# What are the symbol definitions in the equations for finding orbital elements?

I've been looking at the answer to this question on how to convert position and velocity to orbital elements, but some of the symbols are not defined and there are no reference links for those symbols. I also do not know what to put in the search bar to find the answers.

One of the symbols ($\hat K$) is not defined in the body of the answer, but only in the example code as 0, 0, 1. What exactly is the name of $\hat K$? Other symbols that I can't find definitions for are $\vec n$, $n$, $n_I$ and $h_K$. What are the names of these symbols and how are they defined in their equations for finding the orbital elements of an object?

I did find out that $\mu$ is the Standard Gravitational Parmameter ($GM$) where $G$ is the gravitational constant and $M$ is the mass of the body being orbited.

• Re What exactly is the name of $\hat K$? K-hat, of course. The "hat" denotes that this is a unit vector. The $K$ denotes that this unit vector points along the +z axis. One of the standard set of names for unit vectors along the x, y, and z axes is I-hat, J-hat, and K-hat (or, more typically, lower case i, j, and k, with hats). The origin for this nomenclature dates back almost 175 years, when a middle aged scientist decided that the 16th of October 1843 was a good day to scrawl some graffiti on a bridge in Ireland. He wrote: $i^2 = j^2 = k^2 = ijk = -1$. May 15, 2018 at 2:34
• That explains a few things. May 15, 2018 at 2:39
• That the middle-aged scientist was Sir William Rowan Hamilton, one of the most renowned physicists of all time might explain a bit more. That he developed the quaternions explains things even bit more. The $i$, $j$, and $k$ he scrawled into the Broom Bridge represent the three imaginary parts of his quaternions. Unlike most graffiti, his graffiti on the is memorialized with a stone plaque. May 15, 2018 at 2:56
• A battle ensued in the latter part of the 19th century between mathematicians and physicists who strongly favored Hamilton's quaternions and those who strongly hated them, but saw some useful parts. The haters (vectorialists) won the day. The three vectors used so widely in physics with their dot products and cross products represent the only useful parts of quaternions (in the eyes of the vectorialists). One holdover is the widespread use of $\hat i$, $\hat j$, and $\hat k$ instead of $\hat x$, $\hat y$, and $\hat z$. May 15, 2018 at 3:01
• Regarding $\mu$, you do not want to use $GM$, the product of the gravitational constant and an object's mass. Conceptually, $\mu = GM$. In practice, for any body that has other objects orbiting it gravitationally, $\mu$ is known to several more places of accuracy than is the product $GM$. The gravitational parameter $\mu$ is strongly observable. The gravitational constant $G$ is notorious as the least accurately known physical constant. The mass $M$ isn't observable. It instead is computed by $M = \frac{\mu}G$, with almost all the uncertainty in $M$ due to $G$ rather than $\mu$. May 15, 2018 at 3:19

Searching for "node vector" orbital elements leads me to this page, which gives pretty much the same system of equations in a different notation (bold instead of vector or hat markings):

We can find the vector $\boldsymbol n$ by taking the cross product of the angular momentum vector, $\boldsymbol h$, and the unit vector $\boldsymbol K$= <0, 0, 1>: $$\boldsymbol n = \boldsymbol K × \boldsymbol {h}$$

The convention in use here is that the spatial basis is defined by $\hat I$, $\hat J$, $\hat K$, being the unit vectors <1, 0, 0> <0, 1, 0> <0, 0, 1> respectively†. $\hat K$, the positive Z direction, is perpendicular to the equatorial plane of the primary body -- celestial north, in effect. Taking the cross product of $\hat K$ with the angular momentum vector $\vec h$ (which is perpendicular to the orbital plane of the satellite) yields $\hat n$, which is the node vector:

$\boldsymbol n$ is a unit vector on the line of nodes that points in the direction of the ascending node. The ascending node is the spot where the satellite crosses the equatorial plane in a northerly direction. Likewise, the descending node is the point where the satellite crosses the equatorial plane in the southerly direction.

In the equation

$$i=\cos^{-1}{\frac{h_K}{h}}$$

$h_K$ is just the K (i.e. +Z) component of $\vec h$. The corresponding line of pseudocode is i = acos(h(3)/mag(h)) -- h(3) is indexing the third element of $\vec h$ (like it does in ~checks notes~ no programming language in mainstream use today).

• I think that answers most of my question. But what about the $K$ and $I$ being used as a subscript? What does that mean? May 14, 2018 at 22:57
• Yep, hat is unit/normalized vector. Some notational conventions use plain $n$ for the vector and $| n |$ for the magnitude, but this one marks the vector and reserves the unadorned letter for the magnitude. ¯\_(ツ)_/¯ en.wikipedia.org/wiki/Unit_vector May 14, 2018 at 23:15