Question about ellipses described in the two-body problem

[pc2-brazil]

Good afternoon,

Suppose a two-body problem, where both bodies, of masses m₁ and m₂, are under influence of their mutual attractions. I know that, for the motion of one body relative to the other, this body describes a conic section whose eccentricity is given by the magnitude of the following vector:
$$\mathbf{e} = {\mathbf{\left |v \right |}^2 \mathbf{r} \over {\mu}} - {(\mathbf{r} \cdot \mathbf{v} ) \mathbf{v} \over{\mu}} - {\mathbf{r}\over{\left|\mathbf{r}\right|}}$$, where $$\mu$$ = G(m₁ + m₂)
(see http://books.google.com.br/books?id...resnum=2&ved=0CCkQ6AEwAQ#v=onepage&q&f=false", page 26).
Anyway, I know that each body in a two-body system describes its own ellipse with respect to the center of mass of the system.
I would like to know how to calculate the eccentricity vector for each one of these ellipses.

Thank you in advance.

 

[lippyka]

The eccentricity vector for each ellipse can be calculated using the formula: e = {|v|^2 r / μ} - {(r⋅v) v / μ} - {r / |r|}, where v is the relative velocity of the two bodies and r is the relative position vector. The same formula can be used to calculate the eccentricity of each ellipse, with the appropriate values of v and r substituted.