Parabolic trajectory

The orbital velocity ($v$) of a body travelling along a parabolic trajectory can be computed as:

:$v\; =\; \backslash sqrt\{2\backslash mu\; \backslash over\; r\}$

where:

• $r$ is the radial distance of the orbiting body from the central body,
• $\backslash mu$ 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:

:$v\; =\; \backslash sqrt\{2\}\backslash ,\; v\_o$

where:

• $v\_o$ is orbital velocity of a body in circular orbit.

For a body moving along this kind of trajectory the orbital equation is:

:$r\; =\; \{h^2\; \backslash over\; \backslash mu\}\{1\; \backslash over\; \{1\; +\; \backslash cos\backslash nu\}\}$

where:

• $r\backslash ,$ is the radial distance of the orbiting body from the central body,
• $h\backslash ,$ is the specific angular momentum of the orbiting body,
• $\backslash nu\backslash ,$ is the true anomaly of the orbiting body,
• $\backslash mu\backslash ,$ is the standard gravitational parameter.

Under standard assumptions, the specific orbital energy ($\backslash epsilon$) of a parabolic trajectory is zero, so the orbital energy conservation equation for this trajectory takes the form:

:$\backslash epsilon\; =\; \{v^2\; \backslash over\; 2\}\; -\; \{\backslash mu\; \backslash over\; r\}\; =\; 0$

where:

• $v\backslash ,$ is the orbital velocity of the orbiting body,
• $r\backslash ,$ is the radial distance of the orbiting body from the central body,
• $\backslash mu\backslash ,$ 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 relates the time of flight $t$ to the true anomaly $\backslash nu$ of a parabolic trajectory:

{{cite book

| last1 = Bate

| first1 = Roger

| last2 = Mueller

| first2 = Donald

| last3 = White

| first3 = Jerry

| title = Fundamentals of Astrodynamics

| url = https://archive.org/details/fundamentalsofas00bate

| url-access = registration

| publisher = Dover Publications, Inc., New York

| year = 1971

| isbn = 0-486-60061-0

}} p 188

:$t\; -\; T\; =\; \backslash frac\{1\}\{2\}\; \backslash sqrt\{\backslash frac\{p^3\}\{\backslash mu\}\}\; \backslash left(D\; +\; \backslash frac\{1\}\{3\}\; D^3\; \backslash right)$

where:

• $D\; =\; \backslash tan\; \backslash frac\{\backslash nu\}\{2\}$ is an auxiliary variable
• $T$ is the time of periapsis passage
• $\backslash mu$ is the standard gravitational parameter
• $p$ is the semi-latus rectum of the trajectory ($p\; =\; h^2/\backslash mu$ )

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 $r\_p\; =\; p/2$:

:$t\; -\; T\; =\; \backslash sqrt\{\backslash frac\{2\; r\_p^3\}\{\backslash mu\}\}\; \backslash left(D\; +\; \backslash frac\{1\}\{3\}\; D^3\backslash right)$

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

:$\backslash begin\{align\}$

A &= \frac{3}{2} \sqrt{\frac{\mu}{2r_p^3}} (t - T) \\[3pt]

B &= \sqrt[3]{A + \sqrt{A^{2}+1}}

\end{align}

then

: $\backslash nu\; =\; 2\backslash arctan\backslash left(B\; -\; \backslash frac\{1\}\{B\}\backslash right)$

With hyperbolic functions the solution can be also expressed as:{{cite journal

| last1 = Zechmeister

| first1 = Mathias

| title = Solving Kepler's equation with CORDIC double iterations

| journal = MNRAS

| date = 2020

| volume= 500

| issue = 1

| pages = 109–117

| doi = 10.1093/mnras/staa2441

| doi-access = free

| arxiv = 2008.02894

| bibcode = 2021MNRAS.500..109Z

}} Eq.(40) and Appendix C.

: $\backslash nu\; =\; 2\backslash arctan\backslash left(2\backslash sinh\backslash frac\{\backslash mathrm\{arcsinh\}\; \backslash frac\{3M\}\{2\}\}\{3\}\backslash right)$

where

: $M\; =\; \backslash sqrt\{\backslash frac\{\backslash mu\}\{2r\_p^3\}\}\; (t\; -\; T)$

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:

: $r\; =\; \backslash sqrt[3]\{\backslash frac\{9\}\{2\}\; \backslash mu\; t^2\}$

where

• μ is the standard gravitational parameter
• $t\; =\; 0\backslash !\backslash ,$ 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 $t\; =\; 0\backslash !\backslash ,$ is 1.5 times the current speed, i.e. 1.5 times the local escape velocity.

To have $t\; =\; 0\backslash !\backslash ,$ 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.

• Kepler orbit
• Parabola

{{Reflist}}

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