analysis: air resistance for very close approaches, solar pressure for deep space analysis,

and/or a variable mass for the orbiting body for the application of a rocket. Further developments

may also include alteration to accommodate an n-body problem, for which the gravitational

effects of n number of bodies are accounted for.

Works Cited Curtis, H. D. (2014). Orbital Mechanics for Engineering Students. Elsevier.

Missions to our Solar System. (2013, August 15). Retrieved May 2015, from NASA:

http://solarsystem.nasa.gov/missions/profile.cfm?MCode=Ulysses

Sauer, T. (2012). Numerical Analysis. Pearson.