In orbital mechanics, the universal variable formulation is a method used to solve the two-body Kepler problem. It is a generalized form of Kepler's Equation, extending it to apply not only to elliptic orbits, but also parabolic and hyperbolic orbits common for spacecraft departing from a planetary orbit. It is also applicable to ejection of small bodies in Solar System from the vicinity of massive planets, during which processes the approximating two-body orbits can have widely varying eccentricities, almost always
A common problem in orbital mechanics is the following: Given a body in an orbit and a fixed original time
to ,
t~.
Note that the conventional form of Kepler's equation cannot be applied to parabolic and hyperbolic orbits without special adaptions, to accommodate imaginary numbers, since its ordinary form is specifically tailored to sines and cosines; escape trajectories instead use    and    (hyperbolic functions).
E ,
s
\operatornameds | |
\operatornamedt |
=
1 | |
r |
where
r\equivr(t)
r ,
r ,
We can regularize the fundamental equation
\operatornamed2r | |
\operatornamedt2 |
+\mu
r | |
~r3 |
=0 ,
where
~~\mu\equivG\left(m1+m2\right)~~
by applying the change of variable from time
t
s
\operatornamed2r | |
~\operatornameds2 |
+\alpha r=-P
P
\alpha
\alpha\equiv
\mu | |
a |
~.
The equation is the same as the equation for the harmonic oscillator, a well-known equation in both physics and mathematics, however, the unknown constant vector is somewhat inconvenient. Taking the derivative again, we eliminate the constant vector
P ,
\operatornamed3r | |
~\operatornameds3 |
+\alpha
\operatornamedr | |
\operatornameds |
=0
The family of solutions to this differential equation[2] are for convenience written symbolically in terms of the three functions
s c1\left( \alphas2 \right) ,
s2c2\left( \alphas2 \right) ,
s3c3\left( \alphas2 \right) ;
ck(x) ,
\tfrac{\operatornamedt}{ \operatornameds }=r
\int\tilde{t=to
After extensive algebra and back-substitutions, its solution results in[2]
t-to=ro s c1\left( \alphas2 \right)+ro
~\operatornamedro | |
\operatornamedt |
s2c2\left( \alphas2 \right)+\mu s3c3\left( \alphas2 \right)
which is the universal variable formulation of Kepler's equation.
There is no closed analytic solution, but this universal variable form of Kepler's equation can be solved numerically for
s ,
t ~.
s
f
g
f |
g |
\begin{align} f(s)&=1-\left(
\mu | |
~ro |
\right)s2c2\left( \alphas2 \right) ,\\[1.5ex] g(s)&=t-to-\mu s3c3\left( \alphas2 \right) ,\\[1.5ex]
f |
(s)\equiv
\operatornamedf | |
\operatornamedt |
&=-\left(
\mu | |
ror |
\right)s c1\left( \alphas2 \right) ,\\[1.5ex]
g |
(s)\equiv
\operatornamedg | |
\operatornamedt |
&=1-\left(
\mu | |
r |
\right) s2c2\left( \alphas2 \right)~.\\[-1ex]\end{align}
The values of the
f
g
t
In addition the velocity of the body at time
t
f |
(s)
g |
(s)
v(t)=
r | |||
|
(s)+
v | |||
|
(s)
where
r(t)
v(t)
t ,
ro
vo
to~.