All of the different calculations can be combined seamlessly into a single formulation that transcends the distinction between the parabolic, hyperbolic, elliptical and rectilinear cases.
(Edit: The true anomaly is buried in the description below: it's the angle between $𝗿$ and $𝗲$, i.e. $𝗿·𝗲 = re \cos f$.)
The Kepler problem may be defined by the following system, rendered in the language of 3D vector algebra:
$$
m {d𝗿 \over dt} = 𝗽, \hspace 1em {d𝗽 \over dt} = -{μ𝗿 \over r^3},
$$
where $m$ is the mass of the body undergoing orbital motion, and $μ$ is the gravitational mass of the center of attraction and $r = |𝗿|$ (and, more generally, where we use the convention of using the lighter-face letter to denote the magnitude of the vector letter, e.g. $L = |𝐋|$).
If the mass of the center of attraction is $M$, then $μ = GMm$, where $G$ is Newton's constant of gravity.
The constants of motion include:
$$
H ≡ {p^2 \over 2m} - {μ \over r}, \hspace 1em 𝗟 ≡ 𝗿×𝗽, \hspace 1em 𝗲 ≡ {𝗽×𝗟 \over μm} - {𝗿 \over r},
$$
with the following dependencies:
$$
𝗟·𝗲 = 0, \hspace 1em e^2 = {2H \over m} \left({L \over μ}\right)^2 + 1.
$$
The orbits have the following shapes:
$$\begin{matrix}
𝗟 = 𝟎, & e = 1: & \mbox{Linear},\\
𝗟 ≠ 𝟎, & e = 0: & \mbox{Circular},\\
𝗟 ≠ 𝟎, & 0 < e < 1: & \mbox{Elliptical},\\
𝗟 ≠ 𝟎, & e = 1: & \mbox{Parabolic},\\
𝗟 ≠ 𝟎, & e > 1: & \mbox{Hyperbolic}.
\end{matrix}$$
From this, a right-handed orthonormal frame $(𝗶,𝗷,𝗸)$ may be defined by:
- $𝗶 = 𝗲/e$ (for circular orbits, $𝗶$ can be chosen as any unit vector perpendicular to $𝗟$),
- $𝗸 = 𝗟/L$ (for linear orbits, $𝗸$ can be chosen as any unit vector perpendicular to $𝗲$),
- $𝗷 = 𝗸×𝗶$.
The equations have 6 degrees of freedom, and while the constants together comprise 7 parameters in all, the 2 constraints reduce this number to 5, which leaves one degree of freedom unaccounted for - the clock, itself; i.e. an explicitly time-dependent parameter. It's normally cited as the time $t_0$ of nearest passage and - in the case of the circular and elliptical orbits - is only determined up to a multiple of the period, since these orbits are periodic.
The Cases - Separately:
First, we can lay out the cases individually. In the hyperbolic case, $F$ is used, not $H$, because $H$ is already being used as an orbital parameter. For the parabolic and linear cases, $G$ is used, which is also what will also be used generically when the cases are all combined. In the linear case (and when the cases are combined), it appears directly and indirectly in the functions $C$, $S$, $D$ and $T$ of $G$ defined below.
The major and minor axes in the elliptical cases are, respectively $a$ and $b$, while the hyperbolic analogues are denoted here, respectively $A$ and $B$. The parabolic and linear cases use $α$, $β$ and/or $γ$, which will also be used when the cases are all combined.
- Elliptical and Circular (with $b^2 = a^2 (1 - e^2)$):
$$\begin{matrix}
t - t_0 = \sqrt{ma^3 \over μ} (E - e \sin E), &
L^2 = μma\left(1 - e^2\right), &
H = -{μ \over 2a}, \\
𝗿 = a (\cos E - e) 𝗶 + b \sin E 𝗷, &
r = a (1 - e \cos E), &
{dE \over dt} = \sqrt{μ \over ma} {1 \over r}, \\
𝗽 = (b \cos E 𝗷 - a \sin E 𝗶) m {dE \over dt}, &
p^2 = {μm \over a} {1 + e \cos E \over 1 - e \cos E}, &
{d^2E \over dt^2} = -{μe \sin E \over mr^3}.
\end{matrix}$$
- Hyperbolic (with $B^2 = A^2 (e^2 - 1)$):
$$\begin{matrix}
t - t_0 = \sqrt{mA^3 \over μ} (e \sinh F - F), &
L^2 = μmA\left(e^2 - 1\right), &
H = {μ \over 2A}, \\
𝗿 = A (e - \cosh F) 𝗶 + B \sinh F 𝗷, &
r = A (e \cosh F - 1), &
{dF \over dt} = \sqrt{μ \over mA} {1 \over r}, \\
𝗽 = (B \cosh F 𝗷 - A \sinh F 𝗶) m {dF \over dt}, &
p^2 = {μm \over A} {e \cosh F + 1 \over e \cosh F - 1}, &
{d^2F \over dt^2} = -{μe \sinh F \over mr^3}.
\end{matrix}$$
- Parabolic:
$$\begin{matrix}
t - t_0 = \sqrt{2mα^3 \over μ} \left(G + {G^3 \over 3}\right), &
L^2 = 2μmα, &
H = 0, \\
𝗿 = α\left(\left(1 - G^2\right) 𝗶 + 2G 𝗷\right), &
r = α\left(1 + G^2\right), &
{dG \over dt} = \sqrt{μ \over 2mα} {1 \over r}, \\
𝗽 = 2α(𝗷 - G𝗶) m {dG \over dt}, &
p^2 = {2μm \over α\left(1 + G^2\right)}, &
{d^2G \over dt^2} = -{μG \over mr^3}.
\end{matrix}$$
- Linear:
$$\begin{matrix}
t - t_0 = \sqrt{mβ \over μ} βT, &
L^2 = 0, &
H = {μz \over 2β} \\
𝗿 = -βD 𝗶, &
r = βD, &
{dG \over dt} = \sqrt{μ \over mβ} {1 \over r} \\
𝗽 = -βS 𝗶 m {dG \over dt}, &
p^2 = {μm(1 + C) \over r}, &
{d^2G \over dt^2} = -{μS \over mr^3}.
\end{matrix}$$
Combining The Cases:
Define the following family of functions, for $k ≥ 0$:
$$e_k ≡ e_k(z, G) = \sum_{n ≥ 0} {G^{k+2n} \over (k+2n)!} z^n = {G^k \over k!} + z {G^{k+2} \over (k+2)!} + z^2 {G^{k+4} \over (k+4)!} + ⋯,$$
and the following specializations:
$$C ≡ e₀(z, G), \hspace 1em S ≡ e₁(z, G), \hspace 1em D ≡ e₂(z, G), \hspace 1em T ≡ e₃(z, G);$$
where we note the identities
$$
{∂C \over ∂G} = zS, \hspace 1em {∂S \over ∂G} = C, \hspace 1em
{∂D \over ∂G} = S, \hspace 1em {∂T \over ∂G} = D, \\
S^2 - 2CD + zD^2 = 0, \hspace 1em C - zD = 1, \hspace 1em S - zT = G;
$$
which are specializations or rewritings of the following:
$$
{∂e_0 \over ∂G} = ze_1, \hspace 1em {e_1}^2 - 2e_0e_2 + z{e_2}^2 = 0, \\
{∂e_{n+1} \over ∂G} = e_n, \hspace 1em e_n = {G^n \over n!} + z e_{n+2} \hspace 1em (n ≥ 0).
$$
From these, the following identities may be derived:
$$
C^2 - zS^2 = 1, \hspace 1em S^2 - CD = D, \hspace 1em S^2 = 2D + zD^2.
$$
At $G = 0$, the values are fixed as $(C,S,D,T) = (1,0,0,0)$.
The correspondences of the generic (and linear) case, respectively, to the elliptical, hyperbolic and parabolic cases, are:
$$\begin{matrix}
G & C & S & D & T &
α & β & γ \\
{E \over \sqrt{-z}} & \cos E & {\sin E \over \sqrt{-z}} & {\cos E - 1 \over z} & {\sin E - E \over z \sqrt{-z}} &
a(1-e) & -az & a \sqrt{z\left(e^2 - 1\right)} = b \sqrt{-z} \\
{F \over \sqrt{+z}} & \cosh F & {\sinh F \over \sqrt{+z}} & {\cosh F - 1 \over z} & {\sinh F - F \over z \sqrt{+z}} &
A(e-1) & +Az & A \sqrt{z\left(e^2 - 1\right)} = B \sqrt{+z} \\
{G} & 1 & G & {G^2 \over 2} & {G^3 \over 6} & α & 2α & 2α
\end{matrix}$$
with
$$
αz = β(e-1), \hspace 1em γ^2 = αβ(e+1).
$$
The cases for linear orbits are included in each of these cases by $e = 1$ and $α = 0 = γ$. The parabolic version of $E$ and $F$ is the limiting case:
$${E \over \sqrt{-z}} → G ← {F \over \sqrt{+z}}$$
as $z → 0$. In the elliptical case, $z < 0$ and in the hyperbolic case, $z > 0$. The other key limits include the following:
$$
a(1 - e) → α ← A(e - 1),
\hspace 1em
-az → β ← +Az,
\hspace 1em
b \sqrt{-z} → γ ← B \sqrt{+z}.
$$
The extra parameter $z$ can be normalized in any of several ways, each one of which fixes the values of the variables $α$, $β$ and $γ$. This includes each of the following normalizations:
- For $L^2 > 0$: $β = α ⇔ z = e - 1 ⇔ γ = α \sqrt{e + 1}$.
- For $L^2 > 0$: $γ = α ⇔ z = {e - 1 \over e + 1} ⇔ β = {α \over e + 1}$.
- For $z > 0$: $z = +1 ⇒ α = A(e - 1), \hspace 1em β = A, \hspace 1em γ = B = A \sqrt{e^2 - 1}$.
- For $z < 0$: $z = -1 ⇒ α = a(1 - e), \hspace 1em β = a, \hspace 1em γ = b = a \sqrt{1 - e^2}$.
In all cases, $α = r_0 ≡ r(G = 0)$, the distance of nearest approach.
In all cases where $L^2 > 0$, the distance of nearest approach $α > 0$, so one can define the ratios:
$$λ = {β \over α}, \hspace 1em κ = {γ \over α} ≥ 0,$$
in terms of which, we have:
$$
e = {κ^2 \over λ} - 1, \hspace 1em z = κ^2 - 2λ.
$$
For the two $L^2 > 0$ normalizations just described, this reduces to the following:
- $β = α ⇔ λ = 1 ⇔ κ = \sqrt {e+1}$,
- $γ = α ⇔ κ = 1 ⇔ λ = {1 \over e+1}$.
The Mean Anaomaly - Combined:
The mean anomaly for the elliptic case, $M$ and the analogue for the hyperbolic case, which we'll label $N$:
$$
M = \sqrt{μ \over ma^3} \left(t - t_0\right) = E - e \sin E, \hspace 1em
N = \sqrt{μ \over mA^3} \left(t - t_0\right) = e \sinh F - F,
$$
don't transition well across the $e = 1$ divide. When written in terms of $e$, $S$, $G$, $β$ and $z$, using the table of correspondences and the identity $G = S - zT$, they take on the respective forms
$$
M = \sqrt{-μz^3 \over mβ^3} \left(t - t_0\right) = \sqrt{-z}((1 - e)S - zT), \\
N = \sqrt{+μz^3 \over mβ^3} \left(t - t_0\right) = \sqrt{+z}((e - 1) S + zT).
$$
Multiplying both sides by $β$ and using the identity $β(e - 1) = αz$, this can be rewritten as:
$$
βM = \sqrt{-μz^3 \over mβ} \left(t - t_0\right) = \sqrt{-z^3}(αS + βT), \\
βN = \sqrt{+μz^3 \over mβ} \left(t - t_0\right) = \sqrt{+z^3}(αS + βT),
$$
which shows that we should actually be rescaling $M$ and $N$, instead, and combining them as:
$${βM \over \sqrt{-z^3}} → \sqrt{μ \over mβ} \left(t - t_0\right) ← {βN \over \sqrt{+z^3}},$$
with the relevant equation being:
$$\sqrt{μ \over mβ} \left(t - t_0\right) = αS + βT.$$
This also works for the parabolic case and the linear case.
The Cases - Combined:
Thus, the combined form of all the cases may be presented as:
$$\begin{matrix}
t - t_0 = \sqrt{mβ \over μ} (αS + βT), &
L^2 = {μmγ^2 \over β} = μmα(e + 1), &
H = {μz \over 2β}, \\
𝗿 = (α - βD) 𝗶 + γS 𝗷, &
r = αC + βD = α + βeD, &
{dG \over dt} = \sqrt{μ \over mβ} {1 \over r}, \\
𝗽 = (γC 𝗷 - βS 𝗶) m {dG \over dt}, &
p^2 = {μm(1 + eC) \over r} = {μmz \over β} {eC + 1 \over eC - 1}, &
{d^2G \over dt^2} = -{μeS \over mr^3}.
\end{matrix}$$
The constants of motion (in addition to $H$), in terms of these parameters, are:
$$
𝗟 = γ \sqrt{μm \over β} 𝗸, \hspace 1em
𝗲 = \left(1 + {αz \over β}\right) 𝗶, \hspace 1em
t_0 = t - \sqrt{mβ \over μ} (αS + βT),
$$
which rounds out the 5 parameters arising from the independent components of $(H,𝗟,𝗲)$ with the 6th, explicitly time-dependent, parameter $t_0$.
Under the $γ = α$ and $κ = 1$ normalization, where we have $(α, β, γ) = (r_0, λ r_0, r_0)$, with
$$p^0 ≡ p(G = 0) = \sqrt{μm \over β},$$
the solutions reduce to the following:
$$
𝗿 = r_0 (𝗶 (1 - λD) + 𝗷 S), \hspace 1em
r = r_0 (1 + (1 - λ) D), \\
𝗽 = p^0 {𝗷 C - 𝗶 λS \over 1 + (1-λ) D}, \hspace 1em
t = t_0 + {mr_0 \over p^0} (G + (1-λ)T), \hspace 1em
{dG \over dt} = {p^0 \over mr} = \sqrt{μ \over mβ} {1 \over r}.
$$
with $\left(λ, z, r_0, p^0\right)$ given in terms of the orbital parameters as:
$$
λ = {μm \over r_0{p^0}^2} = {1 \over e + 1}, \hspace 1em
z = 1 - 2λ = {e - 1 \over e + 1}, \hspace 1em
r_0 = {μ(e - 1) \over 2H}, \hspace 1em
p^0 = {mμ(e + 1) \over L}.
$$
The constants of motion, expressed in terms of $\left(λ, z, r_0, p^0\right)$, reduce to the following:
$$
H = {z{p^0}^2 \over 2m}, \hspace 1em
𝗟 = r_0 p^0 𝗸, \hspace 1em
𝗲 = \left({r_0{p^0}^2 \over μm} - 1\right) 𝗶 = {1 - λ \over λ} 𝗶, \hspace 1em
t_0 = t - {mr_0 \over p^0} (S + λT).
$$
As $e$ ranges over $0 ⇒ ∞$, $λ$ ranges over $1 ⇒ 0$ and $z$ over $-1 ⇒ +1$; and the different cases being given by:
- Circular: $(e, λ, z) = (0, 1, -1)$;
- Elliptical: $(e, λ, z) = (0, 1, -1) ⇒ (e, λ, z) = (1, ½, 0)$;
- Parabolic: $(e, λ, z) = (1, ½, 0)$;
- Hyperbolic: $(e, λ, z) = (1, ½, 0) ⇒ (e, λ, z) = (∞, 0, +1)$.
