In this video we’re going to be taking a deep dive into the math, astrodynamics, and software engineering required to calculate interplanetary trajectories with gravity assists.

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Gravity assists, planetary flybys, gravitational slingshots, are all names that we give to this physical phenomena, that enables us to use the planets in our solar system to rotate and increase the magnitude of our spacecraft velocity vectors, to in turn explore more planets in our solar system. This allows us to replace the mass of the spacecraft that would be fuel with more scientific instruments to learn more about our planetary neighbors.

And not only do gravity assists allow us to explore our solar system, they also enable us to leave it and explore deeper into our universe, as we have done with the Voyager spacecrafts, and one day, with humans on board

This video will be broken down into 3 different sections. We’ll start with the fundamentals of orbital mechanics, which is why orbits are conic sections in the two-body problem, building up to what happens in a gravity assist event.

Section 2 will cover the intuition behind how increases in the spacecraft velocity vector magnitude are possible given that energy and angular momentum are conserved in a closed system.

And finally section 3 will be going over how to calculate gravity assist trajectories using Lambert’s solvers, root solvers, and its implementation in Python.

So now lets take a look at what happens to a spacecraft during a planetary flyby. We will be using the Voyager 2 flyby of Jupiter to illustrate gravity assists. The first thing to note is that planetary flybys are hyperbolic orbits with respect to the planet that they are flying by. Again, since energy is conserved, the magnitude of the velocity of Voyager 2 before it flies by Jupiter and after is constant (with respect to Jupiter), but the vector is rotated.

We’ve established that a planetary flyby rotates the velocity vector, but does not change its magnitude with respect to the planet. Pure rotations of velocity vectors do change the angular momentum of the spacecraft, but they don’t change the specific mechanical energy.

From this we can conclude that we need an increase in the velocity magnitude with respect to the Sun in order for spacecraft like the Voyagers to reach the outer planets. But how then is an increase in velocity magnitude possible given the results from the two body problem?

The two-body dynamics analysis assumes that the central body is stationary (with respect to the inertial frame). In reality, the planets are in orbit around the sun, and thus have velocity vectors themselves.

We already know that the spacecraft velocity vector will be rotated. In this case, the spacecraft velocity vector is rotated towards Jupiter’s velocity vector. We also know that the magnitude of the spacecraft’s velocity vector with respect to Jupiter’s velocity vector must be equal before and after the flyby.

So if we take a look at the vector diagram on the bottom left, we can observe that if we only rotated the v_arrive vector (without changing its magnitude), the difference between the relative velocity vector of the spacecraft departing Jupiter would be smaller than what it arrived with (this is simply the distance between the tips of the vectors)

Therefore, the only way that we can keep the magnitude of those differences equal is to increase the magnitude of the spacecraft departing velocity vector.

We can use the velocity of the planet itself to increase the spacecraft velocity with respect to the sun. Also, the larger the planet (like Jupiter), the more that the velocity vector can be rotated, thus the larger the increase in magnitude that the velocity vector can have. This is why Jupiter is a great planet to do a flyby around, since its in an orbit between Earth and the outer planets and its so massive.

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#gravityassists #gravitationalslingshots #astrodynamics

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