Ordinary Differential Equations (ODEs) | Fundamentals of Orbital Mechanics 2

The laws of nature in our universe usually express themselves as forces, but in the case of orbital mechanics, we are interested in calculating trajectories, which is the same thing as position of a spacecraft over time. Luckily for us, we can use Newton’s second law to calculate the accelerations due to the forces in nature. We do this because acceleration is the first derivative of velocity and the second derivative of position with respect to time.
Therefore, starting with acceleration, we can integrate twice to calculate orbital trajectories.

Link to Space Engineering Podcast clip on ODE solvers:

In this video we’ll be going over in detail the relationships between acceleration, velocity, and position in detail which will be extremely important for when we implement these differential equations into python and solve them using numerical integrators, which is how all the animations and plots are made in these videos.

From last video we saw that Newton’s universal law of gravitation gives us the gravitational force between the large body and the small body in the two body problem. Then from Newton’s second law, we solved for the acceleration of the small body which gave us the scalar and vector forms of the gravitational acceleration.

The acceleration equation we get from Newton’s universal law of gravity is an ordinary differential equation, because acceleration is the first derivative of velocity and the second derivative of position with respect to time.
And to solve for position, we start with the acceleration and integrate once to get the velocity with respect to time, and then we integrate the velocity to get the position.

Note that position and velocity together create 6 numbers we use to describe the orbital state of the spacecraft (this isn’t the only way to describe the state, there also exist other ways like the Keplerian orbital elements, or equinoctial orbital elements).
And don’t worry, we won’t be solving these integrals by hand, thats why we have computers and ordinary differential equation solvers, which will be the topic of the next video!

Links to the Space Engineering Podcast (YouTube, Spotify, Google Podcasts, SimpleCast):

Link to Orbital Mechanics with Python video series:

Link to Spacecraft Attitude Control with Python video series:

Link a Mecánica Orbital con Python (videos en Español):

Link to Numerical Methods with Python video series:

#ordinarydifferentialequations #odes #orbitalmechanics

[adhip]

Hello Mr Gonzalez, I am 17 and in my attempt to keep away from video games I watch your videos and it truly amazing how you bridge multiple lines of thought ( From looking at how the derivatives and intergerals work to using the acceleration equation used)

I feel like space Oppenheimer!

[ashokdarbhe]

only channel on YouTube with such detail, quality, and easy to understand contents. Thank you so much. Really helping me a lot for my studies🙏

[swainscheps]

At 5:43 or so it would be helpful to slowly animate the orbit and display (numerically, not necessarily on the graph) the X and Y values so we can get a feel for how - at a constant speed - they change over time. Sometimes it’s not 100% intuitive until you see it in motion. In the same way at 9:00 as the orbit animation cycles, having a cycling vertical line moving left to right on the corresponding point on the 3 plots would help tie it all together. Good stuff!

[alex.s]

Awesome video, I wish my university professor explained it in such an easy to understand way!

[alfonsogonzalez-astrodynam]

Also I just realized I left a blooper in there!

[teliosausdenwaldern]

When i try to make a quiverplot at this scale the arrows get awfully distorted(Heads becoming huge in comparison the the rest of the arrows). How do i prevent this from happening?

[rostamr]

The use of Python to create these animations is amazing. So, do you think, Python is an ideal or a good language for these types of simulations? I have started to learn Python and seeing these examples are very encouraging...Thank you

[allurbase]

Great news there will be a spanish version too!