Introduction@

What is gravity anyway?@

Gravity, also called gravitational acceleration or gravitational attraction, in its simplest description is the attraction of two (or more) objects due to their mass.

Sir Isaac Newton developed a model for the attraction of two such objects in his Principia. It says that two objects, one having mass \(m_1\) and the other mass \(m_2\), separated by a distance \(r\) exert a force on each other. This force has a magnitude that can be written as

\[\frac{G m_1 m_2}{r^2},\]

where \(G\) is the gravitational constant, an experimentally determined number that is true for any two objects being considered.

Theories of gravity have been floating around for centuries. This particular one was published in 1687; since then, it has been well-studied, tested, and developed into the modern day where it still serves as a workhorse for many astrodynamics applications.

Einstein would like a word@

Newton’s description of gravity hung around for 230 years until it was seriously challenged. After all, Newtonian gravity is not perfect; it simply offers a (very good) description of motion from a mechanical perspective.

The contender was Albert Einstein’s general theory of relativity, which came to light in 1916. His theory not only describes the motion of objects that have mass (like Newton did), but it describes the motion of objects that have a lot of mass (black holes) and also objects that have no mass (photons).

Newton’s description of “motion” would be position as a function of time, but Einstein’s description of “motion” includes position and time. That is, one takeaway from general relativity is that space and time are actually part of the same thing – spacetime. There is no such thing as absolute time, and there are no inertial reference frames (which are necessary to use Newton’s laws).

The implications coming from general relativity were shattering to all of physics, but it since vastly improved our understanding and has not only stood, but completely dominated the tests of time.

So which one do we use?@

General relativity is widely accepted as correct. But it gets complicated fast. There aren’t really that many analytic solutions, and integrating its equations of motion on a computer can be slow.

We use Newtonian gravity despite it being fundamentally wrong! We do this because it’s much easier than general relativity in terms of comprehension and computability, and it’s “close enough” for nearly all practical applications. That is, any spacecraft we launch and most celestial bodies (stars, planets, moons) will most likely not be interacting near black holes or travelling at any appreciable fraction of the speed of light, so this choice is well-justified.

With that, it’s time to dive into orbital mechanics!