# Tag Info

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One method of calculating the angle involves using the ellipse reflection law. Light from one focus reflects off the ellipse into the other focus. Thus in the picture below (by the author), the radial vector from the focus $F_1$ is reflected at $P$ onto the second focus $F_2$, forming a triangle whose third side is the line between the foci. Your flight ...

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If your ellipse is a circle, the Flight Path Angle is 0. You’re done. Otherwise, for an elliptical orbit, start with the polar equation that relates radial distance $r$, true anomaly $\theta$, semimajor axis $a$, and orbital eccentricity $e$: $$r=\frac{a(1-e^2)}{1+e\cos\theta}$$ Solving for $\theta$ gives us the following: \theta = \arccos\left({\frac{-...

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The flight path angle is simply the angle between the velocity vector and the vector perpendicular to the position vector. An easy way to visualise this: If the orbit was a circle, this angle would be zero. The angle is therefore due to the contribution of the inward/outward motion of the object away from the focal point. The semi major axis ($a$) and ...

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There are really two questions here: Do there exist $n$-body systems with long-term stability? Can a third body (massive or not) be shown, a-priori, to be bounded or to escape—without resorting to numerical simulation? 1. Stability of $n$-body systems It is widely known that $n$-body systems are "chaotic" when $n>2$. However, this must be unpacked ...

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FWIW, Here is the formula for converting the focus-eccentricity-directrix parameter set to the generalized quadratic formula $Ax^2 +Bxy + Cy^2 + Dx + Ey + F = 0$ . This is in the R language. FEDtoA <-function(focus = c(0,0), directrix = c(1,0,1), eccentricity = 0.5 ) { h = focus[1] v = focus[2] da = directrix[1] db = directrix[2] dc = directrix[3] ec ...

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Assume three point masses, Newtonian gravity and no losses. If we can also assume no perturbations of any other kind, and perfect placement of the bodies in the initial conditions, then a 3-body Klemperer rosette -- three bodies of equal mass, in an equilateral triangle, with any rotationally symmetrical initial velocities comfortably below barycentric ...

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