Help with Cartesian State Vectors → Keplerian Orbit Elements C#

Question

EDIT

I am currently attempting to try the standard Cartesian State Vectors to Keplerian Orbit Elements conversion but i am having problems getting my head around the formulas and the use of vectors. My premise for doing this is so i get past the simple calculations like velocity, period that i can do quite easily. The resources i have been using are This, This, This and finaly the textbook "Fundamentals of Astrodynamics" By Bate, Mueller, White

I have calculated by hand and then replicated in C# shown below. With some ballpark starting values:

$$ \mathbf{v} = 7770 \ \hat{\mathbf{y}} \ \text{(m/s)}$$

$$ \mathbf{r} = 6771000 \ \hat{\mathbf{x}} \ \text{(m)}$$

$$ \mu = 4.0\text{E+}14 \ \text{(m³/s²)}$$

I have so far manage to successfully use the following formulas:

$$ \mathbf{H} = \mathbf{r} \times \mathbf{v}$$

$$ M = \frac{1}{2} v^2 - \frac{\mu}{r} $$

$$ p = \frac{H^2}{\mu} $$

$$ a = \frac{-\mu}{2M} $$

$$ e = \sqrt{1 - \frac{p}{a}},$$

where $\mathbf{H}$ is the angular momentum vector, $M$ is the total energy (inertial frame), p is the semi-latus rectum, e is eccentricity, and $\mathbf{r}$ and $\mathbf{v}$ (and $r$ and $v$) are the position and velocity vectors (and magnitudes).

However i am having problems trying to follow up with:

$$ \hat n = \hat K \times \mathbf{h} $$ (as seen in PG61 of the Book mentioned above)

And correctly using: $$\vec e = \frac{\vec v \times \vec h}{\mu} \ - \frac{\vec r}{||r||} $$

IN the PDF it is shown as:

$$ \hat n = \mathbf{(0,0,1)^T} \times \mathbf{h} = (-hy,hx,0)^T$$

I am unsure what to plug in or how to use these types of formulas (or i even have the information)

I have included some of my C# code so you can see my line of thinking.progress.

double V = 7770;
double R = 6771000;
double Mu = 4e+14;

Vector velocity = new Vector(0, V);
Vector Radius = new Vector(R, 0);      

double H = Vector.CrossProduct(Radius, velocity);// Angular Momentum
double M = (Math.Pow(V, 2) / 2) - (Mu / R);//Mechanical Energy
double p = Math.Pow(H, 2) / Mu;// Semi-Latus Recum 

double a = -Mu / (2 * M);//Semi-Major Axis? Comes out only 40km less than 
SLR and no where near the correct SMA.
double test = Vector.Multiply(Radius, velocity);//test dot product.

//double e = Math.Sqrt(1 - p / a);//Eccentricty ? No where near correct 
vaule.
//double e2 = (Math.Pow(V, 2) - Mu / R) * Radius - test * velocity / 
Mu);//Eccentricty ?

I have been fiddling with the code and reading material for around two months and i have gotten some progress but my lack of experience with calculating Vectors is letting me down. Help with examples i can reverse engineer or get a better understanding of what i need to do to get going.

Answer 1

I will post a tentative answer because comments don't allow enough room. I always use the vis-viva equation for things like this. If you are just calculating energies or position and velocity at a fixed point (especially the apses) it's a little faster because it only uses scalars:

$$v^2 = \mu \left(\frac{2}{r} - \frac{1}{a} \right)$$

Flip that around to get:

$$a = \frac{1}{ (2/r - v^2/\mu) }$$

With your staring values, I get a value for $a$ of about 6923046 m, which is larger than your starting radius, so you've started at periapsis. Setting $r_{peri}$ to 6771000 m and using $e = 1 - r_{peri}/a$ I get an eccentricity of about 0.02196.

By definition you've started at one of the apses, since $r$ and $v$ are perpendicular. The task is only to figure out which one.