Parabolic trajectory

In 1609, Galileo wrote in his 102nd folio (MS. Gal 72) about parabolic trajectory calculations, ==Velocity==

The orbital velocity (v) of a body travelling along a parabolic trajectory can be computed as: :v = \sqrt{2\mu \over r} where: • r is the radial distance of the orbiting body from the central body, • \mu is the standard gravitational parameter. At any position the orbiting body has the escape velocity for that position. If a body has an escape velocity with respect to the Earth, this is not enough to escape the Solar System, so near the Earth the orbit resembles a parabola, but further away it bends into an elliptical orbit around the Sun. This velocity (v) is closely related to the orbital velocity of a body in a circular orbit of the radius equal to the radial position of orbiting body on the parabolic trajectory: :v = \sqrt{2}\, v_o where: • v_o is orbital velocity of a body in circular orbit. ==Equation of motion==

For a body moving along this kind of trajectory the orbital equation is: :r = {h^2 \over \mu}{1 \over {1 + \cos\nu}} where: • r\, is the radial distance of the orbiting body from the central body, • h\, is the specific angular momentum of the orbiting body, • \nu\, is the true anomaly of the orbiting body, • \mu\, is the standard gravitational parameter. ==Energy==

Under standard assumptions, the specific orbital energy (\epsilon) of a parabolic trajectory is zero, so the orbital energy conservation equation for this trajectory takes the form: :\epsilon = {v^2 \over 2} - {\mu \over r} = 0 where: • v\, is the orbital velocity of the orbiting body, • r\, is the radial distance of the orbiting body from the central body, • \mu\, is the standard gravitational parameter. This is entirely equivalent to the characteristic energy (square of the speed at infinity) being 0: :C_3 = 0 ==Barker's equation==

Barker's equation relates the time of flight t to the true anomaly \nu of a parabolic trajectory: :t - T = \frac{1}{2} \sqrt{\frac{p^3}{\mu}} \left(D + \frac{1}{3} D^3 \right) where: • D = \tan \frac{\nu}{2} is an auxiliary variable • T is the time of periapsis passage • \mu is the standard gravitational parameter • p is the semi-latus rectum of the trajectory, given by p = h^2/\mu More generally, the time (epoch) between any two points on an orbit is : t_f - t_0 = \frac{1}{2} \sqrt{\frac{p^3}{\mu}} \left(D_f + \frac{1}{3} D_f^3 - D_0 - \frac{1}{3} D_0^3\right) Alternately, the equation can be expressed in terms of periapsis distance, in a parabolic orbit r_p = p/2: :t - T = \sqrt{\frac{2 r_p^3}{\mu}} \left(D + \frac{1}{3} D^3\right) Unlike Kepler's equation, which is used to solve for true anomalies in elliptical and hyperbolic trajectories, the true anomaly in Barker's equation can be solved directly for t. If the following substitutions are made :\begin{align} A &= \frac{3}{2} \sqrt{\frac{\mu}{2r_p^3}} (t - T) \\[3pt] B &= \sqrt[3]{A + \sqrt{A^{2}+1}} \end{align} then : \nu = 2\arctan\left(B - \frac{1}{B}\right) With hyperbolic functions the solution can be also expressed as: : \nu = 2\arctan\left(2\sinh\frac{\mathrm{arcsinh} \frac{3M}{2}}{3}\right) where : M = \sqrt{\frac{\mu}{2r_p^3}} (t - T) ==Radial parabolic trajectory==

A radial parabolic trajectory is a non-periodic trajectory on a straight line where the relative velocity of the two objects is always the escape velocity. There are two cases: the bodies move away from each other or towards each other. There is a rather simple expression for the position as function of time: : r = \sqrt[3]{\frac{9}{2} \mu t^2} where • \mu is the standard gravitational parameter • t = 0\!\, corresponds to the extrapolated time of the fictitious starting or ending at the center of the central body. At any time the average speed from t = 0\!\, is 1.5 times the current speed, i.e. 1.5 times the local escape velocity. To have t = 0\!\, at the surface, apply a time shift; for the Earth (and any other spherically symmetric body with the same average density) as central body this time shift is 6 minutes and 20 seconds; seven of these periods later the height above the surface is three times the radius, etc. ==See also==