Editing Elliptic orbit (section)

===From initial position and velocity===

An [[orbit equation]] defines the path of an [[orbiting body]] <math>m_2\,\!</math> around [[central body]] <math>m_1\,\!</math> relative to <math>m_1\,\!</math>, without specifying position as a function of time.  If the eccentricity is less than 1 then the equation of motion describes an elliptical orbit.  Because [[Kepler's equation]] <math>M = E - e \sin E </math> has no general [[closed-form solution]] for the [[Eccentric anomaly]] (E) [[Kepler's equation#Inverse problem|in terms of the Mean anomaly]] (M), equations of motion as a function of time also have no closed-form solution (although [[Mean anomaly#Formulae|numerical solutions exist]] for both).

However, closed-form time-independent path equations of an elliptic orbit with respect to a central body can be determined from just an initial position (<math>\mathbf{r}</math>) and velocity (<math>\mathbf{v}</math>).


For this case it is convenient to use the following assumptions which differ somewhat from the standard assumptions above:

:# The central body's position is at the origin and is the primary focus (<math>\mathbf{F1}</math>) of the ellipse (alternatively, the center of mass may be used instead if the orbiting body has a significant mass)
:# The central body's mass (m1) is known
:# The orbiting body's initial position(<math>\mathbf{r}</math>) and velocity(<math>\mathbf{v}</math>) are known
:# The ellipse lies within the XY-plane

The fourth assumption can be made without loss of generality because any three points (or vectors) must lie within a common plane.   Under these assumptions the second focus (sometimes called the "empty" focus) must also lie within the XY-plane: <math>\mathbf{F2} = \left(f_x,f_y\right)</math> .

====Using vectors====
The general equation of an ellipse under these assumptions using vectors is:

:<math> |\mathbf{F2} - \mathbf{p}| + |\mathbf{p}| = 2a  \qquad\mid  z=0</math>

where:
*<math>a\,\!</math> is the length of the [[semi-major axis]].
*<math>\mathbf{F2} = \left(f_x,f_y\right)</math> is the second ("empty") focus.
*<math>\mathbf{p} = \left(x,y\right)</math> is any (x,y) value satisfying the equation.


The semi-major axis length (a) can be calculated as:

:<math>a = \frac{\mu |\mathbf{r}|}{2\mu - |\mathbf{r}| \mathbf{v}^2}</math>

where <math>\mu\ = Gm_1</math> is the [[standard gravitational parameter]].


The empty focus (<math>\mathbf{F2} = \left(f_x,f_y\right)</math>) can be found by first determining the [[Eccentricity vector]]:

:<math>\mathbf{e} = \frac{\mathbf{r}}{|\mathbf{r}|} - \frac{\mathbf{v}\times \mathbf{h}}{\mu}</math>

Where <math>\mathbf{h}</math> is the specific angular momentum of the orbiting body:<ref>{{cite book |first1=Roger R. |last1=Bate |first2=Donald D. |last2=Mueller |first3=Jerry E. |last3=White |url=https://books.google.com/books?id=UtJK8cetqGkC&pg=PA17 |title=Fundamentals Of Astrodynamics |date=1971 |publisher=Dover |location=New York |isbn=0-486-60061-0 |page=17 |edition=First}}</ref> 

:<math>\mathbf{h} = \mathbf{r} \times \mathbf{v}</math>

Then

:<math>\mathbf{F2} = -2a\mathbf{e}</math>

====Using XY Coordinates====
This can be done in cartesian coordinates using the following procedure:

The general equation of an ellipse under the assumptions above is:

:<math> \sqrt{ \left(f_x - x\right)^2 + \left(f_y - y\right)^2} + \sqrt{ x^2  + y^2 } = 2a  \qquad\mid  z=0</math>

Given:
:<math>r_x, r_y \quad</math> the initial position coordinates
:<math>v_x, v_y \quad</math> the initial velocity coordinates

and
:<math>\mu = Gm_1 \quad</math> the gravitational parameter

Then:
:<math>h = r_x v_y - r_y v_x \quad</math> specific angular momentum
		
:<math>r = \sqrt{r_x^2 + r_y^2} \quad</math> initial distance from F1 (at the origin)

:<math>a = \frac{\mu r}{2\mu - r \left(v_x^2 + v_y^2 \right)} \quad</math> the semi-major axis length


:<math>e_x = \frac{r_x}{r} - \frac{h v_y}{\mu} \quad</math> the [[Eccentricity vector]] coordinates
:<math>e_y = \frac{r_y}{r} + \frac{h v_x}{\mu} \quad</math>


Finally, the empty focus coordinates
:<math>f_x = - 2 a e_x \quad</math>
:<math>f_y = - 2 a e_y \quad</math>


Now the result values ''fx, fy'' and ''a'' can be applied to the general ellipse equation above.