Hyperbolic Collisions

In addition to circular binaries, we have written routines which set up two stars on a hyperbolic trajectory. As was the case with irrotational binaries, there is no way to perform relaxation, so you will need to create relaxed single-star models first. If the binaries is started from a sufficiently large separation, tidal deformations are expected to be small, and spherical initial conditions are perfectly acceptable.

For stars with mass $M_1$ and $M_2$, we define the total and reduced masses by $M_T\equiv M_1+M_2$ and $\mu=M_1M_2/M_T$. For a system with fixed masses, there are two free parameters which describe the initial orbit. We choose to define out initial conditions via the periastron distance $r_p$ and relative velocity at infinity $v_{\infty}\equiv \vert\vec{v}_1-\vec{v}_2\vert$, set by the parameters ``RP'' and ``VPEAK'', respectively. The initial binary separation $r_0$ is given as usual by SEP0, and determines where on the orbital path we start the calculation. For such a system, we know the conserved total energy must be

$$E_T=\frac{1}{2}\mu v_{\infty}^2,$$ (34)

and by energy conservation the initial relative velocity is

$$v_0=\sqrt{v_{\infty}^2+\frac{2GM_T}{r_0}},$$ (35)

and the velocity at periastron is

$$v_p=\sqrt{v_{\infty}^2+\frac{2GM_T}{r_p}}.$$ (36)

From the latter expression, we find that the conserved system angular momentum is given by

$$J_T=\mu v_p r_p=\mu \sqrt{r_p^2 v_{\infty}^2+2GM_T r_p},$$ (37)

which yields immediately the impact parameter $b$

$$b=\frac{J_T}{\mu v_{\infty}}=\sqrt{r_p^2+\frac{2GM_T r_p}{v_{\infty}^2}}.$$ (38)

At our initial separation $r_0$, we know that the transverse velocity is given by

$$v_t=\frac{J_T}{\mu r_0}=\frac{\sqrt{r_p^2 v_{\infty}^2+2GM_T r_p}}{r_0}=\frac{v_{\infty}b}{r_0},$$ (39)

and the radial velocity by $v_r=\sqrt{v_0^2-v_t^2}$.

In order to use the most precise gravity grid possible, i.e., one that covers the smallest physical volume possible, we start both stars in opposite corners of the grid, rather than separated along the x-axis. The transverse velocity is projected perpendicular to the separation vector in such a way that $v_x=v_y$. Technically, it is possible to have an initial velocity which could take the stars off the gravity cubic gravity grid, but only if $v_t/v_0>\sqrt{1/3}$. If you find this is the case, you may want to start the stars further apart!

Please note that the gravitational force calculated for particles can have a weak dependence on the physical size represented by a grid cell. Since the grid will need to be large in these calculations, you may want to make it larger than normal when constructing single star models.