How can the exit velocity vector of a spacecraft performing a hyperbolic flyby maneuver be calculated?

This extended essay focuses on orbital mechanics, which is the study of the motion and trajectories of artificial satellites and spacecraft under the influence of external forces such as gravitational attraction due to celestial bodies, atmospheric drag or thrust. Specifically, I will analyse a fairly simple manoeuvre: the hyperbolic flyby, visualising it with vectors and calculating its impact on the craft through the use of trigonometry and geometry. My research question is: “What are the mathematical methods involved in defining the trajectory and speed of a spacecraft in a hyperbolic flyby manoeuvre?” This manoeuvre is defined as the hyperbola-shaped trajectory that a spacecraft with enough velocity to escape its gravitational attraction takes around a central body. By the end of this investigation, my aim is to be able to calculate the values of speed and direction of a spacecraft after a hyperbolic flyby, having set specific initial conditions. This investigation also aims to define what are the basic initial parameters that need to be defined to calculate the exit speed of a spacecraft after a hyperbolic flyby.

 

The choice of topic for this essay stems from my passion in aerospace, a subject I would like to study at university. I have always been interested in spacecraft construction and launches, and the idea of analysing hyperbolic flybys came while I was reading about BepiColombo, a scientific probe, named after a scientist and engineer who lived near my home town. The spacecraft, sent to study Mercury, had just performed a gravity assist on the planet, used to increase the speed and change the direction of the probe in relation to Mercury. I already knew about this manoeuvre, because it is commonly used, as it is impractical and impossible to load all the fuel needed to reach high speeds once in orbit. This was the case with the Voyager 2 probe, launched in 1977. Its purpose was to study the outer planets of the solar system and interstellar space. The spacecraft used gravity assists from Jupiter, Saturn, Uranus and Neptune, as well as these planet’s moons, to accelerate and escape from the solar system. Voyager two is currently travelling at about 15km/s, but the highest speed reached (relative to the sun), was of about 35 km/s. These speeds have been obtained respectively from two articles by NASA - “Voyager - Mission Status.”2 and “Voyager 2 - In Depth.”

 

These high speeds cannot be reached even with today’s launch systems, showcasing how flybys are fundamental in interplanetary missions. My initial aim was to focus on these manoeuvres and how they could be used multiple times to increase a spacecraft’s velocity and direct it towards a destination, however, after some research, I discovered that the topic was far too difficult. This is because it would require extensive knowledge on the position of planets in their orbits and a deep understanding of planet-probe interactions. For this reason, I decided to focus only on the manoeuvre itself, as they are the basis of planet-probe interactions and interplanetary travel.

 

The study of these flybys included vectors, a topic which I have had to study on my own, and trigonometry, applying the knowledge I had already acquired in a totally different context, proving how maths can be of great help in modelling and understanding the physical world. This essay will be structured as follows: firstly, an overview of the topic will be given, defining the terms used, the variables and the assumptions made, then the problem will be described and analysed from a mathematical perspective, yielding a set of formulae that will be later put into use in a real-life scenario, obtaining a value for the exit velocity, which will be compared with data taken from the Voyager 2 mission. Finally, conclusions will be made to evaluate the possible errors and improvements to this investigation.

• Hyperbolic excess velocity - the spacecraft’s instantaneous velocity before (inbound) and after (outbound) the hyperbolic flyby. Its notation is \(\vec{V}_{\infty} \) in because it represents the velocity at infinity, meaning that the effect of the gravitational force from the concerned planet is negligible.4
• Heliocentric velocity - the spacecraft’s or planet’s velocity with respect to the sun.
• Periapsis - the point in the hyperbolic orbit at which the probe is the closest to the planet.
• Inbound and outbound crossing - the period in which the spacecraft is approaching (inbound) or retreating from (outbound) the periapsis of its orbit.
• Sphere of influence - the area around a celestial body in which the main gravitational influence on an orbiting object is said body.
• Escape velocity - the velocity an object is required to have in order to escape from the sphere of influence of a certain celestial body.
• Leading side flyby - flyby in which the spacecraft crosses in front of the planet’s direction of heliocentric motion.
• Trailing side flyby - flyby in which the spacecraft crosses behind the planet’s direction of heliocentric motion.

\(\vec{V}_{\infty}\,_{in}\)   - Spacecraft’s inbound hyperbolic excess velocity

 

\(\vec{V}_{\infty}\,_{out}\) - Spacecraft’s outbound hyperbolic excess velocity

 

\(\vec{V}_p\) - Planet’s heliocentric velocityPlanet’s heliocentric velocity

 

\(\vec{V}\,_{sun \,in}\)  - Spacecraft’s inbound heliocentric velocity

 

\(\vec{V}\,_{sun\,out}\)  - Spacecraft’s outbound heliocentric velocity

 

î - base vector in the horizontal direction (⟂ to the direction towards the sun)

 

ĵ - base vector in the vertical direction (direction towards the sun)

 

θ_in - Angle between\( \vec{V} _{∞\, in}\) and \(\vec{V}\,_{ p}\) or î(the horizontal direction)

 

θ_out - Angle between\(\vec {V} _{∞\,out}\) out and \(\vec{V}\,_{ p}\) or î (the horizontal direction)

 

α_in - Angle between\( \vec{V}\, _{sun\, in }\)and \(\vec{V}\,_{ p} \)or î (the horizontal direction)

 

α_{out -} Angle between V⃗ sun out and V⃗ p or î(the horizontal direction)

 

γ - Angle between the inbound hyperbolic excess velocity \(\vec{V}_{ ∞\,in}\) in and \(\vec{V}_{∞\,out}\)

 

All of the angles above are measured counterclockwise.

 

m_p  - Planet’s mass

 

m_s  - Spacecraft’s mass

In order not to complicate the maths and render the calculations extremely complex, the following are the assumptions and simplifications that have to be made for the model to be feasible.

• Assumption 1 - The planet’s heliocentric velocity is constant.

By assuming the heliocentric velocity of the planet to be constant, it is also assumed that the planet’s orbit is perfectly circular, although this is very unlikely to happen naturally. This consequence derives from Kepler’s second law of planetary motion.

• Assumption 2 - The planet’s velocity is perpendicular to the sun.

As a consequence of the previous assumption, the instantaneous heliocentric velocity of the planet is considered to be perpendicular to the direction to the sun. The velocity is tangent to the orbit, perpendicular to the radius, the direction to the sun.

• Assumption 3 - The planet’s atmosphere doesn’t have an impact on the spacecraft.

This model does not account for the air resistance that may result from the probe entering the planet’s exosphere (upper layer of the atmosphere), slowing the spacecraft down.

• Assumption 4 - The planet’s gravitational field is even.

The model does not account for differences in the gravitational attraction of the planet, which in nature occur because of the non-spherical shape of planets and their differences in density. For this reason, it is assumed that the planet is perfectly spherical and its mass is evenly distributed.

• Assumption 5 - There is no gravitational attraction from other nearby planets or moons.

In a real scenario, there might be attraction from the planet’s moons or other planets, affecting the spacecraft’s speed and trajectory. This is not accounted for in the model.

• Assumption 6 - There is no gravitational attraction exerted on the planet by the spacecraft.

The model does not account for this, as it has a minimal impact on the manoeuvre. For this reason, the spacecraft is considered to be point-like

A hyperbolic flyby is a manoeuvre performed by spacecraft that is travelling faster than the escape velocity of the planet it is approaching. The spacecraft’s trajectory can be modelled mathematically as an hyperbola, or a conic section with eccentricity greater than one. Planet X occupies one of the two foci of the hyperbola, while the other is left empty. In orbital mechanics, the trajectory the spacecraft follows is the one nearer to the planet. The other branch would represent the trajectory of a particle if the force exerted by the object on the opposing focus were repulsive. For example, if the object on the focus were to be a positively or negatively charged particle, the opposing branch of the hyperbola would represent the trajectory of a particle of the same charge.

During the inbound crossing, the spacecraft accelerates as a consequence of the gravitational attraction caused by the planet’s gravity well. At periapsis, the spacecraft’s velocity is maximum, and during the outbound crossing the spacecraft decelerates once more. Thus, after the manoeuvre, the spacecraft doesn’t gain speed if observed only in the planet’s frame of reference.

The problem will now be analysed through the use of vectors, both in the planet’s frame of reference and in the sun’s frame of reference, showing how the manoeuvre affects the velocity of the spacecraft, enabling a probe to increase its velocity relative to the sun without the use of propellant.

 

The inbound \((\vec{V}\,_{ ∞ in} )\) and outbound \((\vec{V} ∞\,_{ out} )\) hyperbolic excess velocities of a spacecraft approaching a planet can be visualised as follows. The two velocity vectors lie on the asymptotes of the hyperbola. The analysis will only be focused on the time period in which the spacecraft is in the planet’s sphere of influence.

As seen in the illustration, the velocity vector changes direction as a consequence of the gravitational field of planet X, but its magnitude remains the same.

However, this vector only represents the velocity of the spacecraft relative to the planet, which in turn moves relative to the sun. This movement is represented by the planet’s heliocentric velocity\( \vec{V}_{ p} .\)

Adding these two visualisations, moving to the sun’s frame of reference, the spacecraft’s hyperbolic excess velocities\(\vec{ V}_{∞}\,_ {in}\) and \(\vec{V} _{∞ }\,_{out}\) out are summed with the planet’s heliocentric velocity\( \vec{V}\,_{ p} \), giving the spacecraft’s heliocentric velocities \(\vec{V}_{sun\, in}\) and\( \vec{V}_{sun \,out}\)

 

\(\vec{V}_{ sun\, in }=\vec{ V}_ {p }+ \vec{V} _{∞ \,in}\)

\(\vec {V} _{sun \,out} =\vec{ V }_{p }+ \vec{V} _{∞ out}\)

Thanks to the optimal angle, the spacecraft’s resultant outbound heliocentric velocity \(\vec{V}_{ sun\,out }\) is greater in magnitude than the inbound heliocentric velocity \(\vec{V}_{ sun\, in}\), as seen in figure 7

Having calculated the spacecraft’s inbound and outbound heliocentric velocities\( \vec{V}_{sun\, in}\) and\(\vec{ V}_{ sun\, out}\) , the change in the spacecraft’s heliocentric velocity can be found -

 

\(Δ\vec{V}_ {sun} = \vec{V}_{ sun out} − \vec{V}_{ sun in}\)

 

Substituting for the inbound and outbound heliocentric velocities -

 

\(Δ\vec{V}_{ sun }= (\vec{V}_{ p} +\vec{ V}_{ ∞ \,out} ) − (\vec{V}_ {p }+ \vec{V}_{ ∞\, in} )\)

 

Since \(\vec{V}_{ p}\) is constant (see assumption 1), it can be cancelled out from the equation, giving -

 

\(Δ\vec{V}_ {sun }= \vec{V} _{∞ out} − \vec{V}_{∞ in}\)

 

\(Δ\vec{V}_{ sun} = Δ\vec{V}_{ ∞}\)

 

It has been previously shown that the hyperbolic excess velocities are equal in magnitude, because only the direction of these velocities is changed by the gravity of the planet. For this reason, the change in the spacecraft’s heliocentric velocity is given by the change in direction of the probe’s hyperbolic excess velocity The angle that determines the amount of change in heliocentric velocity is the angle between \(\vec{V}_{∞}\) in and \(\vec{V}_{∞}\) out, as shown in the following diagram.

As seen in figure 8, if angle θ_out is smaller than angle θ_in ,\( \vec{V}_{ sun \,out}\) is larger than \(\vec{V}_{ sun\, in }\) in magnitude, showing how a hyperbolic flyby impacts the heliocentric velocity of a spacecraft. Through this manoeuvre, a probe can gain or lose velocity, according to the direction it approaches the planet, and consequently how the vector sum is arranged. In a leading side flyby, the spacecraft’s velocity is reduced, while in a leading side flyby, as illustrated in the previous diagrams, the velocity is increased.