In astrodynamics and celestial mechanics a radial trajectory is a Kepler orbit with zero angular momentum. Two objects in a radial trajectory move directly towards or away from each other in a straight line.
There are three types of radial trajectories (orbits).^{[1]}
Unlike standard orbits which are classified by their orbital eccentricity, radial orbits are classified by their specific orbital energy, the constant sum of the total kinetic and potential energy, divided by the reduced mass:
\epsilon=
v^{2}  
2 

\mu  
x 
where x is the distance between the centers of the masses, v is the relative velocity, and
\mu={G}\left(m_{1}+m_{2\right)}
Another constant is given by:
w=
1  
x 

v^{2}  
2\mu 
=
\epsilon  
\mu 
style 
 
2\mu 
stylev_{infty}
Given the separation and velocity at any time, and the total mass, it is possible to determine the position at any other time.
The first step is to determine the constant w. Use the sign of w to determine the orbit type.
w=
1  
x_{0} 

 
2\mu 
stylex_{0}
stylev_{0}
t(x)=\sqrt{
2x^{3}  
9\mu 
}
where t is the time from or until the time at which the two masses, if they were point masses, would coincide, and x is the separation.
This equation applies only to radial parabolic trajectories, for general parabolic trajectories see Barker's equation.
t(x,w)=
\arcsin\left(\sqrt{wx  
\right) 
\sqrt{wx (1wx)}}{\sqrt{2\muw^{3}}}
This is the radial Kepler equation.^{[2]}
See also equations for a falling body.
t(x,w)=
\sqrt{(wx)^{2}+wx  
 
ln\left(\sqrt{wx}+\sqrt{1+wx}\right)}{\sqrt{2\muw^{3}}}
The radial Kepler equation can be made "universal" (applicable to all trajectories):
t(x,w)=\lim_{u}
\arcsin\left(\sqrt{ux  
\right) 
\sqrt{ux (1ux)}}{\sqrt{2\muu^{3}}}
t(x,w)=
1  
\sqrt{2\mu 
The problem of finding the separation of two bodies at a given time, given their separation and velocity at another time, is known as the Kepler problem. This section solves the Kepler problem for radial orbits.
The first step is to determine the constant
stylew
stylew
w=
1  
x_{0} 

 
2\mu 
stylex_{0}
stylev_{0}
x(t)=\left(
9  
2 
\mut^{2}
 
\right) 
See also position as function of time in a straight escape orbit.
Two intermediate quantities are used: w, and the separation at time t the bodies would have if they were on a parabolic trajectory, p.
w=
1  
x_{0} 

 
2\mu 
and p=\left(
9  
2 
\mut^{2}
 
\right) 
Where t is the time,
x_{0}
v_{0}
\mu={G}\left(m_{1}+m_{2\right)}
The inverse radial Kepler equation is the solution to the radial Kepler problem:
x(t)=
infty  
\sum  
n=1 
\left(\lim_{r}\left[
w^{n1}p^{n}  
n! 
d^{n1}  
dr^{n1} 
\left(r^{n}\left[
3  
2 
\left(\arcsin\left[\sqrt{r}\right]\sqrt{rr^{2}}\right)
 
\right] 
\right) \right] \right)
Evaluating this yields:
x(t)=p
1  
5 
wp^{2}
3  
175 
w^{2}p^{3}
23  
7875 
w^{3}p^{4}
1894  
3031875 
w^{4}p^{5}
3293  
21896875 
w^{5}p^{6}
2418092  
62077640625 
w^{6}p^{7} …
Power series can be easily differentiated term by term. Repeated differentiation gives the formulas for the velocity, acceleration, jerk, snap, etc.
The orbit inside a radial shaft in a uniform spherical body^{[3]} would be a simple harmonic motion, because gravity inside such a body is proportional to the distance to the center. If the small body enters and/or exits the large body at its surface the orbit changes from or to one of those discussed above. For example, if the shaft extends from surface to surface a closed orbit is possible consisting of parts of two cycles of simple harmonic motion and parts of two different (but symmetric) radial elliptic orbits.